Research ArticleSupply Chain Decisions and Coordination under the CombinedEffect of Overconfidence and Fairness Concern

Zhang Zhijian1 Peng Wang 1 Miyu Wan2 Junhua Guo1 and Jian Liu 2

1School of Transportation and Logistics East China Jiaotong University Nanchang 330013 China2School of Information Technology Jiangxi University of Finance and Economics Nanchang 330013 China

Correspondence should be addressed to Peng Wang wangpeng_wl126com

Received 9 January 2020 Revised 8 March 2020 Accepted 10 April 2020 Published 25 April 2020

Guest Editor Baogui Xin

Copyright copy 2020 Zhang Zhijian et alis is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

e purpose of this study was to examine the joint effect of overconfidence and fairness concern on supply chain decisions anddesign contracts to achieve a win-win situation within the supply chain For this study a centralized supply chain model wasestablished without considering the retailersrsquo overconfidence and fairness concern Furthermore the retailersrsquo overconfidence andfairness concerns were introduced into the decentralized supply chain while the Stackelberg game model between the man-ufacturer and the retailer was built Furthermore an innovative supply chain contract ie buyback contract with promotionalcost sharing was designed to achieve supply chain coordination along with overconfidence and fairness concern Finally anumerical analysis was also conducted to analyze the effect of overconfidence fairness concern and the validity of the contracte principal findings of the study include the positive correlation between retailersrsquo overconfidence and optimal order quantitysales effort expected utility and profit Although the order quantity and sales efforts were not affected by the fairness concern ofthe retailer the contract achieved coordination with a win-win outcome when the level of overconfidence and fairness concernwas moderate

1 Introduction

In a global supply chain the manufacturer sells their productto the retailer at a wholesale price set by the manufacturerand in response the retailer chooses the order quantityRetailers sell goods to consumers through various promo-tional methods Besides the interaction between manufac-turer and retailer a reasonable wholesale price appropriateorder quantity and valid promotion are crucial factors forsupply chain members However players in the supply chainoften tend to show characteristics of bounded rationality incomplex and uncertain environments such as overconfi-dence [1] and fairness concern [2] ese two behaviorpreferences were observed anecdotally in practice For ex-ample Eastman Kodak Company overestimated its technicalability and ignored the planning for the future which led todeclined sales and eventually bankruptcy [3] Gome inMarch 2016 was concerned about distributional fairness andwanted to capitalize on its dominating power in the retailing

industry to extract more profit from the manufacturer [4]Overconfidence is the most widespread cognitive bias in thedecision-making process Fairness concern is the reflectionof sociological emotions which are related to social pref-erences in the supply chain erefore overconfidence andfairness concern could affect decision-making profoundly ina wide range of areas such as ordering pricing and saleseffort

In general the above two behavioral preferences ieoverconfidence and fairness concern coexist in the decision-making process A typical case can be used to illustrate thephenomenon During the Chinese shopping festival ldquoDouble11rdquo of 2014 Alibabasrsquo transaction value exceeded RMB 571billion Before the onset of Double 11 the CEO of Taobaopredicted that the refund rate would be in single-digitpercentage points However the real refund rate was 69and the real complaint rate was higher than usual for ex-ample Haier complaint rate was 542 [5] e significantdiscount on the activity motivated a large number of end

HindawiComplexityVolume 2020 Article ID 3056305 16 pageshttpsdoiorg10115520203056305

customers with irrational shopping behavior which resultedin return orders and complaint rate is case indicated thatoverconfidence and fairness concern coexisted in the de-cision-making process On the one hand the prediction bythe CEO and the high refund rate indicated that a largenumber of people were overconfident Consequently wrongdecisions caused losses which included seller and consumerconcern fairness issues that resulted in a high complaint rateHence the coexistence of overconfidence and fairnessconcern was not occasional but inevitable In the context ofthe supply chain numerous interactions existed betweenoverconfidence and fairness concern Overconfidence on-demand prediction triggered retailer overordering whilefairness concern had an opposite effect on the retailerrsquosdecisions For instance in the ldquoDouble 11rdquo shopping festivala retailer of China believed that he possessed better mar-keting capabilities than others and thus ordered moreproducts from a large manufacturer eg Procter amp Gamble(PampG) However the retailer made lower-than-expectedprofit due to excessive inventory and marketing expensesMeanwhile the retailer reduced order quantity and in-creased retail price because he felt that PampG was unfairlyseizing a disproportionate share of the profit [6] ereforethis paper introduces overconfidence and fairness concerninto the context of the supply chain Based on randommarket demands we analyzed the retailerrsquos behavioralpreferences in decision-making which affects the decision-making ability profits and utility in the supply chain

e decision makers of the decentralized supply chainare sufficiently rational However they suffer when manu-facturers charge a wholesale price higher than the pro-duction cost so that the amount of order quantity in thedecentralized channel is lower in the centralized channelanalog [7] erefore coordination is a critical issue insupply chain management At the same time the membersof the supply chain aim to maximize their profits whichresults in the well-known problem of ldquodouble marginali-zationrdquo In order to solve this problem various coordinatingcontracts were proposed in different supply chain structuresHowever coordinating conditions and the degrees of dif-ficulty during coordination changes depend on behavioralfactors [8] Among many different types of contractscombined contract ie the combination of two or morecontracts has received widespread attention and proved tobe effective in resolving these conflicts [9] As compared tothe centralized supply chain optimal decisions or profits in adecentralized supply chain are always suboptimal when theretailer has behavioral preferences in the decentralizedsupply chain

Based on these challenges two critical scenarios areconsidered (a) the centralized supply chain wherein themanufacturer and retailer make decisions as a whole tomaximize the total expected profit and (b) the decentralizedsupply chain consisting of a rational manufacturer and aretailer with overconfidence and fairness concern whereinmembers are independent decision makers and aim tomaximize their profits We then assessed the effect of be-havioral preferences on the equilibrium outcome for thisdecentralized supply chain and compared these effects to

their centralized channel analogs We observed that the twocognitive biases could trigger the decision makers ofdecentralized supply chain members to deviate from theiroptimal decisions in the centralized supply chain whichreduces the profitability of the decentralized supply chainus a valid buyback contract with promotional costsharing was proposed to coordinate the decentralized supplychain with overconfidence and fairness concern Such acontract is based on two types of conventional contractsbuyback contract and promotional cost-sharing contract Inthe buyback contract the upstream manufacturer commitsto buying back unsold goods from the downstream retailer atthe end of the selling season in order to encourage theretailer to order more e latter contract is designed toshare the cost of retailersrsquo sales efforts on increasing salese buyback contract with promotional cost sharing hasbeen widely applied in various industries such as pro-curement of industrial materials [10] medical devices [11]and retailing [12]

e remainder of the paper is organized as follows therelevant papers are described in Section 2 whereas Section 3formalizes the problem e two important models arepresented and solved ie (a) the centralized supply chainmodel wherein the members are both rational and (b) thedecentralized supply chain model wherein the retailer hasoverconfidence and fairness concerne combined effect ofoverconfidence and fairness concern on supply chain de-cisions has also been discussed In Section 4 the buybackcontract with promotional cost sharing is proposed to co-ordinate supply chain decisions in the presence of cognitivebiases A numerical analysis is conducted to examine thetheoretical models and propositions in Section 5 Lastly theconclusions and directions for future studies are discussed inSection 6

2 Literature Review

ere are three streams of literature related to our paperoverconfidence fairness concern and supply chain coor-dination In this section the recent literature on these threetopics has been reviewed and summarized

21 Overconfidence Some studies have investigated over-confidence in supply chain management (SCM) Crosonet al [13] first introduced overconfidence into the supplychain and analyzed the decisions of overconfident news-vendors Li et al [14] examined the implications and in-fluences of overconfidence in a competitive newsvendorsetting Similarly Liu et al [15] presented a two-periodservice capacity procurement model and found that a dy-namic wholesale price mechanism could eliminate thenegative effect of overconfidence Xu et al [16] explored theeffects of overconfidence on retailers in different games Li[17] identified that overconfidence could reduce the doublemarginalization effect in a decentralized supply chain epapers mentioned above defined the decision makersrsquooverconfidence based on a biased belief that the variance ofdemand was underestimated Hence in this paper the

2 Complexity

retailersrsquo overconfidence had the same definition Besideswe also considered that the overconfident retailer over-estimated the mean of market demand and the effects oftheir own sales effort on demand Moreover the mostexisting papers only focused on the effect of overconfidenceon the ordering or pricing we extended previous researchand analyzed the effect of overconfidence on pricing or-dering and sales effort

22 Fairness Concern Fairness concern has been studiedextensively in SCM Cui et al [18] analytically and experi-mentally evaluated how fairness significantly affected thepricing decisions of the manufacturer and the retailerSimilar issues were considered by Wu et al [19] who ex-amined the influence of fairness concern on the retailersrsquoprofit allocation in the newsvendorsrsquo model A recent paperby Chen et al [20] also analyzed the interactions betweenfairness concern and decisions including pricing and servicelevel in a dual-channel supply chain consisting of a man-ufacturer and two retailers Also Liu et al [21] investigatedthe impact of different fairness concerns on order allocationin the logistics service supply chain Zhang et al [22]revealed the characteristics of price changes when fairnessconcern was categorized into unfavorable and favorabledisutility respectivelye above papers mainly studied howfairness concerns affected decision variables in varioussupply chain structures Motivated by those papers we alsoanalyzed the decision-making problems of members in atwo-echelon supply chain wherein the retailer had fairnessconcerns However we extended previous studies and an-alyzed the impact of fairness concern on pricing orderingand the sales effort while other papers ignored sales effort

23 SupplyChainCoordination It can be seen from Sections21 and 22 that both overconfidence and fairness concernhad a systematic effect on the decision-making and per-formance in supply chains they also cause decision-makingbias erefore in order to pursue a win-win situation andreduce decision biases appropriate coordination mecha-nisms are especially necessary To date various contractshave been developed in behavioral supply chain manage-ment For fairness concern Cui et al [18] incorporatedfairness concern into a conventional supply chain andconsidered its influence on channel coordination between amanufacturer and a retailer Cuirsquos model was then extendedby Liu et al [21] to include a nonlinear demandey furtherdevised a revenue-sharing contract to coordinate a two-echelon CLSC with both membersrsquo fairness concernsNevertheless Zheng et al [4] proposed a cooperative gameapproach to coordinate a three-echelon closed-loop supplychain with fairness concern For overconfidence based onthe principal-agent theory Wang et al [23] designed anoptional contract to achieve a two-echelon supply chain withoverconfidence coordination and Pareto improvementJiang et al [24] studied the effect of supplier overconfidenceon buyback contracts in a financing supply chain esestudies revealed that fairness concerns and overconfidencecould complicate coordination in a supply chain and

existing research along this line is generally confined to asupply chain within a non-cooperative game setting

Based on these critical features of our model we tabu-lated our research in the context of a proper literature review(Table 1) e table revealed that the majority of the liter-ature concentrates on the coordination of different structuresupply chains with single-behavioral preference ie over-confidence or fairness concern However overconfidenceand fairness concern coexist in practice and this paperextends the depth and breadth of related literature aboutbehavioral preferences Besides the two cognitive biases(overconfidence and fairness concern) were simultaneouslytaken into consideration and the buyback contract withpromotional cost sharing was proposed in this paperNevertheless the focus of our paper was to investigate theimpact of retailersrsquo fairness concern and overconfidence onthe coordination results of a two-echelon supply chain byemploying a Stackelberg game approach We examined thevalidity of the contract under conditions that the retailer hadcognitive biases which were aligned with previous research

3 Assumptions and Models

31 Assumptions A two-echelon supply chain was con-sidered which included the rational manufacturer as theleader of the supply chain while the retailer had overcon-fidence and fairness concern e manufacturer producedproducts at unit production cost c and then sold them at unitwholesale price w to the retailer e retailer then sold thoseto customers at unit retail price p e manufacturer and theretailer showed risk neutrality Let s be the unit salvage valuev be the unit stockout cost and q be the order quantity of theretailer According to Ma et al [25] we assumed that thesales effort cost of the retailer was monotonically increasingas convex function ie C(e) denoted the sales effort costwhere C(e) (12)ce2 c(cgt 0) is the sales cost coefficientand e(egt 0) is the retailersrsquo sales effort denoted by thepromotional effort level Without loss of generality weassumed that 0lt slt clt vltp wgt c and qgt 0

In the two-echelon supply chain the manufacturerdetermined the wholesale price w and the retailer deter-mined the order quantity q and sales effort e ie w q and e

were decision variables of supply chain members s v and c

were assumed to be exogenous variables Note that the retailprice p was also assumed to be exogenous

32 Centralized Supply Chain Model In this section weconsidered the scenario wherein the manufacturer and theretailer made decisions as individuals e model with nobehavioral preferences was analyzed as a benchmark Be-sides the goal of the coordination contract in Section 4 wasto allow supply chain performance to achieve the optimalsituation under complete rationality Hence the benchmarkwas kept under complete rationality [8 26] However theonly goal was to maximize the overall profitability of thesupply chain by determining the optimal order quantity q

and sales effort e Previous studies proved that when decisionmakers were entirely rational the equilibrium solutions of

Complexity 3

the centralized supply chain were always better than thedecentralized behavioral supply chain [27 28] In order toexplore research problems and compare optimal decisionvariables to the decentralized supply chain with behavioralpreference analogs we assumed that the decision maker wasrational in the centralized supply chain According to Xuet al [16] the stochastic demand function is as follows

D a minus bp + e + θ (1)

where a signifies the market scale while b is the pricesensitivity coefficient of random demand e above func-tion indicated that sales effort e has a positive influence ondemand D(Dgt 0) e sales price p is exogenous thereforewe assumed that θ denotes the uncertainty of the demandwhich was a random variable following a uniform distri-bution on the interval [minus A A] wherein Agt 0 denotes thevariation range of the random variable For variable θ thecumulative distribution function is F(θ) and the probabilitydensity function is f(θ) According to the assumptionf(θ) (12A) and F(θ) (θ minus A2A) Also to avoid trivialcases we assumed Alt a and a minus bpgt 0 which guaranteedthat the demand is nonnegative

Under settings that did not guarantee market demandwe took q to be more than the demand D ie the decisionmaker faced the situation of overstock Under such a situ-ation we assumed minus Alt θlt q minus (a minus bp + e) In contrastwhen q was less than D q minus (a minus bp + e)lt θltA was ob-tained which indicated that the decision maker faced asituation of stockout We assumed that θ0 q minus (a minus bp + e)

indicated that order quantity was equal to the actual marketdemand D Hence f(θ0) (12A) and F(θ0) ((c minus v)(s minus v))

e profit for the centralized supply chain can beexpressed as

q emax

1113945

c

pE[min q D1113864 1113865] + E[s(q minus D)]+

minus E[v(D minus q)]+

minus cq minus C(e)

(2)

e first term in equation (2) is the expected revenue ofthe supply chain while the second term signifies the ex-pected surplus value if the situation of overstock happenede third is the expected loss provided the decision makersrsquoorder quantity was out of stock e term cq is the totalproduction cost and the last term indicated promotioneffort ie the cost of sales

e first-order derivatives of equation (2) with respect toq and e are

zE( 1113937c1113857

zq (s minus v)F θ0( 1113857 + v minus c 0

zE( 1113937c1113857

ze (p minus v) +(v minus s)F θ0( 1113857 minus ce 0

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(3)

erefore the Hessian matrix is

H(q e) (s minus v)f θ0( 1113857 0

0 (s minus v)f θ0( 1113857 minus c1113888 1113889 (4)

Note that s minus vlt 0 and f(θ0)gt 0 thus the Hessianmatrix Πc is a negative definite for q and e We can obtainoptimal order quantity and sales effort as follows

qclowast (v minus s)[(a minus bp)c +(p minus c)] + Ac(v + s minus 2c)

(v minus s)c

eclowast (p minus c)

c

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(5)

33 Decentralized Supply Chain Model In this section theoverconfidence and fairness concerns were incorporatedinto the two-echelon supply chain wherein the

Table 1 Differences between this paper and relevant studies

Papers Overconfidence Fairness concern Demand function CoordinationCroson et al [13] Stochastic demandLi et al [14] Deterministic demandLiu et al [15] Deterministic demandXu et al [16] Stochastic demandLi [17] Stochastic demandCui et al [18] Deterministic demand Wholesale price contractWu et al [19] Stochastic demandChen et al [20] Deterministic demand Revenue sharing contractLiu et al [21] Deterministic demand Membership-profit share contractZhang et al [22] Deterministic demandZheng et al [4] Deterministic demand Cooperative game mechanismWang et al [23] Stochastic demand Option contractJiang et al [24] Stochastic demand Wholesale price contract with buybackis study Stochastic demand Buyback contract with promotion cost sharing

4 Complexity

manufacturer was reported to be rational while the retailerhad a cognitive bias We assumed an overconfident retailerwho not only underestimated the variance of demand butalso overestimated the mean of demand and the effect ofsales effort on demand According to Chen et al [3] weadopted mean-increasing and variance-reducing transfor-mation of the actual market demand D us the marketdemand can be explained as follows

DO a minus bp +(1 + β)e +(1 minus β)θ a minus bp +(1 + β)e + θ1(6)

wherein β indicates the overconfident level (0lt βlt 1)Moreover the continuous random variable θ1 is equal to(1 minus β)θ which is a random noisy signal and assumed to beuniformly distributed ie θ1 sim U(minus A A) When β 0 theretailer had no overconfidence whereas for 0lt βlt 1 themean of DO increased and the variance of DO decreased ascompared to the mean and variance of D ie E(DO)gtE(D)

and var(DO)lt var(D) Meanwhile the impact of the pro-motion effort ((1 + β)egt e) on market demand was over-estimated ie the influence of e on market demandincreased as β increased

Similarly when market demand was uncertain weconsidered an order quantity q for the retailer withoverconfidence and fairness concern provided q was morethan the random demand Do Such an arrangement in-dicated that the retailer faced an overstock situation In thissituation we have minus Alt θ1 lt q minus [a minus bp + (1 + β)e] Incontrast if q was less than DOq minus [a minus bp + (1 + β)e]lt θ1 ltA was obtained which indi-cated that the retailer faced a stockout situation ie weassumed that θ2 q minus [a minus bp + (1 + β)e] HenceF(θ2) (μ(w minus c) + (1 + μ)(w minus v)(s minus v)(1 + μ)) eutility of the overconfident retailer is as follows

q emax

Uor pE min q Do1113864 11138651113858 1113859 + E s q minus DO( 1113857

+1113858 1113859

minus E v DO minus q( 11138571113858 1113859+

minus wq minus C(e)(7)

e reference framework for a retailer with fairnessconcern is the F-S model [29]

Ufr 1113945

r

minus μ 1113945m

minus 1113945r

0⎞⎠ minus λ 1113945r

minus 1113945m

0⎞⎠⎛⎝⎛⎝ (8)

where μ is the degree of fairness concern and λ is thecoefficient of sympathy μ λge 0 and 0lt λlt 1 e first partof equation (12) reflected the retailerrsquos profit whereas thesecond part evaluated the utility loss from disadvanta-geous inequality for the retailer when πm gt πr and thethird part measured the loss from advantageous inequalityfor the retailer when πm lt πr Note that if the retailerrsquosprofit was worse than the manufacturersrsquo profit (ie

πm gt πr) the retailer will have negative utility under thecondition of unfair aversion In contrast if his profit wasmore significant than the manufacturersrsquo the retailer willalso suffer positive utility because of the sympathyHowever Qin et al [30] demonstrated that the profit ofthe leader in a supply chain often accounted for more thanhalf of the overall profit when the Stackelberg game wascarried out In this paper the retailer is a weak follower ascompared to the manufacturer who served as the leaderAlongside this we found out that the profit of themanufacturer was twice that of the retailer when themanufacturer and the retailer were both unboundedlyrational Figures 4 and 5 validate the result in Section 5when μ 0 β 0 hence the retailer had the negativeutility of unfair aversion erefore according to equation(8) when the retailer had fairness concern only his utilityfunction was corrected to

Ufr 1113945

r

minus μ 1113945m

minus 1113945r

⎞⎠ (1 + μ)1113945r

minus μ1113945m

⎛⎝ (9)

where 1113937r and 1113937m denotes the profit of the rational retailerand the rational manufacturer respectively μgt 0 is thedegree of fairness concern

According to the theories of overconfidence and fairnessconcern the utility of the retailer with the two behavioralpreferences which was replaced by 1113937r in equation (9) withUo

r in equation (7) is as follows

q emax

U 1113945d

r

⎞⎠ (1 + μ)Uor minus μ1113945

d

m

⎛⎝ (10)

e retailer with behavioral preference orders q fromthe manufacturer the rational manufacturer profit func-tion is

wmax

1113945

d

m

(w minus c)q (11)

e Stackelberg game model where the manufacturerwas serving as the leader and the retailer served as thefollower was established in the decentralized supplychain e events of the model are described as followsFirstly the manufacturer decided the wholesale price wen the retailer made decisions about the order quantityq and the sales effort e based on the decision of themanufacturer and market demand To solve the subgameperfect equilibrium the backward induction method wasapplied e equilibrium solutions under the Stackelberggame model were considered when the retailer had acognitive bias e solutions are as follows e proof isgiven in Appendix A

Complexity 5

wdlowast

(a minus bp)(1 + μ)(v minus s)c +(v minus s)(1 + β)2[(1 + μ)p +(1 + 3μ)c] + Ac[(s + v)(1 + μ) + 2(1 + 3μ)c]

2(1 + 2μ) 2Ac +(v minus s)(1 + β)21113960 1113961 (12)

qdlowast

(a minus bp)(v minus s)c +(1 + β)2(v minus s)(p minus c) + Ac(s + v minus 2c)

2(v minus s)c (13)

edlowast

(1 + β) (a minus bp)(s minus v)c +(p minus c)(v minus s)(1 + β)2 + 2Ac(2p minus c minus s minus v)1113960 1113961

2c 2Ac +(1 + β)2(v minus s)1113960 1113961 (14)

Based on the above equilibrium solutions in thedecentralized supply chain the propositions about the effectof overconfidence and fairness concern (β and μ) on decisionvariables (qdlowast edlowast and wdlowast) are shown as in the followingNote that the proof of all propositions can be found inAppendix B

Proposition 1 When the retailer had both overconfidenceand fairness concern the optimal order quantity qdlowast and saleseffort edlowast were both positively correlated with the overcon-fident level β but had nothing to do with the degree of fairnessconcern μ Furthermore qdlowast is a strictly convex function withrespect to β

Proposition 1 indicated that the overconfident drivesthe retailer to increase order volume and improve salesefforts e reason is that overconfidence had two maineffects On the one hand the retailer overestimated theaverage market demand and exaggerated the effectivenessof sales effort which directly motivated the retailer to makea more considerable sales effort (for example increasingpromotions) On the other hand the retailer overestimatedthat the variance of the ldquorandom impact factor θ1rdquo issmaller than the actual level In other words from his pointof view a smaller market demand fluctuation influencedprediction us the two effects made the retailer ordermore product quantities and pay more sales effort for morerevenue Meanwhile it was easy to find that qdlowast and edlowast

were only related to β as per equations (13) and (14) whichrevealed that the retailersrsquo decisions on ordering and salesefforts were not affected by fairness concern

Proposition 2 When the retailer has both overconfidenceand fairness concern the optimal wholesale price wdlowast ispositively proportional to the overconfident level β but in-versely proportional to the degree of fairness concern μ

Proposition 2 illustrated that the cognitive bias of theretailer had an opposite effect on the manufacturersrsquodecision Firstly the moment the rational manufacturerrealized that the overconfident retailer could increase theorder quantity he raised the wholesale price of theproduct to maximize the profit which was consistent withthe anecdotes that merchants in real world could takeadvantage of othersrsquo overconfidence to maximize theirprofit Besides when the manufacturer understood thefeatures of the retailersrsquo fairness concern he lowered the

wholesale price to alleviate the retailersrsquo unfair aversionwhich implied that the retailer could boost his bargainingpower in the supply chain while he was concerned with thefairness erefore the rational manufacturer was re-quired to set a reasonable wholesale price to balance theinfluence of the two cognitive biases Next we comparedthe equilibrium outcomes in the centralized supply chainmodel with those in the decentralized supply chain modelere is a proposition as follows

Proposition 3 qclowastmay be more than or less thanqdlowast eclowastmaybe more than or less than edlowast Interestingly Δq qdlowast minus qclowastand Δe edlowast minus eclowast increases as the overconfidence level βincreases respectively

According to Proposition 3 the effect of overconfidencethe optimal order quantity and sales effort on the decen-tralized supply chain could deviate from the optimalequilibrium solution in the centralized supply chain whichresulted in a loss of the expected profits thereby increasingthe revenue gap between manufacturer and retailer

4 Buyback Contract with PromotionCost Sharing

It can be seen from Section 33 that due to the cognitivebias of the retailer his decisions deviated from the op-timal decisions in the centralized supply chain whichaffected the profit or utility of supply chain membersereby in order to coordinate the profit or utility be-tween the supply chain members and ensure the maxi-mized performance of the two-echelon supply chainaccording to Bai et al [31] the buyback contract withpromotional cost sharing was proposed in this sectionHere the manufacturer served as the leader who buysback the unsold products from the retailer at the buybackprice and shared the promotional cost We denoted thiscontract factor as (wsc r and 1 minus ϕ) where wsc is thewholesale price r is the buyback price and 1 minus ϕ is thefraction of promotional costs that the rational manu-facturer paid while the retailer undertook ϕ portion ofthe cost (0ltϕlt 1) In order to avoid retailersrsquo over-ordering and encourage the retailer to accept the con-tract we assumed 0lt sle rltwsc Based on thesedescriptions and assumptions above the manufacturersrsquoprofit and retailersrsquo utility are given by the followingexpression

6 Complexity

1113945

sc

m

(w minus c)q + E (s minus r) q minus DO( 11138571113858 1113859+

minus (1 minus ϕ)C(e)

(15)

U 1113945sc

r

⎞⎠ (1 + μ) p[(a minus bp +(1 + β)e)] + E r q minus DO( 1113857+

1113858 11138591113864⎛⎝

minus E v DO minus q( 1113857+

1113858 1113859 minus wq minus ϕC(e)1113865 minus μ1113945sc

m

(16)

wherein equation (15) is the profit when the manufacturerbuys back the unsold products at buyback price r from theretailer and thus obtained the surplus value of theseproducts (1 minus ϕ)C(e) is the promotional costs which themanufacturer undertook r(q minus DO)+ in equation (16) is therevenue when the retailer sells the unsold inventory of

products to themanufacturer at the end of the selling seasonϕC(e) is the promotional costs which he pay

According to the first derivations of equation (16) basedon q and e the following expression is obtained

qsc 2A(1 + μ)(v minus w) minus μ(w minus c)

(1 + μ)(v minus r) + μ(s minus r)+ a minus bp +(1 + k)e

scminus A

esc (1 + β)[μ(p + c minus 2w) + p minus w]

(2μϕ minus μ + ϕ)c

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(17)

In the buyback contract with promotion cost sharing toachieve supply chain coordination the optimal decisions ofthe retailer were the same as those under the centralizedsupply chain model that is qsc qclowast and esc eclowast [25 26]erefore equation (17) needs to be equal to equation (5) asthe contract factors (wsc r) were obtained

wsclowast

[β(1 + μ) + 1 minus ϕ(1 + 2μ)]p +[βμ +(1 + 2μ)ϕ]c

(1 + 2μ)(1 + β) (18)

rlowast

(v minus s)(p minus c)(1 + β)(1 + μ)vβ + 2A[cϕ(p minus c)(1 + 2μ) + βcv(1 + μ)] minus 2Ac p(v minus s)[1 + β +(2 + β)βμ] minus sv(1 + β)(1 + μ)1113864 1113865

(1 + 2μ)(1 + β)[β(p minus c)(v minus s) + 2Ac(c minus v)]

(19)

e purpose of the buyback contract with promotioncost sharing was to design a reasonable profit allocationscheme between the rational manufacturer and the retailerwith cognitive bias It should be noted that if the profit ofsupply chain members became lower the contract wasunacceptable e contract was acceptable if and only if1113937

scm ge 1113937

dlowastm and U(1113937

scr )geU(1113937

dlowastr )

Based on the above equilibrium solutions in the contractthe following propositions were obtained Due to the so-phisticated formula of the buyback price this section dis-cussed the effect of overconfidence and fairness concern onwholesale price

Proposition 4 e optimal wholesale price in the buybackcontract with promotion cost sharing was negatively corre-lated with μ and ϕ Nevertheless if ϕlt (μ(1 + 2μ)) thewholesale price wsclowast is positively correlated with β In con-trast if ϕlt (μ(1 + 2μ)) it is negatively correlated withβ

Proposition 4 implied that the contract was adopted andthe negative effect of fairness concern μ on wholesale pricewsclowast remained consistent with Proposition 2 Besides as wsc

increased ϕ decreased and 1 minus ϕ increased accordinglywhich meant that the manufacturer bore more promotionalcost when the wholesale price rose which is in line with thepractice Meanwhile the values of μ and ϕ determined theimpact of β (positive or negative influence) on the wholesaleprice wsc which indicated that fairness concern μ andpromotional cost portion 1 minus ϕ were critical to manufac-turerrsquos decisions on the wholesale price wsc us when the

two types of behavioral preferences coexisted in the contractthe effect of overconfidence on the optimal wholesale pricedepended on the relationship between fairness degree andfraction of promotion cost Since the utility and profitfunctions in the supply chain included a large number ofunknown parameters under three scenarios (ie Sections32 33 and 4) it was difficult to compare and analyze theeffects of the cognitive biasesus numerical methods wereutilized to conduct further analysis in the next section

5 Numerical Analysis

In this section a numerical study approach was adopted toallow a more in-depth insight into the two behavioralpreferences First a numerical experiment was conducted toshow how equilibrium solutions and profit (utility) in thetwo-echelon supply chain changed along with the levels ofoverconfidence and fairness concern (ie β and μ) and toexamine the robustness of propositions en we also de-termined the effect of overconfidence and fairness concernon the buyback contract with promotional cost sharing

In our experiments some parameters were gatheredfrom the existing papers For the parameters related tooverconfidence Ren et al [1] set β isin [0 1] andc 05 06 08 09 to perform their numerical experi-ments and Li et al [7] set β isin (0 1) and c 5 to conducttheir study For the parameters related to fairness concernZheng et al [4] set μge 0 c 20 and a 100 to implementtheir numerical analysis and Guo et al [9] set μge 0 toperform their studies According to the parameters above

Complexity 7

we assumed β isin [0 1) and μ 0 1 2 to reflect the level ofoverconfidence and fairness concern in this paper Whenβ 0 and μ 0 the retailer was considered to be rationalSimilarly other fundamental parameter values were set inexperiments as follows a 50 θ sim U(minus 30 30) c 4p 20v 7 s 2 b 2 and c 1 All the parameterssatisfied the constraint conditions and assumptions of thethree models in Sections 32 33 and 4e parameter valueswere also in line with the real situation of the market Wecould easily obtain qclowast 32 eclowast 16 and 1113937

c 252 in the

centralized supply chain

51 Effect of Overconfidence and Fairness Concern on theSupply Chain e changes in the decision variables (w qand e) manufacturer profit and retailer expected utilitywhen the level of overconfidence and fairness concern (β andμ) changed are shown in Figures 1ndash5

As shown in Figures 1 and 2 when the retailer was ra-tional (eg β 0 and μ 0) in the decentralized supplychain both optimal order quantity and sales effort were lowerthan those under the centralized supply chain Next when theretailer had overconfidence and fairness concern simulta-neously overconfidence could only affect the retailerrsquos opti-mal order quantity and sales effort of the retailer because theoptimal order quantity and sales effort were a function ofoverconfidence in equations (13) and (14) Moreover theorder quantity was monotonically increasing as concavefunction with respect to β In comparison sales effort wasmonotonically increasing as convex function with respect toit Furthermore there existed a threshold (β 083) when βwas less than 083 and the two decision variables were bothless than the optimal values (qclowast 32 eclowast 16) under thecentralized decision scenario Nevertheless once β was largerthan 083 the two decision variables were both more thanthem e above discussion verified the correctness ofProposition 1 In short overconfidence was the main factorthat affected the retailerrsquos optimal strategies of order quantity

and sales effort in the decentralized supply chain Howeverthe above findings contradicted with the conclusions from astudy conducted by Xu et al [16] which examined the effectof retailerrsquos overconfidence on the supply chain and the resultsuggested that a higher level of overconfident resulted in thelower ordering quantity

In Figure 3 we observed that the higher overconfidentlevel resulted in the high wholesale price in contrast ahigh degree of fairness concern resulted in low pricewhich demonstrated that Proposition 2 was robust isfigure implied that when the information for both partieswas symmetric the rational manufacturer knew that theretailer was overconfident that is the retailer was opti-mistic about the market demand and increased the orderquantity As a result the manufacturer also raised thewholesale price to maximize its revenue Also if theoverconfident retailer paid attention to the unfairness ofprofit allocation in the decision-making process the re-tailerrsquos bargaining power was enhanced [32] Hence forbetter performance in the supply chain the leadingmanufacturer was required to lower wholesale prices toease the feeling of unfairness by the retailer which isconsistent with the actual economic phenomenon Fur-thermore these findings are consistent with the resultconcluded by Cui et al [2] which indicated that a highdegree of the retailerrsquos fairness concern could lower thewholesale price Interestingly as the levels of the cognitivebias increased its effect on wholesale price graduallydecreased which highlighted its effect on decreasedwholesale prices

Figure 4 displays the impact of the cognitive bias on theretailerrsquos utility e graph showed that the retailer expectedutility increased gradually with the overconfident level andthe degree of fairness concern which suggested that twocognitive biases were beneficial to the retailer e reasonsfor such a phenomenon were that both cognitive biasescould increase the optimal order quantity and sales effortthus enhancing the utility of the retailer

Order quantity in centralized supply chain

01 02 03 04 05 06 07 08 09 10β

15

20

25

30

35

40

Ord

er q

uant

ity

μ = 012

Figure 1 e effect of the cognitive biases on optimal order quantity

8 Complexity

Figure 5 illustrates the impact of overconfidence andfairness concern on the manufacturerrsquos profit at is theoptimal manufacturerrsquos profit is an increasing function of βbut a decreasing function of μ According to Figure 3 thegraph of themanufacturerrsquos profit was similar to the graph ofthe wholesale price which indicated that the manufacturerrsquosprofit was firmly related to wholesale price Figure 5 alsoillustrates the effect of the cognitive bias β on the manu-facturerrsquos decreasing profit

52 Effect of Overconfidence and Fairness Concern on SupplyChain Coordination As can be seen from Section 51 thereexist deviations between the optimal decisions under the

benchmark and the decisions in the presence of the retailerrsquoscognitive biases erefore the buyback contract withpromotion cost sharing was necessary to coordinate therevenue of supply chain members and optimize the per-formance of the supply chain

Here we discussed the validity of the contract When theoptimal order quantity and sales effort of the retailer underthe contract were the same as the optimal decision variablesin the centralized supply chain ie qsc qclowast 32 andesc eclowast 16 the buyback contract with promotion costsharing achieved supply chain coordination First the effectof overconfidence and fairness concern (β and μ) on contractfactors wsrlowast rlowast and ϕ is shown in Figures 6 and 7 andTable 1 For clearly analyzing the effect of β and μ on wsrlowast

Degree of sales effort in centralized supply chain

01 02 03 04 05 06 07 08 09 10β

μ = 012

4

6

8

10

12

14

16

18

Deg

ree o

f sal

es ef

fort

Figure 2 e effect of the cognitive biases on optimal sales effort

01 02 03 04 05 06 07 08 09 10β

-μ = 0--μ = 1-μ = 2

4

5

6

7

8

9

10

11

12

13

14

Who

lesa

le p

rice

Figure 3 e effect of the cognitive biases on the optimal wholesale price

Complexity 9

and rlowast we let ϕ 1 in equations (18) and (19) which meantthat the manufacturer proposed the buyback contract onlyand the retailer bore all sales effort costs

According to Figure 6 it was revealed that the optimalwholesales price is a strictly increasing function of theoverconfidence level β but a strictly decreasing function ofthe degree of fairness concern μ which confirmed the resultof Proposition 4 Similarly Figure 7 highlights that theoptimal buyback price sharply increased with β but de-creased with μ Moreover in the buyback contract wsrlowast wasalways greater than or equal to 4 and rlowast was always greaterthan or equal to 2 in different cognitive bias levels Hence

the optimal buyback price rlowast satisfied the assumptions 2le r Note that the above characteristics influencedcognitive biases hence contract factors wsrlowast and rlowast werealso correct when ϕ is equal to other values (such asϕ 03 05 08) under conditions in which the buybackcontract had promotion cost sharing

However data from Figure 7 can be compared with thedata in Figure 6 which showed rsrlowast is more significant thanwsrlowast when β was within a specific range (for example04le βlt 1) which did not match the assumption rltwsrerefore the buyback contract did not achieve the coor-dination of the behavioral supply chain

01 02 03 04 05 06 07 08 09 10β

-μ = 0--μ = 1-μ = 2

0

20

40

60

80

100

120

Reta

iler e

xpec

ted

utili

ty

Figure 4 e effect of the cognitive biases on retailer utility

01 02 03 04 05 06 07 08 09 10β

-μ = 0--μ = 1-lowastμ = 2

0

20

40

60

80

100

120

140

160

180

200

Man

ufac

ture

r pro

fit

Figure 5 e effect of the cognitive biases on manufacturer profit

10 Complexity

As described in Section 4 if the supply chain membersaccepted the buyback contract with promotion cost sharingthe assumptions 1113937

scm gemax 1113937

dm and U(1113937

scr )gemaxU(1113937

dr )

had to be satisfied Also according to the assumptions thevalue ϕ must satisfy 1113937

scm gemax 1113937

dm and

U(1113937scr )gemaxU(1113937

dr ) us we kept the values of param-

eters to be the same (ie a 50 θ sim U(minus 30 30) c 4p 20 v 7 s 2 b 2 and c 1) to capture a likelydomain of ϕ

As can be seen from Table 2 the overconfident level βincreased the upper and lower limits of ϕ which had a clearupward trend In contrast the upper and lower limits of ϕ

showed a definite declining trend as the degree of fairnessconcern μ increased ese trends illustrated that in order tocoordinate the revenue of both parties and optimize theperformance of the supply chain the manufacturer needs tobear less promotional costs as β increased but bear morepromotional costs as μ value increased Besides when β and μincreased the interval of sharing cost portion ϕ shrunk whichindicated that the buyback contract with promotional costsharing had certain flexibility to coordinate the revenue be-tween supply chain members However the flexibility couldbe weakened with the rise of the cognitive bias level Forsupply chain members the possibility of negotiation or

01 02 03 04 05 06 07 08 09 10β

-μ = 0--μ = 1-μ = 2

4

5

6

7

8

9

10

11

12

Who

lesa

le p

rice

Figure 6 e effect of the cognitive biases on wholesale price wsrlowast

01 02 03 04 05 06 07 08 09 10β

-μ = 0--μ = 1-μ = 2

2

4

6

8

10

12

14

16

18

20

22

Buyb

ack

pric

e

Figure 7 e effect of the cognitive biases on buyback price rlowast

Complexity 11

cooperation between manufacturers and retailers also de-creased with an increase in retailersrsquo cognitive bias levelWhen the overconfident level was more than 081 the re-tailerrsquos utility from decentralized decisions was greater thanthe utility in the contract environment wherein the twoparties could not reach a cooperation agreement us thesupply chain could not achieve coordination as different levelsof cognitive bias under the contract had the same influence ondecision variables profit and utility Table 2 shows a clearpicture wherein μ is equal to 1 and β falls within the range of(0 08) which were used to analyze the effect of the cognitivebiases on the coordination of the supply chain

e results of the numerical simulation in Table 3 in-dicate that the contract factors ie ϕ r wholesale price theprofit and utility of members existed as flexible intervalswhen β isin (0 08) and μ 1 Besides the order quantity qsc

and the degree of sales effort esc under the contract are equalto qclowast and eclowast in a centralized supply chain respectivelyMeanwhile the buyback price r was less than the wholesaleprice wsc and higher than the unit salvage value s whichsatisfied the constrains sle rltwsclowast us the buybackcontract with promotion cost could achieve the coordinationof the two-echelon supply chain Also the buyback price r

was positively correlated with the portion ϕ of promotioncost sharing Notably the wholesale price wsc dropped as ϕincreased when β was in the range of (0 02) in Table 3However the situation reversed when it was in the range of(02 08) which was inconsistent since the manufacturerlowered the wholesale price when the retailer paid morepromotional costs in the contract e reason for such anobservation was that the wholesale price was not related to ϕbut positively related to the overconfident level β when thedegree of fairness concern was moderate Furthermore theretailer accepted the contract designed by manufacturerssince the term U(1113937

scr )gemaxU(1113937

dr ) was guaranteed in

Table 3

6 Conclusions

61MainConclusions In this paper we analyzed the effect ofcognitive bias on equilibrium solutions including thewholesale price sales price sales effort and expected profitsunder demand uncertainty by introducing both overconfi-dence and fairness concern into the two-echelon supply chaine most important conclusion from the study was that forretailers with the two cognitive biases overconfidence waspositive and the main bias that affected supply chain decisionsand the retailersrsquo utility benefits were not only due to over-confidence but also fairness concern For manufacturersoverconfidence could increase the wholesale price and profitHowever the fairness concern had a negative effect on themFurthermore the decisions of the retailer in the decentralizedsupply chain deviated from the optimal decisions in thecentralized supply chainmodel Especially when the thresholdof overconfidence was more than a fixed value the decisionvariables under decentralized decision-making were actuallyhigher than the optimal variables under centralized decision-making Also the buyback contract with promotional costsharing achieved behavioral supply chain coordination Incontrast the promotional cost-sharing portion and buybackprice both benefited from overconfidence but fairness concernhad a detrimental effect on them However as the retailersrsquooverconfidence level increased and fairness concern was equalto 1 the flexibility of contract coordination was weakenedWhen the overconfidence level wasmore than 08 the contractbecame invalid and members were not able to cooperate

62 Management Insights is paper provided some in-teresting managerial insights for decision makers in thedecentralized supply chain wherein retailers had two pref-erences First overconfidence was beneficial to them interms of decision variables and yet the fairness concern

Table 2 e effect of the cognitive biases on the feasible domain of promotion cost-sharing portion ϕ

μβ

0 02 04 06 08 0810 [02884 04976] [03683 05587] [04528 06199] [06343 07754] [07835 08669] mdash1 [02545 04741] [03310 05331] [04167 05908] [06207 07516] [07800 08560] mdash2 [02352 04600] [03101 05156] [04034 05719] [06094 07321] [07736 08478] mdash

Table 3 e effect of the cognitive biases on the coordination of the supply chain

μ 1β

0 02 04 06 08qsc 32 32 32 32 32esc 16 16 16 16 16ϕ [02545 04741] [03310 05331] [04167 05908] [06207 07516] [07800 08560]r [2100 35363] [28268 39188] [30555 42637] [36209 44215] [40485 49164]wsc [49205 53687] [49016 53522] [54092 69568] [62513 72626] [80407 87453]1113937

dm 164103 269591 461996 772619 1112545

U(1113937scm) [168722 217850] [269600 336882] [463175 500938] [772619 814300] [1112545 1120376]

1113937dr 20333 352396 495161 632422 765317

U(1113937scr ) [20333 243043] [352396 400595] [495161 525161] [632422 657581] [765317 784632]

12 Complexity

only damaged the wholesale price Second the two behav-ioral preferences were both conducive to utilities for theretailer but not necessarily beneficial for the manufacturer toreceive profit and overconfidence helped to increasemanufacturersrsquo profit which led to nonoptimal decisionsthat dragged down the performance of the two-echelonsupply chain Finally for the two behavioral preferences ofthe retailer the manufacturer assessed the degree of over-confidence and fairness concern (especially the overconfi-dence) via historical data and the current order quantity orother indicators In contrast factual data indicated that twobehavioral preferences were in the right range (such asβ isin (0 08) and μ isin (0 2)) e manufacturers introducedthe buyback contract with promotional cost sharing into thepractice of pricing and ordered decentralized supply chainsFurthermore they designed a reasonable promotional cost-sharing portion and buyback price thereby achieving anoptimized decision and utility In practice the manufac-turers could analyze whether the sellers had two behavioralpreferences based on the historical data by cooperating withdownstream enterprises ie if two behavioral preferencesexisted they referred to the papersrsquo conclusions to redesign areasonable contract mechanism

63 Future Research Since cognitive biases could existamong all members of supply chains it is necessary toexplore the manufacturersrsquo cognitive biases and analyze thesupply chain coordination problem in the presence of thetwo behavior preferences Besides the overconfidence andfairness concern considered in our centralized decision are afuture research direction

Appendix

A Decentralized Supply Chain Model

In a decentralized supply chain consisting of a rationalmanufacturer and a retailer with cognitive biases maximumprofit or utility is the only goal e backward inductivemethod can be applied to solve a Stackelberg game whereinthe manufacturer is the leader and the retailer is a followere proof of the equilibrium solutions in equations (12)ndash(14)is as follows

e profit of the manufacturer and expected utility ofretailers are as follows respectively

wmax

1113945

d

m

(w minus c)q (A1)

maxqe

U 1113945d

r

⎞⎠ (1 + μ) p[a minus bp +(1 + β)e] + s 1113946θ2

minus Aq minus a minus bp +(1 + β)e + θ1( 1113857f θ1( 1113857( 1113857dθ11113896⎛⎝

minus v 1113946A

θ2a minus bp +(1 + β)e + θ1 minus q( 1113857f θ1( 1113857( 1113857dθ1 minus wq minus

12

ce21113897 minus μ(w minus c)q

(A2)

Solving the first-order partial derivatives of the retailerrsquosexpected utility concerning qd and ed which equal to 0 weobtained the following equations

zU( 1113937dr 1113873

zq (1 + μ) v minus w + F θ2( 1113857(s minus v)1113858 1113859 minus μ(w minus c) 0

(A3)

zU( 1113937dr 1113873

ze (1 + β) p minus v +(v minus s)F θ2( 11138571113858 1113859 minus ce 0

(A4)

Hessian matrix

H(q e) (s minus v)f θ2( 1113857 (v minus s)(1 + β)f θ2( 1113857

(v minus s)f θ2( 1113857 (s minus v)(1 + β)f θ2( 1113857 minus c1113888 1113889

(A5)

According to the assumptions vgt sgt 0 f(θ2)gt 0 andβ isin [0 1) us the Hessian matrix of U(1113937

dr ) is a negative

definite for q and e which meant that the optimal orderquantity and sales effort of retailers existed in the decen-tralized supply chain erefore we obtained the followingformula of the retailersrsquo order quantity and sales effort byjointly solving equations (A3) and (A4)

qd

(a minus bp)(v minus s)(1 + μ)c +(1 + β)2(v minus s)[(1 + μ)(p minus w) + μc] + Ac[(s + v)(1 + μ) + 2μc minus w]

(v minus s)(1 + μ)c (A6)

ed

(1 + β)[(1 + μ)p + μc minus (1 + 2μ)]w

(1 + μ)c (A7)

Complexity 13

Substituting equations (A6) into (A1) we attained

wmax

1113945

d

m

(w minus c) (a minus bp)(v minus s)(1 + μ)c +(1 + β)2(v minus s)[(1 + μ)p + μc] + Ac[(s + v)(1 + μ) + 2μc] minus (1 + 2μ)w 2Ac +(1 + β)2(v minus s)1113960 11139611113966 1113967

(v minus s)(1 + μ)c

(A8)

Taking into account the first-order condition of equation(A7) with respect to w rounded off to 0 we obtained`

z1113937dm

ze

(a minus bp)(v minus s)(1 + μ)c +(1 + β)2(v minus s)[(1 + μ)p + μc] + Ac[(s + v)(1 + μ) + 2μc] minus (2w minus c)(1 + 2μ) 2Ac +(1 + β)2(v minus s)1113960 1113961

(v minus s)(1 + μ)c

(A9)

Solving equation (A8) we found that the optimalwholesale price of the manufacturer is as follows

wdlowast

(a minus bp)(1 + μ)(v minus s)c +(v minus s)(1 + β)2[(1 + μ)p +(1 + 3μ)c] + Ac[(s + v)(1 + μ) + 2(1 + 3μ)c]

2(1 + 2μ) 2Ac +(v minus s)(1 + β)21113960 1113961 (A10)

By substituting equation (A9) into equations (A6) and(A7) we attained the optimal order quantity and sales effortwhich are displayed in equations (13) and (14)

B Propositions

Proof of Proposition 1 We solved the first derivatives ofoptimal order quantity qdlowast and sales effort edlowast with respect

to β in equations (13) and (14) respectively Meanwhiletaking the second-order derivative of qdlowast with respect to βwe obtained

zqdlowast

zβ

(1 + β)(p minus c)

c

z2qdlowast

zβ2

p minus c

c (B1)

zedlowast

zβ4(p minus c)(v minus s)2β β3 + 4β2 + 6β + 41113872 1113873 +(a minus bp)c(1 + β)2(v minus s)2 + Ac(v minus s)(1 + β)2 (2(p minus c) + v + s) + 2(Ac)2(4p minus c minus s minus v)1113960 1113961

2c (1 + β)2(v minus s) + 2Ac1113960 11139612

(B2)

Since the second derivative of edlowast with respect to β wasvery complicated there was no need to solve it According tothe assumptions pgt vgt cgt sgt 0 β isin (0 1) a minus bpgt 0 andcgt 0 so (zqdlowastzβ)gt 0 (z2qdlowastzβ2)gt 0 and (zedlowastzβ)gt 0

Proof of Proposition 2 Based on the process above we tookthe first-order and second-order partial derivatives of op-timal wholesale price wdlowast with respect to β and μ in equation(12) respectively

zwdlowast

zβ

c(v minus s)(1 + μ)(1 + β)[(a minus bp)(v minus s) + A(2p minus s minus v)]

(1 + 2μ) (1 + β)2(v minus s) + 2Ac1113960 11139612 (B3)

zwdlowast

zμ minus

(a minus bp)(v minus s)c +(p minus c)(v minus s)(1 + β)2 + Ac(s + v minus 2c)1113960 1113961

2(1 + 2μ)2 (v minus s)(1 + β)2 + 2Ac1113960 1113961 (B4)

14 Complexity

According to the optimal result of order quantity inequation (13) and the assumption qgt 0 the condition s +

v minus 2cgt 0 is found According to the assumptions we had(zwdlowastzβ)gt 0 and (zwdlowastzμ)lt 0

Proof of Proposition 3 e gap between qdlowast and qclowastdenoted by Δq is as follows

Δq qdlowast

minus qclowast

(p minus c)(v minus s) (1 + β)2 minus 21113960 1113961 minus Ac(s + v minus 2c) minus (a minus bp)(v minus s)c

2c(v minus s) (B5)

If (p minus c)(v minus s)[(1 + β)2 minus 2] is more thanAc(s + v minus 2c) + (a minus bp)(v minus s)c we have Δqgt 0 ieqdlowast gt qclowast In contrast if (p minus c)(v minus s)[(1 + β)2 minus 2] is less

than Ac(s + v minus 2c) + (a minus bp)(v minus s)c Δqlt 0 ieqdlowast lt qclowast

e gap between edlowast and eclowast denoted by Δe is asfollows

Δe edlowast

minus eclowast

Aβc(4p minus 2c minus s minus v) minus Ac(s + v minus 2c) minus (v minus s)(1 + β) (a minus bp)c + 1 minus β21113872 1113873(p minus c)1113960 1113961

2c 2Ac +(1 + β)2(v minus s)1113960 1113961 (B6)

Similarly if Aβc(4p minus 2c minus s minus v)gtAc(s + v minus 2c)

+(v minus s)(1 + β)[(a minus bp)c + (1 minus β2)(p minus c)] we haveΔe edlowast minus eclowast gt 0 In contrast ifAβc(4p minus 2c minus s minus v)ltAc(s + v minus 2c)+

(v minus s)(1 + β)[(a minus bp)c + (1 minus β2)(p minus c)] we haveΔe edlowast minus eclowast lt 0

Upon solving the first-order derivative of Δq and Δe withrespect to β respectively we obtained

zΔ q

zβ

(1 + β)(p minus c)

c (B7)

zΔ e

zβ4(p minus c)(v minus s)2 β3 + 4β2 + 6β + 41113872 1113873β +(a minus bp)c(1 + β)2(v minus s)2 + Ac(v minus s)(1 + β)2 2(p minus c) +v + s) + 2(Ac)2(4p minus c minus s minus v)1113960 1113961

2c (1 + β)2(v minus s) + 2Ac1113960 11139612

(B8)

According to the assumptions pgt vgt cgt sgt 0 β isin (0 1)cgt 0 and a minus bpgt 0 (zΔ qzβ)gt 0 and (zΔ ezβ)gt 0 can beproved Hence the gaps increased as a function with respectto β

Proof of Proposition 4 e proof for Proposition 4 is similarto Proposition 2 Herein we took first-order and second-order partial derivatives of optimal wholesale price wsclowast withrespect to β μ and ϕ in equation (18) respectively esolving results are as follows

zwsclowast

zβ

[(1 + 2μ)ϕ minus μ](p minus c)

(1 + 2μ)(1 + β)2

zwsclowast

zμ minus

(p minus c)β(1 + β)(1 + 2μ)2

zwsclowast

zϕ minus

p minus c

1 + β

(B9)

(zwsclowastzβ) if ϕgt (μ(1 + 2μ)) is more than 0 otherwiseϕgt (μ(1 + 2μ)) is less than 0 us wsc could be increasingwith respect to β nevertheless decreasing with respect to βBesides according to the assumptions pgt vgt cgt sgt 0β isin (0 1) cgt 0 and a minus bpgt 0 it can be evidently foundthat (zwsclowastzμ)gt 0 and (zwsclowastzϕ)gt 0

Data Availability

e data used to support the findings of this study are in-cluded within the article

Conflicts of Interest

e authors declare no conflicts of interest

Acknowledgments

is work was supported by a major project financed by theNational Social Science Fund of China (Approval nos71461009 71662011 and 71761015)

Complexity 15

References

[1] Y Ren D C Croson and R T A Croson ldquoe overconfidentnewsvendorrdquo Journal of the Operational Research Societyvol 68 no 5 pp 496ndash506 2017

[2] T H Cui and P Mallucci ldquoFairness ideals in distributionchannelsrdquo Journal of Marketing Research vol 53 no 6pp 969ndash987 2016

[3] K Chen X Wang M Huang and W-K Ching ldquoSalesforcecontract design joint pricing and production planning withasymmetric overconfidence sales agentrdquo Journal of Industrialamp Management Optimization vol 13 no 2 pp 873ndash8992017

[4] X X Zheng Z Liu K W Li et al ldquoCooperative game ap-proaches to coordinating a three-echelon closed-loop supplychain with fairness concernsrdquo International Journal of Pro-duction Economic vol 212 pp 92ndash110 2018

[5] China Internet Watch False Prosperity behind Double 11Refund Rate Increased to 69 2014 httpswwwchinainternetwatchcom11643false-prosperity-behind-double-11-refund-rate-increased-to-69

[6] C L Lu ldquoPampGrsquos collaborative supply chain transformationrdquoEnterprise Management vol 10 pp 79-80 2016

[7] M Li N C Petruzzi and J Zhang ldquoOverconfident competingnewsvendorsrdquo Management Science vol 63 no 8pp 2637ndash2646 2016

[8] T Nie ldquoDual-fairness supply chain with quantity discountcontractsrdquo European Journal of Operational Researchvol 258 no 2 pp 491ndash500 2017

[9] X Du and B Jiang ldquoSignaling through price and quality toconsumers with fairness concernsrdquo Journal of MarketingResearch vol 53 no 6 pp 988ndash1000 2016

[10] V Kurz A Orland and K Posadzy ldquoFairness versus effi-ciency how procedural fairness concerns affect coordina-tionrdquo Experimental Economics vol 21 no 3 pp 601ndash6262018

[11] S Villa and J A Castantildeeda ldquoTransshipments in supplychains a behavioral investigationrdquo European Journal of Op-erational Research vol 269 no 2 pp 715ndash729 2018

[12] B Gu Y Fu and Y Li ldquoFresh-keeping effort and channelperformance in a fresh product supply chain with loss-averseconsumersrsquo returnsrdquo Mathematical Problems in Engineeringvol 2018 Article ID 4717094 20 pages 2018

[13] D Croson R Croson and Y Ren How to Manage anOverconfident Newsvendor 2008 httpcbeesutdallasedupapersCrosonRenCmsonMS2008

[14] M Li ldquoOverconfident distribution channelsrdquo Production andOperations Management vol 28 no 6 pp 1347ndash1365 2019

[15] W Liu D Wang O Tang and D Zhu ldquoe impacts oflogistics service integratorrsquos overconfidence behaviour onsupply chain decision under demand surgerdquo EuropeanJournal of Industrial Engineering vol 12 no 4 pp 558ndash5972018

[16] L Xu X Shi P Du K Govindan and Z Zhang ldquoOptimi-zation on pricing and overconfidence problem in a duopo-listic supply chainrdquo Computers amp Operations Researchvol 101 pp 162ndash172 2019

[17] Y Li and M Shan ldquoAdvance selling decisions with over-confident consumersrdquo Journal of Industrial and ManagementOptimization vol 2 no 3 pp 891ndash905 2016

[18] T H Cui J S Raju and Z J Zhang ldquoFairness and channelcoordinationrdquo Management Science vol 53 no 8pp 1303ndash1314 2007

[19] X Wu and J A Niederhoff ldquoFairness in selling to thenewsvendorrdquo Production and Operations Managementvol 23 no 11 pp 2002ndash2022 2014

[20] Z Chen and S-I Ivan Su ldquoSocial welfare maximization withthe least subsidy photovoltaic supply chain equilibrium andcoordination with fairness concernrdquo Renewable Energyvol 132 pp 1332ndash1347 2019

[21] W Liu D Wang X Shen X Yan and W Wei ldquoe impactsof distributional and peer-induced fairness concerns on thedecision-making of order allocation in logistics service supplychainrdquo Transportation Research Part E Logistics and Trans-portation Review vol 116 pp 102ndash122 2018

[22] L Zhang H Zhou Y Liu and R Lu ldquoOptimal environmentalquality and price with consumer environmental awarenessand retailerrsquos fairness concerns in supply chainrdquo Journal ofCleaner Production vol 213 pp 1063ndash1079 2019

[23] X LWang S Q Hu and X B Liu ldquoSupply coordination withoption contract considering overconfidence under randomdemand and yieldsrdquo Computer Integrated ManufacturingSystem vol 24 no 11 pp 2898ndash2908 2018

[24] W Jiang and J Liu ldquoInventory financing with overconfidentsupplier based on supply chain contractrdquo MathematicalProblems in Engineering vol 2018 Article ID 505438712 pages 2018

[25] P Ma H Wang and J Shang ldquoContract design for two-stagesupply chain coordination integrating manufacturer-qualityand retailer-marketing effortsrdquo International Journal ofProduction Economics vol 146 no 2 pp 745ndash755 2013

[26] J Liu W F Jiang and Y P Tang ldquoe supply chain coor-dination based on carbon reduction of overconfident man-ufacturerrdquo Journal of Jiangxi Normal University (NaturalScience) vol 42 no 2 pp 203ndash207 2018

[27] R Dubey N Altay and C Blome ldquoSwift trust and com-mitment the missing links for humanitarian supply chaincoordinationrdquo Annals of Operations Research vol 283 no 1-2 pp 159ndash177 2019

[28] C Cui Z Feng and C Tan ldquoCredibilistic loss aversion nashequilibrium for bimatrix games with triangular fuzzy payoffsrdquoComplexity vol 2018 Article ID 7143586 16 pages 2018

[29] E Fehr and KM Schmidt ldquoA theory of fairness competitionand cooperationrdquo e Quarterly Journal of Economicsvol 114 no 3 pp 817ndash868 1999

[30] Z Qin and J Yang ldquoAnalysis of a revenue-sharing contract insupply chain managementrdquo International Journal of LogisticsResearch and Applications vol 11 no 1 pp 17ndash29 2008

[31] S-Z Bai and J Ni ldquoResearch on buy-back contract with saleseffort cost sharingrdquo in Proceedings of the 2009 Chinese Controland Decision Conference IEEE Guilin China pp 1917ndash1921June 2009

[32] E Karagozoglu and K Keskin ldquoTime-varying fairness con-cerns delay and disagreement in bargainingrdquo Journal ofEconomic Behavior and Organization vol 147 pp 115ndash1282018

16 Complexity

retailersrsquo overconfidence had the same definition Besideswe also considered that the overconfident retailer over-estimated the mean of market demand and the effects oftheir own sales effort on demand Moreover the mostexisting papers only focused on the effect of overconfidenceon the ordering or pricing we extended previous researchand analyzed the effect of overconfidence on pricing or-dering and sales effort

22 Fairness Concern Fairness concern has been studiedextensively in SCM Cui et al [18] analytically and experi-mentally evaluated how fairness significantly affected thepricing decisions of the manufacturer and the retailerSimilar issues were considered by Wu et al [19] who ex-amined the influence of fairness concern on the retailersrsquoprofit allocation in the newsvendorsrsquo model A recent paperby Chen et al [20] also analyzed the interactions betweenfairness concern and decisions including pricing and servicelevel in a dual-channel supply chain consisting of a man-ufacturer and two retailers Also Liu et al [21] investigatedthe impact of different fairness concerns on order allocationin the logistics service supply chain Zhang et al [22]revealed the characteristics of price changes when fairnessconcern was categorized into unfavorable and favorabledisutility respectivelye above papers mainly studied howfairness concerns affected decision variables in varioussupply chain structures Motivated by those papers we alsoanalyzed the decision-making problems of members in atwo-echelon supply chain wherein the retailer had fairnessconcerns However we extended previous studies and an-alyzed the impact of fairness concern on pricing orderingand the sales effort while other papers ignored sales effort

23 SupplyChainCoordination It can be seen from Sections21 and 22 that both overconfidence and fairness concernhad a systematic effect on the decision-making and per-formance in supply chains they also cause decision-makingbias erefore in order to pursue a win-win situation andreduce decision biases appropriate coordination mecha-nisms are especially necessary To date various contractshave been developed in behavioral supply chain manage-ment For fairness concern Cui et al [18] incorporatedfairness concern into a conventional supply chain andconsidered its influence on channel coordination between amanufacturer and a retailer Cuirsquos model was then extendedby Liu et al [21] to include a nonlinear demandey furtherdevised a revenue-sharing contract to coordinate a two-echelon CLSC with both membersrsquo fairness concernsNevertheless Zheng et al [4] proposed a cooperative gameapproach to coordinate a three-echelon closed-loop supplychain with fairness concern For overconfidence based onthe principal-agent theory Wang et al [23] designed anoptional contract to achieve a two-echelon supply chain withoverconfidence coordination and Pareto improvementJiang et al [24] studied the effect of supplier overconfidenceon buyback contracts in a financing supply chain esestudies revealed that fairness concerns and overconfidencecould complicate coordination in a supply chain and

existing research along this line is generally confined to asupply chain within a non-cooperative game setting

Based on these critical features of our model we tabu-lated our research in the context of a proper literature review(Table 1) e table revealed that the majority of the liter-ature concentrates on the coordination of different structuresupply chains with single-behavioral preference ie over-confidence or fairness concern However overconfidenceand fairness concern coexist in practice and this paperextends the depth and breadth of related literature aboutbehavioral preferences Besides the two cognitive biases(overconfidence and fairness concern) were simultaneouslytaken into consideration and the buyback contract withpromotional cost sharing was proposed in this paperNevertheless the focus of our paper was to investigate theimpact of retailersrsquo fairness concern and overconfidence onthe coordination results of a two-echelon supply chain byemploying a Stackelberg game approach We examined thevalidity of the contract under conditions that the retailer hadcognitive biases which were aligned with previous research

3 Assumptions and Models

31 Assumptions A two-echelon supply chain was con-sidered which included the rational manufacturer as theleader of the supply chain while the retailer had overcon-fidence and fairness concern e manufacturer producedproducts at unit production cost c and then sold them at unitwholesale price w to the retailer e retailer then sold thoseto customers at unit retail price p e manufacturer and theretailer showed risk neutrality Let s be the unit salvage valuev be the unit stockout cost and q be the order quantity of theretailer According to Ma et al [25] we assumed that thesales effort cost of the retailer was monotonically increasingas convex function ie C(e) denoted the sales effort costwhere C(e) (12)ce2 c(cgt 0) is the sales cost coefficientand e(egt 0) is the retailersrsquo sales effort denoted by thepromotional effort level Without loss of generality weassumed that 0lt slt clt vltp wgt c and qgt 0

In the two-echelon supply chain the manufacturerdetermined the wholesale price w and the retailer deter-mined the order quantity q and sales effort e ie w q and e

were decision variables of supply chain members s v and c

were assumed to be exogenous variables Note that the retailprice p was also assumed to be exogenous

32 Centralized Supply Chain Model In this section weconsidered the scenario wherein the manufacturer and theretailer made decisions as individuals e model with nobehavioral preferences was analyzed as a benchmark Be-sides the goal of the coordination contract in Section 4 wasto allow supply chain performance to achieve the optimalsituation under complete rationality Hence the benchmarkwas kept under complete rationality [8 26] However theonly goal was to maximize the overall profitability of thesupply chain by determining the optimal order quantity q

and sales effort e Previous studies proved that when decisionmakers were entirely rational the equilibrium solutions of

Complexity 3

the centralized supply chain were always better than thedecentralized behavioral supply chain [27 28] In order toexplore research problems and compare optimal decisionvariables to the decentralized supply chain with behavioralpreference analogs we assumed that the decision maker wasrational in the centralized supply chain According to Xuet al [16] the stochastic demand function is as follows

D a minus bp + e + θ (1)

where a signifies the market scale while b is the pricesensitivity coefficient of random demand e above func-tion indicated that sales effort e has a positive influence ondemand D(Dgt 0) e sales price p is exogenous thereforewe assumed that θ denotes the uncertainty of the demandwhich was a random variable following a uniform distri-bution on the interval [minus A A] wherein Agt 0 denotes thevariation range of the random variable For variable θ thecumulative distribution function is F(θ) and the probabilitydensity function is f(θ) According to the assumptionf(θ) (12A) and F(θ) (θ minus A2A) Also to avoid trivialcases we assumed Alt a and a minus bpgt 0 which guaranteedthat the demand is nonnegative

Under settings that did not guarantee market demandwe took q to be more than the demand D ie the decisionmaker faced the situation of overstock Under such a situ-ation we assumed minus Alt θlt q minus (a minus bp + e) In contrastwhen q was less than D q minus (a minus bp + e)lt θltA was ob-tained which indicated that the decision maker faced asituation of stockout We assumed that θ0 q minus (a minus bp + e)

indicated that order quantity was equal to the actual marketdemand D Hence f(θ0) (12A) and F(θ0) ((c minus v)(s minus v))

e profit for the centralized supply chain can beexpressed as

q emax

1113945

c

pE[min q D1113864 1113865] + E[s(q minus D)]+

minus E[v(D minus q)]+

minus cq minus C(e)

(2)

e first term in equation (2) is the expected revenue ofthe supply chain while the second term signifies the ex-pected surplus value if the situation of overstock happenede third is the expected loss provided the decision makersrsquoorder quantity was out of stock e term cq is the totalproduction cost and the last term indicated promotioneffort ie the cost of sales

e first-order derivatives of equation (2) with respect toq and e are

zE( 1113937c1113857

zq (s minus v)F θ0( 1113857 + v minus c 0

zE( 1113937c1113857

ze (p minus v) +(v minus s)F θ0( 1113857 minus ce 0

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(3)

erefore the Hessian matrix is

H(q e) (s minus v)f θ0( 1113857 0

0 (s minus v)f θ0( 1113857 minus c1113888 1113889 (4)

Note that s minus vlt 0 and f(θ0)gt 0 thus the Hessianmatrix Πc is a negative definite for q and e We can obtainoptimal order quantity and sales effort as follows

qclowast (v minus s)[(a minus bp)c +(p minus c)] + Ac(v + s minus 2c)

(v minus s)c

eclowast (p minus c)

c

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(5)

33 Decentralized Supply Chain Model In this section theoverconfidence and fairness concerns were incorporatedinto the two-echelon supply chain wherein the

Table 1 Differences between this paper and relevant studies

Papers Overconfidence Fairness concern Demand function CoordinationCroson et al [13] Stochastic demandLi et al [14] Deterministic demandLiu et al [15] Deterministic demandXu et al [16] Stochastic demandLi [17] Stochastic demandCui et al [18] Deterministic demand Wholesale price contractWu et al [19] Stochastic demandChen et al [20] Deterministic demand Revenue sharing contractLiu et al [21] Deterministic demand Membership-profit share contractZhang et al [22] Deterministic demandZheng et al [4] Deterministic demand Cooperative game mechanismWang et al [23] Stochastic demand Option contractJiang et al [24] Stochastic demand Wholesale price contract with buybackis study Stochastic demand Buyback contract with promotion cost sharing

4 Complexity

manufacturer was reported to be rational while the retailerhad a cognitive bias We assumed an overconfident retailerwho not only underestimated the variance of demand butalso overestimated the mean of demand and the effect ofsales effort on demand According to Chen et al [3] weadopted mean-increasing and variance-reducing transfor-mation of the actual market demand D us the marketdemand can be explained as follows

DO a minus bp +(1 + β)e +(1 minus β)θ a minus bp +(1 + β)e + θ1(6)

wherein β indicates the overconfident level (0lt βlt 1)Moreover the continuous random variable θ1 is equal to(1 minus β)θ which is a random noisy signal and assumed to beuniformly distributed ie θ1 sim U(minus A A) When β 0 theretailer had no overconfidence whereas for 0lt βlt 1 themean of DO increased and the variance of DO decreased ascompared to the mean and variance of D ie E(DO)gtE(D)

and var(DO)lt var(D) Meanwhile the impact of the pro-motion effort ((1 + β)egt e) on market demand was over-estimated ie the influence of e on market demandincreased as β increased

Similarly when market demand was uncertain weconsidered an order quantity q for the retailer withoverconfidence and fairness concern provided q was morethan the random demand Do Such an arrangement in-dicated that the retailer faced an overstock situation In thissituation we have minus Alt θ1 lt q minus [a minus bp + (1 + β)e] Incontrast if q was less than DOq minus [a minus bp + (1 + β)e]lt θ1 ltA was obtained which indi-cated that the retailer faced a stockout situation ie weassumed that θ2 q minus [a minus bp + (1 + β)e] HenceF(θ2) (μ(w minus c) + (1 + μ)(w minus v)(s minus v)(1 + μ)) eutility of the overconfident retailer is as follows

q emax

Uor pE min q Do1113864 11138651113858 1113859 + E s q minus DO( 1113857

+1113858 1113859

minus E v DO minus q( 11138571113858 1113859+

minus wq minus C(e)(7)

e reference framework for a retailer with fairnessconcern is the F-S model [29]

Ufr 1113945

r

minus μ 1113945m

minus 1113945r

0⎞⎠ minus λ 1113945r

minus 1113945m

0⎞⎠⎛⎝⎛⎝ (8)

where μ is the degree of fairness concern and λ is thecoefficient of sympathy μ λge 0 and 0lt λlt 1 e first partof equation (12) reflected the retailerrsquos profit whereas thesecond part evaluated the utility loss from disadvanta-geous inequality for the retailer when πm gt πr and thethird part measured the loss from advantageous inequalityfor the retailer when πm lt πr Note that if the retailerrsquosprofit was worse than the manufacturersrsquo profit (ie

πm gt πr) the retailer will have negative utility under thecondition of unfair aversion In contrast if his profit wasmore significant than the manufacturersrsquo the retailer willalso suffer positive utility because of the sympathyHowever Qin et al [30] demonstrated that the profit ofthe leader in a supply chain often accounted for more thanhalf of the overall profit when the Stackelberg game wascarried out In this paper the retailer is a weak follower ascompared to the manufacturer who served as the leaderAlongside this we found out that the profit of themanufacturer was twice that of the retailer when themanufacturer and the retailer were both unboundedlyrational Figures 4 and 5 validate the result in Section 5when μ 0 β 0 hence the retailer had the negativeutility of unfair aversion erefore according to equation(8) when the retailer had fairness concern only his utilityfunction was corrected to

Ufr 1113945

r

minus μ 1113945m

minus 1113945r

⎞⎠ (1 + μ)1113945r

minus μ1113945m

⎛⎝ (9)

where 1113937r and 1113937m denotes the profit of the rational retailerand the rational manufacturer respectively μgt 0 is thedegree of fairness concern

According to the theories of overconfidence and fairnessconcern the utility of the retailer with the two behavioralpreferences which was replaced by 1113937r in equation (9) withUo

r in equation (7) is as follows

q emax

U 1113945d

r

⎞⎠ (1 + μ)Uor minus μ1113945

d

m

⎛⎝ (10)

e retailer with behavioral preference orders q fromthe manufacturer the rational manufacturer profit func-tion is

wmax

1113945

d

m

(w minus c)q (11)

e Stackelberg game model where the manufacturerwas serving as the leader and the retailer served as thefollower was established in the decentralized supplychain e events of the model are described as followsFirstly the manufacturer decided the wholesale price wen the retailer made decisions about the order quantityq and the sales effort e based on the decision of themanufacturer and market demand To solve the subgameperfect equilibrium the backward induction method wasapplied e equilibrium solutions under the Stackelberggame model were considered when the retailer had acognitive bias e solutions are as follows e proof isgiven in Appendix A

Complexity 5

wdlowast

(a minus bp)(1 + μ)(v minus s)c +(v minus s)(1 + β)2[(1 + μ)p +(1 + 3μ)c] + Ac[(s + v)(1 + μ) + 2(1 + 3μ)c]

2(1 + 2μ) 2Ac +(v minus s)(1 + β)21113960 1113961 (12)

qdlowast

(a minus bp)(v minus s)c +(1 + β)2(v minus s)(p minus c) + Ac(s + v minus 2c)

2(v minus s)c (13)

edlowast

(1 + β) (a minus bp)(s minus v)c +(p minus c)(v minus s)(1 + β)2 + 2Ac(2p minus c minus s minus v)1113960 1113961

2c 2Ac +(1 + β)2(v minus s)1113960 1113961 (14)

Based on the above equilibrium solutions in thedecentralized supply chain the propositions about the effectof overconfidence and fairness concern (β and μ) on decisionvariables (qdlowast edlowast and wdlowast) are shown as in the followingNote that the proof of all propositions can be found inAppendix B

Proposition 1 When the retailer had both overconfidenceand fairness concern the optimal order quantity qdlowast and saleseffort edlowast were both positively correlated with the overcon-fident level β but had nothing to do with the degree of fairnessconcern μ Furthermore qdlowast is a strictly convex function withrespect to β

Proposition 1 indicated that the overconfident drivesthe retailer to increase order volume and improve salesefforts e reason is that overconfidence had two maineffects On the one hand the retailer overestimated theaverage market demand and exaggerated the effectivenessof sales effort which directly motivated the retailer to makea more considerable sales effort (for example increasingpromotions) On the other hand the retailer overestimatedthat the variance of the ldquorandom impact factor θ1rdquo issmaller than the actual level In other words from his pointof view a smaller market demand fluctuation influencedprediction us the two effects made the retailer ordermore product quantities and pay more sales effort for morerevenue Meanwhile it was easy to find that qdlowast and edlowast

were only related to β as per equations (13) and (14) whichrevealed that the retailersrsquo decisions on ordering and salesefforts were not affected by fairness concern

Proposition 2 When the retailer has both overconfidenceand fairness concern the optimal wholesale price wdlowast ispositively proportional to the overconfident level β but in-versely proportional to the degree of fairness concern μ

Proposition 2 illustrated that the cognitive bias of theretailer had an opposite effect on the manufacturersrsquodecision Firstly the moment the rational manufacturerrealized that the overconfident retailer could increase theorder quantity he raised the wholesale price of theproduct to maximize the profit which was consistent withthe anecdotes that merchants in real world could takeadvantage of othersrsquo overconfidence to maximize theirprofit Besides when the manufacturer understood thefeatures of the retailersrsquo fairness concern he lowered the

wholesale price to alleviate the retailersrsquo unfair aversionwhich implied that the retailer could boost his bargainingpower in the supply chain while he was concerned with thefairness erefore the rational manufacturer was re-quired to set a reasonable wholesale price to balance theinfluence of the two cognitive biases Next we comparedthe equilibrium outcomes in the centralized supply chainmodel with those in the decentralized supply chain modelere is a proposition as follows

Proposition 3 qclowastmay be more than or less thanqdlowast eclowastmaybe more than or less than edlowast Interestingly Δq qdlowast minus qclowastand Δe edlowast minus eclowast increases as the overconfidence level βincreases respectively

According to Proposition 3 the effect of overconfidencethe optimal order quantity and sales effort on the decen-tralized supply chain could deviate from the optimalequilibrium solution in the centralized supply chain whichresulted in a loss of the expected profits thereby increasingthe revenue gap between manufacturer and retailer

4 Buyback Contract with PromotionCost Sharing

It can be seen from Section 33 that due to the cognitivebias of the retailer his decisions deviated from the op-timal decisions in the centralized supply chain whichaffected the profit or utility of supply chain membersereby in order to coordinate the profit or utility be-tween the supply chain members and ensure the maxi-mized performance of the two-echelon supply chainaccording to Bai et al [31] the buyback contract withpromotional cost sharing was proposed in this sectionHere the manufacturer served as the leader who buysback the unsold products from the retailer at the buybackprice and shared the promotional cost We denoted thiscontract factor as (wsc r and 1 minus ϕ) where wsc is thewholesale price r is the buyback price and 1 minus ϕ is thefraction of promotional costs that the rational manu-facturer paid while the retailer undertook ϕ portion ofthe cost (0ltϕlt 1) In order to avoid retailersrsquo over-ordering and encourage the retailer to accept the con-tract we assumed 0lt sle rltwsc Based on thesedescriptions and assumptions above the manufacturersrsquoprofit and retailersrsquo utility are given by the followingexpression

6 Complexity

1113945

sc

m

(w minus c)q + E (s minus r) q minus DO( 11138571113858 1113859+

minus (1 minus ϕ)C(e)

(15)

U 1113945sc

r

⎞⎠ (1 + μ) p[(a minus bp +(1 + β)e)] + E r q minus DO( 1113857+

1113858 11138591113864⎛⎝

minus E v DO minus q( 1113857+

1113858 1113859 minus wq minus ϕC(e)1113865 minus μ1113945sc

m

(16)

wherein equation (15) is the profit when the manufacturerbuys back the unsold products at buyback price r from theretailer and thus obtained the surplus value of theseproducts (1 minus ϕ)C(e) is the promotional costs which themanufacturer undertook r(q minus DO)+ in equation (16) is therevenue when the retailer sells the unsold inventory of

products to themanufacturer at the end of the selling seasonϕC(e) is the promotional costs which he pay

According to the first derivations of equation (16) basedon q and e the following expression is obtained

qsc 2A(1 + μ)(v minus w) minus μ(w minus c)

(1 + μ)(v minus r) + μ(s minus r)+ a minus bp +(1 + k)e

scminus A

esc (1 + β)[μ(p + c minus 2w) + p minus w]

(2μϕ minus μ + ϕ)c

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(17)

In the buyback contract with promotion cost sharing toachieve supply chain coordination the optimal decisions ofthe retailer were the same as those under the centralizedsupply chain model that is qsc qclowast and esc eclowast [25 26]erefore equation (17) needs to be equal to equation (5) asthe contract factors (wsc r) were obtained

wsclowast

[β(1 + μ) + 1 minus ϕ(1 + 2μ)]p +[βμ +(1 + 2μ)ϕ]c

(1 + 2μ)(1 + β) (18)

rlowast

(v minus s)(p minus c)(1 + β)(1 + μ)vβ + 2A[cϕ(p minus c)(1 + 2μ) + βcv(1 + μ)] minus 2Ac p(v minus s)[1 + β +(2 + β)βμ] minus sv(1 + β)(1 + μ)1113864 1113865

(1 + 2μ)(1 + β)[β(p minus c)(v minus s) + 2Ac(c minus v)]

(19)

e purpose of the buyback contract with promotioncost sharing was to design a reasonable profit allocationscheme between the rational manufacturer and the retailerwith cognitive bias It should be noted that if the profit ofsupply chain members became lower the contract wasunacceptable e contract was acceptable if and only if1113937

scm ge 1113937

dlowastm and U(1113937

scr )geU(1113937

dlowastr )

Based on the above equilibrium solutions in the contractthe following propositions were obtained Due to the so-phisticated formula of the buyback price this section dis-cussed the effect of overconfidence and fairness concern onwholesale price

Proposition 4 e optimal wholesale price in the buybackcontract with promotion cost sharing was negatively corre-lated with μ and ϕ Nevertheless if ϕlt (μ(1 + 2μ)) thewholesale price wsclowast is positively correlated with β In con-trast if ϕlt (μ(1 + 2μ)) it is negatively correlated withβ

Proposition 4 implied that the contract was adopted andthe negative effect of fairness concern μ on wholesale pricewsclowast remained consistent with Proposition 2 Besides as wsc

increased ϕ decreased and 1 minus ϕ increased accordinglywhich meant that the manufacturer bore more promotionalcost when the wholesale price rose which is in line with thepractice Meanwhile the values of μ and ϕ determined theimpact of β (positive or negative influence) on the wholesaleprice wsc which indicated that fairness concern μ andpromotional cost portion 1 minus ϕ were critical to manufac-turerrsquos decisions on the wholesale price wsc us when the

two types of behavioral preferences coexisted in the contractthe effect of overconfidence on the optimal wholesale pricedepended on the relationship between fairness degree andfraction of promotion cost Since the utility and profitfunctions in the supply chain included a large number ofunknown parameters under three scenarios (ie Sections32 33 and 4) it was difficult to compare and analyze theeffects of the cognitive biasesus numerical methods wereutilized to conduct further analysis in the next section

5 Numerical Analysis

In this section a numerical study approach was adopted toallow a more in-depth insight into the two behavioralpreferences First a numerical experiment was conducted toshow how equilibrium solutions and profit (utility) in thetwo-echelon supply chain changed along with the levels ofoverconfidence and fairness concern (ie β and μ) and toexamine the robustness of propositions en we also de-termined the effect of overconfidence and fairness concernon the buyback contract with promotional cost sharing

In our experiments some parameters were gatheredfrom the existing papers For the parameters related tooverconfidence Ren et al [1] set β isin [0 1] andc 05 06 08 09 to perform their numerical experi-ments and Li et al [7] set β isin (0 1) and c 5 to conducttheir study For the parameters related to fairness concernZheng et al [4] set μge 0 c 20 and a 100 to implementtheir numerical analysis and Guo et al [9] set μge 0 toperform their studies According to the parameters above

Complexity 7

we assumed β isin [0 1) and μ 0 1 2 to reflect the level ofoverconfidence and fairness concern in this paper Whenβ 0 and μ 0 the retailer was considered to be rationalSimilarly other fundamental parameter values were set inexperiments as follows a 50 θ sim U(minus 30 30) c 4p 20v 7 s 2 b 2 and c 1 All the parameterssatisfied the constraint conditions and assumptions of thethree models in Sections 32 33 and 4e parameter valueswere also in line with the real situation of the market Wecould easily obtain qclowast 32 eclowast 16 and 1113937

c 252 in the

centralized supply chain

51 Effect of Overconfidence and Fairness Concern on theSupply Chain e changes in the decision variables (w qand e) manufacturer profit and retailer expected utilitywhen the level of overconfidence and fairness concern (β andμ) changed are shown in Figures 1ndash5

As shown in Figures 1 and 2 when the retailer was ra-tional (eg β 0 and μ 0) in the decentralized supplychain both optimal order quantity and sales effort were lowerthan those under the centralized supply chain Next when theretailer had overconfidence and fairness concern simulta-neously overconfidence could only affect the retailerrsquos opti-mal order quantity and sales effort of the retailer because theoptimal order quantity and sales effort were a function ofoverconfidence in equations (13) and (14) Moreover theorder quantity was monotonically increasing as concavefunction with respect to β In comparison sales effort wasmonotonically increasing as convex function with respect toit Furthermore there existed a threshold (β 083) when βwas less than 083 and the two decision variables were bothless than the optimal values (qclowast 32 eclowast 16) under thecentralized decision scenario Nevertheless once β was largerthan 083 the two decision variables were both more thanthem e above discussion verified the correctness ofProposition 1 In short overconfidence was the main factorthat affected the retailerrsquos optimal strategies of order quantity

and sales effort in the decentralized supply chain Howeverthe above findings contradicted with the conclusions from astudy conducted by Xu et al [16] which examined the effectof retailerrsquos overconfidence on the supply chain and the resultsuggested that a higher level of overconfident resulted in thelower ordering quantity

In Figure 3 we observed that the higher overconfidentlevel resulted in the high wholesale price in contrast ahigh degree of fairness concern resulted in low pricewhich demonstrated that Proposition 2 was robust isfigure implied that when the information for both partieswas symmetric the rational manufacturer knew that theretailer was overconfident that is the retailer was opti-mistic about the market demand and increased the orderquantity As a result the manufacturer also raised thewholesale price to maximize its revenue Also if theoverconfident retailer paid attention to the unfairness ofprofit allocation in the decision-making process the re-tailerrsquos bargaining power was enhanced [32] Hence forbetter performance in the supply chain the leadingmanufacturer was required to lower wholesale prices toease the feeling of unfairness by the retailer which isconsistent with the actual economic phenomenon Fur-thermore these findings are consistent with the resultconcluded by Cui et al [2] which indicated that a highdegree of the retailerrsquos fairness concern could lower thewholesale price Interestingly as the levels of the cognitivebias increased its effect on wholesale price graduallydecreased which highlighted its effect on decreasedwholesale prices

Figure 4 displays the impact of the cognitive bias on theretailerrsquos utility e graph showed that the retailer expectedutility increased gradually with the overconfident level andthe degree of fairness concern which suggested that twocognitive biases were beneficial to the retailer e reasonsfor such a phenomenon were that both cognitive biasescould increase the optimal order quantity and sales effortthus enhancing the utility of the retailer

Order quantity in centralized supply chain

01 02 03 04 05 06 07 08 09 10β

15

20

25

30

35

40

Ord

er q

uant

ity

μ = 012

Figure 1 e effect of the cognitive biases on optimal order quantity

8 Complexity

Figure 5 illustrates the impact of overconfidence andfairness concern on the manufacturerrsquos profit at is theoptimal manufacturerrsquos profit is an increasing function of βbut a decreasing function of μ According to Figure 3 thegraph of themanufacturerrsquos profit was similar to the graph ofthe wholesale price which indicated that the manufacturerrsquosprofit was firmly related to wholesale price Figure 5 alsoillustrates the effect of the cognitive bias β on the manu-facturerrsquos decreasing profit

52 Effect of Overconfidence and Fairness Concern on SupplyChain Coordination As can be seen from Section 51 thereexist deviations between the optimal decisions under the

benchmark and the decisions in the presence of the retailerrsquoscognitive biases erefore the buyback contract withpromotion cost sharing was necessary to coordinate therevenue of supply chain members and optimize the per-formance of the supply chain

Here we discussed the validity of the contract When theoptimal order quantity and sales effort of the retailer underthe contract were the same as the optimal decision variablesin the centralized supply chain ie qsc qclowast 32 andesc eclowast 16 the buyback contract with promotion costsharing achieved supply chain coordination First the effectof overconfidence and fairness concern (β and μ) on contractfactors wsrlowast rlowast and ϕ is shown in Figures 6 and 7 andTable 1 For clearly analyzing the effect of β and μ on wsrlowast

Degree of sales effort in centralized supply chain

01 02 03 04 05 06 07 08 09 10β

μ = 012

4

6

8

10

12

14

16

18

Deg

ree o

f sal

es ef

fort

Figure 2 e effect of the cognitive biases on optimal sales effort

01 02 03 04 05 06 07 08 09 10β

-μ = 0--μ = 1-μ = 2

4

5

6

7

8

9

10

11

12

13

14

Who

lesa

le p

rice

Figure 3 e effect of the cognitive biases on the optimal wholesale price

Complexity 9

and rlowast we let ϕ 1 in equations (18) and (19) which meantthat the manufacturer proposed the buyback contract onlyand the retailer bore all sales effort costs

According to Figure 6 it was revealed that the optimalwholesales price is a strictly increasing function of theoverconfidence level β but a strictly decreasing function ofthe degree of fairness concern μ which confirmed the resultof Proposition 4 Similarly Figure 7 highlights that theoptimal buyback price sharply increased with β but de-creased with μ Moreover in the buyback contract wsrlowast wasalways greater than or equal to 4 and rlowast was always greaterthan or equal to 2 in different cognitive bias levels Hence

the optimal buyback price rlowast satisfied the assumptions 2le r Note that the above characteristics influencedcognitive biases hence contract factors wsrlowast and rlowast werealso correct when ϕ is equal to other values (such asϕ 03 05 08) under conditions in which the buybackcontract had promotion cost sharing

However data from Figure 7 can be compared with thedata in Figure 6 which showed rsrlowast is more significant thanwsrlowast when β was within a specific range (for example04le βlt 1) which did not match the assumption rltwsrerefore the buyback contract did not achieve the coor-dination of the behavioral supply chain

01 02 03 04 05 06 07 08 09 10β

-μ = 0--μ = 1-μ = 2

0

20

40

60

80

100

120

Reta

iler e

xpec

ted

utili

ty

Figure 4 e effect of the cognitive biases on retailer utility

01 02 03 04 05 06 07 08 09 10β

-μ = 0--μ = 1-lowastμ = 2

0

20

40

60

80

100

120

140

160

180

200

Man

ufac

ture

r pro

fit

Figure 5 e effect of the cognitive biases on manufacturer profit

10 Complexity

As described in Section 4 if the supply chain membersaccepted the buyback contract with promotion cost sharingthe assumptions 1113937

scm gemax 1113937

dm and U(1113937

scr )gemaxU(1113937

dr )

had to be satisfied Also according to the assumptions thevalue ϕ must satisfy 1113937

scm gemax 1113937

dm and

U(1113937scr )gemaxU(1113937

dr ) us we kept the values of param-

eters to be the same (ie a 50 θ sim U(minus 30 30) c 4p 20 v 7 s 2 b 2 and c 1) to capture a likelydomain of ϕ

As can be seen from Table 2 the overconfident level βincreased the upper and lower limits of ϕ which had a clearupward trend In contrast the upper and lower limits of ϕ

showed a definite declining trend as the degree of fairnessconcern μ increased ese trends illustrated that in order tocoordinate the revenue of both parties and optimize theperformance of the supply chain the manufacturer needs tobear less promotional costs as β increased but bear morepromotional costs as μ value increased Besides when β and μincreased the interval of sharing cost portion ϕ shrunk whichindicated that the buyback contract with promotional costsharing had certain flexibility to coordinate the revenue be-tween supply chain members However the flexibility couldbe weakened with the rise of the cognitive bias level Forsupply chain members the possibility of negotiation or

01 02 03 04 05 06 07 08 09 10β

-μ = 0--μ = 1-μ = 2

4

5

6

7

8

9

10

11

12

Who

lesa

le p

rice

Figure 6 e effect of the cognitive biases on wholesale price wsrlowast

01 02 03 04 05 06 07 08 09 10β

-μ = 0--μ = 1-μ = 2

2

4

6

8

10

12

14

16

18

20

22

Buyb

ack

pric

e

Figure 7 e effect of the cognitive biases on buyback price rlowast

Complexity 11

cooperation between manufacturers and retailers also de-creased with an increase in retailersrsquo cognitive bias levelWhen the overconfident level was more than 081 the re-tailerrsquos utility from decentralized decisions was greater thanthe utility in the contract environment wherein the twoparties could not reach a cooperation agreement us thesupply chain could not achieve coordination as different levelsof cognitive bias under the contract had the same influence ondecision variables profit and utility Table 2 shows a clearpicture wherein μ is equal to 1 and β falls within the range of(0 08) which were used to analyze the effect of the cognitivebiases on the coordination of the supply chain

e results of the numerical simulation in Table 3 in-dicate that the contract factors ie ϕ r wholesale price theprofit and utility of members existed as flexible intervalswhen β isin (0 08) and μ 1 Besides the order quantity qsc

and the degree of sales effort esc under the contract are equalto qclowast and eclowast in a centralized supply chain respectivelyMeanwhile the buyback price r was less than the wholesaleprice wsc and higher than the unit salvage value s whichsatisfied the constrains sle rltwsclowast us the buybackcontract with promotion cost could achieve the coordinationof the two-echelon supply chain Also the buyback price r

was positively correlated with the portion ϕ of promotioncost sharing Notably the wholesale price wsc dropped as ϕincreased when β was in the range of (0 02) in Table 3However the situation reversed when it was in the range of(02 08) which was inconsistent since the manufacturerlowered the wholesale price when the retailer paid morepromotional costs in the contract e reason for such anobservation was that the wholesale price was not related to ϕbut positively related to the overconfident level β when thedegree of fairness concern was moderate Furthermore theretailer accepted the contract designed by manufacturerssince the term U(1113937

scr )gemaxU(1113937

dr ) was guaranteed in

Table 3

6 Conclusions

61MainConclusions In this paper we analyzed the effect ofcognitive bias on equilibrium solutions including thewholesale price sales price sales effort and expected profitsunder demand uncertainty by introducing both overconfi-dence and fairness concern into the two-echelon supply chaine most important conclusion from the study was that forretailers with the two cognitive biases overconfidence waspositive and the main bias that affected supply chain decisionsand the retailersrsquo utility benefits were not only due to over-confidence but also fairness concern For manufacturersoverconfidence could increase the wholesale price and profitHowever the fairness concern had a negative effect on themFurthermore the decisions of the retailer in the decentralizedsupply chain deviated from the optimal decisions in thecentralized supply chainmodel Especially when the thresholdof overconfidence was more than a fixed value the decisionvariables under decentralized decision-making were actuallyhigher than the optimal variables under centralized decision-making Also the buyback contract with promotional costsharing achieved behavioral supply chain coordination Incontrast the promotional cost-sharing portion and buybackprice both benefited from overconfidence but fairness concernhad a detrimental effect on them However as the retailersrsquooverconfidence level increased and fairness concern was equalto 1 the flexibility of contract coordination was weakenedWhen the overconfidence level wasmore than 08 the contractbecame invalid and members were not able to cooperate

62 Management Insights is paper provided some in-teresting managerial insights for decision makers in thedecentralized supply chain wherein retailers had two pref-erences First overconfidence was beneficial to them interms of decision variables and yet the fairness concern

Table 2 e effect of the cognitive biases on the feasible domain of promotion cost-sharing portion ϕ

μβ

0 02 04 06 08 0810 [02884 04976] [03683 05587] [04528 06199] [06343 07754] [07835 08669] mdash1 [02545 04741] [03310 05331] [04167 05908] [06207 07516] [07800 08560] mdash2 [02352 04600] [03101 05156] [04034 05719] [06094 07321] [07736 08478] mdash

Table 3 e effect of the cognitive biases on the coordination of the supply chain

μ 1β

0 02 04 06 08qsc 32 32 32 32 32esc 16 16 16 16 16ϕ [02545 04741] [03310 05331] [04167 05908] [06207 07516] [07800 08560]r [2100 35363] [28268 39188] [30555 42637] [36209 44215] [40485 49164]wsc [49205 53687] [49016 53522] [54092 69568] [62513 72626] [80407 87453]1113937

dm 164103 269591 461996 772619 1112545

U(1113937scm) [168722 217850] [269600 336882] [463175 500938] [772619 814300] [1112545 1120376]

1113937dr 20333 352396 495161 632422 765317

U(1113937scr ) [20333 243043] [352396 400595] [495161 525161] [632422 657581] [765317 784632]

12 Complexity

only damaged the wholesale price Second the two behav-ioral preferences were both conducive to utilities for theretailer but not necessarily beneficial for the manufacturer toreceive profit and overconfidence helped to increasemanufacturersrsquo profit which led to nonoptimal decisionsthat dragged down the performance of the two-echelonsupply chain Finally for the two behavioral preferences ofthe retailer the manufacturer assessed the degree of over-confidence and fairness concern (especially the overconfi-dence) via historical data and the current order quantity orother indicators In contrast factual data indicated that twobehavioral preferences were in the right range (such asβ isin (0 08) and μ isin (0 2)) e manufacturers introducedthe buyback contract with promotional cost sharing into thepractice of pricing and ordered decentralized supply chainsFurthermore they designed a reasonable promotional cost-sharing portion and buyback price thereby achieving anoptimized decision and utility In practice the manufac-turers could analyze whether the sellers had two behavioralpreferences based on the historical data by cooperating withdownstream enterprises ie if two behavioral preferencesexisted they referred to the papersrsquo conclusions to redesign areasonable contract mechanism

63 Future Research Since cognitive biases could existamong all members of supply chains it is necessary toexplore the manufacturersrsquo cognitive biases and analyze thesupply chain coordination problem in the presence of thetwo behavior preferences Besides the overconfidence andfairness concern considered in our centralized decision are afuture research direction

Appendix

A Decentralized Supply Chain Model

In a decentralized supply chain consisting of a rationalmanufacturer and a retailer with cognitive biases maximumprofit or utility is the only goal e backward inductivemethod can be applied to solve a Stackelberg game whereinthe manufacturer is the leader and the retailer is a followere proof of the equilibrium solutions in equations (12)ndash(14)is as follows

e profit of the manufacturer and expected utility ofretailers are as follows respectively

wmax

1113945

d

m

(w minus c)q (A1)

maxqe

U 1113945d

r

⎞⎠ (1 + μ) p[a minus bp +(1 + β)e] + s 1113946θ2

minus Aq minus a minus bp +(1 + β)e + θ1( 1113857f θ1( 1113857( 1113857dθ11113896⎛⎝

minus v 1113946A

θ2a minus bp +(1 + β)e + θ1 minus q( 1113857f θ1( 1113857( 1113857dθ1 minus wq minus

12

ce21113897 minus μ(w minus c)q

(A2)

Solving the first-order partial derivatives of the retailerrsquosexpected utility concerning qd and ed which equal to 0 weobtained the following equations

zU( 1113937dr 1113873

zq (1 + μ) v minus w + F θ2( 1113857(s minus v)1113858 1113859 minus μ(w minus c) 0

(A3)

zU( 1113937dr 1113873

ze (1 + β) p minus v +(v minus s)F θ2( 11138571113858 1113859 minus ce 0

(A4)

Hessian matrix

H(q e) (s minus v)f θ2( 1113857 (v minus s)(1 + β)f θ2( 1113857

(v minus s)f θ2( 1113857 (s minus v)(1 + β)f θ2( 1113857 minus c1113888 1113889

(A5)

According to the assumptions vgt sgt 0 f(θ2)gt 0 andβ isin [0 1) us the Hessian matrix of U(1113937

dr ) is a negative

definite for q and e which meant that the optimal orderquantity and sales effort of retailers existed in the decen-tralized supply chain erefore we obtained the followingformula of the retailersrsquo order quantity and sales effort byjointly solving equations (A3) and (A4)

qd

(a minus bp)(v minus s)(1 + μ)c +(1 + β)2(v minus s)[(1 + μ)(p minus w) + μc] + Ac[(s + v)(1 + μ) + 2μc minus w]

(v minus s)(1 + μ)c (A6)

ed

(1 + β)[(1 + μ)p + μc minus (1 + 2μ)]w

(1 + μ)c (A7)

Complexity 13

Substituting equations (A6) into (A1) we attained

wmax

1113945

d

m

(w minus c) (a minus bp)(v minus s)(1 + μ)c +(1 + β)2(v minus s)[(1 + μ)p + μc] + Ac[(s + v)(1 + μ) + 2μc] minus (1 + 2μ)w 2Ac +(1 + β)2(v minus s)1113960 11139611113966 1113967

(v minus s)(1 + μ)c

(A8)

Taking into account the first-order condition of equation(A7) with respect to w rounded off to 0 we obtained`

z1113937dm

ze

(a minus bp)(v minus s)(1 + μ)c +(1 + β)2(v minus s)[(1 + μ)p + μc] + Ac[(s + v)(1 + μ) + 2μc] minus (2w minus c)(1 + 2μ) 2Ac +(1 + β)2(v minus s)1113960 1113961

(v minus s)(1 + μ)c

(A9)

Solving equation (A8) we found that the optimalwholesale price of the manufacturer is as follows

wdlowast

(a minus bp)(1 + μ)(v minus s)c +(v minus s)(1 + β)2[(1 + μ)p +(1 + 3μ)c] + Ac[(s + v)(1 + μ) + 2(1 + 3μ)c]

2(1 + 2μ) 2Ac +(v minus s)(1 + β)21113960 1113961 (A10)

By substituting equation (A9) into equations (A6) and(A7) we attained the optimal order quantity and sales effortwhich are displayed in equations (13) and (14)

B Propositions

Proof of Proposition 1 We solved the first derivatives ofoptimal order quantity qdlowast and sales effort edlowast with respect

to β in equations (13) and (14) respectively Meanwhiletaking the second-order derivative of qdlowast with respect to βwe obtained

zqdlowast

zβ

(1 + β)(p minus c)

c

z2qdlowast

zβ2

p minus c

c (B1)

zedlowast

zβ4(p minus c)(v minus s)2β β3 + 4β2 + 6β + 41113872 1113873 +(a minus bp)c(1 + β)2(v minus s)2 + Ac(v minus s)(1 + β)2 (2(p minus c) + v + s) + 2(Ac)2(4p minus c minus s minus v)1113960 1113961

2c (1 + β)2(v minus s) + 2Ac1113960 11139612

(B2)

Since the second derivative of edlowast with respect to β wasvery complicated there was no need to solve it According tothe assumptions pgt vgt cgt sgt 0 β isin (0 1) a minus bpgt 0 andcgt 0 so (zqdlowastzβ)gt 0 (z2qdlowastzβ2)gt 0 and (zedlowastzβ)gt 0

Proof of Proposition 2 Based on the process above we tookthe first-order and second-order partial derivatives of op-timal wholesale price wdlowast with respect to β and μ in equation(12) respectively

zwdlowast

zβ

c(v minus s)(1 + μ)(1 + β)[(a minus bp)(v minus s) + A(2p minus s minus v)]

(1 + 2μ) (1 + β)2(v minus s) + 2Ac1113960 11139612 (B3)

zwdlowast

zμ minus

(a minus bp)(v minus s)c +(p minus c)(v minus s)(1 + β)2 + Ac(s + v minus 2c)1113960 1113961

2(1 + 2μ)2 (v minus s)(1 + β)2 + 2Ac1113960 1113961 (B4)

14 Complexity

According to the optimal result of order quantity inequation (13) and the assumption qgt 0 the condition s +

v minus 2cgt 0 is found According to the assumptions we had(zwdlowastzβ)gt 0 and (zwdlowastzμ)lt 0

Proof of Proposition 3 e gap between qdlowast and qclowastdenoted by Δq is as follows

Δq qdlowast

minus qclowast

(p minus c)(v minus s) (1 + β)2 minus 21113960 1113961 minus Ac(s + v minus 2c) minus (a minus bp)(v minus s)c

2c(v minus s) (B5)

If (p minus c)(v minus s)[(1 + β)2 minus 2] is more thanAc(s + v minus 2c) + (a minus bp)(v minus s)c we have Δqgt 0 ieqdlowast gt qclowast In contrast if (p minus c)(v minus s)[(1 + β)2 minus 2] is less

than Ac(s + v minus 2c) + (a minus bp)(v minus s)c Δqlt 0 ieqdlowast lt qclowast

e gap between edlowast and eclowast denoted by Δe is asfollows

Δe edlowast

minus eclowast

Aβc(4p minus 2c minus s minus v) minus Ac(s + v minus 2c) minus (v minus s)(1 + β) (a minus bp)c + 1 minus β21113872 1113873(p minus c)1113960 1113961

2c 2Ac +(1 + β)2(v minus s)1113960 1113961 (B6)

Similarly if Aβc(4p minus 2c minus s minus v)gtAc(s + v minus 2c)

+(v minus s)(1 + β)[(a minus bp)c + (1 minus β2)(p minus c)] we haveΔe edlowast minus eclowast gt 0 In contrast ifAβc(4p minus 2c minus s minus v)ltAc(s + v minus 2c)+

(v minus s)(1 + β)[(a minus bp)c + (1 minus β2)(p minus c)] we haveΔe edlowast minus eclowast lt 0

Upon solving the first-order derivative of Δq and Δe withrespect to β respectively we obtained

zΔ q

zβ

(1 + β)(p minus c)

c (B7)

zΔ e

zβ4(p minus c)(v minus s)2 β3 + 4β2 + 6β + 41113872 1113873β +(a minus bp)c(1 + β)2(v minus s)2 + Ac(v minus s)(1 + β)2 2(p minus c) +v + s) + 2(Ac)2(4p minus c minus s minus v)1113960 1113961

2c (1 + β)2(v minus s) + 2Ac1113960 11139612

(B8)

According to the assumptions pgt vgt cgt sgt 0 β isin (0 1)cgt 0 and a minus bpgt 0 (zΔ qzβ)gt 0 and (zΔ ezβ)gt 0 can beproved Hence the gaps increased as a function with respectto β

Proof of Proposition 4 e proof for Proposition 4 is similarto Proposition 2 Herein we took first-order and second-order partial derivatives of optimal wholesale price wsclowast withrespect to β μ and ϕ in equation (18) respectively esolving results are as follows

zwsclowast

zβ

[(1 + 2μ)ϕ minus μ](p minus c)

(1 + 2μ)(1 + β)2

zwsclowast

zμ minus

(p minus c)β(1 + β)(1 + 2μ)2

zwsclowast

zϕ minus

p minus c

1 + β

(B9)

(zwsclowastzβ) if ϕgt (μ(1 + 2μ)) is more than 0 otherwiseϕgt (μ(1 + 2μ)) is less than 0 us wsc could be increasingwith respect to β nevertheless decreasing with respect to βBesides according to the assumptions pgt vgt cgt sgt 0β isin (0 1) cgt 0 and a minus bpgt 0 it can be evidently foundthat (zwsclowastzμ)gt 0 and (zwsclowastzϕ)gt 0

Data Availability

e data used to support the findings of this study are in-cluded within the article

Conflicts of Interest

e authors declare no conflicts of interest

Acknowledgments

is work was supported by a major project financed by theNational Social Science Fund of China (Approval nos71461009 71662011 and 71761015)

Complexity 15

References

[1] Y Ren D C Croson and R T A Croson ldquoe overconfidentnewsvendorrdquo Journal of the Operational Research Societyvol 68 no 5 pp 496ndash506 2017

[2] T H Cui and P Mallucci ldquoFairness ideals in distributionchannelsrdquo Journal of Marketing Research vol 53 no 6pp 969ndash987 2016

[3] K Chen X Wang M Huang and W-K Ching ldquoSalesforcecontract design joint pricing and production planning withasymmetric overconfidence sales agentrdquo Journal of Industrialamp Management Optimization vol 13 no 2 pp 873ndash8992017

[4] X X Zheng Z Liu K W Li et al ldquoCooperative game ap-proaches to coordinating a three-echelon closed-loop supplychain with fairness concernsrdquo International Journal of Pro-duction Economic vol 212 pp 92ndash110 2018

[5] China Internet Watch False Prosperity behind Double 11Refund Rate Increased to 69 2014 httpswwwchinainternetwatchcom11643false-prosperity-behind-double-11-refund-rate-increased-to-69

[6] C L Lu ldquoPampGrsquos collaborative supply chain transformationrdquoEnterprise Management vol 10 pp 79-80 2016

[7] M Li N C Petruzzi and J Zhang ldquoOverconfident competingnewsvendorsrdquo Management Science vol 63 no 8pp 2637ndash2646 2016

[8] T Nie ldquoDual-fairness supply chain with quantity discountcontractsrdquo European Journal of Operational Researchvol 258 no 2 pp 491ndash500 2017

[9] X Du and B Jiang ldquoSignaling through price and quality toconsumers with fairness concernsrdquo Journal of MarketingResearch vol 53 no 6 pp 988ndash1000 2016

[10] V Kurz A Orland and K Posadzy ldquoFairness versus effi-ciency how procedural fairness concerns affect coordina-tionrdquo Experimental Economics vol 21 no 3 pp 601ndash6262018

[11] S Villa and J A Castantildeeda ldquoTransshipments in supplychains a behavioral investigationrdquo European Journal of Op-erational Research vol 269 no 2 pp 715ndash729 2018

[12] B Gu Y Fu and Y Li ldquoFresh-keeping effort and channelperformance in a fresh product supply chain with loss-averseconsumersrsquo returnsrdquo Mathematical Problems in Engineeringvol 2018 Article ID 4717094 20 pages 2018

[13] D Croson R Croson and Y Ren How to Manage anOverconfident Newsvendor 2008 httpcbeesutdallasedupapersCrosonRenCmsonMS2008

[14] M Li ldquoOverconfident distribution channelsrdquo Production andOperations Management vol 28 no 6 pp 1347ndash1365 2019

[15] W Liu D Wang O Tang and D Zhu ldquoe impacts oflogistics service integratorrsquos overconfidence behaviour onsupply chain decision under demand surgerdquo EuropeanJournal of Industrial Engineering vol 12 no 4 pp 558ndash5972018

[16] L Xu X Shi P Du K Govindan and Z Zhang ldquoOptimi-zation on pricing and overconfidence problem in a duopo-listic supply chainrdquo Computers amp Operations Researchvol 101 pp 162ndash172 2019

[17] Y Li and M Shan ldquoAdvance selling decisions with over-confident consumersrdquo Journal of Industrial and ManagementOptimization vol 2 no 3 pp 891ndash905 2016

[18] T H Cui J S Raju and Z J Zhang ldquoFairness and channelcoordinationrdquo Management Science vol 53 no 8pp 1303ndash1314 2007

[19] X Wu and J A Niederhoff ldquoFairness in selling to thenewsvendorrdquo Production and Operations Managementvol 23 no 11 pp 2002ndash2022 2014

[20] Z Chen and S-I Ivan Su ldquoSocial welfare maximization withthe least subsidy photovoltaic supply chain equilibrium andcoordination with fairness concernrdquo Renewable Energyvol 132 pp 1332ndash1347 2019

[21] W Liu D Wang X Shen X Yan and W Wei ldquoe impactsof distributional and peer-induced fairness concerns on thedecision-making of order allocation in logistics service supplychainrdquo Transportation Research Part E Logistics and Trans-portation Review vol 116 pp 102ndash122 2018

[22] L Zhang H Zhou Y Liu and R Lu ldquoOptimal environmentalquality and price with consumer environmental awarenessand retailerrsquos fairness concerns in supply chainrdquo Journal ofCleaner Production vol 213 pp 1063ndash1079 2019

[23] X LWang S Q Hu and X B Liu ldquoSupply coordination withoption contract considering overconfidence under randomdemand and yieldsrdquo Computer Integrated ManufacturingSystem vol 24 no 11 pp 2898ndash2908 2018

[24] W Jiang and J Liu ldquoInventory financing with overconfidentsupplier based on supply chain contractrdquo MathematicalProblems in Engineering vol 2018 Article ID 505438712 pages 2018

[25] P Ma H Wang and J Shang ldquoContract design for two-stagesupply chain coordination integrating manufacturer-qualityand retailer-marketing effortsrdquo International Journal ofProduction Economics vol 146 no 2 pp 745ndash755 2013

[26] J Liu W F Jiang and Y P Tang ldquoe supply chain coor-dination based on carbon reduction of overconfident man-ufacturerrdquo Journal of Jiangxi Normal University (NaturalScience) vol 42 no 2 pp 203ndash207 2018

[27] R Dubey N Altay and C Blome ldquoSwift trust and com-mitment the missing links for humanitarian supply chaincoordinationrdquo Annals of Operations Research vol 283 no 1-2 pp 159ndash177 2019

[28] C Cui Z Feng and C Tan ldquoCredibilistic loss aversion nashequilibrium for bimatrix games with triangular fuzzy payoffsrdquoComplexity vol 2018 Article ID 7143586 16 pages 2018

[29] E Fehr and KM Schmidt ldquoA theory of fairness competitionand cooperationrdquo e Quarterly Journal of Economicsvol 114 no 3 pp 817ndash868 1999

[30] Z Qin and J Yang ldquoAnalysis of a revenue-sharing contract insupply chain managementrdquo International Journal of LogisticsResearch and Applications vol 11 no 1 pp 17ndash29 2008

[31] S-Z Bai and J Ni ldquoResearch on buy-back contract with saleseffort cost sharingrdquo in Proceedings of the 2009 Chinese Controland Decision Conference IEEE Guilin China pp 1917ndash1921June 2009

[32] E Karagozoglu and K Keskin ldquoTime-varying fairness con-cerns delay and disagreement in bargainingrdquo Journal ofEconomic Behavior and Organization vol 147 pp 115ndash1282018

16 Complexity

the centralized supply chain were always better than thedecentralized behavioral supply chain [27 28] In order toexplore research problems and compare optimal decisionvariables to the decentralized supply chain with behavioralpreference analogs we assumed that the decision maker wasrational in the centralized supply chain According to Xuet al [16] the stochastic demand function is as follows

D a minus bp + e + θ (1)

where a signifies the market scale while b is the pricesensitivity coefficient of random demand e above func-tion indicated that sales effort e has a positive influence ondemand D(Dgt 0) e sales price p is exogenous thereforewe assumed that θ denotes the uncertainty of the demandwhich was a random variable following a uniform distri-bution on the interval [minus A A] wherein Agt 0 denotes thevariation range of the random variable For variable θ thecumulative distribution function is F(θ) and the probabilitydensity function is f(θ) According to the assumptionf(θ) (12A) and F(θ) (θ minus A2A) Also to avoid trivialcases we assumed Alt a and a minus bpgt 0 which guaranteedthat the demand is nonnegative

Under settings that did not guarantee market demandwe took q to be more than the demand D ie the decisionmaker faced the situation of overstock Under such a situ-ation we assumed minus Alt θlt q minus (a minus bp + e) In contrastwhen q was less than D q minus (a minus bp + e)lt θltA was ob-tained which indicated that the decision maker faced asituation of stockout We assumed that θ0 q minus (a minus bp + e)

indicated that order quantity was equal to the actual marketdemand D Hence f(θ0) (12A) and F(θ0) ((c minus v)(s minus v))

e profit for the centralized supply chain can beexpressed as

q emax

1113945

c

pE[min q D1113864 1113865] + E[s(q minus D)]+

minus E[v(D minus q)]+

minus cq minus C(e)

(2)

e first term in equation (2) is the expected revenue ofthe supply chain while the second term signifies the ex-pected surplus value if the situation of overstock happenede third is the expected loss provided the decision makersrsquoorder quantity was out of stock e term cq is the totalproduction cost and the last term indicated promotioneffort ie the cost of sales

e first-order derivatives of equation (2) with respect toq and e are

zE( 1113937c1113857

zq (s minus v)F θ0( 1113857 + v minus c 0

zE( 1113937c1113857

ze (p minus v) +(v minus s)F θ0( 1113857 minus ce 0

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(3)

erefore the Hessian matrix is

H(q e) (s minus v)f θ0( 1113857 0

0 (s minus v)f θ0( 1113857 minus c1113888 1113889 (4)

Note that s minus vlt 0 and f(θ0)gt 0 thus the Hessianmatrix Πc is a negative definite for q and e We can obtainoptimal order quantity and sales effort as follows

qclowast (v minus s)[(a minus bp)c +(p minus c)] + Ac(v + s minus 2c)

(v minus s)c

eclowast (p minus c)

c

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(5)

33 Decentralized Supply Chain Model In this section theoverconfidence and fairness concerns were incorporatedinto the two-echelon supply chain wherein the

Table 1 Differences between this paper and relevant studies

Papers Overconfidence Fairness concern Demand function CoordinationCroson et al [13] Stochastic demandLi et al [14] Deterministic demandLiu et al [15] Deterministic demandXu et al [16] Stochastic demandLi [17] Stochastic demandCui et al [18] Deterministic demand Wholesale price contractWu et al [19] Stochastic demandChen et al [20] Deterministic demand Revenue sharing contractLiu et al [21] Deterministic demand Membership-profit share contractZhang et al [22] Deterministic demandZheng et al [4] Deterministic demand Cooperative game mechanismWang et al [23] Stochastic demand Option contractJiang et al [24] Stochastic demand Wholesale price contract with buybackis study Stochastic demand Buyback contract with promotion cost sharing

4 Complexity

manufacturer was reported to be rational while the retailerhad a cognitive bias We assumed an overconfident retailerwho not only underestimated the variance of demand butalso overestimated the mean of demand and the effect ofsales effort on demand According to Chen et al [3] weadopted mean-increasing and variance-reducing transfor-mation of the actual market demand D us the marketdemand can be explained as follows

DO a minus bp +(1 + β)e +(1 minus β)θ a minus bp +(1 + β)e + θ1(6)

wherein β indicates the overconfident level (0lt βlt 1)Moreover the continuous random variable θ1 is equal to(1 minus β)θ which is a random noisy signal and assumed to beuniformly distributed ie θ1 sim U(minus A A) When β 0 theretailer had no overconfidence whereas for 0lt βlt 1 themean of DO increased and the variance of DO decreased ascompared to the mean and variance of D ie E(DO)gtE(D)

and var(DO)lt var(D) Meanwhile the impact of the pro-motion effort ((1 + β)egt e) on market demand was over-estimated ie the influence of e on market demandincreased as β increased

Similarly when market demand was uncertain weconsidered an order quantity q for the retailer withoverconfidence and fairness concern provided q was morethan the random demand Do Such an arrangement in-dicated that the retailer faced an overstock situation In thissituation we have minus Alt θ1 lt q minus [a minus bp + (1 + β)e] Incontrast if q was less than DOq minus [a minus bp + (1 + β)e]lt θ1 ltA was obtained which indi-cated that the retailer faced a stockout situation ie weassumed that θ2 q minus [a minus bp + (1 + β)e] HenceF(θ2) (μ(w minus c) + (1 + μ)(w minus v)(s minus v)(1 + μ)) eutility of the overconfident retailer is as follows

q emax

Uor pE min q Do1113864 11138651113858 1113859 + E s q minus DO( 1113857

+1113858 1113859

minus E v DO minus q( 11138571113858 1113859+

minus wq minus C(e)(7)

e reference framework for a retailer with fairnessconcern is the F-S model [29]

Ufr 1113945

r

minus μ 1113945m

minus 1113945r

0⎞⎠ minus λ 1113945r

minus 1113945m

0⎞⎠⎛⎝⎛⎝ (8)

where μ is the degree of fairness concern and λ is thecoefficient of sympathy μ λge 0 and 0lt λlt 1 e first partof equation (12) reflected the retailerrsquos profit whereas thesecond part evaluated the utility loss from disadvanta-geous inequality for the retailer when πm gt πr and thethird part measured the loss from advantageous inequalityfor the retailer when πm lt πr Note that if the retailerrsquosprofit was worse than the manufacturersrsquo profit (ie

πm gt πr) the retailer will have negative utility under thecondition of unfair aversion In contrast if his profit wasmore significant than the manufacturersrsquo the retailer willalso suffer positive utility because of the sympathyHowever Qin et al [30] demonstrated that the profit ofthe leader in a supply chain often accounted for more thanhalf of the overall profit when the Stackelberg game wascarried out In this paper the retailer is a weak follower ascompared to the manufacturer who served as the leaderAlongside this we found out that the profit of themanufacturer was twice that of the retailer when themanufacturer and the retailer were both unboundedlyrational Figures 4 and 5 validate the result in Section 5when μ 0 β 0 hence the retailer had the negativeutility of unfair aversion erefore according to equation(8) when the retailer had fairness concern only his utilityfunction was corrected to

Ufr 1113945

r

minus μ 1113945m

minus 1113945r

⎞⎠ (1 + μ)1113945r

minus μ1113945m

⎛⎝ (9)

where 1113937r and 1113937m denotes the profit of the rational retailerand the rational manufacturer respectively μgt 0 is thedegree of fairness concern

According to the theories of overconfidence and fairnessconcern the utility of the retailer with the two behavioralpreferences which was replaced by 1113937r in equation (9) withUo

r in equation (7) is as follows

q emax

U 1113945d

r

⎞⎠ (1 + μ)Uor minus μ1113945

d

m

⎛⎝ (10)

e retailer with behavioral preference orders q fromthe manufacturer the rational manufacturer profit func-tion is

wmax

1113945

d

m

(w minus c)q (11)

e Stackelberg game model where the manufacturerwas serving as the leader and the retailer served as thefollower was established in the decentralized supplychain e events of the model are described as followsFirstly the manufacturer decided the wholesale price wen the retailer made decisions about the order quantityq and the sales effort e based on the decision of themanufacturer and market demand To solve the subgameperfect equilibrium the backward induction method wasapplied e equilibrium solutions under the Stackelberggame model were considered when the retailer had acognitive bias e solutions are as follows e proof isgiven in Appendix A

Complexity 5

wdlowast

(a minus bp)(1 + μ)(v minus s)c +(v minus s)(1 + β)2[(1 + μ)p +(1 + 3μ)c] + Ac[(s + v)(1 + μ) + 2(1 + 3μ)c]

2(1 + 2μ) 2Ac +(v minus s)(1 + β)21113960 1113961 (12)

qdlowast

(a minus bp)(v minus s)c +(1 + β)2(v minus s)(p minus c) + Ac(s + v minus 2c)

2(v minus s)c (13)

edlowast

(1 + β) (a minus bp)(s minus v)c +(p minus c)(v minus s)(1 + β)2 + 2Ac(2p minus c minus s minus v)1113960 1113961

2c 2Ac +(1 + β)2(v minus s)1113960 1113961 (14)

Based on the above equilibrium solutions in thedecentralized supply chain the propositions about the effectof overconfidence and fairness concern (β and μ) on decisionvariables (qdlowast edlowast and wdlowast) are shown as in the followingNote that the proof of all propositions can be found inAppendix B

Proposition 1 When the retailer had both overconfidenceand fairness concern the optimal order quantity qdlowast and saleseffort edlowast were both positively correlated with the overcon-fident level β but had nothing to do with the degree of fairnessconcern μ Furthermore qdlowast is a strictly convex function withrespect to β

Proposition 1 indicated that the overconfident drivesthe retailer to increase order volume and improve salesefforts e reason is that overconfidence had two maineffects On the one hand the retailer overestimated theaverage market demand and exaggerated the effectivenessof sales effort which directly motivated the retailer to makea more considerable sales effort (for example increasingpromotions) On the other hand the retailer overestimatedthat the variance of the ldquorandom impact factor θ1rdquo issmaller than the actual level In other words from his pointof view a smaller market demand fluctuation influencedprediction us the two effects made the retailer ordermore product quantities and pay more sales effort for morerevenue Meanwhile it was easy to find that qdlowast and edlowast

were only related to β as per equations (13) and (14) whichrevealed that the retailersrsquo decisions on ordering and salesefforts were not affected by fairness concern

Proposition 2 When the retailer has both overconfidenceand fairness concern the optimal wholesale price wdlowast ispositively proportional to the overconfident level β but in-versely proportional to the degree of fairness concern μ

Proposition 2 illustrated that the cognitive bias of theretailer had an opposite effect on the manufacturersrsquodecision Firstly the moment the rational manufacturerrealized that the overconfident retailer could increase theorder quantity he raised the wholesale price of theproduct to maximize the profit which was consistent withthe anecdotes that merchants in real world could takeadvantage of othersrsquo overconfidence to maximize theirprofit Besides when the manufacturer understood thefeatures of the retailersrsquo fairness concern he lowered the

wholesale price to alleviate the retailersrsquo unfair aversionwhich implied that the retailer could boost his bargainingpower in the supply chain while he was concerned with thefairness erefore the rational manufacturer was re-quired to set a reasonable wholesale price to balance theinfluence of the two cognitive biases Next we comparedthe equilibrium outcomes in the centralized supply chainmodel with those in the decentralized supply chain modelere is a proposition as follows

Proposition 3 qclowastmay be more than or less thanqdlowast eclowastmaybe more than or less than edlowast Interestingly Δq qdlowast minus qclowastand Δe edlowast minus eclowast increases as the overconfidence level βincreases respectively

According to Proposition 3 the effect of overconfidencethe optimal order quantity and sales effort on the decen-tralized supply chain could deviate from the optimalequilibrium solution in the centralized supply chain whichresulted in a loss of the expected profits thereby increasingthe revenue gap between manufacturer and retailer

4 Buyback Contract with PromotionCost Sharing

It can be seen from Section 33 that due to the cognitivebias of the retailer his decisions deviated from the op-timal decisions in the centralized supply chain whichaffected the profit or utility of supply chain membersereby in order to coordinate the profit or utility be-tween the supply chain members and ensure the maxi-mized performance of the two-echelon supply chainaccording to Bai et al [31] the buyback contract withpromotional cost sharing was proposed in this sectionHere the manufacturer served as the leader who buysback the unsold products from the retailer at the buybackprice and shared the promotional cost We denoted thiscontract factor as (wsc r and 1 minus ϕ) where wsc is thewholesale price r is the buyback price and 1 minus ϕ is thefraction of promotional costs that the rational manu-facturer paid while the retailer undertook ϕ portion ofthe cost (0ltϕlt 1) In order to avoid retailersrsquo over-ordering and encourage the retailer to accept the con-tract we assumed 0lt sle rltwsc Based on thesedescriptions and assumptions above the manufacturersrsquoprofit and retailersrsquo utility are given by the followingexpression

6 Complexity

1113945

sc

m

(w minus c)q + E (s minus r) q minus DO( 11138571113858 1113859+

minus (1 minus ϕ)C(e)

(15)

U 1113945sc

r

⎞⎠ (1 + μ) p[(a minus bp +(1 + β)e)] + E r q minus DO( 1113857+

1113858 11138591113864⎛⎝

minus E v DO minus q( 1113857+

1113858 1113859 minus wq minus ϕC(e)1113865 minus μ1113945sc

m

(16)

wherein equation (15) is the profit when the manufacturerbuys back the unsold products at buyback price r from theretailer and thus obtained the surplus value of theseproducts (1 minus ϕ)C(e) is the promotional costs which themanufacturer undertook r(q minus DO)+ in equation (16) is therevenue when the retailer sells the unsold inventory of

products to themanufacturer at the end of the selling seasonϕC(e) is the promotional costs which he pay

According to the first derivations of equation (16) basedon q and e the following expression is obtained

qsc 2A(1 + μ)(v minus w) minus μ(w minus c)

(1 + μ)(v minus r) + μ(s minus r)+ a minus bp +(1 + k)e

scminus A

esc (1 + β)[μ(p + c minus 2w) + p minus w]

(2μϕ minus μ + ϕ)c

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(17)

In the buyback contract with promotion cost sharing toachieve supply chain coordination the optimal decisions ofthe retailer were the same as those under the centralizedsupply chain model that is qsc qclowast and esc eclowast [25 26]erefore equation (17) needs to be equal to equation (5) asthe contract factors (wsc r) were obtained

wsclowast

[β(1 + μ) + 1 minus ϕ(1 + 2μ)]p +[βμ +(1 + 2μ)ϕ]c

(1 + 2μ)(1 + β) (18)

rlowast

(v minus s)(p minus c)(1 + β)(1 + μ)vβ + 2A[cϕ(p minus c)(1 + 2μ) + βcv(1 + μ)] minus 2Ac p(v minus s)[1 + β +(2 + β)βμ] minus sv(1 + β)(1 + μ)1113864 1113865

(1 + 2μ)(1 + β)[β(p minus c)(v minus s) + 2Ac(c minus v)]

(19)

e purpose of the buyback contract with promotioncost sharing was to design a reasonable profit allocationscheme between the rational manufacturer and the retailerwith cognitive bias It should be noted that if the profit ofsupply chain members became lower the contract wasunacceptable e contract was acceptable if and only if1113937

scm ge 1113937

dlowastm and U(1113937

scr )geU(1113937

dlowastr )

Based on the above equilibrium solutions in the contractthe following propositions were obtained Due to the so-phisticated formula of the buyback price this section dis-cussed the effect of overconfidence and fairness concern onwholesale price

Proposition 4 e optimal wholesale price in the buybackcontract with promotion cost sharing was negatively corre-lated with μ and ϕ Nevertheless if ϕlt (μ(1 + 2μ)) thewholesale price wsclowast is positively correlated with β In con-trast if ϕlt (μ(1 + 2μ)) it is negatively correlated withβ

Proposition 4 implied that the contract was adopted andthe negative effect of fairness concern μ on wholesale pricewsclowast remained consistent with Proposition 2 Besides as wsc

increased ϕ decreased and 1 minus ϕ increased accordinglywhich meant that the manufacturer bore more promotionalcost when the wholesale price rose which is in line with thepractice Meanwhile the values of μ and ϕ determined theimpact of β (positive or negative influence) on the wholesaleprice wsc which indicated that fairness concern μ andpromotional cost portion 1 minus ϕ were critical to manufac-turerrsquos decisions on the wholesale price wsc us when the

two types of behavioral preferences coexisted in the contractthe effect of overconfidence on the optimal wholesale pricedepended on the relationship between fairness degree andfraction of promotion cost Since the utility and profitfunctions in the supply chain included a large number ofunknown parameters under three scenarios (ie Sections32 33 and 4) it was difficult to compare and analyze theeffects of the cognitive biasesus numerical methods wereutilized to conduct further analysis in the next section

5 Numerical Analysis

In this section a numerical study approach was adopted toallow a more in-depth insight into the two behavioralpreferences First a numerical experiment was conducted toshow how equilibrium solutions and profit (utility) in thetwo-echelon supply chain changed along with the levels ofoverconfidence and fairness concern (ie β and μ) and toexamine the robustness of propositions en we also de-termined the effect of overconfidence and fairness concernon the buyback contract with promotional cost sharing

In our experiments some parameters were gatheredfrom the existing papers For the parameters related tooverconfidence Ren et al [1] set β isin [0 1] andc 05 06 08 09 to perform their numerical experi-ments and Li et al [7] set β isin (0 1) and c 5 to conducttheir study For the parameters related to fairness concernZheng et al [4] set μge 0 c 20 and a 100 to implementtheir numerical analysis and Guo et al [9] set μge 0 toperform their studies According to the parameters above

Complexity 7

we assumed β isin [0 1) and μ 0 1 2 to reflect the level ofoverconfidence and fairness concern in this paper Whenβ 0 and μ 0 the retailer was considered to be rationalSimilarly other fundamental parameter values were set inexperiments as follows a 50 θ sim U(minus 30 30) c 4p 20v 7 s 2 b 2 and c 1 All the parameterssatisfied the constraint conditions and assumptions of thethree models in Sections 32 33 and 4e parameter valueswere also in line with the real situation of the market Wecould easily obtain qclowast 32 eclowast 16 and 1113937

c 252 in the

centralized supply chain

51 Effect of Overconfidence and Fairness Concern on theSupply Chain e changes in the decision variables (w qand e) manufacturer profit and retailer expected utilitywhen the level of overconfidence and fairness concern (β andμ) changed are shown in Figures 1ndash5

As shown in Figures 1 and 2 when the retailer was ra-tional (eg β 0 and μ 0) in the decentralized supplychain both optimal order quantity and sales effort were lowerthan those under the centralized supply chain Next when theretailer had overconfidence and fairness concern simulta-neously overconfidence could only affect the retailerrsquos opti-mal order quantity and sales effort of the retailer because theoptimal order quantity and sales effort were a function ofoverconfidence in equations (13) and (14) Moreover theorder quantity was monotonically increasing as concavefunction with respect to β In comparison sales effort wasmonotonically increasing as convex function with respect toit Furthermore there existed a threshold (β 083) when βwas less than 083 and the two decision variables were bothless than the optimal values (qclowast 32 eclowast 16) under thecentralized decision scenario Nevertheless once β was largerthan 083 the two decision variables were both more thanthem e above discussion verified the correctness ofProposition 1 In short overconfidence was the main factorthat affected the retailerrsquos optimal strategies of order quantity

and sales effort in the decentralized supply chain Howeverthe above findings contradicted with the conclusions from astudy conducted by Xu et al [16] which examined the effectof retailerrsquos overconfidence on the supply chain and the resultsuggested that a higher level of overconfident resulted in thelower ordering quantity

In Figure 3 we observed that the higher overconfidentlevel resulted in the high wholesale price in contrast ahigh degree of fairness concern resulted in low pricewhich demonstrated that Proposition 2 was robust isfigure implied that when the information for both partieswas symmetric the rational manufacturer knew that theretailer was overconfident that is the retailer was opti-mistic about the market demand and increased the orderquantity As a result the manufacturer also raised thewholesale price to maximize its revenue Also if theoverconfident retailer paid attention to the unfairness ofprofit allocation in the decision-making process the re-tailerrsquos bargaining power was enhanced [32] Hence forbetter performance in the supply chain the leadingmanufacturer was required to lower wholesale prices toease the feeling of unfairness by the retailer which isconsistent with the actual economic phenomenon Fur-thermore these findings are consistent with the resultconcluded by Cui et al [2] which indicated that a highdegree of the retailerrsquos fairness concern could lower thewholesale price Interestingly as the levels of the cognitivebias increased its effect on wholesale price graduallydecreased which highlighted its effect on decreasedwholesale prices

Figure 4 displays the impact of the cognitive bias on theretailerrsquos utility e graph showed that the retailer expectedutility increased gradually with the overconfident level andthe degree of fairness concern which suggested that twocognitive biases were beneficial to the retailer e reasonsfor such a phenomenon were that both cognitive biasescould increase the optimal order quantity and sales effortthus enhancing the utility of the retailer

Order quantity in centralized supply chain

01 02 03 04 05 06 07 08 09 10β

15

20

25

30

35

40

Ord

er q

uant

ity

μ = 012

Figure 1 e effect of the cognitive biases on optimal order quantity

8 Complexity

Figure 5 illustrates the impact of overconfidence andfairness concern on the manufacturerrsquos profit at is theoptimal manufacturerrsquos profit is an increasing function of βbut a decreasing function of μ According to Figure 3 thegraph of themanufacturerrsquos profit was similar to the graph ofthe wholesale price which indicated that the manufacturerrsquosprofit was firmly related to wholesale price Figure 5 alsoillustrates the effect of the cognitive bias β on the manu-facturerrsquos decreasing profit

52 Effect of Overconfidence and Fairness Concern on SupplyChain Coordination As can be seen from Section 51 thereexist deviations between the optimal decisions under the

benchmark and the decisions in the presence of the retailerrsquoscognitive biases erefore the buyback contract withpromotion cost sharing was necessary to coordinate therevenue of supply chain members and optimize the per-formance of the supply chain

Here we discussed the validity of the contract When theoptimal order quantity and sales effort of the retailer underthe contract were the same as the optimal decision variablesin the centralized supply chain ie qsc qclowast 32 andesc eclowast 16 the buyback contract with promotion costsharing achieved supply chain coordination First the effectof overconfidence and fairness concern (β and μ) on contractfactors wsrlowast rlowast and ϕ is shown in Figures 6 and 7 andTable 1 For clearly analyzing the effect of β and μ on wsrlowast

Degree of sales effort in centralized supply chain

01 02 03 04 05 06 07 08 09 10β

μ = 012

4

6

8

10

12

14

16

18

Deg

ree o

f sal

es ef

fort

Figure 2 e effect of the cognitive biases on optimal sales effort

01 02 03 04 05 06 07 08 09 10β

-μ = 0--μ = 1-μ = 2

4

5

6

7

8

9

10

11

12

13

14

Who

lesa

le p

rice

Figure 3 e effect of the cognitive biases on the optimal wholesale price

Complexity 9

and rlowast we let ϕ 1 in equations (18) and (19) which meantthat the manufacturer proposed the buyback contract onlyand the retailer bore all sales effort costs

According to Figure 6 it was revealed that the optimalwholesales price is a strictly increasing function of theoverconfidence level β but a strictly decreasing function ofthe degree of fairness concern μ which confirmed the resultof Proposition 4 Similarly Figure 7 highlights that theoptimal buyback price sharply increased with β but de-creased with μ Moreover in the buyback contract wsrlowast wasalways greater than or equal to 4 and rlowast was always greaterthan or equal to 2 in different cognitive bias levels Hence

the optimal buyback price rlowast satisfied the assumptions 2le r Note that the above characteristics influencedcognitive biases hence contract factors wsrlowast and rlowast werealso correct when ϕ is equal to other values (such asϕ 03 05 08) under conditions in which the buybackcontract had promotion cost sharing

However data from Figure 7 can be compared with thedata in Figure 6 which showed rsrlowast is more significant thanwsrlowast when β was within a specific range (for example04le βlt 1) which did not match the assumption rltwsrerefore the buyback contract did not achieve the coor-dination of the behavioral supply chain

01 02 03 04 05 06 07 08 09 10β

-μ = 0--μ = 1-μ = 2

0

20

40

60

80

100

120

Reta

iler e

xpec

ted

utili

ty

Figure 4 e effect of the cognitive biases on retailer utility

01 02 03 04 05 06 07 08 09 10β

-μ = 0--μ = 1-lowastμ = 2

0

20

40

60

80

100

120

140

160

180

200

Man

ufac

ture

r pro

fit

Figure 5 e effect of the cognitive biases on manufacturer profit

10 Complexity

As described in Section 4 if the supply chain membersaccepted the buyback contract with promotion cost sharingthe assumptions 1113937

scm gemax 1113937

dm and U(1113937

scr )gemaxU(1113937

dr )

had to be satisfied Also according to the assumptions thevalue ϕ must satisfy 1113937

scm gemax 1113937

dm and

U(1113937scr )gemaxU(1113937

dr ) us we kept the values of param-

eters to be the same (ie a 50 θ sim U(minus 30 30) c 4p 20 v 7 s 2 b 2 and c 1) to capture a likelydomain of ϕ

As can be seen from Table 2 the overconfident level βincreased the upper and lower limits of ϕ which had a clearupward trend In contrast the upper and lower limits of ϕ

showed a definite declining trend as the degree of fairnessconcern μ increased ese trends illustrated that in order tocoordinate the revenue of both parties and optimize theperformance of the supply chain the manufacturer needs tobear less promotional costs as β increased but bear morepromotional costs as μ value increased Besides when β and μincreased the interval of sharing cost portion ϕ shrunk whichindicated that the buyback contract with promotional costsharing had certain flexibility to coordinate the revenue be-tween supply chain members However the flexibility couldbe weakened with the rise of the cognitive bias level Forsupply chain members the possibility of negotiation or

01 02 03 04 05 06 07 08 09 10β

-μ = 0--μ = 1-μ = 2

4

5

6

7

8

9

10

11

12

Who

lesa

le p

rice

Figure 6 e effect of the cognitive biases on wholesale price wsrlowast

01 02 03 04 05 06 07 08 09 10β

-μ = 0--μ = 1-μ = 2

2

4

6

8

10

12

14

16

18

20

22

Buyb

ack

pric

e

Figure 7 e effect of the cognitive biases on buyback price rlowast

Complexity 11

cooperation between manufacturers and retailers also de-creased with an increase in retailersrsquo cognitive bias levelWhen the overconfident level was more than 081 the re-tailerrsquos utility from decentralized decisions was greater thanthe utility in the contract environment wherein the twoparties could not reach a cooperation agreement us thesupply chain could not achieve coordination as different levelsof cognitive bias under the contract had the same influence ondecision variables profit and utility Table 2 shows a clearpicture wherein μ is equal to 1 and β falls within the range of(0 08) which were used to analyze the effect of the cognitivebiases on the coordination of the supply chain

e results of the numerical simulation in Table 3 in-dicate that the contract factors ie ϕ r wholesale price theprofit and utility of members existed as flexible intervalswhen β isin (0 08) and μ 1 Besides the order quantity qsc

and the degree of sales effort esc under the contract are equalto qclowast and eclowast in a centralized supply chain respectivelyMeanwhile the buyback price r was less than the wholesaleprice wsc and higher than the unit salvage value s whichsatisfied the constrains sle rltwsclowast us the buybackcontract with promotion cost could achieve the coordinationof the two-echelon supply chain Also the buyback price r

was positively correlated with the portion ϕ of promotioncost sharing Notably the wholesale price wsc dropped as ϕincreased when β was in the range of (0 02) in Table 3However the situation reversed when it was in the range of(02 08) which was inconsistent since the manufacturerlowered the wholesale price when the retailer paid morepromotional costs in the contract e reason for such anobservation was that the wholesale price was not related to ϕbut positively related to the overconfident level β when thedegree of fairness concern was moderate Furthermore theretailer accepted the contract designed by manufacturerssince the term U(1113937

scr )gemaxU(1113937

dr ) was guaranteed in

Table 3

6 Conclusions

61MainConclusions In this paper we analyzed the effect ofcognitive bias on equilibrium solutions including thewholesale price sales price sales effort and expected profitsunder demand uncertainty by introducing both overconfi-dence and fairness concern into the two-echelon supply chaine most important conclusion from the study was that forretailers with the two cognitive biases overconfidence waspositive and the main bias that affected supply chain decisionsand the retailersrsquo utility benefits were not only due to over-confidence but also fairness concern For manufacturersoverconfidence could increase the wholesale price and profitHowever the fairness concern had a negative effect on themFurthermore the decisions of the retailer in the decentralizedsupply chain deviated from the optimal decisions in thecentralized supply chainmodel Especially when the thresholdof overconfidence was more than a fixed value the decisionvariables under decentralized decision-making were actuallyhigher than the optimal variables under centralized decision-making Also the buyback contract with promotional costsharing achieved behavioral supply chain coordination Incontrast the promotional cost-sharing portion and buybackprice both benefited from overconfidence but fairness concernhad a detrimental effect on them However as the retailersrsquooverconfidence level increased and fairness concern was equalto 1 the flexibility of contract coordination was weakenedWhen the overconfidence level wasmore than 08 the contractbecame invalid and members were not able to cooperate

62 Management Insights is paper provided some in-teresting managerial insights for decision makers in thedecentralized supply chain wherein retailers had two pref-erences First overconfidence was beneficial to them interms of decision variables and yet the fairness concern

Table 2 e effect of the cognitive biases on the feasible domain of promotion cost-sharing portion ϕ

μβ

0 02 04 06 08 0810 [02884 04976] [03683 05587] [04528 06199] [06343 07754] [07835 08669] mdash1 [02545 04741] [03310 05331] [04167 05908] [06207 07516] [07800 08560] mdash2 [02352 04600] [03101 05156] [04034 05719] [06094 07321] [07736 08478] mdash

Table 3 e effect of the cognitive biases on the coordination of the supply chain

μ 1β

0 02 04 06 08qsc 32 32 32 32 32esc 16 16 16 16 16ϕ [02545 04741] [03310 05331] [04167 05908] [06207 07516] [07800 08560]r [2100 35363] [28268 39188] [30555 42637] [36209 44215] [40485 49164]wsc [49205 53687] [49016 53522] [54092 69568] [62513 72626] [80407 87453]1113937

dm 164103 269591 461996 772619 1112545

U(1113937scm) [168722 217850] [269600 336882] [463175 500938] [772619 814300] [1112545 1120376]

1113937dr 20333 352396 495161 632422 765317

U(1113937scr ) [20333 243043] [352396 400595] [495161 525161] [632422 657581] [765317 784632]

12 Complexity

only damaged the wholesale price Second the two behav-ioral preferences were both conducive to utilities for theretailer but not necessarily beneficial for the manufacturer toreceive profit and overconfidence helped to increasemanufacturersrsquo profit which led to nonoptimal decisionsthat dragged down the performance of the two-echelonsupply chain Finally for the two behavioral preferences ofthe retailer the manufacturer assessed the degree of over-confidence and fairness concern (especially the overconfi-dence) via historical data and the current order quantity orother indicators In contrast factual data indicated that twobehavioral preferences were in the right range (such asβ isin (0 08) and μ isin (0 2)) e manufacturers introducedthe buyback contract with promotional cost sharing into thepractice of pricing and ordered decentralized supply chainsFurthermore they designed a reasonable promotional cost-sharing portion and buyback price thereby achieving anoptimized decision and utility In practice the manufac-turers could analyze whether the sellers had two behavioralpreferences based on the historical data by cooperating withdownstream enterprises ie if two behavioral preferencesexisted they referred to the papersrsquo conclusions to redesign areasonable contract mechanism

63 Future Research Since cognitive biases could existamong all members of supply chains it is necessary toexplore the manufacturersrsquo cognitive biases and analyze thesupply chain coordination problem in the presence of thetwo behavior preferences Besides the overconfidence andfairness concern considered in our centralized decision are afuture research direction

Appendix

A Decentralized Supply Chain Model

In a decentralized supply chain consisting of a rationalmanufacturer and a retailer with cognitive biases maximumprofit or utility is the only goal e backward inductivemethod can be applied to solve a Stackelberg game whereinthe manufacturer is the leader and the retailer is a followere proof of the equilibrium solutions in equations (12)ndash(14)is as follows

e profit of the manufacturer and expected utility ofretailers are as follows respectively

wmax

1113945

d

m

(w minus c)q (A1)

maxqe

U 1113945d

r

⎞⎠ (1 + μ) p[a minus bp +(1 + β)e] + s 1113946θ2

minus Aq minus a minus bp +(1 + β)e + θ1( 1113857f θ1( 1113857( 1113857dθ11113896⎛⎝

minus v 1113946A

θ2a minus bp +(1 + β)e + θ1 minus q( 1113857f θ1( 1113857( 1113857dθ1 minus wq minus

12

ce21113897 minus μ(w minus c)q

(A2)

Solving the first-order partial derivatives of the retailerrsquosexpected utility concerning qd and ed which equal to 0 weobtained the following equations

zU( 1113937dr 1113873

zq (1 + μ) v minus w + F θ2( 1113857(s minus v)1113858 1113859 minus μ(w minus c) 0

(A3)

zU( 1113937dr 1113873

ze (1 + β) p minus v +(v minus s)F θ2( 11138571113858 1113859 minus ce 0

(A4)

Hessian matrix

H(q e) (s minus v)f θ2( 1113857 (v minus s)(1 + β)f θ2( 1113857

(v minus s)f θ2( 1113857 (s minus v)(1 + β)f θ2( 1113857 minus c1113888 1113889

(A5)

According to the assumptions vgt sgt 0 f(θ2)gt 0 andβ isin [0 1) us the Hessian matrix of U(1113937

dr ) is a negative

definite for q and e which meant that the optimal orderquantity and sales effort of retailers existed in the decen-tralized supply chain erefore we obtained the followingformula of the retailersrsquo order quantity and sales effort byjointly solving equations (A3) and (A4)

qd

(a minus bp)(v minus s)(1 + μ)c +(1 + β)2(v minus s)[(1 + μ)(p minus w) + μc] + Ac[(s + v)(1 + μ) + 2μc minus w]

(v minus s)(1 + μ)c (A6)

ed

(1 + β)[(1 + μ)p + μc minus (1 + 2μ)]w

(1 + μ)c (A7)

Complexity 13

Substituting equations (A6) into (A1) we attained

wmax

1113945

d

m

(w minus c) (a minus bp)(v minus s)(1 + μ)c +(1 + β)2(v minus s)[(1 + μ)p + μc] + Ac[(s + v)(1 + μ) + 2μc] minus (1 + 2μ)w 2Ac +(1 + β)2(v minus s)1113960 11139611113966 1113967

(v minus s)(1 + μ)c

(A8)

Taking into account the first-order condition of equation(A7) with respect to w rounded off to 0 we obtained`

z1113937dm

ze

(a minus bp)(v minus s)(1 + μ)c +(1 + β)2(v minus s)[(1 + μ)p + μc] + Ac[(s + v)(1 + μ) + 2μc] minus (2w minus c)(1 + 2μ) 2Ac +(1 + β)2(v minus s)1113960 1113961

(v minus s)(1 + μ)c

(A9)

Solving equation (A8) we found that the optimalwholesale price of the manufacturer is as follows

wdlowast

(a minus bp)(1 + μ)(v minus s)c +(v minus s)(1 + β)2[(1 + μ)p +(1 + 3μ)c] + Ac[(s + v)(1 + μ) + 2(1 + 3μ)c]

2(1 + 2μ) 2Ac +(v minus s)(1 + β)21113960 1113961 (A10)

By substituting equation (A9) into equations (A6) and(A7) we attained the optimal order quantity and sales effortwhich are displayed in equations (13) and (14)

B Propositions

Proof of Proposition 1 We solved the first derivatives ofoptimal order quantity qdlowast and sales effort edlowast with respect

to β in equations (13) and (14) respectively Meanwhiletaking the second-order derivative of qdlowast with respect to βwe obtained

zqdlowast

zβ

(1 + β)(p minus c)

c

z2qdlowast

zβ2

p minus c

c (B1)

zedlowast

zβ4(p minus c)(v minus s)2β β3 + 4β2 + 6β + 41113872 1113873 +(a minus bp)c(1 + β)2(v minus s)2 + Ac(v minus s)(1 + β)2 (2(p minus c) + v + s) + 2(Ac)2(4p minus c minus s minus v)1113960 1113961

2c (1 + β)2(v minus s) + 2Ac1113960 11139612

(B2)

Since the second derivative of edlowast with respect to β wasvery complicated there was no need to solve it According tothe assumptions pgt vgt cgt sgt 0 β isin (0 1) a minus bpgt 0 andcgt 0 so (zqdlowastzβ)gt 0 (z2qdlowastzβ2)gt 0 and (zedlowastzβ)gt 0

Proof of Proposition 2 Based on the process above we tookthe first-order and second-order partial derivatives of op-timal wholesale price wdlowast with respect to β and μ in equation(12) respectively

zwdlowast

zβ

c(v minus s)(1 + μ)(1 + β)[(a minus bp)(v minus s) + A(2p minus s minus v)]

(1 + 2μ) (1 + β)2(v minus s) + 2Ac1113960 11139612 (B3)

zwdlowast

zμ minus

(a minus bp)(v minus s)c +(p minus c)(v minus s)(1 + β)2 + Ac(s + v minus 2c)1113960 1113961

2(1 + 2μ)2 (v minus s)(1 + β)2 + 2Ac1113960 1113961 (B4)

14 Complexity

According to the optimal result of order quantity inequation (13) and the assumption qgt 0 the condition s +

v minus 2cgt 0 is found According to the assumptions we had(zwdlowastzβ)gt 0 and (zwdlowastzμ)lt 0

Proof of Proposition 3 e gap between qdlowast and qclowastdenoted by Δq is as follows

Δq qdlowast

minus qclowast

(p minus c)(v minus s) (1 + β)2 minus 21113960 1113961 minus Ac(s + v minus 2c) minus (a minus bp)(v minus s)c

2c(v minus s) (B5)

If (p minus c)(v minus s)[(1 + β)2 minus 2] is more thanAc(s + v minus 2c) + (a minus bp)(v minus s)c we have Δqgt 0 ieqdlowast gt qclowast In contrast if (p minus c)(v minus s)[(1 + β)2 minus 2] is less

than Ac(s + v minus 2c) + (a minus bp)(v minus s)c Δqlt 0 ieqdlowast lt qclowast

e gap between edlowast and eclowast denoted by Δe is asfollows

Δe edlowast

minus eclowast

Aβc(4p minus 2c minus s minus v) minus Ac(s + v minus 2c) minus (v minus s)(1 + β) (a minus bp)c + 1 minus β21113872 1113873(p minus c)1113960 1113961

2c 2Ac +(1 + β)2(v minus s)1113960 1113961 (B6)

Similarly if Aβc(4p minus 2c minus s minus v)gtAc(s + v minus 2c)

+(v minus s)(1 + β)[(a minus bp)c + (1 minus β2)(p minus c)] we haveΔe edlowast minus eclowast gt 0 In contrast ifAβc(4p minus 2c minus s minus v)ltAc(s + v minus 2c)+

(v minus s)(1 + β)[(a minus bp)c + (1 minus β2)(p minus c)] we haveΔe edlowast minus eclowast lt 0

Upon solving the first-order derivative of Δq and Δe withrespect to β respectively we obtained

zΔ q

zβ

(1 + β)(p minus c)

c (B7)

zΔ e

zβ4(p minus c)(v minus s)2 β3 + 4β2 + 6β + 41113872 1113873β +(a minus bp)c(1 + β)2(v minus s)2 + Ac(v minus s)(1 + β)2 2(p minus c) +v + s) + 2(Ac)2(4p minus c minus s minus v)1113960 1113961

2c (1 + β)2(v minus s) + 2Ac1113960 11139612

(B8)

According to the assumptions pgt vgt cgt sgt 0 β isin (0 1)cgt 0 and a minus bpgt 0 (zΔ qzβ)gt 0 and (zΔ ezβ)gt 0 can beproved Hence the gaps increased as a function with respectto β

Proof of Proposition 4 e proof for Proposition 4 is similarto Proposition 2 Herein we took first-order and second-order partial derivatives of optimal wholesale price wsclowast withrespect to β μ and ϕ in equation (18) respectively esolving results are as follows

zwsclowast

zβ

[(1 + 2μ)ϕ minus μ](p minus c)

(1 + 2μ)(1 + β)2

zwsclowast

zμ minus

(p minus c)β(1 + β)(1 + 2μ)2

zwsclowast

zϕ minus

p minus c

1 + β

(B9)

(zwsclowastzβ) if ϕgt (μ(1 + 2μ)) is more than 0 otherwiseϕgt (μ(1 + 2μ)) is less than 0 us wsc could be increasingwith respect to β nevertheless decreasing with respect to βBesides according to the assumptions pgt vgt cgt sgt 0β isin (0 1) cgt 0 and a minus bpgt 0 it can be evidently foundthat (zwsclowastzμ)gt 0 and (zwsclowastzϕ)gt 0

Data Availability

e data used to support the findings of this study are in-cluded within the article

Conflicts of Interest

e authors declare no conflicts of interest

Acknowledgments

is work was supported by a major project financed by theNational Social Science Fund of China (Approval nos71461009 71662011 and 71761015)

Complexity 15

References

[1] Y Ren D C Croson and R T A Croson ldquoe overconfidentnewsvendorrdquo Journal of the Operational Research Societyvol 68 no 5 pp 496ndash506 2017

[2] T H Cui and P Mallucci ldquoFairness ideals in distributionchannelsrdquo Journal of Marketing Research vol 53 no 6pp 969ndash987 2016

[3] K Chen X Wang M Huang and W-K Ching ldquoSalesforcecontract design joint pricing and production planning withasymmetric overconfidence sales agentrdquo Journal of Industrialamp Management Optimization vol 13 no 2 pp 873ndash8992017

[4] X X Zheng Z Liu K W Li et al ldquoCooperative game ap-proaches to coordinating a three-echelon closed-loop supplychain with fairness concernsrdquo International Journal of Pro-duction Economic vol 212 pp 92ndash110 2018

[5] China Internet Watch False Prosperity behind Double 11Refund Rate Increased to 69 2014 httpswwwchinainternetwatchcom11643false-prosperity-behind-double-11-refund-rate-increased-to-69

[6] C L Lu ldquoPampGrsquos collaborative supply chain transformationrdquoEnterprise Management vol 10 pp 79-80 2016

[7] M Li N C Petruzzi and J Zhang ldquoOverconfident competingnewsvendorsrdquo Management Science vol 63 no 8pp 2637ndash2646 2016

[8] T Nie ldquoDual-fairness supply chain with quantity discountcontractsrdquo European Journal of Operational Researchvol 258 no 2 pp 491ndash500 2017

[9] X Du and B Jiang ldquoSignaling through price and quality toconsumers with fairness concernsrdquo Journal of MarketingResearch vol 53 no 6 pp 988ndash1000 2016

[10] V Kurz A Orland and K Posadzy ldquoFairness versus effi-ciency how procedural fairness concerns affect coordina-tionrdquo Experimental Economics vol 21 no 3 pp 601ndash6262018

[11] S Villa and J A Castantildeeda ldquoTransshipments in supplychains a behavioral investigationrdquo European Journal of Op-erational Research vol 269 no 2 pp 715ndash729 2018

[12] B Gu Y Fu and Y Li ldquoFresh-keeping effort and channelperformance in a fresh product supply chain with loss-averseconsumersrsquo returnsrdquo Mathematical Problems in Engineeringvol 2018 Article ID 4717094 20 pages 2018

[13] D Croson R Croson and Y Ren How to Manage anOverconfident Newsvendor 2008 httpcbeesutdallasedupapersCrosonRenCmsonMS2008

[14] M Li ldquoOverconfident distribution channelsrdquo Production andOperations Management vol 28 no 6 pp 1347ndash1365 2019

[15] W Liu D Wang O Tang and D Zhu ldquoe impacts oflogistics service integratorrsquos overconfidence behaviour onsupply chain decision under demand surgerdquo EuropeanJournal of Industrial Engineering vol 12 no 4 pp 558ndash5972018

[16] L Xu X Shi P Du K Govindan and Z Zhang ldquoOptimi-zation on pricing and overconfidence problem in a duopo-listic supply chainrdquo Computers amp Operations Researchvol 101 pp 162ndash172 2019

[17] Y Li and M Shan ldquoAdvance selling decisions with over-confident consumersrdquo Journal of Industrial and ManagementOptimization vol 2 no 3 pp 891ndash905 2016

[18] T H Cui J S Raju and Z J Zhang ldquoFairness and channelcoordinationrdquo Management Science vol 53 no 8pp 1303ndash1314 2007

[19] X Wu and J A Niederhoff ldquoFairness in selling to thenewsvendorrdquo Production and Operations Managementvol 23 no 11 pp 2002ndash2022 2014

[20] Z Chen and S-I Ivan Su ldquoSocial welfare maximization withthe least subsidy photovoltaic supply chain equilibrium andcoordination with fairness concernrdquo Renewable Energyvol 132 pp 1332ndash1347 2019

[21] W Liu D Wang X Shen X Yan and W Wei ldquoe impactsof distributional and peer-induced fairness concerns on thedecision-making of order allocation in logistics service supplychainrdquo Transportation Research Part E Logistics and Trans-portation Review vol 116 pp 102ndash122 2018

[22] L Zhang H Zhou Y Liu and R Lu ldquoOptimal environmentalquality and price with consumer environmental awarenessand retailerrsquos fairness concerns in supply chainrdquo Journal ofCleaner Production vol 213 pp 1063ndash1079 2019

[23] X LWang S Q Hu and X B Liu ldquoSupply coordination withoption contract considering overconfidence under randomdemand and yieldsrdquo Computer Integrated ManufacturingSystem vol 24 no 11 pp 2898ndash2908 2018

[24] W Jiang and J Liu ldquoInventory financing with overconfidentsupplier based on supply chain contractrdquo MathematicalProblems in Engineering vol 2018 Article ID 505438712 pages 2018

[25] P Ma H Wang and J Shang ldquoContract design for two-stagesupply chain coordination integrating manufacturer-qualityand retailer-marketing effortsrdquo International Journal ofProduction Economics vol 146 no 2 pp 745ndash755 2013

[26] J Liu W F Jiang and Y P Tang ldquoe supply chain coor-dination based on carbon reduction of overconfident man-ufacturerrdquo Journal of Jiangxi Normal University (NaturalScience) vol 42 no 2 pp 203ndash207 2018

[27] R Dubey N Altay and C Blome ldquoSwift trust and com-mitment the missing links for humanitarian supply chaincoordinationrdquo Annals of Operations Research vol 283 no 1-2 pp 159ndash177 2019

[28] C Cui Z Feng and C Tan ldquoCredibilistic loss aversion nashequilibrium for bimatrix games with triangular fuzzy payoffsrdquoComplexity vol 2018 Article ID 7143586 16 pages 2018

[29] E Fehr and KM Schmidt ldquoA theory of fairness competitionand cooperationrdquo e Quarterly Journal of Economicsvol 114 no 3 pp 817ndash868 1999

[30] Z Qin and J Yang ldquoAnalysis of a revenue-sharing contract insupply chain managementrdquo International Journal of LogisticsResearch and Applications vol 11 no 1 pp 17ndash29 2008

[31] S-Z Bai and J Ni ldquoResearch on buy-back contract with saleseffort cost sharingrdquo in Proceedings of the 2009 Chinese Controland Decision Conference IEEE Guilin China pp 1917ndash1921June 2009

[32] E Karagozoglu and K Keskin ldquoTime-varying fairness con-cerns delay and disagreement in bargainingrdquo Journal ofEconomic Behavior and Organization vol 147 pp 115ndash1282018

16 Complexity

manufacturer was reported to be rational while the retailerhad a cognitive bias We assumed an overconfident retailerwho not only underestimated the variance of demand butalso overestimated the mean of demand and the effect ofsales effort on demand According to Chen et al [3] weadopted mean-increasing and variance-reducing transfor-mation of the actual market demand D us the marketdemand can be explained as follows

DO a minus bp +(1 + β)e +(1 minus β)θ a minus bp +(1 + β)e + θ1(6)

wherein β indicates the overconfident level (0lt βlt 1)Moreover the continuous random variable θ1 is equal to(1 minus β)θ which is a random noisy signal and assumed to beuniformly distributed ie θ1 sim U(minus A A) When β 0 theretailer had no overconfidence whereas for 0lt βlt 1 themean of DO increased and the variance of DO decreased ascompared to the mean and variance of D ie E(DO)gtE(D)

and var(DO)lt var(D) Meanwhile the impact of the pro-motion effort ((1 + β)egt e) on market demand was over-estimated ie the influence of e on market demandincreased as β increased

Similarly when market demand was uncertain weconsidered an order quantity q for the retailer withoverconfidence and fairness concern provided q was morethan the random demand Do Such an arrangement in-dicated that the retailer faced an overstock situation In thissituation we have minus Alt θ1 lt q minus [a minus bp + (1 + β)e] Incontrast if q was less than DOq minus [a minus bp + (1 + β)e]lt θ1 ltA was obtained which indi-cated that the retailer faced a stockout situation ie weassumed that θ2 q minus [a minus bp + (1 + β)e] HenceF(θ2) (μ(w minus c) + (1 + μ)(w minus v)(s minus v)(1 + μ)) eutility of the overconfident retailer is as follows

q emax

Uor pE min q Do1113864 11138651113858 1113859 + E s q minus DO( 1113857

+1113858 1113859

minus E v DO minus q( 11138571113858 1113859+

minus wq minus C(e)(7)

e reference framework for a retailer with fairnessconcern is the F-S model [29]

Ufr 1113945

r

minus μ 1113945m

minus 1113945r

0⎞⎠ minus λ 1113945r

minus 1113945m

0⎞⎠⎛⎝⎛⎝ (8)

where μ is the degree of fairness concern and λ is thecoefficient of sympathy μ λge 0 and 0lt λlt 1 e first partof equation (12) reflected the retailerrsquos profit whereas thesecond part evaluated the utility loss from disadvanta-geous inequality for the retailer when πm gt πr and thethird part measured the loss from advantageous inequalityfor the retailer when πm lt πr Note that if the retailerrsquosprofit was worse than the manufacturersrsquo profit (ie

πm gt πr) the retailer will have negative utility under thecondition of unfair aversion In contrast if his profit wasmore significant than the manufacturersrsquo the retailer willalso suffer positive utility because of the sympathyHowever Qin et al [30] demonstrated that the profit ofthe leader in a supply chain often accounted for more thanhalf of the overall profit when the Stackelberg game wascarried out In this paper the retailer is a weak follower ascompared to the manufacturer who served as the leaderAlongside this we found out that the profit of themanufacturer was twice that of the retailer when themanufacturer and the retailer were both unboundedlyrational Figures 4 and 5 validate the result in Section 5when μ 0 β 0 hence the retailer had the negativeutility of unfair aversion erefore according to equation(8) when the retailer had fairness concern only his utilityfunction was corrected to

Ufr 1113945

r

minus μ 1113945m

minus 1113945r

⎞⎠ (1 + μ)1113945r

minus μ1113945m

⎛⎝ (9)

where 1113937r and 1113937m denotes the profit of the rational retailerand the rational manufacturer respectively μgt 0 is thedegree of fairness concern

According to the theories of overconfidence and fairnessconcern the utility of the retailer with the two behavioralpreferences which was replaced by 1113937r in equation (9) withUo

r in equation (7) is as follows

q emax

U 1113945d

r

⎞⎠ (1 + μ)Uor minus μ1113945

d

m

⎛⎝ (10)

e retailer with behavioral preference orders q fromthe manufacturer the rational manufacturer profit func-tion is

wmax

1113945

d

m

(w minus c)q (11)

e Stackelberg game model where the manufacturerwas serving as the leader and the retailer served as thefollower was established in the decentralized supplychain e events of the model are described as followsFirstly the manufacturer decided the wholesale price wen the retailer made decisions about the order quantityq and the sales effort e based on the decision of themanufacturer and market demand To solve the subgameperfect equilibrium the backward induction method wasapplied e equilibrium solutions under the Stackelberggame model were considered when the retailer had acognitive bias e solutions are as follows e proof isgiven in Appendix A

Complexity 5

wdlowast

(a minus bp)(1 + μ)(v minus s)c +(v minus s)(1 + β)2[(1 + μ)p +(1 + 3μ)c] + Ac[(s + v)(1 + μ) + 2(1 + 3μ)c]

2(1 + 2μ) 2Ac +(v minus s)(1 + β)21113960 1113961 (12)

qdlowast

(a minus bp)(v minus s)c +(1 + β)2(v minus s)(p minus c) + Ac(s + v minus 2c)

2(v minus s)c (13)

edlowast

(1 + β) (a minus bp)(s minus v)c +(p minus c)(v minus s)(1 + β)2 + 2Ac(2p minus c minus s minus v)1113960 1113961

2c 2Ac +(1 + β)2(v minus s)1113960 1113961 (14)

Based on the above equilibrium solutions in thedecentralized supply chain the propositions about the effectof overconfidence and fairness concern (β and μ) on decisionvariables (qdlowast edlowast and wdlowast) are shown as in the followingNote that the proof of all propositions can be found inAppendix B

Proposition 1 When the retailer had both overconfidenceand fairness concern the optimal order quantity qdlowast and saleseffort edlowast were both positively correlated with the overcon-fident level β but had nothing to do with the degree of fairnessconcern μ Furthermore qdlowast is a strictly convex function withrespect to β

Proposition 1 indicated that the overconfident drivesthe retailer to increase order volume and improve salesefforts e reason is that overconfidence had two maineffects On the one hand the retailer overestimated theaverage market demand and exaggerated the effectivenessof sales effort which directly motivated the retailer to makea more considerable sales effort (for example increasingpromotions) On the other hand the retailer overestimatedthat the variance of the ldquorandom impact factor θ1rdquo issmaller than the actual level In other words from his pointof view a smaller market demand fluctuation influencedprediction us the two effects made the retailer ordermore product quantities and pay more sales effort for morerevenue Meanwhile it was easy to find that qdlowast and edlowast

were only related to β as per equations (13) and (14) whichrevealed that the retailersrsquo decisions on ordering and salesefforts were not affected by fairness concern

Proposition 2 When the retailer has both overconfidenceand fairness concern the optimal wholesale price wdlowast ispositively proportional to the overconfident level β but in-versely proportional to the degree of fairness concern μ

Proposition 2 illustrated that the cognitive bias of theretailer had an opposite effect on the manufacturersrsquodecision Firstly the moment the rational manufacturerrealized that the overconfident retailer could increase theorder quantity he raised the wholesale price of theproduct to maximize the profit which was consistent withthe anecdotes that merchants in real world could takeadvantage of othersrsquo overconfidence to maximize theirprofit Besides when the manufacturer understood thefeatures of the retailersrsquo fairness concern he lowered the

wholesale price to alleviate the retailersrsquo unfair aversionwhich implied that the retailer could boost his bargainingpower in the supply chain while he was concerned with thefairness erefore the rational manufacturer was re-quired to set a reasonable wholesale price to balance theinfluence of the two cognitive biases Next we comparedthe equilibrium outcomes in the centralized supply chainmodel with those in the decentralized supply chain modelere is a proposition as follows

Proposition 3 qclowastmay be more than or less thanqdlowast eclowastmaybe more than or less than edlowast Interestingly Δq qdlowast minus qclowastand Δe edlowast minus eclowast increases as the overconfidence level βincreases respectively

According to Proposition 3 the effect of overconfidencethe optimal order quantity and sales effort on the decen-tralized supply chain could deviate from the optimalequilibrium solution in the centralized supply chain whichresulted in a loss of the expected profits thereby increasingthe revenue gap between manufacturer and retailer

4 Buyback Contract with PromotionCost Sharing

It can be seen from Section 33 that due to the cognitivebias of the retailer his decisions deviated from the op-timal decisions in the centralized supply chain whichaffected the profit or utility of supply chain membersereby in order to coordinate the profit or utility be-tween the supply chain members and ensure the maxi-mized performance of the two-echelon supply chainaccording to Bai et al [31] the buyback contract withpromotional cost sharing was proposed in this sectionHere the manufacturer served as the leader who buysback the unsold products from the retailer at the buybackprice and shared the promotional cost We denoted thiscontract factor as (wsc r and 1 minus ϕ) where wsc is thewholesale price r is the buyback price and 1 minus ϕ is thefraction of promotional costs that the rational manu-facturer paid while the retailer undertook ϕ portion ofthe cost (0ltϕlt 1) In order to avoid retailersrsquo over-ordering and encourage the retailer to accept the con-tract we assumed 0lt sle rltwsc Based on thesedescriptions and assumptions above the manufacturersrsquoprofit and retailersrsquo utility are given by the followingexpression

6 Complexity

1113945

sc

m

(w minus c)q + E (s minus r) q minus DO( 11138571113858 1113859+

minus (1 minus ϕ)C(e)

(15)

U 1113945sc

r

⎞⎠ (1 + μ) p[(a minus bp +(1 + β)e)] + E r q minus DO( 1113857+

1113858 11138591113864⎛⎝

minus E v DO minus q( 1113857+

1113858 1113859 minus wq minus ϕC(e)1113865 minus μ1113945sc

m

(16)

wherein equation (15) is the profit when the manufacturerbuys back the unsold products at buyback price r from theretailer and thus obtained the surplus value of theseproducts (1 minus ϕ)C(e) is the promotional costs which themanufacturer undertook r(q minus DO)+ in equation (16) is therevenue when the retailer sells the unsold inventory of

products to themanufacturer at the end of the selling seasonϕC(e) is the promotional costs which he pay

According to the first derivations of equation (16) basedon q and e the following expression is obtained

qsc 2A(1 + μ)(v minus w) minus μ(w minus c)

(1 + μ)(v minus r) + μ(s minus r)+ a minus bp +(1 + k)e

scminus A

esc (1 + β)[μ(p + c minus 2w) + p minus w]

(2μϕ minus μ + ϕ)c

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(17)

In the buyback contract with promotion cost sharing toachieve supply chain coordination the optimal decisions ofthe retailer were the same as those under the centralizedsupply chain model that is qsc qclowast and esc eclowast [25 26]erefore equation (17) needs to be equal to equation (5) asthe contract factors (wsc r) were obtained

wsclowast

[β(1 + μ) + 1 minus ϕ(1 + 2μ)]p +[βμ +(1 + 2μ)ϕ]c

(1 + 2μ)(1 + β) (18)

rlowast

(v minus s)(p minus c)(1 + β)(1 + μ)vβ + 2A[cϕ(p minus c)(1 + 2μ) + βcv(1 + μ)] minus 2Ac p(v minus s)[1 + β +(2 + β)βμ] minus sv(1 + β)(1 + μ)1113864 1113865

(1 + 2μ)(1 + β)[β(p minus c)(v minus s) + 2Ac(c minus v)]

(19)

e purpose of the buyback contract with promotioncost sharing was to design a reasonable profit allocationscheme between the rational manufacturer and the retailerwith cognitive bias It should be noted that if the profit ofsupply chain members became lower the contract wasunacceptable e contract was acceptable if and only if1113937

scm ge 1113937

dlowastm and U(1113937

scr )geU(1113937

dlowastr )

Based on the above equilibrium solutions in the contractthe following propositions were obtained Due to the so-phisticated formula of the buyback price this section dis-cussed the effect of overconfidence and fairness concern onwholesale price

Proposition 4 e optimal wholesale price in the buybackcontract with promotion cost sharing was negatively corre-lated with μ and ϕ Nevertheless if ϕlt (μ(1 + 2μ)) thewholesale price wsclowast is positively correlated with β In con-trast if ϕlt (μ(1 + 2μ)) it is negatively correlated withβ

Proposition 4 implied that the contract was adopted andthe negative effect of fairness concern μ on wholesale pricewsclowast remained consistent with Proposition 2 Besides as wsc

increased ϕ decreased and 1 minus ϕ increased accordinglywhich meant that the manufacturer bore more promotionalcost when the wholesale price rose which is in line with thepractice Meanwhile the values of μ and ϕ determined theimpact of β (positive or negative influence) on the wholesaleprice wsc which indicated that fairness concern μ andpromotional cost portion 1 minus ϕ were critical to manufac-turerrsquos decisions on the wholesale price wsc us when the

two types of behavioral preferences coexisted in the contractthe effect of overconfidence on the optimal wholesale pricedepended on the relationship between fairness degree andfraction of promotion cost Since the utility and profitfunctions in the supply chain included a large number ofunknown parameters under three scenarios (ie Sections32 33 and 4) it was difficult to compare and analyze theeffects of the cognitive biasesus numerical methods wereutilized to conduct further analysis in the next section

5 Numerical Analysis

In this section a numerical study approach was adopted toallow a more in-depth insight into the two behavioralpreferences First a numerical experiment was conducted toshow how equilibrium solutions and profit (utility) in thetwo-echelon supply chain changed along with the levels ofoverconfidence and fairness concern (ie β and μ) and toexamine the robustness of propositions en we also de-termined the effect of overconfidence and fairness concernon the buyback contract with promotional cost sharing

In our experiments some parameters were gatheredfrom the existing papers For the parameters related tooverconfidence Ren et al [1] set β isin [0 1] andc 05 06 08 09 to perform their numerical experi-ments and Li et al [7] set β isin (0 1) and c 5 to conducttheir study For the parameters related to fairness concernZheng et al [4] set μge 0 c 20 and a 100 to implementtheir numerical analysis and Guo et al [9] set μge 0 toperform their studies According to the parameters above

Complexity 7

we assumed β isin [0 1) and μ 0 1 2 to reflect the level ofoverconfidence and fairness concern in this paper Whenβ 0 and μ 0 the retailer was considered to be rationalSimilarly other fundamental parameter values were set inexperiments as follows a 50 θ sim U(minus 30 30) c 4p 20v 7 s 2 b 2 and c 1 All the parameterssatisfied the constraint conditions and assumptions of thethree models in Sections 32 33 and 4e parameter valueswere also in line with the real situation of the market Wecould easily obtain qclowast 32 eclowast 16 and 1113937

c 252 in the

centralized supply chain

51 Effect of Overconfidence and Fairness Concern on theSupply Chain e changes in the decision variables (w qand e) manufacturer profit and retailer expected utilitywhen the level of overconfidence and fairness concern (β andμ) changed are shown in Figures 1ndash5

As shown in Figures 1 and 2 when the retailer was ra-tional (eg β 0 and μ 0) in the decentralized supplychain both optimal order quantity and sales effort were lowerthan those under the centralized supply chain Next when theretailer had overconfidence and fairness concern simulta-neously overconfidence could only affect the retailerrsquos opti-mal order quantity and sales effort of the retailer because theoptimal order quantity and sales effort were a function ofoverconfidence in equations (13) and (14) Moreover theorder quantity was monotonically increasing as concavefunction with respect to β In comparison sales effort wasmonotonically increasing as convex function with respect toit Furthermore there existed a threshold (β 083) when βwas less than 083 and the two decision variables were bothless than the optimal values (qclowast 32 eclowast 16) under thecentralized decision scenario Nevertheless once β was largerthan 083 the two decision variables were both more thanthem e above discussion verified the correctness ofProposition 1 In short overconfidence was the main factorthat affected the retailerrsquos optimal strategies of order quantity

and sales effort in the decentralized supply chain Howeverthe above findings contradicted with the conclusions from astudy conducted by Xu et al [16] which examined the effectof retailerrsquos overconfidence on the supply chain and the resultsuggested that a higher level of overconfident resulted in thelower ordering quantity

In Figure 3 we observed that the higher overconfidentlevel resulted in the high wholesale price in contrast ahigh degree of fairness concern resulted in low pricewhich demonstrated that Proposition 2 was robust isfigure implied that when the information for both partieswas symmetric the rational manufacturer knew that theretailer was overconfident that is the retailer was opti-mistic about the market demand and increased the orderquantity As a result the manufacturer also raised thewholesale price to maximize its revenue Also if theoverconfident retailer paid attention to the unfairness ofprofit allocation in the decision-making process the re-tailerrsquos bargaining power was enhanced [32] Hence forbetter performance in the supply chain the leadingmanufacturer was required to lower wholesale prices toease the feeling of unfairness by the retailer which isconsistent with the actual economic phenomenon Fur-thermore these findings are consistent with the resultconcluded by Cui et al [2] which indicated that a highdegree of the retailerrsquos fairness concern could lower thewholesale price Interestingly as the levels of the cognitivebias increased its effect on wholesale price graduallydecreased which highlighted its effect on decreasedwholesale prices

Figure 4 displays the impact of the cognitive bias on theretailerrsquos utility e graph showed that the retailer expectedutility increased gradually with the overconfident level andthe degree of fairness concern which suggested that twocognitive biases were beneficial to the retailer e reasonsfor such a phenomenon were that both cognitive biasescould increase the optimal order quantity and sales effortthus enhancing the utility of the retailer

Order quantity in centralized supply chain

01 02 03 04 05 06 07 08 09 10β

15

20

25

30

35

40

Ord

er q

uant

ity

μ = 012

Figure 1 e effect of the cognitive biases on optimal order quantity

8 Complexity

Figure 5 illustrates the impact of overconfidence andfairness concern on the manufacturerrsquos profit at is theoptimal manufacturerrsquos profit is an increasing function of βbut a decreasing function of μ According to Figure 3 thegraph of themanufacturerrsquos profit was similar to the graph ofthe wholesale price which indicated that the manufacturerrsquosprofit was firmly related to wholesale price Figure 5 alsoillustrates the effect of the cognitive bias β on the manu-facturerrsquos decreasing profit

52 Effect of Overconfidence and Fairness Concern on SupplyChain Coordination As can be seen from Section 51 thereexist deviations between the optimal decisions under the

benchmark and the decisions in the presence of the retailerrsquoscognitive biases erefore the buyback contract withpromotion cost sharing was necessary to coordinate therevenue of supply chain members and optimize the per-formance of the supply chain

Here we discussed the validity of the contract When theoptimal order quantity and sales effort of the retailer underthe contract were the same as the optimal decision variablesin the centralized supply chain ie qsc qclowast 32 andesc eclowast 16 the buyback contract with promotion costsharing achieved supply chain coordination First the effectof overconfidence and fairness concern (β and μ) on contractfactors wsrlowast rlowast and ϕ is shown in Figures 6 and 7 andTable 1 For clearly analyzing the effect of β and μ on wsrlowast

Degree of sales effort in centralized supply chain

01 02 03 04 05 06 07 08 09 10β

μ = 012

4

6

8

10

12

14

16

18

Deg

ree o

f sal

es ef

fort

Figure 2 e effect of the cognitive biases on optimal sales effort

01 02 03 04 05 06 07 08 09 10β

-μ = 0--μ = 1-μ = 2

4

5

6

7

8

9

10

11

12

13

14

Who

lesa

le p

rice

Figure 3 e effect of the cognitive biases on the optimal wholesale price

Complexity 9

and rlowast we let ϕ 1 in equations (18) and (19) which meantthat the manufacturer proposed the buyback contract onlyand the retailer bore all sales effort costs

According to Figure 6 it was revealed that the optimalwholesales price is a strictly increasing function of theoverconfidence level β but a strictly decreasing function ofthe degree of fairness concern μ which confirmed the resultof Proposition 4 Similarly Figure 7 highlights that theoptimal buyback price sharply increased with β but de-creased with μ Moreover in the buyback contract wsrlowast wasalways greater than or equal to 4 and rlowast was always greaterthan or equal to 2 in different cognitive bias levels Hence

the optimal buyback price rlowast satisfied the assumptions 2le r Note that the above characteristics influencedcognitive biases hence contract factors wsrlowast and rlowast werealso correct when ϕ is equal to other values (such asϕ 03 05 08) under conditions in which the buybackcontract had promotion cost sharing

However data from Figure 7 can be compared with thedata in Figure 6 which showed rsrlowast is more significant thanwsrlowast when β was within a specific range (for example04le βlt 1) which did not match the assumption rltwsrerefore the buyback contract did not achieve the coor-dination of the behavioral supply chain

01 02 03 04 05 06 07 08 09 10β

-μ = 0--μ = 1-μ = 2

0

20

40

60

80

100

120

Reta

iler e

xpec

ted

utili

ty

Figure 4 e effect of the cognitive biases on retailer utility

01 02 03 04 05 06 07 08 09 10β

-μ = 0--μ = 1-lowastμ = 2

0

20

40

60

80

100

120

140

160

180

200

Man

ufac

ture

r pro

fit

Figure 5 e effect of the cognitive biases on manufacturer profit

10 Complexity

As described in Section 4 if the supply chain membersaccepted the buyback contract with promotion cost sharingthe assumptions 1113937

scm gemax 1113937

dm and U(1113937

scr )gemaxU(1113937

dr )

had to be satisfied Also according to the assumptions thevalue ϕ must satisfy 1113937

scm gemax 1113937

dm and

U(1113937scr )gemaxU(1113937

dr ) us we kept the values of param-

eters to be the same (ie a 50 θ sim U(minus 30 30) c 4p 20 v 7 s 2 b 2 and c 1) to capture a likelydomain of ϕ

As can be seen from Table 2 the overconfident level βincreased the upper and lower limits of ϕ which had a clearupward trend In contrast the upper and lower limits of ϕ

showed a definite declining trend as the degree of fairnessconcern μ increased ese trends illustrated that in order tocoordinate the revenue of both parties and optimize theperformance of the supply chain the manufacturer needs tobear less promotional costs as β increased but bear morepromotional costs as μ value increased Besides when β and μincreased the interval of sharing cost portion ϕ shrunk whichindicated that the buyback contract with promotional costsharing had certain flexibility to coordinate the revenue be-tween supply chain members However the flexibility couldbe weakened with the rise of the cognitive bias level Forsupply chain members the possibility of negotiation or

01 02 03 04 05 06 07 08 09 10β

-μ = 0--μ = 1-μ = 2

4

5

6

7

8

9

10

11

12

Who

lesa

le p

rice

Figure 6 e effect of the cognitive biases on wholesale price wsrlowast

01 02 03 04 05 06 07 08 09 10β

-μ = 0--μ = 1-μ = 2

2

4

6

8

10

12

14

16

18

20

22

Buyb

ack

pric

e

Figure 7 e effect of the cognitive biases on buyback price rlowast

Complexity 11

cooperation between manufacturers and retailers also de-creased with an increase in retailersrsquo cognitive bias levelWhen the overconfident level was more than 081 the re-tailerrsquos utility from decentralized decisions was greater thanthe utility in the contract environment wherein the twoparties could not reach a cooperation agreement us thesupply chain could not achieve coordination as different levelsof cognitive bias under the contract had the same influence ondecision variables profit and utility Table 2 shows a clearpicture wherein μ is equal to 1 and β falls within the range of(0 08) which were used to analyze the effect of the cognitivebiases on the coordination of the supply chain

e results of the numerical simulation in Table 3 in-dicate that the contract factors ie ϕ r wholesale price theprofit and utility of members existed as flexible intervalswhen β isin (0 08) and μ 1 Besides the order quantity qsc

and the degree of sales effort esc under the contract are equalto qclowast and eclowast in a centralized supply chain respectivelyMeanwhile the buyback price r was less than the wholesaleprice wsc and higher than the unit salvage value s whichsatisfied the constrains sle rltwsclowast us the buybackcontract with promotion cost could achieve the coordinationof the two-echelon supply chain Also the buyback price r

was positively correlated with the portion ϕ of promotioncost sharing Notably the wholesale price wsc dropped as ϕincreased when β was in the range of (0 02) in Table 3However the situation reversed when it was in the range of(02 08) which was inconsistent since the manufacturerlowered the wholesale price when the retailer paid morepromotional costs in the contract e reason for such anobservation was that the wholesale price was not related to ϕbut positively related to the overconfident level β when thedegree of fairness concern was moderate Furthermore theretailer accepted the contract designed by manufacturerssince the term U(1113937

scr )gemaxU(1113937

dr ) was guaranteed in

Table 3

6 Conclusions

61MainConclusions In this paper we analyzed the effect ofcognitive bias on equilibrium solutions including thewholesale price sales price sales effort and expected profitsunder demand uncertainty by introducing both overconfi-dence and fairness concern into the two-echelon supply chaine most important conclusion from the study was that forretailers with the two cognitive biases overconfidence waspositive and the main bias that affected supply chain decisionsand the retailersrsquo utility benefits were not only due to over-confidence but also fairness concern For manufacturersoverconfidence could increase the wholesale price and profitHowever the fairness concern had a negative effect on themFurthermore the decisions of the retailer in the decentralizedsupply chain deviated from the optimal decisions in thecentralized supply chainmodel Especially when the thresholdof overconfidence was more than a fixed value the decisionvariables under decentralized decision-making were actuallyhigher than the optimal variables under centralized decision-making Also the buyback contract with promotional costsharing achieved behavioral supply chain coordination Incontrast the promotional cost-sharing portion and buybackprice both benefited from overconfidence but fairness concernhad a detrimental effect on them However as the retailersrsquooverconfidence level increased and fairness concern was equalto 1 the flexibility of contract coordination was weakenedWhen the overconfidence level wasmore than 08 the contractbecame invalid and members were not able to cooperate

62 Management Insights is paper provided some in-teresting managerial insights for decision makers in thedecentralized supply chain wherein retailers had two pref-erences First overconfidence was beneficial to them interms of decision variables and yet the fairness concern

Table 2 e effect of the cognitive biases on the feasible domain of promotion cost-sharing portion ϕ

μβ

0 02 04 06 08 0810 [02884 04976] [03683 05587] [04528 06199] [06343 07754] [07835 08669] mdash1 [02545 04741] [03310 05331] [04167 05908] [06207 07516] [07800 08560] mdash2 [02352 04600] [03101 05156] [04034 05719] [06094 07321] [07736 08478] mdash

Table 3 e effect of the cognitive biases on the coordination of the supply chain

μ 1β

0 02 04 06 08qsc 32 32 32 32 32esc 16 16 16 16 16ϕ [02545 04741] [03310 05331] [04167 05908] [06207 07516] [07800 08560]r [2100 35363] [28268 39188] [30555 42637] [36209 44215] [40485 49164]wsc [49205 53687] [49016 53522] [54092 69568] [62513 72626] [80407 87453]1113937

dm 164103 269591 461996 772619 1112545

U(1113937scm) [168722 217850] [269600 336882] [463175 500938] [772619 814300] [1112545 1120376]

1113937dr 20333 352396 495161 632422 765317

U(1113937scr ) [20333 243043] [352396 400595] [495161 525161] [632422 657581] [765317 784632]

12 Complexity

only damaged the wholesale price Second the two behav-ioral preferences were both conducive to utilities for theretailer but not necessarily beneficial for the manufacturer toreceive profit and overconfidence helped to increasemanufacturersrsquo profit which led to nonoptimal decisionsthat dragged down the performance of the two-echelonsupply chain Finally for the two behavioral preferences ofthe retailer the manufacturer assessed the degree of over-confidence and fairness concern (especially the overconfi-dence) via historical data and the current order quantity orother indicators In contrast factual data indicated that twobehavioral preferences were in the right range (such asβ isin (0 08) and μ isin (0 2)) e manufacturers introducedthe buyback contract with promotional cost sharing into thepractice of pricing and ordered decentralized supply chainsFurthermore they designed a reasonable promotional cost-sharing portion and buyback price thereby achieving anoptimized decision and utility In practice the manufac-turers could analyze whether the sellers had two behavioralpreferences based on the historical data by cooperating withdownstream enterprises ie if two behavioral preferencesexisted they referred to the papersrsquo conclusions to redesign areasonable contract mechanism

63 Future Research Since cognitive biases could existamong all members of supply chains it is necessary toexplore the manufacturersrsquo cognitive biases and analyze thesupply chain coordination problem in the presence of thetwo behavior preferences Besides the overconfidence andfairness concern considered in our centralized decision are afuture research direction

Appendix

A Decentralized Supply Chain Model

In a decentralized supply chain consisting of a rationalmanufacturer and a retailer with cognitive biases maximumprofit or utility is the only goal e backward inductivemethod can be applied to solve a Stackelberg game whereinthe manufacturer is the leader and the retailer is a followere proof of the equilibrium solutions in equations (12)ndash(14)is as follows

e profit of the manufacturer and expected utility ofretailers are as follows respectively

wmax

1113945

d

m

(w minus c)q (A1)

maxqe

U 1113945d

r

⎞⎠ (1 + μ) p[a minus bp +(1 + β)e] + s 1113946θ2

minus Aq minus a minus bp +(1 + β)e + θ1( 1113857f θ1( 1113857( 1113857dθ11113896⎛⎝

minus v 1113946A

θ2a minus bp +(1 + β)e + θ1 minus q( 1113857f θ1( 1113857( 1113857dθ1 minus wq minus

12

ce21113897 minus μ(w minus c)q

(A2)

Solving the first-order partial derivatives of the retailerrsquosexpected utility concerning qd and ed which equal to 0 weobtained the following equations

zU( 1113937dr 1113873

zq (1 + μ) v minus w + F θ2( 1113857(s minus v)1113858 1113859 minus μ(w minus c) 0

(A3)

zU( 1113937dr 1113873

ze (1 + β) p minus v +(v minus s)F θ2( 11138571113858 1113859 minus ce 0

(A4)

Hessian matrix

H(q e) (s minus v)f θ2( 1113857 (v minus s)(1 + β)f θ2( 1113857

(v minus s)f θ2( 1113857 (s minus v)(1 + β)f θ2( 1113857 minus c1113888 1113889

(A5)

According to the assumptions vgt sgt 0 f(θ2)gt 0 andβ isin [0 1) us the Hessian matrix of U(1113937

dr ) is a negative

definite for q and e which meant that the optimal orderquantity and sales effort of retailers existed in the decen-tralized supply chain erefore we obtained the followingformula of the retailersrsquo order quantity and sales effort byjointly solving equations (A3) and (A4)

qd

(a minus bp)(v minus s)(1 + μ)c +(1 + β)2(v minus s)[(1 + μ)(p minus w) + μc] + Ac[(s + v)(1 + μ) + 2μc minus w]

(v minus s)(1 + μ)c (A6)

ed

(1 + β)[(1 + μ)p + μc minus (1 + 2μ)]w

(1 + μ)c (A7)

Complexity 13

Substituting equations (A6) into (A1) we attained

wmax

1113945

d

m

(w minus c) (a minus bp)(v minus s)(1 + μ)c +(1 + β)2(v minus s)[(1 + μ)p + μc] + Ac[(s + v)(1 + μ) + 2μc] minus (1 + 2μ)w 2Ac +(1 + β)2(v minus s)1113960 11139611113966 1113967

(v minus s)(1 + μ)c

(A8)

Taking into account the first-order condition of equation(A7) with respect to w rounded off to 0 we obtained`

z1113937dm

ze

(a minus bp)(v minus s)(1 + μ)c +(1 + β)2(v minus s)[(1 + μ)p + μc] + Ac[(s + v)(1 + μ) + 2μc] minus (2w minus c)(1 + 2μ) 2Ac +(1 + β)2(v minus s)1113960 1113961

(v minus s)(1 + μ)c

(A9)

Solving equation (A8) we found that the optimalwholesale price of the manufacturer is as follows

wdlowast

(a minus bp)(1 + μ)(v minus s)c +(v minus s)(1 + β)2[(1 + μ)p +(1 + 3μ)c] + Ac[(s + v)(1 + μ) + 2(1 + 3μ)c]

2(1 + 2μ) 2Ac +(v minus s)(1 + β)21113960 1113961 (A10)

By substituting equation (A9) into equations (A6) and(A7) we attained the optimal order quantity and sales effortwhich are displayed in equations (13) and (14)

B Propositions

Proof of Proposition 1 We solved the first derivatives ofoptimal order quantity qdlowast and sales effort edlowast with respect

to β in equations (13) and (14) respectively Meanwhiletaking the second-order derivative of qdlowast with respect to βwe obtained

zqdlowast

zβ

(1 + β)(p minus c)

c

z2qdlowast

zβ2

p minus c

c (B1)

zedlowast

zβ4(p minus c)(v minus s)2β β3 + 4β2 + 6β + 41113872 1113873 +(a minus bp)c(1 + β)2(v minus s)2 + Ac(v minus s)(1 + β)2 (2(p minus c) + v + s) + 2(Ac)2(4p minus c minus s minus v)1113960 1113961

2c (1 + β)2(v minus s) + 2Ac1113960 11139612

(B2)

Since the second derivative of edlowast with respect to β wasvery complicated there was no need to solve it According tothe assumptions pgt vgt cgt sgt 0 β isin (0 1) a minus bpgt 0 andcgt 0 so (zqdlowastzβ)gt 0 (z2qdlowastzβ2)gt 0 and (zedlowastzβ)gt 0

Proof of Proposition 2 Based on the process above we tookthe first-order and second-order partial derivatives of op-timal wholesale price wdlowast with respect to β and μ in equation(12) respectively

zwdlowast

zβ

c(v minus s)(1 + μ)(1 + β)[(a minus bp)(v minus s) + A(2p minus s minus v)]

(1 + 2μ) (1 + β)2(v minus s) + 2Ac1113960 11139612 (B3)

zwdlowast

zμ minus

(a minus bp)(v minus s)c +(p minus c)(v minus s)(1 + β)2 + Ac(s + v minus 2c)1113960 1113961

2(1 + 2μ)2 (v minus s)(1 + β)2 + 2Ac1113960 1113961 (B4)

14 Complexity

According to the optimal result of order quantity inequation (13) and the assumption qgt 0 the condition s +

v minus 2cgt 0 is found According to the assumptions we had(zwdlowastzβ)gt 0 and (zwdlowastzμ)lt 0

Proof of Proposition 3 e gap between qdlowast and qclowastdenoted by Δq is as follows

Δq qdlowast

minus qclowast

(p minus c)(v minus s) (1 + β)2 minus 21113960 1113961 minus Ac(s + v minus 2c) minus (a minus bp)(v minus s)c

2c(v minus s) (B5)

If (p minus c)(v minus s)[(1 + β)2 minus 2] is more thanAc(s + v minus 2c) + (a minus bp)(v minus s)c we have Δqgt 0 ieqdlowast gt qclowast In contrast if (p minus c)(v minus s)[(1 + β)2 minus 2] is less

than Ac(s + v minus 2c) + (a minus bp)(v minus s)c Δqlt 0 ieqdlowast lt qclowast

e gap between edlowast and eclowast denoted by Δe is asfollows

Δe edlowast

minus eclowast

Aβc(4p minus 2c minus s minus v) minus Ac(s + v minus 2c) minus (v minus s)(1 + β) (a minus bp)c + 1 minus β21113872 1113873(p minus c)1113960 1113961

2c 2Ac +(1 + β)2(v minus s)1113960 1113961 (B6)

Similarly if Aβc(4p minus 2c minus s minus v)gtAc(s + v minus 2c)

+(v minus s)(1 + β)[(a minus bp)c + (1 minus β2)(p minus c)] we haveΔe edlowast minus eclowast gt 0 In contrast ifAβc(4p minus 2c minus s minus v)ltAc(s + v minus 2c)+

(v minus s)(1 + β)[(a minus bp)c + (1 minus β2)(p minus c)] we haveΔe edlowast minus eclowast lt 0

Upon solving the first-order derivative of Δq and Δe withrespect to β respectively we obtained

zΔ q

zβ

(1 + β)(p minus c)

c (B7)

zΔ e

zβ4(p minus c)(v minus s)2 β3 + 4β2 + 6β + 41113872 1113873β +(a minus bp)c(1 + β)2(v minus s)2 + Ac(v minus s)(1 + β)2 2(p minus c) +v + s) + 2(Ac)2(4p minus c minus s minus v)1113960 1113961

2c (1 + β)2(v minus s) + 2Ac1113960 11139612

(B8)

According to the assumptions pgt vgt cgt sgt 0 β isin (0 1)cgt 0 and a minus bpgt 0 (zΔ qzβ)gt 0 and (zΔ ezβ)gt 0 can beproved Hence the gaps increased as a function with respectto β

Proof of Proposition 4 e proof for Proposition 4 is similarto Proposition 2 Herein we took first-order and second-order partial derivatives of optimal wholesale price wsclowast withrespect to β μ and ϕ in equation (18) respectively esolving results are as follows

zwsclowast

zβ

[(1 + 2μ)ϕ minus μ](p minus c)

(1 + 2μ)(1 + β)2

zwsclowast

zμ minus

(p minus c)β(1 + β)(1 + 2μ)2

zwsclowast

zϕ minus

p minus c

1 + β

(B9)

(zwsclowastzβ) if ϕgt (μ(1 + 2μ)) is more than 0 otherwiseϕgt (μ(1 + 2μ)) is less than 0 us wsc could be increasingwith respect to β nevertheless decreasing with respect to βBesides according to the assumptions pgt vgt cgt sgt 0β isin (0 1) cgt 0 and a minus bpgt 0 it can be evidently foundthat (zwsclowastzμ)gt 0 and (zwsclowastzϕ)gt 0

Data Availability

e data used to support the findings of this study are in-cluded within the article

Conflicts of Interest

e authors declare no conflicts of interest

Acknowledgments

is work was supported by a major project financed by theNational Social Science Fund of China (Approval nos71461009 71662011 and 71761015)

Complexity 15

References

[1] Y Ren D C Croson and R T A Croson ldquoe overconfidentnewsvendorrdquo Journal of the Operational Research Societyvol 68 no 5 pp 496ndash506 2017

[2] T H Cui and P Mallucci ldquoFairness ideals in distributionchannelsrdquo Journal of Marketing Research vol 53 no 6pp 969ndash987 2016

[3] K Chen X Wang M Huang and W-K Ching ldquoSalesforcecontract design joint pricing and production planning withasymmetric overconfidence sales agentrdquo Journal of Industrialamp Management Optimization vol 13 no 2 pp 873ndash8992017

[4] X X Zheng Z Liu K W Li et al ldquoCooperative game ap-proaches to coordinating a three-echelon closed-loop supplychain with fairness concernsrdquo International Journal of Pro-duction Economic vol 212 pp 92ndash110 2018

[5] China Internet Watch False Prosperity behind Double 11Refund Rate Increased to 69 2014 httpswwwchinainternetwatchcom11643false-prosperity-behind-double-11-refund-rate-increased-to-69

[6] C L Lu ldquoPampGrsquos collaborative supply chain transformationrdquoEnterprise Management vol 10 pp 79-80 2016

[7] M Li N C Petruzzi and J Zhang ldquoOverconfident competingnewsvendorsrdquo Management Science vol 63 no 8pp 2637ndash2646 2016

[8] T Nie ldquoDual-fairness supply chain with quantity discountcontractsrdquo European Journal of Operational Researchvol 258 no 2 pp 491ndash500 2017

[9] X Du and B Jiang ldquoSignaling through price and quality toconsumers with fairness concernsrdquo Journal of MarketingResearch vol 53 no 6 pp 988ndash1000 2016

[10] V Kurz A Orland and K Posadzy ldquoFairness versus effi-ciency how procedural fairness concerns affect coordina-tionrdquo Experimental Economics vol 21 no 3 pp 601ndash6262018

[11] S Villa and J A Castantildeeda ldquoTransshipments in supplychains a behavioral investigationrdquo European Journal of Op-erational Research vol 269 no 2 pp 715ndash729 2018

[12] B Gu Y Fu and Y Li ldquoFresh-keeping effort and channelperformance in a fresh product supply chain with loss-averseconsumersrsquo returnsrdquo Mathematical Problems in Engineeringvol 2018 Article ID 4717094 20 pages 2018

[13] D Croson R Croson and Y Ren How to Manage anOverconfident Newsvendor 2008 httpcbeesutdallasedupapersCrosonRenCmsonMS2008

[14] M Li ldquoOverconfident distribution channelsrdquo Production andOperations Management vol 28 no 6 pp 1347ndash1365 2019

[15] W Liu D Wang O Tang and D Zhu ldquoe impacts oflogistics service integratorrsquos overconfidence behaviour onsupply chain decision under demand surgerdquo EuropeanJournal of Industrial Engineering vol 12 no 4 pp 558ndash5972018

[16] L Xu X Shi P Du K Govindan and Z Zhang ldquoOptimi-zation on pricing and overconfidence problem in a duopo-listic supply chainrdquo Computers amp Operations Researchvol 101 pp 162ndash172 2019

[17] Y Li and M Shan ldquoAdvance selling decisions with over-confident consumersrdquo Journal of Industrial and ManagementOptimization vol 2 no 3 pp 891ndash905 2016

[18] T H Cui J S Raju and Z J Zhang ldquoFairness and channelcoordinationrdquo Management Science vol 53 no 8pp 1303ndash1314 2007

[19] X Wu and J A Niederhoff ldquoFairness in selling to thenewsvendorrdquo Production and Operations Managementvol 23 no 11 pp 2002ndash2022 2014

[20] Z Chen and S-I Ivan Su ldquoSocial welfare maximization withthe least subsidy photovoltaic supply chain equilibrium andcoordination with fairness concernrdquo Renewable Energyvol 132 pp 1332ndash1347 2019

[21] W Liu D Wang X Shen X Yan and W Wei ldquoe impactsof distributional and peer-induced fairness concerns on thedecision-making of order allocation in logistics service supplychainrdquo Transportation Research Part E Logistics and Trans-portation Review vol 116 pp 102ndash122 2018

[22] L Zhang H Zhou Y Liu and R Lu ldquoOptimal environmentalquality and price with consumer environmental awarenessand retailerrsquos fairness concerns in supply chainrdquo Journal ofCleaner Production vol 213 pp 1063ndash1079 2019

[23] X LWang S Q Hu and X B Liu ldquoSupply coordination withoption contract considering overconfidence under randomdemand and yieldsrdquo Computer Integrated ManufacturingSystem vol 24 no 11 pp 2898ndash2908 2018

[24] W Jiang and J Liu ldquoInventory financing with overconfidentsupplier based on supply chain contractrdquo MathematicalProblems in Engineering vol 2018 Article ID 505438712 pages 2018

[25] P Ma H Wang and J Shang ldquoContract design for two-stagesupply chain coordination integrating manufacturer-qualityand retailer-marketing effortsrdquo International Journal ofProduction Economics vol 146 no 2 pp 745ndash755 2013

[26] J Liu W F Jiang and Y P Tang ldquoe supply chain coor-dination based on carbon reduction of overconfident man-ufacturerrdquo Journal of Jiangxi Normal University (NaturalScience) vol 42 no 2 pp 203ndash207 2018

[27] R Dubey N Altay and C Blome ldquoSwift trust and com-mitment the missing links for humanitarian supply chaincoordinationrdquo Annals of Operations Research vol 283 no 1-2 pp 159ndash177 2019

[28] C Cui Z Feng and C Tan ldquoCredibilistic loss aversion nashequilibrium for bimatrix games with triangular fuzzy payoffsrdquoComplexity vol 2018 Article ID 7143586 16 pages 2018

[29] E Fehr and KM Schmidt ldquoA theory of fairness competitionand cooperationrdquo e Quarterly Journal of Economicsvol 114 no 3 pp 817ndash868 1999

[30] Z Qin and J Yang ldquoAnalysis of a revenue-sharing contract insupply chain managementrdquo International Journal of LogisticsResearch and Applications vol 11 no 1 pp 17ndash29 2008

[31] S-Z Bai and J Ni ldquoResearch on buy-back contract with saleseffort cost sharingrdquo in Proceedings of the 2009 Chinese Controland Decision Conference IEEE Guilin China pp 1917ndash1921June 2009

[32] E Karagozoglu and K Keskin ldquoTime-varying fairness con-cerns delay and disagreement in bargainingrdquo Journal ofEconomic Behavior and Organization vol 147 pp 115ndash1282018

16 Complexity

wdlowast

(a minus bp)(1 + μ)(v minus s)c +(v minus s)(1 + β)2[(1 + μ)p +(1 + 3μ)c] + Ac[(s + v)(1 + μ) + 2(1 + 3μ)c]

2(1 + 2μ) 2Ac +(v minus s)(1 + β)21113960 1113961 (12)

qdlowast

(a minus bp)(v minus s)c +(1 + β)2(v minus s)(p minus c) + Ac(s + v minus 2c)

2(v minus s)c (13)

edlowast

(1 + β) (a minus bp)(s minus v)c +(p minus c)(v minus s)(1 + β)2 + 2Ac(2p minus c minus s minus v)1113960 1113961

2c 2Ac +(1 + β)2(v minus s)1113960 1113961 (14)

Based on the above equilibrium solutions in thedecentralized supply chain the propositions about the effectof overconfidence and fairness concern (β and μ) on decisionvariables (qdlowast edlowast and wdlowast) are shown as in the followingNote that the proof of all propositions can be found inAppendix B

Proposition 1 When the retailer had both overconfidenceand fairness concern the optimal order quantity qdlowast and saleseffort edlowast were both positively correlated with the overcon-fident level β but had nothing to do with the degree of fairnessconcern μ Furthermore qdlowast is a strictly convex function withrespect to β

Proposition 1 indicated that the overconfident drivesthe retailer to increase order volume and improve salesefforts e reason is that overconfidence had two maineffects On the one hand the retailer overestimated theaverage market demand and exaggerated the effectivenessof sales effort which directly motivated the retailer to makea more considerable sales effort (for example increasingpromotions) On the other hand the retailer overestimatedthat the variance of the ldquorandom impact factor θ1rdquo issmaller than the actual level In other words from his pointof view a smaller market demand fluctuation influencedprediction us the two effects made the retailer ordermore product quantities and pay more sales effort for morerevenue Meanwhile it was easy to find that qdlowast and edlowast

were only related to β as per equations (13) and (14) whichrevealed that the retailersrsquo decisions on ordering and salesefforts were not affected by fairness concern

Proposition 2 When the retailer has both overconfidenceand fairness concern the optimal wholesale price wdlowast ispositively proportional to the overconfident level β but in-versely proportional to the degree of fairness concern μ

Proposition 2 illustrated that the cognitive bias of theretailer had an opposite effect on the manufacturersrsquodecision Firstly the moment the rational manufacturerrealized that the overconfident retailer could increase theorder quantity he raised the wholesale price of theproduct to maximize the profit which was consistent withthe anecdotes that merchants in real world could takeadvantage of othersrsquo overconfidence to maximize theirprofit Besides when the manufacturer understood thefeatures of the retailersrsquo fairness concern he lowered the

wholesale price to alleviate the retailersrsquo unfair aversionwhich implied that the retailer could boost his bargainingpower in the supply chain while he was concerned with thefairness erefore the rational manufacturer was re-quired to set a reasonable wholesale price to balance theinfluence of the two cognitive biases Next we comparedthe equilibrium outcomes in the centralized supply chainmodel with those in the decentralized supply chain modelere is a proposition as follows

Proposition 3 qclowastmay be more than or less thanqdlowast eclowastmaybe more than or less than edlowast Interestingly Δq qdlowast minus qclowastand Δe edlowast minus eclowast increases as the overconfidence level βincreases respectively

According to Proposition 3 the effect of overconfidencethe optimal order quantity and sales effort on the decen-tralized supply chain could deviate from the optimalequilibrium solution in the centralized supply chain whichresulted in a loss of the expected profits thereby increasingthe revenue gap between manufacturer and retailer

4 Buyback Contract with PromotionCost Sharing

It can be seen from Section 33 that due to the cognitivebias of the retailer his decisions deviated from the op-timal decisions in the centralized supply chain whichaffected the profit or utility of supply chain membersereby in order to coordinate the profit or utility be-tween the supply chain members and ensure the maxi-mized performance of the two-echelon supply chainaccording to Bai et al [31] the buyback contract withpromotional cost sharing was proposed in this sectionHere the manufacturer served as the leader who buysback the unsold products from the retailer at the buybackprice and shared the promotional cost We denoted thiscontract factor as (wsc r and 1 minus ϕ) where wsc is thewholesale price r is the buyback price and 1 minus ϕ is thefraction of promotional costs that the rational manu-facturer paid while the retailer undertook ϕ portion ofthe cost (0ltϕlt 1) In order to avoid retailersrsquo over-ordering and encourage the retailer to accept the con-tract we assumed 0lt sle rltwsc Based on thesedescriptions and assumptions above the manufacturersrsquoprofit and retailersrsquo utility are given by the followingexpression

6 Complexity

1113945

sc

m

(w minus c)q + E (s minus r) q minus DO( 11138571113858 1113859+

minus (1 minus ϕ)C(e)

(15)

U 1113945sc

r

⎞⎠ (1 + μ) p[(a minus bp +(1 + β)e)] + E r q minus DO( 1113857+

1113858 11138591113864⎛⎝

minus E v DO minus q( 1113857+

1113858 1113859 minus wq minus ϕC(e)1113865 minus μ1113945sc

m

(16)

wherein equation (15) is the profit when the manufacturerbuys back the unsold products at buyback price r from theretailer and thus obtained the surplus value of theseproducts (1 minus ϕ)C(e) is the promotional costs which themanufacturer undertook r(q minus DO)+ in equation (16) is therevenue when the retailer sells the unsold inventory of

products to themanufacturer at the end of the selling seasonϕC(e) is the promotional costs which he pay

According to the first derivations of equation (16) basedon q and e the following expression is obtained

qsc 2A(1 + μ)(v minus w) minus μ(w minus c)

(1 + μ)(v minus r) + μ(s minus r)+ a minus bp +(1 + k)e

scminus A

esc (1 + β)[μ(p + c minus 2w) + p minus w]

(2μϕ minus μ + ϕ)c

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(17)

In the buyback contract with promotion cost sharing toachieve supply chain coordination the optimal decisions ofthe retailer were the same as those under the centralizedsupply chain model that is qsc qclowast and esc eclowast [25 26]erefore equation (17) needs to be equal to equation (5) asthe contract factors (wsc r) were obtained

wsclowast

[β(1 + μ) + 1 minus ϕ(1 + 2μ)]p +[βμ +(1 + 2μ)ϕ]c

(1 + 2μ)(1 + β) (18)

rlowast

(v minus s)(p minus c)(1 + β)(1 + μ)vβ + 2A[cϕ(p minus c)(1 + 2μ) + βcv(1 + μ)] minus 2Ac p(v minus s)[1 + β +(2 + β)βμ] minus sv(1 + β)(1 + μ)1113864 1113865

(1 + 2μ)(1 + β)[β(p minus c)(v minus s) + 2Ac(c minus v)]

(19)

e purpose of the buyback contract with promotioncost sharing was to design a reasonable profit allocationscheme between the rational manufacturer and the retailerwith cognitive bias It should be noted that if the profit ofsupply chain members became lower the contract wasunacceptable e contract was acceptable if and only if1113937

scm ge 1113937

dlowastm and U(1113937

scr )geU(1113937

dlowastr )

Based on the above equilibrium solutions in the contractthe following propositions were obtained Due to the so-phisticated formula of the buyback price this section dis-cussed the effect of overconfidence and fairness concern onwholesale price

Proposition 4 e optimal wholesale price in the buybackcontract with promotion cost sharing was negatively corre-lated with μ and ϕ Nevertheless if ϕlt (μ(1 + 2μ)) thewholesale price wsclowast is positively correlated with β In con-trast if ϕlt (μ(1 + 2μ)) it is negatively correlated withβ

Proposition 4 implied that the contract was adopted andthe negative effect of fairness concern μ on wholesale pricewsclowast remained consistent with Proposition 2 Besides as wsc

increased ϕ decreased and 1 minus ϕ increased accordinglywhich meant that the manufacturer bore more promotionalcost when the wholesale price rose which is in line with thepractice Meanwhile the values of μ and ϕ determined theimpact of β (positive or negative influence) on the wholesaleprice wsc which indicated that fairness concern μ andpromotional cost portion 1 minus ϕ were critical to manufac-turerrsquos decisions on the wholesale price wsc us when the

two types of behavioral preferences coexisted in the contractthe effect of overconfidence on the optimal wholesale pricedepended on the relationship between fairness degree andfraction of promotion cost Since the utility and profitfunctions in the supply chain included a large number ofunknown parameters under three scenarios (ie Sections32 33 and 4) it was difficult to compare and analyze theeffects of the cognitive biasesus numerical methods wereutilized to conduct further analysis in the next section

5 Numerical Analysis

In this section a numerical study approach was adopted toallow a more in-depth insight into the two behavioralpreferences First a numerical experiment was conducted toshow how equilibrium solutions and profit (utility) in thetwo-echelon supply chain changed along with the levels ofoverconfidence and fairness concern (ie β and μ) and toexamine the robustness of propositions en we also de-termined the effect of overconfidence and fairness concernon the buyback contract with promotional cost sharing

In our experiments some parameters were gatheredfrom the existing papers For the parameters related tooverconfidence Ren et al [1] set β isin [0 1] andc 05 06 08 09 to perform their numerical experi-ments and Li et al [7] set β isin (0 1) and c 5 to conducttheir study For the parameters related to fairness concernZheng et al [4] set μge 0 c 20 and a 100 to implementtheir numerical analysis and Guo et al [9] set μge 0 toperform their studies According to the parameters above

Complexity 7

we assumed β isin [0 1) and μ 0 1 2 to reflect the level ofoverconfidence and fairness concern in this paper Whenβ 0 and μ 0 the retailer was considered to be rationalSimilarly other fundamental parameter values were set inexperiments as follows a 50 θ sim U(minus 30 30) c 4p 20v 7 s 2 b 2 and c 1 All the parameterssatisfied the constraint conditions and assumptions of thethree models in Sections 32 33 and 4e parameter valueswere also in line with the real situation of the market Wecould easily obtain qclowast 32 eclowast 16 and 1113937

c 252 in the

centralized supply chain

51 Effect of Overconfidence and Fairness Concern on theSupply Chain e changes in the decision variables (w qand e) manufacturer profit and retailer expected utilitywhen the level of overconfidence and fairness concern (β andμ) changed are shown in Figures 1ndash5

As shown in Figures 1 and 2 when the retailer was ra-tional (eg β 0 and μ 0) in the decentralized supplychain both optimal order quantity and sales effort were lowerthan those under the centralized supply chain Next when theretailer had overconfidence and fairness concern simulta-neously overconfidence could only affect the retailerrsquos opti-mal order quantity and sales effort of the retailer because theoptimal order quantity and sales effort were a function ofoverconfidence in equations (13) and (14) Moreover theorder quantity was monotonically increasing as concavefunction with respect to β In comparison sales effort wasmonotonically increasing as convex function with respect toit Furthermore there existed a threshold (β 083) when βwas less than 083 and the two decision variables were bothless than the optimal values (qclowast 32 eclowast 16) under thecentralized decision scenario Nevertheless once β was largerthan 083 the two decision variables were both more thanthem e above discussion verified the correctness ofProposition 1 In short overconfidence was the main factorthat affected the retailerrsquos optimal strategies of order quantity

and sales effort in the decentralized supply chain Howeverthe above findings contradicted with the conclusions from astudy conducted by Xu et al [16] which examined the effectof retailerrsquos overconfidence on the supply chain and the resultsuggested that a higher level of overconfident resulted in thelower ordering quantity

In Figure 3 we observed that the higher overconfidentlevel resulted in the high wholesale price in contrast ahigh degree of fairness concern resulted in low pricewhich demonstrated that Proposition 2 was robust isfigure implied that when the information for both partieswas symmetric the rational manufacturer knew that theretailer was overconfident that is the retailer was opti-mistic about the market demand and increased the orderquantity As a result the manufacturer also raised thewholesale price to maximize its revenue Also if theoverconfident retailer paid attention to the unfairness ofprofit allocation in the decision-making process the re-tailerrsquos bargaining power was enhanced [32] Hence forbetter performance in the supply chain the leadingmanufacturer was required to lower wholesale prices toease the feeling of unfairness by the retailer which isconsistent with the actual economic phenomenon Fur-thermore these findings are consistent with the resultconcluded by Cui et al [2] which indicated that a highdegree of the retailerrsquos fairness concern could lower thewholesale price Interestingly as the levels of the cognitivebias increased its effect on wholesale price graduallydecreased which highlighted its effect on decreasedwholesale prices

Figure 4 displays the impact of the cognitive bias on theretailerrsquos utility e graph showed that the retailer expectedutility increased gradually with the overconfident level andthe degree of fairness concern which suggested that twocognitive biases were beneficial to the retailer e reasonsfor such a phenomenon were that both cognitive biasescould increase the optimal order quantity and sales effortthus enhancing the utility of the retailer

Order quantity in centralized supply chain

01 02 03 04 05 06 07 08 09 10β

15

20

25

30

35

40

Ord

er q

uant

ity

μ = 012

Figure 1 e effect of the cognitive biases on optimal order quantity

8 Complexity

Figure 5 illustrates the impact of overconfidence andfairness concern on the manufacturerrsquos profit at is theoptimal manufacturerrsquos profit is an increasing function of βbut a decreasing function of μ According to Figure 3 thegraph of themanufacturerrsquos profit was similar to the graph ofthe wholesale price which indicated that the manufacturerrsquosprofit was firmly related to wholesale price Figure 5 alsoillustrates the effect of the cognitive bias β on the manu-facturerrsquos decreasing profit

52 Effect of Overconfidence and Fairness Concern on SupplyChain Coordination As can be seen from Section 51 thereexist deviations between the optimal decisions under the

benchmark and the decisions in the presence of the retailerrsquoscognitive biases erefore the buyback contract withpromotion cost sharing was necessary to coordinate therevenue of supply chain members and optimize the per-formance of the supply chain

Here we discussed the validity of the contract When theoptimal order quantity and sales effort of the retailer underthe contract were the same as the optimal decision variablesin the centralized supply chain ie qsc qclowast 32 andesc eclowast 16 the buyback contract with promotion costsharing achieved supply chain coordination First the effectof overconfidence and fairness concern (β and μ) on contractfactors wsrlowast rlowast and ϕ is shown in Figures 6 and 7 andTable 1 For clearly analyzing the effect of β and μ on wsrlowast

Degree of sales effort in centralized supply chain

01 02 03 04 05 06 07 08 09 10β

μ = 012

4

6

8

10

12

14

16

18

Deg

ree o

f sal

es ef

fort

Figure 2 e effect of the cognitive biases on optimal sales effort

01 02 03 04 05 06 07 08 09 10β

-μ = 0--μ = 1-μ = 2

4

5

6

7

8

9

10

11

12

13

14

Who

lesa

le p

rice

Figure 3 e effect of the cognitive biases on the optimal wholesale price

Complexity 9

and rlowast we let ϕ 1 in equations (18) and (19) which meantthat the manufacturer proposed the buyback contract onlyand the retailer bore all sales effort costs

According to Figure 6 it was revealed that the optimalwholesales price is a strictly increasing function of theoverconfidence level β but a strictly decreasing function ofthe degree of fairness concern μ which confirmed the resultof Proposition 4 Similarly Figure 7 highlights that theoptimal buyback price sharply increased with β but de-creased with μ Moreover in the buyback contract wsrlowast wasalways greater than or equal to 4 and rlowast was always greaterthan or equal to 2 in different cognitive bias levels Hence

the optimal buyback price rlowast satisfied the assumptions 2le r Note that the above characteristics influencedcognitive biases hence contract factors wsrlowast and rlowast werealso correct when ϕ is equal to other values (such asϕ 03 05 08) under conditions in which the buybackcontract had promotion cost sharing

However data from Figure 7 can be compared with thedata in Figure 6 which showed rsrlowast is more significant thanwsrlowast when β was within a specific range (for example04le βlt 1) which did not match the assumption rltwsrerefore the buyback contract did not achieve the coor-dination of the behavioral supply chain

01 02 03 04 05 06 07 08 09 10β

-μ = 0--μ = 1-μ = 2

0

20

40

60

80

100

120

Reta

iler e

xpec

ted

utili

ty

Figure 4 e effect of the cognitive biases on retailer utility

01 02 03 04 05 06 07 08 09 10β

-μ = 0--μ = 1-lowastμ = 2

0

20

40

60

80

100

120

140

160

180

200

Man

ufac

ture

r pro

fit

Figure 5 e effect of the cognitive biases on manufacturer profit

10 Complexity

As described in Section 4 if the supply chain membersaccepted the buyback contract with promotion cost sharingthe assumptions 1113937

scm gemax 1113937

dm and U(1113937

scr )gemaxU(1113937

dr )

had to be satisfied Also according to the assumptions thevalue ϕ must satisfy 1113937

scm gemax 1113937

dm and

U(1113937scr )gemaxU(1113937

dr ) us we kept the values of param-

eters to be the same (ie a 50 θ sim U(minus 30 30) c 4p 20 v 7 s 2 b 2 and c 1) to capture a likelydomain of ϕ

As can be seen from Table 2 the overconfident level βincreased the upper and lower limits of ϕ which had a clearupward trend In contrast the upper and lower limits of ϕ

showed a definite declining trend as the degree of fairnessconcern μ increased ese trends illustrated that in order tocoordinate the revenue of both parties and optimize theperformance of the supply chain the manufacturer needs tobear less promotional costs as β increased but bear morepromotional costs as μ value increased Besides when β and μincreased the interval of sharing cost portion ϕ shrunk whichindicated that the buyback contract with promotional costsharing had certain flexibility to coordinate the revenue be-tween supply chain members However the flexibility couldbe weakened with the rise of the cognitive bias level Forsupply chain members the possibility of negotiation or

01 02 03 04 05 06 07 08 09 10β

-μ = 0--μ = 1-μ = 2

4

5

6

7

8

9

10

11

12

Who

lesa

le p

rice

Figure 6 e effect of the cognitive biases on wholesale price wsrlowast

01 02 03 04 05 06 07 08 09 10β

-μ = 0--μ = 1-μ = 2

2

4

6

8

10

12

14

16

18

20

22

Buyb

ack

pric

e

Figure 7 e effect of the cognitive biases on buyback price rlowast

Complexity 11

cooperation between manufacturers and retailers also de-creased with an increase in retailersrsquo cognitive bias levelWhen the overconfident level was more than 081 the re-tailerrsquos utility from decentralized decisions was greater thanthe utility in the contract environment wherein the twoparties could not reach a cooperation agreement us thesupply chain could not achieve coordination as different levelsof cognitive bias under the contract had the same influence ondecision variables profit and utility Table 2 shows a clearpicture wherein μ is equal to 1 and β falls within the range of(0 08) which were used to analyze the effect of the cognitivebiases on the coordination of the supply chain

e results of the numerical simulation in Table 3 in-dicate that the contract factors ie ϕ r wholesale price theprofit and utility of members existed as flexible intervalswhen β isin (0 08) and μ 1 Besides the order quantity qsc

and the degree of sales effort esc under the contract are equalto qclowast and eclowast in a centralized supply chain respectivelyMeanwhile the buyback price r was less than the wholesaleprice wsc and higher than the unit salvage value s whichsatisfied the constrains sle rltwsclowast us the buybackcontract with promotion cost could achieve the coordinationof the two-echelon supply chain Also the buyback price r

was positively correlated with the portion ϕ of promotioncost sharing Notably the wholesale price wsc dropped as ϕincreased when β was in the range of (0 02) in Table 3However the situation reversed when it was in the range of(02 08) which was inconsistent since the manufacturerlowered the wholesale price when the retailer paid morepromotional costs in the contract e reason for such anobservation was that the wholesale price was not related to ϕbut positively related to the overconfident level β when thedegree of fairness concern was moderate Furthermore theretailer accepted the contract designed by manufacturerssince the term U(1113937

scr )gemaxU(1113937

dr ) was guaranteed in

Table 3

6 Conclusions

61MainConclusions In this paper we analyzed the effect ofcognitive bias on equilibrium solutions including thewholesale price sales price sales effort and expected profitsunder demand uncertainty by introducing both overconfi-dence and fairness concern into the two-echelon supply chaine most important conclusion from the study was that forretailers with the two cognitive biases overconfidence waspositive and the main bias that affected supply chain decisionsand the retailersrsquo utility benefits were not only due to over-confidence but also fairness concern For manufacturersoverconfidence could increase the wholesale price and profitHowever the fairness concern had a negative effect on themFurthermore the decisions of the retailer in the decentralizedsupply chain deviated from the optimal decisions in thecentralized supply chainmodel Especially when the thresholdof overconfidence was more than a fixed value the decisionvariables under decentralized decision-making were actuallyhigher than the optimal variables under centralized decision-making Also the buyback contract with promotional costsharing achieved behavioral supply chain coordination Incontrast the promotional cost-sharing portion and buybackprice both benefited from overconfidence but fairness concernhad a detrimental effect on them However as the retailersrsquooverconfidence level increased and fairness concern was equalto 1 the flexibility of contract coordination was weakenedWhen the overconfidence level wasmore than 08 the contractbecame invalid and members were not able to cooperate

62 Management Insights is paper provided some in-teresting managerial insights for decision makers in thedecentralized supply chain wherein retailers had two pref-erences First overconfidence was beneficial to them interms of decision variables and yet the fairness concern

Table 2 e effect of the cognitive biases on the feasible domain of promotion cost-sharing portion ϕ

μβ

0 02 04 06 08 0810 [02884 04976] [03683 05587] [04528 06199] [06343 07754] [07835 08669] mdash1 [02545 04741] [03310 05331] [04167 05908] [06207 07516] [07800 08560] mdash2 [02352 04600] [03101 05156] [04034 05719] [06094 07321] [07736 08478] mdash

Table 3 e effect of the cognitive biases on the coordination of the supply chain

μ 1β

0 02 04 06 08qsc 32 32 32 32 32esc 16 16 16 16 16ϕ [02545 04741] [03310 05331] [04167 05908] [06207 07516] [07800 08560]r [2100 35363] [28268 39188] [30555 42637] [36209 44215] [40485 49164]wsc [49205 53687] [49016 53522] [54092 69568] [62513 72626] [80407 87453]1113937

dm 164103 269591 461996 772619 1112545

U(1113937scm) [168722 217850] [269600 336882] [463175 500938] [772619 814300] [1112545 1120376]

1113937dr 20333 352396 495161 632422 765317

U(1113937scr ) [20333 243043] [352396 400595] [495161 525161] [632422 657581] [765317 784632]

12 Complexity

only damaged the wholesale price Second the two behav-ioral preferences were both conducive to utilities for theretailer but not necessarily beneficial for the manufacturer toreceive profit and overconfidence helped to increasemanufacturersrsquo profit which led to nonoptimal decisionsthat dragged down the performance of the two-echelonsupply chain Finally for the two behavioral preferences ofthe retailer the manufacturer assessed the degree of over-confidence and fairness concern (especially the overconfi-dence) via historical data and the current order quantity orother indicators In contrast factual data indicated that twobehavioral preferences were in the right range (such asβ isin (0 08) and μ isin (0 2)) e manufacturers introducedthe buyback contract with promotional cost sharing into thepractice of pricing and ordered decentralized supply chainsFurthermore they designed a reasonable promotional cost-sharing portion and buyback price thereby achieving anoptimized decision and utility In practice the manufac-turers could analyze whether the sellers had two behavioralpreferences based on the historical data by cooperating withdownstream enterprises ie if two behavioral preferencesexisted they referred to the papersrsquo conclusions to redesign areasonable contract mechanism

63 Future Research Since cognitive biases could existamong all members of supply chains it is necessary toexplore the manufacturersrsquo cognitive biases and analyze thesupply chain coordination problem in the presence of thetwo behavior preferences Besides the overconfidence andfairness concern considered in our centralized decision are afuture research direction

Appendix

A Decentralized Supply Chain Model

In a decentralized supply chain consisting of a rationalmanufacturer and a retailer with cognitive biases maximumprofit or utility is the only goal e backward inductivemethod can be applied to solve a Stackelberg game whereinthe manufacturer is the leader and the retailer is a followere proof of the equilibrium solutions in equations (12)ndash(14)is as follows

e profit of the manufacturer and expected utility ofretailers are as follows respectively

wmax

1113945

d

m

(w minus c)q (A1)

maxqe

U 1113945d

r

⎞⎠ (1 + μ) p[a minus bp +(1 + β)e] + s 1113946θ2

minus Aq minus a minus bp +(1 + β)e + θ1( 1113857f θ1( 1113857( 1113857dθ11113896⎛⎝

minus v 1113946A

θ2a minus bp +(1 + β)e + θ1 minus q( 1113857f θ1( 1113857( 1113857dθ1 minus wq minus

12

ce21113897 minus μ(w minus c)q

(A2)

Solving the first-order partial derivatives of the retailerrsquosexpected utility concerning qd and ed which equal to 0 weobtained the following equations

zU( 1113937dr 1113873

zq (1 + μ) v minus w + F θ2( 1113857(s minus v)1113858 1113859 minus μ(w minus c) 0

(A3)

zU( 1113937dr 1113873

ze (1 + β) p minus v +(v minus s)F θ2( 11138571113858 1113859 minus ce 0

(A4)

Hessian matrix

H(q e) (s minus v)f θ2( 1113857 (v minus s)(1 + β)f θ2( 1113857

(v minus s)f θ2( 1113857 (s minus v)(1 + β)f θ2( 1113857 minus c1113888 1113889

(A5)

According to the assumptions vgt sgt 0 f(θ2)gt 0 andβ isin [0 1) us the Hessian matrix of U(1113937

dr ) is a negative

definite for q and e which meant that the optimal orderquantity and sales effort of retailers existed in the decen-tralized supply chain erefore we obtained the followingformula of the retailersrsquo order quantity and sales effort byjointly solving equations (A3) and (A4)

qd

(a minus bp)(v minus s)(1 + μ)c +(1 + β)2(v minus s)[(1 + μ)(p minus w) + μc] + Ac[(s + v)(1 + μ) + 2μc minus w]

(v minus s)(1 + μ)c (A6)

ed

(1 + β)[(1 + μ)p + μc minus (1 + 2μ)]w

(1 + μ)c (A7)

Complexity 13

Substituting equations (A6) into (A1) we attained

wmax

1113945

d

m

(w minus c) (a minus bp)(v minus s)(1 + μ)c +(1 + β)2(v minus s)[(1 + μ)p + μc] + Ac[(s + v)(1 + μ) + 2μc] minus (1 + 2μ)w 2Ac +(1 + β)2(v minus s)1113960 11139611113966 1113967

(v minus s)(1 + μ)c

(A8)

Taking into account the first-order condition of equation(A7) with respect to w rounded off to 0 we obtained`

z1113937dm

ze

(a minus bp)(v minus s)(1 + μ)c +(1 + β)2(v minus s)[(1 + μ)p + μc] + Ac[(s + v)(1 + μ) + 2μc] minus (2w minus c)(1 + 2μ) 2Ac +(1 + β)2(v minus s)1113960 1113961

(v minus s)(1 + μ)c

(A9)

Solving equation (A8) we found that the optimalwholesale price of the manufacturer is as follows

wdlowast

(a minus bp)(1 + μ)(v minus s)c +(v minus s)(1 + β)2[(1 + μ)p +(1 + 3μ)c] + Ac[(s + v)(1 + μ) + 2(1 + 3μ)c]

2(1 + 2μ) 2Ac +(v minus s)(1 + β)21113960 1113961 (A10)

By substituting equation (A9) into equations (A6) and(A7) we attained the optimal order quantity and sales effortwhich are displayed in equations (13) and (14)

B Propositions

Proof of Proposition 1 We solved the first derivatives ofoptimal order quantity qdlowast and sales effort edlowast with respect

to β in equations (13) and (14) respectively Meanwhiletaking the second-order derivative of qdlowast with respect to βwe obtained

zqdlowast

zβ

(1 + β)(p minus c)

c

z2qdlowast

zβ2

p minus c

c (B1)

zedlowast

zβ4(p minus c)(v minus s)2β β3 + 4β2 + 6β + 41113872 1113873 +(a minus bp)c(1 + β)2(v minus s)2 + Ac(v minus s)(1 + β)2 (2(p minus c) + v + s) + 2(Ac)2(4p minus c minus s minus v)1113960 1113961

2c (1 + β)2(v minus s) + 2Ac1113960 11139612

(B2)

Since the second derivative of edlowast with respect to β wasvery complicated there was no need to solve it According tothe assumptions pgt vgt cgt sgt 0 β isin (0 1) a minus bpgt 0 andcgt 0 so (zqdlowastzβ)gt 0 (z2qdlowastzβ2)gt 0 and (zedlowastzβ)gt 0

Proof of Proposition 2 Based on the process above we tookthe first-order and second-order partial derivatives of op-timal wholesale price wdlowast with respect to β and μ in equation(12) respectively

zwdlowast

zβ

c(v minus s)(1 + μ)(1 + β)[(a minus bp)(v minus s) + A(2p minus s minus v)]

(1 + 2μ) (1 + β)2(v minus s) + 2Ac1113960 11139612 (B3)

zwdlowast

zμ minus

(a minus bp)(v minus s)c +(p minus c)(v minus s)(1 + β)2 + Ac(s + v minus 2c)1113960 1113961

2(1 + 2μ)2 (v minus s)(1 + β)2 + 2Ac1113960 1113961 (B4)

14 Complexity

According to the optimal result of order quantity inequation (13) and the assumption qgt 0 the condition s +

v minus 2cgt 0 is found According to the assumptions we had(zwdlowastzβ)gt 0 and (zwdlowastzμ)lt 0

Proof of Proposition 3 e gap between qdlowast and qclowastdenoted by Δq is as follows

Δq qdlowast

minus qclowast

(p minus c)(v minus s) (1 + β)2 minus 21113960 1113961 minus Ac(s + v minus 2c) minus (a minus bp)(v minus s)c

2c(v minus s) (B5)

If (p minus c)(v minus s)[(1 + β)2 minus 2] is more thanAc(s + v minus 2c) + (a minus bp)(v minus s)c we have Δqgt 0 ieqdlowast gt qclowast In contrast if (p minus c)(v minus s)[(1 + β)2 minus 2] is less

than Ac(s + v minus 2c) + (a minus bp)(v minus s)c Δqlt 0 ieqdlowast lt qclowast

e gap between edlowast and eclowast denoted by Δe is asfollows

Δe edlowast

minus eclowast

Aβc(4p minus 2c minus s minus v) minus Ac(s + v minus 2c) minus (v minus s)(1 + β) (a minus bp)c + 1 minus β21113872 1113873(p minus c)1113960 1113961

2c 2Ac +(1 + β)2(v minus s)1113960 1113961 (B6)

Similarly if Aβc(4p minus 2c minus s minus v)gtAc(s + v minus 2c)

+(v minus s)(1 + β)[(a minus bp)c + (1 minus β2)(p minus c)] we haveΔe edlowast minus eclowast gt 0 In contrast ifAβc(4p minus 2c minus s minus v)ltAc(s + v minus 2c)+

(v minus s)(1 + β)[(a minus bp)c + (1 minus β2)(p minus c)] we haveΔe edlowast minus eclowast lt 0

Upon solving the first-order derivative of Δq and Δe withrespect to β respectively we obtained

zΔ q

zβ

(1 + β)(p minus c)

c (B7)

zΔ e

zβ4(p minus c)(v minus s)2 β3 + 4β2 + 6β + 41113872 1113873β +(a minus bp)c(1 + β)2(v minus s)2 + Ac(v minus s)(1 + β)2 2(p minus c) +v + s) + 2(Ac)2(4p minus c minus s minus v)1113960 1113961

2c (1 + β)2(v minus s) + 2Ac1113960 11139612

(B8)

According to the assumptions pgt vgt cgt sgt 0 β isin (0 1)cgt 0 and a minus bpgt 0 (zΔ qzβ)gt 0 and (zΔ ezβ)gt 0 can beproved Hence the gaps increased as a function with respectto β

Proof of Proposition 4 e proof for Proposition 4 is similarto Proposition 2 Herein we took first-order and second-order partial derivatives of optimal wholesale price wsclowast withrespect to β μ and ϕ in equation (18) respectively esolving results are as follows

zwsclowast

zβ

[(1 + 2μ)ϕ minus μ](p minus c)

(1 + 2μ)(1 + β)2

zwsclowast

zμ minus

(p minus c)β(1 + β)(1 + 2μ)2

zwsclowast

zϕ minus

p minus c

1 + β

(B9)

(zwsclowastzβ) if ϕgt (μ(1 + 2μ)) is more than 0 otherwiseϕgt (μ(1 + 2μ)) is less than 0 us wsc could be increasingwith respect to β nevertheless decreasing with respect to βBesides according to the assumptions pgt vgt cgt sgt 0β isin (0 1) cgt 0 and a minus bpgt 0 it can be evidently foundthat (zwsclowastzμ)gt 0 and (zwsclowastzϕ)gt 0

Data Availability

e data used to support the findings of this study are in-cluded within the article

Conflicts of Interest

e authors declare no conflicts of interest

Acknowledgments

is work was supported by a major project financed by theNational Social Science Fund of China (Approval nos71461009 71662011 and 71761015)

Complexity 15

References

[1] Y Ren D C Croson and R T A Croson ldquoe overconfidentnewsvendorrdquo Journal of the Operational Research Societyvol 68 no 5 pp 496ndash506 2017

[2] T H Cui and P Mallucci ldquoFairness ideals in distributionchannelsrdquo Journal of Marketing Research vol 53 no 6pp 969ndash987 2016

[3] K Chen X Wang M Huang and W-K Ching ldquoSalesforcecontract design joint pricing and production planning withasymmetric overconfidence sales agentrdquo Journal of Industrialamp Management Optimization vol 13 no 2 pp 873ndash8992017

[4] X X Zheng Z Liu K W Li et al ldquoCooperative game ap-proaches to coordinating a three-echelon closed-loop supplychain with fairness concernsrdquo International Journal of Pro-duction Economic vol 212 pp 92ndash110 2018

[5] China Internet Watch False Prosperity behind Double 11Refund Rate Increased to 69 2014 httpswwwchinainternetwatchcom11643false-prosperity-behind-double-11-refund-rate-increased-to-69

[6] C L Lu ldquoPampGrsquos collaborative supply chain transformationrdquoEnterprise Management vol 10 pp 79-80 2016

[7] M Li N C Petruzzi and J Zhang ldquoOverconfident competingnewsvendorsrdquo Management Science vol 63 no 8pp 2637ndash2646 2016

[8] T Nie ldquoDual-fairness supply chain with quantity discountcontractsrdquo European Journal of Operational Researchvol 258 no 2 pp 491ndash500 2017

[9] X Du and B Jiang ldquoSignaling through price and quality toconsumers with fairness concernsrdquo Journal of MarketingResearch vol 53 no 6 pp 988ndash1000 2016

[10] V Kurz A Orland and K Posadzy ldquoFairness versus effi-ciency how procedural fairness concerns affect coordina-tionrdquo Experimental Economics vol 21 no 3 pp 601ndash6262018

[11] S Villa and J A Castantildeeda ldquoTransshipments in supplychains a behavioral investigationrdquo European Journal of Op-erational Research vol 269 no 2 pp 715ndash729 2018

[12] B Gu Y Fu and Y Li ldquoFresh-keeping effort and channelperformance in a fresh product supply chain with loss-averseconsumersrsquo returnsrdquo Mathematical Problems in Engineeringvol 2018 Article ID 4717094 20 pages 2018

[13] D Croson R Croson and Y Ren How to Manage anOverconfident Newsvendor 2008 httpcbeesutdallasedupapersCrosonRenCmsonMS2008

[14] M Li ldquoOverconfident distribution channelsrdquo Production andOperations Management vol 28 no 6 pp 1347ndash1365 2019

[15] W Liu D Wang O Tang and D Zhu ldquoe impacts oflogistics service integratorrsquos overconfidence behaviour onsupply chain decision under demand surgerdquo EuropeanJournal of Industrial Engineering vol 12 no 4 pp 558ndash5972018

[16] L Xu X Shi P Du K Govindan and Z Zhang ldquoOptimi-zation on pricing and overconfidence problem in a duopo-listic supply chainrdquo Computers amp Operations Researchvol 101 pp 162ndash172 2019

[17] Y Li and M Shan ldquoAdvance selling decisions with over-confident consumersrdquo Journal of Industrial and ManagementOptimization vol 2 no 3 pp 891ndash905 2016

[18] T H Cui J S Raju and Z J Zhang ldquoFairness and channelcoordinationrdquo Management Science vol 53 no 8pp 1303ndash1314 2007

[19] X Wu and J A Niederhoff ldquoFairness in selling to thenewsvendorrdquo Production and Operations Managementvol 23 no 11 pp 2002ndash2022 2014

[20] Z Chen and S-I Ivan Su ldquoSocial welfare maximization withthe least subsidy photovoltaic supply chain equilibrium andcoordination with fairness concernrdquo Renewable Energyvol 132 pp 1332ndash1347 2019

[21] W Liu D Wang X Shen X Yan and W Wei ldquoe impactsof distributional and peer-induced fairness concerns on thedecision-making of order allocation in logistics service supplychainrdquo Transportation Research Part E Logistics and Trans-portation Review vol 116 pp 102ndash122 2018

[22] L Zhang H Zhou Y Liu and R Lu ldquoOptimal environmentalquality and price with consumer environmental awarenessand retailerrsquos fairness concerns in supply chainrdquo Journal ofCleaner Production vol 213 pp 1063ndash1079 2019

[23] X LWang S Q Hu and X B Liu ldquoSupply coordination withoption contract considering overconfidence under randomdemand and yieldsrdquo Computer Integrated ManufacturingSystem vol 24 no 11 pp 2898ndash2908 2018

[24] W Jiang and J Liu ldquoInventory financing with overconfidentsupplier based on supply chain contractrdquo MathematicalProblems in Engineering vol 2018 Article ID 505438712 pages 2018

[25] P Ma H Wang and J Shang ldquoContract design for two-stagesupply chain coordination integrating manufacturer-qualityand retailer-marketing effortsrdquo International Journal ofProduction Economics vol 146 no 2 pp 745ndash755 2013

[26] J Liu W F Jiang and Y P Tang ldquoe supply chain coor-dination based on carbon reduction of overconfident man-ufacturerrdquo Journal of Jiangxi Normal University (NaturalScience) vol 42 no 2 pp 203ndash207 2018

[27] R Dubey N Altay and C Blome ldquoSwift trust and com-mitment the missing links for humanitarian supply chaincoordinationrdquo Annals of Operations Research vol 283 no 1-2 pp 159ndash177 2019

[28] C Cui Z Feng and C Tan ldquoCredibilistic loss aversion nashequilibrium for bimatrix games with triangular fuzzy payoffsrdquoComplexity vol 2018 Article ID 7143586 16 pages 2018

[29] E Fehr and KM Schmidt ldquoA theory of fairness competitionand cooperationrdquo e Quarterly Journal of Economicsvol 114 no 3 pp 817ndash868 1999

[30] Z Qin and J Yang ldquoAnalysis of a revenue-sharing contract insupply chain managementrdquo International Journal of LogisticsResearch and Applications vol 11 no 1 pp 17ndash29 2008

[31] S-Z Bai and J Ni ldquoResearch on buy-back contract with saleseffort cost sharingrdquo in Proceedings of the 2009 Chinese Controland Decision Conference IEEE Guilin China pp 1917ndash1921June 2009

[32] E Karagozoglu and K Keskin ldquoTime-varying fairness con-cerns delay and disagreement in bargainingrdquo Journal ofEconomic Behavior and Organization vol 147 pp 115ndash1282018

16 Complexity

1113945

sc

m

(w minus c)q + E (s minus r) q minus DO( 11138571113858 1113859+

minus (1 minus ϕ)C(e)

(15)

U 1113945sc

r

⎞⎠ (1 + μ) p[(a minus bp +(1 + β)e)] + E r q minus DO( 1113857+

1113858 11138591113864⎛⎝

minus E v DO minus q( 1113857+

1113858 1113859 minus wq minus ϕC(e)1113865 minus μ1113945sc

m

(16)

wherein equation (15) is the profit when the manufacturerbuys back the unsold products at buyback price r from theretailer and thus obtained the surplus value of theseproducts (1 minus ϕ)C(e) is the promotional costs which themanufacturer undertook r(q minus DO)+ in equation (16) is therevenue when the retailer sells the unsold inventory of

products to themanufacturer at the end of the selling seasonϕC(e) is the promotional costs which he pay

According to the first derivations of equation (16) basedon q and e the following expression is obtained

qsc 2A(1 + μ)(v minus w) minus μ(w minus c)

(1 + μ)(v minus r) + μ(s minus r)+ a minus bp +(1 + k)e

scminus A

esc (1 + β)[μ(p + c minus 2w) + p minus w]

(2μϕ minus μ + ϕ)c

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(17)

In the buyback contract with promotion cost sharing toachieve supply chain coordination the optimal decisions ofthe retailer were the same as those under the centralizedsupply chain model that is qsc qclowast and esc eclowast [25 26]erefore equation (17) needs to be equal to equation (5) asthe contract factors (wsc r) were obtained

wsclowast

[β(1 + μ) + 1 minus ϕ(1 + 2μ)]p +[βμ +(1 + 2μ)ϕ]c

(1 + 2μ)(1 + β) (18)

rlowast

(v minus s)(p minus c)(1 + β)(1 + μ)vβ + 2A[cϕ(p minus c)(1 + 2μ) + βcv(1 + μ)] minus 2Ac p(v minus s)[1 + β +(2 + β)βμ] minus sv(1 + β)(1 + μ)1113864 1113865

(1 + 2μ)(1 + β)[β(p minus c)(v minus s) + 2Ac(c minus v)]

(19)

e purpose of the buyback contract with promotioncost sharing was to design a reasonable profit allocationscheme between the rational manufacturer and the retailerwith cognitive bias It should be noted that if the profit ofsupply chain members became lower the contract wasunacceptable e contract was acceptable if and only if1113937

scm ge 1113937

dlowastm and U(1113937

scr )geU(1113937

dlowastr )

Based on the above equilibrium solutions in the contractthe following propositions were obtained Due to the so-phisticated formula of the buyback price this section dis-cussed the effect of overconfidence and fairness concern onwholesale price

Proposition 4 e optimal wholesale price in the buybackcontract with promotion cost sharing was negatively corre-lated with μ and ϕ Nevertheless if ϕlt (μ(1 + 2μ)) thewholesale price wsclowast is positively correlated with β In con-trast if ϕlt (μ(1 + 2μ)) it is negatively correlated withβ

Proposition 4 implied that the contract was adopted andthe negative effect of fairness concern μ on wholesale pricewsclowast remained consistent with Proposition 2 Besides as wsc

increased ϕ decreased and 1 minus ϕ increased accordinglywhich meant that the manufacturer bore more promotionalcost when the wholesale price rose which is in line with thepractice Meanwhile the values of μ and ϕ determined theimpact of β (positive or negative influence) on the wholesaleprice wsc which indicated that fairness concern μ andpromotional cost portion 1 minus ϕ were critical to manufac-turerrsquos decisions on the wholesale price wsc us when the

two types of behavioral preferences coexisted in the contractthe effect of overconfidence on the optimal wholesale pricedepended on the relationship between fairness degree andfraction of promotion cost Since the utility and profitfunctions in the supply chain included a large number ofunknown parameters under three scenarios (ie Sections32 33 and 4) it was difficult to compare and analyze theeffects of the cognitive biasesus numerical methods wereutilized to conduct further analysis in the next section

5 Numerical Analysis

In this section a numerical study approach was adopted toallow a more in-depth insight into the two behavioralpreferences First a numerical experiment was conducted toshow how equilibrium solutions and profit (utility) in thetwo-echelon supply chain changed along with the levels ofoverconfidence and fairness concern (ie β and μ) and toexamine the robustness of propositions en we also de-termined the effect of overconfidence and fairness concernon the buyback contract with promotional cost sharing

In our experiments some parameters were gatheredfrom the existing papers For the parameters related tooverconfidence Ren et al [1] set β isin [0 1] andc 05 06 08 09 to perform their numerical experi-ments and Li et al [7] set β isin (0 1) and c 5 to conducttheir study For the parameters related to fairness concernZheng et al [4] set μge 0 c 20 and a 100 to implementtheir numerical analysis and Guo et al [9] set μge 0 toperform their studies According to the parameters above

Complexity 7

we assumed β isin [0 1) and μ 0 1 2 to reflect the level ofoverconfidence and fairness concern in this paper Whenβ 0 and μ 0 the retailer was considered to be rationalSimilarly other fundamental parameter values were set inexperiments as follows a 50 θ sim U(minus 30 30) c 4p 20v 7 s 2 b 2 and c 1 All the parameterssatisfied the constraint conditions and assumptions of thethree models in Sections 32 33 and 4e parameter valueswere also in line with the real situation of the market Wecould easily obtain qclowast 32 eclowast 16 and 1113937

c 252 in the

centralized supply chain

51 Effect of Overconfidence and Fairness Concern on theSupply Chain e changes in the decision variables (w qand e) manufacturer profit and retailer expected utilitywhen the level of overconfidence and fairness concern (β andμ) changed are shown in Figures 1ndash5

As shown in Figures 1 and 2 when the retailer was ra-tional (eg β 0 and μ 0) in the decentralized supplychain both optimal order quantity and sales effort were lowerthan those under the centralized supply chain Next when theretailer had overconfidence and fairness concern simulta-neously overconfidence could only affect the retailerrsquos opti-mal order quantity and sales effort of the retailer because theoptimal order quantity and sales effort were a function ofoverconfidence in equations (13) and (14) Moreover theorder quantity was monotonically increasing as concavefunction with respect to β In comparison sales effort wasmonotonically increasing as convex function with respect toit Furthermore there existed a threshold (β 083) when βwas less than 083 and the two decision variables were bothless than the optimal values (qclowast 32 eclowast 16) under thecentralized decision scenario Nevertheless once β was largerthan 083 the two decision variables were both more thanthem e above discussion verified the correctness ofProposition 1 In short overconfidence was the main factorthat affected the retailerrsquos optimal strategies of order quantity

and sales effort in the decentralized supply chain Howeverthe above findings contradicted with the conclusions from astudy conducted by Xu et al [16] which examined the effectof retailerrsquos overconfidence on the supply chain and the resultsuggested that a higher level of overconfident resulted in thelower ordering quantity

In Figure 3 we observed that the higher overconfidentlevel resulted in the high wholesale price in contrast ahigh degree of fairness concern resulted in low pricewhich demonstrated that Proposition 2 was robust isfigure implied that when the information for both partieswas symmetric the rational manufacturer knew that theretailer was overconfident that is the retailer was opti-mistic about the market demand and increased the orderquantity As a result the manufacturer also raised thewholesale price to maximize its revenue Also if theoverconfident retailer paid attention to the unfairness ofprofit allocation in the decision-making process the re-tailerrsquos bargaining power was enhanced [32] Hence forbetter performance in the supply chain the leadingmanufacturer was required to lower wholesale prices toease the feeling of unfairness by the retailer which isconsistent with the actual economic phenomenon Fur-thermore these findings are consistent with the resultconcluded by Cui et al [2] which indicated that a highdegree of the retailerrsquos fairness concern could lower thewholesale price Interestingly as the levels of the cognitivebias increased its effect on wholesale price graduallydecreased which highlighted its effect on decreasedwholesale prices

Figure 4 displays the impact of the cognitive bias on theretailerrsquos utility e graph showed that the retailer expectedutility increased gradually with the overconfident level andthe degree of fairness concern which suggested that twocognitive biases were beneficial to the retailer e reasonsfor such a phenomenon were that both cognitive biasescould increase the optimal order quantity and sales effortthus enhancing the utility of the retailer

Order quantity in centralized supply chain

01 02 03 04 05 06 07 08 09 10β

15

20

25

30

35

40

Ord

er q

uant

ity

μ = 012

Figure 1 e effect of the cognitive biases on optimal order quantity

8 Complexity

Figure 5 illustrates the impact of overconfidence andfairness concern on the manufacturerrsquos profit at is theoptimal manufacturerrsquos profit is an increasing function of βbut a decreasing function of μ According to Figure 3 thegraph of themanufacturerrsquos profit was similar to the graph ofthe wholesale price which indicated that the manufacturerrsquosprofit was firmly related to wholesale price Figure 5 alsoillustrates the effect of the cognitive bias β on the manu-facturerrsquos decreasing profit

52 Effect of Overconfidence and Fairness Concern on SupplyChain Coordination As can be seen from Section 51 thereexist deviations between the optimal decisions under the

benchmark and the decisions in the presence of the retailerrsquoscognitive biases erefore the buyback contract withpromotion cost sharing was necessary to coordinate therevenue of supply chain members and optimize the per-formance of the supply chain

Here we discussed the validity of the contract When theoptimal order quantity and sales effort of the retailer underthe contract were the same as the optimal decision variablesin the centralized supply chain ie qsc qclowast 32 andesc eclowast 16 the buyback contract with promotion costsharing achieved supply chain coordination First the effectof overconfidence and fairness concern (β and μ) on contractfactors wsrlowast rlowast and ϕ is shown in Figures 6 and 7 andTable 1 For clearly analyzing the effect of β and μ on wsrlowast

Degree of sales effort in centralized supply chain

01 02 03 04 05 06 07 08 09 10β

μ = 012

4

6

8

10

12

14

16

18

Deg

ree o

f sal

es ef

fort

Figure 2 e effect of the cognitive biases on optimal sales effort

01 02 03 04 05 06 07 08 09 10β

-μ = 0--μ = 1-μ = 2

4

5

6

7

8

9

10

11

12

13

14

Who

lesa

le p

rice

Figure 3 e effect of the cognitive biases on the optimal wholesale price

Complexity 9

and rlowast we let ϕ 1 in equations (18) and (19) which meantthat the manufacturer proposed the buyback contract onlyand the retailer bore all sales effort costs

According to Figure 6 it was revealed that the optimalwholesales price is a strictly increasing function of theoverconfidence level β but a strictly decreasing function ofthe degree of fairness concern μ which confirmed the resultof Proposition 4 Similarly Figure 7 highlights that theoptimal buyback price sharply increased with β but de-creased with μ Moreover in the buyback contract wsrlowast wasalways greater than or equal to 4 and rlowast was always greaterthan or equal to 2 in different cognitive bias levels Hence

the optimal buyback price rlowast satisfied the assumptions 2le r Note that the above characteristics influencedcognitive biases hence contract factors wsrlowast and rlowast werealso correct when ϕ is equal to other values (such asϕ 03 05 08) under conditions in which the buybackcontract had promotion cost sharing

However data from Figure 7 can be compared with thedata in Figure 6 which showed rsrlowast is more significant thanwsrlowast when β was within a specific range (for example04le βlt 1) which did not match the assumption rltwsrerefore the buyback contract did not achieve the coor-dination of the behavioral supply chain

01 02 03 04 05 06 07 08 09 10β

-μ = 0--μ = 1-μ = 2

0

20

40

60

80

100

120

Reta

iler e

xpec

ted

utili

ty

Figure 4 e effect of the cognitive biases on retailer utility

01 02 03 04 05 06 07 08 09 10β

-μ = 0--μ = 1-lowastμ = 2

0

20

40

60

80

100

120

140

160

180

200

Man

ufac

ture

r pro

fit

Figure 5 e effect of the cognitive biases on manufacturer profit

10 Complexity

As described in Section 4 if the supply chain membersaccepted the buyback contract with promotion cost sharingthe assumptions 1113937

scm gemax 1113937

dm and U(1113937

scr )gemaxU(1113937

dr )

had to be satisfied Also according to the assumptions thevalue ϕ must satisfy 1113937

scm gemax 1113937

dm and

U(1113937scr )gemaxU(1113937

dr ) us we kept the values of param-

eters to be the same (ie a 50 θ sim U(minus 30 30) c 4p 20 v 7 s 2 b 2 and c 1) to capture a likelydomain of ϕ

As can be seen from Table 2 the overconfident level βincreased the upper and lower limits of ϕ which had a clearupward trend In contrast the upper and lower limits of ϕ

showed a definite declining trend as the degree of fairnessconcern μ increased ese trends illustrated that in order tocoordinate the revenue of both parties and optimize theperformance of the supply chain the manufacturer needs tobear less promotional costs as β increased but bear morepromotional costs as μ value increased Besides when β and μincreased the interval of sharing cost portion ϕ shrunk whichindicated that the buyback contract with promotional costsharing had certain flexibility to coordinate the revenue be-tween supply chain members However the flexibility couldbe weakened with the rise of the cognitive bias level Forsupply chain members the possibility of negotiation or

01 02 03 04 05 06 07 08 09 10β

-μ = 0--μ = 1-μ = 2

4

5

6

7

8

9

10

11

12

Who

lesa

le p

rice

Figure 6 e effect of the cognitive biases on wholesale price wsrlowast

01 02 03 04 05 06 07 08 09 10β

-μ = 0--μ = 1-μ = 2

2

4

6

8

10

12

14

16

18

20

22

Buyb

ack

pric

e

Figure 7 e effect of the cognitive biases on buyback price rlowast

Complexity 11

cooperation between manufacturers and retailers also de-creased with an increase in retailersrsquo cognitive bias levelWhen the overconfident level was more than 081 the re-tailerrsquos utility from decentralized decisions was greater thanthe utility in the contract environment wherein the twoparties could not reach a cooperation agreement us thesupply chain could not achieve coordination as different levelsof cognitive bias under the contract had the same influence ondecision variables profit and utility Table 2 shows a clearpicture wherein μ is equal to 1 and β falls within the range of(0 08) which were used to analyze the effect of the cognitivebiases on the coordination of the supply chain

e results of the numerical simulation in Table 3 in-dicate that the contract factors ie ϕ r wholesale price theprofit and utility of members existed as flexible intervalswhen β isin (0 08) and μ 1 Besides the order quantity qsc

and the degree of sales effort esc under the contract are equalto qclowast and eclowast in a centralized supply chain respectivelyMeanwhile the buyback price r was less than the wholesaleprice wsc and higher than the unit salvage value s whichsatisfied the constrains sle rltwsclowast us the buybackcontract with promotion cost could achieve the coordinationof the two-echelon supply chain Also the buyback price r

was positively correlated with the portion ϕ of promotioncost sharing Notably the wholesale price wsc dropped as ϕincreased when β was in the range of (0 02) in Table 3However the situation reversed when it was in the range of(02 08) which was inconsistent since the manufacturerlowered the wholesale price when the retailer paid morepromotional costs in the contract e reason for such anobservation was that the wholesale price was not related to ϕbut positively related to the overconfident level β when thedegree of fairness concern was moderate Furthermore theretailer accepted the contract designed by manufacturerssince the term U(1113937

scr )gemaxU(1113937

dr ) was guaranteed in

Table 3

6 Conclusions

61MainConclusions In this paper we analyzed the effect ofcognitive bias on equilibrium solutions including thewholesale price sales price sales effort and expected profitsunder demand uncertainty by introducing both overconfi-dence and fairness concern into the two-echelon supply chaine most important conclusion from the study was that forretailers with the two cognitive biases overconfidence waspositive and the main bias that affected supply chain decisionsand the retailersrsquo utility benefits were not only due to over-confidence but also fairness concern For manufacturersoverconfidence could increase the wholesale price and profitHowever the fairness concern had a negative effect on themFurthermore the decisions of the retailer in the decentralizedsupply chain deviated from the optimal decisions in thecentralized supply chainmodel Especially when the thresholdof overconfidence was more than a fixed value the decisionvariables under decentralized decision-making were actuallyhigher than the optimal variables under centralized decision-making Also the buyback contract with promotional costsharing achieved behavioral supply chain coordination Incontrast the promotional cost-sharing portion and buybackprice both benefited from overconfidence but fairness concernhad a detrimental effect on them However as the retailersrsquooverconfidence level increased and fairness concern was equalto 1 the flexibility of contract coordination was weakenedWhen the overconfidence level wasmore than 08 the contractbecame invalid and members were not able to cooperate

62 Management Insights is paper provided some in-teresting managerial insights for decision makers in thedecentralized supply chain wherein retailers had two pref-erences First overconfidence was beneficial to them interms of decision variables and yet the fairness concern

Table 2 e effect of the cognitive biases on the feasible domain of promotion cost-sharing portion ϕ

μβ

0 02 04 06 08 0810 [02884 04976] [03683 05587] [04528 06199] [06343 07754] [07835 08669] mdash1 [02545 04741] [03310 05331] [04167 05908] [06207 07516] [07800 08560] mdash2 [02352 04600] [03101 05156] [04034 05719] [06094 07321] [07736 08478] mdash

Table 3 e effect of the cognitive biases on the coordination of the supply chain

μ 1β

0 02 04 06 08qsc 32 32 32 32 32esc 16 16 16 16 16ϕ [02545 04741] [03310 05331] [04167 05908] [06207 07516] [07800 08560]r [2100 35363] [28268 39188] [30555 42637] [36209 44215] [40485 49164]wsc [49205 53687] [49016 53522] [54092 69568] [62513 72626] [80407 87453]1113937

dm 164103 269591 461996 772619 1112545

U(1113937scm) [168722 217850] [269600 336882] [463175 500938] [772619 814300] [1112545 1120376]

1113937dr 20333 352396 495161 632422 765317

U(1113937scr ) [20333 243043] [352396 400595] [495161 525161] [632422 657581] [765317 784632]

12 Complexity

only damaged the wholesale price Second the two behav-ioral preferences were both conducive to utilities for theretailer but not necessarily beneficial for the manufacturer toreceive profit and overconfidence helped to increasemanufacturersrsquo profit which led to nonoptimal decisionsthat dragged down the performance of the two-echelonsupply chain Finally for the two behavioral preferences ofthe retailer the manufacturer assessed the degree of over-confidence and fairness concern (especially the overconfi-dence) via historical data and the current order quantity orother indicators In contrast factual data indicated that twobehavioral preferences were in the right range (such asβ isin (0 08) and μ isin (0 2)) e manufacturers introducedthe buyback contract with promotional cost sharing into thepractice of pricing and ordered decentralized supply chainsFurthermore they designed a reasonable promotional cost-sharing portion and buyback price thereby achieving anoptimized decision and utility In practice the manufac-turers could analyze whether the sellers had two behavioralpreferences based on the historical data by cooperating withdownstream enterprises ie if two behavioral preferencesexisted they referred to the papersrsquo conclusions to redesign areasonable contract mechanism

63 Future Research Since cognitive biases could existamong all members of supply chains it is necessary toexplore the manufacturersrsquo cognitive biases and analyze thesupply chain coordination problem in the presence of thetwo behavior preferences Besides the overconfidence andfairness concern considered in our centralized decision are afuture research direction

Appendix

A Decentralized Supply Chain Model

In a decentralized supply chain consisting of a rationalmanufacturer and a retailer with cognitive biases maximumprofit or utility is the only goal e backward inductivemethod can be applied to solve a Stackelberg game whereinthe manufacturer is the leader and the retailer is a followere proof of the equilibrium solutions in equations (12)ndash(14)is as follows

e profit of the manufacturer and expected utility ofretailers are as follows respectively

wmax

1113945

d

m

(w minus c)q (A1)

maxqe

U 1113945d

r

⎞⎠ (1 + μ) p[a minus bp +(1 + β)e] + s 1113946θ2

minus Aq minus a minus bp +(1 + β)e + θ1( 1113857f θ1( 1113857( 1113857dθ11113896⎛⎝

minus v 1113946A

θ2a minus bp +(1 + β)e + θ1 minus q( 1113857f θ1( 1113857( 1113857dθ1 minus wq minus

12

ce21113897 minus μ(w minus c)q

(A2)

Solving the first-order partial derivatives of the retailerrsquosexpected utility concerning qd and ed which equal to 0 weobtained the following equations

zU( 1113937dr 1113873

zq (1 + μ) v minus w + F θ2( 1113857(s minus v)1113858 1113859 minus μ(w minus c) 0

(A3)

zU( 1113937dr 1113873

ze (1 + β) p minus v +(v minus s)F θ2( 11138571113858 1113859 minus ce 0

(A4)

Hessian matrix

H(q e) (s minus v)f θ2( 1113857 (v minus s)(1 + β)f θ2( 1113857

(v minus s)f θ2( 1113857 (s minus v)(1 + β)f θ2( 1113857 minus c1113888 1113889

(A5)

According to the assumptions vgt sgt 0 f(θ2)gt 0 andβ isin [0 1) us the Hessian matrix of U(1113937

dr ) is a negative

definite for q and e which meant that the optimal orderquantity and sales effort of retailers existed in the decen-tralized supply chain erefore we obtained the followingformula of the retailersrsquo order quantity and sales effort byjointly solving equations (A3) and (A4)

qd

(a minus bp)(v minus s)(1 + μ)c +(1 + β)2(v minus s)[(1 + μ)(p minus w) + μc] + Ac[(s + v)(1 + μ) + 2μc minus w]

(v minus s)(1 + μ)c (A6)

ed

(1 + β)[(1 + μ)p + μc minus (1 + 2μ)]w

(1 + μ)c (A7)

Complexity 13

Substituting equations (A6) into (A1) we attained

wmax

1113945

d

m

(w minus c) (a minus bp)(v minus s)(1 + μ)c +(1 + β)2(v minus s)[(1 + μ)p + μc] + Ac[(s + v)(1 + μ) + 2μc] minus (1 + 2μ)w 2Ac +(1 + β)2(v minus s)1113960 11139611113966 1113967

(v minus s)(1 + μ)c

(A8)

Taking into account the first-order condition of equation(A7) with respect to w rounded off to 0 we obtained`

z1113937dm

ze

(a minus bp)(v minus s)(1 + μ)c +(1 + β)2(v minus s)[(1 + μ)p + μc] + Ac[(s + v)(1 + μ) + 2μc] minus (2w minus c)(1 + 2μ) 2Ac +(1 + β)2(v minus s)1113960 1113961

(v minus s)(1 + μ)c

(A9)

Solving equation (A8) we found that the optimalwholesale price of the manufacturer is as follows

wdlowast

(a minus bp)(1 + μ)(v minus s)c +(v minus s)(1 + β)2[(1 + μ)p +(1 + 3μ)c] + Ac[(s + v)(1 + μ) + 2(1 + 3μ)c]

2(1 + 2μ) 2Ac +(v minus s)(1 + β)21113960 1113961 (A10)

By substituting equation (A9) into equations (A6) and(A7) we attained the optimal order quantity and sales effortwhich are displayed in equations (13) and (14)

B Propositions

Proof of Proposition 1 We solved the first derivatives ofoptimal order quantity qdlowast and sales effort edlowast with respect

to β in equations (13) and (14) respectively Meanwhiletaking the second-order derivative of qdlowast with respect to βwe obtained

zqdlowast

zβ

(1 + β)(p minus c)

c

z2qdlowast

zβ2

p minus c

c (B1)

zedlowast

zβ4(p minus c)(v minus s)2β β3 + 4β2 + 6β + 41113872 1113873 +(a minus bp)c(1 + β)2(v minus s)2 + Ac(v minus s)(1 + β)2 (2(p minus c) + v + s) + 2(Ac)2(4p minus c minus s minus v)1113960 1113961

2c (1 + β)2(v minus s) + 2Ac1113960 11139612

(B2)

Since the second derivative of edlowast with respect to β wasvery complicated there was no need to solve it According tothe assumptions pgt vgt cgt sgt 0 β isin (0 1) a minus bpgt 0 andcgt 0 so (zqdlowastzβ)gt 0 (z2qdlowastzβ2)gt 0 and (zedlowastzβ)gt 0

Proof of Proposition 2 Based on the process above we tookthe first-order and second-order partial derivatives of op-timal wholesale price wdlowast with respect to β and μ in equation(12) respectively

zwdlowast

zβ

c(v minus s)(1 + μ)(1 + β)[(a minus bp)(v minus s) + A(2p minus s minus v)]

(1 + 2μ) (1 + β)2(v minus s) + 2Ac1113960 11139612 (B3)

zwdlowast

zμ minus

(a minus bp)(v minus s)c +(p minus c)(v minus s)(1 + β)2 + Ac(s + v minus 2c)1113960 1113961

2(1 + 2μ)2 (v minus s)(1 + β)2 + 2Ac1113960 1113961 (B4)

14 Complexity

According to the optimal result of order quantity inequation (13) and the assumption qgt 0 the condition s +

v minus 2cgt 0 is found According to the assumptions we had(zwdlowastzβ)gt 0 and (zwdlowastzμ)lt 0

Proof of Proposition 3 e gap between qdlowast and qclowastdenoted by Δq is as follows

Δq qdlowast

minus qclowast

(p minus c)(v minus s) (1 + β)2 minus 21113960 1113961 minus Ac(s + v minus 2c) minus (a minus bp)(v minus s)c

2c(v minus s) (B5)

If (p minus c)(v minus s)[(1 + β)2 minus 2] is more thanAc(s + v minus 2c) + (a minus bp)(v minus s)c we have Δqgt 0 ieqdlowast gt qclowast In contrast if (p minus c)(v minus s)[(1 + β)2 minus 2] is less

than Ac(s + v minus 2c) + (a minus bp)(v minus s)c Δqlt 0 ieqdlowast lt qclowast

e gap between edlowast and eclowast denoted by Δe is asfollows

Δe edlowast

minus eclowast

Aβc(4p minus 2c minus s minus v) minus Ac(s + v minus 2c) minus (v minus s)(1 + β) (a minus bp)c + 1 minus β21113872 1113873(p minus c)1113960 1113961

2c 2Ac +(1 + β)2(v minus s)1113960 1113961 (B6)

Similarly if Aβc(4p minus 2c minus s minus v)gtAc(s + v minus 2c)

+(v minus s)(1 + β)[(a minus bp)c + (1 minus β2)(p minus c)] we haveΔe edlowast minus eclowast gt 0 In contrast ifAβc(4p minus 2c minus s minus v)ltAc(s + v minus 2c)+

(v minus s)(1 + β)[(a minus bp)c + (1 minus β2)(p minus c)] we haveΔe edlowast minus eclowast lt 0

Upon solving the first-order derivative of Δq and Δe withrespect to β respectively we obtained

zΔ q

zβ

(1 + β)(p minus c)

c (B7)

zΔ e

zβ4(p minus c)(v minus s)2 β3 + 4β2 + 6β + 41113872 1113873β +(a minus bp)c(1 + β)2(v minus s)2 + Ac(v minus s)(1 + β)2 2(p minus c) +v + s) + 2(Ac)2(4p minus c minus s minus v)1113960 1113961

2c (1 + β)2(v minus s) + 2Ac1113960 11139612

(B8)

According to the assumptions pgt vgt cgt sgt 0 β isin (0 1)cgt 0 and a minus bpgt 0 (zΔ qzβ)gt 0 and (zΔ ezβ)gt 0 can beproved Hence the gaps increased as a function with respectto β

Proof of Proposition 4 e proof for Proposition 4 is similarto Proposition 2 Herein we took first-order and second-order partial derivatives of optimal wholesale price wsclowast withrespect to β μ and ϕ in equation (18) respectively esolving results are as follows

zwsclowast

zβ

[(1 + 2μ)ϕ minus μ](p minus c)

(1 + 2μ)(1 + β)2

zwsclowast

zμ minus

(p minus c)β(1 + β)(1 + 2μ)2

zwsclowast

zϕ minus

p minus c

1 + β

(B9)

(zwsclowastzβ) if ϕgt (μ(1 + 2μ)) is more than 0 otherwiseϕgt (μ(1 + 2μ)) is less than 0 us wsc could be increasingwith respect to β nevertheless decreasing with respect to βBesides according to the assumptions pgt vgt cgt sgt 0β isin (0 1) cgt 0 and a minus bpgt 0 it can be evidently foundthat (zwsclowastzμ)gt 0 and (zwsclowastzϕ)gt 0

Data Availability

e data used to support the findings of this study are in-cluded within the article

Conflicts of Interest

e authors declare no conflicts of interest

Acknowledgments

is work was supported by a major project financed by theNational Social Science Fund of China (Approval nos71461009 71662011 and 71761015)

Complexity 15

References

[1] Y Ren D C Croson and R T A Croson ldquoe overconfidentnewsvendorrdquo Journal of the Operational Research Societyvol 68 no 5 pp 496ndash506 2017

[2] T H Cui and P Mallucci ldquoFairness ideals in distributionchannelsrdquo Journal of Marketing Research vol 53 no 6pp 969ndash987 2016

[3] K Chen X Wang M Huang and W-K Ching ldquoSalesforcecontract design joint pricing and production planning withasymmetric overconfidence sales agentrdquo Journal of Industrialamp Management Optimization vol 13 no 2 pp 873ndash8992017

[4] X X Zheng Z Liu K W Li et al ldquoCooperative game ap-proaches to coordinating a three-echelon closed-loop supplychain with fairness concernsrdquo International Journal of Pro-duction Economic vol 212 pp 92ndash110 2018

[5] China Internet Watch False Prosperity behind Double 11Refund Rate Increased to 69 2014 httpswwwchinainternetwatchcom11643false-prosperity-behind-double-11-refund-rate-increased-to-69

[6] C L Lu ldquoPampGrsquos collaborative supply chain transformationrdquoEnterprise Management vol 10 pp 79-80 2016

[7] M Li N C Petruzzi and J Zhang ldquoOverconfident competingnewsvendorsrdquo Management Science vol 63 no 8pp 2637ndash2646 2016

[8] T Nie ldquoDual-fairness supply chain with quantity discountcontractsrdquo European Journal of Operational Researchvol 258 no 2 pp 491ndash500 2017

[9] X Du and B Jiang ldquoSignaling through price and quality toconsumers with fairness concernsrdquo Journal of MarketingResearch vol 53 no 6 pp 988ndash1000 2016

[10] V Kurz A Orland and K Posadzy ldquoFairness versus effi-ciency how procedural fairness concerns affect coordina-tionrdquo Experimental Economics vol 21 no 3 pp 601ndash6262018

[11] S Villa and J A Castantildeeda ldquoTransshipments in supplychains a behavioral investigationrdquo European Journal of Op-erational Research vol 269 no 2 pp 715ndash729 2018

[12] B Gu Y Fu and Y Li ldquoFresh-keeping effort and channelperformance in a fresh product supply chain with loss-averseconsumersrsquo returnsrdquo Mathematical Problems in Engineeringvol 2018 Article ID 4717094 20 pages 2018

[13] D Croson R Croson and Y Ren How to Manage anOverconfident Newsvendor 2008 httpcbeesutdallasedupapersCrosonRenCmsonMS2008

[14] M Li ldquoOverconfident distribution channelsrdquo Production andOperations Management vol 28 no 6 pp 1347ndash1365 2019

[15] W Liu D Wang O Tang and D Zhu ldquoe impacts oflogistics service integratorrsquos overconfidence behaviour onsupply chain decision under demand surgerdquo EuropeanJournal of Industrial Engineering vol 12 no 4 pp 558ndash5972018

[16] L Xu X Shi P Du K Govindan and Z Zhang ldquoOptimi-zation on pricing and overconfidence problem in a duopo-listic supply chainrdquo Computers amp Operations Researchvol 101 pp 162ndash172 2019

[17] Y Li and M Shan ldquoAdvance selling decisions with over-confident consumersrdquo Journal of Industrial and ManagementOptimization vol 2 no 3 pp 891ndash905 2016

[18] T H Cui J S Raju and Z J Zhang ldquoFairness and channelcoordinationrdquo Management Science vol 53 no 8pp 1303ndash1314 2007

[19] X Wu and J A Niederhoff ldquoFairness in selling to thenewsvendorrdquo Production and Operations Managementvol 23 no 11 pp 2002ndash2022 2014

[20] Z Chen and S-I Ivan Su ldquoSocial welfare maximization withthe least subsidy photovoltaic supply chain equilibrium andcoordination with fairness concernrdquo Renewable Energyvol 132 pp 1332ndash1347 2019

[21] W Liu D Wang X Shen X Yan and W Wei ldquoe impactsof distributional and peer-induced fairness concerns on thedecision-making of order allocation in logistics service supplychainrdquo Transportation Research Part E Logistics and Trans-portation Review vol 116 pp 102ndash122 2018

[22] L Zhang H Zhou Y Liu and R Lu ldquoOptimal environmentalquality and price with consumer environmental awarenessand retailerrsquos fairness concerns in supply chainrdquo Journal ofCleaner Production vol 213 pp 1063ndash1079 2019

[23] X LWang S Q Hu and X B Liu ldquoSupply coordination withoption contract considering overconfidence under randomdemand and yieldsrdquo Computer Integrated ManufacturingSystem vol 24 no 11 pp 2898ndash2908 2018

[24] W Jiang and J Liu ldquoInventory financing with overconfidentsupplier based on supply chain contractrdquo MathematicalProblems in Engineering vol 2018 Article ID 505438712 pages 2018

[25] P Ma H Wang and J Shang ldquoContract design for two-stagesupply chain coordination integrating manufacturer-qualityand retailer-marketing effortsrdquo International Journal ofProduction Economics vol 146 no 2 pp 745ndash755 2013

[26] J Liu W F Jiang and Y P Tang ldquoe supply chain coor-dination based on carbon reduction of overconfident man-ufacturerrdquo Journal of Jiangxi Normal University (NaturalScience) vol 42 no 2 pp 203ndash207 2018

[27] R Dubey N Altay and C Blome ldquoSwift trust and com-mitment the missing links for humanitarian supply chaincoordinationrdquo Annals of Operations Research vol 283 no 1-2 pp 159ndash177 2019

[28] C Cui Z Feng and C Tan ldquoCredibilistic loss aversion nashequilibrium for bimatrix games with triangular fuzzy payoffsrdquoComplexity vol 2018 Article ID 7143586 16 pages 2018

[29] E Fehr and KM Schmidt ldquoA theory of fairness competitionand cooperationrdquo e Quarterly Journal of Economicsvol 114 no 3 pp 817ndash868 1999

[30] Z Qin and J Yang ldquoAnalysis of a revenue-sharing contract insupply chain managementrdquo International Journal of LogisticsResearch and Applications vol 11 no 1 pp 17ndash29 2008

[31] S-Z Bai and J Ni ldquoResearch on buy-back contract with saleseffort cost sharingrdquo in Proceedings of the 2009 Chinese Controland Decision Conference IEEE Guilin China pp 1917ndash1921June 2009

[32] E Karagozoglu and K Keskin ldquoTime-varying fairness con-cerns delay and disagreement in bargainingrdquo Journal ofEconomic Behavior and Organization vol 147 pp 115ndash1282018

16 Complexity

we assumed β isin [0 1) and μ 0 1 2 to reflect the level ofoverconfidence and fairness concern in this paper Whenβ 0 and μ 0 the retailer was considered to be rationalSimilarly other fundamental parameter values were set inexperiments as follows a 50 θ sim U(minus 30 30) c 4p 20v 7 s 2 b 2 and c 1 All the parameterssatisfied the constraint conditions and assumptions of thethree models in Sections 32 33 and 4e parameter valueswere also in line with the real situation of the market Wecould easily obtain qclowast 32 eclowast 16 and 1113937

c 252 in the

centralized supply chain

51 Effect of Overconfidence and Fairness Concern on theSupply Chain e changes in the decision variables (w qand e) manufacturer profit and retailer expected utilitywhen the level of overconfidence and fairness concern (β andμ) changed are shown in Figures 1ndash5

As shown in Figures 1 and 2 when the retailer was ra-tional (eg β 0 and μ 0) in the decentralized supplychain both optimal order quantity and sales effort were lowerthan those under the centralized supply chain Next when theretailer had overconfidence and fairness concern simulta-neously overconfidence could only affect the retailerrsquos opti-mal order quantity and sales effort of the retailer because theoptimal order quantity and sales effort were a function ofoverconfidence in equations (13) and (14) Moreover theorder quantity was monotonically increasing as concavefunction with respect to β In comparison sales effort wasmonotonically increasing as convex function with respect toit Furthermore there existed a threshold (β 083) when βwas less than 083 and the two decision variables were bothless than the optimal values (qclowast 32 eclowast 16) under thecentralized decision scenario Nevertheless once β was largerthan 083 the two decision variables were both more thanthem e above discussion verified the correctness ofProposition 1 In short overconfidence was the main factorthat affected the retailerrsquos optimal strategies of order quantity

and sales effort in the decentralized supply chain Howeverthe above findings contradicted with the conclusions from astudy conducted by Xu et al [16] which examined the effectof retailerrsquos overconfidence on the supply chain and the resultsuggested that a higher level of overconfident resulted in thelower ordering quantity

In Figure 3 we observed that the higher overconfidentlevel resulted in the high wholesale price in contrast ahigh degree of fairness concern resulted in low pricewhich demonstrated that Proposition 2 was robust isfigure implied that when the information for both partieswas symmetric the rational manufacturer knew that theretailer was overconfident that is the retailer was opti-mistic about the market demand and increased the orderquantity As a result the manufacturer also raised thewholesale price to maximize its revenue Also if theoverconfident retailer paid attention to the unfairness ofprofit allocation in the decision-making process the re-tailerrsquos bargaining power was enhanced [32] Hence forbetter performance in the supply chain the leadingmanufacturer was required to lower wholesale prices toease the feeling of unfairness by the retailer which isconsistent with the actual economic phenomenon Fur-thermore these findings are consistent with the resultconcluded by Cui et al [2] which indicated that a highdegree of the retailerrsquos fairness concern could lower thewholesale price Interestingly as the levels of the cognitivebias increased its effect on wholesale price graduallydecreased which highlighted its effect on decreasedwholesale prices

Figure 4 displays the impact of the cognitive bias on theretailerrsquos utility e graph showed that the retailer expectedutility increased gradually with the overconfident level andthe degree of fairness concern which suggested that twocognitive biases were beneficial to the retailer e reasonsfor such a phenomenon were that both cognitive biasescould increase the optimal order quantity and sales effortthus enhancing the utility of the retailer

Order quantity in centralized supply chain

01 02 03 04 05 06 07 08 09 10β

15

20

25

30

35

40

Ord

er q

uant

ity

μ = 012

Figure 1 e effect of the cognitive biases on optimal order quantity

8 Complexity

Figure 5 illustrates the impact of overconfidence andfairness concern on the manufacturerrsquos profit at is theoptimal manufacturerrsquos profit is an increasing function of βbut a decreasing function of μ According to Figure 3 thegraph of themanufacturerrsquos profit was similar to the graph ofthe wholesale price which indicated that the manufacturerrsquosprofit was firmly related to wholesale price Figure 5 alsoillustrates the effect of the cognitive bias β on the manu-facturerrsquos decreasing profit

52 Effect of Overconfidence and Fairness Concern on SupplyChain Coordination As can be seen from Section 51 thereexist deviations between the optimal decisions under the

benchmark and the decisions in the presence of the retailerrsquoscognitive biases erefore the buyback contract withpromotion cost sharing was necessary to coordinate therevenue of supply chain members and optimize the per-formance of the supply chain

Here we discussed the validity of the contract When theoptimal order quantity and sales effort of the retailer underthe contract were the same as the optimal decision variablesin the centralized supply chain ie qsc qclowast 32 andesc eclowast 16 the buyback contract with promotion costsharing achieved supply chain coordination First the effectof overconfidence and fairness concern (β and μ) on contractfactors wsrlowast rlowast and ϕ is shown in Figures 6 and 7 andTable 1 For clearly analyzing the effect of β and μ on wsrlowast

Degree of sales effort in centralized supply chain

01 02 03 04 05 06 07 08 09 10β

μ = 012

4

6

8

10

12

14

16

18

Deg

ree o

f sal

es ef

fort

Figure 2 e effect of the cognitive biases on optimal sales effort

01 02 03 04 05 06 07 08 09 10β

-μ = 0--μ = 1-μ = 2

4

5

6

7

8

9

10

11

12

13

14

Who

lesa

le p

rice

Figure 3 e effect of the cognitive biases on the optimal wholesale price

Complexity 9

and rlowast we let ϕ 1 in equations (18) and (19) which meantthat the manufacturer proposed the buyback contract onlyand the retailer bore all sales effort costs

According to Figure 6 it was revealed that the optimalwholesales price is a strictly increasing function of theoverconfidence level β but a strictly decreasing function ofthe degree of fairness concern μ which confirmed the resultof Proposition 4 Similarly Figure 7 highlights that theoptimal buyback price sharply increased with β but de-creased with μ Moreover in the buyback contract wsrlowast wasalways greater than or equal to 4 and rlowast was always greaterthan or equal to 2 in different cognitive bias levels Hence

the optimal buyback price rlowast satisfied the assumptions 2le r Note that the above characteristics influencedcognitive biases hence contract factors wsrlowast and rlowast werealso correct when ϕ is equal to other values (such asϕ 03 05 08) under conditions in which the buybackcontract had promotion cost sharing

However data from Figure 7 can be compared with thedata in Figure 6 which showed rsrlowast is more significant thanwsrlowast when β was within a specific range (for example04le βlt 1) which did not match the assumption rltwsrerefore the buyback contract did not achieve the coor-dination of the behavioral supply chain

01 02 03 04 05 06 07 08 09 10β

-μ = 0--μ = 1-μ = 2

0

20

40

60

80

100

120

Reta

iler e

xpec

ted

utili

ty

Figure 4 e effect of the cognitive biases on retailer utility

01 02 03 04 05 06 07 08 09 10β

-μ = 0--μ = 1-lowastμ = 2

0

20

40

60

80

100

120

140

160

180

200

Man

ufac

ture

r pro

fit

Figure 5 e effect of the cognitive biases on manufacturer profit

10 Complexity

As described in Section 4 if the supply chain membersaccepted the buyback contract with promotion cost sharingthe assumptions 1113937

scm gemax 1113937

dm and U(1113937

scr )gemaxU(1113937

dr )

had to be satisfied Also according to the assumptions thevalue ϕ must satisfy 1113937

scm gemax 1113937

dm and

U(1113937scr )gemaxU(1113937

dr ) us we kept the values of param-

eters to be the same (ie a 50 θ sim U(minus 30 30) c 4p 20 v 7 s 2 b 2 and c 1) to capture a likelydomain of ϕ

As can be seen from Table 2 the overconfident level βincreased the upper and lower limits of ϕ which had a clearupward trend In contrast the upper and lower limits of ϕ

showed a definite declining trend as the degree of fairnessconcern μ increased ese trends illustrated that in order tocoordinate the revenue of both parties and optimize theperformance of the supply chain the manufacturer needs tobear less promotional costs as β increased but bear morepromotional costs as μ value increased Besides when β and μincreased the interval of sharing cost portion ϕ shrunk whichindicated that the buyback contract with promotional costsharing had certain flexibility to coordinate the revenue be-tween supply chain members However the flexibility couldbe weakened with the rise of the cognitive bias level Forsupply chain members the possibility of negotiation or

01 02 03 04 05 06 07 08 09 10β

-μ = 0--μ = 1-μ = 2

4

5

6

7

8

9

10

11

12

Who

lesa

le p

rice

Figure 6 e effect of the cognitive biases on wholesale price wsrlowast

01 02 03 04 05 06 07 08 09 10β

-μ = 0--μ = 1-μ = 2

2

4

6

8

10

12

14

16

18

20

22

Buyb

ack

pric

e

Figure 7 e effect of the cognitive biases on buyback price rlowast

Complexity 11

cooperation between manufacturers and retailers also de-creased with an increase in retailersrsquo cognitive bias levelWhen the overconfident level was more than 081 the re-tailerrsquos utility from decentralized decisions was greater thanthe utility in the contract environment wherein the twoparties could not reach a cooperation agreement us thesupply chain could not achieve coordination as different levelsof cognitive bias under the contract had the same influence ondecision variables profit and utility Table 2 shows a clearpicture wherein μ is equal to 1 and β falls within the range of(0 08) which were used to analyze the effect of the cognitivebiases on the coordination of the supply chain

e results of the numerical simulation in Table 3 in-dicate that the contract factors ie ϕ r wholesale price theprofit and utility of members existed as flexible intervalswhen β isin (0 08) and μ 1 Besides the order quantity qsc

and the degree of sales effort esc under the contract are equalto qclowast and eclowast in a centralized supply chain respectivelyMeanwhile the buyback price r was less than the wholesaleprice wsc and higher than the unit salvage value s whichsatisfied the constrains sle rltwsclowast us the buybackcontract with promotion cost could achieve the coordinationof the two-echelon supply chain Also the buyback price r

was positively correlated with the portion ϕ of promotioncost sharing Notably the wholesale price wsc dropped as ϕincreased when β was in the range of (0 02) in Table 3However the situation reversed when it was in the range of(02 08) which was inconsistent since the manufacturerlowered the wholesale price when the retailer paid morepromotional costs in the contract e reason for such anobservation was that the wholesale price was not related to ϕbut positively related to the overconfident level β when thedegree of fairness concern was moderate Furthermore theretailer accepted the contract designed by manufacturerssince the term U(1113937

scr )gemaxU(1113937

dr ) was guaranteed in

Table 3

6 Conclusions

61MainConclusions In this paper we analyzed the effect ofcognitive bias on equilibrium solutions including thewholesale price sales price sales effort and expected profitsunder demand uncertainty by introducing both overconfi-dence and fairness concern into the two-echelon supply chaine most important conclusion from the study was that forretailers with the two cognitive biases overconfidence waspositive and the main bias that affected supply chain decisionsand the retailersrsquo utility benefits were not only due to over-confidence but also fairness concern For manufacturersoverconfidence could increase the wholesale price and profitHowever the fairness concern had a negative effect on themFurthermore the decisions of the retailer in the decentralizedsupply chain deviated from the optimal decisions in thecentralized supply chainmodel Especially when the thresholdof overconfidence was more than a fixed value the decisionvariables under decentralized decision-making were actuallyhigher than the optimal variables under centralized decision-making Also the buyback contract with promotional costsharing achieved behavioral supply chain coordination Incontrast the promotional cost-sharing portion and buybackprice both benefited from overconfidence but fairness concernhad a detrimental effect on them However as the retailersrsquooverconfidence level increased and fairness concern was equalto 1 the flexibility of contract coordination was weakenedWhen the overconfidence level wasmore than 08 the contractbecame invalid and members were not able to cooperate

62 Management Insights is paper provided some in-teresting managerial insights for decision makers in thedecentralized supply chain wherein retailers had two pref-erences First overconfidence was beneficial to them interms of decision variables and yet the fairness concern

Table 2 e effect of the cognitive biases on the feasible domain of promotion cost-sharing portion ϕ

μβ

0 02 04 06 08 0810 [02884 04976] [03683 05587] [04528 06199] [06343 07754] [07835 08669] mdash1 [02545 04741] [03310 05331] [04167 05908] [06207 07516] [07800 08560] mdash2 [02352 04600] [03101 05156] [04034 05719] [06094 07321] [07736 08478] mdash

Table 3 e effect of the cognitive biases on the coordination of the supply chain

μ 1β

0 02 04 06 08qsc 32 32 32 32 32esc 16 16 16 16 16ϕ [02545 04741] [03310 05331] [04167 05908] [06207 07516] [07800 08560]r [2100 35363] [28268 39188] [30555 42637] [36209 44215] [40485 49164]wsc [49205 53687] [49016 53522] [54092 69568] [62513 72626] [80407 87453]1113937

dm 164103 269591 461996 772619 1112545

U(1113937scm) [168722 217850] [269600 336882] [463175 500938] [772619 814300] [1112545 1120376]

1113937dr 20333 352396 495161 632422 765317

U(1113937scr ) [20333 243043] [352396 400595] [495161 525161] [632422 657581] [765317 784632]

12 Complexity

only damaged the wholesale price Second the two behav-ioral preferences were both conducive to utilities for theretailer but not necessarily beneficial for the manufacturer toreceive profit and overconfidence helped to increasemanufacturersrsquo profit which led to nonoptimal decisionsthat dragged down the performance of the two-echelonsupply chain Finally for the two behavioral preferences ofthe retailer the manufacturer assessed the degree of over-confidence and fairness concern (especially the overconfi-dence) via historical data and the current order quantity orother indicators In contrast factual data indicated that twobehavioral preferences were in the right range (such asβ isin (0 08) and μ isin (0 2)) e manufacturers introducedthe buyback contract with promotional cost sharing into thepractice of pricing and ordered decentralized supply chainsFurthermore they designed a reasonable promotional cost-sharing portion and buyback price thereby achieving anoptimized decision and utility In practice the manufac-turers could analyze whether the sellers had two behavioralpreferences based on the historical data by cooperating withdownstream enterprises ie if two behavioral preferencesexisted they referred to the papersrsquo conclusions to redesign areasonable contract mechanism

63 Future Research Since cognitive biases could existamong all members of supply chains it is necessary toexplore the manufacturersrsquo cognitive biases and analyze thesupply chain coordination problem in the presence of thetwo behavior preferences Besides the overconfidence andfairness concern considered in our centralized decision are afuture research direction

Appendix

A Decentralized Supply Chain Model

In a decentralized supply chain consisting of a rationalmanufacturer and a retailer with cognitive biases maximumprofit or utility is the only goal e backward inductivemethod can be applied to solve a Stackelberg game whereinthe manufacturer is the leader and the retailer is a followere proof of the equilibrium solutions in equations (12)ndash(14)is as follows

e profit of the manufacturer and expected utility ofretailers are as follows respectively

wmax

1113945

d

m

(w minus c)q (A1)

maxqe

U 1113945d

r

⎞⎠ (1 + μ) p[a minus bp +(1 + β)e] + s 1113946θ2

minus Aq minus a minus bp +(1 + β)e + θ1( 1113857f θ1( 1113857( 1113857dθ11113896⎛⎝

minus v 1113946A

θ2a minus bp +(1 + β)e + θ1 minus q( 1113857f θ1( 1113857( 1113857dθ1 minus wq minus

12

ce21113897 minus μ(w minus c)q

(A2)

Solving the first-order partial derivatives of the retailerrsquosexpected utility concerning qd and ed which equal to 0 weobtained the following equations

zU( 1113937dr 1113873

zq (1 + μ) v minus w + F θ2( 1113857(s minus v)1113858 1113859 minus μ(w minus c) 0

(A3)

zU( 1113937dr 1113873

ze (1 + β) p minus v +(v minus s)F θ2( 11138571113858 1113859 minus ce 0

(A4)

Hessian matrix

H(q e) (s minus v)f θ2( 1113857 (v minus s)(1 + β)f θ2( 1113857

(v minus s)f θ2( 1113857 (s minus v)(1 + β)f θ2( 1113857 minus c1113888 1113889

(A5)

According to the assumptions vgt sgt 0 f(θ2)gt 0 andβ isin [0 1) us the Hessian matrix of U(1113937

dr ) is a negative

definite for q and e which meant that the optimal orderquantity and sales effort of retailers existed in the decen-tralized supply chain erefore we obtained the followingformula of the retailersrsquo order quantity and sales effort byjointly solving equations (A3) and (A4)

qd

(a minus bp)(v minus s)(1 + μ)c +(1 + β)2(v minus s)[(1 + μ)(p minus w) + μc] + Ac[(s + v)(1 + μ) + 2μc minus w]

(v minus s)(1 + μ)c (A6)

ed

(1 + β)[(1 + μ)p + μc minus (1 + 2μ)]w

(1 + μ)c (A7)

Complexity 13

Substituting equations (A6) into (A1) we attained

wmax

1113945

d

m

(w minus c) (a minus bp)(v minus s)(1 + μ)c +(1 + β)2(v minus s)[(1 + μ)p + μc] + Ac[(s + v)(1 + μ) + 2μc] minus (1 + 2μ)w 2Ac +(1 + β)2(v minus s)1113960 11139611113966 1113967

(v minus s)(1 + μ)c

(A8)

Taking into account the first-order condition of equation(A7) with respect to w rounded off to 0 we obtained`

z1113937dm

ze

(a minus bp)(v minus s)(1 + μ)c +(1 + β)2(v minus s)[(1 + μ)p + μc] + Ac[(s + v)(1 + μ) + 2μc] minus (2w minus c)(1 + 2μ) 2Ac +(1 + β)2(v minus s)1113960 1113961

(v minus s)(1 + μ)c

(A9)

Solving equation (A8) we found that the optimalwholesale price of the manufacturer is as follows

wdlowast

(a minus bp)(1 + μ)(v minus s)c +(v minus s)(1 + β)2[(1 + μ)p +(1 + 3μ)c] + Ac[(s + v)(1 + μ) + 2(1 + 3μ)c]

2(1 + 2μ) 2Ac +(v minus s)(1 + β)21113960 1113961 (A10)

By substituting equation (A9) into equations (A6) and(A7) we attained the optimal order quantity and sales effortwhich are displayed in equations (13) and (14)

B Propositions

Proof of Proposition 1 We solved the first derivatives ofoptimal order quantity qdlowast and sales effort edlowast with respect

to β in equations (13) and (14) respectively Meanwhiletaking the second-order derivative of qdlowast with respect to βwe obtained

zqdlowast

zβ

(1 + β)(p minus c)

c

z2qdlowast

zβ2

p minus c

c (B1)

zedlowast

zβ4(p minus c)(v minus s)2β β3 + 4β2 + 6β + 41113872 1113873 +(a minus bp)c(1 + β)2(v minus s)2 + Ac(v minus s)(1 + β)2 (2(p minus c) + v + s) + 2(Ac)2(4p minus c minus s minus v)1113960 1113961

2c (1 + β)2(v minus s) + 2Ac1113960 11139612

(B2)

Since the second derivative of edlowast with respect to β wasvery complicated there was no need to solve it According tothe assumptions pgt vgt cgt sgt 0 β isin (0 1) a minus bpgt 0 andcgt 0 so (zqdlowastzβ)gt 0 (z2qdlowastzβ2)gt 0 and (zedlowastzβ)gt 0

Proof of Proposition 2 Based on the process above we tookthe first-order and second-order partial derivatives of op-timal wholesale price wdlowast with respect to β and μ in equation(12) respectively

zwdlowast

zβ

c(v minus s)(1 + μ)(1 + β)[(a minus bp)(v minus s) + A(2p minus s minus v)]

(1 + 2μ) (1 + β)2(v minus s) + 2Ac1113960 11139612 (B3)

zwdlowast

zμ minus

(a minus bp)(v minus s)c +(p minus c)(v minus s)(1 + β)2 + Ac(s + v minus 2c)1113960 1113961

2(1 + 2μ)2 (v minus s)(1 + β)2 + 2Ac1113960 1113961 (B4)

14 Complexity

According to the optimal result of order quantity inequation (13) and the assumption qgt 0 the condition s +

v minus 2cgt 0 is found According to the assumptions we had(zwdlowastzβ)gt 0 and (zwdlowastzμ)lt 0

Proof of Proposition 3 e gap between qdlowast and qclowastdenoted by Δq is as follows

Δq qdlowast

minus qclowast

(p minus c)(v minus s) (1 + β)2 minus 21113960 1113961 minus Ac(s + v minus 2c) minus (a minus bp)(v minus s)c

2c(v minus s) (B5)

If (p minus c)(v minus s)[(1 + β)2 minus 2] is more thanAc(s + v minus 2c) + (a minus bp)(v minus s)c we have Δqgt 0 ieqdlowast gt qclowast In contrast if (p minus c)(v minus s)[(1 + β)2 minus 2] is less

than Ac(s + v minus 2c) + (a minus bp)(v minus s)c Δqlt 0 ieqdlowast lt qclowast

e gap between edlowast and eclowast denoted by Δe is asfollows

Δe edlowast

minus eclowast

Aβc(4p minus 2c minus s minus v) minus Ac(s + v minus 2c) minus (v minus s)(1 + β) (a minus bp)c + 1 minus β21113872 1113873(p minus c)1113960 1113961

2c 2Ac +(1 + β)2(v minus s)1113960 1113961 (B6)

Similarly if Aβc(4p minus 2c minus s minus v)gtAc(s + v minus 2c)

+(v minus s)(1 + β)[(a minus bp)c + (1 minus β2)(p minus c)] we haveΔe edlowast minus eclowast gt 0 In contrast ifAβc(4p minus 2c minus s minus v)ltAc(s + v minus 2c)+

(v minus s)(1 + β)[(a minus bp)c + (1 minus β2)(p minus c)] we haveΔe edlowast minus eclowast lt 0

Upon solving the first-order derivative of Δq and Δe withrespect to β respectively we obtained

zΔ q

zβ

(1 + β)(p minus c)

c (B7)

zΔ e

zβ4(p minus c)(v minus s)2 β3 + 4β2 + 6β + 41113872 1113873β +(a minus bp)c(1 + β)2(v minus s)2 + Ac(v minus s)(1 + β)2 2(p minus c) +v + s) + 2(Ac)2(4p minus c minus s minus v)1113960 1113961

2c (1 + β)2(v minus s) + 2Ac1113960 11139612

(B8)

According to the assumptions pgt vgt cgt sgt 0 β isin (0 1)cgt 0 and a minus bpgt 0 (zΔ qzβ)gt 0 and (zΔ ezβ)gt 0 can beproved Hence the gaps increased as a function with respectto β

Proof of Proposition 4 e proof for Proposition 4 is similarto Proposition 2 Herein we took first-order and second-order partial derivatives of optimal wholesale price wsclowast withrespect to β μ and ϕ in equation (18) respectively esolving results are as follows

zwsclowast

zβ

[(1 + 2μ)ϕ minus μ](p minus c)

(1 + 2μ)(1 + β)2

zwsclowast

zμ minus

(p minus c)β(1 + β)(1 + 2μ)2

zwsclowast

zϕ minus

p minus c

1 + β

(B9)

(zwsclowastzβ) if ϕgt (μ(1 + 2μ)) is more than 0 otherwiseϕgt (μ(1 + 2μ)) is less than 0 us wsc could be increasingwith respect to β nevertheless decreasing with respect to βBesides according to the assumptions pgt vgt cgt sgt 0β isin (0 1) cgt 0 and a minus bpgt 0 it can be evidently foundthat (zwsclowastzμ)gt 0 and (zwsclowastzϕ)gt 0

Data Availability

e data used to support the findings of this study are in-cluded within the article

Conflicts of Interest

e authors declare no conflicts of interest

Acknowledgments

is work was supported by a major project financed by theNational Social Science Fund of China (Approval nos71461009 71662011 and 71761015)

Complexity 15

References

[1] Y Ren D C Croson and R T A Croson ldquoe overconfidentnewsvendorrdquo Journal of the Operational Research Societyvol 68 no 5 pp 496ndash506 2017

[2] T H Cui and P Mallucci ldquoFairness ideals in distributionchannelsrdquo Journal of Marketing Research vol 53 no 6pp 969ndash987 2016

[3] K Chen X Wang M Huang and W-K Ching ldquoSalesforcecontract design joint pricing and production planning withasymmetric overconfidence sales agentrdquo Journal of Industrialamp Management Optimization vol 13 no 2 pp 873ndash8992017

[4] X X Zheng Z Liu K W Li et al ldquoCooperative game ap-proaches to coordinating a three-echelon closed-loop supplychain with fairness concernsrdquo International Journal of Pro-duction Economic vol 212 pp 92ndash110 2018

[5] China Internet Watch False Prosperity behind Double 11Refund Rate Increased to 69 2014 httpswwwchinainternetwatchcom11643false-prosperity-behind-double-11-refund-rate-increased-to-69

[6] C L Lu ldquoPampGrsquos collaborative supply chain transformationrdquoEnterprise Management vol 10 pp 79-80 2016

[7] M Li N C Petruzzi and J Zhang ldquoOverconfident competingnewsvendorsrdquo Management Science vol 63 no 8pp 2637ndash2646 2016

[8] T Nie ldquoDual-fairness supply chain with quantity discountcontractsrdquo European Journal of Operational Researchvol 258 no 2 pp 491ndash500 2017

[9] X Du and B Jiang ldquoSignaling through price and quality toconsumers with fairness concernsrdquo Journal of MarketingResearch vol 53 no 6 pp 988ndash1000 2016

[10] V Kurz A Orland and K Posadzy ldquoFairness versus effi-ciency how procedural fairness concerns affect coordina-tionrdquo Experimental Economics vol 21 no 3 pp 601ndash6262018

[11] S Villa and J A Castantildeeda ldquoTransshipments in supplychains a behavioral investigationrdquo European Journal of Op-erational Research vol 269 no 2 pp 715ndash729 2018

[12] B Gu Y Fu and Y Li ldquoFresh-keeping effort and channelperformance in a fresh product supply chain with loss-averseconsumersrsquo returnsrdquo Mathematical Problems in Engineeringvol 2018 Article ID 4717094 20 pages 2018

[13] D Croson R Croson and Y Ren How to Manage anOverconfident Newsvendor 2008 httpcbeesutdallasedupapersCrosonRenCmsonMS2008

[14] M Li ldquoOverconfident distribution channelsrdquo Production andOperations Management vol 28 no 6 pp 1347ndash1365 2019

[15] W Liu D Wang O Tang and D Zhu ldquoe impacts oflogistics service integratorrsquos overconfidence behaviour onsupply chain decision under demand surgerdquo EuropeanJournal of Industrial Engineering vol 12 no 4 pp 558ndash5972018

[16] L Xu X Shi P Du K Govindan and Z Zhang ldquoOptimi-zation on pricing and overconfidence problem in a duopo-listic supply chainrdquo Computers amp Operations Researchvol 101 pp 162ndash172 2019

[17] Y Li and M Shan ldquoAdvance selling decisions with over-confident consumersrdquo Journal of Industrial and ManagementOptimization vol 2 no 3 pp 891ndash905 2016

[18] T H Cui J S Raju and Z J Zhang ldquoFairness and channelcoordinationrdquo Management Science vol 53 no 8pp 1303ndash1314 2007

[19] X Wu and J A Niederhoff ldquoFairness in selling to thenewsvendorrdquo Production and Operations Managementvol 23 no 11 pp 2002ndash2022 2014

[20] Z Chen and S-I Ivan Su ldquoSocial welfare maximization withthe least subsidy photovoltaic supply chain equilibrium andcoordination with fairness concernrdquo Renewable Energyvol 132 pp 1332ndash1347 2019

[21] W Liu D Wang X Shen X Yan and W Wei ldquoe impactsof distributional and peer-induced fairness concerns on thedecision-making of order allocation in logistics service supplychainrdquo Transportation Research Part E Logistics and Trans-portation Review vol 116 pp 102ndash122 2018

[22] L Zhang H Zhou Y Liu and R Lu ldquoOptimal environmentalquality and price with consumer environmental awarenessand retailerrsquos fairness concerns in supply chainrdquo Journal ofCleaner Production vol 213 pp 1063ndash1079 2019

[23] X LWang S Q Hu and X B Liu ldquoSupply coordination withoption contract considering overconfidence under randomdemand and yieldsrdquo Computer Integrated ManufacturingSystem vol 24 no 11 pp 2898ndash2908 2018

[24] W Jiang and J Liu ldquoInventory financing with overconfidentsupplier based on supply chain contractrdquo MathematicalProblems in Engineering vol 2018 Article ID 505438712 pages 2018

[25] P Ma H Wang and J Shang ldquoContract design for two-stagesupply chain coordination integrating manufacturer-qualityand retailer-marketing effortsrdquo International Journal ofProduction Economics vol 146 no 2 pp 745ndash755 2013

[26] J Liu W F Jiang and Y P Tang ldquoe supply chain coor-dination based on carbon reduction of overconfident man-ufacturerrdquo Journal of Jiangxi Normal University (NaturalScience) vol 42 no 2 pp 203ndash207 2018

[27] R Dubey N Altay and C Blome ldquoSwift trust and com-mitment the missing links for humanitarian supply chaincoordinationrdquo Annals of Operations Research vol 283 no 1-2 pp 159ndash177 2019

[28] C Cui Z Feng and C Tan ldquoCredibilistic loss aversion nashequilibrium for bimatrix games with triangular fuzzy payoffsrdquoComplexity vol 2018 Article ID 7143586 16 pages 2018

[29] E Fehr and KM Schmidt ldquoA theory of fairness competitionand cooperationrdquo e Quarterly Journal of Economicsvol 114 no 3 pp 817ndash868 1999

[30] Z Qin and J Yang ldquoAnalysis of a revenue-sharing contract insupply chain managementrdquo International Journal of LogisticsResearch and Applications vol 11 no 1 pp 17ndash29 2008

[31] S-Z Bai and J Ni ldquoResearch on buy-back contract with saleseffort cost sharingrdquo in Proceedings of the 2009 Chinese Controland Decision Conference IEEE Guilin China pp 1917ndash1921June 2009

[32] E Karagozoglu and K Keskin ldquoTime-varying fairness con-cerns delay and disagreement in bargainingrdquo Journal ofEconomic Behavior and Organization vol 147 pp 115ndash1282018

16 Complexity

Figure 5 illustrates the impact of overconfidence andfairness concern on the manufacturerrsquos profit at is theoptimal manufacturerrsquos profit is an increasing function of βbut a decreasing function of μ According to Figure 3 thegraph of themanufacturerrsquos profit was similar to the graph ofthe wholesale price which indicated that the manufacturerrsquosprofit was firmly related to wholesale price Figure 5 alsoillustrates the effect of the cognitive bias β on the manu-facturerrsquos decreasing profit

52 Effect of Overconfidence and Fairness Concern on SupplyChain Coordination As can be seen from Section 51 thereexist deviations between the optimal decisions under the

benchmark and the decisions in the presence of the retailerrsquoscognitive biases erefore the buyback contract withpromotion cost sharing was necessary to coordinate therevenue of supply chain members and optimize the per-formance of the supply chain

Here we discussed the validity of the contract When theoptimal order quantity and sales effort of the retailer underthe contract were the same as the optimal decision variablesin the centralized supply chain ie qsc qclowast 32 andesc eclowast 16 the buyback contract with promotion costsharing achieved supply chain coordination First the effectof overconfidence and fairness concern (β and μ) on contractfactors wsrlowast rlowast and ϕ is shown in Figures 6 and 7 andTable 1 For clearly analyzing the effect of β and μ on wsrlowast

Degree of sales effort in centralized supply chain

01 02 03 04 05 06 07 08 09 10β

μ = 012

4

6

8

10

12

14

16

18

Deg

ree o

f sal

es ef

fort

Figure 2 e effect of the cognitive biases on optimal sales effort

01 02 03 04 05 06 07 08 09 10β

-μ = 0--μ = 1-μ = 2

4

5

6

7

8

9

10

11

12

13

14

Who

lesa

le p

rice

Figure 3 e effect of the cognitive biases on the optimal wholesale price

Complexity 9

and rlowast we let ϕ 1 in equations (18) and (19) which meantthat the manufacturer proposed the buyback contract onlyand the retailer bore all sales effort costs

According to Figure 6 it was revealed that the optimalwholesales price is a strictly increasing function of theoverconfidence level β but a strictly decreasing function ofthe degree of fairness concern μ which confirmed the resultof Proposition 4 Similarly Figure 7 highlights that theoptimal buyback price sharply increased with β but de-creased with μ Moreover in the buyback contract wsrlowast wasalways greater than or equal to 4 and rlowast was always greaterthan or equal to 2 in different cognitive bias levels Hence

the optimal buyback price rlowast satisfied the assumptions 2le r Note that the above characteristics influencedcognitive biases hence contract factors wsrlowast and rlowast werealso correct when ϕ is equal to other values (such asϕ 03 05 08) under conditions in which the buybackcontract had promotion cost sharing

However data from Figure 7 can be compared with thedata in Figure 6 which showed rsrlowast is more significant thanwsrlowast when β was within a specific range (for example04le βlt 1) which did not match the assumption rltwsrerefore the buyback contract did not achieve the coor-dination of the behavioral supply chain

01 02 03 04 05 06 07 08 09 10β

-μ = 0--μ = 1-μ = 2

0

20

40

60

80

100

120

Reta

iler e

xpec

ted

utili

ty

Figure 4 e effect of the cognitive biases on retailer utility

01 02 03 04 05 06 07 08 09 10β

-μ = 0--μ = 1-lowastμ = 2

0

20

40

60

80

100

120

140

160

180

200

Man

ufac

ture

r pro

fit

Figure 5 e effect of the cognitive biases on manufacturer profit

10 Complexity

As described in Section 4 if the supply chain membersaccepted the buyback contract with promotion cost sharingthe assumptions 1113937

scm gemax 1113937

dm and U(1113937

scr )gemaxU(1113937

dr )

had to be satisfied Also according to the assumptions thevalue ϕ must satisfy 1113937

scm gemax 1113937

dm and

U(1113937scr )gemaxU(1113937

dr ) us we kept the values of param-

eters to be the same (ie a 50 θ sim U(minus 30 30) c 4p 20 v 7 s 2 b 2 and c 1) to capture a likelydomain of ϕ

As can be seen from Table 2 the overconfident level βincreased the upper and lower limits of ϕ which had a clearupward trend In contrast the upper and lower limits of ϕ

showed a definite declining trend as the degree of fairnessconcern μ increased ese trends illustrated that in order tocoordinate the revenue of both parties and optimize theperformance of the supply chain the manufacturer needs tobear less promotional costs as β increased but bear morepromotional costs as μ value increased Besides when β and μincreased the interval of sharing cost portion ϕ shrunk whichindicated that the buyback contract with promotional costsharing had certain flexibility to coordinate the revenue be-tween supply chain members However the flexibility couldbe weakened with the rise of the cognitive bias level Forsupply chain members the possibility of negotiation or

01 02 03 04 05 06 07 08 09 10β

-μ = 0--μ = 1-μ = 2

4

5

6

7

8

9

10

11

12

Who

lesa

le p

rice

Figure 6 e effect of the cognitive biases on wholesale price wsrlowast

01 02 03 04 05 06 07 08 09 10β

-μ = 0--μ = 1-μ = 2

2

4

6

8

10

12

14

16

18

20

22

Buyb

ack

pric

e

Figure 7 e effect of the cognitive biases on buyback price rlowast

Complexity 11

cooperation between manufacturers and retailers also de-creased with an increase in retailersrsquo cognitive bias levelWhen the overconfident level was more than 081 the re-tailerrsquos utility from decentralized decisions was greater thanthe utility in the contract environment wherein the twoparties could not reach a cooperation agreement us thesupply chain could not achieve coordination as different levelsof cognitive bias under the contract had the same influence ondecision variables profit and utility Table 2 shows a clearpicture wherein μ is equal to 1 and β falls within the range of(0 08) which were used to analyze the effect of the cognitivebiases on the coordination of the supply chain

e results of the numerical simulation in Table 3 in-dicate that the contract factors ie ϕ r wholesale price theprofit and utility of members existed as flexible intervalswhen β isin (0 08) and μ 1 Besides the order quantity qsc

and the degree of sales effort esc under the contract are equalto qclowast and eclowast in a centralized supply chain respectivelyMeanwhile the buyback price r was less than the wholesaleprice wsc and higher than the unit salvage value s whichsatisfied the constrains sle rltwsclowast us the buybackcontract with promotion cost could achieve the coordinationof the two-echelon supply chain Also the buyback price r

was positively correlated with the portion ϕ of promotioncost sharing Notably the wholesale price wsc dropped as ϕincreased when β was in the range of (0 02) in Table 3However the situation reversed when it was in the range of(02 08) which was inconsistent since the manufacturerlowered the wholesale price when the retailer paid morepromotional costs in the contract e reason for such anobservation was that the wholesale price was not related to ϕbut positively related to the overconfident level β when thedegree of fairness concern was moderate Furthermore theretailer accepted the contract designed by manufacturerssince the term U(1113937

scr )gemaxU(1113937

dr ) was guaranteed in

Table 3

6 Conclusions

61MainConclusions In this paper we analyzed the effect ofcognitive bias on equilibrium solutions including thewholesale price sales price sales effort and expected profitsunder demand uncertainty by introducing both overconfi-dence and fairness concern into the two-echelon supply chaine most important conclusion from the study was that forretailers with the two cognitive biases overconfidence waspositive and the main bias that affected supply chain decisionsand the retailersrsquo utility benefits were not only due to over-confidence but also fairness concern For manufacturersoverconfidence could increase the wholesale price and profitHowever the fairness concern had a negative effect on themFurthermore the decisions of the retailer in the decentralizedsupply chain deviated from the optimal decisions in thecentralized supply chainmodel Especially when the thresholdof overconfidence was more than a fixed value the decisionvariables under decentralized decision-making were actuallyhigher than the optimal variables under centralized decision-making Also the buyback contract with promotional costsharing achieved behavioral supply chain coordination Incontrast the promotional cost-sharing portion and buybackprice both benefited from overconfidence but fairness concernhad a detrimental effect on them However as the retailersrsquooverconfidence level increased and fairness concern was equalto 1 the flexibility of contract coordination was weakenedWhen the overconfidence level wasmore than 08 the contractbecame invalid and members were not able to cooperate

62 Management Insights is paper provided some in-teresting managerial insights for decision makers in thedecentralized supply chain wherein retailers had two pref-erences First overconfidence was beneficial to them interms of decision variables and yet the fairness concern

Table 2 e effect of the cognitive biases on the feasible domain of promotion cost-sharing portion ϕ

μβ

0 02 04 06 08 0810 [02884 04976] [03683 05587] [04528 06199] [06343 07754] [07835 08669] mdash1 [02545 04741] [03310 05331] [04167 05908] [06207 07516] [07800 08560] mdash2 [02352 04600] [03101 05156] [04034 05719] [06094 07321] [07736 08478] mdash

Table 3 e effect of the cognitive biases on the coordination of the supply chain

μ 1β

0 02 04 06 08qsc 32 32 32 32 32esc 16 16 16 16 16ϕ [02545 04741] [03310 05331] [04167 05908] [06207 07516] [07800 08560]r [2100 35363] [28268 39188] [30555 42637] [36209 44215] [40485 49164]wsc [49205 53687] [49016 53522] [54092 69568] [62513 72626] [80407 87453]1113937

dm 164103 269591 461996 772619 1112545

U(1113937scm) [168722 217850] [269600 336882] [463175 500938] [772619 814300] [1112545 1120376]

1113937dr 20333 352396 495161 632422 765317

U(1113937scr ) [20333 243043] [352396 400595] [495161 525161] [632422 657581] [765317 784632]

12 Complexity

only damaged the wholesale price Second the two behav-ioral preferences were both conducive to utilities for theretailer but not necessarily beneficial for the manufacturer toreceive profit and overconfidence helped to increasemanufacturersrsquo profit which led to nonoptimal decisionsthat dragged down the performance of the two-echelonsupply chain Finally for the two behavioral preferences ofthe retailer the manufacturer assessed the degree of over-confidence and fairness concern (especially the overconfi-dence) via historical data and the current order quantity orother indicators In contrast factual data indicated that twobehavioral preferences were in the right range (such asβ isin (0 08) and μ isin (0 2)) e manufacturers introducedthe buyback contract with promotional cost sharing into thepractice of pricing and ordered decentralized supply chainsFurthermore they designed a reasonable promotional cost-sharing portion and buyback price thereby achieving anoptimized decision and utility In practice the manufac-turers could analyze whether the sellers had two behavioralpreferences based on the historical data by cooperating withdownstream enterprises ie if two behavioral preferencesexisted they referred to the papersrsquo conclusions to redesign areasonable contract mechanism

63 Future Research Since cognitive biases could existamong all members of supply chains it is necessary toexplore the manufacturersrsquo cognitive biases and analyze thesupply chain coordination problem in the presence of thetwo behavior preferences Besides the overconfidence andfairness concern considered in our centralized decision are afuture research direction

Appendix

A Decentralized Supply Chain Model

In a decentralized supply chain consisting of a rationalmanufacturer and a retailer with cognitive biases maximumprofit or utility is the only goal e backward inductivemethod can be applied to solve a Stackelberg game whereinthe manufacturer is the leader and the retailer is a followere proof of the equilibrium solutions in equations (12)ndash(14)is as follows

e profit of the manufacturer and expected utility ofretailers are as follows respectively

wmax

1113945

d

m

(w minus c)q (A1)

maxqe

U 1113945d

r

⎞⎠ (1 + μ) p[a minus bp +(1 + β)e] + s 1113946θ2

minus Aq minus a minus bp +(1 + β)e + θ1( 1113857f θ1( 1113857( 1113857dθ11113896⎛⎝

minus v 1113946A

θ2a minus bp +(1 + β)e + θ1 minus q( 1113857f θ1( 1113857( 1113857dθ1 minus wq minus

12

ce21113897 minus μ(w minus c)q

(A2)

Solving the first-order partial derivatives of the retailerrsquosexpected utility concerning qd and ed which equal to 0 weobtained the following equations

zU( 1113937dr 1113873

zq (1 + μ) v minus w + F θ2( 1113857(s minus v)1113858 1113859 minus μ(w minus c) 0

(A3)

zU( 1113937dr 1113873

ze (1 + β) p minus v +(v minus s)F θ2( 11138571113858 1113859 minus ce 0

(A4)

Hessian matrix

H(q e) (s minus v)f θ2( 1113857 (v minus s)(1 + β)f θ2( 1113857

(v minus s)f θ2( 1113857 (s minus v)(1 + β)f θ2( 1113857 minus c1113888 1113889

(A5)

According to the assumptions vgt sgt 0 f(θ2)gt 0 andβ isin [0 1) us the Hessian matrix of U(1113937

dr ) is a negative

definite for q and e which meant that the optimal orderquantity and sales effort of retailers existed in the decen-tralized supply chain erefore we obtained the followingformula of the retailersrsquo order quantity and sales effort byjointly solving equations (A3) and (A4)

qd

(a minus bp)(v minus s)(1 + μ)c +(1 + β)2(v minus s)[(1 + μ)(p minus w) + μc] + Ac[(s + v)(1 + μ) + 2μc minus w]

(v minus s)(1 + μ)c (A6)

ed

(1 + β)[(1 + μ)p + μc minus (1 + 2μ)]w

(1 + μ)c (A7)

Complexity 13

Substituting equations (A6) into (A1) we attained

wmax

1113945

d

m

(w minus c) (a minus bp)(v minus s)(1 + μ)c +(1 + β)2(v minus s)[(1 + μ)p + μc] + Ac[(s + v)(1 + μ) + 2μc] minus (1 + 2μ)w 2Ac +(1 + β)2(v minus s)1113960 11139611113966 1113967

(v minus s)(1 + μ)c

(A8)

Taking into account the first-order condition of equation(A7) with respect to w rounded off to 0 we obtained`

z1113937dm

ze

(a minus bp)(v minus s)(1 + μ)c +(1 + β)2(v minus s)[(1 + μ)p + μc] + Ac[(s + v)(1 + μ) + 2μc] minus (2w minus c)(1 + 2μ) 2Ac +(1 + β)2(v minus s)1113960 1113961

(v minus s)(1 + μ)c

(A9)

Solving equation (A8) we found that the optimalwholesale price of the manufacturer is as follows

wdlowast

(a minus bp)(1 + μ)(v minus s)c +(v minus s)(1 + β)2[(1 + μ)p +(1 + 3μ)c] + Ac[(s + v)(1 + μ) + 2(1 + 3μ)c]

2(1 + 2μ) 2Ac +(v minus s)(1 + β)21113960 1113961 (A10)

By substituting equation (A9) into equations (A6) and(A7) we attained the optimal order quantity and sales effortwhich are displayed in equations (13) and (14)

B Propositions

Proof of Proposition 1 We solved the first derivatives ofoptimal order quantity qdlowast and sales effort edlowast with respect

to β in equations (13) and (14) respectively Meanwhiletaking the second-order derivative of qdlowast with respect to βwe obtained

zqdlowast

zβ

(1 + β)(p minus c)

c

z2qdlowast

zβ2

p minus c

c (B1)

zedlowast

zβ4(p minus c)(v minus s)2β β3 + 4β2 + 6β + 41113872 1113873 +(a minus bp)c(1 + β)2(v minus s)2 + Ac(v minus s)(1 + β)2 (2(p minus c) + v + s) + 2(Ac)2(4p minus c minus s minus v)1113960 1113961

2c (1 + β)2(v minus s) + 2Ac1113960 11139612

(B2)

Since the second derivative of edlowast with respect to β wasvery complicated there was no need to solve it According tothe assumptions pgt vgt cgt sgt 0 β isin (0 1) a minus bpgt 0 andcgt 0 so (zqdlowastzβ)gt 0 (z2qdlowastzβ2)gt 0 and (zedlowastzβ)gt 0

Proof of Proposition 2 Based on the process above we tookthe first-order and second-order partial derivatives of op-timal wholesale price wdlowast with respect to β and μ in equation(12) respectively

zwdlowast

zβ

c(v minus s)(1 + μ)(1 + β)[(a minus bp)(v minus s) + A(2p minus s minus v)]

(1 + 2μ) (1 + β)2(v minus s) + 2Ac1113960 11139612 (B3)

zwdlowast

zμ minus

(a minus bp)(v minus s)c +(p minus c)(v minus s)(1 + β)2 + Ac(s + v minus 2c)1113960 1113961

2(1 + 2μ)2 (v minus s)(1 + β)2 + 2Ac1113960 1113961 (B4)

14 Complexity

According to the optimal result of order quantity inequation (13) and the assumption qgt 0 the condition s +

v minus 2cgt 0 is found According to the assumptions we had(zwdlowastzβ)gt 0 and (zwdlowastzμ)lt 0

Proof of Proposition 3 e gap between qdlowast and qclowastdenoted by Δq is as follows

Δq qdlowast

minus qclowast

(p minus c)(v minus s) (1 + β)2 minus 21113960 1113961 minus Ac(s + v minus 2c) minus (a minus bp)(v minus s)c

2c(v minus s) (B5)

If (p minus c)(v minus s)[(1 + β)2 minus 2] is more thanAc(s + v minus 2c) + (a minus bp)(v minus s)c we have Δqgt 0 ieqdlowast gt qclowast In contrast if (p minus c)(v minus s)[(1 + β)2 minus 2] is less

than Ac(s + v minus 2c) + (a minus bp)(v minus s)c Δqlt 0 ieqdlowast lt qclowast

e gap between edlowast and eclowast denoted by Δe is asfollows

Δe edlowast

minus eclowast

Aβc(4p minus 2c minus s minus v) minus Ac(s + v minus 2c) minus (v minus s)(1 + β) (a minus bp)c + 1 minus β21113872 1113873(p minus c)1113960 1113961

2c 2Ac +(1 + β)2(v minus s)1113960 1113961 (B6)

Similarly if Aβc(4p minus 2c minus s minus v)gtAc(s + v minus 2c)

+(v minus s)(1 + β)[(a minus bp)c + (1 minus β2)(p minus c)] we haveΔe edlowast minus eclowast gt 0 In contrast ifAβc(4p minus 2c minus s minus v)ltAc(s + v minus 2c)+

(v minus s)(1 + β)[(a minus bp)c + (1 minus β2)(p minus c)] we haveΔe edlowast minus eclowast lt 0

Upon solving the first-order derivative of Δq and Δe withrespect to β respectively we obtained

zΔ q

zβ

(1 + β)(p minus c)

c (B7)

zΔ e

zβ4(p minus c)(v minus s)2 β3 + 4β2 + 6β + 41113872 1113873β +(a minus bp)c(1 + β)2(v minus s)2 + Ac(v minus s)(1 + β)2 2(p minus c) +v + s) + 2(Ac)2(4p minus c minus s minus v)1113960 1113961

2c (1 + β)2(v minus s) + 2Ac1113960 11139612

(B8)

According to the assumptions pgt vgt cgt sgt 0 β isin (0 1)cgt 0 and a minus bpgt 0 (zΔ qzβ)gt 0 and (zΔ ezβ)gt 0 can beproved Hence the gaps increased as a function with respectto β

Proof of Proposition 4 e proof for Proposition 4 is similarto Proposition 2 Herein we took first-order and second-order partial derivatives of optimal wholesale price wsclowast withrespect to β μ and ϕ in equation (18) respectively esolving results are as follows

zwsclowast

zβ

[(1 + 2μ)ϕ minus μ](p minus c)

(1 + 2μ)(1 + β)2

zwsclowast

zμ minus

(p minus c)β(1 + β)(1 + 2μ)2

zwsclowast

zϕ minus

p minus c

1 + β

(B9)

(zwsclowastzβ) if ϕgt (μ(1 + 2μ)) is more than 0 otherwiseϕgt (μ(1 + 2μ)) is less than 0 us wsc could be increasingwith respect to β nevertheless decreasing with respect to βBesides according to the assumptions pgt vgt cgt sgt 0β isin (0 1) cgt 0 and a minus bpgt 0 it can be evidently foundthat (zwsclowastzμ)gt 0 and (zwsclowastzϕ)gt 0

Data Availability

e data used to support the findings of this study are in-cluded within the article

Conflicts of Interest

e authors declare no conflicts of interest

Acknowledgments

is work was supported by a major project financed by theNational Social Science Fund of China (Approval nos71461009 71662011 and 71761015)

Complexity 15

References

[1] Y Ren D C Croson and R T A Croson ldquoe overconfidentnewsvendorrdquo Journal of the Operational Research Societyvol 68 no 5 pp 496ndash506 2017

[2] T H Cui and P Mallucci ldquoFairness ideals in distributionchannelsrdquo Journal of Marketing Research vol 53 no 6pp 969ndash987 2016

[3] K Chen X Wang M Huang and W-K Ching ldquoSalesforcecontract design joint pricing and production planning withasymmetric overconfidence sales agentrdquo Journal of Industrialamp Management Optimization vol 13 no 2 pp 873ndash8992017

[4] X X Zheng Z Liu K W Li et al ldquoCooperative game ap-proaches to coordinating a three-echelon closed-loop supplychain with fairness concernsrdquo International Journal of Pro-duction Economic vol 212 pp 92ndash110 2018

[5] China Internet Watch False Prosperity behind Double 11Refund Rate Increased to 69 2014 httpswwwchinainternetwatchcom11643false-prosperity-behind-double-11-refund-rate-increased-to-69

[6] C L Lu ldquoPampGrsquos collaborative supply chain transformationrdquoEnterprise Management vol 10 pp 79-80 2016

[7] M Li N C Petruzzi and J Zhang ldquoOverconfident competingnewsvendorsrdquo Management Science vol 63 no 8pp 2637ndash2646 2016

[8] T Nie ldquoDual-fairness supply chain with quantity discountcontractsrdquo European Journal of Operational Researchvol 258 no 2 pp 491ndash500 2017

[9] X Du and B Jiang ldquoSignaling through price and quality toconsumers with fairness concernsrdquo Journal of MarketingResearch vol 53 no 6 pp 988ndash1000 2016

[10] V Kurz A Orland and K Posadzy ldquoFairness versus effi-ciency how procedural fairness concerns affect coordina-tionrdquo Experimental Economics vol 21 no 3 pp 601ndash6262018

[11] S Villa and J A Castantildeeda ldquoTransshipments in supplychains a behavioral investigationrdquo European Journal of Op-erational Research vol 269 no 2 pp 715ndash729 2018

[12] B Gu Y Fu and Y Li ldquoFresh-keeping effort and channelperformance in a fresh product supply chain with loss-averseconsumersrsquo returnsrdquo Mathematical Problems in Engineeringvol 2018 Article ID 4717094 20 pages 2018

[13] D Croson R Croson and Y Ren How to Manage anOverconfident Newsvendor 2008 httpcbeesutdallasedupapersCrosonRenCmsonMS2008

[14] M Li ldquoOverconfident distribution channelsrdquo Production andOperations Management vol 28 no 6 pp 1347ndash1365 2019

[15] W Liu D Wang O Tang and D Zhu ldquoe impacts oflogistics service integratorrsquos overconfidence behaviour onsupply chain decision under demand surgerdquo EuropeanJournal of Industrial Engineering vol 12 no 4 pp 558ndash5972018

[16] L Xu X Shi P Du K Govindan and Z Zhang ldquoOptimi-zation on pricing and overconfidence problem in a duopo-listic supply chainrdquo Computers amp Operations Researchvol 101 pp 162ndash172 2019

[17] Y Li and M Shan ldquoAdvance selling decisions with over-confident consumersrdquo Journal of Industrial and ManagementOptimization vol 2 no 3 pp 891ndash905 2016

[18] T H Cui J S Raju and Z J Zhang ldquoFairness and channelcoordinationrdquo Management Science vol 53 no 8pp 1303ndash1314 2007

[19] X Wu and J A Niederhoff ldquoFairness in selling to thenewsvendorrdquo Production and Operations Managementvol 23 no 11 pp 2002ndash2022 2014

[20] Z Chen and S-I Ivan Su ldquoSocial welfare maximization withthe least subsidy photovoltaic supply chain equilibrium andcoordination with fairness concernrdquo Renewable Energyvol 132 pp 1332ndash1347 2019

[21] W Liu D Wang X Shen X Yan and W Wei ldquoe impactsof distributional and peer-induced fairness concerns on thedecision-making of order allocation in logistics service supplychainrdquo Transportation Research Part E Logistics and Trans-portation Review vol 116 pp 102ndash122 2018

[22] L Zhang H Zhou Y Liu and R Lu ldquoOptimal environmentalquality and price with consumer environmental awarenessand retailerrsquos fairness concerns in supply chainrdquo Journal ofCleaner Production vol 213 pp 1063ndash1079 2019

[23] X LWang S Q Hu and X B Liu ldquoSupply coordination withoption contract considering overconfidence under randomdemand and yieldsrdquo Computer Integrated ManufacturingSystem vol 24 no 11 pp 2898ndash2908 2018

[24] W Jiang and J Liu ldquoInventory financing with overconfidentsupplier based on supply chain contractrdquo MathematicalProblems in Engineering vol 2018 Article ID 505438712 pages 2018

[25] P Ma H Wang and J Shang ldquoContract design for two-stagesupply chain coordination integrating manufacturer-qualityand retailer-marketing effortsrdquo International Journal ofProduction Economics vol 146 no 2 pp 745ndash755 2013

[26] J Liu W F Jiang and Y P Tang ldquoe supply chain coor-dination based on carbon reduction of overconfident man-ufacturerrdquo Journal of Jiangxi Normal University (NaturalScience) vol 42 no 2 pp 203ndash207 2018

[27] R Dubey N Altay and C Blome ldquoSwift trust and com-mitment the missing links for humanitarian supply chaincoordinationrdquo Annals of Operations Research vol 283 no 1-2 pp 159ndash177 2019

[28] C Cui Z Feng and C Tan ldquoCredibilistic loss aversion nashequilibrium for bimatrix games with triangular fuzzy payoffsrdquoComplexity vol 2018 Article ID 7143586 16 pages 2018

[29] E Fehr and KM Schmidt ldquoA theory of fairness competitionand cooperationrdquo e Quarterly Journal of Economicsvol 114 no 3 pp 817ndash868 1999

[30] Z Qin and J Yang ldquoAnalysis of a revenue-sharing contract insupply chain managementrdquo International Journal of LogisticsResearch and Applications vol 11 no 1 pp 17ndash29 2008

[31] S-Z Bai and J Ni ldquoResearch on buy-back contract with saleseffort cost sharingrdquo in Proceedings of the 2009 Chinese Controland Decision Conference IEEE Guilin China pp 1917ndash1921June 2009

[32] E Karagozoglu and K Keskin ldquoTime-varying fairness con-cerns delay and disagreement in bargainingrdquo Journal ofEconomic Behavior and Organization vol 147 pp 115ndash1282018

16 Complexity

and rlowast we let ϕ 1 in equations (18) and (19) which meantthat the manufacturer proposed the buyback contract onlyand the retailer bore all sales effort costs

According to Figure 6 it was revealed that the optimalwholesales price is a strictly increasing function of theoverconfidence level β but a strictly decreasing function ofthe degree of fairness concern μ which confirmed the resultof Proposition 4 Similarly Figure 7 highlights that theoptimal buyback price sharply increased with β but de-creased with μ Moreover in the buyback contract wsrlowast wasalways greater than or equal to 4 and rlowast was always greaterthan or equal to 2 in different cognitive bias levels Hence

the optimal buyback price rlowast satisfied the assumptions 2le r Note that the above characteristics influencedcognitive biases hence contract factors wsrlowast and rlowast werealso correct when ϕ is equal to other values (such asϕ 03 05 08) under conditions in which the buybackcontract had promotion cost sharing

However data from Figure 7 can be compared with thedata in Figure 6 which showed rsrlowast is more significant thanwsrlowast when β was within a specific range (for example04le βlt 1) which did not match the assumption rltwsrerefore the buyback contract did not achieve the coor-dination of the behavioral supply chain

01 02 03 04 05 06 07 08 09 10β

-μ = 0--μ = 1-μ = 2

0

20

40

60

80

100

120

Reta

iler e

xpec

ted

utili

ty

Figure 4 e effect of the cognitive biases on retailer utility

01 02 03 04 05 06 07 08 09 10β

-μ = 0--μ = 1-lowastμ = 2

0

20

40

60

80

100

120

140

160

180

200

Man

ufac

ture

r pro

fit

Figure 5 e effect of the cognitive biases on manufacturer profit

10 Complexity

As described in Section 4 if the supply chain membersaccepted the buyback contract with promotion cost sharingthe assumptions 1113937

scm gemax 1113937

dm and U(1113937

scr )gemaxU(1113937

dr )

had to be satisfied Also according to the assumptions thevalue ϕ must satisfy 1113937

scm gemax 1113937

dm and

U(1113937scr )gemaxU(1113937

dr ) us we kept the values of param-

eters to be the same (ie a 50 θ sim U(minus 30 30) c 4p 20 v 7 s 2 b 2 and c 1) to capture a likelydomain of ϕ

As can be seen from Table 2 the overconfident level βincreased the upper and lower limits of ϕ which had a clearupward trend In contrast the upper and lower limits of ϕ

showed a definite declining trend as the degree of fairnessconcern μ increased ese trends illustrated that in order tocoordinate the revenue of both parties and optimize theperformance of the supply chain the manufacturer needs tobear less promotional costs as β increased but bear morepromotional costs as μ value increased Besides when β and μincreased the interval of sharing cost portion ϕ shrunk whichindicated that the buyback contract with promotional costsharing had certain flexibility to coordinate the revenue be-tween supply chain members However the flexibility couldbe weakened with the rise of the cognitive bias level Forsupply chain members the possibility of negotiation or

01 02 03 04 05 06 07 08 09 10β

-μ = 0--μ = 1-μ = 2

4

5

6

7

8

9

10

11

12

Who

lesa

le p

rice

Figure 6 e effect of the cognitive biases on wholesale price wsrlowast

01 02 03 04 05 06 07 08 09 10β

-μ = 0--μ = 1-μ = 2

2

4

6

8

10

12

14

16

18

20

22

Buyb

ack

pric

e

Figure 7 e effect of the cognitive biases on buyback price rlowast

Complexity 11

cooperation between manufacturers and retailers also de-creased with an increase in retailersrsquo cognitive bias levelWhen the overconfident level was more than 081 the re-tailerrsquos utility from decentralized decisions was greater thanthe utility in the contract environment wherein the twoparties could not reach a cooperation agreement us thesupply chain could not achieve coordination as different levelsof cognitive bias under the contract had the same influence ondecision variables profit and utility Table 2 shows a clearpicture wherein μ is equal to 1 and β falls within the range of(0 08) which were used to analyze the effect of the cognitivebiases on the coordination of the supply chain

e results of the numerical simulation in Table 3 in-dicate that the contract factors ie ϕ r wholesale price theprofit and utility of members existed as flexible intervalswhen β isin (0 08) and μ 1 Besides the order quantity qsc

and the degree of sales effort esc under the contract are equalto qclowast and eclowast in a centralized supply chain respectivelyMeanwhile the buyback price r was less than the wholesaleprice wsc and higher than the unit salvage value s whichsatisfied the constrains sle rltwsclowast us the buybackcontract with promotion cost could achieve the coordinationof the two-echelon supply chain Also the buyback price r

was positively correlated with the portion ϕ of promotioncost sharing Notably the wholesale price wsc dropped as ϕincreased when β was in the range of (0 02) in Table 3However the situation reversed when it was in the range of(02 08) which was inconsistent since the manufacturerlowered the wholesale price when the retailer paid morepromotional costs in the contract e reason for such anobservation was that the wholesale price was not related to ϕbut positively related to the overconfident level β when thedegree of fairness concern was moderate Furthermore theretailer accepted the contract designed by manufacturerssince the term U(1113937

scr )gemaxU(1113937

dr ) was guaranteed in

Table 3

6 Conclusions

61MainConclusions In this paper we analyzed the effect ofcognitive bias on equilibrium solutions including thewholesale price sales price sales effort and expected profitsunder demand uncertainty by introducing both overconfi-dence and fairness concern into the two-echelon supply chaine most important conclusion from the study was that forretailers with the two cognitive biases overconfidence waspositive and the main bias that affected supply chain decisionsand the retailersrsquo utility benefits were not only due to over-confidence but also fairness concern For manufacturersoverconfidence could increase the wholesale price and profitHowever the fairness concern had a negative effect on themFurthermore the decisions of the retailer in the decentralizedsupply chain deviated from the optimal decisions in thecentralized supply chainmodel Especially when the thresholdof overconfidence was more than a fixed value the decisionvariables under decentralized decision-making were actuallyhigher than the optimal variables under centralized decision-making Also the buyback contract with promotional costsharing achieved behavioral supply chain coordination Incontrast the promotional cost-sharing portion and buybackprice both benefited from overconfidence but fairness concernhad a detrimental effect on them However as the retailersrsquooverconfidence level increased and fairness concern was equalto 1 the flexibility of contract coordination was weakenedWhen the overconfidence level wasmore than 08 the contractbecame invalid and members were not able to cooperate

62 Management Insights is paper provided some in-teresting managerial insights for decision makers in thedecentralized supply chain wherein retailers had two pref-erences First overconfidence was beneficial to them interms of decision variables and yet the fairness concern

Table 2 e effect of the cognitive biases on the feasible domain of promotion cost-sharing portion ϕ

μβ

0 02 04 06 08 0810 [02884 04976] [03683 05587] [04528 06199] [06343 07754] [07835 08669] mdash1 [02545 04741] [03310 05331] [04167 05908] [06207 07516] [07800 08560] mdash2 [02352 04600] [03101 05156] [04034 05719] [06094 07321] [07736 08478] mdash

Table 3 e effect of the cognitive biases on the coordination of the supply chain

μ 1β

0 02 04 06 08qsc 32 32 32 32 32esc 16 16 16 16 16ϕ [02545 04741] [03310 05331] [04167 05908] [06207 07516] [07800 08560]r [2100 35363] [28268 39188] [30555 42637] [36209 44215] [40485 49164]wsc [49205 53687] [49016 53522] [54092 69568] [62513 72626] [80407 87453]1113937

dm 164103 269591 461996 772619 1112545

U(1113937scm) [168722 217850] [269600 336882] [463175 500938] [772619 814300] [1112545 1120376]

1113937dr 20333 352396 495161 632422 765317

U(1113937scr ) [20333 243043] [352396 400595] [495161 525161] [632422 657581] [765317 784632]

12 Complexity

only damaged the wholesale price Second the two behav-ioral preferences were both conducive to utilities for theretailer but not necessarily beneficial for the manufacturer toreceive profit and overconfidence helped to increasemanufacturersrsquo profit which led to nonoptimal decisionsthat dragged down the performance of the two-echelonsupply chain Finally for the two behavioral preferences ofthe retailer the manufacturer assessed the degree of over-confidence and fairness concern (especially the overconfi-dence) via historical data and the current order quantity orother indicators In contrast factual data indicated that twobehavioral preferences were in the right range (such asβ isin (0 08) and μ isin (0 2)) e manufacturers introducedthe buyback contract with promotional cost sharing into thepractice of pricing and ordered decentralized supply chainsFurthermore they designed a reasonable promotional cost-sharing portion and buyback price thereby achieving anoptimized decision and utility In practice the manufac-turers could analyze whether the sellers had two behavioralpreferences based on the historical data by cooperating withdownstream enterprises ie if two behavioral preferencesexisted they referred to the papersrsquo conclusions to redesign areasonable contract mechanism

63 Future Research Since cognitive biases could existamong all members of supply chains it is necessary toexplore the manufacturersrsquo cognitive biases and analyze thesupply chain coordination problem in the presence of thetwo behavior preferences Besides the overconfidence andfairness concern considered in our centralized decision are afuture research direction

Appendix

A Decentralized Supply Chain Model

In a decentralized supply chain consisting of a rationalmanufacturer and a retailer with cognitive biases maximumprofit or utility is the only goal e backward inductivemethod can be applied to solve a Stackelberg game whereinthe manufacturer is the leader and the retailer is a followere proof of the equilibrium solutions in equations (12)ndash(14)is as follows

e profit of the manufacturer and expected utility ofretailers are as follows respectively

wmax

1113945

d

m

(w minus c)q (A1)

maxqe

U 1113945d

r

⎞⎠ (1 + μ) p[a minus bp +(1 + β)e] + s 1113946θ2

minus Aq minus a minus bp +(1 + β)e + θ1( 1113857f θ1( 1113857( 1113857dθ11113896⎛⎝

minus v 1113946A

θ2a minus bp +(1 + β)e + θ1 minus q( 1113857f θ1( 1113857( 1113857dθ1 minus wq minus

12

ce21113897 minus μ(w minus c)q

(A2)

Solving the first-order partial derivatives of the retailerrsquosexpected utility concerning qd and ed which equal to 0 weobtained the following equations

zU( 1113937dr 1113873

zq (1 + μ) v minus w + F θ2( 1113857(s minus v)1113858 1113859 minus μ(w minus c) 0

(A3)

zU( 1113937dr 1113873

ze (1 + β) p minus v +(v minus s)F θ2( 11138571113858 1113859 minus ce 0

(A4)

Hessian matrix

H(q e) (s minus v)f θ2( 1113857 (v minus s)(1 + β)f θ2( 1113857

(v minus s)f θ2( 1113857 (s minus v)(1 + β)f θ2( 1113857 minus c1113888 1113889

(A5)

According to the assumptions vgt sgt 0 f(θ2)gt 0 andβ isin [0 1) us the Hessian matrix of U(1113937

dr ) is a negative

definite for q and e which meant that the optimal orderquantity and sales effort of retailers existed in the decen-tralized supply chain erefore we obtained the followingformula of the retailersrsquo order quantity and sales effort byjointly solving equations (A3) and (A4)

qd

(a minus bp)(v minus s)(1 + μ)c +(1 + β)2(v minus s)[(1 + μ)(p minus w) + μc] + Ac[(s + v)(1 + μ) + 2μc minus w]

(v minus s)(1 + μ)c (A6)

ed

(1 + β)[(1 + μ)p + μc minus (1 + 2μ)]w

(1 + μ)c (A7)

Complexity 13

Substituting equations (A6) into (A1) we attained

wmax

1113945

d

m

(w minus c) (a minus bp)(v minus s)(1 + μ)c +(1 + β)2(v minus s)[(1 + μ)p + μc] + Ac[(s + v)(1 + μ) + 2μc] minus (1 + 2μ)w 2Ac +(1 + β)2(v minus s)1113960 11139611113966 1113967

(v minus s)(1 + μ)c

(A8)

Taking into account the first-order condition of equation(A7) with respect to w rounded off to 0 we obtained`

z1113937dm

ze

(a minus bp)(v minus s)(1 + μ)c +(1 + β)2(v minus s)[(1 + μ)p + μc] + Ac[(s + v)(1 + μ) + 2μc] minus (2w minus c)(1 + 2μ) 2Ac +(1 + β)2(v minus s)1113960 1113961

(v minus s)(1 + μ)c

(A9)

Solving equation (A8) we found that the optimalwholesale price of the manufacturer is as follows

wdlowast

(a minus bp)(1 + μ)(v minus s)c +(v minus s)(1 + β)2[(1 + μ)p +(1 + 3μ)c] + Ac[(s + v)(1 + μ) + 2(1 + 3μ)c]

2(1 + 2μ) 2Ac +(v minus s)(1 + β)21113960 1113961 (A10)

By substituting equation (A9) into equations (A6) and(A7) we attained the optimal order quantity and sales effortwhich are displayed in equations (13) and (14)

B Propositions

Proof of Proposition 1 We solved the first derivatives ofoptimal order quantity qdlowast and sales effort edlowast with respect

to β in equations (13) and (14) respectively Meanwhiletaking the second-order derivative of qdlowast with respect to βwe obtained

zqdlowast

zβ

(1 + β)(p minus c)

c

z2qdlowast

zβ2

p minus c

c (B1)

zedlowast

zβ4(p minus c)(v minus s)2β β3 + 4β2 + 6β + 41113872 1113873 +(a minus bp)c(1 + β)2(v minus s)2 + Ac(v minus s)(1 + β)2 (2(p minus c) + v + s) + 2(Ac)2(4p minus c minus s minus v)1113960 1113961

2c (1 + β)2(v minus s) + 2Ac1113960 11139612

(B2)

Since the second derivative of edlowast with respect to β wasvery complicated there was no need to solve it According tothe assumptions pgt vgt cgt sgt 0 β isin (0 1) a minus bpgt 0 andcgt 0 so (zqdlowastzβ)gt 0 (z2qdlowastzβ2)gt 0 and (zedlowastzβ)gt 0

Proof of Proposition 2 Based on the process above we tookthe first-order and second-order partial derivatives of op-timal wholesale price wdlowast with respect to β and μ in equation(12) respectively

zwdlowast

zβ

c(v minus s)(1 + μ)(1 + β)[(a minus bp)(v minus s) + A(2p minus s minus v)]

(1 + 2μ) (1 + β)2(v minus s) + 2Ac1113960 11139612 (B3)

zwdlowast

zμ minus

(a minus bp)(v minus s)c +(p minus c)(v minus s)(1 + β)2 + Ac(s + v minus 2c)1113960 1113961

2(1 + 2μ)2 (v minus s)(1 + β)2 + 2Ac1113960 1113961 (B4)

14 Complexity

According to the optimal result of order quantity inequation (13) and the assumption qgt 0 the condition s +

v minus 2cgt 0 is found According to the assumptions we had(zwdlowastzβ)gt 0 and (zwdlowastzμ)lt 0

Proof of Proposition 3 e gap between qdlowast and qclowastdenoted by Δq is as follows

Δq qdlowast

minus qclowast

(p minus c)(v minus s) (1 + β)2 minus 21113960 1113961 minus Ac(s + v minus 2c) minus (a minus bp)(v minus s)c

2c(v minus s) (B5)

If (p minus c)(v minus s)[(1 + β)2 minus 2] is more thanAc(s + v minus 2c) + (a minus bp)(v minus s)c we have Δqgt 0 ieqdlowast gt qclowast In contrast if (p minus c)(v minus s)[(1 + β)2 minus 2] is less

than Ac(s + v minus 2c) + (a minus bp)(v minus s)c Δqlt 0 ieqdlowast lt qclowast

e gap between edlowast and eclowast denoted by Δe is asfollows

Δe edlowast

minus eclowast

Aβc(4p minus 2c minus s minus v) minus Ac(s + v minus 2c) minus (v minus s)(1 + β) (a minus bp)c + 1 minus β21113872 1113873(p minus c)1113960 1113961

2c 2Ac +(1 + β)2(v minus s)1113960 1113961 (B6)

Similarly if Aβc(4p minus 2c minus s minus v)gtAc(s + v minus 2c)

+(v minus s)(1 + β)[(a minus bp)c + (1 minus β2)(p minus c)] we haveΔe edlowast minus eclowast gt 0 In contrast ifAβc(4p minus 2c minus s minus v)ltAc(s + v minus 2c)+

(v minus s)(1 + β)[(a minus bp)c + (1 minus β2)(p minus c)] we haveΔe edlowast minus eclowast lt 0

Upon solving the first-order derivative of Δq and Δe withrespect to β respectively we obtained

zΔ q

zβ

(1 + β)(p minus c)

c (B7)

zΔ e

zβ4(p minus c)(v minus s)2 β3 + 4β2 + 6β + 41113872 1113873β +(a minus bp)c(1 + β)2(v minus s)2 + Ac(v minus s)(1 + β)2 2(p minus c) +v + s) + 2(Ac)2(4p minus c minus s minus v)1113960 1113961

2c (1 + β)2(v minus s) + 2Ac1113960 11139612

(B8)

According to the assumptions pgt vgt cgt sgt 0 β isin (0 1)cgt 0 and a minus bpgt 0 (zΔ qzβ)gt 0 and (zΔ ezβ)gt 0 can beproved Hence the gaps increased as a function with respectto β

Proof of Proposition 4 e proof for Proposition 4 is similarto Proposition 2 Herein we took first-order and second-order partial derivatives of optimal wholesale price wsclowast withrespect to β μ and ϕ in equation (18) respectively esolving results are as follows

zwsclowast

zβ

[(1 + 2μ)ϕ minus μ](p minus c)

(1 + 2μ)(1 + β)2

zwsclowast

zμ minus

(p minus c)β(1 + β)(1 + 2μ)2

zwsclowast

zϕ minus

p minus c

1 + β

(B9)

(zwsclowastzβ) if ϕgt (μ(1 + 2μ)) is more than 0 otherwiseϕgt (μ(1 + 2μ)) is less than 0 us wsc could be increasingwith respect to β nevertheless decreasing with respect to βBesides according to the assumptions pgt vgt cgt sgt 0β isin (0 1) cgt 0 and a minus bpgt 0 it can be evidently foundthat (zwsclowastzμ)gt 0 and (zwsclowastzϕ)gt 0

Data Availability

e data used to support the findings of this study are in-cluded within the article

Conflicts of Interest

e authors declare no conflicts of interest

Acknowledgments

is work was supported by a major project financed by theNational Social Science Fund of China (Approval nos71461009 71662011 and 71761015)

Complexity 15

References

[1] Y Ren D C Croson and R T A Croson ldquoe overconfidentnewsvendorrdquo Journal of the Operational Research Societyvol 68 no 5 pp 496ndash506 2017

[2] T H Cui and P Mallucci ldquoFairness ideals in distributionchannelsrdquo Journal of Marketing Research vol 53 no 6pp 969ndash987 2016

[3] K Chen X Wang M Huang and W-K Ching ldquoSalesforcecontract design joint pricing and production planning withasymmetric overconfidence sales agentrdquo Journal of Industrialamp Management Optimization vol 13 no 2 pp 873ndash8992017

[4] X X Zheng Z Liu K W Li et al ldquoCooperative game ap-proaches to coordinating a three-echelon closed-loop supplychain with fairness concernsrdquo International Journal of Pro-duction Economic vol 212 pp 92ndash110 2018

[5] China Internet Watch False Prosperity behind Double 11Refund Rate Increased to 69 2014 httpswwwchinainternetwatchcom11643false-prosperity-behind-double-11-refund-rate-increased-to-69

[6] C L Lu ldquoPampGrsquos collaborative supply chain transformationrdquoEnterprise Management vol 10 pp 79-80 2016

[7] M Li N C Petruzzi and J Zhang ldquoOverconfident competingnewsvendorsrdquo Management Science vol 63 no 8pp 2637ndash2646 2016

[8] T Nie ldquoDual-fairness supply chain with quantity discountcontractsrdquo European Journal of Operational Researchvol 258 no 2 pp 491ndash500 2017

[9] X Du and B Jiang ldquoSignaling through price and quality toconsumers with fairness concernsrdquo Journal of MarketingResearch vol 53 no 6 pp 988ndash1000 2016

[10] V Kurz A Orland and K Posadzy ldquoFairness versus effi-ciency how procedural fairness concerns affect coordina-tionrdquo Experimental Economics vol 21 no 3 pp 601ndash6262018

[11] S Villa and J A Castantildeeda ldquoTransshipments in supplychains a behavioral investigationrdquo European Journal of Op-erational Research vol 269 no 2 pp 715ndash729 2018

[12] B Gu Y Fu and Y Li ldquoFresh-keeping effort and channelperformance in a fresh product supply chain with loss-averseconsumersrsquo returnsrdquo Mathematical Problems in Engineeringvol 2018 Article ID 4717094 20 pages 2018

[13] D Croson R Croson and Y Ren How to Manage anOverconfident Newsvendor 2008 httpcbeesutdallasedupapersCrosonRenCmsonMS2008

[14] M Li ldquoOverconfident distribution channelsrdquo Production andOperations Management vol 28 no 6 pp 1347ndash1365 2019

[15] W Liu D Wang O Tang and D Zhu ldquoe impacts oflogistics service integratorrsquos overconfidence behaviour onsupply chain decision under demand surgerdquo EuropeanJournal of Industrial Engineering vol 12 no 4 pp 558ndash5972018

[16] L Xu X Shi P Du K Govindan and Z Zhang ldquoOptimi-zation on pricing and overconfidence problem in a duopo-listic supply chainrdquo Computers amp Operations Researchvol 101 pp 162ndash172 2019

[17] Y Li and M Shan ldquoAdvance selling decisions with over-confident consumersrdquo Journal of Industrial and ManagementOptimization vol 2 no 3 pp 891ndash905 2016

[18] T H Cui J S Raju and Z J Zhang ldquoFairness and channelcoordinationrdquo Management Science vol 53 no 8pp 1303ndash1314 2007

[19] X Wu and J A Niederhoff ldquoFairness in selling to thenewsvendorrdquo Production and Operations Managementvol 23 no 11 pp 2002ndash2022 2014

[20] Z Chen and S-I Ivan Su ldquoSocial welfare maximization withthe least subsidy photovoltaic supply chain equilibrium andcoordination with fairness concernrdquo Renewable Energyvol 132 pp 1332ndash1347 2019

[21] W Liu D Wang X Shen X Yan and W Wei ldquoe impactsof distributional and peer-induced fairness concerns on thedecision-making of order allocation in logistics service supplychainrdquo Transportation Research Part E Logistics and Trans-portation Review vol 116 pp 102ndash122 2018

[22] L Zhang H Zhou Y Liu and R Lu ldquoOptimal environmentalquality and price with consumer environmental awarenessand retailerrsquos fairness concerns in supply chainrdquo Journal ofCleaner Production vol 213 pp 1063ndash1079 2019

[23] X LWang S Q Hu and X B Liu ldquoSupply coordination withoption contract considering overconfidence under randomdemand and yieldsrdquo Computer Integrated ManufacturingSystem vol 24 no 11 pp 2898ndash2908 2018

[24] W Jiang and J Liu ldquoInventory financing with overconfidentsupplier based on supply chain contractrdquo MathematicalProblems in Engineering vol 2018 Article ID 505438712 pages 2018

[25] P Ma H Wang and J Shang ldquoContract design for two-stagesupply chain coordination integrating manufacturer-qualityand retailer-marketing effortsrdquo International Journal ofProduction Economics vol 146 no 2 pp 745ndash755 2013

[26] J Liu W F Jiang and Y P Tang ldquoe supply chain coor-dination based on carbon reduction of overconfident man-ufacturerrdquo Journal of Jiangxi Normal University (NaturalScience) vol 42 no 2 pp 203ndash207 2018

[27] R Dubey N Altay and C Blome ldquoSwift trust and com-mitment the missing links for humanitarian supply chaincoordinationrdquo Annals of Operations Research vol 283 no 1-2 pp 159ndash177 2019

[28] C Cui Z Feng and C Tan ldquoCredibilistic loss aversion nashequilibrium for bimatrix games with triangular fuzzy payoffsrdquoComplexity vol 2018 Article ID 7143586 16 pages 2018

[29] E Fehr and KM Schmidt ldquoA theory of fairness competitionand cooperationrdquo e Quarterly Journal of Economicsvol 114 no 3 pp 817ndash868 1999

[30] Z Qin and J Yang ldquoAnalysis of a revenue-sharing contract insupply chain managementrdquo International Journal of LogisticsResearch and Applications vol 11 no 1 pp 17ndash29 2008

[31] S-Z Bai and J Ni ldquoResearch on buy-back contract with saleseffort cost sharingrdquo in Proceedings of the 2009 Chinese Controland Decision Conference IEEE Guilin China pp 1917ndash1921June 2009

[32] E Karagozoglu and K Keskin ldquoTime-varying fairness con-cerns delay and disagreement in bargainingrdquo Journal ofEconomic Behavior and Organization vol 147 pp 115ndash1282018

16 Complexity

As described in Section 4 if the supply chain membersaccepted the buyback contract with promotion cost sharingthe assumptions 1113937

scm gemax 1113937

dm and U(1113937

scr )gemaxU(1113937

dr )

had to be satisfied Also according to the assumptions thevalue ϕ must satisfy 1113937

scm gemax 1113937

dm and

U(1113937scr )gemaxU(1113937

dr ) us we kept the values of param-

eters to be the same (ie a 50 θ sim U(minus 30 30) c 4p 20 v 7 s 2 b 2 and c 1) to capture a likelydomain of ϕ

As can be seen from Table 2 the overconfident level βincreased the upper and lower limits of ϕ which had a clearupward trend In contrast the upper and lower limits of ϕ

showed a definite declining trend as the degree of fairnessconcern μ increased ese trends illustrated that in order tocoordinate the revenue of both parties and optimize theperformance of the supply chain the manufacturer needs tobear less promotional costs as β increased but bear morepromotional costs as μ value increased Besides when β and μincreased the interval of sharing cost portion ϕ shrunk whichindicated that the buyback contract with promotional costsharing had certain flexibility to coordinate the revenue be-tween supply chain members However the flexibility couldbe weakened with the rise of the cognitive bias level Forsupply chain members the possibility of negotiation or

01 02 03 04 05 06 07 08 09 10β

-μ = 0--μ = 1-μ = 2

4

5

6

7

8

9

10

11

12

Who

lesa

le p

rice

Figure 6 e effect of the cognitive biases on wholesale price wsrlowast

01 02 03 04 05 06 07 08 09 10β

-μ = 0--μ = 1-μ = 2

2

4

6

8

10

12

14

16

18

20

22

Buyb

ack

pric

e

Figure 7 e effect of the cognitive biases on buyback price rlowast

Complexity 11

cooperation between manufacturers and retailers also de-creased with an increase in retailersrsquo cognitive bias levelWhen the overconfident level was more than 081 the re-tailerrsquos utility from decentralized decisions was greater thanthe utility in the contract environment wherein the twoparties could not reach a cooperation agreement us thesupply chain could not achieve coordination as different levelsof cognitive bias under the contract had the same influence ondecision variables profit and utility Table 2 shows a clearpicture wherein μ is equal to 1 and β falls within the range of(0 08) which were used to analyze the effect of the cognitivebiases on the coordination of the supply chain

e results of the numerical simulation in Table 3 in-dicate that the contract factors ie ϕ r wholesale price theprofit and utility of members existed as flexible intervalswhen β isin (0 08) and μ 1 Besides the order quantity qsc

and the degree of sales effort esc under the contract are equalto qclowast and eclowast in a centralized supply chain respectivelyMeanwhile the buyback price r was less than the wholesaleprice wsc and higher than the unit salvage value s whichsatisfied the constrains sle rltwsclowast us the buybackcontract with promotion cost could achieve the coordinationof the two-echelon supply chain Also the buyback price r

was positively correlated with the portion ϕ of promotioncost sharing Notably the wholesale price wsc dropped as ϕincreased when β was in the range of (0 02) in Table 3However the situation reversed when it was in the range of(02 08) which was inconsistent since the manufacturerlowered the wholesale price when the retailer paid morepromotional costs in the contract e reason for such anobservation was that the wholesale price was not related to ϕbut positively related to the overconfident level β when thedegree of fairness concern was moderate Furthermore theretailer accepted the contract designed by manufacturerssince the term U(1113937

scr )gemaxU(1113937

dr ) was guaranteed in

Table 3

6 Conclusions

61MainConclusions In this paper we analyzed the effect ofcognitive bias on equilibrium solutions including thewholesale price sales price sales effort and expected profitsunder demand uncertainty by introducing both overconfi-dence and fairness concern into the two-echelon supply chaine most important conclusion from the study was that forretailers with the two cognitive biases overconfidence waspositive and the main bias that affected supply chain decisionsand the retailersrsquo utility benefits were not only due to over-confidence but also fairness concern For manufacturersoverconfidence could increase the wholesale price and profitHowever the fairness concern had a negative effect on themFurthermore the decisions of the retailer in the decentralizedsupply chain deviated from the optimal decisions in thecentralized supply chainmodel Especially when the thresholdof overconfidence was more than a fixed value the decisionvariables under decentralized decision-making were actuallyhigher than the optimal variables under centralized decision-making Also the buyback contract with promotional costsharing achieved behavioral supply chain coordination Incontrast the promotional cost-sharing portion and buybackprice both benefited from overconfidence but fairness concernhad a detrimental effect on them However as the retailersrsquooverconfidence level increased and fairness concern was equalto 1 the flexibility of contract coordination was weakenedWhen the overconfidence level wasmore than 08 the contractbecame invalid and members were not able to cooperate

62 Management Insights is paper provided some in-teresting managerial insights for decision makers in thedecentralized supply chain wherein retailers had two pref-erences First overconfidence was beneficial to them interms of decision variables and yet the fairness concern

Table 2 e effect of the cognitive biases on the feasible domain of promotion cost-sharing portion ϕ

μβ

0 02 04 06 08 0810 [02884 04976] [03683 05587] [04528 06199] [06343 07754] [07835 08669] mdash1 [02545 04741] [03310 05331] [04167 05908] [06207 07516] [07800 08560] mdash2 [02352 04600] [03101 05156] [04034 05719] [06094 07321] [07736 08478] mdash

Table 3 e effect of the cognitive biases on the coordination of the supply chain

μ 1β

0 02 04 06 08qsc 32 32 32 32 32esc 16 16 16 16 16ϕ [02545 04741] [03310 05331] [04167 05908] [06207 07516] [07800 08560]r [2100 35363] [28268 39188] [30555 42637] [36209 44215] [40485 49164]wsc [49205 53687] [49016 53522] [54092 69568] [62513 72626] [80407 87453]1113937

dm 164103 269591 461996 772619 1112545

U(1113937scm) [168722 217850] [269600 336882] [463175 500938] [772619 814300] [1112545 1120376]

1113937dr 20333 352396 495161 632422 765317

U(1113937scr ) [20333 243043] [352396 400595] [495161 525161] [632422 657581] [765317 784632]

12 Complexity

only damaged the wholesale price Second the two behav-ioral preferences were both conducive to utilities for theretailer but not necessarily beneficial for the manufacturer toreceive profit and overconfidence helped to increasemanufacturersrsquo profit which led to nonoptimal decisionsthat dragged down the performance of the two-echelonsupply chain Finally for the two behavioral preferences ofthe retailer the manufacturer assessed the degree of over-confidence and fairness concern (especially the overconfi-dence) via historical data and the current order quantity orother indicators In contrast factual data indicated that twobehavioral preferences were in the right range (such asβ isin (0 08) and μ isin (0 2)) e manufacturers introducedthe buyback contract with promotional cost sharing into thepractice of pricing and ordered decentralized supply chainsFurthermore they designed a reasonable promotional cost-sharing portion and buyback price thereby achieving anoptimized decision and utility In practice the manufac-turers could analyze whether the sellers had two behavioralpreferences based on the historical data by cooperating withdownstream enterprises ie if two behavioral preferencesexisted they referred to the papersrsquo conclusions to redesign areasonable contract mechanism

63 Future Research Since cognitive biases could existamong all members of supply chains it is necessary toexplore the manufacturersrsquo cognitive biases and analyze thesupply chain coordination problem in the presence of thetwo behavior preferences Besides the overconfidence andfairness concern considered in our centralized decision are afuture research direction

Appendix

A Decentralized Supply Chain Model

In a decentralized supply chain consisting of a rationalmanufacturer and a retailer with cognitive biases maximumprofit or utility is the only goal e backward inductivemethod can be applied to solve a Stackelberg game whereinthe manufacturer is the leader and the retailer is a followere proof of the equilibrium solutions in equations (12)ndash(14)is as follows

e profit of the manufacturer and expected utility ofretailers are as follows respectively

wmax

1113945

d

m

(w minus c)q (A1)

maxqe

U 1113945d

r

⎞⎠ (1 + μ) p[a minus bp +(1 + β)e] + s 1113946θ2

minus Aq minus a minus bp +(1 + β)e + θ1( 1113857f θ1( 1113857( 1113857dθ11113896⎛⎝

minus v 1113946A

θ2a minus bp +(1 + β)e + θ1 minus q( 1113857f θ1( 1113857( 1113857dθ1 minus wq minus

12

ce21113897 minus μ(w minus c)q

(A2)

Solving the first-order partial derivatives of the retailerrsquosexpected utility concerning qd and ed which equal to 0 weobtained the following equations

zU( 1113937dr 1113873

zq (1 + μ) v minus w + F θ2( 1113857(s minus v)1113858 1113859 minus μ(w minus c) 0

(A3)

zU( 1113937dr 1113873

ze (1 + β) p minus v +(v minus s)F θ2( 11138571113858 1113859 minus ce 0

(A4)

Hessian matrix

H(q e) (s minus v)f θ2( 1113857 (v minus s)(1 + β)f θ2( 1113857

(v minus s)f θ2( 1113857 (s minus v)(1 + β)f θ2( 1113857 minus c1113888 1113889

(A5)

According to the assumptions vgt sgt 0 f(θ2)gt 0 andβ isin [0 1) us the Hessian matrix of U(1113937

dr ) is a negative

definite for q and e which meant that the optimal orderquantity and sales effort of retailers existed in the decen-tralized supply chain erefore we obtained the followingformula of the retailersrsquo order quantity and sales effort byjointly solving equations (A3) and (A4)

qd

(a minus bp)(v minus s)(1 + μ)c +(1 + β)2(v minus s)[(1 + μ)(p minus w) + μc] + Ac[(s + v)(1 + μ) + 2μc minus w]

(v minus s)(1 + μ)c (A6)

ed

(1 + β)[(1 + μ)p + μc minus (1 + 2μ)]w

(1 + μ)c (A7)

Complexity 13

Substituting equations (A6) into (A1) we attained

wmax

1113945

d

m

(w minus c) (a minus bp)(v minus s)(1 + μ)c +(1 + β)2(v minus s)[(1 + μ)p + μc] + Ac[(s + v)(1 + μ) + 2μc] minus (1 + 2μ)w 2Ac +(1 + β)2(v minus s)1113960 11139611113966 1113967

(v minus s)(1 + μ)c

(A8)

Taking into account the first-order condition of equation(A7) with respect to w rounded off to 0 we obtained`

z1113937dm

ze

(a minus bp)(v minus s)(1 + μ)c +(1 + β)2(v minus s)[(1 + μ)p + μc] + Ac[(s + v)(1 + μ) + 2μc] minus (2w minus c)(1 + 2μ) 2Ac +(1 + β)2(v minus s)1113960 1113961

(v minus s)(1 + μ)c

(A9)

Solving equation (A8) we found that the optimalwholesale price of the manufacturer is as follows

wdlowast

(a minus bp)(1 + μ)(v minus s)c +(v minus s)(1 + β)2[(1 + μ)p +(1 + 3μ)c] + Ac[(s + v)(1 + μ) + 2(1 + 3μ)c]

2(1 + 2μ) 2Ac +(v minus s)(1 + β)21113960 1113961 (A10)

By substituting equation (A9) into equations (A6) and(A7) we attained the optimal order quantity and sales effortwhich are displayed in equations (13) and (14)

B Propositions

Proof of Proposition 1 We solved the first derivatives ofoptimal order quantity qdlowast and sales effort edlowast with respect

to β in equations (13) and (14) respectively Meanwhiletaking the second-order derivative of qdlowast with respect to βwe obtained

zqdlowast

zβ

(1 + β)(p minus c)

c

z2qdlowast

zβ2

p minus c

c (B1)

zedlowast

zβ4(p minus c)(v minus s)2β β3 + 4β2 + 6β + 41113872 1113873 +(a minus bp)c(1 + β)2(v minus s)2 + Ac(v minus s)(1 + β)2 (2(p minus c) + v + s) + 2(Ac)2(4p minus c minus s minus v)1113960 1113961

2c (1 + β)2(v minus s) + 2Ac1113960 11139612

(B2)

Since the second derivative of edlowast with respect to β wasvery complicated there was no need to solve it According tothe assumptions pgt vgt cgt sgt 0 β isin (0 1) a minus bpgt 0 andcgt 0 so (zqdlowastzβ)gt 0 (z2qdlowastzβ2)gt 0 and (zedlowastzβ)gt 0

Proof of Proposition 2 Based on the process above we tookthe first-order and second-order partial derivatives of op-timal wholesale price wdlowast with respect to β and μ in equation(12) respectively

zwdlowast

zβ

c(v minus s)(1 + μ)(1 + β)[(a minus bp)(v minus s) + A(2p minus s minus v)]

(1 + 2μ) (1 + β)2(v minus s) + 2Ac1113960 11139612 (B3)

zwdlowast

zμ minus

(a minus bp)(v minus s)c +(p minus c)(v minus s)(1 + β)2 + Ac(s + v minus 2c)1113960 1113961

2(1 + 2μ)2 (v minus s)(1 + β)2 + 2Ac1113960 1113961 (B4)

14 Complexity

According to the optimal result of order quantity inequation (13) and the assumption qgt 0 the condition s +

v minus 2cgt 0 is found According to the assumptions we had(zwdlowastzβ)gt 0 and (zwdlowastzμ)lt 0

Proof of Proposition 3 e gap between qdlowast and qclowastdenoted by Δq is as follows

Δq qdlowast

minus qclowast

(p minus c)(v minus s) (1 + β)2 minus 21113960 1113961 minus Ac(s + v minus 2c) minus (a minus bp)(v minus s)c

2c(v minus s) (B5)

If (p minus c)(v minus s)[(1 + β)2 minus 2] is more thanAc(s + v minus 2c) + (a minus bp)(v minus s)c we have Δqgt 0 ieqdlowast gt qclowast In contrast if (p minus c)(v minus s)[(1 + β)2 minus 2] is less

than Ac(s + v minus 2c) + (a minus bp)(v minus s)c Δqlt 0 ieqdlowast lt qclowast

e gap between edlowast and eclowast denoted by Δe is asfollows

Δe edlowast

minus eclowast

Aβc(4p minus 2c minus s minus v) minus Ac(s + v minus 2c) minus (v minus s)(1 + β) (a minus bp)c + 1 minus β21113872 1113873(p minus c)1113960 1113961

2c 2Ac +(1 + β)2(v minus s)1113960 1113961 (B6)

Similarly if Aβc(4p minus 2c minus s minus v)gtAc(s + v minus 2c)

+(v minus s)(1 + β)[(a minus bp)c + (1 minus β2)(p minus c)] we haveΔe edlowast minus eclowast gt 0 In contrast ifAβc(4p minus 2c minus s minus v)ltAc(s + v minus 2c)+

(v minus s)(1 + β)[(a minus bp)c + (1 minus β2)(p minus c)] we haveΔe edlowast minus eclowast lt 0

Upon solving the first-order derivative of Δq and Δe withrespect to β respectively we obtained

zΔ q

zβ

(1 + β)(p minus c)

c (B7)

zΔ e

zβ4(p minus c)(v minus s)2 β3 + 4β2 + 6β + 41113872 1113873β +(a minus bp)c(1 + β)2(v minus s)2 + Ac(v minus s)(1 + β)2 2(p minus c) +v + s) + 2(Ac)2(4p minus c minus s minus v)1113960 1113961

2c (1 + β)2(v minus s) + 2Ac1113960 11139612

(B8)

According to the assumptions pgt vgt cgt sgt 0 β isin (0 1)cgt 0 and a minus bpgt 0 (zΔ qzβ)gt 0 and (zΔ ezβ)gt 0 can beproved Hence the gaps increased as a function with respectto β

Proof of Proposition 4 e proof for Proposition 4 is similarto Proposition 2 Herein we took first-order and second-order partial derivatives of optimal wholesale price wsclowast withrespect to β μ and ϕ in equation (18) respectively esolving results are as follows

zwsclowast

zβ

[(1 + 2μ)ϕ minus μ](p minus c)

(1 + 2μ)(1 + β)2

zwsclowast

zμ minus

(p minus c)β(1 + β)(1 + 2μ)2

zwsclowast

zϕ minus

p minus c

1 + β

(B9)

(zwsclowastzβ) if ϕgt (μ(1 + 2μ)) is more than 0 otherwiseϕgt (μ(1 + 2μ)) is less than 0 us wsc could be increasingwith respect to β nevertheless decreasing with respect to βBesides according to the assumptions pgt vgt cgt sgt 0β isin (0 1) cgt 0 and a minus bpgt 0 it can be evidently foundthat (zwsclowastzμ)gt 0 and (zwsclowastzϕ)gt 0

Data Availability

e data used to support the findings of this study are in-cluded within the article

Conflicts of Interest

e authors declare no conflicts of interest

Acknowledgments

is work was supported by a major project financed by theNational Social Science Fund of China (Approval nos71461009 71662011 and 71761015)

Complexity 15

References

[1] Y Ren D C Croson and R T A Croson ldquoe overconfidentnewsvendorrdquo Journal of the Operational Research Societyvol 68 no 5 pp 496ndash506 2017

[2] T H Cui and P Mallucci ldquoFairness ideals in distributionchannelsrdquo Journal of Marketing Research vol 53 no 6pp 969ndash987 2016

[3] K Chen X Wang M Huang and W-K Ching ldquoSalesforcecontract design joint pricing and production planning withasymmetric overconfidence sales agentrdquo Journal of Industrialamp Management Optimization vol 13 no 2 pp 873ndash8992017

[4] X X Zheng Z Liu K W Li et al ldquoCooperative game ap-proaches to coordinating a three-echelon closed-loop supplychain with fairness concernsrdquo International Journal of Pro-duction Economic vol 212 pp 92ndash110 2018

[5] China Internet Watch False Prosperity behind Double 11Refund Rate Increased to 69 2014 httpswwwchinainternetwatchcom11643false-prosperity-behind-double-11-refund-rate-increased-to-69

[6] C L Lu ldquoPampGrsquos collaborative supply chain transformationrdquoEnterprise Management vol 10 pp 79-80 2016

[7] M Li N C Petruzzi and J Zhang ldquoOverconfident competingnewsvendorsrdquo Management Science vol 63 no 8pp 2637ndash2646 2016

[8] T Nie ldquoDual-fairness supply chain with quantity discountcontractsrdquo European Journal of Operational Researchvol 258 no 2 pp 491ndash500 2017

[9] X Du and B Jiang ldquoSignaling through price and quality toconsumers with fairness concernsrdquo Journal of MarketingResearch vol 53 no 6 pp 988ndash1000 2016

[10] V Kurz A Orland and K Posadzy ldquoFairness versus effi-ciency how procedural fairness concerns affect coordina-tionrdquo Experimental Economics vol 21 no 3 pp 601ndash6262018

[11] S Villa and J A Castantildeeda ldquoTransshipments in supplychains a behavioral investigationrdquo European Journal of Op-erational Research vol 269 no 2 pp 715ndash729 2018

[12] B Gu Y Fu and Y Li ldquoFresh-keeping effort and channelperformance in a fresh product supply chain with loss-averseconsumersrsquo returnsrdquo Mathematical Problems in Engineeringvol 2018 Article ID 4717094 20 pages 2018

[13] D Croson R Croson and Y Ren How to Manage anOverconfident Newsvendor 2008 httpcbeesutdallasedupapersCrosonRenCmsonMS2008

[14] M Li ldquoOverconfident distribution channelsrdquo Production andOperations Management vol 28 no 6 pp 1347ndash1365 2019

[15] W Liu D Wang O Tang and D Zhu ldquoe impacts oflogistics service integratorrsquos overconfidence behaviour onsupply chain decision under demand surgerdquo EuropeanJournal of Industrial Engineering vol 12 no 4 pp 558ndash5972018

[16] L Xu X Shi P Du K Govindan and Z Zhang ldquoOptimi-zation on pricing and overconfidence problem in a duopo-listic supply chainrdquo Computers amp Operations Researchvol 101 pp 162ndash172 2019

[17] Y Li and M Shan ldquoAdvance selling decisions with over-confident consumersrdquo Journal of Industrial and ManagementOptimization vol 2 no 3 pp 891ndash905 2016

[18] T H Cui J S Raju and Z J Zhang ldquoFairness and channelcoordinationrdquo Management Science vol 53 no 8pp 1303ndash1314 2007

[19] X Wu and J A Niederhoff ldquoFairness in selling to thenewsvendorrdquo Production and Operations Managementvol 23 no 11 pp 2002ndash2022 2014

[20] Z Chen and S-I Ivan Su ldquoSocial welfare maximization withthe least subsidy photovoltaic supply chain equilibrium andcoordination with fairness concernrdquo Renewable Energyvol 132 pp 1332ndash1347 2019

[21] W Liu D Wang X Shen X Yan and W Wei ldquoe impactsof distributional and peer-induced fairness concerns on thedecision-making of order allocation in logistics service supplychainrdquo Transportation Research Part E Logistics and Trans-portation Review vol 116 pp 102ndash122 2018

[22] L Zhang H Zhou Y Liu and R Lu ldquoOptimal environmentalquality and price with consumer environmental awarenessand retailerrsquos fairness concerns in supply chainrdquo Journal ofCleaner Production vol 213 pp 1063ndash1079 2019

[23] X LWang S Q Hu and X B Liu ldquoSupply coordination withoption contract considering overconfidence under randomdemand and yieldsrdquo Computer Integrated ManufacturingSystem vol 24 no 11 pp 2898ndash2908 2018

[24] W Jiang and J Liu ldquoInventory financing with overconfidentsupplier based on supply chain contractrdquo MathematicalProblems in Engineering vol 2018 Article ID 505438712 pages 2018

[25] P Ma H Wang and J Shang ldquoContract design for two-stagesupply chain coordination integrating manufacturer-qualityand retailer-marketing effortsrdquo International Journal ofProduction Economics vol 146 no 2 pp 745ndash755 2013

[26] J Liu W F Jiang and Y P Tang ldquoe supply chain coor-dination based on carbon reduction of overconfident man-ufacturerrdquo Journal of Jiangxi Normal University (NaturalScience) vol 42 no 2 pp 203ndash207 2018

[27] R Dubey N Altay and C Blome ldquoSwift trust and com-mitment the missing links for humanitarian supply chaincoordinationrdquo Annals of Operations Research vol 283 no 1-2 pp 159ndash177 2019

[28] C Cui Z Feng and C Tan ldquoCredibilistic loss aversion nashequilibrium for bimatrix games with triangular fuzzy payoffsrdquoComplexity vol 2018 Article ID 7143586 16 pages 2018

[29] E Fehr and KM Schmidt ldquoA theory of fairness competitionand cooperationrdquo e Quarterly Journal of Economicsvol 114 no 3 pp 817ndash868 1999

[30] Z Qin and J Yang ldquoAnalysis of a revenue-sharing contract insupply chain managementrdquo International Journal of LogisticsResearch and Applications vol 11 no 1 pp 17ndash29 2008

[31] S-Z Bai and J Ni ldquoResearch on buy-back contract with saleseffort cost sharingrdquo in Proceedings of the 2009 Chinese Controland Decision Conference IEEE Guilin China pp 1917ndash1921June 2009

[32] E Karagozoglu and K Keskin ldquoTime-varying fairness con-cerns delay and disagreement in bargainingrdquo Journal ofEconomic Behavior and Organization vol 147 pp 115ndash1282018

16 Complexity

cooperation between manufacturers and retailers also de-creased with an increase in retailersrsquo cognitive bias levelWhen the overconfident level was more than 081 the re-tailerrsquos utility from decentralized decisions was greater thanthe utility in the contract environment wherein the twoparties could not reach a cooperation agreement us thesupply chain could not achieve coordination as different levelsof cognitive bias under the contract had the same influence ondecision variables profit and utility Table 2 shows a clearpicture wherein μ is equal to 1 and β falls within the range of(0 08) which were used to analyze the effect of the cognitivebiases on the coordination of the supply chain

e results of the numerical simulation in Table 3 in-dicate that the contract factors ie ϕ r wholesale price theprofit and utility of members existed as flexible intervalswhen β isin (0 08) and μ 1 Besides the order quantity qsc

and the degree of sales effort esc under the contract are equalto qclowast and eclowast in a centralized supply chain respectivelyMeanwhile the buyback price r was less than the wholesaleprice wsc and higher than the unit salvage value s whichsatisfied the constrains sle rltwsclowast us the buybackcontract with promotion cost could achieve the coordinationof the two-echelon supply chain Also the buyback price r

was positively correlated with the portion ϕ of promotioncost sharing Notably the wholesale price wsc dropped as ϕincreased when β was in the range of (0 02) in Table 3However the situation reversed when it was in the range of(02 08) which was inconsistent since the manufacturerlowered the wholesale price when the retailer paid morepromotional costs in the contract e reason for such anobservation was that the wholesale price was not related to ϕbut positively related to the overconfident level β when thedegree of fairness concern was moderate Furthermore theretailer accepted the contract designed by manufacturerssince the term U(1113937

scr )gemaxU(1113937

dr ) was guaranteed in

Table 3

6 Conclusions

61MainConclusions In this paper we analyzed the effect ofcognitive bias on equilibrium solutions including thewholesale price sales price sales effort and expected profitsunder demand uncertainty by introducing both overconfi-dence and fairness concern into the two-echelon supply chaine most important conclusion from the study was that forretailers with the two cognitive biases overconfidence waspositive and the main bias that affected supply chain decisionsand the retailersrsquo utility benefits were not only due to over-confidence but also fairness concern For manufacturersoverconfidence could increase the wholesale price and profitHowever the fairness concern had a negative effect on themFurthermore the decisions of the retailer in the decentralizedsupply chain deviated from the optimal decisions in thecentralized supply chainmodel Especially when the thresholdof overconfidence was more than a fixed value the decisionvariables under decentralized decision-making were actuallyhigher than the optimal variables under centralized decision-making Also the buyback contract with promotional costsharing achieved behavioral supply chain coordination Incontrast the promotional cost-sharing portion and buybackprice both benefited from overconfidence but fairness concernhad a detrimental effect on them However as the retailersrsquooverconfidence level increased and fairness concern was equalto 1 the flexibility of contract coordination was weakenedWhen the overconfidence level wasmore than 08 the contractbecame invalid and members were not able to cooperate

62 Management Insights is paper provided some in-teresting managerial insights for decision makers in thedecentralized supply chain wherein retailers had two pref-erences First overconfidence was beneficial to them interms of decision variables and yet the fairness concern

Table 2 e effect of the cognitive biases on the feasible domain of promotion cost-sharing portion ϕ

μβ

0 02 04 06 08 0810 [02884 04976] [03683 05587] [04528 06199] [06343 07754] [07835 08669] mdash1 [02545 04741] [03310 05331] [04167 05908] [06207 07516] [07800 08560] mdash2 [02352 04600] [03101 05156] [04034 05719] [06094 07321] [07736 08478] mdash

Table 3 e effect of the cognitive biases on the coordination of the supply chain

μ 1β

0 02 04 06 08qsc 32 32 32 32 32esc 16 16 16 16 16ϕ [02545 04741] [03310 05331] [04167 05908] [06207 07516] [07800 08560]r [2100 35363] [28268 39188] [30555 42637] [36209 44215] [40485 49164]wsc [49205 53687] [49016 53522] [54092 69568] [62513 72626] [80407 87453]1113937

dm 164103 269591 461996 772619 1112545

U(1113937scm) [168722 217850] [269600 336882] [463175 500938] [772619 814300] [1112545 1120376]

1113937dr 20333 352396 495161 632422 765317

U(1113937scr ) [20333 243043] [352396 400595] [495161 525161] [632422 657581] [765317 784632]

12 Complexity

only damaged the wholesale price Second the two behav-ioral preferences were both conducive to utilities for theretailer but not necessarily beneficial for the manufacturer toreceive profit and overconfidence helped to increasemanufacturersrsquo profit which led to nonoptimal decisionsthat dragged down the performance of the two-echelonsupply chain Finally for the two behavioral preferences ofthe retailer the manufacturer assessed the degree of over-confidence and fairness concern (especially the overconfi-dence) via historical data and the current order quantity orother indicators In contrast factual data indicated that twobehavioral preferences were in the right range (such asβ isin (0 08) and μ isin (0 2)) e manufacturers introducedthe buyback contract with promotional cost sharing into thepractice of pricing and ordered decentralized supply chainsFurthermore they designed a reasonable promotional cost-sharing portion and buyback price thereby achieving anoptimized decision and utility In practice the manufac-turers could analyze whether the sellers had two behavioralpreferences based on the historical data by cooperating withdownstream enterprises ie if two behavioral preferencesexisted they referred to the papersrsquo conclusions to redesign areasonable contract mechanism

63 Future Research Since cognitive biases could existamong all members of supply chains it is necessary toexplore the manufacturersrsquo cognitive biases and analyze thesupply chain coordination problem in the presence of thetwo behavior preferences Besides the overconfidence andfairness concern considered in our centralized decision are afuture research direction

Appendix

A Decentralized Supply Chain Model

In a decentralized supply chain consisting of a rationalmanufacturer and a retailer with cognitive biases maximumprofit or utility is the only goal e backward inductivemethod can be applied to solve a Stackelberg game whereinthe manufacturer is the leader and the retailer is a followere proof of the equilibrium solutions in equations (12)ndash(14)is as follows

e profit of the manufacturer and expected utility ofretailers are as follows respectively

wmax

1113945

d

m

(w minus c)q (A1)

maxqe

U 1113945d

r

⎞⎠ (1 + μ) p[a minus bp +(1 + β)e] + s 1113946θ2

minus Aq minus a minus bp +(1 + β)e + θ1( 1113857f θ1( 1113857( 1113857dθ11113896⎛⎝

minus v 1113946A

θ2a minus bp +(1 + β)e + θ1 minus q( 1113857f θ1( 1113857( 1113857dθ1 minus wq minus

12

ce21113897 minus μ(w minus c)q

(A2)

Solving the first-order partial derivatives of the retailerrsquosexpected utility concerning qd and ed which equal to 0 weobtained the following equations

zU( 1113937dr 1113873

zq (1 + μ) v minus w + F θ2( 1113857(s minus v)1113858 1113859 minus μ(w minus c) 0

(A3)

zU( 1113937dr 1113873

ze (1 + β) p minus v +(v minus s)F θ2( 11138571113858 1113859 minus ce 0

(A4)

Hessian matrix

H(q e) (s minus v)f θ2( 1113857 (v minus s)(1 + β)f θ2( 1113857

(v minus s)f θ2( 1113857 (s minus v)(1 + β)f θ2( 1113857 minus c1113888 1113889

(A5)

According to the assumptions vgt sgt 0 f(θ2)gt 0 andβ isin [0 1) us the Hessian matrix of U(1113937

dr ) is a negative

definite for q and e which meant that the optimal orderquantity and sales effort of retailers existed in the decen-tralized supply chain erefore we obtained the followingformula of the retailersrsquo order quantity and sales effort byjointly solving equations (A3) and (A4)

qd

(a minus bp)(v minus s)(1 + μ)c +(1 + β)2(v minus s)[(1 + μ)(p minus w) + μc] + Ac[(s + v)(1 + μ) + 2μc minus w]

(v minus s)(1 + μ)c (A6)

ed

(1 + β)[(1 + μ)p + μc minus (1 + 2μ)]w

(1 + μ)c (A7)

Complexity 13

Substituting equations (A6) into (A1) we attained

wmax

1113945

d

m

(w minus c) (a minus bp)(v minus s)(1 + μ)c +(1 + β)2(v minus s)[(1 + μ)p + μc] + Ac[(s + v)(1 + μ) + 2μc] minus (1 + 2μ)w 2Ac +(1 + β)2(v minus s)1113960 11139611113966 1113967

(v minus s)(1 + μ)c

(A8)

Taking into account the first-order condition of equation(A7) with respect to w rounded off to 0 we obtained`

z1113937dm

ze

(a minus bp)(v minus s)(1 + μ)c +(1 + β)2(v minus s)[(1 + μ)p + μc] + Ac[(s + v)(1 + μ) + 2μc] minus (2w minus c)(1 + 2μ) 2Ac +(1 + β)2(v minus s)1113960 1113961

(v minus s)(1 + μ)c

(A9)

Solving equation (A8) we found that the optimalwholesale price of the manufacturer is as follows

wdlowast

(a minus bp)(1 + μ)(v minus s)c +(v minus s)(1 + β)2[(1 + μ)p +(1 + 3μ)c] + Ac[(s + v)(1 + μ) + 2(1 + 3μ)c]

2(1 + 2μ) 2Ac +(v minus s)(1 + β)21113960 1113961 (A10)

By substituting equation (A9) into equations (A6) and(A7) we attained the optimal order quantity and sales effortwhich are displayed in equations (13) and (14)

B Propositions

Proof of Proposition 1 We solved the first derivatives ofoptimal order quantity qdlowast and sales effort edlowast with respect

to β in equations (13) and (14) respectively Meanwhiletaking the second-order derivative of qdlowast with respect to βwe obtained

zqdlowast

zβ

(1 + β)(p minus c)

c

z2qdlowast

zβ2

p minus c

c (B1)

zedlowast

zβ4(p minus c)(v minus s)2β β3 + 4β2 + 6β + 41113872 1113873 +(a minus bp)c(1 + β)2(v minus s)2 + Ac(v minus s)(1 + β)2 (2(p minus c) + v + s) + 2(Ac)2(4p minus c minus s minus v)1113960 1113961

2c (1 + β)2(v minus s) + 2Ac1113960 11139612

(B2)

Since the second derivative of edlowast with respect to β wasvery complicated there was no need to solve it According tothe assumptions pgt vgt cgt sgt 0 β isin (0 1) a minus bpgt 0 andcgt 0 so (zqdlowastzβ)gt 0 (z2qdlowastzβ2)gt 0 and (zedlowastzβ)gt 0

Proof of Proposition 2 Based on the process above we tookthe first-order and second-order partial derivatives of op-timal wholesale price wdlowast with respect to β and μ in equation(12) respectively

zwdlowast

zβ

c(v minus s)(1 + μ)(1 + β)[(a minus bp)(v minus s) + A(2p minus s minus v)]

(1 + 2μ) (1 + β)2(v minus s) + 2Ac1113960 11139612 (B3)

zwdlowast

zμ minus

(a minus bp)(v minus s)c +(p minus c)(v minus s)(1 + β)2 + Ac(s + v minus 2c)1113960 1113961

2(1 + 2μ)2 (v minus s)(1 + β)2 + 2Ac1113960 1113961 (B4)

14 Complexity

According to the optimal result of order quantity inequation (13) and the assumption qgt 0 the condition s +

v minus 2cgt 0 is found According to the assumptions we had(zwdlowastzβ)gt 0 and (zwdlowastzμ)lt 0

Proof of Proposition 3 e gap between qdlowast and qclowastdenoted by Δq is as follows

Δq qdlowast

minus qclowast

(p minus c)(v minus s) (1 + β)2 minus 21113960 1113961 minus Ac(s + v minus 2c) minus (a minus bp)(v minus s)c

2c(v minus s) (B5)

If (p minus c)(v minus s)[(1 + β)2 minus 2] is more thanAc(s + v minus 2c) + (a minus bp)(v minus s)c we have Δqgt 0 ieqdlowast gt qclowast In contrast if (p minus c)(v minus s)[(1 + β)2 minus 2] is less

than Ac(s + v minus 2c) + (a minus bp)(v minus s)c Δqlt 0 ieqdlowast lt qclowast

e gap between edlowast and eclowast denoted by Δe is asfollows

Δe edlowast

minus eclowast

Aβc(4p minus 2c minus s minus v) minus Ac(s + v minus 2c) minus (v minus s)(1 + β) (a minus bp)c + 1 minus β21113872 1113873(p minus c)1113960 1113961

2c 2Ac +(1 + β)2(v minus s)1113960 1113961 (B6)

Similarly if Aβc(4p minus 2c minus s minus v)gtAc(s + v minus 2c)

+(v minus s)(1 + β)[(a minus bp)c + (1 minus β2)(p minus c)] we haveΔe edlowast minus eclowast gt 0 In contrast ifAβc(4p minus 2c minus s minus v)ltAc(s + v minus 2c)+

(v minus s)(1 + β)[(a minus bp)c + (1 minus β2)(p minus c)] we haveΔe edlowast minus eclowast lt 0

Upon solving the first-order derivative of Δq and Δe withrespect to β respectively we obtained

zΔ q

zβ

(1 + β)(p minus c)

c (B7)

zΔ e

zβ4(p minus c)(v minus s)2 β3 + 4β2 + 6β + 41113872 1113873β +(a minus bp)c(1 + β)2(v minus s)2 + Ac(v minus s)(1 + β)2 2(p minus c) +v + s) + 2(Ac)2(4p minus c minus s minus v)1113960 1113961

2c (1 + β)2(v minus s) + 2Ac1113960 11139612

(B8)

According to the assumptions pgt vgt cgt sgt 0 β isin (0 1)cgt 0 and a minus bpgt 0 (zΔ qzβ)gt 0 and (zΔ ezβ)gt 0 can beproved Hence the gaps increased as a function with respectto β

Proof of Proposition 4 e proof for Proposition 4 is similarto Proposition 2 Herein we took first-order and second-order partial derivatives of optimal wholesale price wsclowast withrespect to β μ and ϕ in equation (18) respectively esolving results are as follows

zwsclowast

zβ

[(1 + 2μ)ϕ minus μ](p minus c)

(1 + 2μ)(1 + β)2

zwsclowast

zμ minus

(p minus c)β(1 + β)(1 + 2μ)2

zwsclowast

zϕ minus

p minus c

1 + β

(B9)

(zwsclowastzβ) if ϕgt (μ(1 + 2μ)) is more than 0 otherwiseϕgt (μ(1 + 2μ)) is less than 0 us wsc could be increasingwith respect to β nevertheless decreasing with respect to βBesides according to the assumptions pgt vgt cgt sgt 0β isin (0 1) cgt 0 and a minus bpgt 0 it can be evidently foundthat (zwsclowastzμ)gt 0 and (zwsclowastzϕ)gt 0

Data Availability

e data used to support the findings of this study are in-cluded within the article

Conflicts of Interest

e authors declare no conflicts of interest

Acknowledgments

is work was supported by a major project financed by theNational Social Science Fund of China (Approval nos71461009 71662011 and 71761015)

Complexity 15

References

[1] Y Ren D C Croson and R T A Croson ldquoe overconfidentnewsvendorrdquo Journal of the Operational Research Societyvol 68 no 5 pp 496ndash506 2017

[2] T H Cui and P Mallucci ldquoFairness ideals in distributionchannelsrdquo Journal of Marketing Research vol 53 no 6pp 969ndash987 2016

[3] K Chen X Wang M Huang and W-K Ching ldquoSalesforcecontract design joint pricing and production planning withasymmetric overconfidence sales agentrdquo Journal of Industrialamp Management Optimization vol 13 no 2 pp 873ndash8992017

[4] X X Zheng Z Liu K W Li et al ldquoCooperative game ap-proaches to coordinating a three-echelon closed-loop supplychain with fairness concernsrdquo International Journal of Pro-duction Economic vol 212 pp 92ndash110 2018

[5] China Internet Watch False Prosperity behind Double 11Refund Rate Increased to 69 2014 httpswwwchinainternetwatchcom11643false-prosperity-behind-double-11-refund-rate-increased-to-69

[6] C L Lu ldquoPampGrsquos collaborative supply chain transformationrdquoEnterprise Management vol 10 pp 79-80 2016

[7] M Li N C Petruzzi and J Zhang ldquoOverconfident competingnewsvendorsrdquo Management Science vol 63 no 8pp 2637ndash2646 2016

[8] T Nie ldquoDual-fairness supply chain with quantity discountcontractsrdquo European Journal of Operational Researchvol 258 no 2 pp 491ndash500 2017

[9] X Du and B Jiang ldquoSignaling through price and quality toconsumers with fairness concernsrdquo Journal of MarketingResearch vol 53 no 6 pp 988ndash1000 2016

[10] V Kurz A Orland and K Posadzy ldquoFairness versus effi-ciency how procedural fairness concerns affect coordina-tionrdquo Experimental Economics vol 21 no 3 pp 601ndash6262018

[11] S Villa and J A Castantildeeda ldquoTransshipments in supplychains a behavioral investigationrdquo European Journal of Op-erational Research vol 269 no 2 pp 715ndash729 2018

[12] B Gu Y Fu and Y Li ldquoFresh-keeping effort and channelperformance in a fresh product supply chain with loss-averseconsumersrsquo returnsrdquo Mathematical Problems in Engineeringvol 2018 Article ID 4717094 20 pages 2018

[13] D Croson R Croson and Y Ren How to Manage anOverconfident Newsvendor 2008 httpcbeesutdallasedupapersCrosonRenCmsonMS2008

[14] M Li ldquoOverconfident distribution channelsrdquo Production andOperations Management vol 28 no 6 pp 1347ndash1365 2019

[15] W Liu D Wang O Tang and D Zhu ldquoe impacts oflogistics service integratorrsquos overconfidence behaviour onsupply chain decision under demand surgerdquo EuropeanJournal of Industrial Engineering vol 12 no 4 pp 558ndash5972018

[16] L Xu X Shi P Du K Govindan and Z Zhang ldquoOptimi-zation on pricing and overconfidence problem in a duopo-listic supply chainrdquo Computers amp Operations Researchvol 101 pp 162ndash172 2019

[17] Y Li and M Shan ldquoAdvance selling decisions with over-confident consumersrdquo Journal of Industrial and ManagementOptimization vol 2 no 3 pp 891ndash905 2016

[18] T H Cui J S Raju and Z J Zhang ldquoFairness and channelcoordinationrdquo Management Science vol 53 no 8pp 1303ndash1314 2007

[19] X Wu and J A Niederhoff ldquoFairness in selling to thenewsvendorrdquo Production and Operations Managementvol 23 no 11 pp 2002ndash2022 2014

[20] Z Chen and S-I Ivan Su ldquoSocial welfare maximization withthe least subsidy photovoltaic supply chain equilibrium andcoordination with fairness concernrdquo Renewable Energyvol 132 pp 1332ndash1347 2019

[21] W Liu D Wang X Shen X Yan and W Wei ldquoe impactsof distributional and peer-induced fairness concerns on thedecision-making of order allocation in logistics service supplychainrdquo Transportation Research Part E Logistics and Trans-portation Review vol 116 pp 102ndash122 2018

[22] L Zhang H Zhou Y Liu and R Lu ldquoOptimal environmentalquality and price with consumer environmental awarenessand retailerrsquos fairness concerns in supply chainrdquo Journal ofCleaner Production vol 213 pp 1063ndash1079 2019

[23] X LWang S Q Hu and X B Liu ldquoSupply coordination withoption contract considering overconfidence under randomdemand and yieldsrdquo Computer Integrated ManufacturingSystem vol 24 no 11 pp 2898ndash2908 2018

[24] W Jiang and J Liu ldquoInventory financing with overconfidentsupplier based on supply chain contractrdquo MathematicalProblems in Engineering vol 2018 Article ID 505438712 pages 2018

[25] P Ma H Wang and J Shang ldquoContract design for two-stagesupply chain coordination integrating manufacturer-qualityand retailer-marketing effortsrdquo International Journal ofProduction Economics vol 146 no 2 pp 745ndash755 2013

[26] J Liu W F Jiang and Y P Tang ldquoe supply chain coor-dination based on carbon reduction of overconfident man-ufacturerrdquo Journal of Jiangxi Normal University (NaturalScience) vol 42 no 2 pp 203ndash207 2018

[27] R Dubey N Altay and C Blome ldquoSwift trust and com-mitment the missing links for humanitarian supply chaincoordinationrdquo Annals of Operations Research vol 283 no 1-2 pp 159ndash177 2019

[28] C Cui Z Feng and C Tan ldquoCredibilistic loss aversion nashequilibrium for bimatrix games with triangular fuzzy payoffsrdquoComplexity vol 2018 Article ID 7143586 16 pages 2018

[29] E Fehr and KM Schmidt ldquoA theory of fairness competitionand cooperationrdquo e Quarterly Journal of Economicsvol 114 no 3 pp 817ndash868 1999

[30] Z Qin and J Yang ldquoAnalysis of a revenue-sharing contract insupply chain managementrdquo International Journal of LogisticsResearch and Applications vol 11 no 1 pp 17ndash29 2008

[31] S-Z Bai and J Ni ldquoResearch on buy-back contract with saleseffort cost sharingrdquo in Proceedings of the 2009 Chinese Controland Decision Conference IEEE Guilin China pp 1917ndash1921June 2009

[32] E Karagozoglu and K Keskin ldquoTime-varying fairness con-cerns delay and disagreement in bargainingrdquo Journal ofEconomic Behavior and Organization vol 147 pp 115ndash1282018

16 Complexity

only damaged the wholesale price Second the two behav-ioral preferences were both conducive to utilities for theretailer but not necessarily beneficial for the manufacturer toreceive profit and overconfidence helped to increasemanufacturersrsquo profit which led to nonoptimal decisionsthat dragged down the performance of the two-echelonsupply chain Finally for the two behavioral preferences ofthe retailer the manufacturer assessed the degree of over-confidence and fairness concern (especially the overconfi-dence) via historical data and the current order quantity orother indicators In contrast factual data indicated that twobehavioral preferences were in the right range (such asβ isin (0 08) and μ isin (0 2)) e manufacturers introducedthe buyback contract with promotional cost sharing into thepractice of pricing and ordered decentralized supply chainsFurthermore they designed a reasonable promotional cost-sharing portion and buyback price thereby achieving anoptimized decision and utility In practice the manufac-turers could analyze whether the sellers had two behavioralpreferences based on the historical data by cooperating withdownstream enterprises ie if two behavioral preferencesexisted they referred to the papersrsquo conclusions to redesign areasonable contract mechanism

63 Future Research Since cognitive biases could existamong all members of supply chains it is necessary toexplore the manufacturersrsquo cognitive biases and analyze thesupply chain coordination problem in the presence of thetwo behavior preferences Besides the overconfidence andfairness concern considered in our centralized decision are afuture research direction

Appendix

A Decentralized Supply Chain Model

In a decentralized supply chain consisting of a rationalmanufacturer and a retailer with cognitive biases maximumprofit or utility is the only goal e backward inductivemethod can be applied to solve a Stackelberg game whereinthe manufacturer is the leader and the retailer is a followere proof of the equilibrium solutions in equations (12)ndash(14)is as follows

e profit of the manufacturer and expected utility ofretailers are as follows respectively

wmax

1113945

d

m

(w minus c)q (A1)

maxqe

U 1113945d

r

⎞⎠ (1 + μ) p[a minus bp +(1 + β)e] + s 1113946θ2

minus Aq minus a minus bp +(1 + β)e + θ1( 1113857f θ1( 1113857( 1113857dθ11113896⎛⎝

minus v 1113946A

θ2a minus bp +(1 + β)e + θ1 minus q( 1113857f θ1( 1113857( 1113857dθ1 minus wq minus

12

ce21113897 minus μ(w minus c)q

(A2)

Solving the first-order partial derivatives of the retailerrsquosexpected utility concerning qd and ed which equal to 0 weobtained the following equations

zU( 1113937dr 1113873

zq (1 + μ) v minus w + F θ2( 1113857(s minus v)1113858 1113859 minus μ(w minus c) 0

(A3)

zU( 1113937dr 1113873

ze (1 + β) p minus v +(v minus s)F θ2( 11138571113858 1113859 minus ce 0

(A4)

Hessian matrix

H(q e) (s minus v)f θ2( 1113857 (v minus s)(1 + β)f θ2( 1113857

(v minus s)f θ2( 1113857 (s minus v)(1 + β)f θ2( 1113857 minus c1113888 1113889

(A5)

According to the assumptions vgt sgt 0 f(θ2)gt 0 andβ isin [0 1) us the Hessian matrix of U(1113937

dr ) is a negative

definite for q and e which meant that the optimal orderquantity and sales effort of retailers existed in the decen-tralized supply chain erefore we obtained the followingformula of the retailersrsquo order quantity and sales effort byjointly solving equations (A3) and (A4)

qd

(a minus bp)(v minus s)(1 + μ)c +(1 + β)2(v minus s)[(1 + μ)(p minus w) + μc] + Ac[(s + v)(1 + μ) + 2μc minus w]

(v minus s)(1 + μ)c (A6)

ed

(1 + β)[(1 + μ)p + μc minus (1 + 2μ)]w

(1 + μ)c (A7)

Complexity 13

Substituting equations (A6) into (A1) we attained

wmax

1113945

d

m

(w minus c) (a minus bp)(v minus s)(1 + μ)c +(1 + β)2(v minus s)[(1 + μ)p + μc] + Ac[(s + v)(1 + μ) + 2μc] minus (1 + 2μ)w 2Ac +(1 + β)2(v minus s)1113960 11139611113966 1113967

(v minus s)(1 + μ)c

(A8)

Taking into account the first-order condition of equation(A7) with respect to w rounded off to 0 we obtained`

z1113937dm

ze

(a minus bp)(v minus s)(1 + μ)c +(1 + β)2(v minus s)[(1 + μ)p + μc] + Ac[(s + v)(1 + μ) + 2μc] minus (2w minus c)(1 + 2μ) 2Ac +(1 + β)2(v minus s)1113960 1113961

(v minus s)(1 + μ)c

(A9)

Solving equation (A8) we found that the optimalwholesale price of the manufacturer is as follows

wdlowast

(a minus bp)(1 + μ)(v minus s)c +(v minus s)(1 + β)2[(1 + μ)p +(1 + 3μ)c] + Ac[(s + v)(1 + μ) + 2(1 + 3μ)c]

2(1 + 2μ) 2Ac +(v minus s)(1 + β)21113960 1113961 (A10)

By substituting equation (A9) into equations (A6) and(A7) we attained the optimal order quantity and sales effortwhich are displayed in equations (13) and (14)

B Propositions

Proof of Proposition 1 We solved the first derivatives ofoptimal order quantity qdlowast and sales effort edlowast with respect

to β in equations (13) and (14) respectively Meanwhiletaking the second-order derivative of qdlowast with respect to βwe obtained

zqdlowast

zβ

(1 + β)(p minus c)

c

z2qdlowast

zβ2

p minus c

c (B1)

zedlowast

zβ4(p minus c)(v minus s)2β β3 + 4β2 + 6β + 41113872 1113873 +(a minus bp)c(1 + β)2(v minus s)2 + Ac(v minus s)(1 + β)2 (2(p minus c) + v + s) + 2(Ac)2(4p minus c minus s minus v)1113960 1113961

2c (1 + β)2(v minus s) + 2Ac1113960 11139612

(B2)

Since the second derivative of edlowast with respect to β wasvery complicated there was no need to solve it According tothe assumptions pgt vgt cgt sgt 0 β isin (0 1) a minus bpgt 0 andcgt 0 so (zqdlowastzβ)gt 0 (z2qdlowastzβ2)gt 0 and (zedlowastzβ)gt 0

Proof of Proposition 2 Based on the process above we tookthe first-order and second-order partial derivatives of op-timal wholesale price wdlowast with respect to β and μ in equation(12) respectively

zwdlowast

zβ

c(v minus s)(1 + μ)(1 + β)[(a minus bp)(v minus s) + A(2p minus s minus v)]

(1 + 2μ) (1 + β)2(v minus s) + 2Ac1113960 11139612 (B3)

zwdlowast

zμ minus

(a minus bp)(v minus s)c +(p minus c)(v minus s)(1 + β)2 + Ac(s + v minus 2c)1113960 1113961

2(1 + 2μ)2 (v minus s)(1 + β)2 + 2Ac1113960 1113961 (B4)

14 Complexity

According to the optimal result of order quantity inequation (13) and the assumption qgt 0 the condition s +

v minus 2cgt 0 is found According to the assumptions we had(zwdlowastzβ)gt 0 and (zwdlowastzμ)lt 0

Proof of Proposition 3 e gap between qdlowast and qclowastdenoted by Δq is as follows

Δq qdlowast

minus qclowast

(p minus c)(v minus s) (1 + β)2 minus 21113960 1113961 minus Ac(s + v minus 2c) minus (a minus bp)(v minus s)c

2c(v minus s) (B5)

If (p minus c)(v minus s)[(1 + β)2 minus 2] is more thanAc(s + v minus 2c) + (a minus bp)(v minus s)c we have Δqgt 0 ieqdlowast gt qclowast In contrast if (p minus c)(v minus s)[(1 + β)2 minus 2] is less

than Ac(s + v minus 2c) + (a minus bp)(v minus s)c Δqlt 0 ieqdlowast lt qclowast

e gap between edlowast and eclowast denoted by Δe is asfollows

Δe edlowast

minus eclowast

Aβc(4p minus 2c minus s minus v) minus Ac(s + v minus 2c) minus (v minus s)(1 + β) (a minus bp)c + 1 minus β21113872 1113873(p minus c)1113960 1113961

2c 2Ac +(1 + β)2(v minus s)1113960 1113961 (B6)

Similarly if Aβc(4p minus 2c minus s minus v)gtAc(s + v minus 2c)

+(v minus s)(1 + β)[(a minus bp)c + (1 minus β2)(p minus c)] we haveΔe edlowast minus eclowast gt 0 In contrast ifAβc(4p minus 2c minus s minus v)ltAc(s + v minus 2c)+

(v minus s)(1 + β)[(a minus bp)c + (1 minus β2)(p minus c)] we haveΔe edlowast minus eclowast lt 0

Upon solving the first-order derivative of Δq and Δe withrespect to β respectively we obtained

zΔ q

zβ

(1 + β)(p minus c)

c (B7)

zΔ e

zβ4(p minus c)(v minus s)2 β3 + 4β2 + 6β + 41113872 1113873β +(a minus bp)c(1 + β)2(v minus s)2 + Ac(v minus s)(1 + β)2 2(p minus c) +v + s) + 2(Ac)2(4p minus c minus s minus v)1113960 1113961

2c (1 + β)2(v minus s) + 2Ac1113960 11139612

(B8)

According to the assumptions pgt vgt cgt sgt 0 β isin (0 1)cgt 0 and a minus bpgt 0 (zΔ qzβ)gt 0 and (zΔ ezβ)gt 0 can beproved Hence the gaps increased as a function with respectto β

Proof of Proposition 4 e proof for Proposition 4 is similarto Proposition 2 Herein we took first-order and second-order partial derivatives of optimal wholesale price wsclowast withrespect to β μ and ϕ in equation (18) respectively esolving results are as follows

zwsclowast

zβ

[(1 + 2μ)ϕ minus μ](p minus c)

(1 + 2μ)(1 + β)2

zwsclowast

zμ minus

(p minus c)β(1 + β)(1 + 2μ)2

zwsclowast

zϕ minus

p minus c

1 + β

(B9)

(zwsclowastzβ) if ϕgt (μ(1 + 2μ)) is more than 0 otherwiseϕgt (μ(1 + 2μ)) is less than 0 us wsc could be increasingwith respect to β nevertheless decreasing with respect to βBesides according to the assumptions pgt vgt cgt sgt 0β isin (0 1) cgt 0 and a minus bpgt 0 it can be evidently foundthat (zwsclowastzμ)gt 0 and (zwsclowastzϕ)gt 0

Data Availability

e data used to support the findings of this study are in-cluded within the article

Conflicts of Interest

e authors declare no conflicts of interest

Acknowledgments

is work was supported by a major project financed by theNational Social Science Fund of China (Approval nos71461009 71662011 and 71761015)

Complexity 15

References

[1] Y Ren D C Croson and R T A Croson ldquoe overconfidentnewsvendorrdquo Journal of the Operational Research Societyvol 68 no 5 pp 496ndash506 2017

[2] T H Cui and P Mallucci ldquoFairness ideals in distributionchannelsrdquo Journal of Marketing Research vol 53 no 6pp 969ndash987 2016

[3] K Chen X Wang M Huang and W-K Ching ldquoSalesforcecontract design joint pricing and production planning withasymmetric overconfidence sales agentrdquo Journal of Industrialamp Management Optimization vol 13 no 2 pp 873ndash8992017

[4] X X Zheng Z Liu K W Li et al ldquoCooperative game ap-proaches to coordinating a three-echelon closed-loop supplychain with fairness concernsrdquo International Journal of Pro-duction Economic vol 212 pp 92ndash110 2018

[5] China Internet Watch False Prosperity behind Double 11Refund Rate Increased to 69 2014 httpswwwchinainternetwatchcom11643false-prosperity-behind-double-11-refund-rate-increased-to-69

[6] C L Lu ldquoPampGrsquos collaborative supply chain transformationrdquoEnterprise Management vol 10 pp 79-80 2016

[7] M Li N C Petruzzi and J Zhang ldquoOverconfident competingnewsvendorsrdquo Management Science vol 63 no 8pp 2637ndash2646 2016

[8] T Nie ldquoDual-fairness supply chain with quantity discountcontractsrdquo European Journal of Operational Researchvol 258 no 2 pp 491ndash500 2017

[9] X Du and B Jiang ldquoSignaling through price and quality toconsumers with fairness concernsrdquo Journal of MarketingResearch vol 53 no 6 pp 988ndash1000 2016

[10] V Kurz A Orland and K Posadzy ldquoFairness versus effi-ciency how procedural fairness concerns affect coordina-tionrdquo Experimental Economics vol 21 no 3 pp 601ndash6262018

[11] S Villa and J A Castantildeeda ldquoTransshipments in supplychains a behavioral investigationrdquo European Journal of Op-erational Research vol 269 no 2 pp 715ndash729 2018

[12] B Gu Y Fu and Y Li ldquoFresh-keeping effort and channelperformance in a fresh product supply chain with loss-averseconsumersrsquo returnsrdquo Mathematical Problems in Engineeringvol 2018 Article ID 4717094 20 pages 2018

[13] D Croson R Croson and Y Ren How to Manage anOverconfident Newsvendor 2008 httpcbeesutdallasedupapersCrosonRenCmsonMS2008

[14] M Li ldquoOverconfident distribution channelsrdquo Production andOperations Management vol 28 no 6 pp 1347ndash1365 2019

[15] W Liu D Wang O Tang and D Zhu ldquoe impacts oflogistics service integratorrsquos overconfidence behaviour onsupply chain decision under demand surgerdquo EuropeanJournal of Industrial Engineering vol 12 no 4 pp 558ndash5972018

[16] L Xu X Shi P Du K Govindan and Z Zhang ldquoOptimi-zation on pricing and overconfidence problem in a duopo-listic supply chainrdquo Computers amp Operations Researchvol 101 pp 162ndash172 2019

[17] Y Li and M Shan ldquoAdvance selling decisions with over-confident consumersrdquo Journal of Industrial and ManagementOptimization vol 2 no 3 pp 891ndash905 2016

[18] T H Cui J S Raju and Z J Zhang ldquoFairness and channelcoordinationrdquo Management Science vol 53 no 8pp 1303ndash1314 2007

[19] X Wu and J A Niederhoff ldquoFairness in selling to thenewsvendorrdquo Production and Operations Managementvol 23 no 11 pp 2002ndash2022 2014

[20] Z Chen and S-I Ivan Su ldquoSocial welfare maximization withthe least subsidy photovoltaic supply chain equilibrium andcoordination with fairness concernrdquo Renewable Energyvol 132 pp 1332ndash1347 2019

[21] W Liu D Wang X Shen X Yan and W Wei ldquoe impactsof distributional and peer-induced fairness concerns on thedecision-making of order allocation in logistics service supplychainrdquo Transportation Research Part E Logistics and Trans-portation Review vol 116 pp 102ndash122 2018

[22] L Zhang H Zhou Y Liu and R Lu ldquoOptimal environmentalquality and price with consumer environmental awarenessand retailerrsquos fairness concerns in supply chainrdquo Journal ofCleaner Production vol 213 pp 1063ndash1079 2019

[23] X LWang S Q Hu and X B Liu ldquoSupply coordination withoption contract considering overconfidence under randomdemand and yieldsrdquo Computer Integrated ManufacturingSystem vol 24 no 11 pp 2898ndash2908 2018

[24] W Jiang and J Liu ldquoInventory financing with overconfidentsupplier based on supply chain contractrdquo MathematicalProblems in Engineering vol 2018 Article ID 505438712 pages 2018

[25] P Ma H Wang and J Shang ldquoContract design for two-stagesupply chain coordination integrating manufacturer-qualityand retailer-marketing effortsrdquo International Journal ofProduction Economics vol 146 no 2 pp 745ndash755 2013

[26] J Liu W F Jiang and Y P Tang ldquoe supply chain coor-dination based on carbon reduction of overconfident man-ufacturerrdquo Journal of Jiangxi Normal University (NaturalScience) vol 42 no 2 pp 203ndash207 2018

[27] R Dubey N Altay and C Blome ldquoSwift trust and com-mitment the missing links for humanitarian supply chaincoordinationrdquo Annals of Operations Research vol 283 no 1-2 pp 159ndash177 2019

[28] C Cui Z Feng and C Tan ldquoCredibilistic loss aversion nashequilibrium for bimatrix games with triangular fuzzy payoffsrdquoComplexity vol 2018 Article ID 7143586 16 pages 2018

[29] E Fehr and KM Schmidt ldquoA theory of fairness competitionand cooperationrdquo e Quarterly Journal of Economicsvol 114 no 3 pp 817ndash868 1999

[30] Z Qin and J Yang ldquoAnalysis of a revenue-sharing contract insupply chain managementrdquo International Journal of LogisticsResearch and Applications vol 11 no 1 pp 17ndash29 2008

[31] S-Z Bai and J Ni ldquoResearch on buy-back contract with saleseffort cost sharingrdquo in Proceedings of the 2009 Chinese Controland Decision Conference IEEE Guilin China pp 1917ndash1921June 2009

[32] E Karagozoglu and K Keskin ldquoTime-varying fairness con-cerns delay and disagreement in bargainingrdquo Journal ofEconomic Behavior and Organization vol 147 pp 115ndash1282018

16 Complexity

Substituting equations (A6) into (A1) we attained

wmax

1113945

d

m

(w minus c) (a minus bp)(v minus s)(1 + μ)c +(1 + β)2(v minus s)[(1 + μ)p + μc] + Ac[(s + v)(1 + μ) + 2μc] minus (1 + 2μ)w 2Ac +(1 + β)2(v minus s)1113960 11139611113966 1113967

(v minus s)(1 + μ)c

(A8)

Taking into account the first-order condition of equation(A7) with respect to w rounded off to 0 we obtained`

z1113937dm

ze

(a minus bp)(v minus s)(1 + μ)c +(1 + β)2(v minus s)[(1 + μ)p + μc] + Ac[(s + v)(1 + μ) + 2μc] minus (2w minus c)(1 + 2μ) 2Ac +(1 + β)2(v minus s)1113960 1113961

(v minus s)(1 + μ)c

(A9)

Solving equation (A8) we found that the optimalwholesale price of the manufacturer is as follows

wdlowast

(a minus bp)(1 + μ)(v minus s)c +(v minus s)(1 + β)2[(1 + μ)p +(1 + 3μ)c] + Ac[(s + v)(1 + μ) + 2(1 + 3μ)c]

2(1 + 2μ) 2Ac +(v minus s)(1 + β)21113960 1113961 (A10)

By substituting equation (A9) into equations (A6) and(A7) we attained the optimal order quantity and sales effortwhich are displayed in equations (13) and (14)

B Propositions

Proof of Proposition 1 We solved the first derivatives ofoptimal order quantity qdlowast and sales effort edlowast with respect

to β in equations (13) and (14) respectively Meanwhiletaking the second-order derivative of qdlowast with respect to βwe obtained

zqdlowast

zβ

(1 + β)(p minus c)

c

z2qdlowast

zβ2

p minus c

c (B1)

zedlowast

zβ4(p minus c)(v minus s)2β β3 + 4β2 + 6β + 41113872 1113873 +(a minus bp)c(1 + β)2(v minus s)2 + Ac(v minus s)(1 + β)2 (2(p minus c) + v + s) + 2(Ac)2(4p minus c minus s minus v)1113960 1113961

2c (1 + β)2(v minus s) + 2Ac1113960 11139612

(B2)

Since the second derivative of edlowast with respect to β wasvery complicated there was no need to solve it According tothe assumptions pgt vgt cgt sgt 0 β isin (0 1) a minus bpgt 0 andcgt 0 so (zqdlowastzβ)gt 0 (z2qdlowastzβ2)gt 0 and (zedlowastzβ)gt 0

Proof of Proposition 2 Based on the process above we tookthe first-order and second-order partial derivatives of op-timal wholesale price wdlowast with respect to β and μ in equation(12) respectively

zwdlowast

zβ

c(v minus s)(1 + μ)(1 + β)[(a minus bp)(v minus s) + A(2p minus s minus v)]

(1 + 2μ) (1 + β)2(v minus s) + 2Ac1113960 11139612 (B3)

zwdlowast

zμ minus

(a minus bp)(v minus s)c +(p minus c)(v minus s)(1 + β)2 + Ac(s + v minus 2c)1113960 1113961

2(1 + 2μ)2 (v minus s)(1 + β)2 + 2Ac1113960 1113961 (B4)

14 Complexity

According to the optimal result of order quantity inequation (13) and the assumption qgt 0 the condition s +

v minus 2cgt 0 is found According to the assumptions we had(zwdlowastzβ)gt 0 and (zwdlowastzμ)lt 0

Proof of Proposition 3 e gap between qdlowast and qclowastdenoted by Δq is as follows

Δq qdlowast

minus qclowast

(p minus c)(v minus s) (1 + β)2 minus 21113960 1113961 minus Ac(s + v minus 2c) minus (a minus bp)(v minus s)c

2c(v minus s) (B5)

If (p minus c)(v minus s)[(1 + β)2 minus 2] is more thanAc(s + v minus 2c) + (a minus bp)(v minus s)c we have Δqgt 0 ieqdlowast gt qclowast In contrast if (p minus c)(v minus s)[(1 + β)2 minus 2] is less

than Ac(s + v minus 2c) + (a minus bp)(v minus s)c Δqlt 0 ieqdlowast lt qclowast

e gap between edlowast and eclowast denoted by Δe is asfollows

Δe edlowast

minus eclowast

Aβc(4p minus 2c minus s minus v) minus Ac(s + v minus 2c) minus (v minus s)(1 + β) (a minus bp)c + 1 minus β21113872 1113873(p minus c)1113960 1113961

2c 2Ac +(1 + β)2(v minus s)1113960 1113961 (B6)

Similarly if Aβc(4p minus 2c minus s minus v)gtAc(s + v minus 2c)

+(v minus s)(1 + β)[(a minus bp)c + (1 minus β2)(p minus c)] we haveΔe edlowast minus eclowast gt 0 In contrast ifAβc(4p minus 2c minus s minus v)ltAc(s + v minus 2c)+

(v minus s)(1 + β)[(a minus bp)c + (1 minus β2)(p minus c)] we haveΔe edlowast minus eclowast lt 0

Upon solving the first-order derivative of Δq and Δe withrespect to β respectively we obtained

zΔ q

zβ

(1 + β)(p minus c)

c (B7)

zΔ e

zβ4(p minus c)(v minus s)2 β3 + 4β2 + 6β + 41113872 1113873β +(a minus bp)c(1 + β)2(v minus s)2 + Ac(v minus s)(1 + β)2 2(p minus c) +v + s) + 2(Ac)2(4p minus c minus s minus v)1113960 1113961

2c (1 + β)2(v minus s) + 2Ac1113960 11139612

(B8)

According to the assumptions pgt vgt cgt sgt 0 β isin (0 1)cgt 0 and a minus bpgt 0 (zΔ qzβ)gt 0 and (zΔ ezβ)gt 0 can beproved Hence the gaps increased as a function with respectto β

Proof of Proposition 4 e proof for Proposition 4 is similarto Proposition 2 Herein we took first-order and second-order partial derivatives of optimal wholesale price wsclowast withrespect to β μ and ϕ in equation (18) respectively esolving results are as follows

zwsclowast

zβ

[(1 + 2μ)ϕ minus μ](p minus c)

(1 + 2μ)(1 + β)2

zwsclowast

zμ minus

(p minus c)β(1 + β)(1 + 2μ)2

zwsclowast

zϕ minus

p minus c

1 + β

(B9)

(zwsclowastzβ) if ϕgt (μ(1 + 2μ)) is more than 0 otherwiseϕgt (μ(1 + 2μ)) is less than 0 us wsc could be increasingwith respect to β nevertheless decreasing with respect to βBesides according to the assumptions pgt vgt cgt sgt 0β isin (0 1) cgt 0 and a minus bpgt 0 it can be evidently foundthat (zwsclowastzμ)gt 0 and (zwsclowastzϕ)gt 0

Data Availability

e data used to support the findings of this study are in-cluded within the article

Conflicts of Interest

e authors declare no conflicts of interest

Acknowledgments

is work was supported by a major project financed by theNational Social Science Fund of China (Approval nos71461009 71662011 and 71761015)

Complexity 15

References

[1] Y Ren D C Croson and R T A Croson ldquoe overconfidentnewsvendorrdquo Journal of the Operational Research Societyvol 68 no 5 pp 496ndash506 2017

[2] T H Cui and P Mallucci ldquoFairness ideals in distributionchannelsrdquo Journal of Marketing Research vol 53 no 6pp 969ndash987 2016

[3] K Chen X Wang M Huang and W-K Ching ldquoSalesforcecontract design joint pricing and production planning withasymmetric overconfidence sales agentrdquo Journal of Industrialamp Management Optimization vol 13 no 2 pp 873ndash8992017

[4] X X Zheng Z Liu K W Li et al ldquoCooperative game ap-proaches to coordinating a three-echelon closed-loop supplychain with fairness concernsrdquo International Journal of Pro-duction Economic vol 212 pp 92ndash110 2018

[5] China Internet Watch False Prosperity behind Double 11Refund Rate Increased to 69 2014 httpswwwchinainternetwatchcom11643false-prosperity-behind-double-11-refund-rate-increased-to-69

[6] C L Lu ldquoPampGrsquos collaborative supply chain transformationrdquoEnterprise Management vol 10 pp 79-80 2016

[7] M Li N C Petruzzi and J Zhang ldquoOverconfident competingnewsvendorsrdquo Management Science vol 63 no 8pp 2637ndash2646 2016

[8] T Nie ldquoDual-fairness supply chain with quantity discountcontractsrdquo European Journal of Operational Researchvol 258 no 2 pp 491ndash500 2017

[9] X Du and B Jiang ldquoSignaling through price and quality toconsumers with fairness concernsrdquo Journal of MarketingResearch vol 53 no 6 pp 988ndash1000 2016

[10] V Kurz A Orland and K Posadzy ldquoFairness versus effi-ciency how procedural fairness concerns affect coordina-tionrdquo Experimental Economics vol 21 no 3 pp 601ndash6262018

[11] S Villa and J A Castantildeeda ldquoTransshipments in supplychains a behavioral investigationrdquo European Journal of Op-erational Research vol 269 no 2 pp 715ndash729 2018

[12] B Gu Y Fu and Y Li ldquoFresh-keeping effort and channelperformance in a fresh product supply chain with loss-averseconsumersrsquo returnsrdquo Mathematical Problems in Engineeringvol 2018 Article ID 4717094 20 pages 2018

[13] D Croson R Croson and Y Ren How to Manage anOverconfident Newsvendor 2008 httpcbeesutdallasedupapersCrosonRenCmsonMS2008

[14] M Li ldquoOverconfident distribution channelsrdquo Production andOperations Management vol 28 no 6 pp 1347ndash1365 2019

[15] W Liu D Wang O Tang and D Zhu ldquoe impacts oflogistics service integratorrsquos overconfidence behaviour onsupply chain decision under demand surgerdquo EuropeanJournal of Industrial Engineering vol 12 no 4 pp 558ndash5972018

[16] L Xu X Shi P Du K Govindan and Z Zhang ldquoOptimi-zation on pricing and overconfidence problem in a duopo-listic supply chainrdquo Computers amp Operations Researchvol 101 pp 162ndash172 2019

[17] Y Li and M Shan ldquoAdvance selling decisions with over-confident consumersrdquo Journal of Industrial and ManagementOptimization vol 2 no 3 pp 891ndash905 2016

[18] T H Cui J S Raju and Z J Zhang ldquoFairness and channelcoordinationrdquo Management Science vol 53 no 8pp 1303ndash1314 2007

[19] X Wu and J A Niederhoff ldquoFairness in selling to thenewsvendorrdquo Production and Operations Managementvol 23 no 11 pp 2002ndash2022 2014

[20] Z Chen and S-I Ivan Su ldquoSocial welfare maximization withthe least subsidy photovoltaic supply chain equilibrium andcoordination with fairness concernrdquo Renewable Energyvol 132 pp 1332ndash1347 2019

[21] W Liu D Wang X Shen X Yan and W Wei ldquoe impactsof distributional and peer-induced fairness concerns on thedecision-making of order allocation in logistics service supplychainrdquo Transportation Research Part E Logistics and Trans-portation Review vol 116 pp 102ndash122 2018

[22] L Zhang H Zhou Y Liu and R Lu ldquoOptimal environmentalquality and price with consumer environmental awarenessand retailerrsquos fairness concerns in supply chainrdquo Journal ofCleaner Production vol 213 pp 1063ndash1079 2019

[23] X LWang S Q Hu and X B Liu ldquoSupply coordination withoption contract considering overconfidence under randomdemand and yieldsrdquo Computer Integrated ManufacturingSystem vol 24 no 11 pp 2898ndash2908 2018

[24] W Jiang and J Liu ldquoInventory financing with overconfidentsupplier based on supply chain contractrdquo MathematicalProblems in Engineering vol 2018 Article ID 505438712 pages 2018

[25] P Ma H Wang and J Shang ldquoContract design for two-stagesupply chain coordination integrating manufacturer-qualityand retailer-marketing effortsrdquo International Journal ofProduction Economics vol 146 no 2 pp 745ndash755 2013

[26] J Liu W F Jiang and Y P Tang ldquoe supply chain coor-dination based on carbon reduction of overconfident man-ufacturerrdquo Journal of Jiangxi Normal University (NaturalScience) vol 42 no 2 pp 203ndash207 2018

[27] R Dubey N Altay and C Blome ldquoSwift trust and com-mitment the missing links for humanitarian supply chaincoordinationrdquo Annals of Operations Research vol 283 no 1-2 pp 159ndash177 2019

[28] C Cui Z Feng and C Tan ldquoCredibilistic loss aversion nashequilibrium for bimatrix games with triangular fuzzy payoffsrdquoComplexity vol 2018 Article ID 7143586 16 pages 2018

[29] E Fehr and KM Schmidt ldquoA theory of fairness competitionand cooperationrdquo e Quarterly Journal of Economicsvol 114 no 3 pp 817ndash868 1999

[30] Z Qin and J Yang ldquoAnalysis of a revenue-sharing contract insupply chain managementrdquo International Journal of LogisticsResearch and Applications vol 11 no 1 pp 17ndash29 2008

[31] S-Z Bai and J Ni ldquoResearch on buy-back contract with saleseffort cost sharingrdquo in Proceedings of the 2009 Chinese Controland Decision Conference IEEE Guilin China pp 1917ndash1921June 2009

[32] E Karagozoglu and K Keskin ldquoTime-varying fairness con-cerns delay and disagreement in bargainingrdquo Journal ofEconomic Behavior and Organization vol 147 pp 115ndash1282018

16 Complexity

According to the optimal result of order quantity inequation (13) and the assumption qgt 0 the condition s +

v minus 2cgt 0 is found According to the assumptions we had(zwdlowastzβ)gt 0 and (zwdlowastzμ)lt 0

Proof of Proposition 3 e gap between qdlowast and qclowastdenoted by Δq is as follows

Δq qdlowast

minus qclowast

(p minus c)(v minus s) (1 + β)2 minus 21113960 1113961 minus Ac(s + v minus 2c) minus (a minus bp)(v minus s)c

2c(v minus s) (B5)

If (p minus c)(v minus s)[(1 + β)2 minus 2] is more thanAc(s + v minus 2c) + (a minus bp)(v minus s)c we have Δqgt 0 ieqdlowast gt qclowast In contrast if (p minus c)(v minus s)[(1 + β)2 minus 2] is less

than Ac(s + v minus 2c) + (a minus bp)(v minus s)c Δqlt 0 ieqdlowast lt qclowast

e gap between edlowast and eclowast denoted by Δe is asfollows

Δe edlowast

minus eclowast

Aβc(4p minus 2c minus s minus v) minus Ac(s + v minus 2c) minus (v minus s)(1 + β) (a minus bp)c + 1 minus β21113872 1113873(p minus c)1113960 1113961

2c 2Ac +(1 + β)2(v minus s)1113960 1113961 (B6)

Similarly if Aβc(4p minus 2c minus s minus v)gtAc(s + v minus 2c)

+(v minus s)(1 + β)[(a minus bp)c + (1 minus β2)(p minus c)] we haveΔe edlowast minus eclowast gt 0 In contrast ifAβc(4p minus 2c minus s minus v)ltAc(s + v minus 2c)+

(v minus s)(1 + β)[(a minus bp)c + (1 minus β2)(p minus c)] we haveΔe edlowast minus eclowast lt 0

Upon solving the first-order derivative of Δq and Δe withrespect to β respectively we obtained

zΔ q

zβ

(1 + β)(p minus c)

c (B7)

zΔ e

zβ4(p minus c)(v minus s)2 β3 + 4β2 + 6β + 41113872 1113873β +(a minus bp)c(1 + β)2(v minus s)2 + Ac(v minus s)(1 + β)2 2(p minus c) +v + s) + 2(Ac)2(4p minus c minus s minus v)1113960 1113961

2c (1 + β)2(v minus s) + 2Ac1113960 11139612

(B8)

According to the assumptions pgt vgt cgt sgt 0 β isin (0 1)cgt 0 and a minus bpgt 0 (zΔ qzβ)gt 0 and (zΔ ezβ)gt 0 can beproved Hence the gaps increased as a function with respectto β

Proof of Proposition 4 e proof for Proposition 4 is similarto Proposition 2 Herein we took first-order and second-order partial derivatives of optimal wholesale price wsclowast withrespect to β μ and ϕ in equation (18) respectively esolving results are as follows

zwsclowast

zβ

[(1 + 2μ)ϕ minus μ](p minus c)

(1 + 2μ)(1 + β)2

zwsclowast

zμ minus

(p minus c)β(1 + β)(1 + 2μ)2

zwsclowast

zϕ minus

p minus c

1 + β

(B9)

(zwsclowastzβ) if ϕgt (μ(1 + 2μ)) is more than 0 otherwiseϕgt (μ(1 + 2μ)) is less than 0 us wsc could be increasingwith respect to β nevertheless decreasing with respect to βBesides according to the assumptions pgt vgt cgt sgt 0β isin (0 1) cgt 0 and a minus bpgt 0 it can be evidently foundthat (zwsclowastzμ)gt 0 and (zwsclowastzϕ)gt 0

Data Availability

e data used to support the findings of this study are in-cluded within the article

Conflicts of Interest

e authors declare no conflicts of interest

Acknowledgments

is work was supported by a major project financed by theNational Social Science Fund of China (Approval nos71461009 71662011 and 71761015)

Complexity 15

References

[1] Y Ren D C Croson and R T A Croson ldquoe overconfidentnewsvendorrdquo Journal of the Operational Research Societyvol 68 no 5 pp 496ndash506 2017

[2] T H Cui and P Mallucci ldquoFairness ideals in distributionchannelsrdquo Journal of Marketing Research vol 53 no 6pp 969ndash987 2016

[3] K Chen X Wang M Huang and W-K Ching ldquoSalesforcecontract design joint pricing and production planning withasymmetric overconfidence sales agentrdquo Journal of Industrialamp Management Optimization vol 13 no 2 pp 873ndash8992017

[4] X X Zheng Z Liu K W Li et al ldquoCooperative game ap-proaches to coordinating a three-echelon closed-loop supplychain with fairness concernsrdquo International Journal of Pro-duction Economic vol 212 pp 92ndash110 2018

[5] China Internet Watch False Prosperity behind Double 11Refund Rate Increased to 69 2014 httpswwwchinainternetwatchcom11643false-prosperity-behind-double-11-refund-rate-increased-to-69

[6] C L Lu ldquoPampGrsquos collaborative supply chain transformationrdquoEnterprise Management vol 10 pp 79-80 2016

[7] M Li N C Petruzzi and J Zhang ldquoOverconfident competingnewsvendorsrdquo Management Science vol 63 no 8pp 2637ndash2646 2016

[8] T Nie ldquoDual-fairness supply chain with quantity discountcontractsrdquo European Journal of Operational Researchvol 258 no 2 pp 491ndash500 2017

[9] X Du and B Jiang ldquoSignaling through price and quality toconsumers with fairness concernsrdquo Journal of MarketingResearch vol 53 no 6 pp 988ndash1000 2016

[10] V Kurz A Orland and K Posadzy ldquoFairness versus effi-ciency how procedural fairness concerns affect coordina-tionrdquo Experimental Economics vol 21 no 3 pp 601ndash6262018

[11] S Villa and J A Castantildeeda ldquoTransshipments in supplychains a behavioral investigationrdquo European Journal of Op-erational Research vol 269 no 2 pp 715ndash729 2018

[12] B Gu Y Fu and Y Li ldquoFresh-keeping effort and channelperformance in a fresh product supply chain with loss-averseconsumersrsquo returnsrdquo Mathematical Problems in Engineeringvol 2018 Article ID 4717094 20 pages 2018

[13] D Croson R Croson and Y Ren How to Manage anOverconfident Newsvendor 2008 httpcbeesutdallasedupapersCrosonRenCmsonMS2008

[14] M Li ldquoOverconfident distribution channelsrdquo Production andOperations Management vol 28 no 6 pp 1347ndash1365 2019

[15] W Liu D Wang O Tang and D Zhu ldquoe impacts oflogistics service integratorrsquos overconfidence behaviour onsupply chain decision under demand surgerdquo EuropeanJournal of Industrial Engineering vol 12 no 4 pp 558ndash5972018

[16] L Xu X Shi P Du K Govindan and Z Zhang ldquoOptimi-zation on pricing and overconfidence problem in a duopo-listic supply chainrdquo Computers amp Operations Researchvol 101 pp 162ndash172 2019

[17] Y Li and M Shan ldquoAdvance selling decisions with over-confident consumersrdquo Journal of Industrial and ManagementOptimization vol 2 no 3 pp 891ndash905 2016

[18] T H Cui J S Raju and Z J Zhang ldquoFairness and channelcoordinationrdquo Management Science vol 53 no 8pp 1303ndash1314 2007

[19] X Wu and J A Niederhoff ldquoFairness in selling to thenewsvendorrdquo Production and Operations Managementvol 23 no 11 pp 2002ndash2022 2014

[20] Z Chen and S-I Ivan Su ldquoSocial welfare maximization withthe least subsidy photovoltaic supply chain equilibrium andcoordination with fairness concernrdquo Renewable Energyvol 132 pp 1332ndash1347 2019

[21] W Liu D Wang X Shen X Yan and W Wei ldquoe impactsof distributional and peer-induced fairness concerns on thedecision-making of order allocation in logistics service supplychainrdquo Transportation Research Part E Logistics and Trans-portation Review vol 116 pp 102ndash122 2018

[22] L Zhang H Zhou Y Liu and R Lu ldquoOptimal environmentalquality and price with consumer environmental awarenessand retailerrsquos fairness concerns in supply chainrdquo Journal ofCleaner Production vol 213 pp 1063ndash1079 2019

[23] X LWang S Q Hu and X B Liu ldquoSupply coordination withoption contract considering overconfidence under randomdemand and yieldsrdquo Computer Integrated ManufacturingSystem vol 24 no 11 pp 2898ndash2908 2018

[24] W Jiang and J Liu ldquoInventory financing with overconfidentsupplier based on supply chain contractrdquo MathematicalProblems in Engineering vol 2018 Article ID 505438712 pages 2018

[25] P Ma H Wang and J Shang ldquoContract design for two-stagesupply chain coordination integrating manufacturer-qualityand retailer-marketing effortsrdquo International Journal ofProduction Economics vol 146 no 2 pp 745ndash755 2013

[26] J Liu W F Jiang and Y P Tang ldquoe supply chain coor-dination based on carbon reduction of overconfident man-ufacturerrdquo Journal of Jiangxi Normal University (NaturalScience) vol 42 no 2 pp 203ndash207 2018

[27] R Dubey N Altay and C Blome ldquoSwift trust and com-mitment the missing links for humanitarian supply chaincoordinationrdquo Annals of Operations Research vol 283 no 1-2 pp 159ndash177 2019

[28] C Cui Z Feng and C Tan ldquoCredibilistic loss aversion nashequilibrium for bimatrix games with triangular fuzzy payoffsrdquoComplexity vol 2018 Article ID 7143586 16 pages 2018

[29] E Fehr and KM Schmidt ldquoA theory of fairness competitionand cooperationrdquo e Quarterly Journal of Economicsvol 114 no 3 pp 817ndash868 1999

[30] Z Qin and J Yang ldquoAnalysis of a revenue-sharing contract insupply chain managementrdquo International Journal of LogisticsResearch and Applications vol 11 no 1 pp 17ndash29 2008

[31] S-Z Bai and J Ni ldquoResearch on buy-back contract with saleseffort cost sharingrdquo in Proceedings of the 2009 Chinese Controland Decision Conference IEEE Guilin China pp 1917ndash1921June 2009

[32] E Karagozoglu and K Keskin ldquoTime-varying fairness con-cerns delay and disagreement in bargainingrdquo Journal ofEconomic Behavior and Organization vol 147 pp 115ndash1282018

16 Complexity

References

[1] Y Ren D C Croson and R T A Croson ldquoe overconfidentnewsvendorrdquo Journal of the Operational Research Societyvol 68 no 5 pp 496ndash506 2017

[2] T H Cui and P Mallucci ldquoFairness ideals in distributionchannelsrdquo Journal of Marketing Research vol 53 no 6pp 969ndash987 2016

[3] K Chen X Wang M Huang and W-K Ching ldquoSalesforcecontract design joint pricing and production planning withasymmetric overconfidence sales agentrdquo Journal of Industrialamp Management Optimization vol 13 no 2 pp 873ndash8992017

[4] X X Zheng Z Liu K W Li et al ldquoCooperative game ap-proaches to coordinating a three-echelon closed-loop supplychain with fairness concernsrdquo International Journal of Pro-duction Economic vol 212 pp 92ndash110 2018

[5] China Internet Watch False Prosperity behind Double 11Refund Rate Increased to 69 2014 httpswwwchinainternetwatchcom11643false-prosperity-behind-double-11-refund-rate-increased-to-69

[6] C L Lu ldquoPampGrsquos collaborative supply chain transformationrdquoEnterprise Management vol 10 pp 79-80 2016

[7] M Li N C Petruzzi and J Zhang ldquoOverconfident competingnewsvendorsrdquo Management Science vol 63 no 8pp 2637ndash2646 2016

[8] T Nie ldquoDual-fairness supply chain with quantity discountcontractsrdquo European Journal of Operational Researchvol 258 no 2 pp 491ndash500 2017

[9] X Du and B Jiang ldquoSignaling through price and quality toconsumers with fairness concernsrdquo Journal of MarketingResearch vol 53 no 6 pp 988ndash1000 2016

[10] V Kurz A Orland and K Posadzy ldquoFairness versus effi-ciency how procedural fairness concerns affect coordina-tionrdquo Experimental Economics vol 21 no 3 pp 601ndash6262018

[11] S Villa and J A Castantildeeda ldquoTransshipments in supplychains a behavioral investigationrdquo European Journal of Op-erational Research vol 269 no 2 pp 715ndash729 2018

[12] B Gu Y Fu and Y Li ldquoFresh-keeping effort and channelperformance in a fresh product supply chain with loss-averseconsumersrsquo returnsrdquo Mathematical Problems in Engineeringvol 2018 Article ID 4717094 20 pages 2018

[13] D Croson R Croson and Y Ren How to Manage anOverconfident Newsvendor 2008 httpcbeesutdallasedupapersCrosonRenCmsonMS2008

[14] M Li ldquoOverconfident distribution channelsrdquo Production andOperations Management vol 28 no 6 pp 1347ndash1365 2019

[15] W Liu D Wang O Tang and D Zhu ldquoe impacts oflogistics service integratorrsquos overconfidence behaviour onsupply chain decision under demand surgerdquo EuropeanJournal of Industrial Engineering vol 12 no 4 pp 558ndash5972018

[16] L Xu X Shi P Du K Govindan and Z Zhang ldquoOptimi-zation on pricing and overconfidence problem in a duopo-listic supply chainrdquo Computers amp Operations Researchvol 101 pp 162ndash172 2019

[17] Y Li and M Shan ldquoAdvance selling decisions with over-confident consumersrdquo Journal of Industrial and ManagementOptimization vol 2 no 3 pp 891ndash905 2016

[18] T H Cui J S Raju and Z J Zhang ldquoFairness and channelcoordinationrdquo Management Science vol 53 no 8pp 1303ndash1314 2007

[19] X Wu and J A Niederhoff ldquoFairness in selling to thenewsvendorrdquo Production and Operations Managementvol 23 no 11 pp 2002ndash2022 2014

[20] Z Chen and S-I Ivan Su ldquoSocial welfare maximization withthe least subsidy photovoltaic supply chain equilibrium andcoordination with fairness concernrdquo Renewable Energyvol 132 pp 1332ndash1347 2019

[21] W Liu D Wang X Shen X Yan and W Wei ldquoe impactsof distributional and peer-induced fairness concerns on thedecision-making of order allocation in logistics service supplychainrdquo Transportation Research Part E Logistics and Trans-portation Review vol 116 pp 102ndash122 2018

[22] L Zhang H Zhou Y Liu and R Lu ldquoOptimal environmentalquality and price with consumer environmental awarenessand retailerrsquos fairness concerns in supply chainrdquo Journal ofCleaner Production vol 213 pp 1063ndash1079 2019

[23] X LWang S Q Hu and X B Liu ldquoSupply coordination withoption contract considering overconfidence under randomdemand and yieldsrdquo Computer Integrated ManufacturingSystem vol 24 no 11 pp 2898ndash2908 2018

[24] W Jiang and J Liu ldquoInventory financing with overconfidentsupplier based on supply chain contractrdquo MathematicalProblems in Engineering vol 2018 Article ID 505438712 pages 2018

[25] P Ma H Wang and J Shang ldquoContract design for two-stagesupply chain coordination integrating manufacturer-qualityand retailer-marketing effortsrdquo International Journal ofProduction Economics vol 146 no 2 pp 745ndash755 2013

[26] J Liu W F Jiang and Y P Tang ldquoe supply chain coor-dination based on carbon reduction of overconfident man-ufacturerrdquo Journal of Jiangxi Normal University (NaturalScience) vol 42 no 2 pp 203ndash207 2018

[27] R Dubey N Altay and C Blome ldquoSwift trust and com-mitment the missing links for humanitarian supply chaincoordinationrdquo Annals of Operations Research vol 283 no 1-2 pp 159ndash177 2019

[28] C Cui Z Feng and C Tan ldquoCredibilistic loss aversion nashequilibrium for bimatrix games with triangular fuzzy payoffsrdquoComplexity vol 2018 Article ID 7143586 16 pages 2018

[29] E Fehr and KM Schmidt ldquoA theory of fairness competitionand cooperationrdquo e Quarterly Journal of Economicsvol 114 no 3 pp 817ndash868 1999

[30] Z Qin and J Yang ldquoAnalysis of a revenue-sharing contract insupply chain managementrdquo International Journal of LogisticsResearch and Applications vol 11 no 1 pp 17ndash29 2008

[31] S-Z Bai and J Ni ldquoResearch on buy-back contract with saleseffort cost sharingrdquo in Proceedings of the 2009 Chinese Controland Decision Conference IEEE Guilin China pp 1917ndash1921June 2009

[32] E Karagozoglu and K Keskin ldquoTime-varying fairness con-cerns delay and disagreement in bargainingrdquo Journal ofEconomic Behavior and Organization vol 147 pp 115ndash1282018

16 Complexity