a mathematical model for production planning in hybrid manufacturing-remanufacturing systems

ORIGINAL ARTICLE

A mathematical model for production planning in hybridmanufacturing-remanufacturing systems

M. Chen & P. Abrishami

Received: 14 October 2013 /Accepted: 5 December 2013 /Published online: 3 January 2014# Springer-Verlag London 2014

Abstract In recent years, environmental legislation, societalpressure, and economic opportunities have motivated manyfirms to integrate remanufacturing activities into the regularproduction environment. This presents many new challengesinvolving the collection, disassembly, refurbishing of usedproducts, and incorporation of remanufacturing activities intonew product manufacturing. In this paper, we present a mixedinteger programming model addressing production planningproblems in such hybrid systems. A Lagrangian decomposi-tion based method is developed to solve the problem efficient-ly. A numerical example is presented to illustrate the proposedmodel and the developed solution procedure.

Keywords Mathematical model . Production planning .

Manufacturing and remanufacturing . Lagrangian relaxation

1 Introduction

Manufacturing products using components from end-of-lifeproducts has been in practice for many years. In practice,recovery of used products and components can be a profitablebusiness. Thierry et al. [12] presented various product recov-ery options as direct reuse, resale, repair, refurbishing,remanufacturing, cannibalization, and recycling. These op-tions can be reclassified into three broad categories such asreuse, recycling, and remanufacturing as discussed in Kim

et al. [4]. Remanufacturing is an industrial process in whichworn-out products are restored to like-new conditions. Suchprocesses are widely used in, for example, automotive andelectronic part and component manufacturing, among manyother industries. In a typical remanufacturing process, end-of-life products are completely disassembled through a series ofindustrial processes in factory environment. Usable parts arecleaned, refurbished, and put into part inventory. Then thenew product may be reassembled from the old and, wherenecessary, new parts to produce a fully equivalent and some-times superior in performance and expected lifetime to theoriginal new product [7]. Remanufacturing systems havecommon and different features and characteristics from regu-lar manufacturing systems. In recent years, growing environ-mental concerns as well as government guidelines and regu-lations have resulted in increasing attention andmuch researchin remanufacturing and reverse logistics. Li et al. [6] usedstochastic dynamic programming to solve production plan-ning problems in a remanufacturing system, considering un-certainty issues in such production systems. Amin and Zhang[2] proposed mathematical model for closed-loop networkconfiguration considering product life cycle. They used off-shelf optimization software to solve the developed model andperformed sensitivity analysis. Yuan and Gao [13] developeda model and a solution method for solving inventory problemsconsidering both manufacturer and retailer in a closed-loopsupply chain. In addition to dedicated remanufacturing sys-tems, remanufacturing can be done in systems where theoriginal new products are produced. Manufacturing new prod-ucts and remanufacturing returned products can be performedin a samemanufacturing facility with shared resources in suchhybrid manufacturing-remanufacturing systems (HMRS). Inthese systems, the original manufacturer is in charge ofcollecting, refurbishing, and remanufacturing of used

M. Chen (*) : P. AbrishamiDepartment of Mechanical and Industrial Engineering, ConcordiaUniversity, 1455 de Maisonneuve W., Montreal, QC,Canada H3G 1M8e-mail: [email protected]

Int J Adv Manuf Technol (2014) 71:1187–1196DOI 10.1007/s00170-013-5538-0

products. Researchers have studied various aspects of HMRS.Laan et al. [5] investigated the influence of lead-time durationand lead-time variability on total expected costs in a systemwith manufacturing and remanufacturing operations. AWagner-Whitin type method was developed to solve the prob-lem. Huang et al. [3] proposed a robustH∞ control strategy forproduction control of HMRS systems and presented analyticalresults based on extensive simulation runs. Zanoni et al. [14]compared different inventory control policies in HMRSwheredemand, return rate, and lead times are stochastic. In addition,different mathematical programming models have been de-veloped for modeling and solving different HMRS productionplanning problems. A mixed integer programming modelformulating lot sizing problems in HMRS was presented inTeunter et al. [10]. Optimization and heuristic solutionprocedures were developed to solve the discussed problemunder various conditions. Teunter et al. [11] also studiedeconomic lot scheduling problem with manufacturing andremanufacturing on separate dedicated lines and developedan algorithm to determine the optimal common cycle time.Pan et al. [8] presented an optimization model for capacitatedlot sizing problem in HMRS where the system has separateproduction capacities for new products and returned products.Zhang et al. [15] also considered an HMRS problem withseparate manufacturing and remanufacturing capacities. Agenetic algorithm based solution method was developed tosolve the considered problem. Rubio and Corominas [9]proposed a production-management model in HMRS anddeveloped a solution procedure to optimally allocatemanufacturing and remanufacturing capacities in such sys-tems. A recent and extensive review on various models forsolving planning and inventory problems in closed-loop sup-ply chains can be found in Akçalı and Çetinkaya [1]. Thequantitative models, with many of them consideringmanufacturing and remanufacturing activities, were placedand reviewed in deterministic models and stochasticcategories. The authors identified that interactions ofdifferent activities in such systems require more in-depthstudy and research. As can be observed from the literature,research using deterministic models to solve production plan-ning problems in HMRS has been conducted by severalresearchers. However, modeling and solution methods ad-dressing interactions of manufacturing and remanufacturingactivities sharing common resources have been limited. In thispaper, we present a mixed integer programming model forproduction planning involving returned product disassembly,core remanufacturing, and new part manufacturing. We alsoassume that manufacturing and remanufacturing activities inthe considered HMRS system share a common resource as thenew and returned products are similar. Section 2 presents theproblem details andmodel development. A solution procedurebased on Lagrangian decomposition is proposed in Sect. 3.Numerical examples are presented in Sect. 4 to illustrate the

developed model and solution procedure with concludingremarks draw in Sect. 5.

2 Problem description and modeling

In this paper, we consider a production planning problem inHMRS systems. The considered problem is of deterministicnature, similar to some of the reverse supply chain modelsdiscussed in Akçalı and Çetinkaya [1]. The system manufac-tures parts from new materials to satisfy the given demands. Italso remanufactures similar parts from returned products afterdisassembling these products. We assume that the demandsfor the remanufactured parts are known and separate fromthose for the new parts. Manufacturing and remanufacturingare associated with production cost, system setup cost, andpart inventory cost in multiple time periods of the planninghorizon for both new and remanufactured parts. Similarly,there are costs associated with system setup, disassembling,and inventory of the returned products. In addition, there arecosts to obtain or purchase the returned products to retrieve theparts for remanufacturing. In this problem, we consider thatmanufacturing and remanufacturing share a same critical andlimited resource. Relaxation of this assumption will bediscussed later in this paper. The purpose of solving theconsidered problem is to minimize the total cost while meet-ing the given demands of the new and remanufactured parts. Amixed integer linear programming (MILP) model for solvingthe above-described problem is formulated. We first presentthe notations and variables used in the proposedMILP model.

Parameters

Pi,t Unit manufacturing cost of new component i intime period t

S i,t Setup cost for manufacturing new component i intime period t

Vi,t Unit inventory cost for new component i in timeperiod t

Pi;t Unit remanufacturing cost of recovered componenti in time period t

Si;t Setup cost for remanufacturing recoveredcomponent i in time period t

V i;t Unit inventory cost for remanufactured componenti in time period t

Di,t Demand for new component i in time period tDi;t Demand for remanufactured component i in time

period tB i,j Number of component i contained in product jURi Average recovering rate of component i from all

returned productsAQj ,t Unit cost to acquire returned product j in time

period t

1188 Int J Adv Manuf Technol (2014) 71:1187–1196

RDj,t Unit cost to disassemble returned product j in timeperiod t

SDj,t Setup cost for disassembling product j in timeperiod t

INj,t Unit inventory cost for storing returned product j intime period t

ACAPt Available production time in time period tASTi Production time for manufacturing new

component iSTi Setup time for manufacturing new component iASRi Production time for remanufacturing returned

component iSRi Setup time for remanufacturing returned

component iM A large positive number

Decision variables

x i,t Number of new component i to produce in timeperiod t

e i,t Number of new component i in inventory at the end oftime period t

xi;t Number of remanufactured component i to produce intime period t

ei;t Number of remanufactured component i in inventory atthe end of time period t

d j,t Number of returned product j to disassemble in timeperiod t

r j,t Number of returned product j to acquire in time period tf j,t Number of returned product j in inventory at the end of

time period t

θi;t ¼1; If the system is set up to make

new component i in time period t;0; otherwise:

(

θi;t ¼1; If the system is set up to remanufacture

returned component i in time period t;0; otherwise:

(

δ j;t ¼1; If returned product j will be disassembled

in time period t;0; otherwise:

(

2.1 MILP model

Using the notations given above, the objective function andthe constraints of the proposed MILP model for HMRS pro-duction planning problem are presented next.

M-model

minZ ¼Xt¼1

T Xi¼1

I

ðPi;txi;t þ Si;tθi;t þ V i;tei;t þ Pi;txi;t

þ Si;tθi;t þ V i;tei;tÞ

þXt¼1

T Xj¼1

J

AQj;tr j;t þ SDj;tδ j;t þ RDj;td j;t þ IN j;t f j;t

� �ð1Þ

Subject to:

ei;t−1 þ xi;t−ei;t ¼ Di;t ð2Þ

xi;t ≤Mθi;t ð3Þ



f j;t þ d j;t− f j;t−1 ¼ r j;t ð6Þ

d j;t ≤Mδ j;t ð7Þ

Xi¼1

I

ASTixi;t þ STiθi;t þ ASRixi;t þ SRiθi;t� �

≤ACAPt ð8Þ

xi;t ≤URi

Xj¼1

J

Bi; jd j;t ð9Þ

ei;0 ¼ ei;0 ¼ f j;0 ¼ 0

xi;t; ei;t; xi;t; ei;t; fj;t; d j;t; r j;t ≥0

θi;t; θi;t; δ j;t ¼ 0; 1f g

ð10Þ

The first part of the objective function in Eq. (1) isto minimize the production cost, setup cost, and inven-tory cost for parts manufactured from new materialsand parts from returned and disassembled products.The second part includes the cost of acquiring thereturned products, setup cost for disassembly opera-tions, disassembly cost, and inventory cost of thereturned products. Equations (2) and (3) are the rela-tions among new parts demands, inventory, production,and system setups. Equations (4) and (5) are similar

Int J Adv Manuf Technol (2014) 71:1187–1196 1189

requirements for parts from returned products.Equations (6) and (7) are relations among returnedproducts, disassembled products, disassembly setup,and product inventory. Equation (8) is the shared re-source constraints for both manufacturing andremanufacturing parts. Equation (9) gives the limit ofthe parts obtained from the returned products based onthe quality level and bill of materials. Equation (10) isthe assumption that the initial inventories are all 0, withoutloss of generality. The rest are non-negative and integer re-quirement of the variables. The above model is similar to thestandard capacitated lot sizing problem which is NP-hard. Inthis paper, we use a decomposition method based on; due tothe special structure of this model, we propose a de-composition method based on Lagrangian relaxation tosolve the above-presented problem, more efficiently.Details of the proposed solution method are presented in thenext section.

3 Solution methodology

In solving the developed MILP model to obtain optimal ornear optimal solutions of the considered HMRS problem, wepropose a solution method based on Lagrangian relaxation.Lagrangian relaxation and decomposition methods have beenwidely used in solving different types of integer programmingproblems. In solving the MILP model presented in the previ-ous section, we relax the two constraints functions in Eqs. (8)and (9) in the M-model to decompose it into several sub-models of much smaller sizes. The first two sets of the sub-models are then solved byWagner-Whitin method. We devel-op a heuristic algorithm to solve the third sub-model. Standardsub-gradient search is used to update the Lagrangian multi-pliers in searching for an optimal solution of theM-model. Wefirst present model decomposition and the resulted sub-models. As mentioned earlier, we apply Lagrangian relaxationto Eqs. (8) and (9). The original mathematical model thenbecomes as follows:

Table 1 Cost and demand for component of new materials

Time period Cost and demand Component type

1 2 3 4 5 6

1 Production cost 100 80 90 90 75 105

Setup cost 600 600 600 600 600 600

Inventory cost 100 100 100 100 100 100

Demand 50 80 40 45 75 62

2 Production cost 90 100 90 85 110 90

Setup cost 700 700 700 700 700 700

Inventory cost 120 120 120 120 120 120

Demand 40 35 55 51 43 50

3 Production cost 110 80 95 70 85 95

Setup cost 600 600 600 600 600 600

Inventory cost 105 105 105 105 105 105

Demand 50 20 45 48 27 43

4 Production cost 95 80 100 80 80 110

Setup cost 550 550 550 550 550 550

Inventory cost 115 115 115 115 115 115

Demand 55 40 45 42 38 39

5 Production cost 85 100 100 90 110 80

Setup cost 600 600 600 600 600 600

Inventory cost 100 100 100 100 100 100

Demand 35 40 30 30 44 33

6 Production cost 90 95 100 105 90 90

Setup cost 700 700 700 700 700 700

Inventory cost 110 110 110 110 110 110

Demand 35 45 55 30 39 51

Table 2 Cost and demand for component from returned product

Time period Cost and demand Component type

1 2 3 4 5 6

1 Production cost 30 20 40 15 20 50

Setup cost 400 400 400 400 400 400

Inventory cost 50 50 50 50 50 50

Demand 10 15 10 7 11 12


Setup cost 500 500 500 500 500 500


Demand 20 10 10 18 13 8


Setup cost 450 450 450 450 450 450


Demand 12 9 10 11 10 8


Setup cost 350 350 350 350 350 350


Demand 10 15 10 12 11 9


Setup cost 400 400 400 400 400 400


Demand 8 10 12 10 9 11


Setup cost 470 470 470 470 470 470


Demand 12 18 12 10 15 11


R-model

minZ λ; γð Þ ¼Xt¼1

T Xi¼1

I

Pi;txi;t þ Si;tθi;t þ V i;tei;t þ Pi;txi;t þ Si;tθi;t þ V i;tei;t� �

þXt¼1

T Xj¼1

J

AQj;tr j;t þ SDj;tδ j;t þ RDj;td j;t þ IN j;t f j;t

� �

þXt¼1

T

λt

Xi¼1

I

ASTi xi;t þ STi θi;t þ ASRi xi;t þ SRi θi;t� �

−ACAPt

" #

þXt¼1

T Xi¼1

I

γi;t xi;t−URi

Xj¼1

J

Bi; jd j;t

!

ð11Þ

Subject to:






d j;t ≤Mδ j;t ð7Þ

ei;0 ¼ ei;0 ¼ fj;0 ¼ 0

xi;t; ei;t; xi;t; ei;t; fj;t; d j;t; r j;t ≥0

θi;t; θi;t; δ j;t ¼ 0; 1f gλt; γi;t ≥0

ð10Þ

The relaxed model, R-model, can be decomposed intothree sets of sub-models based on new parts, remanufacturedparts, and returned products, respectively. After placing thecoefficients of the same variables together, the three groups ofsub-models, SM1-i and SM2-i , i =1,…,I , and SM3-j , j =1,…,J , are presented below.

SM1-i , i =1,…, I :

minZ1 λð Þ ¼Xt¼1

T

Pi;t þ λtASTi

� �xi;t þ Si;t þ λtST

� �θi;t þ V i;tei;t

� �−Xt¼1

T

λtACAPt

ð12Þ

Table 3 Data for returned products

Time period Cost

Product 1 Product 2 Product 3

Disas. Setup. Inven. Acq. Disas. Setup. Inven. Acq. Disas. Setup. Inven. Acq.

1 30 22 40 25 25 35 50 35 20 30 30 25

2 35 30 40 15 30 25 50 20 18 28 30 28

3 20 33 40 20 15 35 50 30 30 30 30 30

4 15 22 40 21 22 30 50 30 18 32 30 28

5 27 33 40 18 28 27 50 22 14 27 30 30

6 22 38 40 21 18 35 50 28 28 31 30 32

Table 4 Number of parts contained in the returned products

Component type 1 2 3 4 5 6Product type

1 10 10 8 13 8 11

2 12 12 10 12 15 16

3 15 11 3 11 2 9


Subject to:

ei;t−1 þ xi;t−ei;t ¼ Di;t ð2Þxi;t ≤Mθi;t

ei;0 ¼ 0; xi;t; ei;t ≥0

θi;t ¼ 0; 1f gð3Þ

SM2-i , i =1,…, I :

minZ2 λ; γð Þ ¼Xt¼1

T

Pi;t þ λtASRi þ γi;t� �

xi;t þ Si;t þ λtSRi

� �θi;t þ V i;tei;t

h ið13Þ

Subject to:


xi;t ≤Mθi;t

ei;0 ¼ 0; xi;t; ei;t ≥0

θi;t ¼ 0; 1f g

ð5Þ

and

SM3-j , j =1,…, J :

minZ3 γð Þ ¼Xt¼1

T

AQj;tr j;t þ SDj;tδ j;t þ RDj;t−Xi¼1

I

Bi; jγi;tURi

!" #d j;t þ IN j;t f j;t

( )ð14Þ

Subject to


d j;t ≤Mδ j;t

f j;0 ¼ 0; f j;t; d j;t; r j;t ≥0

δ j;t ¼ 0; 1f gð7Þ

Each of the SM1-i and SM2-i sub-models is reduced to aproduction-inventory problem of multiple time periods. Thesesub-problems can be solved using Wagner-Whitin method.Since the purpose of the sub-problem solving is mainly to find

improved feasible solutions in an iterative solution procedure,we use Silver-Meal heuristic method to find near-optimalsolutions of these sub-problems. Sub-model SM3-j cannotbe solved by Wagner-Whitin method or Silver-Meal heuristicas they have a different structure. We develop the followingsimple heuristic to obtain a feasible and near optimal solutionfor each SM3-j . The heuristic method is based on the fact thatEq. (14) will be minimized with all decision variables being 0if all coefficients of dj,t are non-negative. When some of thesecoefficients are negative with sufficient values to offset theobjective function increase due to Eq. (6), Z3(γ ) in Eq. (14)can be reduced if the corresponding dj,t can take the maxi-mum allowable value. The step-by-step procedure of theheuristic method is given below.

Table 6 Resource availabilityTime period 1 2 3 4 5 6

Resource available units 140,000 20,000 40,000 79,000 30,000 30,000

Table 5 Resource time require-ment and quality ratio Component type

1 2 3 4 5 6

New component production time 100 150 100 120 110 100

New component setup time 50 50 40 30 40 50

Remanuf. component production time 80 80 75 80 65 80

Remanuf. component setup time 30 30 35 30 30 25

Quality ratio 0.5 0.5 0.6 0.2 0.7 0.6


3.1 Heuristic method for solving SM3-j

Step 1 Let exi;t be the current solution of sub-problem

2. Let ed j;t be the minimum values satisfying exi;t ≤URi

∑j¼1

JBi; jed j;t . Define eRj;t ¼ RDj;t− ∑

i¼1

IBi; jγi;tURi

� .

Let eAj;1 ¼ AQj;1 . Let d j;1 ¼ r j;1 ¼ ed j;1 , δ j,1=1, and

let t=2 go to step 2.Step 2 Recursively calculate: eAj;t ¼ min AQj;t; IN j;t−1þ

neAj;t−1g .

Step 2.1 If eRj;t ≥− eAj;t � ed j;t þ SDj;t

� �, let dj,t=δ j,

t =rj,t=f j,t=0, go to step 3; otherwise, let

δ j,t=1 and d j;t ¼ ed j;t , go to step 2.2.

Step 2.2 If AQj;t ≤ IN j;t−1 þ eAj;t−1 , let r j;t ¼ ed j;t ,go to step 3.

Otherwise, let s be the last time periodwhen r j,t≠0 and let

r j;s ¼ r j;s þed j;t

fj;k ¼ fj;k þed j;t ; k ¼ s;…; t;

d j;t ¼ δ j;t ¼ 0

Go to step 3.

Step 3 If t =T, stop; otherwise, let t =t +1, go to step 2.After obtaining the solutions of SM1-i , SM2-

i , i =1,…,I , and SM3-j , j =1,…,J , we followthe standard sub-gradient search to maximizethe Z function with regard to Lagrangian multipliersλ and γ .

3.1.1 Summary of the solution procedure to solve the M-model

Step 1 Identify a feasible solution of the original M-model. Let the feasible solution be the incum-bent and the corresponding objective functionbe Zub .

Step 2 Arbitrarily select the first set of Lagrangianmultipliers, λ , γ . Convert the M-model to theR-model by relaxing Eqs. (7) and (8) as shownin Eq. (11) subject to Eqs. (2)–(7), Eq. (10).Decompose R-model into SM1, SM2, andSM3-j , j =1,…,J .

Step 3 Solve SM1, SM2, and SM3 for the current set ofLagrangian multipliers.

If the solution is feasible to M-model andcorresponding Z <Zub, let Zub=Z and the cur-rent solution be the incumbent, go to step 4.

If the solution is infeasible to M-model but thecorresponding Z <Zub, let Zub=Z , go to step 4.

Function Convergence

-2500000

-2000000

-1500000

-1000000

-500000

0

500000

1000000

0 20 40 60 80 100 120 140 160 180 200

Z ValuesL Values

Fig. 1 Convergence of L and Zfunctions

Function Convergence at Ending Iterations

100000

120000

140000

160000

180000

200000

220000

240000

260000

280000

150 160 170 180 190 200

Z Values

L Values

Fig. 2 Convergence of L and Zfunctions at later iterations


Step 4 If stopping criterion is reached, stop; otherwise, usesub-gradient search to update Lagrangian multipliersλ , γ . Go to step 3.

The solution procedure was implementedusing PythonTM software. A non-optimal feasi-ble solution of the original M-model was ob-tained using a simple heuristic method. Thecorresponding objective function value was used

as the initial upper bound required by the sub-gradient search. All computations in testing thesolution procedure were performed on a PCplatform.

4 Numerical examples

Several numerical examples of the considered HMRSproduction planning problem were used to test thesolution procedure presented in the previous sectionwith details of one example problem with six timeperiods presented below to illustrate the developedmodel and solution method. We first present the dataof the problem followed by the computational resultsand analysis.

4.1 Example problem and data

The considered HMRS system producing six differenttypes of components produced from new materials andfrom remanufactured components. The remanufacturedcomponents come from three different types ofreturned products after disassembly. Tables 1 and 2p re s en t the cos t s and demands o f new andremanufactured components, respectively. Table 3 pre-sents similar data for the returned products. The num-bers of components contained in the different productsare shown in Table 4. Table 5 gives the time require-ments to produce the new and remanufactured compo-nents for the shared resource. It also gives the ratio ofeach type component that is of good quality and canbe remanufactured, out of the total retrieved. Table 6 isthe available time for setting up the system and pro-ducing the components by the shared resource in thesix time periods.

4.2 Results and analysis

In solving this example problem, we used 200 iterationsin the sub-gradient search in updating Lagrangian mul-tiplier. The convergence of the Lagrangian functionL (λ , γ ) composed of objective function values of allthe sub-models is apparent. Figure 1 also shows thevalues of the objective function Z of the original M-model. These values are obtained based on the sub-model solutions with corresponding Lagrangian

Table 7 Production quantities of new and remanufactured components

Period Component type

1 2 3 4 5 6

1 New material 50 80 40 45 75 62

Remanufactured 25 0 10 0 0

2 New material 5 0 55 94 0 50


3 New material 50 20 45 48 27 43


4 New material 55 40 45 42 38 39


5 New material 75 0 30 107 0 0


6 New material 35 45 55 30 39 51


Table 8 Inventory levels of new and remanufactured components

Period Component type

1 2 3 4 5 6

1 New material 0 35 0 0 40 0

Remanufactured 15 10 9 15 10 18

2 New material 0 0 0 0 0 0




4 New material 0 43 0 0 77 0







multipliers. More detailed convergence behavior at thelater iterations is repeated in Fig. 2.

The Z value of the best feasible solution obtained withinthe 200 iterations of search is 220,101. The correspondingLagrangian lower bound 212,087. The relative error is lessthan 4 %. Details of the best feasible solution are presented inTables 7, 8, and 9.

It took several seconds on the PC computer to com-plete the 200 iterations for this example problem. Wealso test several other example problems and observedsimilar convergence behavior with comparable computa-tional efforts.

5 Summary and conclusions

Production planning problems in hybrid manufacturing-remanufacturing systems (HMRS) are discussed. Optimaldecisions on system setup, production, and inventory to pro-duce new components and remanufactured components arerequired in solving such problems. The optimal productiondecisions are to be coordinated with decisions for purchasing,disassembly, and inventory of returned products to retrieve thecomponents for remanufacturing. A mixed integer linear pro-gramming (MILP) is developed to obtain optimal solution ofthe considered problem. To efficiently solve the MILP model,a solution procedure based on Lagrangian decomposition isdeveloped so that the original model can be solved throughsolving a set of much smaller sub-problems. The model andsolution procedure were tested using several numerical exam-ples with details of one example problem presented in thepaper. The testing results show that the proposed solutionprocedure can reach optimal or very close to optimal solutionsin short computational time. Our future research includes

extending the model to cover more HMRS features,experimenting with larger size problems and applying theresults in real case applications.

Acknowledgments This research is supported by Discovery Grantfrom NSERC of Canada and by Graduate Studies Support Program(GSSP) from the Faculty of Engineering and Computer Science,Concordia University, Montreal, Quebec, Canada. The authors alsowould like to thank Mr. M. Towhidi at École Polytechnique de Montréalfor his help in coding the developed solution procedure.

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Table 9 Returned productacquisition and disassembly Time period Returned product type

1 2 3

Acquisition Disassembly Acquisition Disassembly Acquisition Disassembly

1 9 9 1 1 0 0

2 0 0 0 0 0 0

3 3 3 0 0 1 1

4 8 8 1 1 0 0

5 0 0 0 0 0 0

6 3 3 1 1 0 0


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14. Zanoni S, Ferretti I, Tang O (2006) Cost performance and bullwhipeffect in a hybrid manufacturing and remanufacturing systems withdifferent control policies. Int J Prod Res 44:3847–3862

15. Zhang J, Liu X, Tu YL (2011) A capacitated production planningproblem for closed-loop supply chain with remanufacturing. Int JAdv Manuf Technol 54:757–766


a mathematical model for production planning in hybrid manufacturing-remanufacturing systems

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