# A mathematical model for production planning in hybrid manufacturing-remanufacturing systems

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<ul><li><p>ORIGINAL ARTICLE</p><p>A mathematical model for production planning in hybridmanufacturing-remanufacturing systems</p><p>M. Chen & P. Abrishami</p><p>Received: 14 October 2013 /Accepted: 5 December 2013 /Published online: 3 January 2014# Springer-Verlag London 2014</p><p>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.</p><p>Keywords Mathematical model . Production planning .</p><p>Manufacturing and remanufacturing . Lagrangian relaxation</p><p>1 Introduction</p><p>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</p><p>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</p><p>M. Chen (*) : P. AbrishamiDepartment of Mechanical and Industrial Engineering, ConcordiaUniversity, 1455 de Maisonneuve W., Montreal, QC,Canada H3G 1M8e-mail: mychen@encs.concordia.ca</p><p>Int J Adv Manuf Technol (2014) 71:11871196DOI 10.1007/s00170-013-5538-0</p></li><li><p>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 Akal and etinkaya [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</p><p>developed model and solution procedure with concludingremarks draw in Sect. 5.</p><p>2 Problem description and modeling</p><p>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 Akal and etinkaya [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.</p><p>Parameters</p><p>Pi,t Unit manufacturing cost of new component i intime period t</p><p>S i,t Setup cost for manufacturing new component i intime period t</p><p>Vi,t Unit inventory cost for new component i in timeperiod t</p><p>Pi;t Unit remanufacturing cost of recovered componenti in time period t</p><p>Si;t Setup cost for remanufacturing recoveredcomponent i in time period t</p><p>V i;t Unit inventory cost for remanufactured componenti in time period t</p><p>Di,t Demand for new component i in time period tDi;t Demand for remanufactured component i in time</p><p>period tB i,j Number of component i contained in product jURi Average recovering rate of component i from all</p><p>returned productsAQj ,t Unit cost to acquire returned product j in time</p><p>period t</p><p>1188 Int J Adv Manuf Technol (2014) 71:11871196</p></li><li><p>RDj,t Unit cost to disassemble returned product j in timeperiod t</p><p>SDj,t Setup cost for disassembling product j in timeperiod t</p><p>INj,t Unit inventory cost for storing returned product j intime period t</p><p>ACAPt Available production time in time period tASTi Production time for manufacturing new</p><p>component iSTi Setup time for manufacturing new component iASRi Production time for remanufacturing returned</p><p>component iSRi Setup time for remanufacturing returned</p><p>component iM A large positive number</p><p>Decision variables</p><p>x i,t Number of new component i to produce in timeperiod t</p><p>e i,t Number of new component i in inventory at the end oftime period t</p><p>xi;t Number of remanufactured component i to produce intime period t</p><p>ei;t Number of remanufactured component i in inventory atthe end of time period t</p><p>d j,t Number of returned product j to disassemble in timeperiod t</p><p>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</p><p>time period t</p><p>i;t 1; If the system is set up to make</p><p>new component i in time period t;0; otherwise:</p><p>(</p><p>i;t 1; If the system is set up to remanufacture</p><p>returned component i in time period t;0; otherwise:</p><p>(</p><p> j;t 1; If returned product j will be disassembled</p><p>in time period t;0; otherwise:</p><p>(</p><p>2.1 MILP model</p><p>Using the notations given above, the objective function andthe constraints of the proposed MILP model for HMRS pro-duction planning problem are presented next.</p><p>M-model</p><p>minZ Xt1</p><p>T Xi1</p><p>I</p><p>Pi;txi;t Si;ti;t V i;tei;t Pi;txi;t Si;ti;t V i;tei;t</p><p>Xt1</p><p>T Xj1</p><p>J</p><p>AQj;tr j;t SDj;t j;t RDj;td j;t IN j;t f j;t </p><p>1</p><p>Subject to:</p><p>ei;t1 xi;tei;t Di;t 2</p><p>xi;t Mi;t 3</p><p>ei;t1 xi;tei;t Di;t 4</p><p>xi;t Mi;t 5</p><p>f j;t d j;t f j;t1 r j;t 6</p><p>d j;t M j;t 7</p><p>Xi1</p><p>I</p><p>ASTixi;t STii;t ASRixi;t SRii;t </p><p>ACAPt 8</p><p>xi;t URiXj1</p><p>J</p><p>Bi; jd j;t 9</p><p>ei;0 ei;0 f j;0 0</p><p>xi;t; ei;t; xi;t; ei;t; fj;t; d j;t; r j;t 0</p><p>i;t; i;t; j;t 0; 1f g</p><p>10</p><p>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</p><p>Int J Adv Manuf Technol (2014) 71:11871196 1189</p></li><li><p>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.</p><p>3 Solution methodology</p><p>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:</p><p>Table 1 Cost and demand for component of new materials</p><p>Time period Cost and demand Component type</p><p>1 2 3 4 5 6</p><p>1 Production cost 100 80 90 90 75 105</p><p>Setup cost 600 600 600 600 600 600</p><p>Inventory cost 100 100 100 100 100 100</p><p>Demand 50 80 40 45 75 62</p><p>2 Production cost 90 100 90 85 110 90</p><p>Setup cost 700 700 700 700 700 700</p><p>Inventory cost 120 120 120 120 120 120</p><p>Demand 40 35 55 51 43 50</p><p>3 Production cost 110 80 95 70 85 95</p><p>Setup cost 600 600 600 600 600 600</p><p>Inventory cost 105 105 105 105 105 105</p><p>Demand 50 20 45 48 27 43</p><p>4 Production cost 95 80 100 80 80 110</p><p>Setup cost 550 550 550 550 550 550</p><p>Inventory cost 115 115 115 115 115 115</p><p>Demand 55 40 45 42 38 39</p><p>5 Production cost 85 100 100 90 110 80</p><p>Setup cost 600 600 600 600 600 600</p><p>Inventory cost 100 100 100 100 100 100</p><p>Demand 35 40 30 30 44 33</p><p>6 Production cost 90 95 100 105 90 90</p><p>Setup cost 700 700 700 700 700 700</p><p>Inventory cost 110 110 110 110 110 110</p><p>Demand 35 45 55 30 39 51</p><p>Table 2 Cost and demand for component from returned product</p><p>Time period Cost and demand Component type</p><p>1 2 3 4 5 6</p><p>1 Production cost 30 20 40 15 20 50</p><p>Setup cost 400 400 400 400 400 400</p><p>Inventory cost 50 50 50 50 50 50</p><p>Demand 10 15 10 7 11 12</p><p>2 Production cost 40 50 30 30 55 35</p><p>Setup cost 500 500 500 500 500 500</p><p>Inventory cost 80 80 80 80 80 80</p><p>Demand 20 10 10 18 13 8</p><p>3 Production cost 20 15 18 18 20 22</p><p>Setup cost 450 450 450 450 450 450</p><p>Inventory cost 60 60 60 60 60 60</p><p>Demand 12 9 10 11 10 8</p><p>4 Production cost 15 30 25 22 32 19</p><p>Setup cost 350 350 350 350 350 350</p><p>Inventory cost 75 75 75...</p></li></ul>