BuR -- Business Research Volume6|Issue1|May2013 | 55--75 A ... · Short-Term Production Planning Problem in Closed-Loop Supply Chains Florian Sahling, Institute of Production Management,

BuR -- Business ResearchOfficial Open Access Journal of VHBGerman Academic Association for Business Research (VHB)Volume 6 | Issue 1 | May 2013 | 55--75

A Column-Generation Approach for aShort-Term Production Planning Problem inClosed-Loop Supply ChainsFlorian Sahling, Institute of Production Management, Leibniz Universität Hannover, Germany, E-mail: [email protected]

AbstractWe present a new model formulation for a multi-product lot-sizing problem with product returns andremanufacturing subject to a capacity constraint. The given external demand of the products has tobe satisfied by remanufactured or newly produced goods. The objective is to determine a feasibleproduction plan, which minimizes production, holding, and setup costs. As the LP relaxation of amodel formulation based on the well-known CLSP leads to very poor lower bounds, we propose acolumn-generation approach to determine tighter bounds. The lower bound obtained by columngeneration can be easily transferred into a feasible solution by a truncated branch-and-boundapproach using CPLEX. The results of an extensive numerical study show the high solution quality ofthe proposed solution approach.

JEL-classification: C61, M11

Keywords: closed-loop supply chains, remanufacturing, column-generation

Manuscript received April 12, 2012, accepted by Karl Inderfurth (Operations and InformationSystems), October 24, 2012.

1 IntroductionIn the last decade, many researchers havebeen attracted by production planning aspectsin closed-loop supply chains (Fleischmann,Bloemhof Ruwaard, Dekker, van der Laan,van Nunen, and van Wassenhove 1997; Guideand Van Wassenhove 2001; Rubio, Chamorro,and Miranda 2008; Sbihi and Eglese 2010). Inaddition to forward-oriented supply chains fromthe manufacturer to the customer, the reversedirection also has to be taken into account in thefield of closed-loop supply chains. This meansthat customers return old and used products tothe manufacturer at the product’s end-of-use.Well-known examples are printer cartridges andsingle-use cameras (Akcali and Cetinkaya 2011:2376).These returned products can be remanufactured tobecomeproducts, which are as good as new. There-fore, remanufactured products can also be used tofulfill customers’ demand. Returned products are

denoted as recoverables, i.e., these items have tobe remanufactured. After (re)manufacturing prod-ucts are denoted as serviceables. The products canbe (re)manufactured on a capacity-restricted pro-duction system. For (re)manufacturing products asetup is necessary,which causes setup costs and/orsetup times. Hence, a multi-product capacitatedlot-sizing problem arises, as the respective prod-ucts compete for the production system’s scarcecapacity. Most of the established model formu-lations neglect either the capacity constraints orthe interactions of multiple products. Therefore, anew model formulation for the multi-product ca-pacitated lot-sizing problem with product returnsand remanufacturing (CLSP-RM) is presented.Furthermore, a solution approach by combiningcolumn generation and a truncated branch-and-bound approach is proposed.The remainder of this paper is organized as fol-lows: In section 2 we provide a literature review oflot-sizing problems in closed-loop supply chains.Afterwards, we present a new model formulation

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for a capacitated lot-sizing problem with remanu-facturing in section 3. In this section, we alsointroduce valid inequalities to tighten the lowerbounds of the LP relaxation. In section 4 a column-generation approach is proposed to determine alower bound close to the optimal solution. To gen-erate a feasible production plan based on this lowerbound, a truncated branch-and-bound method isapplied. Numerical results are reported in sec-tion 5. The paper ends with a summary and anoutlook for further research in section 6.

2 Literature ReviewReviews of the literature on dynamic lot sizing ingeneral are given, e.g., by Karimi, Fatemi Ghomi,and Wilson (2003), Quadt and Kuhn (2008) andBuschkühl, Sahling, Helber, and Tempelmeier(2010). In this section, we focus only on dynamiclot-sizing problems with product returns andremanufacturing. However, the literature onlot-sizing problems in closed-loop supply chainsis rather limited.One of the first papers on dynamic lot-sizing prob-lems in closed-loop supply chainswas presented byRichter and Sombrutzki (2000). They suggested amodel formulation for the reverse Wagner-Whitinproblem (Wagner and Whitin 1958) by assumingthat the demand can be totally satisfied by reman-ufacturing returned products. Richter and Weber(2001) extended the reverseWagner-Whitin prob-lem by introducing variable manufacturing andremanufacturing costs. Richter and Sombrutzki(2000) aswell as Richter andWeber (2001) provedthat the reverse Wagner-Whitin problem can usu-ally be solved in polynomial time.Teunter, Bayindir, and van den Heuvel (2006)described model formulations for the single-leveluncapacitated lot-sizing problem with remanufac-turing (SLULSP-RM) based on theWagner-Whitinproblem. Two cases are distinguished: In the firstcase, the products are manufactured and remanu-factured on the same (uncapacitated) productionsystem and only one joint setup is necessary forboth remanufacturing and manufacturing. In thesecond case, it is assumed that manufacturing andremanufacturing take place on different (uncapac-itated) production systems. Thus, a separate setupis required on each production system.Teunter, Bayindir, and van den Heuvel (2006)proved that in the case of joint setups the SLULSP-

RM can be solved in polynomial time using anadapted version of Wagner and Whitin’s dynamicprogramming approach. Furthermore, van denHeuvel (2004) showed that the SLULSP-RM isNP-hard in the case of separate setups.To solve both cases of the SLULSP-RM, Teunter,Bayindir, and vandenHeuvel (2006) adaptedwell-known heuristics for the Wagner-Whitin problem,namely the least-unit cost approach, the Silver-Meal heuristic and the part-period-balancing ap-proach. They pointed out that the adapted Silver-Meal heuristic outperforms the other approaches.Schulz (2011) introduced further variants of theSilver-Meal heuristic to solve the SLULSP-RMwith separate setup costs. Furthermore, he pro-posed improvement steps yielding a higher solu-tion quality than the Silver-Meal heuristic alone.To strengthen the lower bound of the LP relax-ation of the SLULSP-RM, Retel Helmrich, Jans,van den Heuvel, and Wagelmans (2010) adaptedvalid inequalities proposed by Barany, van Roy,andWolsey (1984) for theWagner-Whitinproblemand present reformulations based on the shortest-path problem for the SLULSP-RM.Quariguasi Frota Neto, Walther, Bloemhof, vanNunen, and Spengler (2009) integrated sustain-able aspects into the SLULSP-RM by taking the re-quired energy for (re)manufacturing into account.TheSLULSP-RMwith joint setupswas extendedbySchwarz, Buscher, andRudert (2009) and Schwarz(2010) by allowing disposals of the returned prod-ucts. The dynamic programming approach by Te-unter, Bayindir, and van den Heuvel (2006) wasadapted to solve the SLULSP-RM with disposalswithout an increaseof thenumerical effort. Pineyroand Viera (2010) also considered the possibility todispose returned products. A tabu search heuris-tic was suggested to solve this kind of lot-sizingproblem. Li, Chen, and Cai (2006) presented anuncapacitated multi-product lot-sizing problem.However, the interactions of these products bycompeting for the scarce capacity are neglected.To the best of our knowledge, only a few papers ad-dressed capacitated lot-sizing problemswith prod-uct returns and remanufacturing. Li, Chen, andCai (2007) described a model formulation for alot-sizing problem with product substitution, i.e.,the remanufactured product serves as a substi-tute for the serviceable. The production capacityis limited, while setup times are neglected. A ge-netic algorithm is applied to determine a produc-

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Figure 1: Dynamic Capacitated Lot-Sizing with Product Returns and Remanufacturing(Schulz 2011: 2521)

rktY

Customers

Remanufacturing

Manufacturing

Serviceablesinventory

ktY

Recoverablesinventory

rktQ

ktQ ktd

ktr

tion plan. Pan, Tang, and Liu (2009) investigateda single-product capacitated lot-sizing problem.Here, the number of produced, remanufacturedand disposed items are restricted in each period.Akcali and Cetinkaya (2011) pointed out that mostof the lot-sizing model formulations with prod-uct returns and remanufacturing in the existingliterature only investigate single-item problems.Therefore, the interactions between several itemsproduced on the same production system are to-tally neglected. Neglecting these interactions oftenleads to an infeasible production plan if theseitems compete for the scarce capacities of the pro-duction system. Hence, we present in this paper anew model formulation for the multi-product ca-pacitated lot-sizing problem with product returnsand remanufacturing. This CLSP-RM is solved bycombining a column-generation approach and atruncated branch-and-bound method.

3 Dynamic Capacitated Lot-Sizingwith Product Returns andRemanufacturing

3.1 Model Formulation of the CLSP-RMThe objective of the capacitated lot-sizing problemwith product returns and remanufacturing is todetermine a production schedule for K products(k = 1, . . . ,K) which minimizes the sum of setup,production, remanufacturing and holding costsover a planning horizon of T periods (t = 1, . . . ,T).

This production schedule includes the productionquantities Qkt and the remanufacturing quantitiesQr

kt as well as the end-of-period inventory levels ofthe serviceables Ykt and of the recoverables Yr

kt ofproduct k in period t.For each product k the external demand dkt has tobe fulfilled in the respective period t, while back-logging is not allowed. Serviceables of product kcan be held in stock (Ykt) to satisfy the demandin later periods at holding costs hck per unit andperiod. The quantities of returned items rkt ofproduct k in period t are also known in advance.The returned items can be remanufactured (Qr

kt)to become as good as new, i.e. serviceables. Fur-thermore, returned products can also be held instock as recoverables (Yr

kt) at holding costs hcrk per

unit and period to be remanufactured in later pe-riods. The variable production costs are pck andthe remanufacturing costs are pcrk for one unit ofproduct k.The products are (re)manufactured on a produc-tion system, which consists of two separated re-sources: On the first resource, the products canonly be manufactured, while remanufacturing islocated on the second resource. For this reason,a (separate) setup is required for producing andanother setup for remanufacturing. For manufac-turing product k, a setup leads to setup costs sckand setup times tsk. If product k ismanufactured inperiod t, the setup variable γkt equals 1, otherwiseγkt = 0. For remanufacturing product k, differentsetup costs scrk and setup times tsrk are considered.

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A further setup variable γrkt is used to model asetup for the recoverables of product k in period t.Analogously, the setup variable γrkt of product k inperiod t equals 1, if product k is remanufacturedin the respective period, otherwise γrkt = 0. Thecapacity ct of the manufacturing system and thecapacity crt of the remanufacturing systemare givenin each period t. The production times are tpk andthe remanufacturing times are tpr

k for product k.The described production process is visualized inFigure 1.The CLSP-RM with separate setups (CLSP-RM-SS) can be stated using the notation in Table 1:

Model CLSP-RM-SS:

(1) min Z =∑k∈K

∑t∈T

(hck · Ykt + hcrk · Y

rkt)

min Z +∑k∈K

∑t∈T

(pck · Qkt + pcrk · Q

rkt)

min Z +∑k∈K

∑t∈T

(sck · γkt + scrk · γ

rkt)

subject to

(2) Yk,t−1 + Qkt + Qrkt = dkt + Ykt ∀ k, t

(3) Yrk,t−1 + rkt = Qr

kt + Yrkt ∀ k, t

(4)∑k∈K

(tpk · Qkt + tsk · γkt) ≤ ct ∀ t

(5)∑k∈K

(tprk · Q

rkt + tsrk · γ

rkt) ≤ crt ∀ t

(6) Qkt ≤ Mkt · γkt ∀ k, t

(7) Qrkt ≤ Mkt · γrkt ∀ k, t

(8) Qkt,Qrkt,Ykt,Yr

kt ≥ 0 ∀ k, t

(9) γkt, γrkt ∈ {0,1} ∀ k, t

The objective function (1)minimizes the sum of in-ventory holding, production, remanufacturing and

Table 1: Notation used for the CLSP-RM

Indices and index sets:K set of products (k ∈ {1, . . . ,K})T set of periods (t ∈ {1, . . . ,T})Parameters:ct available capacity of the resource for man-

ufacturing in period tcrt available capacity of the resource for re-

manufacturing in period tdkt external demand of product k in period thck holding cost of product k per unit and pe-

riodhcrk holding cost for returns of product k per

unit and periodMkt big number for product k in period tpck production cost of product k per unitpcrk remanufacturing cost of product k per unitrkt returns of product k in period tsck setup cost of product kscrk setup cost of returned product ktpk production time for one unit of product ktpr

k remanufacturing time for one unit of prod-uct k

tsk setup time of product ktsrk setup time of returned product kDecision variables:Qkt production quantity of product k in period tQr

kt remanufacturing quantity of product k inperiod t

Ykt inventory of product k at the end of period tY rkt inventory of recoverables of product k at

the end of period tγkt binary (joint) setup variable of product k in

period tγrkt binary setup variable for remanufacturing

product k in period t

setup costs. Equations (2) and (3) represent theinventory balance constraints for the serviceables,respectively the recoverables. Equations (2) ensurethat the given demand dkt for product k is satisfiedcompletely in each period t. Analogously, restric-tions (3) guarantee that the returned items areeither held in stock or remanufactured in the re-spective period. Constraints (4) and (5) contain thecapacity restrictions. In both cases, the production

58


(re)manufacturing times and also the setup timesmust not exceed the given capacity. Constraints (6)and (7) link the production variables Qkt and Qr

ktto the respective binary setup variable γkt or γrktfor product k in period t. Constraints (8) are thenon-negativity constraints. The setup variables aredefined to be binary, according to constraints (9).To tighten the linking constraints (6) and (7), theparameterMkt can be determined as follows:

Mkt =max{ct − tsk, crt − tsrk}

min{tpk, tprk}

∀ k, t

which describes the maximum (re)manufacturingquantity of product k that can possibly be(re)manufactured in period t due to the availablecapacity.In the next step, we present a small numeri-cal example for the CLSP-RM-SS with 4 prod-ucts and 5 periods. The underlying parametersettings are given in Appendix 6. The optimal(re)manufacturing quantities are shown in Fig-ure 2. A Gantt chart and the optimal productionplan are given in Appendix 6 (Figure 7(a) andTable 10).

Figure 2: Numerical example for theCLSP-RM-SS with 4 products and 5 periods

050

100150200250300350400

t=1 t=2 t=3 t=4 t=5

Production�Quantity�k=1 Remanufacturing�Quantity�k=1




The capacity of the resource for manufacturingis almost completely used. With the exception ofperiod 3, products are either manufactured or re-manufactured. Only one product is manufacturedand remanufactured as well, e.g., product 2 in thethird period. The total costs of the production planamount to 9620.In the case of joint setups, the products are(re)manufactured on the same resource with agiven capacity ct in each period t. A (joint) setupis necessary to prepare the production system tomanufacture and/or remanufacture product k in

period t. This setup leads to setup costs sck andsetup times tsk. If product k is produced and/orremanufactured in period t, the respective setupvariable γkt equals 1, otherwise γkt = 0.The CLSP-RM with joint setups (CLSP-RM-JS)can be modeled as follows.

Model CLSP-RM-JS:

(10) min Z =∑k∈K

∑t∈T


rkt)

min Z +∑k∈K

∑t∈T


rkt)

min Z +∑k∈K

∑t∈T

sck · γkt

subject to

(2), (3), (8), (9)

(11)∑k∈K

(tpk · Qkt + tprk · Q

rkt + tsk · γkt) ≤ ct ∀ t

(12) Qkt + Qrkt ≤ Mkt · γkt ∀ k, t

In the objective function (10) of the CLSP-RM-JS, the setup costs scrk can be omitted comparedto (1). Due to the consideration of a commonproduction system, only one capacity constraint(11) is required. Finally, the linking constraints (6)and (7) can be combined into constraints (12).

Figure 3: Numerical example for theCLSP-RM-JS with 4 products and 5 periods

050

100150200250300350400

t=1 t=2 t=3 t=4 t=5





Finally, the numerical example is also solvedin the case of the CLSP-RM-JS. The optimal(re)manufacturing quantities are illustrated inFigure 3. AGantt chart and the optimal production

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plan are also provided in Appendix 6 (Figure 7(b)and Table 11).In the case of the CLSP-RM-JS, the total costsamount to 6090 and are, therefore, lowercompared to the CLSP-RM-SS. However, pleasenote that the objective function values are notcomparable as only one setup is required for(re)manufacturing in the case of the CLSP-RM-JS.

3.2 Complexity ResultsTheorem 1 The CLSP-RM isNP-hard.

Proof By setting rkt = 0 for all products k andperiods t the CLSP-RM can be reduced to the well-known CLSP. As Florian, Lenstra, and Kan (1980)showed that the CLSP is NP-hard, the CLSP-RMis alsoNP-hard. �

Theorem 2 For the CLSP-RM, the proof of theexistence of a feasible solution is NP-complete inthe case of positive setup times (tsk > 0).

Proof As explained above, the CLSP-RM can beeasily reduced to the CLSP. Florian, Lenstra, andKan (1980) also proved that it is NP-complete toprove the existence of a feasible solution for theCLSP in the case of positive setup times. Therefore,the same holds for the CLSP-RM. �

3.3 Model Extensions of the CLSP-RMPractical extensions of the CLSP-RM can be, e.g.,the use of overtime or allowing back-orders. Ineach period the available capacity ct of the resourcefor manufacturing can be extended by the useof overtime (Ot). The overtime costs are oc foreach required unit of overtime. Analogously, theamount of overtime required on the resource forremanufacturing is denoted as Or

t . Furthermore,in the case of a stock-out, the demand can beback-ordered; in this case a backlog Bkt has to betaken into account. The backlog costs are bck forproduct k.In the following we present a model formulationfor the CLSP-RM-SS with backlogs and overtimeby using the additional notation presented inTable 2.

Model CLSP-RM-SS-BO-Ov:

(13) min Z =∑k∈K

∑t∈T


rkt)

min Z +∑k∈K

∑t∈T


rkt)

min Z +∑k∈K

∑t∈T


rkt)

min Z+∑k∈K

∑t∈T

bck ·Bkt+∑t∈T

oc·(Ot + Or

t)

subject to

(3), (6) -- (9)

(14) Yk,t−1 + Qkt + Qrkt + Bkt =

dkt + Ykt + Bk,t−1 ∀ k, t

(15)∑k∈K

(tpk · Qkt + tsk · γkt) ≤ ct + Ot ∀ t

(16)∑k∈K

(tprk · Q

rkt + tsrk · γ

rkt) ≤ crt + Or

t ∀ t

(17) Bkt, Ot, Ort ≥ 0 ∀ k, t

In the objective function (1) the backlog and over-time costs have to be taken into account, see (13).Furthermore, backlogs have to be considered inthe inventory balance equation (14). The capacityrestrictions (4) and (5) are extended by the useof overtime, see (15) and (16). Finally, backlogsand the amount of overtime are non-negative ac-cording to (17). The CLSP-RM-JS can be extendedanalogously.It is worth mentioning that the CLSP-RM-SS-BO-Ov is no longer NP−complete since backloggingand the use of overtime are not restricted.

Table 2: Additional notation used for theCLSP-RM with back-orders and overtime

Parameters:bck backlog cost of product k per unit and pe-

riodoc overtime cost per unitDecision variables:Bkt backlog of product k in period tOt amount of overtime used by the resource

for manufacturing in period tOr

t amount of overtime used by the resourcefor remanufacturing in period t

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3.4 Valid InequalitiesAs the well-known CLSP is a special case of theCLSP-RM, it is not surprising that first numericalexperiments have shown that the LP relaxationof the proposed model formulations for the CLSP-RMyields very poor lower bounds. Thus, includingvalid inequalities in themodel formulation is oftenpromising for accelerating the solution process.Therefore, we use the adapted valid inequalitiesfor the SLULSP-RM proposed by Retel Helmrich,Jans, van den Heuvel, and Wagelmans (2010).In the case of separate setups, the valid inequalitiesare defined as follows:

(18) Ykt ≥t+p∑

s=t+1

dks −t+p∑

s=t+1

Mks · (γks + γrks)

∀ k, t = 1, . . . ,T − 1,p = 1, . . . ,T − t

(19) Yrkt ≥

t∑s=t−p

rks −t∑

s=t−p

Mks · γrks

∀ k, t = 2, . . . ,T ,p = 1, . . . , t − 1

The valid inequalities (18) canbe interpreted as fol-lows: If we assume no setup activities for product kin the periods t + 1, . . . , t + p, the end-of-periodinventory Ykt must be high enough to fulfill thetotal demand in the periods t + 1, . . . , t + p. Oth-erwise, if the end-of-period inventory Ykt is notsufficient at least one setup is required in the peri-ods t + 1, . . . , t + p. Analogously, inequalities (19)are defined for the recoverables.If joint setups are considered, the valid inequalitiescan be defined as follows:

(20) Ykt ≥t+p∑

s=t+1

dks −t+p∑

s=t+1

Mks · γks

∀ k, t = 1, . . . ,T − 1,p = 1, . . . ,T − t

(21) Yrkt ≥

t∑s=t−p

rks −t∑

s=t−p

Mks · γks

∀ k, t = 2, . . . ,T ,p = 1, . . . , t − 1

The interpretation of the valid inequalities (20)and (21) is similar to the explanation of (18) and

(19). Obviously, the valid inequalities are also validfor the CLSP-RM, as the demand has to be fulfilledwithout backlogging and disposals are not allowed.The adaptation of the VI in the case of back-orderscan be found in Appendix 6.Due to the NP-hardness of the CLSP-RM it isnearly impossible to solve problem instances ofpractical size to optimality within a reasonabletime. Therefore, we propose anMP-based solutionapproach to determine high-quality solutions. Atfirst, a column-generation (CG) approach (Haase2005) is applied to generate tight lower boundsfor the CLSP-RM. Afterwards, a truncated branch-and-boundapproach is used to construct a solutionbased on the lower bound provided by CG.

4 A Column-Generation Approach4.1 Basic IdeaThe general idea of a column-generation approachcan be described as follows (Bahl 1983; Cattrysse,Maes, and van Wassenhove 1990): A reformula-tion of the CLSP-RM is used as a so-called masterproblem. At first, the master problem includesonly a very limited number of columns. Iteratively,new columns are generated for themaster problemby solving a respective subproblem. If the new col-umnpromises a reduction of the objective functionvalue, this new column is included in the masterproblem. New columns are generated iteratively aslong as they reduce the current objective function’svalue of the master problem.In the following, we only present the proceedingof the CG approach for the CLSP-RM with sepa-rate setups. Analogously, the CG approach can beadapted to theCLSP-RMwith joint setups straight-forwardly.

4.2 The Master Problem -- A SetPartitioning Problem

For the CG approach, we use a reformulation of theCLSP-RM, namely the Set Partitioning Problem(SPP, Garfinkel and Nemhauser 1969; Cattrysse,Maes, and vanWassenhove 1990) with capacity re-strictions, as the LP relaxation of the SPP providesa substantially tighter lower bound (Haase 2005)than the LP relaxation of the presented modelformulation.The objective of the SPP is to choose exactly oneproduction schedule for each product k at minimal

61


costs so that the capacity constraints are met ineach period. Therefore, a set of production sched-ules Sk exists for each product k which has to begenerated in advance. For each production sched-ule n ∈ Sk of product k the total costs nckn (includ-ing setup, (re)manufacturing and holding costs)are known. Furthermore, the capacity consump-tions tpktn (including manufacturing and setuptimes) and tpr

ktn (including remanufacturing andsetup times) can also be derived. For the choice ofa schedule n for product k a binary variable δkn isused. Here, δkn equals 1, if schedule n is chosen forproduct k and otherwise δkn = 0.

Table 3: Additional notation used for theSPP

Indices and index sets:Sk set of schedules of product k (n ∈

{1, . . . ,Sk})Parameters:nckn cost of schedule n of product ktpktn capacity consumption of schedule n for

manufacturing product k in period ttpr

ktn capacity consumption of schedule n forremanufacturing product k in period t

Decision variables:δkn binary variable for choosing schedule n of

product k

In the following, we present themodel formulationof the SPP using the additional notation given inTable 3.

Model SPP:

(22) min Z =∑k∈K

∑n∈Sk

nckn · δkn

subject to

(23)∑n∈Sk

δkn = 1 ∀ k

(24)∑k∈K

∑n∈Sk

tpktn · δkn ≤ ct ∀ t

(25)∑k∈K

∑n∈Sk

tprktn · δkn ≤ crt ∀ t

(26) δkn ∈ {0,1} ∀ k,n ∈ Sk

The objective function (22) minimizes the totalcosts. Equation (23) guarantees that exactly oneschedule is chosen for each product k. The capacityconstraints (24) and (25) ensure that the capacityconsumption of all products does not exceed thegiven capacity in period t. Constraints (26) definethe variables δkn to be binary. It is worth mention-ing that the optimal solution of the SPP is also theoptimal solution of the CLSP-RM if all possibleproduction schedules are known in advance.As the generation of all production schedules isquite time consuming, new schedules are deter-mined iteratively and included in the SPP after-wards. Therefore, in each iteration a subproblemis solved in order to generate a new schedule n foreach product k. This new production schedule n isincluded in the SPP, i.e. in the set Sk, if it promisesa reduction of the current objective function value,i.e., the reduced costs are negative. However, if thenumber of schedules increases, the solution timeto solve the SPP to optimality also increases sub-stantially. Hence, to reduce the numerical effort,we only solve the LP relaxation of the SPP and thus,the CG approach terminates with a lower boundfor the CLSP-RM.

4.3 The Subproblem -- An SLULSP-RMAs a subproblem in the CG approach, we solvean SLULSP-RM (Teunter, Bayindir, and vandenHeuvel 2006) to optimality for each product k.Numerical investigations have shown that thenumerical effort to solve an instance of theSLULSP-RM with separate (or joint) setups tooptimality is almost negligible using CPLEX withthe VIs (18) and (19), respectively (20) and (21).The SLULSP-RMk can be stated for a respectiveproduct k using the dual variables σk, πt,and πr

t of the relaxed SPP. Here, σk denotesthe dual variable which corresponds to equa-tion (23) for product k. The dual variables (orshadow prices) πt and πr

t correspond to the ca-pacity constraint (24), respectively (25) of period t.

Model SLULSP-RMk:

(27) min Zk =∑t∈T


rkt)

min Zk +∑t∈T


rkt)

62


min Zk +∑t∈T


rkt)− σk

min Zk −∑t∈T

πt · (tpk · Qkt + tsk · γkt)

min Zk −∑t∈T

πrt ·

(tpr

k · Qrkt + tsrk · γ

rkt)

subject to

(2), (3), (6) -- (9)

The objective function (27) of the SLULSP-RMktends to minimize the reduced costs of the poten-tially new schedule for product k. The remainingconstraints equal those of the CLSP-RM for therespective product k except for the capacity con-straints (11). Each SLULSP-RMk is solved to opti-mality using CPLEX. A new production schedulefor product k is only included in the set Sk if the re-duced costs of this new schedule are negative, i.e.,the objective function value of the SLULSP-RMk isnegative.

4.4 Outline of the Column-GenerationApproach

To initialize the column-generation approach westart with a dummy schedule for each product k.Within a dummy schedule, no production and nosetups take place, i.e., qkt = 0 for all products kand all periods t. Hence, there exists no capacityconsumption, i.e., tpktn = tpr

ktn = 0. However, toavoid the choice of these schedules in the optimalsolution, they are penalized at very high costs nckn.In each iteration, the LP relaxation of the masterproblem is solved first. Afterwards, the optimalsolution of the related SLULSP-RMk is determinedfor each product k by taking the dual variables σk,πt, and πr

t into account (see (27)). If the reducedcosts of the corresponding new schedule, i.e., theobjective function value of the related SLULSP-RMk, are negative this new schedule is included inthe set Sk. The algorithm terminates if the reducedcosts are non-negative for all products in the cur-rent iteration. The LP relaxation of the SPP yieldsa lower bound for the CLSP-RM. Note that no fea-sible lower bound exists if any dummy schedule isselected in the solution of the LP relaxation afterthe column-generation approach terminates.In the next step, the lower bound obtained by thecolumn-generation approach is used to generate

a feasible solution for the CLSP-RM by a trun-cated branch-and-bound approach using CPLEX.To minimize the numerical effort all binary setupvariables are fixed for those products k with aninteger solution in the LP relaxation of the SPP,i.e., ∃n ∈ Sk with δkn = 1. The respective setupvariables γkt and γrkt of product k are fixed relatedto the setup pattern, which corresponds to the pro-duction schedule nwith δkn = 1. After a given timelimit the branch-and-bound approach (B&B) ter-minates with a production plan for the CLSP-RM.A flow chart of the CG approach is presented inFigure 5.In Figure 4, the course of the objective functionvalue of the relaxed SPP is displayed for the nu-merical example in section 3.1 in the case of theCLSP-RM-SS. After the generation of 9 produc-tion schedules for each product the CG approachterminates with a feasible lower bound.

Figure 4: Course of the objective functionvalue of the master problem during the CGapproach compared to the optimalsolution

20000

25000

30000

35000

40000

45000

5000

10000

15000

1 2 3 4 5 6 7 8 9

Objective�function�value�of�master�problem Optimal�solution

In Appendix 6 we present necessary adaptationsof the solution approach in the case of back-ordersand overtime.

5 Numerical Experiments5.1 Description of the Test DesignWe analyze the quality of our solution approach bydefining five problem classes (PC) by varying thenumber of products and periods. Each problemclass consists of 216 test instances (TI) leading to1080 TI. Table 4 gives an overview of these fiveproblem classes.We vary different parameters to define the TI,e.g., the time between orders (TBO) to determinesetup costs, the utilization Util of the production

63


Figure 5: Flow chart of Column-Generation Approach

Initialization

Solve master problem SPP

At least one production plan

with negative reduced costs

Solve for each product k a subproblem SLULSP-RMk

yes

Insert production plans with negative reduced costs in SPP

Solve reduced CLSP-RM via Branch-and-Bound

no

Table 4: Dimensions of the five problemclasses

K T #TIClass 1 8 16 216Class 2 10 24 216Class 3 20 24 216Class 4 40 24 216Class 5 100 24 216

system, the consideration of setup times ts and theportion of holding costs hcr for returned productscompared to theholding costs of the serviceables. Adetailed description of the TI including calculationrules is given in Appendix 6. Table 5 shows a list ofthese parameters and their range of values.We implemented the model formulations and thesolution approach in GAMS 23.7. For the solutionof the subproblems and for the determination ofreference values we used CPLEX 12.3. The MP-

Table 5: Varying parameters

TBO pattern TBO ∈ {1,2,4}resource utilization Util ∈ {0.7,0.8,0.9}setup time ts ∈ {0,20}holding costs of re-coverables

hcr ∈ {0.5,0.7,0.9}

based heuristic ran on an Intel Xeon CPU witha 2.93 GHz processor and 8 GB of RAM usingtwo threads. The reference values are determinedon the cluster TANE of the RRZN in Hannoverusing 4 parallel threads, each with a 2.93 GHz pro-cessor and 36 GB of RAM (http://www.rrzn.uni-hannover.de/tane.html).

5.2 Solution Quality of CPLEX ReferenceValues

The valid inequalities (18) and (19) are included inthe CLSP-RM-SS to speed up the solution process

64


Table 6: Solution quality of CPLEX reference values

CLSP-RM-JS TCPUCPX TLimCPX AvgGAPCPX OptSolCPX

Class 1 37s 1h 0.00% 100.00%Class 2 798s 2h 0.06% 93.06%Class 3 3,085s 4h 0.03% 83.33%Class 4 6,206s 8h 0.02% 79.63%Class 5 13,712s 12h 0.02% 68.98%

CLSP-RM-SS TCPUCPX TLimCPX AvgGAPCPX OptSolCPX

Class 1 2,989s 1h 1.71% 22.69%Class 2 7,180s 2h 3.21% 0.00%Class 3 14,366s 4h 3.02% 0.00%Class 4 28,803s 8h 3.37% 0.00%Class 5 42,733s 12h 3.90% 0.00%

TCPUCPX Average solution time AvgGAPCPX Average integrality gapTLimCPX Given time limit OptSolCPX Portion of TIwith a proven optimal solution

of CPLEX. The valid inequalities (20) and (21)are used in the case of joint setups, respectively. InTable 6 the solution quality of the CPLEX referencevalues is reported, for both the case of joint andseparate setups. The CPLEX reference values andthe numerical results of the solution approach areprovided for all TI online in the supplementarymaterial.A (feasible) solution has been found for all TI.The average solution time in seconds is given in‘‘TCPUCPX ’’. As we do not expect to solve all TIto optimality within a reasonable time, each TI issolved by a truncated branch-and-bound approachwithin a given time limit (column ‘‘TLimCPX ’’).The average integrality gap is reported in column‘‘AvgGAPCPX ’’, whereGAPCPX

TI is derived as follows:

GAPCPXTI =

(SolCPXTI − LowBCPXTI )

SolCPXTI· 100%.

For each TI, SolCPXTI describes the best objectivefunction value found. LowBCPX

TI denotes the bestlower bound obtained by CPLEX within the giventime limit.In the case of joint setups, all TI of PC 1 and a largenumber of TI of the remaining problem classescould be solved to optimality within the given timelimit using the cluster TANE as reported in column‘‘OptSolCPX ’’. However, in the case of separate se-tups, CPLEXoften fails to find the optimal solutionwithin thegiven time limitdue to the largernumberof binary setup variables. In the following, the ref-

erence solutions obtained by CPLEX are denotedas CPLEX solutions or values.

5.3 Numerical Analysis of the LowerBounds obtained byColumn-Generation

To get an impression regarding the quality of thelower bound obtained by the column-generationapproach, we first compare this lower bound withthe CPLEX reference values. In Table 7, the aver-age solution time in seconds to generate a feasiblelower bound for the CLSP-RM is reported in col-umn ‘‘TCPUCG’’. For each TI the deviation GAPCG

TIof the CPLEX reference solution SolCPXTI from thelower bound LowBCG

TI obtained by CG can be de-termined as follows:

GAPCGTI =

(SolCPXTI − LowBCGTI )

LowBCGTI

· 100%.

The average deviation is given in column‘‘AvgGAPCG’’. Additionally, column ‘‘OptSolCG’’indicates the portion of TI whose lower boundequals the optimal solution. Finally, the portionof products with an integer solution in the lowerbound is given in column ‘‘KFixed’’. Table 7 givesan overview of the numerical results of the CGapproach for all five problem classes.For all TI a feasible lower bound is obtained bythe column-generation approach. Here, a lowerbound is called feasible if the LP relaxation doesnot include any dummy schedule. As expected,

65


Table 7: Numerical results of the lower bounds obtained by CG

CLSP-RM-JS TCPUCG AvgGAPCG OptSolCG KFixedClass 1 2.4s 0.46% 25.46% 65.63%Class 2 9.0s 0.26% 25.00% 66.44%Class 3 16.0s 0.07% 27.78% 80.58%Class 4 31.3s 0.03% 33.33% 91.28%Class 5 82.9s 0.01% 35.65% 96.44%

CLSP-RM-SS TCPUCG AvgGAPCG OptSolCG KFixedClass 1 15.5s 1.21% 5.09% 43.11%Class 2 179.0s 1.14% 4.17% 39.72%Class 3 299.9s 0.67% 6.48% 64.28%Class 4 594.5s 0.64% 2.31% 82.70%Class 5 1,574.3s 1.00% 0.00% 93.35%

TCPUCG Average solution time OptSolCG Portion of optimally solved TIAvgGAPCG GAP of lower bound obtained by CG KFixed Portion of products with integer solution

the run time increases with respect to the risingnumber of products.In the case of joint setups, the average integralityGAP does not exceed 0.5%,while the average lowerbound even amounts to only 0.01% in the case ofthe largest PC 5. Thus, the lower bound obtainedby the CG approach is very close to the optimalsolution. At least 25% of the TI could already besolved tooptimality viaCG.Theportionofproductswith an integer solution (∃n ∈ Sk with δkn = 1)is quite high at more than 65%. The portion evenrises to 96% for PC 5.In the case of separate setups, the average solu-tion time increases substantially due to the NP-hardness of the SLULSP-RMwith separate setups.The average integrality gap is almost comparableto the lower bound obtained for the case of jointsetups. The portion of products with an integer so-lution (δkn = 1) is only slightly lower compared tothe portion of products in the case of joint setups.A detailed evaluation of our numerical resultsshows that an increase of setup costs and, there-fore, of the TBO leads to a significant increase ofthe solution time. The integrality GAP also deteri-orates with respect to the increase in setup costs. Itis worthmentioning that at least 75% of the TI withlow setup costs (TBO = 1) could even be solvedto optimality with the CG approach in the caseof joint setups. In summary, the solution qualitydecreases slightly if the setup cost rises. The nu-merical results of the CG approach with respect to

the varying setup costs can be found in Table 12.In the case of scarce capacities, i.e., a higher uti-lization, the numerical effort rises as the solutiontimes in column ‘‘TCPUCG’’ shows. Simultaneously,the number of iterations increases as well. The av-erage lower bound also rises in the case of a higherutilization. The numerical results with respect tothe varying utilization of the production systemarepresented in Table 13. We observed that the im-pact of the remaining parameters on the solutionquality is rather limited.

5.4 Numerical Analysis of the UpperBounds obtained by the truncatedBranch-and-Bound Method

In the next step, we investigate the solution qualityof the production plans obtained by the truncatedB&B method. Although a setup decision has al-ready been fixed for a large number of products inthe lower bound obtained by the CG approach, wedo not expect to solve the remaining problem tooptimality within a reasonable time. Therefore, atime limit is used for the truncated B&B approach.In Table 8 the numerical results of the upperbound obtained by the B&B approach are reported.In contrast to Table 7, the column ‘‘TCPUCG’’ issubstituted by ‘‘TCPUB&B’’. Here, the entries incolumn ‘‘TCPUB&B’’ show the run time in secondsof the truncated B&B approach. For each TI thedeviation DevB&BTI of the CPLEX reference solutionSolCPXTI from the solution SolB&BTI obtained by the

66


Table 8: Numerical results of the upper bounds obtained by the truncated B&B

CLSP-RM-JS TCPUB&B AvgDevB&B AvgGAPB&B OptSolB&B TLimB&B

Class 1 4.8s 0.12% 0.58% 55.09% 50sClass 2 19.9s 0.10% 0.37% 42.13% 100sClass 3 26.9s 0.06% 0.13% 41.20% 150sClass 4 34.8s 0.03% 0.05% 39.35% 300sClass 5 59.8s 0.01% 0.02% 41.20% 600s

CLSP-RM-SS TCPUB&B AvgDevB&B AvgGAPB&B BetSolB&B TLimB&B

Class 1 34.3s 0.66% 1.88% 30.09% 50sClass 2 87.1s 1.60% 2.79% 18.06% 100sClass 3 122.7s 1.78% 2.49% 43.52% 150sClass 4 207.1s 0.29% 0.94% 78.70% 300sClass 5 379.3s -0.80% 0.19% 100.00% 600s

TCPUB&B Solution time of truncated B&B OptSolB&B Portion of optimally solved TIAvgDevB&B Average deviation from CPLEX solution TLimB&B Given time limitAvgGAPB&B Average deviation from CG lower bound BetSolB&B Portion of better solved TI

truncated B&B approach can be determined asfollows:

DevB&BTI =(SolCPXTI − SolB&BTI )

SolCPXTI· 100%.

In column ‘‘AvgDevB&B’’ the average deviationfrom the CPLEX reference solution is reported.Furthermore, we report the average gap based onthe lower bound obtained by the CG approach incolumn ‘‘AvgGAPB&B’’, whereGAPB&B is defined asfollows:

GAPB&BTI =

(SolB&BTI − LowBCGTI )

LowBCGTI

· 100%.

In the case of joint setups, the portion of optimallysolved TI are given in ‘‘OptSolB&B’’. As the numberof optimally solved TI is rather small in the case ofseparate setups, the column ‘‘OptSolB&B’’ is substi-tuted by the portion of TI which are solved at leastas well as by CPLEX or even better. Therefore, thecolumn is named ‘‘BetSolB&B’’. The given time limitis reported in ‘‘TLimB&B’’Due to the very small number of remaining freebinary variables the numerical effort is rather lim-ited. In the case of joint setups, themean deviationfrom the CPLEX reference values ranges between0.01% and 0.58%. This can be explained by thelarge portion of already fixed products after the CGapproach (Table 7). More than 39% of all TI couldbe solved to optimality.

In the case of separate setups, the average devia-tion from the CPLEX reference values is also verymoderate. For more than 18% of the TI the trun-catedB&Bmethod finds a production planwhich isat least as good as the CPLEX solution or even bet-ter. Furthermore, the truncated B&B determinesfor all TI a better solution than CPLEX in the caseof PC 5. The low average GAP compared to thelower bound obtained by CG also demonstratesthe high solution quality of the combined solutionapproach.The numerical results of the truncated B&Bmethod based on the different TBO patternsare reported in Table 14. Analogous to theobservations regarding the CG approach, thesolution time rises with respect to increasingsetup costs. The same holds true for the solutionquality. In the case of low setup costs (TBO = 1)the average deviation from the CPLEX referencevalues is lower compared to the average deviationin the case of high setup costs (TBO = 4). Forlow setup costs, at least 87.50% of all TI couldbe solved to optimality. This number decreasessubstantially to 0% with increasing setup costs.Furthermore, the total solution time of bothapproaches also grows according to rising setupcosts.For the sakeof completeness,wegive thenumericalresults of the truncated B&B method with respectto the utilization of the production system in Ta-

67


ble 15. The rise of solution times is also noticeablein the case of a higher utilization, while the averagedeviation from the CPLEX reference values deteri-orates only marginally. Worth mentioning are theresults of PC 5: Contrary to the results of the otherclasses, the solution quality does not decline in thecases of higher setup costs and/or higher utiliza-tion. This is due to the fact that CPLEX fails to findbetter solutions within the given time limit.We also investigated the solution quality of theproposed solution approach by allowing overtimeand back-orders. The solution quality and the so-lution time are nearly comparable. Therefore, weabstain from showing these numerical results.

6 Summary and OutlookIn this paper, we presented a new model formu-lation for a capacitated lot-sizing problem withproduct returns and remanufacturing. In litera-ture, only one-product problems have been con-sidered so that the influence of different productson the same production systems are completely ig-nored. Furthermore, most of these approaches inliterature neglect capacity restrictions. Thus, theseproduction plans are of limited value as the ca-pacity restrictions are often violated. Obviously, arescheduling is necessary in these cases.The proposed solution approach yields a very highsolution quality. The lower bounds found by theCG approach are very close to the optimal solutionas our numerical investigation shows. The combi-nation of CG and a truncated branch-and-boundmethod leads to a very high solution quality andthe solutions are very close to the CPLEX referencevalues, but with substantially reduced numericaleffort. Furthermore, we have shown that an inte-grated (re)manufacturing planning can yield a costreduction.Future research should address the adaptationof further solution approaches, e.g., the Fix-and-Optimize heuristic by Helber and Sahling (2010).Possible extensions of the CLSP-RM will be linkedlot sizeswhere the setup state of one product can becarried over to a subsequent period. Furthermore,different quality levels of the returned productsshould be taken into account. Different quality lev-els will result in varying remanufacturing costs andtimes.

AppendicesAppendix A: Description of the NumericalExampleFor the numerical example in section 3.1, the setupcosts sck and scrk equal 500 for each product k.The holding costs are hck = 1 and hcrk = 0.5. The(re)manufacturing times tpk and tpr

k are assumedto be 1 and the setup times tsk and tsrk are equalto 20 for each product k. The demand and returnsare given in Table 9.

Table 9: Demand and return data of thenumerical example in section 3.1

k \ t dk1 dk2 dk3 dk4 dk5

1 40 70 60 70 602 100 60 180 170 1103 80 90 90 140 1404 30 80 50 80 90

k \ t rk1 rk2 rk3 rk4 rk51 20 30 30 20 302 50 40 60 40 403 40 60 50 30 404 30 30 30 50 20

In the case of the CLSP-RM-SS, the capacities ctand crt amount to 300 capacity units in each period.In Table 10, the optimal production plan is givenfor the CLSP-RM-SS.

Table 10: Optimal production plan of thenumerical example for CLSP-RM-SS

k \ t qk1 qk2 qk3 qk4 qk51 40 130 0 0 02 100 0 90 280 03 80 0 140 0 1404 0 130 0 0 90

k \ t qrk1 qrk2 qrk3 qrk4 qrk51 0 0 0 70 602 0 60 90 0 03 0 90 0 90 04 30 0 0 80 0

In the case of the CLSP-RM-JS, the capacity ctamounts to 600 capacity units in each period. InTable 11, the optimal production plan is given for

68


Figure 6: Gantt charts of the numerical example with 4 products and 5 periods insection 3.1. Note that the sequences of the products presented in the Gantt charts arechosen arbitrarily since the sequences are not determined by the CLSP-RM.

2 t = 2t = 1m = 1

m = 2

c1 = 300 c2 = 300 c3 = 300

t = 3

c4 = 300 c5 = 300

t = 4 t = 5

(a) CLSP-RM-SS

2 t = 2t = 1m = 1

c1 = 600 c2 = 600 c3 = 600

t = 3

c4 = 600 c5 = 600

t = 4 t = 5

(b) CLSP-RM-JS

Q1 Qr1 Q2 Q3 Q4Qr

2 Qr3 Qr

4

the CLSP-RM-JS.

Table 11: Optimal production plan of thenumerical example for CLSP-RM-JS

k \ t qk1 qk2 qk3 qk4 qk51 20 230 0 0 02 110 0 360 0 03 40 120 0 200 04 130 0 0 60 0

k \ t qrk1 qrk2 qrk3 qrk4 qrk51 20 30 0 0 02 50 0 100 0 03 40 60 0 80 04 30 0 0 110 0

Appendix B: Adaptations in the Case ofBack-Orders and OvertimeThe valid inequalities (18) for the CLSP-RM-SShave to be adapted by taking backlogs into consid-eration.

(28) Ykt−Bkt ≥t+p∑

s=t+1

dks−Bk,t+p−t+p∑

s=t+1

Mks·(γks+γrks)

∀ k, t = 1, . . . ,T − 1,p = 1, . . . ,T − t

Analogously, theVI (20) for the CLSP-RM-JSmustbe changed.

For the solution approach, the use of overtimehas to be considered only in the master problem.Therefore, the objective function (22), the capacityconstraints (24) and (25) have to be extended.

Model SPP-Ov:

(29) min Z =∑k∈K

∑n∈Sk

nckn ·δkn+∑t∈T

oct ·(Ot + Or

t)

subject to

(23), (26)

(30)∑k∈K

∑n∈Sk

tpktn · δkn ≤ ct + Ot ∀ t

(31)∑k∈K

∑n∈Sk

tprktn · δkn ≤ crt + Or

t ∀ t

Backlogs have to be considered only in thesubproblem.

Model SLULSP-RM-BOk:

(32) min Zk =∑t∈T


rkt)

min Zk +∑t∈T


rkt)

min Zk +∑t∈T

(bck · Bkt)

69


min Zk −∑t∈T

πt · (tpk · Qkt + tsk · γkt)

min Zk −∑t∈T

πrt ·

(tpr

k · Qrkt + tsrk · γ

rkt)

subject to

(3), (6) -- (9), (14), (17)

However, further adaptations of the solution ap-proach are not necessary.

Appendix C: Description of the TestInstancesAt first, for each product k the average demand dkis generated randomly by following a uniform dis-tribution in the interval [50,150]. Based on the av-erage demand dk, two demand series are providedbased on a normal distribution with a different co-efficient of variation for each series. Analogously,two return series for each product k are generated.However, the average return rk equals one-thirdof the average demand dk of product k. The de-mand and return series are provided online in thesupplementary material.The holding costs hck of the serviceables, the pro-cessing times tpk and the remanufacturing timestpr

k are assumed to be 1 for each product k. Theholding costs hcrk of the recoverables are assumedto be lower than the holding costs hck of the ser-viceables. For the holding costs hcrk three scenariosare defined (Table 5). The production and reman-ufacturing costs pck and pcrk are assumed to beequal and therefore not crucial.The setup costs sck are determined based on theaverage demand dk of product k and the respectivetime between orders (TBO):

sck =dk · TBO2 · hck

2∀ k.

In the case of separate setups, the setup costs scrkfor remanufacturing equals the setup costs sck.The setups times tsk (and tsrk) are assumed to bezero or equal to 20 for all products. To derive thecapacity ct in the case of joint setups, we assume asetup activity in each period t for all products andallow the production of the average demand dk;this leads to

ct =∑k

(max{tpk, tpr

k} · dk + tsk)

∀ t.

In the case of separate setups, we also assume asetup activity on both production systems in eachperiod t for all products. However, the averagedemand dk is reduced by the average return rk.This leads to

ct =∑k

(tpk · (dk − rk) + tsk

)∀ t.

Additionally, the capacity crt is defined accordingto the average return rk and allowing also a setupactivity in each period:

crt =∑k

(tpr

k · rk + tsrk)

∀ t.

To guarantee a large amount of feasible solutions,the available capacity is then extendedwith respectto the given utilization Util

ct =ctUtil

∀ t.

The capacity crt of the remanufacturing system isextended analogously.

70


Table 12: Numerical results of CG approach related to TBOs

CLSP-RM-JS TCPUCG AvgGAPCG OptSolCG KFixedClass 1TBO = 1 0.6s 0.10% 75.00% 90.45%TBO = 2 1.5s 0.20% 1.39% 67.01%TBO = 4 5.0s 1.08% 0.00% 39.41%

Class 2TBO = 1 1.1s 0.03% 75.00% 95.83%TBO = 2 4.2s 0.11% 0.00% 64.44%TBO = 4 21.7s 0.64% 0.00% 39.03%




CLSP-RM-SS TCPUCG AvgGAPCG OptSolCG KFixedClass 1TBO = 1 5.5s 0.24% 15.28% 65.10%TBO = 2 14.9s 0.76% 0.00% 41.84%TBO = 4 26.1s 2.63% 0.00% 22.40%



Class 4TBO = 1 94.9s 0.03% 6.94% 93.96%TBO = 2 541.7s 0.38% 0.00% 84.55%TBO = 4 1,146.7s 1.51% 0.00% 69.58%

Class 5TBO = 1 187.7s 0.06% 0.00% 98.71%TBO = 2 1,614.9s 0.78% 0.00% 94.17%TBO = 4 2,920.1s 2.16% 0.00% 87.17%

71


Table 13: Numerical results of CG approach related to the resource utilization

CLSP-RM-JS TCPUCG AvgGAPCG OptSolCG KFixedClass 1Util = 0.7 1.7s 0.16% 34.72% 77.08%Util = 0.8 2.2s 0.32% 33.33% 68.58%Util = 0.9 3.2s 0.91% 8.33% 51.22%

Class 2Util = 0.7 6.4s 0.08% 33.33% 75.69%Util = 0.8 8.9s 0.18% 33.33% 68.47%Util = 0.9 11.7s 0.52% 8.33% 55.14%




CLSP-RM-SS TCPUCG AvgGAPCG OptSolCG KFixedClass 1Util = 0.7 10.9s 0.61% 13.89% 57.81%Util = 0.8 15.0s 1.08% 1.39% 41.84%Util = 0.9 20.6s 1.93% 0.00% 29.69%




Class 5Util = 0.7 1,442.6s 0.68% 0.00% 96.33%Util = 0.8 1,559.7s 0.93% 0.00% 94.15%Util = 0.9 1,720.5s 1.39% 0.00% 89.56%

72


Table 14: Numerical results of the upper bounds obtained by the truncated B&B related toTBOs

CLSP-RM-JS TCPUB&B AvgDevB&B AvgGAPB&B OptSolB&B






CLSP-RM-SS TCPUB&B AvgDevB&B AvgGAPB&B BetSolB&B




Class 4TBO = 1 88.3s -0.02% 0.01% 87.50%TBO = 2 233.0s -0.15% 0.23% 86.11%TBO = 4 300.1s 1.04% 2.58% 62.50%

Class 5TBO = 1 98.0s -0.06% 0.00% 100.00%TBO = 2 439.8s -0.71% 0.06% 100.00%TBO = 4 600.2s -1.62% 0.50% 100.00%

73


Table 15: Numerical results of the upper bounds obtained by the truncated B&B related tothe resource utilization

CLSP-RM-JS TCPUB&B AvgDevB&B AvgGAPB&B OptSolB&B






CLSP-RM-SS TCPUB&B AvgDevB&B AvgGAPB&B BetSolB&B




Class 4Util = 0.7 155.1s -0.20% 0.18% 94.44%Util = 0.8 215.7s 0.16% 0.75% 84.72%Util = 0.9 250.7s 0.91% 1.88% 56.94%

Class 5Util = 0.7 286.9s -0.63% 0.05% 100.00%Util = 0.8 353.6s -0.78% 0.14% 100.00%Util = 0.9 497.5s -0.99% 0.37% 100.00%

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AcknowledgementsI would like to thank both referees and the de-partment editor for their very helpful commentswhich helped me to improve this paper, and PaulCochrane for his valuable support during the nu-merical study.

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BiographyFlorian Sahling holds the position of an Assistant Professorfor "Quantitative Supply Chain Management" at the Depart-ment of Production Management at the Leibniz UniversitätHannover. He studied Mathematics and Management Scienceat the University of Hamburg and the University of Bath. Hereceived his doctoral degree in 2009. His main area of researchis production planning and scheduling.

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BuR -- Business Research Volume6|Issue1|May2013 | 55--75 A ... · Short-Term Production Planning Problem in Closed-Loop Supply Chains Florian Sahling, Institute of Production Management,