Hindawi Publishing CorporationAdvances in Decision SciencesVolume 2013 Article ID 915657 10 pageshttpdxdoiorg1011552013915657

Research ArticleAn Inventory Decision Model When DemandFollows Innovation Diffusion Process under Effect ofTechnological Substitution

K K Aggarwal Chandra K Jaggi and Alok Kumar

Department of Operational Research Faculty of Mathematical Sciences New Academic Block University of Delhi Delhi 110007 India

Correspondence should be addressed to Alok Kumar alok 20 oryahoocoin

Received 14 April 2013 Revised 3 September 2013 Accepted 10 September 2013

Academic Editor S Dempe

Copyright copy 2013 K K Aggarwal et alThis 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

The concept of marketing literature especially innovation diffusion concept plays a pivotal role in developing EOQ models inthe field of inventory management The integration of marketing parameters especially the idea of diffusion of new products withthe inventory models makes the models more realistic which is most essential while building the economic ordering policies ofthe products Also because of rapid technological development the diffusion of technology can also be viewed as an evolutionaryprocess of replacement of an old technology by a new oneTherefore the effect of technological substitution alongwith the diffusionof new products must be taken into account while formulating economic ordering policies in an inventory model In this paper amathematical model has been developed for obtaining the Economic Order Quantity (EOQ) in which the demand of the product isassumed to follow an innovation diffusion process as proposed by Fourt and WoodLock (1960) The idea of effect of technologicalsubstitution of products has been incorporated in the demand model to make the economic ordering policies more realistic Anumerical example with sensitivity analysis of the optimal solution with respect to different parameters of the system is performedto illustrate the effectiveness of the model

1 Introduction

Technological breakthroughs are continuously being experi-enced in every field of business management and because ofthis new products are constantly entering into the marketsystem Also the penetration of new products into themarketsystem generally come in successive generations This hap-pens due to speedy technological progress because of rapiddevelopment of information and communication technologymarket The impact of globalization plays one of the pivotalroles to spread awareness about the new products amongsocieties Also the changing needs of the society increasesthe demand of new products which encourage innovations ofthe products Therefore the theory of innovation diffusion ishighly desired for attentive management of the new productsin order to minimize the total cost and maximize thebenefits To make the business fascinating and demandingthe importance of innovations in business and industry ishighly significant The innovations especially technological

innovations have made the firms to upgrade their productsskillfully for surviving in the market Diffusion is definedas the process by which an innovation is communicatedthrough certain channels over time among members of asocial system (Rogers [1]) The various diffusion processeshave been studied in the literature which are as followsThe pure innovative model has been considered by Fourtand WoodLock [2] whereas Fisher and Pry [3] explain thepure imitative model The Bass Model [4] admits both theinnovative and imitative aspects of product adoption TheBass Model [4] introduced over three decades ago has beenwidely used in marketing by Peres et al [5] Parker [6]Rogers [1 7] and Peres et al [8] and Mahajan et al [9] Theperiodic review inventory model with demand influenced bypromotion decisions has been considered by Cheng and Sethi[10] Kalish [11] has discussed an adoption model of a NewProduct with Price advertising and uncertainty Mahajan andRobert [12] have studied Innovation Diffusion in a DynamicPotential Adopter Population Sharif andRamanathan [13 14]

2 Advances in Decision Sciences

Cum

ulat

ive p

erce

ntag

e of b

uyer

s

Time

Figure 1 Fourt and WoodLock Curve (Source Lilien et al [38])

Adop

tion

rate

Adoption time

Generation oneGeneration two

Generation three

Figure 2 Series of technological generations (Source Norton andBass [15])

have considered the application of dynamic potential adopterdiffusion model through their study of diffusion of oralcontraceptives in Thailand The diffusion process in termsof number of customers who have bought the product bytime ldquo119905rdquo has been explained by a modified exponential curve(Figure 1) by Fourt and WoodLock [2] This model capturesthe innovative characteristic with its coefficient ldquo119901rdquo as thecoefficient of innovation

Also diffusion of new products cannot be justified with-out taking into account the effects of successive generationsof products because an important feature of most modernnew technologies is that they come in successive generationsNew products are substituted with more advanced productsand technological generations which creates heterogeneity inthe adopting population Moreover it is observed that anygiven generation of technology will end up being replacedby a new generation The old products are not straight awayreplaced by the new products because of constant innovationbut new products intend to substitute and start competingwith the old products which creates parallel diffusion of boththe old and the new products in the market This makes thedecreasing pattern of diffusion of old products The series oftechnological generations have been well explained throughthe Figure 2 There are lots of models in the marketing liter-ature which have considered the importance of generation ofproducts Norton and Bass [15] model is a standard exampleof multiple generation model which assumes that the coef-ficients of innovation and imitation remain unchanged fromgeneration to generation of technology Norton and Bass [15]is basically based upon the Bass model The Bass model isalso extended by Mahajan and Muller [16] who proposed amodel to capture substitution and diffusion patterns for eachsuccessive generation of technological products Speece and

Maclachlan [17] and Danaher et al [18] incorporated price asan explanatory variable in their model Islam andMeade [19]proposed that the coefficients of later generation technologyare constant incrementdecrement over the coefficients ofthe first generation Kim et al [20] developed a model andtried to include intergenerational linkages among successivegenerations within a product category Bayus [21] developeda model for consumer sales of second-generation productsby incorporating replacement behavior of first generationadopters and suggested different dynamic pricing policies ofsecond-generation consumer durables Bardhan and Chanda[22] have developed a mathematical model for adoptionof successive generations of a high technology productChanda and Bardhan [23] developed a model on dynamicoptimal advertising expenditure strategies of two successivegenerations of consumer durables Bardhan and Chanda[24] also developed a mathematical model for diffusion ofproducts with multiple generations of a high technologyproduct The above discussed models are well explained inthe marketing literature and are extremely significant in theestimation of new products having successive generationfeatures These models are highly useful in making theEconomic inventory policy of any organization as far as newproducts along with its successive generations are concernedUnfortunately the concept of the above marketing modelshas not been integrated with traditional inventory modelswhich is highly desired while making the economic orderingpolicy for new products with successive generation featuresThough there are some inventory models described belowwhich have incorporated the concept of innovation diffusionin their demand functions but these models do not considerthe effect of diminishing nature of demand rate when thesubstitute products are entered into the market becauseof technological innovations which are popularly knownas the successive generations of the products Chanda andKumar [25] have explained the economic order quantitymodel with demand influenced by dynamic innovation effectAggarwal et al [26] have considered the EOQ model withinnovation diffusion criterion having dynamic potentialmar-ket size Also Chanda and Kumar [27] explain the EOQmodel having demand influenced by innovation diffusioncriterion under inflationary condition Chern et al [28] haveformulated the economic order quantity model in which thedemand follows innovation diffusion criterion as consideredby Bass [4] Aggarwal et al [29] formulated an inventorymodel for new product when demand follows dynamicinnovation process having dynamic potential market sizeKumar et al [30] discussed economic order quantity modelunder fuzzy sensewhen the demand follows Bassrsquos innovationdiffusion process Aggarwal et al [31] have developed aneconomic order quantity model under fuzzy sense whendemand follows innovation diffusion process having dynamicpotential market size The approach in this paper is to focuson the effect of innovation factor on adoption behaviorof consumers with the effect of technological substitutionof the products The uniqueness of this model is to makean economic inventory policy of a new product under thecondition of its diminishing demand after a certain intervalof time when its substitute product enters into the market

Advances in Decision Sciences 3

because of technological innovation The paper is dividedinto the model development special case numerical exam-ples with sensitivity analysis observations and managerialimplications Finally the paper concludes with a discussionon the application extension and limitations of the modelAssumptions Inventory model has been developed under thefollowing assumptions

(i) The replenishment rate is infinite and thus replen-ishments are instantaneous

(ii) Lead time is zero(iii) The planning horizon is finite(iv) Shortages are not allowed(v) Demand rate is time dependent governed by innova-

tion diffusion process and is affected by the introduc-tion of substitute products because of technologicalinnovations

(vi) The size of the potential market of total number ofadopters remains constant Here the potentialmarketsize includes the number of initial purchases for thetime interval for which replacement purchases areexcluded

(vii) The coefficient of innovation remains constant thatis the likelihood that somebody who is not yet usingthe product will start using it because of mass mediacoverage or other external factors will act as constantthroughout the cycle length

(viii) There is only one product bought per new adopter(ix) The innovationrsquos sales are confined to a single geo-

graphic area(x) The impact of marketing strategies by the innovator is

adequately captured by the modelrsquos parameters

Notations Notations used in the modeling framework are asfollows

119860 ordering cost119862 unit cost119868 inventory carrying charge119868119862 inventory carrying cost119879 length of the replenishment cycle1198791 time of introduction of second generation product

120579 rate of diminishing demand of first generationproduct119901 coefficient of innovation119876 number of items received at the beginning of theperiod119870(119879) the total cost of the system per unit time119868(119905) on hand inventory at any time 119905119899(119905) = 120582(119905) = 119878(119905) the number of adoptions at time 119905that is Demand at time 119905119873 potential market size

119873(119905) cumulative number of adopters at time 119905119891(119905) the likelihood of purchase at time 119905119865(119905) the cumulative fraction of adopters at time 119905

2 Mathematical Model

The basic assumption considered by different researchersin marketing literature for a fundamental diffusion modelis that the rate of diffusion or the number of adoptersat any given point in time is directly proportional to thenumber of remaining potential adopters at that momentMathematically this can be represented as follows

119899 (119905) =119889119873 (119905)

119889119905= 119892 (119905) 119873 minus 119873 (119905) (1)

where 119892(119905) is known as the rate of adoption or individualprobability of adoption

It has also been assumed that 119892(119905) depends on timethrough a linear function of 119873(119905) (Mahajan and Peterson[32]) Hence

119899 (119905) =119889119873 (119905)

119889119905= (119886 + 119887119873 (119905)) 119873 minus 119873 (119905) 119886 ge 0 119887 ge 0

(2)

Depending on the importance of each source of influencedifferent versions can be derived from the fundamentaldiffusion model (Mahajan and Peterson [32])

When 119887 = 0 the model only considers external influencewhen 119886 = 0 the model only considers internal influenceWhen 119886 = 0 and 119887 = 0 the resulting model is called a mixedinfluence diffusion model (Ruiz-Conde et al [33])

The basic assumptions used in the Bass Model are thatthe adoption of a new product spreads through a populationprimarily due to contact with prior adopters Hence theprobability that an initial purchase occurs at time 119905 given thatno purchase has occurred is a linear function of the numberof previous buyers that is

119891 (119905)

1 minus 119865 (119905)= 119901 + 119902119865 (119905) (3)

If we define 119899(119905) = 119873119891(119905) and 119873(119905) = 119873119865(119905) we canexpress (3) as follows

119899 (119905) =119889119873 (119905)

119889119905= (119901 + 119902

119873 (119905)

119873

) 119873 minus 119873 (119905) (4)

Equation (4) is a mixed influence diffusion model Considera case of diffusion model of external influence only (taking119902 = 0) as proposed by Fourt and WoodLock [2] (4) can bedefined as follows

119899 (119905) =119889119873 (119905)

119889119905= 119901 119873 minus 119873 (119905) (5)

The basic demand model that has been used in the EOQmodel is based on the following assumptions

4 Advances in Decision Sciences

(i) Adoptions take place due to innovation-diffusioneffect and it is influenced by the innovation-effect(mass media) that is external influence only

(ii) Adoptions of the first generation product is dimin-ishing by the introduction of the second generationproduct

Here the diffusion of new products which is spreadthrough the external influence has been considered for mak-ing economic ordering policies of the products The demandof first generation products that is products which aresupposed to be introduced at the beginning of the planningperiod also known as old products is affected (diminished)due to introduction of the second generation products thatis products which are supposed to be new in comparison tothe first generation products and affecting the demand of oldproducts because of technological innovations after a certaininterval of time Therefore using the above assumptionsexplanations and (5) the following demand function hasbeen constructed for our model120582 (119905)

=

119901119873 minus 119873 (119905) 0 le 119905 lt 1198791 119901 ge 120579

119901 119873 minus 119873 (119905) minus 120579 119873 minus 119873 (119905) 1198791le 119905 le 119879 119901 ge 120579

(6)The above demand model shows that the inventory level isdepleted wholly due to demand of the old products with aparticular pace depending on the parameters associated withit up to time period 119879

1and after time 119879

1and up to time

period 119879 the rate of decreasing inventory level is somewhatless than what was experienced before time 119879

1because some

demand of the old products are satisfied (substituted) by thenew products introduced after time period 119879

1because of

technological innovations The demand usage 120582(119905) which isa function of time plays pivotal role to shrink the inventorysize over a period of time If in the time interval (119905 119905 + 119889119905)the inventory size is dipping at the rate 120582(119905)119889119905 then the totalreduction in the inventory size during the time interval119889119905 canbe given by minus119889119868(119905) = 120582(119905)119889119905 Thus the differential equationdescribing the instantaneous state of the inventory level at anytime 119905 119868(119905) in the interval (0 119879) is given by

119889119868 (119905)

119889119905= minus120582 (119905) 0 le 119905 le 119879 (7)

where120582 (119905)

= 119899 (119905) =119889119873 (119905)

119889119905

=

119901119873 minus 119873 (119905) 0 le 119905 lt 1198791 119901 ge 120579

119901 119873 minus 119873 (119905) minus 120579 119873 minus 119873 (119905) 1198791le 119905 le 119879 119901 ge 120579

997904rArr 119873(119905) =

119873 (1 minus 119890minus119901119905) 0 le 119905 lt 119879

1

119873 minus 119873 minus 119873(1198791) 119890(119901minus120579)(119879

1minus119905) 1198791le 119905 le 119879

(8)

Now by model assumption replenishment is instanta-neous and shortages are not allowedThus the inventory levelat the initial point of the planning horizon can be assumed tobe the cumulative adoption of the product during the cycletime 119879 Hence

119868 (0) = 119876 = int

119879

0

120582 (119905) 119889119905 (9)

997904rArr 119868 (0) = 119876 = 119873(1 minus 119890minus1199011198791) + [119873 (119879

1) minus 119873]

times (119890(119901minus120579)(119879

1minus119879)

minus 1)

(10)

The solution of the differential equation (7) is

119868 (119905) =

119868 (0) minus 119873 (1 minus 119890minus119901119905) 0 le 119905 lt 119879

1

[119873 minus 119873 (1198791)]

times [119890(119901minus120579)(119879

1minus119905)minus 119890minus(119901minus120579)(119879

1minus119879)] 119879

1le 119905 le 119879

(11)

Now

int

119879

0

119868 (119905) 119889119905 = int

1198791

0

119868 (119905) 119889119905 + int

119879

1198791

119868 (119905) 119889119905 (12)

997904rArr int

119879

0

119868 (119905) 119889119905 = 1198791119868 (0) +

119873

119901[1 minus 119890

minus1199011198791]

minus 1198731198791+ [119873 minus 119873(119879

1)]

times[119890(119901minus120579)(119879

1minus119879)

minus 1

(120579 minus 119901)+(1198791minus 119879) 119890

(119901minus120579)(1198791minus119879)]

(13)

The different cost elements involved in the inventorysystem per unit time can be defined as

Ordering cost per unit time = 119860

119879 (14)

Material cost per unit time = 119876119862

119879 (15)

=119862

119879119873(1 minus 119890

minus1199011198791)

+ (119862

119879) [119873 (119879

1) minus 119873] (119890

(119901minus120579)(1198791minus119879)

minus 1)

(16)

Cost of carrying inventory per unit time

=119868119862

119879int

119879

0

119868 (119905) 119889119905

(17)

Advances in Decision Sciences 5

= (119868119862

119879)1198791119868 (0) +

119868119862119873

119901119879[1 minus 119890

minus1199011198791]

minus (119868119862

119879)1198731198791

+ (119868119862

119879) [119873 minus 119873(119879

1)]

times [119890(119901minus120579)(119879

1minus119879)

minus 1

(120579 minus 119901)+ (1198791minus 119879) 119890

(119901minus120579)(1198791minus119879)]

(18)

Thus the total cost of the inventory system per unit time119870(119879) is

119870 (119879) = (14) + (16) + (18)

997904rArr 119870 (119879) =119860

119879+ [119873 (119879

1) minus 119873] (119890

(119901minus120579)(1198791minus119879)

minus 1)

times (119862

119879) (1 + 119868119879

1) + 119873 (1 minus 119890

minus1199011198791) (

119862

119879)

times (1 + 1198681198791) +

119868119862119873

119901119879[1 minus 119890

minus1199011198791] minus (

119868119862

119879)1198731198791

+ (119868119862

119879) [119873 minus 119873(119879

1)]

times [119890(119901minus120579)(119879

1minus119879)

minus 1

(120579 minus 119901)+ (1198791minus 119879) 119890

(119901minus120579)(1198791minus119879)]

(19)

The necessary criterion for 119870(119879) to be minimum is

119889119870 (119879)

119889119879= 0 (20)

997904rArrminus119860

1198792+ [119873 (119879

1) minus 119873] (119890

(119901minus120579)(1198791minus119879)

minus 1) (minus119862

1198792) (1 + 119868119879

1)

minus (119862

1198792) (1 + 119868119879

1)119873 (1 minus 119890

minus1199011198791)

+ (119862

119879) (1 + 119868119879

1) [119873 (119879

1) minus 119873] (120579 minus 119901) 119890

(119901minus120579)(1198791minus119879)

minus119868119862119873

1199011198792[1 minus 119890

minus1199011198791] + (

119868119862

1198792)1198731198791

+ (minus119868119862

1198792) [119873 minus 119873(119879

1)]

times [119890(119901minus120579)(119879

1minus119879)

minus 1

(120579 minus 119901)+ (1198791minus 119879) 119890

(119901minus120579)(1198791minus119879)]

+ (119868119862

119879) [119873 minus 119873(119879

1)]

times (1198791minus 119879) (120579 minus 119901) 119890

(119901minus120579)(1198791minus119879)

= 0

(21)

Now for119870(119879) to be convex

1198892119870 (119879)

1198891198792

100381610038161003816100381610038161003816100381610038161003816119879=119879lowast

gt 0

997904rArr2119860

119879lowast3+ (

2119862

119879lowast3) (1 + 119868119879

1)119873 (1 minus 119890

minus1199011198791)

+ [119873 (1198791) minus 119873] (119890

(119901minus120579)(1198791minus119879lowast)minus 1) (

2119862

1198793) (1 + 119868119879

1)

+ (119862

119879lowast) (1 + 119868119879

1) [119873 (119879

1) minus 119873] (120579 minus 119901)

2

119890(119901minus120579)(119879

1minus119879lowast)

+2119868119862119873

119901119879lowast3

[1 minus 119890minus1199011198791] + (

2119868119862

119879lowast3) [119873 minus 119873(119879

1)]

times [119890(119901minus120579)(119879

1minus119879lowast)minus 1

(120579 minus 119901)+ (1198791minus 119879lowast) 119890(119901minus120579)(119879

1minus119879lowast)]

+ (119868119862

119879lowast) [119873 minus 119873(119879

1)] (1198791minus 119879lowast)

times (120579 minus 119901)2

119890(119901minus120579)(119879

1minus119879lowast)

gt (2119862

119879lowast2) (1 + 119868119879

1) [119873 (119879

1) minus 119873]

times (120579 minus 119901) 119890(119901minus120579)(119879

1minus119879)

+ (2119868119862

119879lowast3)1198731198791

+ (2119868119862

119879lowast2) [119873 minus 119873(119879

1)]

times [(1198791minus 119879lowast) (120579 minus 119901) 119890

(119901minus120579)(1198791minus119879lowast)]

+ (119868119862

119879lowast) [119873 minus 119873(119879

1)] [(120579 minus 119901) 119890

(119901minus120579)(1198791minus119879lowast)]

(22)

The solution of the equation 119889119870(119879)119889119879 = 0 gives theoptimum value of 119879 provided it satisfies the condition1198892119870(119879)119889119879

2gt 0 Since the above cost equation (21) is highly

nonlinear the problem has been solved numerically for givenparameter valuesThe solution gives the optimumvalue119879lowast ofthe replenishment cycle time 119879 Once 119879lowast is known the valueof optimum order quantity 119876lowast and the optimum cost 119870(119879lowast)can easily be obtained from (10) and (19) respectively Thenumerical solution for the given base value has been obtainedby using software packages Lingo and Excel-Solver

21 Special Case When 120579 becomes negligible that is equalto zero which means here we are not considering the effectof introduction of new products on the old one after timeperiod 119879

1 That is demand rate of the old product is the

same throughout the planning period (0 119879) and for this casethe adoption rate that is demand rate that is 120582(119905) = 119899(119905)reduces to the classical Fourt and WoodLock model [2] Theobjective here is to study the different managerial policiesfor the model which is under external influence only andits demand is not affected by the introduction of any other

6 Advances in Decision Sciences

substitute products In that case the total cost per unit time119870(119879) is given by

119870 (119879) =119860

119879+119873119862

119879(1 minus 119890

minus119901119879) +

119873119868119862

119901119879(1 minus 119890

minus119901119879) minus 119873119868119862119890

minus119901119879

(23)

119868 (0) = 119876 = int

119879

0

120582 (119905) 119889119905 = 119873 (1 minus 119890minus119901119879

) (24)

For optimum total cost the necessary criterion is

119889119870 (119879)

119889119879= 0 (25)

997904rArrminus119860

1198792+119873119862119901119890

minus119901119879

119879minus

119873119862(1 minus 119890minus119901119879

)

1198792

+119873119868119862119890

minus119901119879

119879

minus

119873119868119862 (1 minus 119890minus119901119879

)

1199011198792

+ 119873119868119862119901119890minus119901119879

= 0

(26)

Now for119870(119879) to be convex

1198892119870 (119879)

1198891198792

100381610038161003816100381610038161003816100381610038161003816119879=119879lowast

gt 0

997904rArr2119860

119879lowast3+

2119873119862(1 minus 119890minus119901119879lowast

)

119879lowast3

+

2119873119868119862 (1 minus 119890minus119901119879lowast

)

119901119879lowast3

gt1198731198621199012119890minus119901119879lowast

119879lowast

+119873119868119901119862119890

minus119901119879lowast

119879lowast2

+ 1198731198681198621199012119890minus119901119879lowast

(27)

The solution of the equation 119889119870(119879)119889119879 = 0 gives theoptimum value of 119879 provided it satisfies the condition1198892119870(119879)119889119879

2gt 0 Since cost equation (26) is highly nonlinear

the problem has been solved numerically for given parametervalues The solution gives the optimum value 119879lowast of thereplenishment cycle time 119879 Once 119879lowast is known the valuesof optimum order quantity 119876lowast and the optimum cost 119870(119879lowast)can easily be obtained from (24) and (23) respectively Thenumerical solution for the given base value has been obtainedby using software packages LINGO and Excel-Solver

3 Solution Procedure

The solution procedure has been summarized in the follow-ing algorithmStep 1 Input all parameter values such as unit cost coefficientof innovation potential market size and ordering cost foreach case separatelyStep 2 Compute all possible values of ldquo119879rdquo separately for (21)and (26) as the case may beStep 3 select the appropriate value of ldquo119879rdquo say 119879lowast for eachcase by using (19) and (23) and by satisfying the above statedcondition 1198892119870(119879)1198891198792 gt 0Step 4Compute119870(119879lowast) from (19) and (23) amp119876(119879lowast) from (10)and (24) for each case separately

The above steps are used for all replenishment cyclesusing appropriate parameter values In order to obtain theappropriate values of ldquo119879rdquo we need to follow the aboveprocedure with the help of LINGO and Excel-Solver softwarepackages

4 Numerical Examples

The effectiveness of the proposed model has been shown bythe following numerical examples A hypothetical examplehas the following parameter values in appropriate units

119860 =$1100order

119862 =$300unit

119868 = 025 119873 = 50000

(28)

The results have been well presented in the followingdifferent tables Also to prove the validity of the modelnumerically and to get the appropriate parameter values thereferences have been considered as Chanda and Kumar [25]Chandrasekaran and Tellis [34] Sultan et al [35] Talukdaret al [36] Van den Bulte and Stremersch [37] Chanda andKumar [27] Aggarwal et al [26 31] Aggarwal et al [29] andKumar et al [30]

(i) The mean value of the coefficient of innovation fora new product usually lies between 00007 and 003(Sultan et al [35] Talukdar et al [36] Van den Bulteand Stremersch [37])

(ii) The mean value of the coefficient of innovation fora new product is usually 001for developed countriesand 00003 for developing countries (Talukdar et al[36])

41 Special Case A hypothetical example has the followingparameter values in appropriate units

119860 =$1100order

119862 =$300unit

119868 = 025 119873 = 50000 120579 = 0

(29)

5 Observations

The results obtained from different numerical tables inSection 4 explain the effect of changes in the system param-eters on the optimal values of total cost 119870(119879lowast) the optimalcycle length 119879

lowast and the optimal order quantity 119876(119879lowast)

We have observed the following relationship during thenumerical exercise

(a) As the coefficient of innovation increases keepingother parameters constant then the optimal cyclelength 119879

lowast decreases while both the optimal orderquantity 119876(119879

lowast) and the optimal total cost 119870(119879lowast)

increases as depicted in Tables 1 and 4 This isconsistent with the reality as more investment onpromotion will increase the diffusion of a productin the market resulting in shrinkage of the optimalreorder cycle time as a result optimal cost is increased

Advances in Decision Sciences 7

Table 1 Sensitivity analysis on coefficient of innovation ldquo119901rdquo

119901 119879lowast

119870(119879lowast) 119876(119879

lowast)

0001 086 16400 390002 059 32705 570003 048 48684 700004 041 64499 810005 037 80211 910006 034 95848 1010007 031 111430 1090008 029 126967 1170009 027 142468 124001 026 157938 131002 019 311593 190For 1198791 = 01 120579 = 00001

(b) As the cycle length 1198791 that is the time of intro-

duction of second product increases keeping otherparameters constant then the optimal cycle length119879lowastoptimal order quantity 119876(119879lowast) and the optimal totalcost 119870(119879lowast) all these three values increase as depictedin Table 2 This is true with the fact that as the timeof introduction of second product increases meansmaximum demand is satisfied by more number ofprevious that is old products and it is also obviousthat the demand is a decreasing function of timewhich implies that less demand for more time periodleads to increase in optimal cycle length And sinceoptimal cycle length increases therefore it forcesthe inventory manager to keep more number ofinventories to satisfy decreasing demand for longertime period and because more number of inventoriesare kept for longer time period it leads to increase inthe optimal cost (Figure 3)

(c) As the value of 120579 increases keeping other parametersconstant then the optimal cycle length 119879lowast increaseswhereas optimal order quantity 119876(119879lowast) and the opti-mal total cost119870(119879lowast) decreases as depicted in Table 3This is also true with the fact that as 120579 increasesmeans demand of the first product decreases morebecause more demand is satisfied by its substituteproduct which is introduced after a certain interval oftime and therefore less inventories are kept for longerperiod which leads to decrease in the optimal cost

6 Managerial Implications

The goodwill of any organization is entirely dependent onits effective management and this is possible only when theproblemof the organization is analyzed properly from all per-spectives Here our problem is concerned with the effectivemanagement of products which are newly introduced in themarket and along with this the effect of second generationproducts on the demand of first generation products is alsorealized the idea included here is that the effect of substituteproducts decreases the demand of old products after a certain

Table 2 Sensitivity analysis on ldquo1198791rdquo

1198791

119879lowast

119870(119879lowast) 119876(119879

lowast)

010 059 68778 124011 061 69032 128012 063 69279 131013 064 69520 135014 066 69756 139015 067 69987 143016 069 70213 146017 07 70435 150018 072 70653 153019 073 70866 157020 075 71076 160For 120579 = 0001 119901 = 0005

Table 3 Sensitivity analysis on ldquo120579rdquo

120579 119879lowast

119870(119879lowast) 119876(119879

lowast)

0019 038 1802327 22280020 041 1649195 21800021 044 1495228 21210022 048 1340344 20500023 052 1184432 19650024 058 1027337 18650025 065 868835 17470026 073 708581 16050027 086 545998 14330028 107 379973 12170029 154 207669 918For 1198791 = 001 119901 = 003119873 = 500000

Table 4 Sensitivity analysis on coefficient of innovation ldquo119901rdquo

119901 119879lowast

119870(119879lowast) 119876(119879

lowast)

0001 076 17865 380002 054 34044 540003 044 49942 670004 038 65695 770005 034 81354 860006 031 96946 950007 029 112487 1020008 027 127987 1100009 026 143453 117001 024 158891 123002 017 312305 178Special case

interval of time because of technological innovations andother related factors Therefore to overcome this problem amathematical model has been developed to know the actualsituation of the economic ordering policies which will helpthe inventory manager to take effective action to maintainthe optimal cost The results obtained in the model create apool of knowledge on the diffusion processes of innovations

8 Advances in Decision Sciences

48625

48675

48725

48775

48825

48875

Time

Cos

t

038 042 048 052 056 06

Cost-time graphFor Tlowast

= 048 K(Tlowast) = 48684

(a)

69745

69755

69765

69775

69785

69795

Time

Cos

t

061 063 065 067 069 071

Cost-time graphFor Tlowast

= 066 K(Tlowast) = 69756

(b)

868700

868800

868900

869000

869100

869200

Time

Cos

t

061 063 065 067 069

Cost-time graphFor Tlowast

= 065 K(Tlowast) = 868835

(c)

33000

33500

34000

34500

35000

35500

36000

Time

Cos

t

024 034 044 054 064 074 084 094 104

Cost-time graph (special case)For Tlowast

= 054 K(Tlowast) = 34044

(d)

Figure 3 Cost-time graphs showing convexity of the cost functions

and its effective management which are most valuable toboth researchers and managers For managers this poolof knowledge is important because it provides useful toolsand useful information for managing and scheduling theinventories of new products when its demand is affectedby the generation of other new products which substituteproducts after a certain time period The uniqueness of thismodel is that how the inventory manager should keep theoptimal cycle time of any fixed lot size of one product enteringinto the inventory system when its substitute product entersinto the system after a fixed interval so that optimal costis maintained and the cost of inventory obsolescence isminimized

7 Conclusion

The technological breakthrough is constantly being experi-enced in various products across the globe which is neces-sary for the products to gain competitive advantage Thereare various marketing strategies to promote and establishproducts which are newly introduced into the market butkeeping its inventories efficiently and effectively play a greaterrole Here the role of inventory models becomes significantThe primary objective of developing inventory models isto take efficient action concerned with economic orderingpolicies and to understand well the behaviour of parametersassociated with it In this paper a mathematical model has

Advances in Decision Sciences 9

been developed for obtaining the Economic Order Quantity(EOQ) in which the demand of the product is assumed tofollow an innovation diffusion process as proposed by Fourtand WoodLock [2] and along with this the demand of theproduct decreases after a certain interval of time because ofthe introduction of its substitute product Here the modelanalyses the situation when demand of the previous productis affected by introducing second generation product Thereare few inventory models such as Chern et al [28] Chandaand Kumar [25] Aggarwal et al [26] and Chanda andKumar [27] which capture the sensitive nature of marketingparameters but have not incorporated the idea of the effectof introduction of second generation product This paperattempts to bridge this gap The approach of this paper is toobtain economic ordering policies of a new product whenits substitute product enters into the market after a certaintime period A special case has also been discussed whichexcludes the effect of substitute products on the inventorypolicies of old products Numerical examples with sensitivityanalyses are presented to illustrate the model properly Theresults obtained are largely consistent with what we observeempirically A simple solution procedure in the form ofalgorithm is presented to determine the optimal cycle timeand optimal order quantity of the cost functionTheproposedmodel can be extended for backorders quantity discountpartial lost sales and so forth and it can also be extendedto solve by different approaches such as geometric program-ming approach and genetic algorithm approach At the sametime future research should also focus on developing somealternative approach to obtain the optimal analytical solutionof the highly nonlinear cost function which we have solvednumerically

Acknowledgments

The research presented in this paper was carried out at theDepartment ofOperational Reseach Faculty ofMathematicalSciences University of Delhi The commercial identitiesmentioned in the paper such as LINGO and Excel-Solversoftware have only been used for the numerical analysis of themodel discussed in the paper which is only for the academicpurpose The authors of this paper do not have any directfinancial relation with the commercial identities mentionedin the paper and there is no conflict of interests regardingfinancial gains for all the authors

References

[1] E Rogers Diffusion of Innovations Free Press New York NYUSA 3rd edition 1983

[2] L A Fourt and J W WoodLock ldquoEarly prediction of Marketsuccess for new grocery productsrdquo Journal of Marketing vol 26no 2 pp 31ndash38 1960

[3] J C Fisher and R H Pry ldquoA simple substitution modelof technological changerdquo Technological Forecasting and SocialChange vol 3 no C pp 75ndash88 1971

[4] F M Bass ldquoA new product growth for model consumerdurablesrdquo Management Science vol 50 no 12 pp 1825ndash18322004

[5] R Peres E Muller and V Mahajan ldquoInnovation diffusion andnew product growth models a critical review and researchdirectionsrdquo International Journal of Research in Marketing vol27 no 2 pp 91ndash106 2010

[6] PM Parker ldquoAggregate diffusion forecastingmodels inmarket-ing a critical reviewrdquo International Journal of Forecasting vol10 no 2 pp 353ndash380 1994

[7] E M Rogers Diffusion of Innovations Free Press New YorkNY USA 4th edition 1995

[8] R Peres E Muller and V Mahajan ldquoInnovation diffusion andnew product growth models a critical review and researchdirectionsrdquo International Journal of Research in Marketing vol27 no 2 pp 91ndash106 2010

[9] V Mahajan E Muller and Y Wind New-Product DiffusionModels Kluwer Academic Publishers Boston Mass USA2000

[10] F Cheng and S P Sethi ldquoPeriodic review inventory modelwith demand influenced by promotion decisionsrdquoManagementScience vol 45 no 11 pp 1510ndash1523 1999

[11] S Kalish ldquoAnewproduct adoptionmodelwith price advertisingand uncertaintyrdquoManagement Science vol 31 no 12 pp 1569ndash1585 1985

[12] VMahajan andA P Robert ldquoInnovation diffusion in a dynamicpotential adopter populationrdquoManagement Science vol 24 no15 pp 1589ndash1597 1978

[13] M N Sharif and K Ramanathan ldquoPolynomial innovationdiffusion modelsrdquo Technological Forecasting and Social Changevol 21 no 4 pp 301ndash323 1982

[14] M N Sharif and K Ramanathan ldquoBinomial innovation dif-fusion models with dynamic potential adopter populationrdquoTechnological Forecasting and Social Change vol 20 no 1 pp63ndash87 1981

[15] J A Norton and F M Bass ldquoA diffusion theory model ofadoption and substitution for successive generation of high-technology productsrdquo Management Science vol 33 no 9 pp1069ndash1086 1987

[16] V Mahajan and E Muller ldquoTiming diffusion and substitutionof successive generations of technological innovations the IBMmainframe caserdquo Technological Forecasting and Social Changevol 51 no 2 pp 109ndash132 1996

[17] M W Speece and D L Maclachlan ldquoApplication of a multi-generation diffusion model to milk container technologyrdquoTechnological Forecasting and Social Change vol 49 no 3 pp281ndash295 1995

[18] P J Danaher B G S Hardie and W P Putsis Jr ldquoMarketing-mix variables and the diffusion of successive generations of atechnological innovationrdquo Journal of Marketing Research vol38 no 4 pp 501ndash514 2001

[19] T Islam and NMeade ldquoThe diffusion of successive generationsof a technology a more general modelrdquo Technological Forecast-ing and Social Change vol 56 no 1 pp 49ndash60 1997

[20] N Kim D R Chang and A D Shocker ldquoModeling intercat-egory and generational dynamics for a growing informationtechnology industryrdquo Management Science vol 46 no 4 pp496ndash512 2000

[21] B Bayus ldquoThe dynamic pricing of next generation consumerdurablesrdquoMarketing Science vol 11 no 3 pp 251ndash265 1992

[22] A K Bardhan and U Chanda ldquoA model for first and sub-stitution adoption of successive generations of a productrdquoInternational Journal of Modelling and Simulation vol 28 no4 pp 487ndash494 2008

10 Advances in Decision Sciences

[23] U Chanda and A K Bardhan ldquoDynamic optimal advertisingexpenditure strategies of two successive generations of con-sumer durablesrdquo in Proceedings of the International Confer-ence on Modelling Computation and Optimization held duringJanuary 2008 in the venue SQC amp OR Unit Indian StatisticalInstitute New Delhi India 2008

[24] K A Bardhan and U Chanda ldquoA mathematical model fordiffusion of products with multiple generationsrdquo InternationalJournal of Operational Research vol 4 no 2 pp 214ndash229 2009

[25] U Chanda and A Kumar ldquoEconomic order quantity modelwith demand influenced by dynamic innovation effectrdquo Interna-tional Journal of Operational Research vol 11 no 2 pp 193ndash2152011

[26] K K Aggarwal C K Jaggi and A Kumar ldquoEconomic orderquantity model with innovation diffusion criterion havingdynamic potential market sizerdquo International Journal of AppliedDecision Sciences vol 4 no 3 pp 280ndash303 2011

[27] U Chanda and A Kumar ldquoEconomic order quantity model oninflationary conditions with demand influenced by innovationdiffusion criterionrdquo International Journal of Procurement Man-agement vol 5 no 2 pp 160ndash177 2012

[28] M-S Chern J-T Teng and H-L Yang ldquoInventory lot-sizepolicies for the bass diffusion demand models of new durableproductsrdquo Journal of the Chinese Institute of Engineers vol 24no 2 pp 237ndash244 2001

[29] K K Aggarwal and A Kumar ldquoAn inventory model for newproduct when demand follows dynamic innovation processhaving dynamic potential market sizerdquo International Journal ofStrategic Decision Sciences vol 3 no 4 pp 78ndash99 2012

[30] A Kumar K K Aggarwal and U Chanda ldquoEconomic orderquantity model under fuzzy sense with demand follows Bassrsquosinnovation diffusion processrdquo International Journal of AdvancedOperations Management vol 5 no 3 pp 261ndash281 2013

[31] K K Aggarwal C K Jaggi and A Kumar ldquoEconomic orderquantity model under fuzzy sense when demand follows inno-vation diffusion process having dynamic potential market sizerdquoInternational Journal of Services and Operations Managementvol 13 no 3 pp 361ndash391 2012

[32] V Mahajan and R A PetersonModels of Innovation DiffusionSage 1985

[33] E Ruiz-Conde P S H Leeflang and J EWieringa ldquoMarketingvariables in macro-level diffusion modelsrdquo Journal fur Betrieb-swirtschaft vol 56 no 3 pp 155ndash183 2006

[34] D Chandrasekaran and G J Tellis ldquoA critical review ofmarketing research on diffusion of new productsrdquo Review ofMarketing Research pp 39ndash80 2007

[35] F Sultan J U Farley and D R Lehmann ldquoA meta-analysis ofdiffusion modelsrdquo International Journal of Forecasting vol 27no 1 pp 70ndash77 1990

[36] D Talukdar K Sudhir and A Ainslie ldquoInvestigating newproduct diffusion across products and countriesrdquo MarketingScience vol 21 no 1 pp 97ndash114 2002

[37] C van den Bulte and S Stremersch ldquoSocial contagion andincome heterogeneity in new product diffusion ameta-analytictestrdquoMarketing Science vol 23 no 4 pp 530ndash544 2004

[38] G L Lilien P Kotler and K S Moorthy Marketing ModelsPrentice-Hall of India Private New Delhi India 1999

Submit your manuscripts athttpwwwhindawicom

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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Journal of

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Mathematical PhysicsAdvances in

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OptimizationJournal of

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International Journal of

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Operations ResearchAdvances in

Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Advances in Decision Sciences

Cum

ulat

ive p

erce

ntag

e of b

uyer

s

Time

Figure 1 Fourt and WoodLock Curve (Source Lilien et al [38])

Adop

tion

rate

Adoption time

Generation oneGeneration two

Generation three

Figure 2 Series of technological generations (Source Norton andBass [15])

have considered the application of dynamic potential adopterdiffusion model through their study of diffusion of oralcontraceptives in Thailand The diffusion process in termsof number of customers who have bought the product bytime ldquo119905rdquo has been explained by a modified exponential curve(Figure 1) by Fourt and WoodLock [2] This model capturesthe innovative characteristic with its coefficient ldquo119901rdquo as thecoefficient of innovation

Also diffusion of new products cannot be justified with-out taking into account the effects of successive generationsof products because an important feature of most modernnew technologies is that they come in successive generationsNew products are substituted with more advanced productsand technological generations which creates heterogeneity inthe adopting population Moreover it is observed that anygiven generation of technology will end up being replacedby a new generation The old products are not straight awayreplaced by the new products because of constant innovationbut new products intend to substitute and start competingwith the old products which creates parallel diffusion of boththe old and the new products in the market This makes thedecreasing pattern of diffusion of old products The series oftechnological generations have been well explained throughthe Figure 2 There are lots of models in the marketing liter-ature which have considered the importance of generation ofproducts Norton and Bass [15] model is a standard exampleof multiple generation model which assumes that the coef-ficients of innovation and imitation remain unchanged fromgeneration to generation of technology Norton and Bass [15]is basically based upon the Bass model The Bass model isalso extended by Mahajan and Muller [16] who proposed amodel to capture substitution and diffusion patterns for eachsuccessive generation of technological products Speece and

Maclachlan [17] and Danaher et al [18] incorporated price asan explanatory variable in their model Islam andMeade [19]proposed that the coefficients of later generation technologyare constant incrementdecrement over the coefficients ofthe first generation Kim et al [20] developed a model andtried to include intergenerational linkages among successivegenerations within a product category Bayus [21] developeda model for consumer sales of second-generation productsby incorporating replacement behavior of first generationadopters and suggested different dynamic pricing policies ofsecond-generation consumer durables Bardhan and Chanda[22] have developed a mathematical model for adoptionof successive generations of a high technology productChanda and Bardhan [23] developed a model on dynamicoptimal advertising expenditure strategies of two successivegenerations of consumer durables Bardhan and Chanda[24] also developed a mathematical model for diffusion ofproducts with multiple generations of a high technologyproduct The above discussed models are well explained inthe marketing literature and are extremely significant in theestimation of new products having successive generationfeatures These models are highly useful in making theEconomic inventory policy of any organization as far as newproducts along with its successive generations are concernedUnfortunately the concept of the above marketing modelshas not been integrated with traditional inventory modelswhich is highly desired while making the economic orderingpolicy for new products with successive generation featuresThough there are some inventory models described belowwhich have incorporated the concept of innovation diffusionin their demand functions but these models do not considerthe effect of diminishing nature of demand rate when thesubstitute products are entered into the market becauseof technological innovations which are popularly knownas the successive generations of the products Chanda andKumar [25] have explained the economic order quantitymodel with demand influenced by dynamic innovation effectAggarwal et al [26] have considered the EOQ model withinnovation diffusion criterion having dynamic potentialmar-ket size Also Chanda and Kumar [27] explain the EOQmodel having demand influenced by innovation diffusioncriterion under inflationary condition Chern et al [28] haveformulated the economic order quantity model in which thedemand follows innovation diffusion criterion as consideredby Bass [4] Aggarwal et al [29] formulated an inventorymodel for new product when demand follows dynamicinnovation process having dynamic potential market sizeKumar et al [30] discussed economic order quantity modelunder fuzzy sensewhen the demand follows Bassrsquos innovationdiffusion process Aggarwal et al [31] have developed aneconomic order quantity model under fuzzy sense whendemand follows innovation diffusion process having dynamicpotential market size The approach in this paper is to focuson the effect of innovation factor on adoption behaviorof consumers with the effect of technological substitutionof the products The uniqueness of this model is to makean economic inventory policy of a new product under thecondition of its diminishing demand after a certain intervalof time when its substitute product enters into the market

Advances in Decision Sciences 3

because of technological innovation The paper is dividedinto the model development special case numerical exam-ples with sensitivity analysis observations and managerialimplications Finally the paper concludes with a discussionon the application extension and limitations of the modelAssumptions Inventory model has been developed under thefollowing assumptions

(i) The replenishment rate is infinite and thus replen-ishments are instantaneous

(ii) Lead time is zero(iii) The planning horizon is finite(iv) Shortages are not allowed(v) Demand rate is time dependent governed by innova-

tion diffusion process and is affected by the introduc-tion of substitute products because of technologicalinnovations

(vi) The size of the potential market of total number ofadopters remains constant Here the potentialmarketsize includes the number of initial purchases for thetime interval for which replacement purchases areexcluded

(vii) The coefficient of innovation remains constant thatis the likelihood that somebody who is not yet usingthe product will start using it because of mass mediacoverage or other external factors will act as constantthroughout the cycle length

(viii) There is only one product bought per new adopter(ix) The innovationrsquos sales are confined to a single geo-

graphic area(x) The impact of marketing strategies by the innovator is

adequately captured by the modelrsquos parameters

Notations Notations used in the modeling framework are asfollows

119860 ordering cost119862 unit cost119868 inventory carrying charge119868119862 inventory carrying cost119879 length of the replenishment cycle1198791 time of introduction of second generation product

120579 rate of diminishing demand of first generationproduct119901 coefficient of innovation119876 number of items received at the beginning of theperiod119870(119879) the total cost of the system per unit time119868(119905) on hand inventory at any time 119905119899(119905) = 120582(119905) = 119878(119905) the number of adoptions at time 119905that is Demand at time 119905119873 potential market size

119873(119905) cumulative number of adopters at time 119905119891(119905) the likelihood of purchase at time 119905119865(119905) the cumulative fraction of adopters at time 119905

2 Mathematical Model

The basic assumption considered by different researchersin marketing literature for a fundamental diffusion modelis that the rate of diffusion or the number of adoptersat any given point in time is directly proportional to thenumber of remaining potential adopters at that momentMathematically this can be represented as follows

119899 (119905) =119889119873 (119905)

119889119905= 119892 (119905) 119873 minus 119873 (119905) (1)

where 119892(119905) is known as the rate of adoption or individualprobability of adoption

It has also been assumed that 119892(119905) depends on timethrough a linear function of 119873(119905) (Mahajan and Peterson[32]) Hence

119899 (119905) =119889119873 (119905)

119889119905= (119886 + 119887119873 (119905)) 119873 minus 119873 (119905) 119886 ge 0 119887 ge 0

(2)

Depending on the importance of each source of influencedifferent versions can be derived from the fundamentaldiffusion model (Mahajan and Peterson [32])

When 119887 = 0 the model only considers external influencewhen 119886 = 0 the model only considers internal influenceWhen 119886 = 0 and 119887 = 0 the resulting model is called a mixedinfluence diffusion model (Ruiz-Conde et al [33])

The basic assumptions used in the Bass Model are thatthe adoption of a new product spreads through a populationprimarily due to contact with prior adopters Hence theprobability that an initial purchase occurs at time 119905 given thatno purchase has occurred is a linear function of the numberof previous buyers that is

119891 (119905)

1 minus 119865 (119905)= 119901 + 119902119865 (119905) (3)

If we define 119899(119905) = 119873119891(119905) and 119873(119905) = 119873119865(119905) we canexpress (3) as follows

119899 (119905) =119889119873 (119905)

119889119905= (119901 + 119902

119873 (119905)

119873

) 119873 minus 119873 (119905) (4)

Equation (4) is a mixed influence diffusion model Considera case of diffusion model of external influence only (taking119902 = 0) as proposed by Fourt and WoodLock [2] (4) can bedefined as follows

119899 (119905) =119889119873 (119905)

119889119905= 119901 119873 minus 119873 (119905) (5)

The basic demand model that has been used in the EOQmodel is based on the following assumptions

4 Advances in Decision Sciences

(i) Adoptions take place due to innovation-diffusioneffect and it is influenced by the innovation-effect(mass media) that is external influence only

(ii) Adoptions of the first generation product is dimin-ishing by the introduction of the second generationproduct

Here the diffusion of new products which is spreadthrough the external influence has been considered for mak-ing economic ordering policies of the products The demandof first generation products that is products which aresupposed to be introduced at the beginning of the planningperiod also known as old products is affected (diminished)due to introduction of the second generation products thatis products which are supposed to be new in comparison tothe first generation products and affecting the demand of oldproducts because of technological innovations after a certaininterval of time Therefore using the above assumptionsexplanations and (5) the following demand function hasbeen constructed for our model120582 (119905)

=

119901119873 minus 119873 (119905) 0 le 119905 lt 1198791 119901 ge 120579

119901 119873 minus 119873 (119905) minus 120579 119873 minus 119873 (119905) 1198791le 119905 le 119879 119901 ge 120579

(6)The above demand model shows that the inventory level isdepleted wholly due to demand of the old products with aparticular pace depending on the parameters associated withit up to time period 119879

1and after time 119879

1and up to time

period 119879 the rate of decreasing inventory level is somewhatless than what was experienced before time 119879

1because some

demand of the old products are satisfied (substituted) by thenew products introduced after time period 119879

1because of

technological innovations The demand usage 120582(119905) which isa function of time plays pivotal role to shrink the inventorysize over a period of time If in the time interval (119905 119905 + 119889119905)the inventory size is dipping at the rate 120582(119905)119889119905 then the totalreduction in the inventory size during the time interval119889119905 canbe given by minus119889119868(119905) = 120582(119905)119889119905 Thus the differential equationdescribing the instantaneous state of the inventory level at anytime 119905 119868(119905) in the interval (0 119879) is given by

119889119868 (119905)

119889119905= minus120582 (119905) 0 le 119905 le 119879 (7)

where120582 (119905)

= 119899 (119905) =119889119873 (119905)

119889119905

=

119901119873 minus 119873 (119905) 0 le 119905 lt 1198791 119901 ge 120579

119901 119873 minus 119873 (119905) minus 120579 119873 minus 119873 (119905) 1198791le 119905 le 119879 119901 ge 120579

997904rArr 119873(119905) =

119873 (1 minus 119890minus119901119905) 0 le 119905 lt 119879

1

119873 minus 119873 minus 119873(1198791) 119890(119901minus120579)(119879

1minus119905) 1198791le 119905 le 119879

(8)

Now by model assumption replenishment is instanta-neous and shortages are not allowedThus the inventory levelat the initial point of the planning horizon can be assumed tobe the cumulative adoption of the product during the cycletime 119879 Hence

119868 (0) = 119876 = int

119879

0

120582 (119905) 119889119905 (9)

997904rArr 119868 (0) = 119876 = 119873(1 minus 119890minus1199011198791) + [119873 (119879

1) minus 119873]

times (119890(119901minus120579)(119879

1minus119879)

minus 1)

(10)

The solution of the differential equation (7) is

119868 (119905) =

119868 (0) minus 119873 (1 minus 119890minus119901119905) 0 le 119905 lt 119879

1

[119873 minus 119873 (1198791)]

times [119890(119901minus120579)(119879

1minus119905)minus 119890minus(119901minus120579)(119879

1minus119879)] 119879

1le 119905 le 119879

(11)

Now

int

119879

0

119868 (119905) 119889119905 = int

1198791

0

119868 (119905) 119889119905 + int

119879

1198791

119868 (119905) 119889119905 (12)

997904rArr int

119879

0

119868 (119905) 119889119905 = 1198791119868 (0) +

119873

119901[1 minus 119890

minus1199011198791]

minus 1198731198791+ [119873 minus 119873(119879

1)]

times[119890(119901minus120579)(119879

1minus119879)

minus 1

(120579 minus 119901)+(1198791minus 119879) 119890

(119901minus120579)(1198791minus119879)]

(13)

The different cost elements involved in the inventorysystem per unit time can be defined as

Ordering cost per unit time = 119860

119879 (14)

Material cost per unit time = 119876119862

119879 (15)

=119862

119879119873(1 minus 119890

minus1199011198791)

+ (119862

119879) [119873 (119879

1) minus 119873] (119890

(119901minus120579)(1198791minus119879)

minus 1)

(16)

Cost of carrying inventory per unit time

=119868119862

119879int

119879

0

119868 (119905) 119889119905

(17)

Advances in Decision Sciences 5

= (119868119862

119879)1198791119868 (0) +

119868119862119873

119901119879[1 minus 119890

minus1199011198791]

minus (119868119862

119879)1198731198791

+ (119868119862

119879) [119873 minus 119873(119879

1)]

times [119890(119901minus120579)(119879

1minus119879)

minus 1

(120579 minus 119901)+ (1198791minus 119879) 119890

(119901minus120579)(1198791minus119879)]

(18)

Thus the total cost of the inventory system per unit time119870(119879) is

119870 (119879) = (14) + (16) + (18)

997904rArr 119870 (119879) =119860

119879+ [119873 (119879

1) minus 119873] (119890

(119901minus120579)(1198791minus119879)

minus 1)

times (119862

119879) (1 + 119868119879

1) + 119873 (1 minus 119890

minus1199011198791) (

119862

119879)

times (1 + 1198681198791) +

119868119862119873

119901119879[1 minus 119890

minus1199011198791] minus (

119868119862

119879)1198731198791

+ (119868119862

119879) [119873 minus 119873(119879

1)]

times [119890(119901minus120579)(119879

1minus119879)

minus 1

(120579 minus 119901)+ (1198791minus 119879) 119890

(119901minus120579)(1198791minus119879)]

(19)

The necessary criterion for 119870(119879) to be minimum is

119889119870 (119879)

119889119879= 0 (20)

997904rArrminus119860

1198792+ [119873 (119879

1) minus 119873] (119890

(119901minus120579)(1198791minus119879)

minus 1) (minus119862

1198792) (1 + 119868119879

1)

minus (119862

1198792) (1 + 119868119879

1)119873 (1 minus 119890

minus1199011198791)

+ (119862

119879) (1 + 119868119879

1) [119873 (119879

1) minus 119873] (120579 minus 119901) 119890

(119901minus120579)(1198791minus119879)

minus119868119862119873

1199011198792[1 minus 119890

minus1199011198791] + (

119868119862

1198792)1198731198791

+ (minus119868119862

1198792) [119873 minus 119873(119879

1)]

times [119890(119901minus120579)(119879

1minus119879)

minus 1

(120579 minus 119901)+ (1198791minus 119879) 119890

(119901minus120579)(1198791minus119879)]

+ (119868119862

119879) [119873 minus 119873(119879

1)]

times (1198791minus 119879) (120579 minus 119901) 119890

(119901minus120579)(1198791minus119879)

= 0

(21)

Now for119870(119879) to be convex

1198892119870 (119879)

1198891198792

100381610038161003816100381610038161003816100381610038161003816119879=119879lowast

gt 0

997904rArr2119860

119879lowast3+ (

2119862

119879lowast3) (1 + 119868119879

1)119873 (1 minus 119890

minus1199011198791)

+ [119873 (1198791) minus 119873] (119890

(119901minus120579)(1198791minus119879lowast)minus 1) (

2119862

1198793) (1 + 119868119879

1)

+ (119862

119879lowast) (1 + 119868119879

1) [119873 (119879

1) minus 119873] (120579 minus 119901)

2

119890(119901minus120579)(119879

1minus119879lowast)

+2119868119862119873

119901119879lowast3

[1 minus 119890minus1199011198791] + (

2119868119862

119879lowast3) [119873 minus 119873(119879

1)]

times [119890(119901minus120579)(119879

1minus119879lowast)minus 1

(120579 minus 119901)+ (1198791minus 119879lowast) 119890(119901minus120579)(119879

1minus119879lowast)]

+ (119868119862

119879lowast) [119873 minus 119873(119879

1)] (1198791minus 119879lowast)

times (120579 minus 119901)2

119890(119901minus120579)(119879

1minus119879lowast)

gt (2119862

119879lowast2) (1 + 119868119879

1) [119873 (119879

1) minus 119873]

times (120579 minus 119901) 119890(119901minus120579)(119879

1minus119879)

+ (2119868119862

119879lowast3)1198731198791

+ (2119868119862

119879lowast2) [119873 minus 119873(119879

1)]

times [(1198791minus 119879lowast) (120579 minus 119901) 119890

(119901minus120579)(1198791minus119879lowast)]

+ (119868119862

119879lowast) [119873 minus 119873(119879

1)] [(120579 minus 119901) 119890

(119901minus120579)(1198791minus119879lowast)]

(22)

The solution of the equation 119889119870(119879)119889119879 = 0 gives theoptimum value of 119879 provided it satisfies the condition1198892119870(119879)119889119879

2gt 0 Since the above cost equation (21) is highly

nonlinear the problem has been solved numerically for givenparameter valuesThe solution gives the optimumvalue119879lowast ofthe replenishment cycle time 119879 Once 119879lowast is known the valueof optimum order quantity 119876lowast and the optimum cost 119870(119879lowast)can easily be obtained from (10) and (19) respectively Thenumerical solution for the given base value has been obtainedby using software packages Lingo and Excel-Solver

21 Special Case When 120579 becomes negligible that is equalto zero which means here we are not considering the effectof introduction of new products on the old one after timeperiod 119879

1 That is demand rate of the old product is the

same throughout the planning period (0 119879) and for this casethe adoption rate that is demand rate that is 120582(119905) = 119899(119905)reduces to the classical Fourt and WoodLock model [2] Theobjective here is to study the different managerial policiesfor the model which is under external influence only andits demand is not affected by the introduction of any other

6 Advances in Decision Sciences

substitute products In that case the total cost per unit time119870(119879) is given by

119870 (119879) =119860

119879+119873119862

119879(1 minus 119890

minus119901119879) +

119873119868119862

119901119879(1 minus 119890

minus119901119879) minus 119873119868119862119890

minus119901119879

(23)

119868 (0) = 119876 = int

119879

0

120582 (119905) 119889119905 = 119873 (1 minus 119890minus119901119879

) (24)

For optimum total cost the necessary criterion is

119889119870 (119879)

119889119879= 0 (25)

997904rArrminus119860

1198792+119873119862119901119890

minus119901119879

119879minus

119873119862(1 minus 119890minus119901119879

)

1198792

+119873119868119862119890

minus119901119879

119879

minus

119873119868119862 (1 minus 119890minus119901119879

)

1199011198792

+ 119873119868119862119901119890minus119901119879

= 0

(26)

Now for119870(119879) to be convex

1198892119870 (119879)

1198891198792

100381610038161003816100381610038161003816100381610038161003816119879=119879lowast

gt 0

997904rArr2119860

119879lowast3+

2119873119862(1 minus 119890minus119901119879lowast

)

119879lowast3

+

2119873119868119862 (1 minus 119890minus119901119879lowast

)

119901119879lowast3

gt1198731198621199012119890minus119901119879lowast

119879lowast

+119873119868119901119862119890

minus119901119879lowast

119879lowast2

+ 1198731198681198621199012119890minus119901119879lowast

(27)

The solution of the equation 119889119870(119879)119889119879 = 0 gives theoptimum value of 119879 provided it satisfies the condition1198892119870(119879)119889119879

2gt 0 Since cost equation (26) is highly nonlinear

the problem has been solved numerically for given parametervalues The solution gives the optimum value 119879lowast of thereplenishment cycle time 119879 Once 119879lowast is known the valuesof optimum order quantity 119876lowast and the optimum cost 119870(119879lowast)can easily be obtained from (24) and (23) respectively Thenumerical solution for the given base value has been obtainedby using software packages LINGO and Excel-Solver

3 Solution Procedure

The solution procedure has been summarized in the follow-ing algorithmStep 1 Input all parameter values such as unit cost coefficientof innovation potential market size and ordering cost foreach case separatelyStep 2 Compute all possible values of ldquo119879rdquo separately for (21)and (26) as the case may beStep 3 select the appropriate value of ldquo119879rdquo say 119879lowast for eachcase by using (19) and (23) and by satisfying the above statedcondition 1198892119870(119879)1198891198792 gt 0Step 4Compute119870(119879lowast) from (19) and (23) amp119876(119879lowast) from (10)and (24) for each case separately

The above steps are used for all replenishment cyclesusing appropriate parameter values In order to obtain theappropriate values of ldquo119879rdquo we need to follow the aboveprocedure with the help of LINGO and Excel-Solver softwarepackages

4 Numerical Examples

The effectiveness of the proposed model has been shown bythe following numerical examples A hypothetical examplehas the following parameter values in appropriate units

119860 =$1100order

119862 =$300unit

119868 = 025 119873 = 50000

(28)

The results have been well presented in the followingdifferent tables Also to prove the validity of the modelnumerically and to get the appropriate parameter values thereferences have been considered as Chanda and Kumar [25]Chandrasekaran and Tellis [34] Sultan et al [35] Talukdaret al [36] Van den Bulte and Stremersch [37] Chanda andKumar [27] Aggarwal et al [26 31] Aggarwal et al [29] andKumar et al [30]

(i) The mean value of the coefficient of innovation fora new product usually lies between 00007 and 003(Sultan et al [35] Talukdar et al [36] Van den Bulteand Stremersch [37])

(ii) The mean value of the coefficient of innovation fora new product is usually 001for developed countriesand 00003 for developing countries (Talukdar et al[36])

41 Special Case A hypothetical example has the followingparameter values in appropriate units

119860 =$1100order

119862 =$300unit

119868 = 025 119873 = 50000 120579 = 0

(29)

5 Observations

The results obtained from different numerical tables inSection 4 explain the effect of changes in the system param-eters on the optimal values of total cost 119870(119879lowast) the optimalcycle length 119879

lowast and the optimal order quantity 119876(119879lowast)

We have observed the following relationship during thenumerical exercise

(a) As the coefficient of innovation increases keepingother parameters constant then the optimal cyclelength 119879

lowast decreases while both the optimal orderquantity 119876(119879

lowast) and the optimal total cost 119870(119879lowast)

increases as depicted in Tables 1 and 4 This isconsistent with the reality as more investment onpromotion will increase the diffusion of a productin the market resulting in shrinkage of the optimalreorder cycle time as a result optimal cost is increased

Advances in Decision Sciences 7

Table 1 Sensitivity analysis on coefficient of innovation ldquo119901rdquo

119901 119879lowast

119870(119879lowast) 119876(119879

lowast)

0001 086 16400 390002 059 32705 570003 048 48684 700004 041 64499 810005 037 80211 910006 034 95848 1010007 031 111430 1090008 029 126967 1170009 027 142468 124001 026 157938 131002 019 311593 190For 1198791 = 01 120579 = 00001

(b) As the cycle length 1198791 that is the time of intro-

duction of second product increases keeping otherparameters constant then the optimal cycle length119879lowastoptimal order quantity 119876(119879lowast) and the optimal totalcost 119870(119879lowast) all these three values increase as depictedin Table 2 This is true with the fact that as the timeof introduction of second product increases meansmaximum demand is satisfied by more number ofprevious that is old products and it is also obviousthat the demand is a decreasing function of timewhich implies that less demand for more time periodleads to increase in optimal cycle length And sinceoptimal cycle length increases therefore it forcesthe inventory manager to keep more number ofinventories to satisfy decreasing demand for longertime period and because more number of inventoriesare kept for longer time period it leads to increase inthe optimal cost (Figure 3)

(c) As the value of 120579 increases keeping other parametersconstant then the optimal cycle length 119879lowast increaseswhereas optimal order quantity 119876(119879lowast) and the opti-mal total cost119870(119879lowast) decreases as depicted in Table 3This is also true with the fact that as 120579 increasesmeans demand of the first product decreases morebecause more demand is satisfied by its substituteproduct which is introduced after a certain interval oftime and therefore less inventories are kept for longerperiod which leads to decrease in the optimal cost

6 Managerial Implications

The goodwill of any organization is entirely dependent onits effective management and this is possible only when theproblemof the organization is analyzed properly from all per-spectives Here our problem is concerned with the effectivemanagement of products which are newly introduced in themarket and along with this the effect of second generationproducts on the demand of first generation products is alsorealized the idea included here is that the effect of substituteproducts decreases the demand of old products after a certain

Table 2 Sensitivity analysis on ldquo1198791rdquo

1198791

119879lowast

119870(119879lowast) 119876(119879

lowast)

010 059 68778 124011 061 69032 128012 063 69279 131013 064 69520 135014 066 69756 139015 067 69987 143016 069 70213 146017 07 70435 150018 072 70653 153019 073 70866 157020 075 71076 160For 120579 = 0001 119901 = 0005

Table 3 Sensitivity analysis on ldquo120579rdquo

120579 119879lowast

119870(119879lowast) 119876(119879

lowast)

0019 038 1802327 22280020 041 1649195 21800021 044 1495228 21210022 048 1340344 20500023 052 1184432 19650024 058 1027337 18650025 065 868835 17470026 073 708581 16050027 086 545998 14330028 107 379973 12170029 154 207669 918For 1198791 = 001 119901 = 003119873 = 500000

Table 4 Sensitivity analysis on coefficient of innovation ldquo119901rdquo

119901 119879lowast

119870(119879lowast) 119876(119879

lowast)

0001 076 17865 380002 054 34044 540003 044 49942 670004 038 65695 770005 034 81354 860006 031 96946 950007 029 112487 1020008 027 127987 1100009 026 143453 117001 024 158891 123002 017 312305 178Special case

interval of time because of technological innovations andother related factors Therefore to overcome this problem amathematical model has been developed to know the actualsituation of the economic ordering policies which will helpthe inventory manager to take effective action to maintainthe optimal cost The results obtained in the model create apool of knowledge on the diffusion processes of innovations

8 Advances in Decision Sciences

48625

48675

48725

48775

48825

48875

Time

Cos

t

038 042 048 052 056 06

Cost-time graphFor Tlowast

= 048 K(Tlowast) = 48684

(a)

69745

69755

69765

69775

69785

69795

Time

Cos

t

061 063 065 067 069 071

Cost-time graphFor Tlowast

= 066 K(Tlowast) = 69756

(b)

868700

868800

868900

869000

869100

869200

Time

Cos

t

061 063 065 067 069

Cost-time graphFor Tlowast

= 065 K(Tlowast) = 868835

(c)

33000

33500

34000

34500

35000

35500

36000

Time

Cos

t

024 034 044 054 064 074 084 094 104

Cost-time graph (special case)For Tlowast

= 054 K(Tlowast) = 34044

(d)

Figure 3 Cost-time graphs showing convexity of the cost functions

and its effective management which are most valuable toboth researchers and managers For managers this poolof knowledge is important because it provides useful toolsand useful information for managing and scheduling theinventories of new products when its demand is affectedby the generation of other new products which substituteproducts after a certain time period The uniqueness of thismodel is that how the inventory manager should keep theoptimal cycle time of any fixed lot size of one product enteringinto the inventory system when its substitute product entersinto the system after a fixed interval so that optimal costis maintained and the cost of inventory obsolescence isminimized

7 Conclusion

The technological breakthrough is constantly being experi-enced in various products across the globe which is neces-sary for the products to gain competitive advantage Thereare various marketing strategies to promote and establishproducts which are newly introduced into the market butkeeping its inventories efficiently and effectively play a greaterrole Here the role of inventory models becomes significantThe primary objective of developing inventory models isto take efficient action concerned with economic orderingpolicies and to understand well the behaviour of parametersassociated with it In this paper a mathematical model has

Advances in Decision Sciences 9

been developed for obtaining the Economic Order Quantity(EOQ) in which the demand of the product is assumed tofollow an innovation diffusion process as proposed by Fourtand WoodLock [2] and along with this the demand of theproduct decreases after a certain interval of time because ofthe introduction of its substitute product Here the modelanalyses the situation when demand of the previous productis affected by introducing second generation product Thereare few inventory models such as Chern et al [28] Chandaand Kumar [25] Aggarwal et al [26] and Chanda andKumar [27] which capture the sensitive nature of marketingparameters but have not incorporated the idea of the effectof introduction of second generation product This paperattempts to bridge this gap The approach of this paper is toobtain economic ordering policies of a new product whenits substitute product enters into the market after a certaintime period A special case has also been discussed whichexcludes the effect of substitute products on the inventorypolicies of old products Numerical examples with sensitivityanalyses are presented to illustrate the model properly Theresults obtained are largely consistent with what we observeempirically A simple solution procedure in the form ofalgorithm is presented to determine the optimal cycle timeand optimal order quantity of the cost functionTheproposedmodel can be extended for backorders quantity discountpartial lost sales and so forth and it can also be extendedto solve by different approaches such as geometric program-ming approach and genetic algorithm approach At the sametime future research should also focus on developing somealternative approach to obtain the optimal analytical solutionof the highly nonlinear cost function which we have solvednumerically

Acknowledgments

The research presented in this paper was carried out at theDepartment ofOperational Reseach Faculty ofMathematicalSciences University of Delhi The commercial identitiesmentioned in the paper such as LINGO and Excel-Solversoftware have only been used for the numerical analysis of themodel discussed in the paper which is only for the academicpurpose The authors of this paper do not have any directfinancial relation with the commercial identities mentionedin the paper and there is no conflict of interests regardingfinancial gains for all the authors

References

[1] E Rogers Diffusion of Innovations Free Press New York NYUSA 3rd edition 1983

[2] L A Fourt and J W WoodLock ldquoEarly prediction of Marketsuccess for new grocery productsrdquo Journal of Marketing vol 26no 2 pp 31ndash38 1960

[3] J C Fisher and R H Pry ldquoA simple substitution modelof technological changerdquo Technological Forecasting and SocialChange vol 3 no C pp 75ndash88 1971

[4] F M Bass ldquoA new product growth for model consumerdurablesrdquo Management Science vol 50 no 12 pp 1825ndash18322004

[5] R Peres E Muller and V Mahajan ldquoInnovation diffusion andnew product growth models a critical review and researchdirectionsrdquo International Journal of Research in Marketing vol27 no 2 pp 91ndash106 2010

[6] PM Parker ldquoAggregate diffusion forecastingmodels inmarket-ing a critical reviewrdquo International Journal of Forecasting vol10 no 2 pp 353ndash380 1994

[7] E M Rogers Diffusion of Innovations Free Press New YorkNY USA 4th edition 1995

[8] R Peres E Muller and V Mahajan ldquoInnovation diffusion andnew product growth models a critical review and researchdirectionsrdquo International Journal of Research in Marketing vol27 no 2 pp 91ndash106 2010

[9] V Mahajan E Muller and Y Wind New-Product DiffusionModels Kluwer Academic Publishers Boston Mass USA2000

[10] F Cheng and S P Sethi ldquoPeriodic review inventory modelwith demand influenced by promotion decisionsrdquoManagementScience vol 45 no 11 pp 1510ndash1523 1999

[11] S Kalish ldquoAnewproduct adoptionmodelwith price advertisingand uncertaintyrdquoManagement Science vol 31 no 12 pp 1569ndash1585 1985

[12] VMahajan andA P Robert ldquoInnovation diffusion in a dynamicpotential adopter populationrdquoManagement Science vol 24 no15 pp 1589ndash1597 1978

[13] M N Sharif and K Ramanathan ldquoPolynomial innovationdiffusion modelsrdquo Technological Forecasting and Social Changevol 21 no 4 pp 301ndash323 1982

[14] M N Sharif and K Ramanathan ldquoBinomial innovation dif-fusion models with dynamic potential adopter populationrdquoTechnological Forecasting and Social Change vol 20 no 1 pp63ndash87 1981

[15] J A Norton and F M Bass ldquoA diffusion theory model ofadoption and substitution for successive generation of high-technology productsrdquo Management Science vol 33 no 9 pp1069ndash1086 1987

[16] V Mahajan and E Muller ldquoTiming diffusion and substitutionof successive generations of technological innovations the IBMmainframe caserdquo Technological Forecasting and Social Changevol 51 no 2 pp 109ndash132 1996

[17] M W Speece and D L Maclachlan ldquoApplication of a multi-generation diffusion model to milk container technologyrdquoTechnological Forecasting and Social Change vol 49 no 3 pp281ndash295 1995

[18] P J Danaher B G S Hardie and W P Putsis Jr ldquoMarketing-mix variables and the diffusion of successive generations of atechnological innovationrdquo Journal of Marketing Research vol38 no 4 pp 501ndash514 2001

[19] T Islam and NMeade ldquoThe diffusion of successive generationsof a technology a more general modelrdquo Technological Forecast-ing and Social Change vol 56 no 1 pp 49ndash60 1997

[20] N Kim D R Chang and A D Shocker ldquoModeling intercat-egory and generational dynamics for a growing informationtechnology industryrdquo Management Science vol 46 no 4 pp496ndash512 2000

[21] B Bayus ldquoThe dynamic pricing of next generation consumerdurablesrdquoMarketing Science vol 11 no 3 pp 251ndash265 1992

[22] A K Bardhan and U Chanda ldquoA model for first and sub-stitution adoption of successive generations of a productrdquoInternational Journal of Modelling and Simulation vol 28 no4 pp 487ndash494 2008

10 Advances in Decision Sciences

[23] U Chanda and A K Bardhan ldquoDynamic optimal advertisingexpenditure strategies of two successive generations of con-sumer durablesrdquo in Proceedings of the International Confer-ence on Modelling Computation and Optimization held duringJanuary 2008 in the venue SQC amp OR Unit Indian StatisticalInstitute New Delhi India 2008

[24] K A Bardhan and U Chanda ldquoA mathematical model fordiffusion of products with multiple generationsrdquo InternationalJournal of Operational Research vol 4 no 2 pp 214ndash229 2009

[25] U Chanda and A Kumar ldquoEconomic order quantity modelwith demand influenced by dynamic innovation effectrdquo Interna-tional Journal of Operational Research vol 11 no 2 pp 193ndash2152011

[26] K K Aggarwal C K Jaggi and A Kumar ldquoEconomic orderquantity model with innovation diffusion criterion havingdynamic potential market sizerdquo International Journal of AppliedDecision Sciences vol 4 no 3 pp 280ndash303 2011

[27] U Chanda and A Kumar ldquoEconomic order quantity model oninflationary conditions with demand influenced by innovationdiffusion criterionrdquo International Journal of Procurement Man-agement vol 5 no 2 pp 160ndash177 2012

[28] M-S Chern J-T Teng and H-L Yang ldquoInventory lot-sizepolicies for the bass diffusion demand models of new durableproductsrdquo Journal of the Chinese Institute of Engineers vol 24no 2 pp 237ndash244 2001

[29] K K Aggarwal and A Kumar ldquoAn inventory model for newproduct when demand follows dynamic innovation processhaving dynamic potential market sizerdquo International Journal ofStrategic Decision Sciences vol 3 no 4 pp 78ndash99 2012

[30] A Kumar K K Aggarwal and U Chanda ldquoEconomic orderquantity model under fuzzy sense with demand follows Bassrsquosinnovation diffusion processrdquo International Journal of AdvancedOperations Management vol 5 no 3 pp 261ndash281 2013

[31] K K Aggarwal C K Jaggi and A Kumar ldquoEconomic orderquantity model under fuzzy sense when demand follows inno-vation diffusion process having dynamic potential market sizerdquoInternational Journal of Services and Operations Managementvol 13 no 3 pp 361ndash391 2012

[32] V Mahajan and R A PetersonModels of Innovation DiffusionSage 1985

[33] E Ruiz-Conde P S H Leeflang and J EWieringa ldquoMarketingvariables in macro-level diffusion modelsrdquo Journal fur Betrieb-swirtschaft vol 56 no 3 pp 155ndash183 2006

[34] D Chandrasekaran and G J Tellis ldquoA critical review ofmarketing research on diffusion of new productsrdquo Review ofMarketing Research pp 39ndash80 2007

[35] F Sultan J U Farley and D R Lehmann ldquoA meta-analysis ofdiffusion modelsrdquo International Journal of Forecasting vol 27no 1 pp 70ndash77 1990

[36] D Talukdar K Sudhir and A Ainslie ldquoInvestigating newproduct diffusion across products and countriesrdquo MarketingScience vol 21 no 1 pp 97ndash114 2002

[37] C van den Bulte and S Stremersch ldquoSocial contagion andincome heterogeneity in new product diffusion ameta-analytictestrdquoMarketing Science vol 23 no 4 pp 530ndash544 2004

[38] G L Lilien P Kotler and K S Moorthy Marketing ModelsPrentice-Hall of India Private New Delhi India 1999

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Advances in Decision Sciences 3

because of technological innovation The paper is dividedinto the model development special case numerical exam-ples with sensitivity analysis observations and managerialimplications Finally the paper concludes with a discussionon the application extension and limitations of the modelAssumptions Inventory model has been developed under thefollowing assumptions

(i) The replenishment rate is infinite and thus replen-ishments are instantaneous

(ii) Lead time is zero(iii) The planning horizon is finite(iv) Shortages are not allowed(v) Demand rate is time dependent governed by innova-

tion diffusion process and is affected by the introduc-tion of substitute products because of technologicalinnovations

(vi) The size of the potential market of total number ofadopters remains constant Here the potentialmarketsize includes the number of initial purchases for thetime interval for which replacement purchases areexcluded

(vii) The coefficient of innovation remains constant thatis the likelihood that somebody who is not yet usingthe product will start using it because of mass mediacoverage or other external factors will act as constantthroughout the cycle length

(viii) There is only one product bought per new adopter(ix) The innovationrsquos sales are confined to a single geo-

graphic area(x) The impact of marketing strategies by the innovator is

adequately captured by the modelrsquos parameters

Notations Notations used in the modeling framework are asfollows

119860 ordering cost119862 unit cost119868 inventory carrying charge119868119862 inventory carrying cost119879 length of the replenishment cycle1198791 time of introduction of second generation product

120579 rate of diminishing demand of first generationproduct119901 coefficient of innovation119876 number of items received at the beginning of theperiod119870(119879) the total cost of the system per unit time119868(119905) on hand inventory at any time 119905119899(119905) = 120582(119905) = 119878(119905) the number of adoptions at time 119905that is Demand at time 119905119873 potential market size

119873(119905) cumulative number of adopters at time 119905119891(119905) the likelihood of purchase at time 119905119865(119905) the cumulative fraction of adopters at time 119905

2 Mathematical Model

The basic assumption considered by different researchersin marketing literature for a fundamental diffusion modelis that the rate of diffusion or the number of adoptersat any given point in time is directly proportional to thenumber of remaining potential adopters at that momentMathematically this can be represented as follows

119899 (119905) =119889119873 (119905)

119889119905= 119892 (119905) 119873 minus 119873 (119905) (1)

where 119892(119905) is known as the rate of adoption or individualprobability of adoption

It has also been assumed that 119892(119905) depends on timethrough a linear function of 119873(119905) (Mahajan and Peterson[32]) Hence

119899 (119905) =119889119873 (119905)

119889119905= (119886 + 119887119873 (119905)) 119873 minus 119873 (119905) 119886 ge 0 119887 ge 0

(2)

Depending on the importance of each source of influencedifferent versions can be derived from the fundamentaldiffusion model (Mahajan and Peterson [32])

When 119887 = 0 the model only considers external influencewhen 119886 = 0 the model only considers internal influenceWhen 119886 = 0 and 119887 = 0 the resulting model is called a mixedinfluence diffusion model (Ruiz-Conde et al [33])

The basic assumptions used in the Bass Model are thatthe adoption of a new product spreads through a populationprimarily due to contact with prior adopters Hence theprobability that an initial purchase occurs at time 119905 given thatno purchase has occurred is a linear function of the numberof previous buyers that is

119891 (119905)

1 minus 119865 (119905)= 119901 + 119902119865 (119905) (3)

If we define 119899(119905) = 119873119891(119905) and 119873(119905) = 119873119865(119905) we canexpress (3) as follows

119899 (119905) =119889119873 (119905)

119889119905= (119901 + 119902

119873 (119905)

119873

) 119873 minus 119873 (119905) (4)

Equation (4) is a mixed influence diffusion model Considera case of diffusion model of external influence only (taking119902 = 0) as proposed by Fourt and WoodLock [2] (4) can bedefined as follows

119899 (119905) =119889119873 (119905)

119889119905= 119901 119873 minus 119873 (119905) (5)

The basic demand model that has been used in the EOQmodel is based on the following assumptions

4 Advances in Decision Sciences

(i) Adoptions take place due to innovation-diffusioneffect and it is influenced by the innovation-effect(mass media) that is external influence only

(ii) Adoptions of the first generation product is dimin-ishing by the introduction of the second generationproduct

Here the diffusion of new products which is spreadthrough the external influence has been considered for mak-ing economic ordering policies of the products The demandof first generation products that is products which aresupposed to be introduced at the beginning of the planningperiod also known as old products is affected (diminished)due to introduction of the second generation products thatis products which are supposed to be new in comparison tothe first generation products and affecting the demand of oldproducts because of technological innovations after a certaininterval of time Therefore using the above assumptionsexplanations and (5) the following demand function hasbeen constructed for our model120582 (119905)

=

119901119873 minus 119873 (119905) 0 le 119905 lt 1198791 119901 ge 120579

119901 119873 minus 119873 (119905) minus 120579 119873 minus 119873 (119905) 1198791le 119905 le 119879 119901 ge 120579

(6)The above demand model shows that the inventory level isdepleted wholly due to demand of the old products with aparticular pace depending on the parameters associated withit up to time period 119879

1and after time 119879

1and up to time

period 119879 the rate of decreasing inventory level is somewhatless than what was experienced before time 119879

1because some

demand of the old products are satisfied (substituted) by thenew products introduced after time period 119879

1because of

technological innovations The demand usage 120582(119905) which isa function of time plays pivotal role to shrink the inventorysize over a period of time If in the time interval (119905 119905 + 119889119905)the inventory size is dipping at the rate 120582(119905)119889119905 then the totalreduction in the inventory size during the time interval119889119905 canbe given by minus119889119868(119905) = 120582(119905)119889119905 Thus the differential equationdescribing the instantaneous state of the inventory level at anytime 119905 119868(119905) in the interval (0 119879) is given by

119889119868 (119905)

119889119905= minus120582 (119905) 0 le 119905 le 119879 (7)

where120582 (119905)

= 119899 (119905) =119889119873 (119905)

119889119905

=

119901119873 minus 119873 (119905) 0 le 119905 lt 1198791 119901 ge 120579

119901 119873 minus 119873 (119905) minus 120579 119873 minus 119873 (119905) 1198791le 119905 le 119879 119901 ge 120579

997904rArr 119873(119905) =

119873 (1 minus 119890minus119901119905) 0 le 119905 lt 119879

1

119873 minus 119873 minus 119873(1198791) 119890(119901minus120579)(119879

1minus119905) 1198791le 119905 le 119879

(8)

Now by model assumption replenishment is instanta-neous and shortages are not allowedThus the inventory levelat the initial point of the planning horizon can be assumed tobe the cumulative adoption of the product during the cycletime 119879 Hence

119868 (0) = 119876 = int

119879

0

120582 (119905) 119889119905 (9)

997904rArr 119868 (0) = 119876 = 119873(1 minus 119890minus1199011198791) + [119873 (119879

1) minus 119873]

times (119890(119901minus120579)(119879

1minus119879)

minus 1)

(10)

The solution of the differential equation (7) is

119868 (119905) =

119868 (0) minus 119873 (1 minus 119890minus119901119905) 0 le 119905 lt 119879

1

[119873 minus 119873 (1198791)]

times [119890(119901minus120579)(119879

1minus119905)minus 119890minus(119901minus120579)(119879

1minus119879)] 119879

1le 119905 le 119879

(11)

Now

int

119879

0

119868 (119905) 119889119905 = int

1198791

0

119868 (119905) 119889119905 + int

119879

1198791

119868 (119905) 119889119905 (12)

997904rArr int

119879

0

119868 (119905) 119889119905 = 1198791119868 (0) +

119873

119901[1 minus 119890

minus1199011198791]

minus 1198731198791+ [119873 minus 119873(119879

1)]

times[119890(119901minus120579)(119879

1minus119879)

minus 1

(120579 minus 119901)+(1198791minus 119879) 119890

(119901minus120579)(1198791minus119879)]

(13)

The different cost elements involved in the inventorysystem per unit time can be defined as

Ordering cost per unit time = 119860

119879 (14)

Material cost per unit time = 119876119862

119879 (15)

=119862

119879119873(1 minus 119890

minus1199011198791)

+ (119862

119879) [119873 (119879

1) minus 119873] (119890

(119901minus120579)(1198791minus119879)

minus 1)

(16)

Cost of carrying inventory per unit time

=119868119862

119879int

119879

0

119868 (119905) 119889119905

(17)

Advances in Decision Sciences 5

= (119868119862

119879)1198791119868 (0) +

119868119862119873

119901119879[1 minus 119890

minus1199011198791]

minus (119868119862

119879)1198731198791

+ (119868119862

119879) [119873 minus 119873(119879

1)]

times [119890(119901minus120579)(119879

1minus119879)

minus 1

(120579 minus 119901)+ (1198791minus 119879) 119890

(119901minus120579)(1198791minus119879)]

(18)

Thus the total cost of the inventory system per unit time119870(119879) is

119870 (119879) = (14) + (16) + (18)

997904rArr 119870 (119879) =119860

119879+ [119873 (119879

1) minus 119873] (119890

(119901minus120579)(1198791minus119879)

minus 1)

times (119862

119879) (1 + 119868119879

1) + 119873 (1 minus 119890

minus1199011198791) (

119862

119879)

times (1 + 1198681198791) +

119868119862119873

119901119879[1 minus 119890

minus1199011198791] minus (

119868119862

119879)1198731198791

+ (119868119862

119879) [119873 minus 119873(119879

1)]

times [119890(119901minus120579)(119879

1minus119879)

minus 1

(120579 minus 119901)+ (1198791minus 119879) 119890

(119901minus120579)(1198791minus119879)]

(19)

The necessary criterion for 119870(119879) to be minimum is

119889119870 (119879)

119889119879= 0 (20)

997904rArrminus119860

1198792+ [119873 (119879

1) minus 119873] (119890

(119901minus120579)(1198791minus119879)

minus 1) (minus119862

1198792) (1 + 119868119879

1)

minus (119862

1198792) (1 + 119868119879

1)119873 (1 minus 119890

minus1199011198791)

+ (119862

119879) (1 + 119868119879

1) [119873 (119879

1) minus 119873] (120579 minus 119901) 119890

(119901minus120579)(1198791minus119879)

minus119868119862119873

1199011198792[1 minus 119890

minus1199011198791] + (

119868119862

1198792)1198731198791

+ (minus119868119862

1198792) [119873 minus 119873(119879

1)]

times [119890(119901minus120579)(119879

1minus119879)

minus 1

(120579 minus 119901)+ (1198791minus 119879) 119890

(119901minus120579)(1198791minus119879)]

+ (119868119862

119879) [119873 minus 119873(119879

1)]

times (1198791minus 119879) (120579 minus 119901) 119890

(119901minus120579)(1198791minus119879)

= 0

(21)

Now for119870(119879) to be convex

1198892119870 (119879)

1198891198792

100381610038161003816100381610038161003816100381610038161003816119879=119879lowast

gt 0

997904rArr2119860

119879lowast3+ (

2119862

119879lowast3) (1 + 119868119879

1)119873 (1 minus 119890

minus1199011198791)

+ [119873 (1198791) minus 119873] (119890

(119901minus120579)(1198791minus119879lowast)minus 1) (

2119862

1198793) (1 + 119868119879

1)

+ (119862

119879lowast) (1 + 119868119879

1) [119873 (119879

1) minus 119873] (120579 minus 119901)

2

119890(119901minus120579)(119879

1minus119879lowast)

+2119868119862119873

119901119879lowast3

[1 minus 119890minus1199011198791] + (

2119868119862

119879lowast3) [119873 minus 119873(119879

1)]

times [119890(119901minus120579)(119879

1minus119879lowast)minus 1

(120579 minus 119901)+ (1198791minus 119879lowast) 119890(119901minus120579)(119879

1minus119879lowast)]

+ (119868119862

119879lowast) [119873 minus 119873(119879

1)] (1198791minus 119879lowast)

times (120579 minus 119901)2

119890(119901minus120579)(119879

1minus119879lowast)

gt (2119862

119879lowast2) (1 + 119868119879

1) [119873 (119879

1) minus 119873]

times (120579 minus 119901) 119890(119901minus120579)(119879

1minus119879)

+ (2119868119862

119879lowast3)1198731198791

+ (2119868119862

119879lowast2) [119873 minus 119873(119879

1)]

times [(1198791minus 119879lowast) (120579 minus 119901) 119890

(119901minus120579)(1198791minus119879lowast)]

+ (119868119862

119879lowast) [119873 minus 119873(119879

1)] [(120579 minus 119901) 119890

(119901minus120579)(1198791minus119879lowast)]

(22)

The solution of the equation 119889119870(119879)119889119879 = 0 gives theoptimum value of 119879 provided it satisfies the condition1198892119870(119879)119889119879

2gt 0 Since the above cost equation (21) is highly

nonlinear the problem has been solved numerically for givenparameter valuesThe solution gives the optimumvalue119879lowast ofthe replenishment cycle time 119879 Once 119879lowast is known the valueof optimum order quantity 119876lowast and the optimum cost 119870(119879lowast)can easily be obtained from (10) and (19) respectively Thenumerical solution for the given base value has been obtainedby using software packages Lingo and Excel-Solver

21 Special Case When 120579 becomes negligible that is equalto zero which means here we are not considering the effectof introduction of new products on the old one after timeperiod 119879

1 That is demand rate of the old product is the

same throughout the planning period (0 119879) and for this casethe adoption rate that is demand rate that is 120582(119905) = 119899(119905)reduces to the classical Fourt and WoodLock model [2] Theobjective here is to study the different managerial policiesfor the model which is under external influence only andits demand is not affected by the introduction of any other

6 Advances in Decision Sciences

substitute products In that case the total cost per unit time119870(119879) is given by

119870 (119879) =119860

119879+119873119862

119879(1 minus 119890

minus119901119879) +

119873119868119862

119901119879(1 minus 119890

minus119901119879) minus 119873119868119862119890

minus119901119879

(23)

119868 (0) = 119876 = int

119879

0

120582 (119905) 119889119905 = 119873 (1 minus 119890minus119901119879

) (24)

For optimum total cost the necessary criterion is

119889119870 (119879)

119889119879= 0 (25)

997904rArrminus119860

1198792+119873119862119901119890

minus119901119879

119879minus

119873119862(1 minus 119890minus119901119879

)

1198792

+119873119868119862119890

minus119901119879

119879

minus

119873119868119862 (1 minus 119890minus119901119879

)

1199011198792

+ 119873119868119862119901119890minus119901119879

= 0

(26)

Now for119870(119879) to be convex

1198892119870 (119879)

1198891198792

100381610038161003816100381610038161003816100381610038161003816119879=119879lowast

gt 0

997904rArr2119860

119879lowast3+

2119873119862(1 minus 119890minus119901119879lowast

)

119879lowast3

+

2119873119868119862 (1 minus 119890minus119901119879lowast

)

119901119879lowast3

gt1198731198621199012119890minus119901119879lowast

119879lowast

+119873119868119901119862119890

minus119901119879lowast

119879lowast2

+ 1198731198681198621199012119890minus119901119879lowast

(27)

The solution of the equation 119889119870(119879)119889119879 = 0 gives theoptimum value of 119879 provided it satisfies the condition1198892119870(119879)119889119879

2gt 0 Since cost equation (26) is highly nonlinear

the problem has been solved numerically for given parametervalues The solution gives the optimum value 119879lowast of thereplenishment cycle time 119879 Once 119879lowast is known the valuesof optimum order quantity 119876lowast and the optimum cost 119870(119879lowast)can easily be obtained from (24) and (23) respectively Thenumerical solution for the given base value has been obtainedby using software packages LINGO and Excel-Solver

3 Solution Procedure

The solution procedure has been summarized in the follow-ing algorithmStep 1 Input all parameter values such as unit cost coefficientof innovation potential market size and ordering cost foreach case separatelyStep 2 Compute all possible values of ldquo119879rdquo separately for (21)and (26) as the case may beStep 3 select the appropriate value of ldquo119879rdquo say 119879lowast for eachcase by using (19) and (23) and by satisfying the above statedcondition 1198892119870(119879)1198891198792 gt 0Step 4Compute119870(119879lowast) from (19) and (23) amp119876(119879lowast) from (10)and (24) for each case separately

The above steps are used for all replenishment cyclesusing appropriate parameter values In order to obtain theappropriate values of ldquo119879rdquo we need to follow the aboveprocedure with the help of LINGO and Excel-Solver softwarepackages

4 Numerical Examples

The effectiveness of the proposed model has been shown bythe following numerical examples A hypothetical examplehas the following parameter values in appropriate units

119860 =$1100order

119862 =$300unit

119868 = 025 119873 = 50000

(28)

The results have been well presented in the followingdifferent tables Also to prove the validity of the modelnumerically and to get the appropriate parameter values thereferences have been considered as Chanda and Kumar [25]Chandrasekaran and Tellis [34] Sultan et al [35] Talukdaret al [36] Van den Bulte and Stremersch [37] Chanda andKumar [27] Aggarwal et al [26 31] Aggarwal et al [29] andKumar et al [30]

(i) The mean value of the coefficient of innovation fora new product usually lies between 00007 and 003(Sultan et al [35] Talukdar et al [36] Van den Bulteand Stremersch [37])

(ii) The mean value of the coefficient of innovation fora new product is usually 001for developed countriesand 00003 for developing countries (Talukdar et al[36])

41 Special Case A hypothetical example has the followingparameter values in appropriate units

119860 =$1100order

119862 =$300unit

119868 = 025 119873 = 50000 120579 = 0

(29)

5 Observations

The results obtained from different numerical tables inSection 4 explain the effect of changes in the system param-eters on the optimal values of total cost 119870(119879lowast) the optimalcycle length 119879

lowast and the optimal order quantity 119876(119879lowast)

We have observed the following relationship during thenumerical exercise

(a) As the coefficient of innovation increases keepingother parameters constant then the optimal cyclelength 119879

lowast decreases while both the optimal orderquantity 119876(119879

lowast) and the optimal total cost 119870(119879lowast)

increases as depicted in Tables 1 and 4 This isconsistent with the reality as more investment onpromotion will increase the diffusion of a productin the market resulting in shrinkage of the optimalreorder cycle time as a result optimal cost is increased

Advances in Decision Sciences 7

Table 1 Sensitivity analysis on coefficient of innovation ldquo119901rdquo

119901 119879lowast

119870(119879lowast) 119876(119879

lowast)

0001 086 16400 390002 059 32705 570003 048 48684 700004 041 64499 810005 037 80211 910006 034 95848 1010007 031 111430 1090008 029 126967 1170009 027 142468 124001 026 157938 131002 019 311593 190For 1198791 = 01 120579 = 00001

(b) As the cycle length 1198791 that is the time of intro-

duction of second product increases keeping otherparameters constant then the optimal cycle length119879lowastoptimal order quantity 119876(119879lowast) and the optimal totalcost 119870(119879lowast) all these three values increase as depictedin Table 2 This is true with the fact that as the timeof introduction of second product increases meansmaximum demand is satisfied by more number ofprevious that is old products and it is also obviousthat the demand is a decreasing function of timewhich implies that less demand for more time periodleads to increase in optimal cycle length And sinceoptimal cycle length increases therefore it forcesthe inventory manager to keep more number ofinventories to satisfy decreasing demand for longertime period and because more number of inventoriesare kept for longer time period it leads to increase inthe optimal cost (Figure 3)

(c) As the value of 120579 increases keeping other parametersconstant then the optimal cycle length 119879lowast increaseswhereas optimal order quantity 119876(119879lowast) and the opti-mal total cost119870(119879lowast) decreases as depicted in Table 3This is also true with the fact that as 120579 increasesmeans demand of the first product decreases morebecause more demand is satisfied by its substituteproduct which is introduced after a certain interval oftime and therefore less inventories are kept for longerperiod which leads to decrease in the optimal cost

6 Managerial Implications

The goodwill of any organization is entirely dependent onits effective management and this is possible only when theproblemof the organization is analyzed properly from all per-spectives Here our problem is concerned with the effectivemanagement of products which are newly introduced in themarket and along with this the effect of second generationproducts on the demand of first generation products is alsorealized the idea included here is that the effect of substituteproducts decreases the demand of old products after a certain

Table 2 Sensitivity analysis on ldquo1198791rdquo

1198791

119879lowast

119870(119879lowast) 119876(119879

lowast)

010 059 68778 124011 061 69032 128012 063 69279 131013 064 69520 135014 066 69756 139015 067 69987 143016 069 70213 146017 07 70435 150018 072 70653 153019 073 70866 157020 075 71076 160For 120579 = 0001 119901 = 0005

Table 3 Sensitivity analysis on ldquo120579rdquo

120579 119879lowast

119870(119879lowast) 119876(119879

lowast)

0019 038 1802327 22280020 041 1649195 21800021 044 1495228 21210022 048 1340344 20500023 052 1184432 19650024 058 1027337 18650025 065 868835 17470026 073 708581 16050027 086 545998 14330028 107 379973 12170029 154 207669 918For 1198791 = 001 119901 = 003119873 = 500000

Table 4 Sensitivity analysis on coefficient of innovation ldquo119901rdquo

119901 119879lowast

119870(119879lowast) 119876(119879

lowast)

0001 076 17865 380002 054 34044 540003 044 49942 670004 038 65695 770005 034 81354 860006 031 96946 950007 029 112487 1020008 027 127987 1100009 026 143453 117001 024 158891 123002 017 312305 178Special case

interval of time because of technological innovations andother related factors Therefore to overcome this problem amathematical model has been developed to know the actualsituation of the economic ordering policies which will helpthe inventory manager to take effective action to maintainthe optimal cost The results obtained in the model create apool of knowledge on the diffusion processes of innovations

8 Advances in Decision Sciences

48625

48675

48725

48775

48825

48875

Time

Cos

t

038 042 048 052 056 06

Cost-time graphFor Tlowast

= 048 K(Tlowast) = 48684

(a)

69745

69755

69765

69775

69785

69795

Time

Cos

t

061 063 065 067 069 071

Cost-time graphFor Tlowast

= 066 K(Tlowast) = 69756

(b)

868700

868800

868900

869000

869100

869200

Time

Cos

t

061 063 065 067 069

Cost-time graphFor Tlowast

= 065 K(Tlowast) = 868835

(c)

33000

33500

34000

34500

35000

35500

36000

Time

Cos

t

024 034 044 054 064 074 084 094 104

Cost-time graph (special case)For Tlowast

= 054 K(Tlowast) = 34044

(d)

Figure 3 Cost-time graphs showing convexity of the cost functions

and its effective management which are most valuable toboth researchers and managers For managers this poolof knowledge is important because it provides useful toolsand useful information for managing and scheduling theinventories of new products when its demand is affectedby the generation of other new products which substituteproducts after a certain time period The uniqueness of thismodel is that how the inventory manager should keep theoptimal cycle time of any fixed lot size of one product enteringinto the inventory system when its substitute product entersinto the system after a fixed interval so that optimal costis maintained and the cost of inventory obsolescence isminimized

7 Conclusion

The technological breakthrough is constantly being experi-enced in various products across the globe which is neces-sary for the products to gain competitive advantage Thereare various marketing strategies to promote and establishproducts which are newly introduced into the market butkeeping its inventories efficiently and effectively play a greaterrole Here the role of inventory models becomes significantThe primary objective of developing inventory models isto take efficient action concerned with economic orderingpolicies and to understand well the behaviour of parametersassociated with it In this paper a mathematical model has

Advances in Decision Sciences 9

been developed for obtaining the Economic Order Quantity(EOQ) in which the demand of the product is assumed tofollow an innovation diffusion process as proposed by Fourtand WoodLock [2] and along with this the demand of theproduct decreases after a certain interval of time because ofthe introduction of its substitute product Here the modelanalyses the situation when demand of the previous productis affected by introducing second generation product Thereare few inventory models such as Chern et al [28] Chandaand Kumar [25] Aggarwal et al [26] and Chanda andKumar [27] which capture the sensitive nature of marketingparameters but have not incorporated the idea of the effectof introduction of second generation product This paperattempts to bridge this gap The approach of this paper is toobtain economic ordering policies of a new product whenits substitute product enters into the market after a certaintime period A special case has also been discussed whichexcludes the effect of substitute products on the inventorypolicies of old products Numerical examples with sensitivityanalyses are presented to illustrate the model properly Theresults obtained are largely consistent with what we observeempirically A simple solution procedure in the form ofalgorithm is presented to determine the optimal cycle timeand optimal order quantity of the cost functionTheproposedmodel can be extended for backorders quantity discountpartial lost sales and so forth and it can also be extendedto solve by different approaches such as geometric program-ming approach and genetic algorithm approach At the sametime future research should also focus on developing somealternative approach to obtain the optimal analytical solutionof the highly nonlinear cost function which we have solvednumerically

Acknowledgments

The research presented in this paper was carried out at theDepartment ofOperational Reseach Faculty ofMathematicalSciences University of Delhi The commercial identitiesmentioned in the paper such as LINGO and Excel-Solversoftware have only been used for the numerical analysis of themodel discussed in the paper which is only for the academicpurpose The authors of this paper do not have any directfinancial relation with the commercial identities mentionedin the paper and there is no conflict of interests regardingfinancial gains for all the authors

References

[1] E Rogers Diffusion of Innovations Free Press New York NYUSA 3rd edition 1983

[2] L A Fourt and J W WoodLock ldquoEarly prediction of Marketsuccess for new grocery productsrdquo Journal of Marketing vol 26no 2 pp 31ndash38 1960

[3] J C Fisher and R H Pry ldquoA simple substitution modelof technological changerdquo Technological Forecasting and SocialChange vol 3 no C pp 75ndash88 1971

[4] F M Bass ldquoA new product growth for model consumerdurablesrdquo Management Science vol 50 no 12 pp 1825ndash18322004

[5] R Peres E Muller and V Mahajan ldquoInnovation diffusion andnew product growth models a critical review and researchdirectionsrdquo International Journal of Research in Marketing vol27 no 2 pp 91ndash106 2010

[6] PM Parker ldquoAggregate diffusion forecastingmodels inmarket-ing a critical reviewrdquo International Journal of Forecasting vol10 no 2 pp 353ndash380 1994

[7] E M Rogers Diffusion of Innovations Free Press New YorkNY USA 4th edition 1995

[8] R Peres E Muller and V Mahajan ldquoInnovation diffusion andnew product growth models a critical review and researchdirectionsrdquo International Journal of Research in Marketing vol27 no 2 pp 91ndash106 2010

[9] V Mahajan E Muller and Y Wind New-Product DiffusionModels Kluwer Academic Publishers Boston Mass USA2000

[10] F Cheng and S P Sethi ldquoPeriodic review inventory modelwith demand influenced by promotion decisionsrdquoManagementScience vol 45 no 11 pp 1510ndash1523 1999

[11] S Kalish ldquoAnewproduct adoptionmodelwith price advertisingand uncertaintyrdquoManagement Science vol 31 no 12 pp 1569ndash1585 1985

[12] VMahajan andA P Robert ldquoInnovation diffusion in a dynamicpotential adopter populationrdquoManagement Science vol 24 no15 pp 1589ndash1597 1978

[13] M N Sharif and K Ramanathan ldquoPolynomial innovationdiffusion modelsrdquo Technological Forecasting and Social Changevol 21 no 4 pp 301ndash323 1982

[14] M N Sharif and K Ramanathan ldquoBinomial innovation dif-fusion models with dynamic potential adopter populationrdquoTechnological Forecasting and Social Change vol 20 no 1 pp63ndash87 1981

[15] J A Norton and F M Bass ldquoA diffusion theory model ofadoption and substitution for successive generation of high-technology productsrdquo Management Science vol 33 no 9 pp1069ndash1086 1987

[16] V Mahajan and E Muller ldquoTiming diffusion and substitutionof successive generations of technological innovations the IBMmainframe caserdquo Technological Forecasting and Social Changevol 51 no 2 pp 109ndash132 1996

[17] M W Speece and D L Maclachlan ldquoApplication of a multi-generation diffusion model to milk container technologyrdquoTechnological Forecasting and Social Change vol 49 no 3 pp281ndash295 1995

[18] P J Danaher B G S Hardie and W P Putsis Jr ldquoMarketing-mix variables and the diffusion of successive generations of atechnological innovationrdquo Journal of Marketing Research vol38 no 4 pp 501ndash514 2001

[19] T Islam and NMeade ldquoThe diffusion of successive generationsof a technology a more general modelrdquo Technological Forecast-ing and Social Change vol 56 no 1 pp 49ndash60 1997

[20] N Kim D R Chang and A D Shocker ldquoModeling intercat-egory and generational dynamics for a growing informationtechnology industryrdquo Management Science vol 46 no 4 pp496ndash512 2000

[21] B Bayus ldquoThe dynamic pricing of next generation consumerdurablesrdquoMarketing Science vol 11 no 3 pp 251ndash265 1992

[22] A K Bardhan and U Chanda ldquoA model for first and sub-stitution adoption of successive generations of a productrdquoInternational Journal of Modelling and Simulation vol 28 no4 pp 487ndash494 2008

10 Advances in Decision Sciences

[23] U Chanda and A K Bardhan ldquoDynamic optimal advertisingexpenditure strategies of two successive generations of con-sumer durablesrdquo in Proceedings of the International Confer-ence on Modelling Computation and Optimization held duringJanuary 2008 in the venue SQC amp OR Unit Indian StatisticalInstitute New Delhi India 2008

[24] K A Bardhan and U Chanda ldquoA mathematical model fordiffusion of products with multiple generationsrdquo InternationalJournal of Operational Research vol 4 no 2 pp 214ndash229 2009

[25] U Chanda and A Kumar ldquoEconomic order quantity modelwith demand influenced by dynamic innovation effectrdquo Interna-tional Journal of Operational Research vol 11 no 2 pp 193ndash2152011

[26] K K Aggarwal C K Jaggi and A Kumar ldquoEconomic orderquantity model with innovation diffusion criterion havingdynamic potential market sizerdquo International Journal of AppliedDecision Sciences vol 4 no 3 pp 280ndash303 2011

[27] U Chanda and A Kumar ldquoEconomic order quantity model oninflationary conditions with demand influenced by innovationdiffusion criterionrdquo International Journal of Procurement Man-agement vol 5 no 2 pp 160ndash177 2012

[28] M-S Chern J-T Teng and H-L Yang ldquoInventory lot-sizepolicies for the bass diffusion demand models of new durableproductsrdquo Journal of the Chinese Institute of Engineers vol 24no 2 pp 237ndash244 2001

[29] K K Aggarwal and A Kumar ldquoAn inventory model for newproduct when demand follows dynamic innovation processhaving dynamic potential market sizerdquo International Journal ofStrategic Decision Sciences vol 3 no 4 pp 78ndash99 2012

[30] A Kumar K K Aggarwal and U Chanda ldquoEconomic orderquantity model under fuzzy sense with demand follows Bassrsquosinnovation diffusion processrdquo International Journal of AdvancedOperations Management vol 5 no 3 pp 261ndash281 2013

[31] K K Aggarwal C K Jaggi and A Kumar ldquoEconomic orderquantity model under fuzzy sense when demand follows inno-vation diffusion process having dynamic potential market sizerdquoInternational Journal of Services and Operations Managementvol 13 no 3 pp 361ndash391 2012

[32] V Mahajan and R A PetersonModels of Innovation DiffusionSage 1985

[33] E Ruiz-Conde P S H Leeflang and J EWieringa ldquoMarketingvariables in macro-level diffusion modelsrdquo Journal fur Betrieb-swirtschaft vol 56 no 3 pp 155ndash183 2006

[34] D Chandrasekaran and G J Tellis ldquoA critical review ofmarketing research on diffusion of new productsrdquo Review ofMarketing Research pp 39ndash80 2007

[35] F Sultan J U Farley and D R Lehmann ldquoA meta-analysis ofdiffusion modelsrdquo International Journal of Forecasting vol 27no 1 pp 70ndash77 1990

[36] D Talukdar K Sudhir and A Ainslie ldquoInvestigating newproduct diffusion across products and countriesrdquo MarketingScience vol 21 no 1 pp 97ndash114 2002

[37] C van den Bulte and S Stremersch ldquoSocial contagion andincome heterogeneity in new product diffusion ameta-analytictestrdquoMarketing Science vol 23 no 4 pp 530ndash544 2004

[38] G L Lilien P Kotler and K S Moorthy Marketing ModelsPrentice-Hall of India Private New Delhi India 1999

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Journal of

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Mathematical PhysicsAdvances in

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Advances in Decision Sciences

(i) Adoptions take place due to innovation-diffusioneffect and it is influenced by the innovation-effect(mass media) that is external influence only

(ii) Adoptions of the first generation product is dimin-ishing by the introduction of the second generationproduct

Here the diffusion of new products which is spreadthrough the external influence has been considered for mak-ing economic ordering policies of the products The demandof first generation products that is products which aresupposed to be introduced at the beginning of the planningperiod also known as old products is affected (diminished)due to introduction of the second generation products thatis products which are supposed to be new in comparison tothe first generation products and affecting the demand of oldproducts because of technological innovations after a certaininterval of time Therefore using the above assumptionsexplanations and (5) the following demand function hasbeen constructed for our model120582 (119905)

=

119901119873 minus 119873 (119905) 0 le 119905 lt 1198791 119901 ge 120579

119901 119873 minus 119873 (119905) minus 120579 119873 minus 119873 (119905) 1198791le 119905 le 119879 119901 ge 120579

(6)The above demand model shows that the inventory level isdepleted wholly due to demand of the old products with aparticular pace depending on the parameters associated withit up to time period 119879

1and after time 119879

1and up to time

period 119879 the rate of decreasing inventory level is somewhatless than what was experienced before time 119879

1because some

demand of the old products are satisfied (substituted) by thenew products introduced after time period 119879

1because of

technological innovations The demand usage 120582(119905) which isa function of time plays pivotal role to shrink the inventorysize over a period of time If in the time interval (119905 119905 + 119889119905)the inventory size is dipping at the rate 120582(119905)119889119905 then the totalreduction in the inventory size during the time interval119889119905 canbe given by minus119889119868(119905) = 120582(119905)119889119905 Thus the differential equationdescribing the instantaneous state of the inventory level at anytime 119905 119868(119905) in the interval (0 119879) is given by

119889119868 (119905)

119889119905= minus120582 (119905) 0 le 119905 le 119879 (7)

where120582 (119905)

= 119899 (119905) =119889119873 (119905)

119889119905

=

119901119873 minus 119873 (119905) 0 le 119905 lt 1198791 119901 ge 120579

119901 119873 minus 119873 (119905) minus 120579 119873 minus 119873 (119905) 1198791le 119905 le 119879 119901 ge 120579

997904rArr 119873(119905) =

119873 (1 minus 119890minus119901119905) 0 le 119905 lt 119879

1

119873 minus 119873 minus 119873(1198791) 119890(119901minus120579)(119879

1minus119905) 1198791le 119905 le 119879

(8)

Now by model assumption replenishment is instanta-neous and shortages are not allowedThus the inventory levelat the initial point of the planning horizon can be assumed tobe the cumulative adoption of the product during the cycletime 119879 Hence

119868 (0) = 119876 = int

119879

0

120582 (119905) 119889119905 (9)

997904rArr 119868 (0) = 119876 = 119873(1 minus 119890minus1199011198791) + [119873 (119879

1) minus 119873]

times (119890(119901minus120579)(119879

1minus119879)

minus 1)

(10)

The solution of the differential equation (7) is

119868 (119905) =

119868 (0) minus 119873 (1 minus 119890minus119901119905) 0 le 119905 lt 119879

1

[119873 minus 119873 (1198791)]

times [119890(119901minus120579)(119879

1minus119905)minus 119890minus(119901minus120579)(119879

1minus119879)] 119879

1le 119905 le 119879

(11)

Now

int

119879

0

119868 (119905) 119889119905 = int

1198791

0

119868 (119905) 119889119905 + int

119879

1198791

119868 (119905) 119889119905 (12)

997904rArr int

119879

0

119868 (119905) 119889119905 = 1198791119868 (0) +

119873

119901[1 minus 119890

minus1199011198791]

minus 1198731198791+ [119873 minus 119873(119879

1)]

times[119890(119901minus120579)(119879

1minus119879)

minus 1

(120579 minus 119901)+(1198791minus 119879) 119890

(119901minus120579)(1198791minus119879)]

(13)

The different cost elements involved in the inventorysystem per unit time can be defined as

Ordering cost per unit time = 119860

119879 (14)

Material cost per unit time = 119876119862

119879 (15)

=119862

119879119873(1 minus 119890

minus1199011198791)

+ (119862

119879) [119873 (119879

1) minus 119873] (119890

(119901minus120579)(1198791minus119879)

minus 1)

(16)

Cost of carrying inventory per unit time

=119868119862

119879int

119879

0

119868 (119905) 119889119905

(17)

Advances in Decision Sciences 5

= (119868119862

119879)1198791119868 (0) +

119868119862119873

119901119879[1 minus 119890

minus1199011198791]

minus (119868119862

119879)1198731198791

+ (119868119862

119879) [119873 minus 119873(119879

1)]

times [119890(119901minus120579)(119879

1minus119879)

minus 1

(120579 minus 119901)+ (1198791minus 119879) 119890

(119901minus120579)(1198791minus119879)]

(18)

Thus the total cost of the inventory system per unit time119870(119879) is

119870 (119879) = (14) + (16) + (18)

997904rArr 119870 (119879) =119860

119879+ [119873 (119879

1) minus 119873] (119890

(119901minus120579)(1198791minus119879)

minus 1)

times (119862

119879) (1 + 119868119879

1) + 119873 (1 minus 119890

minus1199011198791) (

119862

119879)

times (1 + 1198681198791) +

119868119862119873

119901119879[1 minus 119890

minus1199011198791] minus (

119868119862

119879)1198731198791

+ (119868119862

119879) [119873 minus 119873(119879

1)]

times [119890(119901minus120579)(119879

1minus119879)

minus 1

(120579 minus 119901)+ (1198791minus 119879) 119890

(119901minus120579)(1198791minus119879)]

(19)

The necessary criterion for 119870(119879) to be minimum is

119889119870 (119879)

119889119879= 0 (20)

997904rArrminus119860

1198792+ [119873 (119879

1) minus 119873] (119890

(119901minus120579)(1198791minus119879)

minus 1) (minus119862

1198792) (1 + 119868119879

1)

minus (119862

1198792) (1 + 119868119879

1)119873 (1 minus 119890

minus1199011198791)

+ (119862

119879) (1 + 119868119879

1) [119873 (119879

1) minus 119873] (120579 minus 119901) 119890

(119901minus120579)(1198791minus119879)

minus119868119862119873

1199011198792[1 minus 119890

minus1199011198791] + (

119868119862

1198792)1198731198791

+ (minus119868119862

1198792) [119873 minus 119873(119879

1)]

times [119890(119901minus120579)(119879

1minus119879)

minus 1

(120579 minus 119901)+ (1198791minus 119879) 119890

(119901minus120579)(1198791minus119879)]

+ (119868119862

119879) [119873 minus 119873(119879

1)]

times (1198791minus 119879) (120579 minus 119901) 119890

(119901minus120579)(1198791minus119879)

= 0

(21)

Now for119870(119879) to be convex

1198892119870 (119879)

1198891198792

100381610038161003816100381610038161003816100381610038161003816119879=119879lowast

gt 0

997904rArr2119860

119879lowast3+ (

2119862

119879lowast3) (1 + 119868119879

1)119873 (1 minus 119890

minus1199011198791)

+ [119873 (1198791) minus 119873] (119890

(119901minus120579)(1198791minus119879lowast)minus 1) (

2119862

1198793) (1 + 119868119879

1)

+ (119862

119879lowast) (1 + 119868119879

1) [119873 (119879

1) minus 119873] (120579 minus 119901)

2

119890(119901minus120579)(119879

1minus119879lowast)

+2119868119862119873

119901119879lowast3

[1 minus 119890minus1199011198791] + (

2119868119862

119879lowast3) [119873 minus 119873(119879

1)]

times [119890(119901minus120579)(119879

1minus119879lowast)minus 1

(120579 minus 119901)+ (1198791minus 119879lowast) 119890(119901minus120579)(119879

1minus119879lowast)]

+ (119868119862

119879lowast) [119873 minus 119873(119879

1)] (1198791minus 119879lowast)

times (120579 minus 119901)2

119890(119901minus120579)(119879

1minus119879lowast)

gt (2119862

119879lowast2) (1 + 119868119879

1) [119873 (119879

1) minus 119873]

times (120579 minus 119901) 119890(119901minus120579)(119879

1minus119879)

+ (2119868119862

119879lowast3)1198731198791

+ (2119868119862

119879lowast2) [119873 minus 119873(119879

1)]

times [(1198791minus 119879lowast) (120579 minus 119901) 119890

(119901minus120579)(1198791minus119879lowast)]

+ (119868119862

119879lowast) [119873 minus 119873(119879

1)] [(120579 minus 119901) 119890

(119901minus120579)(1198791minus119879lowast)]

(22)

The solution of the equation 119889119870(119879)119889119879 = 0 gives theoptimum value of 119879 provided it satisfies the condition1198892119870(119879)119889119879

2gt 0 Since the above cost equation (21) is highly

nonlinear the problem has been solved numerically for givenparameter valuesThe solution gives the optimumvalue119879lowast ofthe replenishment cycle time 119879 Once 119879lowast is known the valueof optimum order quantity 119876lowast and the optimum cost 119870(119879lowast)can easily be obtained from (10) and (19) respectively Thenumerical solution for the given base value has been obtainedby using software packages Lingo and Excel-Solver

21 Special Case When 120579 becomes negligible that is equalto zero which means here we are not considering the effectof introduction of new products on the old one after timeperiod 119879

1 That is demand rate of the old product is the

same throughout the planning period (0 119879) and for this casethe adoption rate that is demand rate that is 120582(119905) = 119899(119905)reduces to the classical Fourt and WoodLock model [2] Theobjective here is to study the different managerial policiesfor the model which is under external influence only andits demand is not affected by the introduction of any other

6 Advances in Decision Sciences

substitute products In that case the total cost per unit time119870(119879) is given by

119870 (119879) =119860

119879+119873119862

119879(1 minus 119890

minus119901119879) +

119873119868119862

119901119879(1 minus 119890

minus119901119879) minus 119873119868119862119890

minus119901119879

(23)

119868 (0) = 119876 = int

119879

0

120582 (119905) 119889119905 = 119873 (1 minus 119890minus119901119879

) (24)

For optimum total cost the necessary criterion is

119889119870 (119879)

119889119879= 0 (25)

997904rArrminus119860

1198792+119873119862119901119890

minus119901119879

119879minus

119873119862(1 minus 119890minus119901119879

)

1198792

+119873119868119862119890

minus119901119879

119879

minus

119873119868119862 (1 minus 119890minus119901119879

)

1199011198792

+ 119873119868119862119901119890minus119901119879

= 0

(26)

Now for119870(119879) to be convex

1198892119870 (119879)

1198891198792

100381610038161003816100381610038161003816100381610038161003816119879=119879lowast

gt 0

997904rArr2119860

119879lowast3+

2119873119862(1 minus 119890minus119901119879lowast

)

119879lowast3

+

2119873119868119862 (1 minus 119890minus119901119879lowast

)

119901119879lowast3

gt1198731198621199012119890minus119901119879lowast

119879lowast

+119873119868119901119862119890

minus119901119879lowast

119879lowast2

+ 1198731198681198621199012119890minus119901119879lowast

(27)

The solution of the equation 119889119870(119879)119889119879 = 0 gives theoptimum value of 119879 provided it satisfies the condition1198892119870(119879)119889119879

2gt 0 Since cost equation (26) is highly nonlinear

the problem has been solved numerically for given parametervalues The solution gives the optimum value 119879lowast of thereplenishment cycle time 119879 Once 119879lowast is known the valuesof optimum order quantity 119876lowast and the optimum cost 119870(119879lowast)can easily be obtained from (24) and (23) respectively Thenumerical solution for the given base value has been obtainedby using software packages LINGO and Excel-Solver

3 Solution Procedure

The solution procedure has been summarized in the follow-ing algorithmStep 1 Input all parameter values such as unit cost coefficientof innovation potential market size and ordering cost foreach case separatelyStep 2 Compute all possible values of ldquo119879rdquo separately for (21)and (26) as the case may beStep 3 select the appropriate value of ldquo119879rdquo say 119879lowast for eachcase by using (19) and (23) and by satisfying the above statedcondition 1198892119870(119879)1198891198792 gt 0Step 4Compute119870(119879lowast) from (19) and (23) amp119876(119879lowast) from (10)and (24) for each case separately

The above steps are used for all replenishment cyclesusing appropriate parameter values In order to obtain theappropriate values of ldquo119879rdquo we need to follow the aboveprocedure with the help of LINGO and Excel-Solver softwarepackages

4 Numerical Examples

The effectiveness of the proposed model has been shown bythe following numerical examples A hypothetical examplehas the following parameter values in appropriate units

119860 =$1100order

119862 =$300unit

119868 = 025 119873 = 50000

(28)

The results have been well presented in the followingdifferent tables Also to prove the validity of the modelnumerically and to get the appropriate parameter values thereferences have been considered as Chanda and Kumar [25]Chandrasekaran and Tellis [34] Sultan et al [35] Talukdaret al [36] Van den Bulte and Stremersch [37] Chanda andKumar [27] Aggarwal et al [26 31] Aggarwal et al [29] andKumar et al [30]

(i) The mean value of the coefficient of innovation fora new product usually lies between 00007 and 003(Sultan et al [35] Talukdar et al [36] Van den Bulteand Stremersch [37])

(ii) The mean value of the coefficient of innovation fora new product is usually 001for developed countriesand 00003 for developing countries (Talukdar et al[36])

41 Special Case A hypothetical example has the followingparameter values in appropriate units

119860 =$1100order

119862 =$300unit

119868 = 025 119873 = 50000 120579 = 0

(29)

5 Observations

The results obtained from different numerical tables inSection 4 explain the effect of changes in the system param-eters on the optimal values of total cost 119870(119879lowast) the optimalcycle length 119879

lowast and the optimal order quantity 119876(119879lowast)

We have observed the following relationship during thenumerical exercise

(a) As the coefficient of innovation increases keepingother parameters constant then the optimal cyclelength 119879

lowast decreases while both the optimal orderquantity 119876(119879

lowast) and the optimal total cost 119870(119879lowast)

increases as depicted in Tables 1 and 4 This isconsistent with the reality as more investment onpromotion will increase the diffusion of a productin the market resulting in shrinkage of the optimalreorder cycle time as a result optimal cost is increased

Advances in Decision Sciences 7

Table 1 Sensitivity analysis on coefficient of innovation ldquo119901rdquo

119901 119879lowast

119870(119879lowast) 119876(119879

lowast)

0001 086 16400 390002 059 32705 570003 048 48684 700004 041 64499 810005 037 80211 910006 034 95848 1010007 031 111430 1090008 029 126967 1170009 027 142468 124001 026 157938 131002 019 311593 190For 1198791 = 01 120579 = 00001

(b) As the cycle length 1198791 that is the time of intro-

duction of second product increases keeping otherparameters constant then the optimal cycle length119879lowastoptimal order quantity 119876(119879lowast) and the optimal totalcost 119870(119879lowast) all these three values increase as depictedin Table 2 This is true with the fact that as the timeof introduction of second product increases meansmaximum demand is satisfied by more number ofprevious that is old products and it is also obviousthat the demand is a decreasing function of timewhich implies that less demand for more time periodleads to increase in optimal cycle length And sinceoptimal cycle length increases therefore it forcesthe inventory manager to keep more number ofinventories to satisfy decreasing demand for longertime period and because more number of inventoriesare kept for longer time period it leads to increase inthe optimal cost (Figure 3)

(c) As the value of 120579 increases keeping other parametersconstant then the optimal cycle length 119879lowast increaseswhereas optimal order quantity 119876(119879lowast) and the opti-mal total cost119870(119879lowast) decreases as depicted in Table 3This is also true with the fact that as 120579 increasesmeans demand of the first product decreases morebecause more demand is satisfied by its substituteproduct which is introduced after a certain interval oftime and therefore less inventories are kept for longerperiod which leads to decrease in the optimal cost

6 Managerial Implications

The goodwill of any organization is entirely dependent onits effective management and this is possible only when theproblemof the organization is analyzed properly from all per-spectives Here our problem is concerned with the effectivemanagement of products which are newly introduced in themarket and along with this the effect of second generationproducts on the demand of first generation products is alsorealized the idea included here is that the effect of substituteproducts decreases the demand of old products after a certain

Table 2 Sensitivity analysis on ldquo1198791rdquo

1198791

119879lowast

119870(119879lowast) 119876(119879

lowast)

010 059 68778 124011 061 69032 128012 063 69279 131013 064 69520 135014 066 69756 139015 067 69987 143016 069 70213 146017 07 70435 150018 072 70653 153019 073 70866 157020 075 71076 160For 120579 = 0001 119901 = 0005

Table 3 Sensitivity analysis on ldquo120579rdquo

120579 119879lowast

119870(119879lowast) 119876(119879

lowast)

0019 038 1802327 22280020 041 1649195 21800021 044 1495228 21210022 048 1340344 20500023 052 1184432 19650024 058 1027337 18650025 065 868835 17470026 073 708581 16050027 086 545998 14330028 107 379973 12170029 154 207669 918For 1198791 = 001 119901 = 003119873 = 500000

Table 4 Sensitivity analysis on coefficient of innovation ldquo119901rdquo

119901 119879lowast

119870(119879lowast) 119876(119879

lowast)

0001 076 17865 380002 054 34044 540003 044 49942 670004 038 65695 770005 034 81354 860006 031 96946 950007 029 112487 1020008 027 127987 1100009 026 143453 117001 024 158891 123002 017 312305 178Special case

interval of time because of technological innovations andother related factors Therefore to overcome this problem amathematical model has been developed to know the actualsituation of the economic ordering policies which will helpthe inventory manager to take effective action to maintainthe optimal cost The results obtained in the model create apool of knowledge on the diffusion processes of innovations

8 Advances in Decision Sciences

48625

48675

48725

48775

48825

48875

Time

Cos

t

038 042 048 052 056 06

Cost-time graphFor Tlowast

= 048 K(Tlowast) = 48684

(a)

69745

69755

69765

69775

69785

69795

Time

Cos

t

061 063 065 067 069 071

Cost-time graphFor Tlowast

= 066 K(Tlowast) = 69756

(b)

868700

868800

868900

869000

869100

869200

Time

Cos

t

061 063 065 067 069

Cost-time graphFor Tlowast

= 065 K(Tlowast) = 868835

(c)

33000

33500

34000

34500

35000

35500

36000

Time

Cos

t

024 034 044 054 064 074 084 094 104

Cost-time graph (special case)For Tlowast

= 054 K(Tlowast) = 34044

(d)

Figure 3 Cost-time graphs showing convexity of the cost functions

and its effective management which are most valuable toboth researchers and managers For managers this poolof knowledge is important because it provides useful toolsand useful information for managing and scheduling theinventories of new products when its demand is affectedby the generation of other new products which substituteproducts after a certain time period The uniqueness of thismodel is that how the inventory manager should keep theoptimal cycle time of any fixed lot size of one product enteringinto the inventory system when its substitute product entersinto the system after a fixed interval so that optimal costis maintained and the cost of inventory obsolescence isminimized

7 Conclusion

The technological breakthrough is constantly being experi-enced in various products across the globe which is neces-sary for the products to gain competitive advantage Thereare various marketing strategies to promote and establishproducts which are newly introduced into the market butkeeping its inventories efficiently and effectively play a greaterrole Here the role of inventory models becomes significantThe primary objective of developing inventory models isto take efficient action concerned with economic orderingpolicies and to understand well the behaviour of parametersassociated with it In this paper a mathematical model has

Advances in Decision Sciences 9

been developed for obtaining the Economic Order Quantity(EOQ) in which the demand of the product is assumed tofollow an innovation diffusion process as proposed by Fourtand WoodLock [2] and along with this the demand of theproduct decreases after a certain interval of time because ofthe introduction of its substitute product Here the modelanalyses the situation when demand of the previous productis affected by introducing second generation product Thereare few inventory models such as Chern et al [28] Chandaand Kumar [25] Aggarwal et al [26] and Chanda andKumar [27] which capture the sensitive nature of marketingparameters but have not incorporated the idea of the effectof introduction of second generation product This paperattempts to bridge this gap The approach of this paper is toobtain economic ordering policies of a new product whenits substitute product enters into the market after a certaintime period A special case has also been discussed whichexcludes the effect of substitute products on the inventorypolicies of old products Numerical examples with sensitivityanalyses are presented to illustrate the model properly Theresults obtained are largely consistent with what we observeempirically A simple solution procedure in the form ofalgorithm is presented to determine the optimal cycle timeand optimal order quantity of the cost functionTheproposedmodel can be extended for backorders quantity discountpartial lost sales and so forth and it can also be extendedto solve by different approaches such as geometric program-ming approach and genetic algorithm approach At the sametime future research should also focus on developing somealternative approach to obtain the optimal analytical solutionof the highly nonlinear cost function which we have solvednumerically

Acknowledgments

The research presented in this paper was carried out at theDepartment ofOperational Reseach Faculty ofMathematicalSciences University of Delhi The commercial identitiesmentioned in the paper such as LINGO and Excel-Solversoftware have only been used for the numerical analysis of themodel discussed in the paper which is only for the academicpurpose The authors of this paper do not have any directfinancial relation with the commercial identities mentionedin the paper and there is no conflict of interests regardingfinancial gains for all the authors

References

[1] E Rogers Diffusion of Innovations Free Press New York NYUSA 3rd edition 1983

[2] L A Fourt and J W WoodLock ldquoEarly prediction of Marketsuccess for new grocery productsrdquo Journal of Marketing vol 26no 2 pp 31ndash38 1960

[3] J C Fisher and R H Pry ldquoA simple substitution modelof technological changerdquo Technological Forecasting and SocialChange vol 3 no C pp 75ndash88 1971

[4] F M Bass ldquoA new product growth for model consumerdurablesrdquo Management Science vol 50 no 12 pp 1825ndash18322004

[5] R Peres E Muller and V Mahajan ldquoInnovation diffusion andnew product growth models a critical review and researchdirectionsrdquo International Journal of Research in Marketing vol27 no 2 pp 91ndash106 2010

[6] PM Parker ldquoAggregate diffusion forecastingmodels inmarket-ing a critical reviewrdquo International Journal of Forecasting vol10 no 2 pp 353ndash380 1994

[7] E M Rogers Diffusion of Innovations Free Press New YorkNY USA 4th edition 1995

[8] R Peres E Muller and V Mahajan ldquoInnovation diffusion andnew product growth models a critical review and researchdirectionsrdquo International Journal of Research in Marketing vol27 no 2 pp 91ndash106 2010

[9] V Mahajan E Muller and Y Wind New-Product DiffusionModels Kluwer Academic Publishers Boston Mass USA2000

[10] F Cheng and S P Sethi ldquoPeriodic review inventory modelwith demand influenced by promotion decisionsrdquoManagementScience vol 45 no 11 pp 1510ndash1523 1999

[11] S Kalish ldquoAnewproduct adoptionmodelwith price advertisingand uncertaintyrdquoManagement Science vol 31 no 12 pp 1569ndash1585 1985

[12] VMahajan andA P Robert ldquoInnovation diffusion in a dynamicpotential adopter populationrdquoManagement Science vol 24 no15 pp 1589ndash1597 1978

[13] M N Sharif and K Ramanathan ldquoPolynomial innovationdiffusion modelsrdquo Technological Forecasting and Social Changevol 21 no 4 pp 301ndash323 1982

[14] M N Sharif and K Ramanathan ldquoBinomial innovation dif-fusion models with dynamic potential adopter populationrdquoTechnological Forecasting and Social Change vol 20 no 1 pp63ndash87 1981

[15] J A Norton and F M Bass ldquoA diffusion theory model ofadoption and substitution for successive generation of high-technology productsrdquo Management Science vol 33 no 9 pp1069ndash1086 1987

[16] V Mahajan and E Muller ldquoTiming diffusion and substitutionof successive generations of technological innovations the IBMmainframe caserdquo Technological Forecasting and Social Changevol 51 no 2 pp 109ndash132 1996

[17] M W Speece and D L Maclachlan ldquoApplication of a multi-generation diffusion model to milk container technologyrdquoTechnological Forecasting and Social Change vol 49 no 3 pp281ndash295 1995

[18] P J Danaher B G S Hardie and W P Putsis Jr ldquoMarketing-mix variables and the diffusion of successive generations of atechnological innovationrdquo Journal of Marketing Research vol38 no 4 pp 501ndash514 2001

[19] T Islam and NMeade ldquoThe diffusion of successive generationsof a technology a more general modelrdquo Technological Forecast-ing and Social Change vol 56 no 1 pp 49ndash60 1997

[20] N Kim D R Chang and A D Shocker ldquoModeling intercat-egory and generational dynamics for a growing informationtechnology industryrdquo Management Science vol 46 no 4 pp496ndash512 2000

[21] B Bayus ldquoThe dynamic pricing of next generation consumerdurablesrdquoMarketing Science vol 11 no 3 pp 251ndash265 1992

[22] A K Bardhan and U Chanda ldquoA model for first and sub-stitution adoption of successive generations of a productrdquoInternational Journal of Modelling and Simulation vol 28 no4 pp 487ndash494 2008

10 Advances in Decision Sciences

[23] U Chanda and A K Bardhan ldquoDynamic optimal advertisingexpenditure strategies of two successive generations of con-sumer durablesrdquo in Proceedings of the International Confer-ence on Modelling Computation and Optimization held duringJanuary 2008 in the venue SQC amp OR Unit Indian StatisticalInstitute New Delhi India 2008

[24] K A Bardhan and U Chanda ldquoA mathematical model fordiffusion of products with multiple generationsrdquo InternationalJournal of Operational Research vol 4 no 2 pp 214ndash229 2009

[25] U Chanda and A Kumar ldquoEconomic order quantity modelwith demand influenced by dynamic innovation effectrdquo Interna-tional Journal of Operational Research vol 11 no 2 pp 193ndash2152011

[26] K K Aggarwal C K Jaggi and A Kumar ldquoEconomic orderquantity model with innovation diffusion criterion havingdynamic potential market sizerdquo International Journal of AppliedDecision Sciences vol 4 no 3 pp 280ndash303 2011

[27] U Chanda and A Kumar ldquoEconomic order quantity model oninflationary conditions with demand influenced by innovationdiffusion criterionrdquo International Journal of Procurement Man-agement vol 5 no 2 pp 160ndash177 2012

[28] M-S Chern J-T Teng and H-L Yang ldquoInventory lot-sizepolicies for the bass diffusion demand models of new durableproductsrdquo Journal of the Chinese Institute of Engineers vol 24no 2 pp 237ndash244 2001

[29] K K Aggarwal and A Kumar ldquoAn inventory model for newproduct when demand follows dynamic innovation processhaving dynamic potential market sizerdquo International Journal ofStrategic Decision Sciences vol 3 no 4 pp 78ndash99 2012

[30] A Kumar K K Aggarwal and U Chanda ldquoEconomic orderquantity model under fuzzy sense with demand follows Bassrsquosinnovation diffusion processrdquo International Journal of AdvancedOperations Management vol 5 no 3 pp 261ndash281 2013

[31] K K Aggarwal C K Jaggi and A Kumar ldquoEconomic orderquantity model under fuzzy sense when demand follows inno-vation diffusion process having dynamic potential market sizerdquoInternational Journal of Services and Operations Managementvol 13 no 3 pp 361ndash391 2012

[32] V Mahajan and R A PetersonModels of Innovation DiffusionSage 1985

[33] E Ruiz-Conde P S H Leeflang and J EWieringa ldquoMarketingvariables in macro-level diffusion modelsrdquo Journal fur Betrieb-swirtschaft vol 56 no 3 pp 155ndash183 2006

[34] D Chandrasekaran and G J Tellis ldquoA critical review ofmarketing research on diffusion of new productsrdquo Review ofMarketing Research pp 39ndash80 2007

[35] F Sultan J U Farley and D R Lehmann ldquoA meta-analysis ofdiffusion modelsrdquo International Journal of Forecasting vol 27no 1 pp 70ndash77 1990

[36] D Talukdar K Sudhir and A Ainslie ldquoInvestigating newproduct diffusion across products and countriesrdquo MarketingScience vol 21 no 1 pp 97ndash114 2002

[37] C van den Bulte and S Stremersch ldquoSocial contagion andincome heterogeneity in new product diffusion ameta-analytictestrdquoMarketing Science vol 23 no 4 pp 530ndash544 2004

[38] G L Lilien P Kotler and K S Moorthy Marketing ModelsPrentice-Hall of India Private New Delhi India 1999

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Advances in Decision Sciences 5

= (119868119862

119879)1198791119868 (0) +

119868119862119873

119901119879[1 minus 119890

minus1199011198791]

minus (119868119862

119879)1198731198791

+ (119868119862

119879) [119873 minus 119873(119879

1)]

times [119890(119901minus120579)(119879

1minus119879)

minus 1

(120579 minus 119901)+ (1198791minus 119879) 119890

(119901minus120579)(1198791minus119879)]

(18)

Thus the total cost of the inventory system per unit time119870(119879) is

119870 (119879) = (14) + (16) + (18)

997904rArr 119870 (119879) =119860

119879+ [119873 (119879

1) minus 119873] (119890

(119901minus120579)(1198791minus119879)

minus 1)

times (119862

119879) (1 + 119868119879

1) + 119873 (1 minus 119890

minus1199011198791) (

119862

119879)

times (1 + 1198681198791) +

119868119862119873

119901119879[1 minus 119890

minus1199011198791] minus (

119868119862

119879)1198731198791

+ (119868119862

119879) [119873 minus 119873(119879

1)]

times [119890(119901minus120579)(119879

1minus119879)

minus 1

(120579 minus 119901)+ (1198791minus 119879) 119890

(119901minus120579)(1198791minus119879)]

(19)

The necessary criterion for 119870(119879) to be minimum is

119889119870 (119879)

119889119879= 0 (20)

997904rArrminus119860

1198792+ [119873 (119879

1) minus 119873] (119890

(119901minus120579)(1198791minus119879)

minus 1) (minus119862

1198792) (1 + 119868119879

1)

minus (119862

1198792) (1 + 119868119879

1)119873 (1 minus 119890

minus1199011198791)

+ (119862

119879) (1 + 119868119879

1) [119873 (119879

1) minus 119873] (120579 minus 119901) 119890

(119901minus120579)(1198791minus119879)

minus119868119862119873

1199011198792[1 minus 119890

minus1199011198791] + (

119868119862

1198792)1198731198791

+ (minus119868119862

1198792) [119873 minus 119873(119879

1)]

times [119890(119901minus120579)(119879

1minus119879)

minus 1

(120579 minus 119901)+ (1198791minus 119879) 119890

(119901minus120579)(1198791minus119879)]

+ (119868119862

119879) [119873 minus 119873(119879

1)]

times (1198791minus 119879) (120579 minus 119901) 119890

(119901minus120579)(1198791minus119879)

= 0

(21)

Now for119870(119879) to be convex

1198892119870 (119879)

1198891198792

100381610038161003816100381610038161003816100381610038161003816119879=119879lowast

gt 0

997904rArr2119860

119879lowast3+ (

2119862

119879lowast3) (1 + 119868119879

1)119873 (1 minus 119890

minus1199011198791)

+ [119873 (1198791) minus 119873] (119890

(119901minus120579)(1198791minus119879lowast)minus 1) (

2119862

1198793) (1 + 119868119879

1)

+ (119862

119879lowast) (1 + 119868119879

1) [119873 (119879

1) minus 119873] (120579 minus 119901)

2

119890(119901minus120579)(119879

1minus119879lowast)

+2119868119862119873

119901119879lowast3

[1 minus 119890minus1199011198791] + (

2119868119862

119879lowast3) [119873 minus 119873(119879

1)]

times [119890(119901minus120579)(119879

1minus119879lowast)minus 1

(120579 minus 119901)+ (1198791minus 119879lowast) 119890(119901minus120579)(119879

1minus119879lowast)]

+ (119868119862

119879lowast) [119873 minus 119873(119879

1)] (1198791minus 119879lowast)

times (120579 minus 119901)2

119890(119901minus120579)(119879

1minus119879lowast)

gt (2119862

119879lowast2) (1 + 119868119879

1) [119873 (119879

1) minus 119873]

times (120579 minus 119901) 119890(119901minus120579)(119879

1minus119879)

+ (2119868119862

119879lowast3)1198731198791

+ (2119868119862

119879lowast2) [119873 minus 119873(119879

1)]

times [(1198791minus 119879lowast) (120579 minus 119901) 119890

(119901minus120579)(1198791minus119879lowast)]

+ (119868119862

119879lowast) [119873 minus 119873(119879

1)] [(120579 minus 119901) 119890

(119901minus120579)(1198791minus119879lowast)]

(22)

The solution of the equation 119889119870(119879)119889119879 = 0 gives theoptimum value of 119879 provided it satisfies the condition1198892119870(119879)119889119879

2gt 0 Since the above cost equation (21) is highly

nonlinear the problem has been solved numerically for givenparameter valuesThe solution gives the optimumvalue119879lowast ofthe replenishment cycle time 119879 Once 119879lowast is known the valueof optimum order quantity 119876lowast and the optimum cost 119870(119879lowast)can easily be obtained from (10) and (19) respectively Thenumerical solution for the given base value has been obtainedby using software packages Lingo and Excel-Solver

21 Special Case When 120579 becomes negligible that is equalto zero which means here we are not considering the effectof introduction of new products on the old one after timeperiod 119879

1 That is demand rate of the old product is the

same throughout the planning period (0 119879) and for this casethe adoption rate that is demand rate that is 120582(119905) = 119899(119905)reduces to the classical Fourt and WoodLock model [2] Theobjective here is to study the different managerial policiesfor the model which is under external influence only andits demand is not affected by the introduction of any other

6 Advances in Decision Sciences

substitute products In that case the total cost per unit time119870(119879) is given by

119870 (119879) =119860

119879+119873119862

119879(1 minus 119890

minus119901119879) +

119873119868119862

119901119879(1 minus 119890

minus119901119879) minus 119873119868119862119890

minus119901119879

(23)

119868 (0) = 119876 = int

119879

0

120582 (119905) 119889119905 = 119873 (1 minus 119890minus119901119879

) (24)

For optimum total cost the necessary criterion is

119889119870 (119879)

119889119879= 0 (25)

997904rArrminus119860

1198792+119873119862119901119890

minus119901119879

119879minus

119873119862(1 minus 119890minus119901119879

)

1198792

+119873119868119862119890

minus119901119879

119879

minus

119873119868119862 (1 minus 119890minus119901119879

)

1199011198792

+ 119873119868119862119901119890minus119901119879

= 0

(26)

Now for119870(119879) to be convex

1198892119870 (119879)

1198891198792

100381610038161003816100381610038161003816100381610038161003816119879=119879lowast

gt 0

997904rArr2119860

119879lowast3+

2119873119862(1 minus 119890minus119901119879lowast

)

119879lowast3

+

2119873119868119862 (1 minus 119890minus119901119879lowast

)

119901119879lowast3

gt1198731198621199012119890minus119901119879lowast

119879lowast

+119873119868119901119862119890

minus119901119879lowast

119879lowast2

+ 1198731198681198621199012119890minus119901119879lowast

(27)

The solution of the equation 119889119870(119879)119889119879 = 0 gives theoptimum value of 119879 provided it satisfies the condition1198892119870(119879)119889119879

2gt 0 Since cost equation (26) is highly nonlinear

the problem has been solved numerically for given parametervalues The solution gives the optimum value 119879lowast of thereplenishment cycle time 119879 Once 119879lowast is known the valuesof optimum order quantity 119876lowast and the optimum cost 119870(119879lowast)can easily be obtained from (24) and (23) respectively Thenumerical solution for the given base value has been obtainedby using software packages LINGO and Excel-Solver

3 Solution Procedure

The solution procedure has been summarized in the follow-ing algorithmStep 1 Input all parameter values such as unit cost coefficientof innovation potential market size and ordering cost foreach case separatelyStep 2 Compute all possible values of ldquo119879rdquo separately for (21)and (26) as the case may beStep 3 select the appropriate value of ldquo119879rdquo say 119879lowast for eachcase by using (19) and (23) and by satisfying the above statedcondition 1198892119870(119879)1198891198792 gt 0Step 4Compute119870(119879lowast) from (19) and (23) amp119876(119879lowast) from (10)and (24) for each case separately

The above steps are used for all replenishment cyclesusing appropriate parameter values In order to obtain theappropriate values of ldquo119879rdquo we need to follow the aboveprocedure with the help of LINGO and Excel-Solver softwarepackages

4 Numerical Examples

The effectiveness of the proposed model has been shown bythe following numerical examples A hypothetical examplehas the following parameter values in appropriate units

119860 =$1100order

119862 =$300unit

119868 = 025 119873 = 50000

(28)

The results have been well presented in the followingdifferent tables Also to prove the validity of the modelnumerically and to get the appropriate parameter values thereferences have been considered as Chanda and Kumar [25]Chandrasekaran and Tellis [34] Sultan et al [35] Talukdaret al [36] Van den Bulte and Stremersch [37] Chanda andKumar [27] Aggarwal et al [26 31] Aggarwal et al [29] andKumar et al [30]

(i) The mean value of the coefficient of innovation fora new product usually lies between 00007 and 003(Sultan et al [35] Talukdar et al [36] Van den Bulteand Stremersch [37])

(ii) The mean value of the coefficient of innovation fora new product is usually 001for developed countriesand 00003 for developing countries (Talukdar et al[36])

41 Special Case A hypothetical example has the followingparameter values in appropriate units

119860 =$1100order

119862 =$300unit

119868 = 025 119873 = 50000 120579 = 0

(29)

5 Observations

The results obtained from different numerical tables inSection 4 explain the effect of changes in the system param-eters on the optimal values of total cost 119870(119879lowast) the optimalcycle length 119879

lowast and the optimal order quantity 119876(119879lowast)

We have observed the following relationship during thenumerical exercise

(a) As the coefficient of innovation increases keepingother parameters constant then the optimal cyclelength 119879

lowast decreases while both the optimal orderquantity 119876(119879

lowast) and the optimal total cost 119870(119879lowast)

increases as depicted in Tables 1 and 4 This isconsistent with the reality as more investment onpromotion will increase the diffusion of a productin the market resulting in shrinkage of the optimalreorder cycle time as a result optimal cost is increased

Advances in Decision Sciences 7

Table 1 Sensitivity analysis on coefficient of innovation ldquo119901rdquo

119901 119879lowast

119870(119879lowast) 119876(119879

lowast)

0001 086 16400 390002 059 32705 570003 048 48684 700004 041 64499 810005 037 80211 910006 034 95848 1010007 031 111430 1090008 029 126967 1170009 027 142468 124001 026 157938 131002 019 311593 190For 1198791 = 01 120579 = 00001

(b) As the cycle length 1198791 that is the time of intro-

duction of second product increases keeping otherparameters constant then the optimal cycle length119879lowastoptimal order quantity 119876(119879lowast) and the optimal totalcost 119870(119879lowast) all these three values increase as depictedin Table 2 This is true with the fact that as the timeof introduction of second product increases meansmaximum demand is satisfied by more number ofprevious that is old products and it is also obviousthat the demand is a decreasing function of timewhich implies that less demand for more time periodleads to increase in optimal cycle length And sinceoptimal cycle length increases therefore it forcesthe inventory manager to keep more number ofinventories to satisfy decreasing demand for longertime period and because more number of inventoriesare kept for longer time period it leads to increase inthe optimal cost (Figure 3)

(c) As the value of 120579 increases keeping other parametersconstant then the optimal cycle length 119879lowast increaseswhereas optimal order quantity 119876(119879lowast) and the opti-mal total cost119870(119879lowast) decreases as depicted in Table 3This is also true with the fact that as 120579 increasesmeans demand of the first product decreases morebecause more demand is satisfied by its substituteproduct which is introduced after a certain interval oftime and therefore less inventories are kept for longerperiod which leads to decrease in the optimal cost

6 Managerial Implications

The goodwill of any organization is entirely dependent onits effective management and this is possible only when theproblemof the organization is analyzed properly from all per-spectives Here our problem is concerned with the effectivemanagement of products which are newly introduced in themarket and along with this the effect of second generationproducts on the demand of first generation products is alsorealized the idea included here is that the effect of substituteproducts decreases the demand of old products after a certain

Table 2 Sensitivity analysis on ldquo1198791rdquo

1198791

119879lowast

119870(119879lowast) 119876(119879

lowast)

010 059 68778 124011 061 69032 128012 063 69279 131013 064 69520 135014 066 69756 139015 067 69987 143016 069 70213 146017 07 70435 150018 072 70653 153019 073 70866 157020 075 71076 160For 120579 = 0001 119901 = 0005

Table 3 Sensitivity analysis on ldquo120579rdquo

120579 119879lowast

119870(119879lowast) 119876(119879

lowast)

0019 038 1802327 22280020 041 1649195 21800021 044 1495228 21210022 048 1340344 20500023 052 1184432 19650024 058 1027337 18650025 065 868835 17470026 073 708581 16050027 086 545998 14330028 107 379973 12170029 154 207669 918For 1198791 = 001 119901 = 003119873 = 500000

Table 4 Sensitivity analysis on coefficient of innovation ldquo119901rdquo

119901 119879lowast

119870(119879lowast) 119876(119879

lowast)

0001 076 17865 380002 054 34044 540003 044 49942 670004 038 65695 770005 034 81354 860006 031 96946 950007 029 112487 1020008 027 127987 1100009 026 143453 117001 024 158891 123002 017 312305 178Special case

interval of time because of technological innovations andother related factors Therefore to overcome this problem amathematical model has been developed to know the actualsituation of the economic ordering policies which will helpthe inventory manager to take effective action to maintainthe optimal cost The results obtained in the model create apool of knowledge on the diffusion processes of innovations

8 Advances in Decision Sciences

48625

48675

48725

48775

48825

48875

Time

Cos

t

038 042 048 052 056 06

Cost-time graphFor Tlowast

= 048 K(Tlowast) = 48684

(a)

69745

69755

69765

69775

69785

69795

Time

Cos

t

061 063 065 067 069 071

Cost-time graphFor Tlowast

= 066 K(Tlowast) = 69756

(b)

868700

868800

868900

869000

869100

869200

Time

Cos

t

061 063 065 067 069

Cost-time graphFor Tlowast

= 065 K(Tlowast) = 868835

(c)

33000

33500

34000

34500

35000

35500

36000

Time

Cos

t

024 034 044 054 064 074 084 094 104

Cost-time graph (special case)For Tlowast

= 054 K(Tlowast) = 34044

(d)

Figure 3 Cost-time graphs showing convexity of the cost functions

and its effective management which are most valuable toboth researchers and managers For managers this poolof knowledge is important because it provides useful toolsand useful information for managing and scheduling theinventories of new products when its demand is affectedby the generation of other new products which substituteproducts after a certain time period The uniqueness of thismodel is that how the inventory manager should keep theoptimal cycle time of any fixed lot size of one product enteringinto the inventory system when its substitute product entersinto the system after a fixed interval so that optimal costis maintained and the cost of inventory obsolescence isminimized

7 Conclusion

The technological breakthrough is constantly being experi-enced in various products across the globe which is neces-sary for the products to gain competitive advantage Thereare various marketing strategies to promote and establishproducts which are newly introduced into the market butkeeping its inventories efficiently and effectively play a greaterrole Here the role of inventory models becomes significantThe primary objective of developing inventory models isto take efficient action concerned with economic orderingpolicies and to understand well the behaviour of parametersassociated with it In this paper a mathematical model has

Advances in Decision Sciences 9

been developed for obtaining the Economic Order Quantity(EOQ) in which the demand of the product is assumed tofollow an innovation diffusion process as proposed by Fourtand WoodLock [2] and along with this the demand of theproduct decreases after a certain interval of time because ofthe introduction of its substitute product Here the modelanalyses the situation when demand of the previous productis affected by introducing second generation product Thereare few inventory models such as Chern et al [28] Chandaand Kumar [25] Aggarwal et al [26] and Chanda andKumar [27] which capture the sensitive nature of marketingparameters but have not incorporated the idea of the effectof introduction of second generation product This paperattempts to bridge this gap The approach of this paper is toobtain economic ordering policies of a new product whenits substitute product enters into the market after a certaintime period A special case has also been discussed whichexcludes the effect of substitute products on the inventorypolicies of old products Numerical examples with sensitivityanalyses are presented to illustrate the model properly Theresults obtained are largely consistent with what we observeempirically A simple solution procedure in the form ofalgorithm is presented to determine the optimal cycle timeand optimal order quantity of the cost functionTheproposedmodel can be extended for backorders quantity discountpartial lost sales and so forth and it can also be extendedto solve by different approaches such as geometric program-ming approach and genetic algorithm approach At the sametime future research should also focus on developing somealternative approach to obtain the optimal analytical solutionof the highly nonlinear cost function which we have solvednumerically

Acknowledgments

The research presented in this paper was carried out at theDepartment ofOperational Reseach Faculty ofMathematicalSciences University of Delhi The commercial identitiesmentioned in the paper such as LINGO and Excel-Solversoftware have only been used for the numerical analysis of themodel discussed in the paper which is only for the academicpurpose The authors of this paper do not have any directfinancial relation with the commercial identities mentionedin the paper and there is no conflict of interests regardingfinancial gains for all the authors

References

[1] E Rogers Diffusion of Innovations Free Press New York NYUSA 3rd edition 1983

[2] L A Fourt and J W WoodLock ldquoEarly prediction of Marketsuccess for new grocery productsrdquo Journal of Marketing vol 26no 2 pp 31ndash38 1960

[3] J C Fisher and R H Pry ldquoA simple substitution modelof technological changerdquo Technological Forecasting and SocialChange vol 3 no C pp 75ndash88 1971

[4] F M Bass ldquoA new product growth for model consumerdurablesrdquo Management Science vol 50 no 12 pp 1825ndash18322004

[5] R Peres E Muller and V Mahajan ldquoInnovation diffusion andnew product growth models a critical review and researchdirectionsrdquo International Journal of Research in Marketing vol27 no 2 pp 91ndash106 2010

[6] PM Parker ldquoAggregate diffusion forecastingmodels inmarket-ing a critical reviewrdquo International Journal of Forecasting vol10 no 2 pp 353ndash380 1994

[7] E M Rogers Diffusion of Innovations Free Press New YorkNY USA 4th edition 1995

[8] R Peres E Muller and V Mahajan ldquoInnovation diffusion andnew product growth models a critical review and researchdirectionsrdquo International Journal of Research in Marketing vol27 no 2 pp 91ndash106 2010

[9] V Mahajan E Muller and Y Wind New-Product DiffusionModels Kluwer Academic Publishers Boston Mass USA2000

[10] F Cheng and S P Sethi ldquoPeriodic review inventory modelwith demand influenced by promotion decisionsrdquoManagementScience vol 45 no 11 pp 1510ndash1523 1999

[11] S Kalish ldquoAnewproduct adoptionmodelwith price advertisingand uncertaintyrdquoManagement Science vol 31 no 12 pp 1569ndash1585 1985

[12] VMahajan andA P Robert ldquoInnovation diffusion in a dynamicpotential adopter populationrdquoManagement Science vol 24 no15 pp 1589ndash1597 1978

[13] M N Sharif and K Ramanathan ldquoPolynomial innovationdiffusion modelsrdquo Technological Forecasting and Social Changevol 21 no 4 pp 301ndash323 1982

[14] M N Sharif and K Ramanathan ldquoBinomial innovation dif-fusion models with dynamic potential adopter populationrdquoTechnological Forecasting and Social Change vol 20 no 1 pp63ndash87 1981

[15] J A Norton and F M Bass ldquoA diffusion theory model ofadoption and substitution for successive generation of high-technology productsrdquo Management Science vol 33 no 9 pp1069ndash1086 1987

[16] V Mahajan and E Muller ldquoTiming diffusion and substitutionof successive generations of technological innovations the IBMmainframe caserdquo Technological Forecasting and Social Changevol 51 no 2 pp 109ndash132 1996

[17] M W Speece and D L Maclachlan ldquoApplication of a multi-generation diffusion model to milk container technologyrdquoTechnological Forecasting and Social Change vol 49 no 3 pp281ndash295 1995

[18] P J Danaher B G S Hardie and W P Putsis Jr ldquoMarketing-mix variables and the diffusion of successive generations of atechnological innovationrdquo Journal of Marketing Research vol38 no 4 pp 501ndash514 2001

[19] T Islam and NMeade ldquoThe diffusion of successive generationsof a technology a more general modelrdquo Technological Forecast-ing and Social Change vol 56 no 1 pp 49ndash60 1997

[20] N Kim D R Chang and A D Shocker ldquoModeling intercat-egory and generational dynamics for a growing informationtechnology industryrdquo Management Science vol 46 no 4 pp496ndash512 2000

[21] B Bayus ldquoThe dynamic pricing of next generation consumerdurablesrdquoMarketing Science vol 11 no 3 pp 251ndash265 1992

[22] A K Bardhan and U Chanda ldquoA model for first and sub-stitution adoption of successive generations of a productrdquoInternational Journal of Modelling and Simulation vol 28 no4 pp 487ndash494 2008

10 Advances in Decision Sciences

[23] U Chanda and A K Bardhan ldquoDynamic optimal advertisingexpenditure strategies of two successive generations of con-sumer durablesrdquo in Proceedings of the International Confer-ence on Modelling Computation and Optimization held duringJanuary 2008 in the venue SQC amp OR Unit Indian StatisticalInstitute New Delhi India 2008

[24] K A Bardhan and U Chanda ldquoA mathematical model fordiffusion of products with multiple generationsrdquo InternationalJournal of Operational Research vol 4 no 2 pp 214ndash229 2009

[25] U Chanda and A Kumar ldquoEconomic order quantity modelwith demand influenced by dynamic innovation effectrdquo Interna-tional Journal of Operational Research vol 11 no 2 pp 193ndash2152011

[26] K K Aggarwal C K Jaggi and A Kumar ldquoEconomic orderquantity model with innovation diffusion criterion havingdynamic potential market sizerdquo International Journal of AppliedDecision Sciences vol 4 no 3 pp 280ndash303 2011

[27] U Chanda and A Kumar ldquoEconomic order quantity model oninflationary conditions with demand influenced by innovationdiffusion criterionrdquo International Journal of Procurement Man-agement vol 5 no 2 pp 160ndash177 2012

[28] M-S Chern J-T Teng and H-L Yang ldquoInventory lot-sizepolicies for the bass diffusion demand models of new durableproductsrdquo Journal of the Chinese Institute of Engineers vol 24no 2 pp 237ndash244 2001

[29] K K Aggarwal and A Kumar ldquoAn inventory model for newproduct when demand follows dynamic innovation processhaving dynamic potential market sizerdquo International Journal ofStrategic Decision Sciences vol 3 no 4 pp 78ndash99 2012

[30] A Kumar K K Aggarwal and U Chanda ldquoEconomic orderquantity model under fuzzy sense with demand follows Bassrsquosinnovation diffusion processrdquo International Journal of AdvancedOperations Management vol 5 no 3 pp 261ndash281 2013

[31] K K Aggarwal C K Jaggi and A Kumar ldquoEconomic orderquantity model under fuzzy sense when demand follows inno-vation diffusion process having dynamic potential market sizerdquoInternational Journal of Services and Operations Managementvol 13 no 3 pp 361ndash391 2012

[32] V Mahajan and R A PetersonModels of Innovation DiffusionSage 1985

[33] E Ruiz-Conde P S H Leeflang and J EWieringa ldquoMarketingvariables in macro-level diffusion modelsrdquo Journal fur Betrieb-swirtschaft vol 56 no 3 pp 155ndash183 2006

[34] D Chandrasekaran and G J Tellis ldquoA critical review ofmarketing research on diffusion of new productsrdquo Review ofMarketing Research pp 39ndash80 2007

[35] F Sultan J U Farley and D R Lehmann ldquoA meta-analysis ofdiffusion modelsrdquo International Journal of Forecasting vol 27no 1 pp 70ndash77 1990

[36] D Talukdar K Sudhir and A Ainslie ldquoInvestigating newproduct diffusion across products and countriesrdquo MarketingScience vol 21 no 1 pp 97ndash114 2002

[37] C van den Bulte and S Stremersch ldquoSocial contagion andincome heterogeneity in new product diffusion ameta-analytictestrdquoMarketing Science vol 23 no 4 pp 530ndash544 2004

[38] G L Lilien P Kotler and K S Moorthy Marketing ModelsPrentice-Hall of India Private New Delhi India 1999

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Advances in Decision Sciences

substitute products In that case the total cost per unit time119870(119879) is given by

119870 (119879) =119860

119879+119873119862

119879(1 minus 119890

minus119901119879) +

119873119868119862

119901119879(1 minus 119890

minus119901119879) minus 119873119868119862119890

minus119901119879

(23)

119868 (0) = 119876 = int

119879

0

120582 (119905) 119889119905 = 119873 (1 minus 119890minus119901119879

) (24)

For optimum total cost the necessary criterion is

119889119870 (119879)

119889119879= 0 (25)

997904rArrminus119860

1198792+119873119862119901119890

minus119901119879

119879minus

119873119862(1 minus 119890minus119901119879

)

1198792

+119873119868119862119890

minus119901119879

119879

minus

119873119868119862 (1 minus 119890minus119901119879

)

1199011198792

+ 119873119868119862119901119890minus119901119879

= 0

(26)

Now for119870(119879) to be convex

1198892119870 (119879)

1198891198792

100381610038161003816100381610038161003816100381610038161003816119879=119879lowast

gt 0

997904rArr2119860

119879lowast3+

2119873119862(1 minus 119890minus119901119879lowast

)

119879lowast3

+

2119873119868119862 (1 minus 119890minus119901119879lowast

)

119901119879lowast3

gt1198731198621199012119890minus119901119879lowast

119879lowast

+119873119868119901119862119890

minus119901119879lowast

119879lowast2

+ 1198731198681198621199012119890minus119901119879lowast

(27)

The solution of the equation 119889119870(119879)119889119879 = 0 gives theoptimum value of 119879 provided it satisfies the condition1198892119870(119879)119889119879

2gt 0 Since cost equation (26) is highly nonlinear

the problem has been solved numerically for given parametervalues The solution gives the optimum value 119879lowast of thereplenishment cycle time 119879 Once 119879lowast is known the valuesof optimum order quantity 119876lowast and the optimum cost 119870(119879lowast)can easily be obtained from (24) and (23) respectively Thenumerical solution for the given base value has been obtainedby using software packages LINGO and Excel-Solver

3 Solution Procedure

The solution procedure has been summarized in the follow-ing algorithmStep 1 Input all parameter values such as unit cost coefficientof innovation potential market size and ordering cost foreach case separatelyStep 2 Compute all possible values of ldquo119879rdquo separately for (21)and (26) as the case may beStep 3 select the appropriate value of ldquo119879rdquo say 119879lowast for eachcase by using (19) and (23) and by satisfying the above statedcondition 1198892119870(119879)1198891198792 gt 0Step 4Compute119870(119879lowast) from (19) and (23) amp119876(119879lowast) from (10)and (24) for each case separately

The above steps are used for all replenishment cyclesusing appropriate parameter values In order to obtain theappropriate values of ldquo119879rdquo we need to follow the aboveprocedure with the help of LINGO and Excel-Solver softwarepackages

4 Numerical Examples

The effectiveness of the proposed model has been shown bythe following numerical examples A hypothetical examplehas the following parameter values in appropriate units

119860 =$1100order

119862 =$300unit

119868 = 025 119873 = 50000

(28)

The results have been well presented in the followingdifferent tables Also to prove the validity of the modelnumerically and to get the appropriate parameter values thereferences have been considered as Chanda and Kumar [25]Chandrasekaran and Tellis [34] Sultan et al [35] Talukdaret al [36] Van den Bulte and Stremersch [37] Chanda andKumar [27] Aggarwal et al [26 31] Aggarwal et al [29] andKumar et al [30]

(i) The mean value of the coefficient of innovation fora new product usually lies between 00007 and 003(Sultan et al [35] Talukdar et al [36] Van den Bulteand Stremersch [37])

(ii) The mean value of the coefficient of innovation fora new product is usually 001for developed countriesand 00003 for developing countries (Talukdar et al[36])

41 Special Case A hypothetical example has the followingparameter values in appropriate units

119860 =$1100order

119862 =$300unit

119868 = 025 119873 = 50000 120579 = 0

(29)

5 Observations

The results obtained from different numerical tables inSection 4 explain the effect of changes in the system param-eters on the optimal values of total cost 119870(119879lowast) the optimalcycle length 119879

lowast and the optimal order quantity 119876(119879lowast)

We have observed the following relationship during thenumerical exercise

(a) As the coefficient of innovation increases keepingother parameters constant then the optimal cyclelength 119879

lowast decreases while both the optimal orderquantity 119876(119879

lowast) and the optimal total cost 119870(119879lowast)

increases as depicted in Tables 1 and 4 This isconsistent with the reality as more investment onpromotion will increase the diffusion of a productin the market resulting in shrinkage of the optimalreorder cycle time as a result optimal cost is increased

Advances in Decision Sciences 7

Table 1 Sensitivity analysis on coefficient of innovation ldquo119901rdquo

119901 119879lowast

119870(119879lowast) 119876(119879

lowast)

0001 086 16400 390002 059 32705 570003 048 48684 700004 041 64499 810005 037 80211 910006 034 95848 1010007 031 111430 1090008 029 126967 1170009 027 142468 124001 026 157938 131002 019 311593 190For 1198791 = 01 120579 = 00001

(b) As the cycle length 1198791 that is the time of intro-

duction of second product increases keeping otherparameters constant then the optimal cycle length119879lowastoptimal order quantity 119876(119879lowast) and the optimal totalcost 119870(119879lowast) all these three values increase as depictedin Table 2 This is true with the fact that as the timeof introduction of second product increases meansmaximum demand is satisfied by more number ofprevious that is old products and it is also obviousthat the demand is a decreasing function of timewhich implies that less demand for more time periodleads to increase in optimal cycle length And sinceoptimal cycle length increases therefore it forcesthe inventory manager to keep more number ofinventories to satisfy decreasing demand for longertime period and because more number of inventoriesare kept for longer time period it leads to increase inthe optimal cost (Figure 3)

(c) As the value of 120579 increases keeping other parametersconstant then the optimal cycle length 119879lowast increaseswhereas optimal order quantity 119876(119879lowast) and the opti-mal total cost119870(119879lowast) decreases as depicted in Table 3This is also true with the fact that as 120579 increasesmeans demand of the first product decreases morebecause more demand is satisfied by its substituteproduct which is introduced after a certain interval oftime and therefore less inventories are kept for longerperiod which leads to decrease in the optimal cost

6 Managerial Implications

The goodwill of any organization is entirely dependent onits effective management and this is possible only when theproblemof the organization is analyzed properly from all per-spectives Here our problem is concerned with the effectivemanagement of products which are newly introduced in themarket and along with this the effect of second generationproducts on the demand of first generation products is alsorealized the idea included here is that the effect of substituteproducts decreases the demand of old products after a certain

Table 2 Sensitivity analysis on ldquo1198791rdquo

1198791

119879lowast

119870(119879lowast) 119876(119879

lowast)

010 059 68778 124011 061 69032 128012 063 69279 131013 064 69520 135014 066 69756 139015 067 69987 143016 069 70213 146017 07 70435 150018 072 70653 153019 073 70866 157020 075 71076 160For 120579 = 0001 119901 = 0005

Table 3 Sensitivity analysis on ldquo120579rdquo

120579 119879lowast

119870(119879lowast) 119876(119879

lowast)

0019 038 1802327 22280020 041 1649195 21800021 044 1495228 21210022 048 1340344 20500023 052 1184432 19650024 058 1027337 18650025 065 868835 17470026 073 708581 16050027 086 545998 14330028 107 379973 12170029 154 207669 918For 1198791 = 001 119901 = 003119873 = 500000

Table 4 Sensitivity analysis on coefficient of innovation ldquo119901rdquo

119901 119879lowast

119870(119879lowast) 119876(119879

lowast)

0001 076 17865 380002 054 34044 540003 044 49942 670004 038 65695 770005 034 81354 860006 031 96946 950007 029 112487 1020008 027 127987 1100009 026 143453 117001 024 158891 123002 017 312305 178Special case

interval of time because of technological innovations andother related factors Therefore to overcome this problem amathematical model has been developed to know the actualsituation of the economic ordering policies which will helpthe inventory manager to take effective action to maintainthe optimal cost The results obtained in the model create apool of knowledge on the diffusion processes of innovations

8 Advances in Decision Sciences

48625

48675

48725

48775

48825

48875

Time

Cos

t

038 042 048 052 056 06

Cost-time graphFor Tlowast

= 048 K(Tlowast) = 48684

(a)

69745

69755

69765

69775

69785

69795

Time

Cos

t

061 063 065 067 069 071

Cost-time graphFor Tlowast

= 066 K(Tlowast) = 69756

(b)

868700

868800

868900

869000

869100

869200

Time

Cos

t

061 063 065 067 069

Cost-time graphFor Tlowast

= 065 K(Tlowast) = 868835

(c)

33000

33500

34000

34500

35000

35500

36000

Time

Cos

t

024 034 044 054 064 074 084 094 104

Cost-time graph (special case)For Tlowast

= 054 K(Tlowast) = 34044

(d)

Figure 3 Cost-time graphs showing convexity of the cost functions

and its effective management which are most valuable toboth researchers and managers For managers this poolof knowledge is important because it provides useful toolsand useful information for managing and scheduling theinventories of new products when its demand is affectedby the generation of other new products which substituteproducts after a certain time period The uniqueness of thismodel is that how the inventory manager should keep theoptimal cycle time of any fixed lot size of one product enteringinto the inventory system when its substitute product entersinto the system after a fixed interval so that optimal costis maintained and the cost of inventory obsolescence isminimized

7 Conclusion

The technological breakthrough is constantly being experi-enced in various products across the globe which is neces-sary for the products to gain competitive advantage Thereare various marketing strategies to promote and establishproducts which are newly introduced into the market butkeeping its inventories efficiently and effectively play a greaterrole Here the role of inventory models becomes significantThe primary objective of developing inventory models isto take efficient action concerned with economic orderingpolicies and to understand well the behaviour of parametersassociated with it In this paper a mathematical model has

Advances in Decision Sciences 9

been developed for obtaining the Economic Order Quantity(EOQ) in which the demand of the product is assumed tofollow an innovation diffusion process as proposed by Fourtand WoodLock [2] and along with this the demand of theproduct decreases after a certain interval of time because ofthe introduction of its substitute product Here the modelanalyses the situation when demand of the previous productis affected by introducing second generation product Thereare few inventory models such as Chern et al [28] Chandaand Kumar [25] Aggarwal et al [26] and Chanda andKumar [27] which capture the sensitive nature of marketingparameters but have not incorporated the idea of the effectof introduction of second generation product This paperattempts to bridge this gap The approach of this paper is toobtain economic ordering policies of a new product whenits substitute product enters into the market after a certaintime period A special case has also been discussed whichexcludes the effect of substitute products on the inventorypolicies of old products Numerical examples with sensitivityanalyses are presented to illustrate the model properly Theresults obtained are largely consistent with what we observeempirically A simple solution procedure in the form ofalgorithm is presented to determine the optimal cycle timeand optimal order quantity of the cost functionTheproposedmodel can be extended for backorders quantity discountpartial lost sales and so forth and it can also be extendedto solve by different approaches such as geometric program-ming approach and genetic algorithm approach At the sametime future research should also focus on developing somealternative approach to obtain the optimal analytical solutionof the highly nonlinear cost function which we have solvednumerically

Acknowledgments

The research presented in this paper was carried out at theDepartment ofOperational Reseach Faculty ofMathematicalSciences University of Delhi The commercial identitiesmentioned in the paper such as LINGO and Excel-Solversoftware have only been used for the numerical analysis of themodel discussed in the paper which is only for the academicpurpose The authors of this paper do not have any directfinancial relation with the commercial identities mentionedin the paper and there is no conflict of interests regardingfinancial gains for all the authors

References

[1] E Rogers Diffusion of Innovations Free Press New York NYUSA 3rd edition 1983

[2] L A Fourt and J W WoodLock ldquoEarly prediction of Marketsuccess for new grocery productsrdquo Journal of Marketing vol 26no 2 pp 31ndash38 1960

[3] J C Fisher and R H Pry ldquoA simple substitution modelof technological changerdquo Technological Forecasting and SocialChange vol 3 no C pp 75ndash88 1971

[4] F M Bass ldquoA new product growth for model consumerdurablesrdquo Management Science vol 50 no 12 pp 1825ndash18322004

[5] R Peres E Muller and V Mahajan ldquoInnovation diffusion andnew product growth models a critical review and researchdirectionsrdquo International Journal of Research in Marketing vol27 no 2 pp 91ndash106 2010

[6] PM Parker ldquoAggregate diffusion forecastingmodels inmarket-ing a critical reviewrdquo International Journal of Forecasting vol10 no 2 pp 353ndash380 1994

[7] E M Rogers Diffusion of Innovations Free Press New YorkNY USA 4th edition 1995

[8] R Peres E Muller and V Mahajan ldquoInnovation diffusion andnew product growth models a critical review and researchdirectionsrdquo International Journal of Research in Marketing vol27 no 2 pp 91ndash106 2010

[9] V Mahajan E Muller and Y Wind New-Product DiffusionModels Kluwer Academic Publishers Boston Mass USA2000

[10] F Cheng and S P Sethi ldquoPeriodic review inventory modelwith demand influenced by promotion decisionsrdquoManagementScience vol 45 no 11 pp 1510ndash1523 1999

[11] S Kalish ldquoAnewproduct adoptionmodelwith price advertisingand uncertaintyrdquoManagement Science vol 31 no 12 pp 1569ndash1585 1985

[12] VMahajan andA P Robert ldquoInnovation diffusion in a dynamicpotential adopter populationrdquoManagement Science vol 24 no15 pp 1589ndash1597 1978

[13] M N Sharif and K Ramanathan ldquoPolynomial innovationdiffusion modelsrdquo Technological Forecasting and Social Changevol 21 no 4 pp 301ndash323 1982

[14] M N Sharif and K Ramanathan ldquoBinomial innovation dif-fusion models with dynamic potential adopter populationrdquoTechnological Forecasting and Social Change vol 20 no 1 pp63ndash87 1981

[15] J A Norton and F M Bass ldquoA diffusion theory model ofadoption and substitution for successive generation of high-technology productsrdquo Management Science vol 33 no 9 pp1069ndash1086 1987

[16] V Mahajan and E Muller ldquoTiming diffusion and substitutionof successive generations of technological innovations the IBMmainframe caserdquo Technological Forecasting and Social Changevol 51 no 2 pp 109ndash132 1996

[17] M W Speece and D L Maclachlan ldquoApplication of a multi-generation diffusion model to milk container technologyrdquoTechnological Forecasting and Social Change vol 49 no 3 pp281ndash295 1995

[18] P J Danaher B G S Hardie and W P Putsis Jr ldquoMarketing-mix variables and the diffusion of successive generations of atechnological innovationrdquo Journal of Marketing Research vol38 no 4 pp 501ndash514 2001

[19] T Islam and NMeade ldquoThe diffusion of successive generationsof a technology a more general modelrdquo Technological Forecast-ing and Social Change vol 56 no 1 pp 49ndash60 1997

[20] N Kim D R Chang and A D Shocker ldquoModeling intercat-egory and generational dynamics for a growing informationtechnology industryrdquo Management Science vol 46 no 4 pp496ndash512 2000

[21] B Bayus ldquoThe dynamic pricing of next generation consumerdurablesrdquoMarketing Science vol 11 no 3 pp 251ndash265 1992

[22] A K Bardhan and U Chanda ldquoA model for first and sub-stitution adoption of successive generations of a productrdquoInternational Journal of Modelling and Simulation vol 28 no4 pp 487ndash494 2008

10 Advances in Decision Sciences

[23] U Chanda and A K Bardhan ldquoDynamic optimal advertisingexpenditure strategies of two successive generations of con-sumer durablesrdquo in Proceedings of the International Confer-ence on Modelling Computation and Optimization held duringJanuary 2008 in the venue SQC amp OR Unit Indian StatisticalInstitute New Delhi India 2008

[24] K A Bardhan and U Chanda ldquoA mathematical model fordiffusion of products with multiple generationsrdquo InternationalJournal of Operational Research vol 4 no 2 pp 214ndash229 2009

[25] U Chanda and A Kumar ldquoEconomic order quantity modelwith demand influenced by dynamic innovation effectrdquo Interna-tional Journal of Operational Research vol 11 no 2 pp 193ndash2152011

[26] K K Aggarwal C K Jaggi and A Kumar ldquoEconomic orderquantity model with innovation diffusion criterion havingdynamic potential market sizerdquo International Journal of AppliedDecision Sciences vol 4 no 3 pp 280ndash303 2011

[27] U Chanda and A Kumar ldquoEconomic order quantity model oninflationary conditions with demand influenced by innovationdiffusion criterionrdquo International Journal of Procurement Man-agement vol 5 no 2 pp 160ndash177 2012

[28] M-S Chern J-T Teng and H-L Yang ldquoInventory lot-sizepolicies for the bass diffusion demand models of new durableproductsrdquo Journal of the Chinese Institute of Engineers vol 24no 2 pp 237ndash244 2001

[29] K K Aggarwal and A Kumar ldquoAn inventory model for newproduct when demand follows dynamic innovation processhaving dynamic potential market sizerdquo International Journal ofStrategic Decision Sciences vol 3 no 4 pp 78ndash99 2012

[30] A Kumar K K Aggarwal and U Chanda ldquoEconomic orderquantity model under fuzzy sense with demand follows Bassrsquosinnovation diffusion processrdquo International Journal of AdvancedOperations Management vol 5 no 3 pp 261ndash281 2013

[31] K K Aggarwal C K Jaggi and A Kumar ldquoEconomic orderquantity model under fuzzy sense when demand follows inno-vation diffusion process having dynamic potential market sizerdquoInternational Journal of Services and Operations Managementvol 13 no 3 pp 361ndash391 2012

[32] V Mahajan and R A PetersonModels of Innovation DiffusionSage 1985

[33] E Ruiz-Conde P S H Leeflang and J EWieringa ldquoMarketingvariables in macro-level diffusion modelsrdquo Journal fur Betrieb-swirtschaft vol 56 no 3 pp 155ndash183 2006

[34] D Chandrasekaran and G J Tellis ldquoA critical review ofmarketing research on diffusion of new productsrdquo Review ofMarketing Research pp 39ndash80 2007

[35] F Sultan J U Farley and D R Lehmann ldquoA meta-analysis ofdiffusion modelsrdquo International Journal of Forecasting vol 27no 1 pp 70ndash77 1990

[36] D Talukdar K Sudhir and A Ainslie ldquoInvestigating newproduct diffusion across products and countriesrdquo MarketingScience vol 21 no 1 pp 97ndash114 2002

[37] C van den Bulte and S Stremersch ldquoSocial contagion andincome heterogeneity in new product diffusion ameta-analytictestrdquoMarketing Science vol 23 no 4 pp 530ndash544 2004

[38] G L Lilien P Kotler and K S Moorthy Marketing ModelsPrentice-Hall of India Private New Delhi India 1999

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Advances in Decision Sciences 7

Table 1 Sensitivity analysis on coefficient of innovation ldquo119901rdquo

119901 119879lowast

119870(119879lowast) 119876(119879

lowast)

0001 086 16400 390002 059 32705 570003 048 48684 700004 041 64499 810005 037 80211 910006 034 95848 1010007 031 111430 1090008 029 126967 1170009 027 142468 124001 026 157938 131002 019 311593 190For 1198791 = 01 120579 = 00001

(b) As the cycle length 1198791 that is the time of intro-

duction of second product increases keeping otherparameters constant then the optimal cycle length119879lowastoptimal order quantity 119876(119879lowast) and the optimal totalcost 119870(119879lowast) all these three values increase as depictedin Table 2 This is true with the fact that as the timeof introduction of second product increases meansmaximum demand is satisfied by more number ofprevious that is old products and it is also obviousthat the demand is a decreasing function of timewhich implies that less demand for more time periodleads to increase in optimal cycle length And sinceoptimal cycle length increases therefore it forcesthe inventory manager to keep more number ofinventories to satisfy decreasing demand for longertime period and because more number of inventoriesare kept for longer time period it leads to increase inthe optimal cost (Figure 3)

(c) As the value of 120579 increases keeping other parametersconstant then the optimal cycle length 119879lowast increaseswhereas optimal order quantity 119876(119879lowast) and the opti-mal total cost119870(119879lowast) decreases as depicted in Table 3This is also true with the fact that as 120579 increasesmeans demand of the first product decreases morebecause more demand is satisfied by its substituteproduct which is introduced after a certain interval oftime and therefore less inventories are kept for longerperiod which leads to decrease in the optimal cost

6 Managerial Implications

The goodwill of any organization is entirely dependent onits effective management and this is possible only when theproblemof the organization is analyzed properly from all per-spectives Here our problem is concerned with the effectivemanagement of products which are newly introduced in themarket and along with this the effect of second generationproducts on the demand of first generation products is alsorealized the idea included here is that the effect of substituteproducts decreases the demand of old products after a certain

Table 2 Sensitivity analysis on ldquo1198791rdquo

1198791

119879lowast

119870(119879lowast) 119876(119879

lowast)

010 059 68778 124011 061 69032 128012 063 69279 131013 064 69520 135014 066 69756 139015 067 69987 143016 069 70213 146017 07 70435 150018 072 70653 153019 073 70866 157020 075 71076 160For 120579 = 0001 119901 = 0005

Table 3 Sensitivity analysis on ldquo120579rdquo

120579 119879lowast

119870(119879lowast) 119876(119879

lowast)

0019 038 1802327 22280020 041 1649195 21800021 044 1495228 21210022 048 1340344 20500023 052 1184432 19650024 058 1027337 18650025 065 868835 17470026 073 708581 16050027 086 545998 14330028 107 379973 12170029 154 207669 918For 1198791 = 001 119901 = 003119873 = 500000

Table 4 Sensitivity analysis on coefficient of innovation ldquo119901rdquo

119901 119879lowast

119870(119879lowast) 119876(119879

lowast)

0001 076 17865 380002 054 34044 540003 044 49942 670004 038 65695 770005 034 81354 860006 031 96946 950007 029 112487 1020008 027 127987 1100009 026 143453 117001 024 158891 123002 017 312305 178Special case

interval of time because of technological innovations andother related factors Therefore to overcome this problem amathematical model has been developed to know the actualsituation of the economic ordering policies which will helpthe inventory manager to take effective action to maintainthe optimal cost The results obtained in the model create apool of knowledge on the diffusion processes of innovations

8 Advances in Decision Sciences

48625

48675

48725

48775

48825

48875

Time

Cos

t

038 042 048 052 056 06

Cost-time graphFor Tlowast

= 048 K(Tlowast) = 48684

(a)

69745

69755

69765

69775

69785

69795

Time

Cos

t

061 063 065 067 069 071

Cost-time graphFor Tlowast

= 066 K(Tlowast) = 69756

(b)

868700

868800

868900

869000

869100

869200

Time

Cos

t

061 063 065 067 069

Cost-time graphFor Tlowast

= 065 K(Tlowast) = 868835

(c)

33000

33500

34000

34500

35000

35500

36000

Time

Cos

t

024 034 044 054 064 074 084 094 104

Cost-time graph (special case)For Tlowast

= 054 K(Tlowast) = 34044

(d)

Figure 3 Cost-time graphs showing convexity of the cost functions

and its effective management which are most valuable toboth researchers and managers For managers this poolof knowledge is important because it provides useful toolsand useful information for managing and scheduling theinventories of new products when its demand is affectedby the generation of other new products which substituteproducts after a certain time period The uniqueness of thismodel is that how the inventory manager should keep theoptimal cycle time of any fixed lot size of one product enteringinto the inventory system when its substitute product entersinto the system after a fixed interval so that optimal costis maintained and the cost of inventory obsolescence isminimized

7 Conclusion

The technological breakthrough is constantly being experi-enced in various products across the globe which is neces-sary for the products to gain competitive advantage Thereare various marketing strategies to promote and establishproducts which are newly introduced into the market butkeeping its inventories efficiently and effectively play a greaterrole Here the role of inventory models becomes significantThe primary objective of developing inventory models isto take efficient action concerned with economic orderingpolicies and to understand well the behaviour of parametersassociated with it In this paper a mathematical model has

Advances in Decision Sciences 9

been developed for obtaining the Economic Order Quantity(EOQ) in which the demand of the product is assumed tofollow an innovation diffusion process as proposed by Fourtand WoodLock [2] and along with this the demand of theproduct decreases after a certain interval of time because ofthe introduction of its substitute product Here the modelanalyses the situation when demand of the previous productis affected by introducing second generation product Thereare few inventory models such as Chern et al [28] Chandaand Kumar [25] Aggarwal et al [26] and Chanda andKumar [27] which capture the sensitive nature of marketingparameters but have not incorporated the idea of the effectof introduction of second generation product This paperattempts to bridge this gap The approach of this paper is toobtain economic ordering policies of a new product whenits substitute product enters into the market after a certaintime period A special case has also been discussed whichexcludes the effect of substitute products on the inventorypolicies of old products Numerical examples with sensitivityanalyses are presented to illustrate the model properly Theresults obtained are largely consistent with what we observeempirically A simple solution procedure in the form ofalgorithm is presented to determine the optimal cycle timeand optimal order quantity of the cost functionTheproposedmodel can be extended for backorders quantity discountpartial lost sales and so forth and it can also be extendedto solve by different approaches such as geometric program-ming approach and genetic algorithm approach At the sametime future research should also focus on developing somealternative approach to obtain the optimal analytical solutionof the highly nonlinear cost function which we have solvednumerically

Acknowledgments

The research presented in this paper was carried out at theDepartment ofOperational Reseach Faculty ofMathematicalSciences University of Delhi The commercial identitiesmentioned in the paper such as LINGO and Excel-Solversoftware have only been used for the numerical analysis of themodel discussed in the paper which is only for the academicpurpose The authors of this paper do not have any directfinancial relation with the commercial identities mentionedin the paper and there is no conflict of interests regardingfinancial gains for all the authors

References

[1] E Rogers Diffusion of Innovations Free Press New York NYUSA 3rd edition 1983

[2] L A Fourt and J W WoodLock ldquoEarly prediction of Marketsuccess for new grocery productsrdquo Journal of Marketing vol 26no 2 pp 31ndash38 1960

[3] J C Fisher and R H Pry ldquoA simple substitution modelof technological changerdquo Technological Forecasting and SocialChange vol 3 no C pp 75ndash88 1971

[4] F M Bass ldquoA new product growth for model consumerdurablesrdquo Management Science vol 50 no 12 pp 1825ndash18322004

[5] R Peres E Muller and V Mahajan ldquoInnovation diffusion andnew product growth models a critical review and researchdirectionsrdquo International Journal of Research in Marketing vol27 no 2 pp 91ndash106 2010

[6] PM Parker ldquoAggregate diffusion forecastingmodels inmarket-ing a critical reviewrdquo International Journal of Forecasting vol10 no 2 pp 353ndash380 1994

[7] E M Rogers Diffusion of Innovations Free Press New YorkNY USA 4th edition 1995

[8] R Peres E Muller and V Mahajan ldquoInnovation diffusion andnew product growth models a critical review and researchdirectionsrdquo International Journal of Research in Marketing vol27 no 2 pp 91ndash106 2010

[9] V Mahajan E Muller and Y Wind New-Product DiffusionModels Kluwer Academic Publishers Boston Mass USA2000

[10] F Cheng and S P Sethi ldquoPeriodic review inventory modelwith demand influenced by promotion decisionsrdquoManagementScience vol 45 no 11 pp 1510ndash1523 1999

[11] S Kalish ldquoAnewproduct adoptionmodelwith price advertisingand uncertaintyrdquoManagement Science vol 31 no 12 pp 1569ndash1585 1985

[12] VMahajan andA P Robert ldquoInnovation diffusion in a dynamicpotential adopter populationrdquoManagement Science vol 24 no15 pp 1589ndash1597 1978

[13] M N Sharif and K Ramanathan ldquoPolynomial innovationdiffusion modelsrdquo Technological Forecasting and Social Changevol 21 no 4 pp 301ndash323 1982

[14] M N Sharif and K Ramanathan ldquoBinomial innovation dif-fusion models with dynamic potential adopter populationrdquoTechnological Forecasting and Social Change vol 20 no 1 pp63ndash87 1981

[15] J A Norton and F M Bass ldquoA diffusion theory model ofadoption and substitution for successive generation of high-technology productsrdquo Management Science vol 33 no 9 pp1069ndash1086 1987

[16] V Mahajan and E Muller ldquoTiming diffusion and substitutionof successive generations of technological innovations the IBMmainframe caserdquo Technological Forecasting and Social Changevol 51 no 2 pp 109ndash132 1996

[17] M W Speece and D L Maclachlan ldquoApplication of a multi-generation diffusion model to milk container technologyrdquoTechnological Forecasting and Social Change vol 49 no 3 pp281ndash295 1995

[18] P J Danaher B G S Hardie and W P Putsis Jr ldquoMarketing-mix variables and the diffusion of successive generations of atechnological innovationrdquo Journal of Marketing Research vol38 no 4 pp 501ndash514 2001

[19] T Islam and NMeade ldquoThe diffusion of successive generationsof a technology a more general modelrdquo Technological Forecast-ing and Social Change vol 56 no 1 pp 49ndash60 1997

[20] N Kim D R Chang and A D Shocker ldquoModeling intercat-egory and generational dynamics for a growing informationtechnology industryrdquo Management Science vol 46 no 4 pp496ndash512 2000

[21] B Bayus ldquoThe dynamic pricing of next generation consumerdurablesrdquoMarketing Science vol 11 no 3 pp 251ndash265 1992

[22] A K Bardhan and U Chanda ldquoA model for first and sub-stitution adoption of successive generations of a productrdquoInternational Journal of Modelling and Simulation vol 28 no4 pp 487ndash494 2008

10 Advances in Decision Sciences

[23] U Chanda and A K Bardhan ldquoDynamic optimal advertisingexpenditure strategies of two successive generations of con-sumer durablesrdquo in Proceedings of the International Confer-ence on Modelling Computation and Optimization held duringJanuary 2008 in the venue SQC amp OR Unit Indian StatisticalInstitute New Delhi India 2008

[24] K A Bardhan and U Chanda ldquoA mathematical model fordiffusion of products with multiple generationsrdquo InternationalJournal of Operational Research vol 4 no 2 pp 214ndash229 2009

[25] U Chanda and A Kumar ldquoEconomic order quantity modelwith demand influenced by dynamic innovation effectrdquo Interna-tional Journal of Operational Research vol 11 no 2 pp 193ndash2152011

[26] K K Aggarwal C K Jaggi and A Kumar ldquoEconomic orderquantity model with innovation diffusion criterion havingdynamic potential market sizerdquo International Journal of AppliedDecision Sciences vol 4 no 3 pp 280ndash303 2011

[27] U Chanda and A Kumar ldquoEconomic order quantity model oninflationary conditions with demand influenced by innovationdiffusion criterionrdquo International Journal of Procurement Man-agement vol 5 no 2 pp 160ndash177 2012

[28] M-S Chern J-T Teng and H-L Yang ldquoInventory lot-sizepolicies for the bass diffusion demand models of new durableproductsrdquo Journal of the Chinese Institute of Engineers vol 24no 2 pp 237ndash244 2001

[29] K K Aggarwal and A Kumar ldquoAn inventory model for newproduct when demand follows dynamic innovation processhaving dynamic potential market sizerdquo International Journal ofStrategic Decision Sciences vol 3 no 4 pp 78ndash99 2012

[30] A Kumar K K Aggarwal and U Chanda ldquoEconomic orderquantity model under fuzzy sense with demand follows Bassrsquosinnovation diffusion processrdquo International Journal of AdvancedOperations Management vol 5 no 3 pp 261ndash281 2013

[31] K K Aggarwal C K Jaggi and A Kumar ldquoEconomic orderquantity model under fuzzy sense when demand follows inno-vation diffusion process having dynamic potential market sizerdquoInternational Journal of Services and Operations Managementvol 13 no 3 pp 361ndash391 2012

[32] V Mahajan and R A PetersonModels of Innovation DiffusionSage 1985

[33] E Ruiz-Conde P S H Leeflang and J EWieringa ldquoMarketingvariables in macro-level diffusion modelsrdquo Journal fur Betrieb-swirtschaft vol 56 no 3 pp 155ndash183 2006

[34] D Chandrasekaran and G J Tellis ldquoA critical review ofmarketing research on diffusion of new productsrdquo Review ofMarketing Research pp 39ndash80 2007

[35] F Sultan J U Farley and D R Lehmann ldquoA meta-analysis ofdiffusion modelsrdquo International Journal of Forecasting vol 27no 1 pp 70ndash77 1990

[36] D Talukdar K Sudhir and A Ainslie ldquoInvestigating newproduct diffusion across products and countriesrdquo MarketingScience vol 21 no 1 pp 97ndash114 2002

[37] C van den Bulte and S Stremersch ldquoSocial contagion andincome heterogeneity in new product diffusion ameta-analytictestrdquoMarketing Science vol 23 no 4 pp 530ndash544 2004

[38] G L Lilien P Kotler and K S Moorthy Marketing ModelsPrentice-Hall of India Private New Delhi India 1999

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Advances in Decision Sciences

48625

48675

48725

48775

48825

48875

Time

Cos

t

038 042 048 052 056 06

Cost-time graphFor Tlowast

= 048 K(Tlowast) = 48684

(a)

69745

69755

69765

69775

69785

69795

Time

Cos

t

061 063 065 067 069 071

Cost-time graphFor Tlowast

= 066 K(Tlowast) = 69756

(b)

868700

868800

868900

869000

869100

869200

Time

Cos

t

061 063 065 067 069

Cost-time graphFor Tlowast

= 065 K(Tlowast) = 868835

(c)

33000

33500

34000

34500

35000

35500

36000

Time

Cos

t

024 034 044 054 064 074 084 094 104

Cost-time graph (special case)For Tlowast

= 054 K(Tlowast) = 34044

(d)

Figure 3 Cost-time graphs showing convexity of the cost functions

and its effective management which are most valuable toboth researchers and managers For managers this poolof knowledge is important because it provides useful toolsand useful information for managing and scheduling theinventories of new products when its demand is affectedby the generation of other new products which substituteproducts after a certain time period The uniqueness of thismodel is that how the inventory manager should keep theoptimal cycle time of any fixed lot size of one product enteringinto the inventory system when its substitute product entersinto the system after a fixed interval so that optimal costis maintained and the cost of inventory obsolescence isminimized

7 Conclusion

The technological breakthrough is constantly being experi-enced in various products across the globe which is neces-sary for the products to gain competitive advantage Thereare various marketing strategies to promote and establishproducts which are newly introduced into the market butkeeping its inventories efficiently and effectively play a greaterrole Here the role of inventory models becomes significantThe primary objective of developing inventory models isto take efficient action concerned with economic orderingpolicies and to understand well the behaviour of parametersassociated with it In this paper a mathematical model has

Advances in Decision Sciences 9

been developed for obtaining the Economic Order Quantity(EOQ) in which the demand of the product is assumed tofollow an innovation diffusion process as proposed by Fourtand WoodLock [2] and along with this the demand of theproduct decreases after a certain interval of time because ofthe introduction of its substitute product Here the modelanalyses the situation when demand of the previous productis affected by introducing second generation product Thereare few inventory models such as Chern et al [28] Chandaand Kumar [25] Aggarwal et al [26] and Chanda andKumar [27] which capture the sensitive nature of marketingparameters but have not incorporated the idea of the effectof introduction of second generation product This paperattempts to bridge this gap The approach of this paper is toobtain economic ordering policies of a new product whenits substitute product enters into the market after a certaintime period A special case has also been discussed whichexcludes the effect of substitute products on the inventorypolicies of old products Numerical examples with sensitivityanalyses are presented to illustrate the model properly Theresults obtained are largely consistent with what we observeempirically A simple solution procedure in the form ofalgorithm is presented to determine the optimal cycle timeand optimal order quantity of the cost functionTheproposedmodel can be extended for backorders quantity discountpartial lost sales and so forth and it can also be extendedto solve by different approaches such as geometric program-ming approach and genetic algorithm approach At the sametime future research should also focus on developing somealternative approach to obtain the optimal analytical solutionof the highly nonlinear cost function which we have solvednumerically

Acknowledgments

The research presented in this paper was carried out at theDepartment ofOperational Reseach Faculty ofMathematicalSciences University of Delhi The commercial identitiesmentioned in the paper such as LINGO and Excel-Solversoftware have only been used for the numerical analysis of themodel discussed in the paper which is only for the academicpurpose The authors of this paper do not have any directfinancial relation with the commercial identities mentionedin the paper and there is no conflict of interests regardingfinancial gains for all the authors

References

[1] E Rogers Diffusion of Innovations Free Press New York NYUSA 3rd edition 1983

[2] L A Fourt and J W WoodLock ldquoEarly prediction of Marketsuccess for new grocery productsrdquo Journal of Marketing vol 26no 2 pp 31ndash38 1960

[3] J C Fisher and R H Pry ldquoA simple substitution modelof technological changerdquo Technological Forecasting and SocialChange vol 3 no C pp 75ndash88 1971

[4] F M Bass ldquoA new product growth for model consumerdurablesrdquo Management Science vol 50 no 12 pp 1825ndash18322004

[5] R Peres E Muller and V Mahajan ldquoInnovation diffusion andnew product growth models a critical review and researchdirectionsrdquo International Journal of Research in Marketing vol27 no 2 pp 91ndash106 2010

[6] PM Parker ldquoAggregate diffusion forecastingmodels inmarket-ing a critical reviewrdquo International Journal of Forecasting vol10 no 2 pp 353ndash380 1994

[7] E M Rogers Diffusion of Innovations Free Press New YorkNY USA 4th edition 1995

[8] R Peres E Muller and V Mahajan ldquoInnovation diffusion andnew product growth models a critical review and researchdirectionsrdquo International Journal of Research in Marketing vol27 no 2 pp 91ndash106 2010

[9] V Mahajan E Muller and Y Wind New-Product DiffusionModels Kluwer Academic Publishers Boston Mass USA2000

[10] F Cheng and S P Sethi ldquoPeriodic review inventory modelwith demand influenced by promotion decisionsrdquoManagementScience vol 45 no 11 pp 1510ndash1523 1999

[11] S Kalish ldquoAnewproduct adoptionmodelwith price advertisingand uncertaintyrdquoManagement Science vol 31 no 12 pp 1569ndash1585 1985

[12] VMahajan andA P Robert ldquoInnovation diffusion in a dynamicpotential adopter populationrdquoManagement Science vol 24 no15 pp 1589ndash1597 1978

[13] M N Sharif and K Ramanathan ldquoPolynomial innovationdiffusion modelsrdquo Technological Forecasting and Social Changevol 21 no 4 pp 301ndash323 1982

[14] M N Sharif and K Ramanathan ldquoBinomial innovation dif-fusion models with dynamic potential adopter populationrdquoTechnological Forecasting and Social Change vol 20 no 1 pp63ndash87 1981

[15] J A Norton and F M Bass ldquoA diffusion theory model ofadoption and substitution for successive generation of high-technology productsrdquo Management Science vol 33 no 9 pp1069ndash1086 1987

[16] V Mahajan and E Muller ldquoTiming diffusion and substitutionof successive generations of technological innovations the IBMmainframe caserdquo Technological Forecasting and Social Changevol 51 no 2 pp 109ndash132 1996

[17] M W Speece and D L Maclachlan ldquoApplication of a multi-generation diffusion model to milk container technologyrdquoTechnological Forecasting and Social Change vol 49 no 3 pp281ndash295 1995

[18] P J Danaher B G S Hardie and W P Putsis Jr ldquoMarketing-mix variables and the diffusion of successive generations of atechnological innovationrdquo Journal of Marketing Research vol38 no 4 pp 501ndash514 2001

[19] T Islam and NMeade ldquoThe diffusion of successive generationsof a technology a more general modelrdquo Technological Forecast-ing and Social Change vol 56 no 1 pp 49ndash60 1997

[20] N Kim D R Chang and A D Shocker ldquoModeling intercat-egory and generational dynamics for a growing informationtechnology industryrdquo Management Science vol 46 no 4 pp496ndash512 2000

[21] B Bayus ldquoThe dynamic pricing of next generation consumerdurablesrdquoMarketing Science vol 11 no 3 pp 251ndash265 1992

[22] A K Bardhan and U Chanda ldquoA model for first and sub-stitution adoption of successive generations of a productrdquoInternational Journal of Modelling and Simulation vol 28 no4 pp 487ndash494 2008

10 Advances in Decision Sciences

[23] U Chanda and A K Bardhan ldquoDynamic optimal advertisingexpenditure strategies of two successive generations of con-sumer durablesrdquo in Proceedings of the International Confer-ence on Modelling Computation and Optimization held duringJanuary 2008 in the venue SQC amp OR Unit Indian StatisticalInstitute New Delhi India 2008

[24] K A Bardhan and U Chanda ldquoA mathematical model fordiffusion of products with multiple generationsrdquo InternationalJournal of Operational Research vol 4 no 2 pp 214ndash229 2009

[25] U Chanda and A Kumar ldquoEconomic order quantity modelwith demand influenced by dynamic innovation effectrdquo Interna-tional Journal of Operational Research vol 11 no 2 pp 193ndash2152011

[26] K K Aggarwal C K Jaggi and A Kumar ldquoEconomic orderquantity model with innovation diffusion criterion havingdynamic potential market sizerdquo International Journal of AppliedDecision Sciences vol 4 no 3 pp 280ndash303 2011

[27] U Chanda and A Kumar ldquoEconomic order quantity model oninflationary conditions with demand influenced by innovationdiffusion criterionrdquo International Journal of Procurement Man-agement vol 5 no 2 pp 160ndash177 2012

[28] M-S Chern J-T Teng and H-L Yang ldquoInventory lot-sizepolicies for the bass diffusion demand models of new durableproductsrdquo Journal of the Chinese Institute of Engineers vol 24no 2 pp 237ndash244 2001

[29] K K Aggarwal and A Kumar ldquoAn inventory model for newproduct when demand follows dynamic innovation processhaving dynamic potential market sizerdquo International Journal ofStrategic Decision Sciences vol 3 no 4 pp 78ndash99 2012

[30] A Kumar K K Aggarwal and U Chanda ldquoEconomic orderquantity model under fuzzy sense with demand follows Bassrsquosinnovation diffusion processrdquo International Journal of AdvancedOperations Management vol 5 no 3 pp 261ndash281 2013

[31] K K Aggarwal C K Jaggi and A Kumar ldquoEconomic orderquantity model under fuzzy sense when demand follows inno-vation diffusion process having dynamic potential market sizerdquoInternational Journal of Services and Operations Managementvol 13 no 3 pp 361ndash391 2012

[32] V Mahajan and R A PetersonModels of Innovation DiffusionSage 1985

[33] E Ruiz-Conde P S H Leeflang and J EWieringa ldquoMarketingvariables in macro-level diffusion modelsrdquo Journal fur Betrieb-swirtschaft vol 56 no 3 pp 155ndash183 2006

[34] D Chandrasekaran and G J Tellis ldquoA critical review ofmarketing research on diffusion of new productsrdquo Review ofMarketing Research pp 39ndash80 2007

[35] F Sultan J U Farley and D R Lehmann ldquoA meta-analysis ofdiffusion modelsrdquo International Journal of Forecasting vol 27no 1 pp 70ndash77 1990

[36] D Talukdar K Sudhir and A Ainslie ldquoInvestigating newproduct diffusion across products and countriesrdquo MarketingScience vol 21 no 1 pp 97ndash114 2002

[37] C van den Bulte and S Stremersch ldquoSocial contagion andincome heterogeneity in new product diffusion ameta-analytictestrdquoMarketing Science vol 23 no 4 pp 530ndash544 2004

[38] G L Lilien P Kotler and K S Moorthy Marketing ModelsPrentice-Hall of India Private New Delhi India 1999

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Advances in Decision Sciences 9

been developed for obtaining the Economic Order Quantity(EOQ) in which the demand of the product is assumed tofollow an innovation diffusion process as proposed by Fourtand WoodLock [2] and along with this the demand of theproduct decreases after a certain interval of time because ofthe introduction of its substitute product Here the modelanalyses the situation when demand of the previous productis affected by introducing second generation product Thereare few inventory models such as Chern et al [28] Chandaand Kumar [25] Aggarwal et al [26] and Chanda andKumar [27] which capture the sensitive nature of marketingparameters but have not incorporated the idea of the effectof introduction of second generation product This paperattempts to bridge this gap The approach of this paper is toobtain economic ordering policies of a new product whenits substitute product enters into the market after a certaintime period A special case has also been discussed whichexcludes the effect of substitute products on the inventorypolicies of old products Numerical examples with sensitivityanalyses are presented to illustrate the model properly Theresults obtained are largely consistent with what we observeempirically A simple solution procedure in the form ofalgorithm is presented to determine the optimal cycle timeand optimal order quantity of the cost functionTheproposedmodel can be extended for backorders quantity discountpartial lost sales and so forth and it can also be extendedto solve by different approaches such as geometric program-ming approach and genetic algorithm approach At the sametime future research should also focus on developing somealternative approach to obtain the optimal analytical solutionof the highly nonlinear cost function which we have solvednumerically

Acknowledgments

The research presented in this paper was carried out at theDepartment ofOperational Reseach Faculty ofMathematicalSciences University of Delhi The commercial identitiesmentioned in the paper such as LINGO and Excel-Solversoftware have only been used for the numerical analysis of themodel discussed in the paper which is only for the academicpurpose The authors of this paper do not have any directfinancial relation with the commercial identities mentionedin the paper and there is no conflict of interests regardingfinancial gains for all the authors

References

[1] E Rogers Diffusion of Innovations Free Press New York NYUSA 3rd edition 1983

[2] L A Fourt and J W WoodLock ldquoEarly prediction of Marketsuccess for new grocery productsrdquo Journal of Marketing vol 26no 2 pp 31ndash38 1960

[3] J C Fisher and R H Pry ldquoA simple substitution modelof technological changerdquo Technological Forecasting and SocialChange vol 3 no C pp 75ndash88 1971

[4] F M Bass ldquoA new product growth for model consumerdurablesrdquo Management Science vol 50 no 12 pp 1825ndash18322004

[5] R Peres E Muller and V Mahajan ldquoInnovation diffusion andnew product growth models a critical review and researchdirectionsrdquo International Journal of Research in Marketing vol27 no 2 pp 91ndash106 2010

[6] PM Parker ldquoAggregate diffusion forecastingmodels inmarket-ing a critical reviewrdquo International Journal of Forecasting vol10 no 2 pp 353ndash380 1994

[7] E M Rogers Diffusion of Innovations Free Press New YorkNY USA 4th edition 1995

[8] R Peres E Muller and V Mahajan ldquoInnovation diffusion andnew product growth models a critical review and researchdirectionsrdquo International Journal of Research in Marketing vol27 no 2 pp 91ndash106 2010

[9] V Mahajan E Muller and Y Wind New-Product DiffusionModels Kluwer Academic Publishers Boston Mass USA2000

[10] F Cheng and S P Sethi ldquoPeriodic review inventory modelwith demand influenced by promotion decisionsrdquoManagementScience vol 45 no 11 pp 1510ndash1523 1999

[11] S Kalish ldquoAnewproduct adoptionmodelwith price advertisingand uncertaintyrdquoManagement Science vol 31 no 12 pp 1569ndash1585 1985

[12] VMahajan andA P Robert ldquoInnovation diffusion in a dynamicpotential adopter populationrdquoManagement Science vol 24 no15 pp 1589ndash1597 1978

[13] M N Sharif and K Ramanathan ldquoPolynomial innovationdiffusion modelsrdquo Technological Forecasting and Social Changevol 21 no 4 pp 301ndash323 1982

[14] M N Sharif and K Ramanathan ldquoBinomial innovation dif-fusion models with dynamic potential adopter populationrdquoTechnological Forecasting and Social Change vol 20 no 1 pp63ndash87 1981

[15] J A Norton and F M Bass ldquoA diffusion theory model ofadoption and substitution for successive generation of high-technology productsrdquo Management Science vol 33 no 9 pp1069ndash1086 1987

[16] V Mahajan and E Muller ldquoTiming diffusion and substitutionof successive generations of technological innovations the IBMmainframe caserdquo Technological Forecasting and Social Changevol 51 no 2 pp 109ndash132 1996

[17] M W Speece and D L Maclachlan ldquoApplication of a multi-generation diffusion model to milk container technologyrdquoTechnological Forecasting and Social Change vol 49 no 3 pp281ndash295 1995

[18] P J Danaher B G S Hardie and W P Putsis Jr ldquoMarketing-mix variables and the diffusion of successive generations of atechnological innovationrdquo Journal of Marketing Research vol38 no 4 pp 501ndash514 2001

[19] T Islam and NMeade ldquoThe diffusion of successive generationsof a technology a more general modelrdquo Technological Forecast-ing and Social Change vol 56 no 1 pp 49ndash60 1997

[20] N Kim D R Chang and A D Shocker ldquoModeling intercat-egory and generational dynamics for a growing informationtechnology industryrdquo Management Science vol 46 no 4 pp496ndash512 2000

[21] B Bayus ldquoThe dynamic pricing of next generation consumerdurablesrdquoMarketing Science vol 11 no 3 pp 251ndash265 1992

[22] A K Bardhan and U Chanda ldquoA model for first and sub-stitution adoption of successive generations of a productrdquoInternational Journal of Modelling and Simulation vol 28 no4 pp 487ndash494 2008

10 Advances in Decision Sciences

[23] U Chanda and A K Bardhan ldquoDynamic optimal advertisingexpenditure strategies of two successive generations of con-sumer durablesrdquo in Proceedings of the International Confer-ence on Modelling Computation and Optimization held duringJanuary 2008 in the venue SQC amp OR Unit Indian StatisticalInstitute New Delhi India 2008

[24] K A Bardhan and U Chanda ldquoA mathematical model fordiffusion of products with multiple generationsrdquo InternationalJournal of Operational Research vol 4 no 2 pp 214ndash229 2009

[25] U Chanda and A Kumar ldquoEconomic order quantity modelwith demand influenced by dynamic innovation effectrdquo Interna-tional Journal of Operational Research vol 11 no 2 pp 193ndash2152011

[26] K K Aggarwal C K Jaggi and A Kumar ldquoEconomic orderquantity model with innovation diffusion criterion havingdynamic potential market sizerdquo International Journal of AppliedDecision Sciences vol 4 no 3 pp 280ndash303 2011

[27] U Chanda and A Kumar ldquoEconomic order quantity model oninflationary conditions with demand influenced by innovationdiffusion criterionrdquo International Journal of Procurement Man-agement vol 5 no 2 pp 160ndash177 2012

[28] M-S Chern J-T Teng and H-L Yang ldquoInventory lot-sizepolicies for the bass diffusion demand models of new durableproductsrdquo Journal of the Chinese Institute of Engineers vol 24no 2 pp 237ndash244 2001

[29] K K Aggarwal and A Kumar ldquoAn inventory model for newproduct when demand follows dynamic innovation processhaving dynamic potential market sizerdquo International Journal ofStrategic Decision Sciences vol 3 no 4 pp 78ndash99 2012

[30] A Kumar K K Aggarwal and U Chanda ldquoEconomic orderquantity model under fuzzy sense with demand follows Bassrsquosinnovation diffusion processrdquo International Journal of AdvancedOperations Management vol 5 no 3 pp 261ndash281 2013

[31] K K Aggarwal C K Jaggi and A Kumar ldquoEconomic orderquantity model under fuzzy sense when demand follows inno-vation diffusion process having dynamic potential market sizerdquoInternational Journal of Services and Operations Managementvol 13 no 3 pp 361ndash391 2012

[32] V Mahajan and R A PetersonModels of Innovation DiffusionSage 1985

[33] E Ruiz-Conde P S H Leeflang and J EWieringa ldquoMarketingvariables in macro-level diffusion modelsrdquo Journal fur Betrieb-swirtschaft vol 56 no 3 pp 155ndash183 2006

[34] D Chandrasekaran and G J Tellis ldquoA critical review ofmarketing research on diffusion of new productsrdquo Review ofMarketing Research pp 39ndash80 2007

[35] F Sultan J U Farley and D R Lehmann ldquoA meta-analysis ofdiffusion modelsrdquo International Journal of Forecasting vol 27no 1 pp 70ndash77 1990

[36] D Talukdar K Sudhir and A Ainslie ldquoInvestigating newproduct diffusion across products and countriesrdquo MarketingScience vol 21 no 1 pp 97ndash114 2002

[37] C van den Bulte and S Stremersch ldquoSocial contagion andincome heterogeneity in new product diffusion ameta-analytictestrdquoMarketing Science vol 23 no 4 pp 530ndash544 2004

[38] G L Lilien P Kotler and K S Moorthy Marketing ModelsPrentice-Hall of India Private New Delhi India 1999

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Advances in Decision Sciences

[23] U Chanda and A K Bardhan ldquoDynamic optimal advertisingexpenditure strategies of two successive generations of con-sumer durablesrdquo in Proceedings of the International Confer-ence on Modelling Computation and Optimization held duringJanuary 2008 in the venue SQC amp OR Unit Indian StatisticalInstitute New Delhi India 2008

[24] K A Bardhan and U Chanda ldquoA mathematical model fordiffusion of products with multiple generationsrdquo InternationalJournal of Operational Research vol 4 no 2 pp 214ndash229 2009

[25] U Chanda and A Kumar ldquoEconomic order quantity modelwith demand influenced by dynamic innovation effectrdquo Interna-tional Journal of Operational Research vol 11 no 2 pp 193ndash2152011

[26] K K Aggarwal C K Jaggi and A Kumar ldquoEconomic orderquantity model with innovation diffusion criterion havingdynamic potential market sizerdquo International Journal of AppliedDecision Sciences vol 4 no 3 pp 280ndash303 2011

[27] U Chanda and A Kumar ldquoEconomic order quantity model oninflationary conditions with demand influenced by innovationdiffusion criterionrdquo International Journal of Procurement Man-agement vol 5 no 2 pp 160ndash177 2012

[28] M-S Chern J-T Teng and H-L Yang ldquoInventory lot-sizepolicies for the bass diffusion demand models of new durableproductsrdquo Journal of the Chinese Institute of Engineers vol 24no 2 pp 237ndash244 2001

[29] K K Aggarwal and A Kumar ldquoAn inventory model for newproduct when demand follows dynamic innovation processhaving dynamic potential market sizerdquo International Journal ofStrategic Decision Sciences vol 3 no 4 pp 78ndash99 2012

[30] A Kumar K K Aggarwal and U Chanda ldquoEconomic orderquantity model under fuzzy sense with demand follows Bassrsquosinnovation diffusion processrdquo International Journal of AdvancedOperations Management vol 5 no 3 pp 261ndash281 2013

[31] K K Aggarwal C K Jaggi and A Kumar ldquoEconomic orderquantity model under fuzzy sense when demand follows inno-vation diffusion process having dynamic potential market sizerdquoInternational Journal of Services and Operations Managementvol 13 no 3 pp 361ndash391 2012

[32] V Mahajan and R A PetersonModels of Innovation DiffusionSage 1985

[33] E Ruiz-Conde P S H Leeflang and J EWieringa ldquoMarketingvariables in macro-level diffusion modelsrdquo Journal fur Betrieb-swirtschaft vol 56 no 3 pp 155ndash183 2006

[34] D Chandrasekaran and G J Tellis ldquoA critical review ofmarketing research on diffusion of new productsrdquo Review ofMarketing Research pp 39ndash80 2007

[35] F Sultan J U Farley and D R Lehmann ldquoA meta-analysis ofdiffusion modelsrdquo International Journal of Forecasting vol 27no 1 pp 70ndash77 1990

[36] D Talukdar K Sudhir and A Ainslie ldquoInvestigating newproduct diffusion across products and countriesrdquo MarketingScience vol 21 no 1 pp 97ndash114 2002

[37] C van den Bulte and S Stremersch ldquoSocial contagion andincome heterogeneity in new product diffusion ameta-analytictestrdquoMarketing Science vol 23 no 4 pp 530ndash544 2004

[38] G L Lilien P Kotler and K S Moorthy Marketing ModelsPrentice-Hall of India Private New Delhi India 1999

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of