Computers & Industrial Engineering 72 (2014) 72–83

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Computers & Industrial Engineering

journal homepage: www.elsevier .com/ locate/caie

Fuzzy multi-objective recoverable remanufacturing planning decisionsinvolving multiple components and multiple machines q

http://dx.doi.org/10.1016/j.cie.2014.03.0070360-8352/� 2014 Elsevier Ltd. All rights reserved.

q This manuscript was processed by Area Editor Manoj Tiwari.⇑ Tel.: +886 08 7703202x7771; fax: +886 08 7740321.

E-mail address: tyson@mail.npust.edu.tw

Tai-Sheng Su ⇑Department of Industrial Management, National Pingtung University of Science and Technology, 1 Hsueh-Fu Rd., Nei Pu Hsiang, Pingtung 912, Taiwan

a r t i c l e i n f o

Article history:Received 13 August 2013Received in revised form 4 February 2014Accepted 5 March 2014Available online 18 March 2014

Keywords:Lot-sizingRecycled materialsRemanufacturingFuzzy multi-objective linear programmingCO2 emissions

a b s t r a c t

The demand for customization is gradually increasing given that recycled materials no longer meet cus-tomer demands for new products. Meeting the needs of a particular customer may require release ofnumerous new materials and recycled materials to minimize both total production costs and total CO2

emissions. The lot-sizing production-to-order problem is to optimize the lot size for each potential lotrelease. This study focuses on the relationship between new materials and recycled materials under vary-ing production cost, machine yield and capacity and energy consumption. Fuzzy multi-objective linearprogramming (FMOLP) models are used to analyze factors in the relative cost-effectiveness and CO2 emis-sions. The proposed model evaluates cost-effectiveness and CO2 emissions and integrates multi-compo-nent and multi-machine functions for remanufacturing systems. The analytical results can help managersduring decision making by enabling systematic analysis of the potential cost-effectiveness of recoverableremanufacturing.

� 2014 Elsevier Ltd. All rights reserved.

1. Introduction

In recent years, resource depletion and waste have increasedrapidly. Therefore, many countries have begun considering envi-ronmental issues such as reverse logistics and the recycling, reuseand remanufacturing of products. Aside from adhering to strictenvironmental laws, businesses must actively strive to lower theenvironmental impacts of their products. The most effective ap-proach is implementing effective methods of product recycling, re-use and remanufacturing. Reverse logistics enables user-endrecycling of obsolete products by dismantling, cleaning, and restor-ing them in a repetitive cycle of reuse and remanufacture. Forexample, end-user recycling by Hewlett–Packard recovers emptytoner cartridges and prepares them for repeated use (Jorjani, Leu,& Scott, 2004). Therefore, including appropriate use and remanu-facturing of recyclable materials in the design of production sys-tems is an important contemporary issue that must be solved.

As the trend in customization increases, some recycled materi-als can no longer meet customer demands. Another considerationis creating value-added products by combining new and recycledmaterials in remanufacturing methods (Li, Chen, & Cai, 2007).Remanufacturing processes require a tradeoff between the cost

and CO2 emissions of new and recycled materials. That is, newmaterials have a high purchasing cost but a short processing time,which reduces energy use. In contrast, recyclable materials havelow purchasing costs but a long processing time, which results inhigh energy use. Therefore, when companies consider the differ-ences in cost and CO2 emissions they compromise by simulta-neously considering the cost of new materials and the cost ofremanufacturing recycled materials.

In practice, decision makers must simultaneously consider mul-tiple objectives such as lowest total cost and lowest total CO2 emis-sions. However, most research on this topic tends to focus on asingle objective. In actual real lot-sizing production-to-order prob-lems involving recoverable remanufacturing systems, input data orparameters such as forecasting demand, resources, costs andobjective function are often imprecise or fuzzy because some infor-mation is incomplete, unavailable, or unobtainable. These factorsresult in a fuzzy objective function. For example, the objectivefunction of annual production costs may be $0.5 million, and theobjective function of annual CO2 emissions may be 5 metric tons.To minimize the effects of this imprecision, a set of fuzzy multi-objective models is needed to produce a set of compromisesolutions.

In this study, a novel FMOLP model is also used to solve the lot-sizing problem of recoverable remanufacturing with multiple com-ponents and multiple stages in uncertain environments. The pro-posed FMOLP model simultaneously optimizes both total

T.-S. Su / Computers & Industrial Engineering 72 (2014) 72–83 73

production costs and CO2 emissions considering customer demand,machine capacity, lot sizes released and CO2 emissions constraints.That is, the model simultaneously achieves the targets of low car-bon emissions and low cost. The original FMOLP model simulta-neously minimizes total production costs and CO2 emissions withreference to multiple components and multiple machines.

The remainder of the paper is organized as follows: Section 2presents a literature review; Section 3 formulates the fuzzy mul-ti-objective recoverable remanufacturing planning decision model;Section 4 assesses the feasibility of the proposed model in the caseof remanufactured toner cartridges. Finally, Section 5 concludesthe study.

2. Literature review

Remanufacturing is a major line of research in the green opera-tions literature. The techniques used in this study to minimize en-ergy and resources for flow systems and to minimize the use ofvirgin materials are based on the green manufacturing and reman-ufacturing literature (Srivastava, 2007).

2.1. Recoverable manufacturing/remanufacturing

Recoverable manufacturing systems minimize the environmen-tal impact of industry by reusing materials, reducing energy use,and reducing the amount of industrial products contained in land-fill (Guide, Jayaraman, Srivastava, & Benton, 2000). Richter andWeber (2001) formulated the reverse Wagner/Whitin model withadditional variable manufacturing and remanufacturing costs tooptimize inventory. Koh, Hwang, Shon, and Ko (2002) proposed amodel for analyzing an inventory system with recycled productsand newly purchased products for procurement as well as the opti-mal inventory level of recoverable items. Konstantaras and Papa-christos (2008) proposed a closed form expression of the optimalsolution for the Koh et al. (2002) model. Teunter and Haneveld(2002) considered the appliance remanufacturing problem of con-trolling parts inventory in the final phase of service. They also pro-posed a partly graphical method of calculating the optimalinventory policy.

Bhattacharya and Guide (2006) examined the problem of opti-mal retail order quantities from the perspective of a manufacturerof new products and from the prospective of a remanufacturer ofused and unsold products. Jayaraman (2006) proposed an analyti-cal approach to production planning and control for closed-loopsupply chains with product recovery and reuse. The approach in-cluded a linear programming model called the remanufacturingaggregate production planning model for aggregate productionplanning and control. Choi, Hwang, and Koh (2007) developed amathematical model of an inventory system in which stationarydemand can be satisfied by recovered products and newly pur-chased products.

Chung and Wee (2008) analyzed how green product design, thenew technology evolution, and remanufacturing affect production-inventory policy. They also developed an integrated deterioratinginventory model that considered the value of the green-componentlife-cycle during remanufacturing in semi-closed supply chains.Zhou and Wang (2008) designed a reverse logistics network modelthat simultaneously considered both repairing and remanufactur-ing options. Bao, Tang, and Ji (2008) investigated the bimodal prop-erties of disassembly/purchasing lead time distribution in aremanufacturing system. They proposed a minimum relative en-tropy method of estimating how bimodal distribution affects sys-tem performance. Behret and Korugan (2009) analyzed a hybridremanufacturing-manufacturing system that allows for differentquality levels for return flows. They also proposed a multi-stage

inventory control model that considered uncertainties in remanu-factured products, including quality, return rates, and return time.

Konstantaras, Skouri, and Jaber (2010) developed an inventorysystem that combined inspection, sorting, recovery (remanufactur-ing), and ordering of new items for a recoverable product in a re-verse logistics environment. Ahiska and King (2010b) analyzedinventory control for a single-product recoverable manufacturingsystem for the entire product life cycle. Piñeyro and Viera (2010)investigated the lot-sizing problem using different demandstreams for new and remanufactured items. They provided a math-ematical model for solving the problem and developed a Tabu-search-based method for finding the near-optimal solution.

For production-remanufacturing inventory systems, Konstantarasand Skouri (2010) developed inventory models for cases of shortageand no shortage for a single product recovery system with variablesetup numbers. Chung and Wee (2011) developed an integrated pro-duction-inventory deteriorating model that considered the greeningoperation process and life-cycles in a green supply chain inventorycontrol system. Teunter, Douwe, and Flapper (2011) studied prob-lems and decision making related to acquisition and remanufactur-ing. Lot acquisition decisions were optimized under consideration ofuncertainty, multiple quality and multinomial quality distribution.Feng and Viswanathan (2011) evaluated a deterministic model ofproduct recovery in a manufacturing system. They proposed twolot-sizing policies and developed two heuristics for solving the prob-lem. In a mathematical model for green product mix decisions, Tsaiet al. (2012) incorporated capacity expansion features by using alinear programming technique that references machine hour con-straints, direct labor constraints, direct material constraints, CO2

emission constraints and product-level constraints. Amin and Zhang(2012) designed a multi-objective mixed-integer linear program-ming model for optimizing a closed-loop supply chain network ofdisassembly, refurbishing and disposal sites. The objective of net-work configuration was to optimize the number of products andmaterials in each section of the network.

2.2. Remanufacturing planning decisions problem

Van der Laan and Salomon (1997) developed a stochastic inven-tory system with production, remanufacturing, and disposal oper-ations to solve inventory control problem. Their stochasticinventory system attempts to minimize the total expected systemcosts. Van der Laan, Salomon, and Dekker (1999) later examinedhow lead-time duration and lead-time variability affect total ex-pected operational costs in a system with manufacturing/remanu-facturing operations. Guide, Kraus, and Srivastava (1997) designeda simulation model with remanufacturing facility, disassembly re-lease mechanisms, and priority dispatching rules to solve a disas-sembly scheduling problem. Guide (1997) derived a simulationmodel for solving the scheduling policies problem with prioritydispatching rules and drum-buffer-rope in a recoverable manufac-turing system. Richter and Sombrutzki (2000) proposed a reverseWagner/Whitin’s model to solve a remanufacturing planning andinventory control problem, which attempts to minimize the totalproduction costs.

Dobos (2003) developed a two-store reverse logistics modelwith continuous disposal to solve a production and inventory con-trol problem in order to minimize the sum of the holding costs inretail stores and costs of manufacturing, remanufacturing and dis-posal. Kiesmüller (2003) designed a mathematical model with leadtime, deterministic and dynamic demand, and return rates to solvea inventory control problem in a single product recovery system.Seliger, Franke, Ciupek, and Basdere (2004) developed a discrete-event simulation model for solving the process and facilityplanning problem in mobile phone remanufacturing, with theobjective of maximizing the profit margin. Kim, Song, Kim, and

74 T.-S. Su / Computers & Industrial Engineering 72 (2014) 72–83

Jeong (2006) proposed a general framework for remanufacturingsystems and a mathematical model for maximizing total cost sav-ings when making decisions about quantity in a reverse logisticsenvironment. Franke, Basdere, Ciupek, and Seliger (2006) designeda mathematical model with capacity and remanufacturing pro-gram planning to solve a combinatorial optimization problem,the objective is to maximize the profit margin in the planning ofa remanufacturing factory for mobile phones. Lebreton and Tuma(2006) developed a linear programming model to assess the prof-itability problem of car and truck tire remanufacturing. That modelfocuses largely on maximizing the profit margin.

Tang, Grubbström, and Zanoni (2007) formulated a newsboymodel to analyze a disassembly�remanufacturing system for solv-ing a manufacturing planning and control problem in order toimprove the performance of remanufacturing systems. This modelfocuses mainly on minimizing the expected total cost. The totalcost denotes the sum of inventory and out-of-stock costs. Li et al.(2007) analyzed a capacitated dynamic lot-sizing problem withsubstitutions and return products. That study also developed acapacitated multi-period two-product production-planning modelthat considers both remanufacturing and substitution anddetermines the relative quantity levels for manufacturing/remanufacturing.

Sutherland, Jenkins, and Haapala (2010) designed a remanufac-turing facility cost model for solving the optimal lot-size problemin a diesel engine remanufacturing facility. That model focusesmainly on minimizing the total annual cost, including product,operational, inventory, and transportation-related costs. Ahiskaand King (2010a) derived an optimal manufacturing/remanufac-turing inventory model using Markov decision processes to solvea inventory optimization problem in a single product recoverablemanufacturing system. That model focuses largely on minimizingthe expected total cost, including manufacturing cost, remanufac-turing cost, holding cost for serviceable inventory, holding costwork in process, holding cost for recoverable inventory, backorder-ing cost, disposal cost, and lost sales costs.

Shi, Zhang, and Sha (2011) developed a nonlinear programmingmodel for a multi-product closed loop system with uncertain de-mand and return to solve a production planning problem by usinga Lagrangian relaxation based approach. The model focuses mainlyon maximizing the manufacturer’s expected profit by jointly deter-mining the production quantities. Wei, Li, and Cai (2011) designeda linear programming model with uncertain product returns anddemand to solve an inventory and production planning problem.That model focuses largely minimizing the expected total cost.Smith, Smith, and Chen (2012) developed a disassembly sequencestructure graph model for solving a multi-target selective disas-sembly sequence planning problem by using a genetic algorithmto obtain optimal solutions. Li, Xia, Gao, and Chao (2013) designeda selective disassembly planning method to solve a disassemblyplanning problem by using a novel constraint handling algorithmon liquid crystal display (LCD) televisions. Georgiadis and Athana-siou (2013) designed a flexible capacity planning model for close-loop supply chain with remanufacturing and uncertain demand tosolve a capacity management problem. That model focuses mainlyon maximizing the net present value of total supply chain profit fora long-term horizon.

2.3. Current research limitations

In summary, although many studies have explored recoverableremanufacture planning, few have explored how costs and energyaffect the decision-making process when using a combination ofnew and recycled materials. Therefore, the following problemsrequire further study:

1. In practice, the decision-making parameters regarding recover-able remanufacture planning contain uncertainties, includingcost and energy use, which result in a fuzzy objective function.Conflicts among different objectives may also occur. Furtherdevelopment of the FMOLP model is needed to reflect the think-ing processes of decision-makers. Currently, however, researchin this area is very limited.

2. Currently, FMOLP concepts are rarely used to solve lot releasesizes problems regarding recoverable materials and new mate-rials used in remanufacturing systems.

3. Additionally, few of the available models address multiple com-ponents and multiple machines for recoverable remanufactur-ing systems.

3. Problem formulation

3.1. Problem description

A product life cycle is characterized by specific stages, includingresearch, development, introduction, maturity, decline and obso-lescence. Each stage is affected by changes in the flow of raw mate-rials, manufacturing, market distribution, and environmentalissues related to upstream suppliers of a material and downstreammanufacturers for a product that use recycled materials and reducewaste of limited resources. A product manufacturing system thatincludes materials recovery can minimize its environmental im-pact (Guide et al., 2000). A material-recovery system, referred toherein as a recoverable-remanufacturing system (Fig. 1), includesthe following stages: raw material, remanufacturing, reassembly,delivery and use, recycle and waste. Further details can be foundin the remanufacturing process proposed by Ilgin and Gupta(2012), which describes the major decision points in each of thefollowing stages:

1. Raw material: Product A consists of multiple parts (A1, A2, . . . ,AI). New materials and recycled materials can be used for thematerials of each part. These materials are outsourced from dif-ferent suppliers and recyclers. The procurement costs and CO2

emission quantities differ for each supplier and recycler. Forexample, a higher procurement cost of new material partsimplies a lower energy consumption for logistics and distribu-tion. Additionally, a lower procurement cost of recovery mate-rial parts implies a higher energy consumption for logisticsand distribution. Therefore, trade-offs occur between costsand CO2 emission quantities. Procurement staffs must selectthe most appropriate lot size while minimizing costs and CO2

emission quantities;2. Remanufacturing: The same components are processed in the

same machine. Different material sources (i.e. new materialsand recycled materials) are processed in the same machine,resulting in different amounts of energy consumptions and,ultimately, different CO2 emission quantities. For instance,energy consumed by a machine is relatively low, owing to theshort processing time for new material parts in order to ensurelow remanufacturing cost. However, recovery material partsmust be disassembled, repaired, and cleared before use. Thesetup time and processing time for the machine are long. More-over, the machine consumes a relatively high amount of energy,leading to high remanufacturing costs. Production plan mem-bers must select the most appropriate lot size to minimize bothcosts and CO2 emission quantities;

3. Reassembly: The yield, assembly cost, and energy consumptionfor an assembly machine must be considered to determine themost appropriate lot size when the part-subassembly is incor-porated in different assembly machines. The products afterreassembly must also be examined, explaining why the

Fig. 1. Recoverable-remanufacturing system.

T.-S. Su / Computers & Industrial Engineering 72 (2014) 72–83 75

defective products are disassembled and repaired, followed byreturning usable parts to the production line; non-defectiveproducts are packed and delivered to the warehouse forstorage.

4. Recycle and waste: Recycled products are disassembled. Disas-sembled parts must be examined as well. Defective parts aredelivered to the resource recycler or reported for disposal;non-defective products are delivered to the material warehousefor storage.

3.2. Fuzzy multi-objective linear programming (FMOLP) model

While environmentally friendly and profitable, recoverable-manufacturing systems are adopted in numerous products, includ-ing copiers, automobile materials, computers, office furniture, steelmaterials and tires (Jorjani et al., 2004; Li et al., 2007; Srivastava,2007). Moreover, the demand for customization is graduallyincreasing because recoverable materials fail to satisfy the con-sumer demand for new products. To meet the demands of a partic-ular customer, production systems may be necessary to use manynew and recoverable materials. The lot-sizing problem must ac-count for trade-offs such as cost and CO2 emissions of new materi-als and recoverable materials. For instance, new materials have ahigh purchase price while the remanufacturing process has low en-ergy consumption; recoverable materials have a low purchaseprice while the remanufacturing process has high energy con-sumption. Therefore, this study simultaneously considers the dif-ferences in cost and CO2 emissions that they compromise byconsidering the cost of new materials and the cost of remanufac-turing recycled materials simultaneously. The lot-sizing produc-tion-to-order decision-making problem involves determining thelot-size for each possible lot release.

Decision makers must consider the trade-offs among multiplefuzzy objectives. For instance, the objective function for annualproduction costs may be $0.5 million, but annual CO2 emissions

may be 5 tons. This imprecision requires a set of fuzzy goals and aset of compromised solutions. The proposed fuzzy mathematicalprogramming model assumes the following:

1. A single product and demand are known, and all demand mustbe satisfied.

2. The cost and CO2 emissions are known for different processingand reassembly machinery, for different stages of production,and for different components.

3. Each remanufacturing/reassembling station has a limitedcapacity.

4. An inspection operation is performed after the end of eachremanufacturing or assembling station to examine the wholelot produced.

NotationsIndices

i number of components, i = 1, 2, . . . , I j number of remanufacturing machines Mj, j = 1, 2, . . . , J k number of reassembling machines Sk, k = 1, 2, . . . , K Parameters

að1Þij

setup cost of Mj for the ith new materials released tothe jth remanufacturing machine ($/unit time)

�að1Þij

setup cost of Mj for the ith recycled materials releasedto the jth remanufacturing machine ($/unit time)

að2Þk

setup cost of Sk for the WIP (work-in-process) releasedto the kth reassembling machine ($/unit time)

bð1Þij

unit cost of variable production for the ith newmaterials released to the jth remanufacturingmachine ($/unit)

�bð1Þij

unit cost of variable production for the ith recycledmaterials released to the jth remanufacturingmachine ($/unit)

(continued on next page)

76 T.-S. Su / Computers & Industrial Engineering 72 (2014) 72–83

bð2Þk

unit cost of variable production for the WIP (work-in-process) released to the kth reassembling machine ($/unit)

rð1Þij

unit cost of materials for the ith new materialsreleased to the jth remanufacturing machine ($/unit)

�rð1Þij

unit cost of materials for the ith recycled materials tothe jth remanufacturing machine ($/unit)

nqð1Þj

unit cost of disposal for defective products producedby the jth remanufacturing machine ($/unit)

nqð2Þk

unit cost of disposal for defective products producedby the kth reassembling machine ($/unit)

hð1Þj

yield parameter of the jth remanufacturing machine,

0 < hð1Þj < 1

hð2Þk

yield parameter of the kth reassembling machine,

0 < hð2Þk < 1

D customer demand M a very large value Mmax

j

maximal capacity of the jth remanufacturing machine

Nð1Þj

total released lot-sizing for the ith componentsreleased to the jth remanufacturing machine,j = 1, 2, . . . , J

Smaxk

maximal capacity of the kth reassembling machine

Yj

expected non-defect quantity of the jthremanufacturing machine

Yminj

denotes the non-defect quantity of the WIP for jthremanufacturing machine released to the kth

reassembling machine, where Yminj ¼minfYjg,

j = 1, 2, . . . , J (unit)

cqð1Þij

CO2 emissions per unit for the ith new materialsreleased to the jth remanufacturing machine (kg/unit)

�c�qð1Þij

CO2 emissions per unit for the ith recycled materialsreleased to the jth remanufacturing machine (kg/unit)

cqð2Þk

CO2 emissions per unit for the WIP released to the kthreassembling machine (kg/unit)

COmax2

maximal CO2 emissions (kg)

COnew2

CO2 emissions for new materials released to the jth

remanufacturing machine (kg)

Decision variables

nð1Þij

lot size for the ith new materials released to the jthremanufacturing machine

�nð1Þij

lot size for the ith recycled materials released to thejth remanufacturing machine

nð2Þk

lot size of the WIP released to the kth reassemblingmachine

Wð1Þj

binary variable indicating the need for a setup for theith new materials released to the jth remanufacturing

machine Wð1Þj ¼

0 if nð1Þij ¼ 0

1 if nð1Þij > 0:

(

Wð1Þj

binary variable indicating whether or not a setup isrequired for the ith recycled materials released to thejth remanufacturing machine

Wð1Þj ¼

0 if �nð1Þij ¼ 0

1 if �nð1Þij > 0:

(

Wð2Þk

binary variable indicating whether or not a setup isrequired for the WIP released to the kth reassembling

machine Wð2Þk ¼

0 if nð2Þk ¼ 01 if nð2Þk > 0:

(

3.2.1. Objective functionsThis study selected the multi-objective function to solve the

recoverable remanufacturing lot-sizing decision-making problemin practical situations. In practice, goals and related input parame-

ters are often uncertain. For instance, costs, energy consumption,capacity and the objective function are often imprecise or fuzzydue to incomplete, unavailable, or unavailable information. There-fore, this study simultaneously considers the following two fuzzyobjective functions in the design of the original FMOLP model:

1. Minimize total production costs

Z1ffiXI

i¼1

XJ

j¼1

að1Þij W ð1Þj

� �þ bð1Þij nð1Þij

� �þ rð1Þij nð1Þij

� �h in

þ �að1Þij W ð1Þj

� �þ �bð1Þij

�nð1Þij

� �þ �rð1Þij

�nð1Þij

� �h i

þ nqð1Þj

� �1�hð1Þj

� �Nð1Þj

� �h ioþXK

k¼1

að2Þk W ð2Þk þbð2Þk nð2Þk

� �h in

þ nqð2Þk

� �1�hð2Þk

� �nð2Þk

� �h ioð1Þ

2. Minimize total CO2 emissions

Z2 ffiXI

i¼1

XJ

j¼1

cqð1Þij nð1Þij

� �þ �c�qð1Þij

�nð1Þij

� �h iþXK

k¼1

cqð2Þk nð2Þk

� �ð2Þ

Symbol ‘‘ffi’’ in Eqs. (1) and (2) is the fuzzified version of ‘‘=’’ andrefers to the fuzzification of aspiration levels. For each objectivefunction of the proposed FMOLP model, the decision maker (DM)is assumed to have a fuzzy objective. For example, for annual totalproduction costs, the objective may be $0.5 million; for annual totalCO2 emissions, the objective may be 5 tons. Thus, Eqs. (1) and (2) arefuzzy and have imprecise aspiration levels. They also incorporatevariations of the DM judgments regarding solutions for the fuzzyoptimization problem. These fuzzy goals require simultaneous opti-mization by a DM in the framework of a fuzzy aspiration levels.

3.2.2. Constraints

XK

k¼1

hð2Þk nð2Þk PD ð3Þ

Nð1Þj ¼nð1Þij þ�nð1Þij ; i¼ j; j¼1;2; . .. ;J ð4Þ

Nð1Þj PD

hð1Þj �hð2Þk

; j¼1;2;. . .;J; k¼1;2;. . .K ð5Þ

Yj¼hð1Þj Nð1Þj ; Yj PXK

k¼1

nð2Þk ; j¼1;2; . .. ;J ð6Þ

Yminj ¼minfYjg; j¼1;2; .. .J ð7Þ

Nð1Þj 6Mmaxj ; j¼1;2; .. . ;J ð8Þ

nð2Þk 6Smaxk ; k¼1;2;. . .;K ð9Þ

nð1Þij 6M �W ð1Þj ð10Þ

�nð1Þij 6M �W ð1Þj ð11Þ

nð2Þk 6M �W ð2Þk ð12Þ

XI

i¼1

XJ

j¼1

ðcqð1Þij nð1Þij Þþ �c�qð1Þij�nð1Þij

� �h iþXK

k¼1

ðcqð2Þk nð2Þk Þ6COmax2 ð13Þ

XI

i¼1

cqð1Þij nð1Þij 6COnew2 ð14Þ

W ð1Þj ;W ð1Þ

j ;W ð2Þk 2f0;1g; j¼1;2; .. . ;J; k¼1;2;. . .;K ð15Þ

nð1Þij ;�nð1Þij ;n

ð2Þk P0 and integer; i¼1;2; . .. ;I; j¼1;2; . . .;J; k¼1;2; . .. ;K ð16Þ

Eq. (3) ensures that the total quantity of non-defective products iseither equivalent to, or is greater than the demand quantity. Eq.(4) indicates that the total amount of lot release sizes includes thelot size of new material and recycled materials. Eq. (5) reveals thatthe total amount of lot release size for varying yields of a machinemust satisfy the demand quantity. Eq. (6) represents the minimumnumber of non-defective products that a remanufacturing machinemust produce in terms of lot release sizes of reassembling ma-

T.-S. Su / Computers & Industrial Engineering 72 (2014) 72–83 77

chines. Eq. (7) shows the non-defect quantity produced by theremanufacturing machine for part j. Non-defective quantity pro-duced by the remanufacturing machine belongs to the work in pro-cess (WIP) in the current stage. Only acceptable WIPs are allowed tobe incorporated into the reassembly machine, due to constraints ofthe lot release size for assembly operation. Eqs. (8) and (9) repre-sent the limits of lot release sizes if the remanufacturing and reas-sembling machine cannot exceed the maximum available machinecapacity, respectively. Eqs. (10)–(12) represent the constraints of lotrelease sizes for the remanufacturing and reassembling machines,respectively. Eqs. (13) and (14) represent the limit of annual totalCO2 emissions resulting from new material released to remanufac-turing machines, respectively; Eqs. (15) and (16) model the binaryand decision variable constraints, respectively.

3.3. Solving the FMOLP model

The original FMOLP model for solving previous problems canapply the piecewise linear membership function developed byHannan (1981). Hannan proposed that piecewise linear member-ship functions should be specified to represent the fuzzy scenariotogether with the fuzzy decision-making of Bellman and Zadeh(1970). Here, a minimum operator is used to integrate the fuzzyset and to transform the original FMOLP model into a single objec-tive linear programming model, and the auxiliary variable L(0 6 L 6 1) is introduced as a measure of satisfaction with the deci-sion-making. The derivation is detailed in Appendix A.

3.4. Solution procedure for the FMOLP model

The proposed interactive FMOLP method uses the followingprocedure to solve the recoverable remanufacturing lot-sizingdecision-making problem.

Step 1: Formulate the original FMLOP model for solving therecoverable remanufacturing lot-sizing decision-mak-ing problem according to Eqs. (1)–(16).

Step 2: Derive a membership function fg(zg) for each of theobjective functions zg (g = 1, 2).

Step 3: Connect the points in the discrete membership func-tion with line segments (fg(zg)).

Step 4: Formulate the piecewise linear equations for eachfuzzy objective function using Eqs. (A.1)–(A.7).

Step 5: Introduce the auxiliary variable L (0 6 L 6 1), whichtransforms the original FMOLP model for the recover-able remanufacturing lot-sizing decision-making prob-lem into an equivalently crisp LP form by using theminimum operator to aggregate fuzzy sets.

Step 6: Repeat, implement, and modify the above process untilthe solution is satisfactory.

3.5. Solution procedure for the weighted additive FMOLP model

In this model, a linear weighted utility function is obtained bymultiplying each membership function of the fuzzy objectives bytheir corresponding weights and then adding the results. If theimportance of each objective function differs, a weight is required.The weighted additive models proposed by Tiwari, Dharmahr, andRao (1987) are expressed as:

lDðZgÞ ¼XG

g¼1

wg � fgðZgÞ ð17Þ

XG

g¼1

wg ¼ 1; 0 6 wg 6 1 ð18Þ

where wg is the weighting coefficient that presents the relativeimportance among the fuzzy objectives and lD(Zg) is the member-ship function for Zg. The following crisp single objective program-ming is equivalent to the above model:

MaxXG

g¼1

wg � Lg ð19Þ

s:t: Lg 6 fgðZgÞ; g ¼ 1;2; . . . ;G ð20Þ0 6 Lg 6 1 8g ð21Þ0 6 wg 6 1 8g ð22ÞXG

g¼1

wg ¼ 1 ð23Þ

Eqs. (3)–(15).

4. Model implementation

4.1. Case descriptions

The company examined in the case study manufactures envi-ronmentally-friendly toner cartridges. Located in central Taiwan,the company produces toner cartridges for brands such as Hew-lett–Packard and Epson. The main target markets for their productsare Africa, Asia, Australia, Europe, North America, and the MiddleEast. Factors considered during decision making for the lot size re-lease production plan include the cost and quality of new materialsversus recycled materials, the difference in energy use of differentmaterial sources when used with factors such as processingmachinery and machinery yield. Production managers attempt tominimize both total production costs and CO2 emissions. Produc-tion managers estimate lot release amounts based on their experi-ence, often leading to lot release sizes that are either too large ortoo small. The proposed FMOLP model provides production manag-ers with an effective means of optimizing lot release sizes undermultiple fuzzy and conflicting objectives.

A supporting expert panel comprising the managers of the pro-curement, production, general services, and accounting depart-ments is organized to provide the production manager withrelevant information. The panel reviews information obtained di-rectly from managers. Data of production statistics are accumu-lated from January to June in 2011, including electric bills,machine processing time, production batch for new materials,and recycled materials to estimate average new materials andrecycled materials per each unit, as well as the unit average powerconsumption in order to incorporate them in the remanufacturingmachines and reassembly machines. Next, the average unit powerconsumption is multiplied by the carbon dioxide emission coeffi-cient to obtain the new materials and recycled materials per unit,the CO2 emissions when incorporating them in the remanufactur-ing machines and reassembly machines.

Data collection was completed in early 2011, and the FMLOPmodel for solving the recoverable remanufacturing lot-sizing deci-sion-making problem was completed later in the same year. Table 1summarizes the detailed data for remanufacturing and reassem-bling machines for the firm, including costs, CO2 emissions, capac-ity limit and yield rate. The problem focuses on satisfying thecustomer demand for 1200 units (D = 1200) without surpassingthe limit on annual total CO2 emissions (29,700 kg) or the limiton CO2 emissions resulting from new material released to reman-ufacturing machines (20,000 kg).

4.2. Solution procedure for the company case

The recoverable remanufacturing lot-sizing decision-makingproblem in the Company case can be solved using the procedure

Table 1Related parameters of remanufacturing and reassembling machines.

Components A1 A2 A3

Remanufacturing machines M1 M2 M3 Reassembling machines S1 S2

að1Þij1100 800 600 að2Þk

800 800

�að1Þij1100 800 600 bð2Þk

10 10

bð1Þij3 3 3 hð2Þk

0.9 0.9

�bð1Þij6 6 6 nqð2Þk

2 2

rð1Þij20 16 16 cqð2Þk

4 4

�rð1Þij10 8 2

nqð1Þj4 3 2

hð1Þj0.9 0.85 0.8

Mmaxj 2000 2000 2000

cqð1Þij4 2 1

�c�qð1Þij8 4 2

($) 0.0

0.2

0.4

0.6

0.8

1.0

0 80000 110000 125000 132500 132500

Fig. 2. Curve for piecewise linear membership function (z1, f1(z1)).

0.8

1.0

78 T.-S. Su / Computers & Industrial Engineering 72 (2014) 72–83

presented in Section 3.4. Imprecise recoverable remanufacturinglot-sizing can be formulated as a fuzzy multi-objective problemby applying piecewise linear functions (Hannan, 1981) as follows:

Step 1: Use the conventional LP model to obtain initial solutionsfor each objective function. Here, the solutions arez1 = $78,201 and z2 = 16,126 kg. Next, use the initialsolutions to formulate the FMOLP model. Table 2 liststhe piecewise linear membership functions of the pro-posed model.

Step 2: With reference to Table 1, plot the piecewise linearmembership functions (z1, f1(z1)) and (z2, f2(z2)) (Figs. 2and 3, respectively). The curves corresponding to thesegments of the fuzzy objective with the Hannanmembership function are nearly linear.

Step 3: Express the piecewise linear membership functions inthe following format:

TabPiec

zf

zf

(kg)0.0

0.2

0.4

0.6

0 16650 17650 18650 19650 19650

Fig. 3. Curve for piecewise linear membership function (z2, f2(z2)).

f1ðz1Þ ¼ �0:000023333jz1 � 110;000j � 0:00000667jz1

� 125;000j � 0:00003667 � z1 þ 5:1833 ð24Þ

f2ðz2Þ ¼ �0:0001jz2 � 17;650j � 0:00005jz2

� 18;650j � 0:00035 � z2 þ 7:0775 ð25Þ

Step 4: Introduce the nonnegative deviational variables, whichyields

z1 þ d�11 � dþ11 ¼ 110;000 ð26Þ

z1 þ d�12 � dþ12 ¼ 125; 000 ð27Þ

z2 þ d�21 � dþ21 ¼ 17;650 ð28Þ

z2 þ d�22 � dþ22 ¼ 18;650 ð29Þ

le 2ewise membership functions.

1 >132,500 132,500 125,000 110,000 80,000 <80,0001(z1) 0 0 0.5 0.8 1.0 1.0

2 >19,650 19,650 18,650 17,650 16,650 <16,6502(z2) 0 0 0.5 0.8 1.0 1.0

Step 5: Formulate the piecewise linear Eqs. for each member-ship function fgðzgÞ, g ¼ 1;2.

f1ðz1Þ ¼ �0:00002333ðd�11 � dþ11Þ � 0:00000667ðd�12 � dþ12Þ� 0:00003667 � z1 þ 5:1833 ð30Þ

f2ðz2Þ ¼ �0:0001ðd�21 � dþ21Þ � 0:00005ðd�22 � dþ22Þ� 0:00035 � z2 þ 7:0775 ð31Þ

Step 6: Introduce the auxiliary variable L ð0 6 L 6 1Þ, and trans-form the original FMOLP problems into an equivalentlycrisp LP form by using the minimum operator to aggre-gate fuzzy sets. The resulting equivalent crisp LP formfor solving the fuzzy multi-objective recoverableremanufacturing lot-sizing decision-making problemcan be formulated as follows.

Table 3Compromise solutions for the a company by the proposed FMOLP approach.

Item Output solutions

nð1Þij , �nð1Þij , Nð1Þj , nð2Þk nð1Þ11 ¼ 1490, �nð1Þ11 ¼ 0 , Nð1Þ1 ¼ 1490,

nð2Þ1 ¼ 1334

nð1Þ22 ¼ 1580, �nð1Þ22 ¼ 0, Nð1Þ2 ¼ 1580,

nð2Þ2 ¼ 0

nð1Þ33 ¼ 419, �nð1Þ33 ¼ 1251, Nð1Þ3 ¼ 1670

L and objective values L = 0.8546,z1 = $101,740,z2 = 17,377 kg

T.-S. Su / Computers & Industrial Engineering 72 (2014) 72–83 79

Max L

s:t: L6�0:00002333 d�11�dþ11

� ��0:00000667

d�12�dþ12

� ��0:00003667 � z1þ5:1833

L6�0:0001 d�21�dþ21

� ��0:00005 d�22�dþ22

� ��0:00035 � z2þ7:0775

z1þd�11�dþ11 ¼ 110;000

z1þd�12�dþ12 ¼ 125;000

z2þd�21�dþ21 ¼ 17;650

z2þd�22�dþ22 ¼ 18;650

Eqs. (3)–(15)

L; d�1h;dþ1h;d

�2h;d

þ2h;n

ð1Þij ; �n

ð1Þij ;n

ð2Þk P 0 8h; 8i; 8j; 8k

Table 4Data for Scenario 1.

Run Variation rate z2 >19,650 19,f2(z2) 0 0

1 �10% z1 >119,250 119f1(z1) 0 0

2 �5% z1 >125,875 125f1(z1) 0 0

3 0% (Original levels) z1 >132,500 132f1(z1) 0 0

4 5% z1 >139,125 139f1(z1) 0 0

5 10% z1 >145,750 145f1(z1) 0 0

Table 5Data for Scenario 2.

Run Variation rate z1 >132,500 132f1(z1) 0 0

1 �10% z2 >17,685 17,f2(z2) 0 0

2 �5% z2 >18,668 18,f2(z2) 0 0

3 0% (Original levels) z2 >19,650 19,f2(z2) 0 0

4 5% z2 >20,633 20,f2(z2) 0 0

5 10% z2 >21,615 21,f2(z2) 0 0

Table 6Data for Scenarios 3–5.

Scenarios Item Run 1 Run 2

Scenario 3 bð1Þij ($/unit) 2.4 2.7

Scenario 4 �bð1Þij ($/unit) 4.8 5.4

Scenario 5 COmax2 (kg) 16,335 16,632

For the ordinary single-objective LP model for fuzzy multi-objec-tives with a recoverable remanufacturing lot-sizing decision-mak-ing problem, running the above Eqs. in the linear programmingsoftware LINGO version 11.0 obtains z1 = $101,740, andz2 = 17,377 kg. Overall satisfaction with the determined goal valuesis 0.8546. Table 3 reveals that the proposed model yields anefficient compromise solution for recoverable remanufacturinglot-sizing decisions.

4.3. Sensitivity analysis

The sensitivity of decision parameters in the FMOLP model isanalyzed in the following five scenarios involving three compo-nents, three remanufacturing machines, and two reassemblingmachines:

1. Scenario 1: The (z2, f2(z2)) are set to their original values in thenumerical example, and only (z1, f1(z1)). are varied. Table 4 pre-sents the data used to implement Scenario 1.

2. Scenario 2: The (z1, f1(z1)) are set to their original values in thenumerical example, and only (z2, f2(z2)). are varied. Table 5 pre-sents the data used to implement Scenario 2.

3. Scenario 3: Sensitivity is analyzed by changing the variable pro-duction cost per unit for the new materials under the decisionconditions of the preceding numerical example.

650 18,650 17,650 16,650 <16,6500.5 0.8 1.0 1.0

,250 112,500 99,000 72,000 <72000.5 0.8 1.0 1.0

,875 118,750 104,500 76,000 <76,0000.5 0.8 1.0 1.0

,500 125,000 110,000 80,000 <80,0000.5 0.8 1.0 1.0

,125 131,250 115,500 84,000 <84,0000.5 0.8 1.0 1.0

,750 137,500 121,000 88,000 <88,0000.5 0.8 1.0 1.0

,500 125,000 110,000 80,000 <80,0000.5 0.8 1.0 1.0

685 16,785 15,885 14,985 <14,9850.5 0.8 1.0 1.0

668 17,718 16,768 15,818 <15,8180.5 0.8 1.0 1.0

650 18,650 17,650 16,650 <16,6500.5 0.8 1.0 1.0

633 19,583 18,533 17,483 <17,4830.5 0.8 1.0 1.0

615 20,515 19,415 18,315 <18,3150.5 0.8 1.0 1.0

Run 3 Run 4 Run 5

3 (Original levels) 3.3 3.6

6 (Original levels) 6.6 7.2

16,929 17,226 29,700 (Original levels)

Table 7Summary of implementation results for five scenarios.

Scenario Objective Run 1 Run 2 Run 3 Run 4 Run 5

Scenario 1 z1 96,026 100,332 101,740 103,082 104,314z2 18,318 17,505 17,377 17,255 17,143L(%) 59.92% 82.86% 85.46% 87.89% 90.14%

Scenario 2 z1 114,902 108,825 101,740 96,531 94,692z2 16,126 16,733 17,377 17,796 18,852L 70.15% 80.74% 85.46% 88.94% 90.17%

Scenario 3 z1 100,188 100,972 101,740 102,505 103,251z2 17,325 17,351 17,377 17,402 17,427L 86.50% 85.97% 85.46% 84.96% 84.46%

Scenario 4 z1 100,679 101,211 101,740 102,304 102,879z2 17,341 17,358 17,377 17,395 17,414L 86.17% 85.82% 85.46% 85.09% 84.70%

Scenario 5 z1 113,203 109,936 106,669 103,402 101,740z2 16,335 16,632 16,929 17,226 17,377L 73.55% 80.00% 82.18% 84.36% 85.46%

0.84

0.86

0.88

0.9

0.92

0.94

0.96

0.98

1

0

20000

40000

60000

80000

100000

120000 (L)

Fig. 4. Total production costs, total CO2 emissions and degree of satisfaction forvarious weights.

80 T.-S. Su / Computers & Industrial Engineering 72 (2014) 72–83

4. Scenario 4: Analyze sensitivity by changing the variable produc-tion cost per unit for the recycled materials under the decisionconditions of the preceding numerical example.

5. Scenario 5: Analyze sensitivity by changing the maximal CO2

emissions under the decision conditions of the precedingnumerical example.

6. Table 6 presents the implementation data for Scenarios 3–5.

4.4. Analytical results

4.4.1. Sensitivity analysis for variable scenarioTable 7 summarizes the results obtained after implementing

the above five scenarios.Several important management implications are identified:

1. Comparisons of Scenario 1 and Scenario 2 with the numericalexample (run 3 in Scenario 1) revealed the conflicts andtrade-offs among dependent objective functions. Thus, the pro-posed FMOLP model meets the practical requirement of simul-taneously minimizing total production costs and CO2 emissions.

2. The sensitivity analysis of unit new materials cost and unitrecycled materials cost in Scenarios 3 and 4 reveals that thechange in each cost category affects the objective functionsand L values. Specifically, the cost of additional units of newmaterials (recycled materials) may increase the objective func-tion values and simultaneously decrease satisfaction. Forinstance, satisfaction decreased from 0.8617 to 0.8470 in Sce-nario 4.

3. The Scenario 5 results indicate that the maximal CO2 emissionsaffect the objective functions and L values. Specifically, thedecreased CO2 emissions resulted in slightly higher objectivefunction values relative to production costs; however, overallsatisfaction increased from 0.7355 to 0.8546 in Scenario 5.

Table 8Test of weight ratios.

Case Weights (w1, w2) Z1 L1

1 (0.9, 0.1) 96,532 0.88942 (0.8, 0.2) 96,532 0.88943 (0.7, 0.3) 109,738 0.80144 (0.6, 0.4) 109,738 0.80145 (0.5, 0.5) 109,738 0.80146 (0.4, 0.6) 109,738 0.80147 (0.3, 0.7) 109,738 0.80148 (0.2, 0.8) 109,738 0.80149 (0.1, 0.9) 109,738 0.8014

4.4.2. Analysis of various weightsExactly how the weights impact each objective function and

satisfaction degree was examined by using 9 test cases. Each testcase was a specific combination of (w1, w2), where w1 e {0.9, 0.8,0.7, 0.6, 0.5, 0.4, 0.3, 0.2, 0.1} and w2 e {0.1, 0.2, 0.3, 0.4, 0.5, 0.6,0.7, 0.8, 0.9}. Table 8 displays the solution for changing the weightwith the objection values and satisfaction degree. Analysis resultsindicate that various weights for total production costs and totalCO2 emissions affect the objective functions (Z1, Z2) and degree ofsatisfaction (L) (Fig. 4).

Table 8 and Fig. 4 reveal that the satisfaction degree is reducedfrom 0.8761 to 0.8609 when the weight for total production costs(Z1) is reduced from 0.9 to 0.7. This situation is in contrast to theincrease of the satisfaction degree from 0.8808 to 0.9801 whenthe weight for total CO2 emissions (Z2) is increased from 0.4 to0.9. These results reflect actual circumstances.

5. Conclusions

This work presents a novel fuzzy multi-objective recoverableremanufacturing planning decision-making model with multi-components and multi-machines. In contrast with previous works,the proposed model elucidates the relationship between newmaterials and recycled materials with a specific production cost,machine yield, and CO2 emissions. The proposed model considersthe factors involved when integrating multi-components and mul-ti-machines, including lot-sizing decisions, machine yield andcapacity, energy consumption, and costs. Satisfaction is used toevaluate the fuzzification of the results when they are used fordecision making.

The proposed FMOLP model simultaneously optimizes totalproduction costs and total CO2 emissions under the considerationsof demand, lot-sizing decisions, machine yield and capacity, energyconsumption and related production costs. However, flexible

Z2 L2 Satisfaction degree (L)

17,796 0.7562 0.876117,796 0.7562 0.862816,650 1.0000 0.860916,650 1.0000 0.880816,650 1.0000 0.900716,650 1.0000 0.920516,650 1.0000 0.940416,650 1.0000 0.960316,650 1.0000 0.9801

Table A1Membership function fg(zg).

z1 >X10 X10 X11 . . . X1H X1,H+1 <X1,H+1

f1(z1) 0 0 q11 . . . q1H 1.0 1.0z2 >X20 X20 X21 . . . X2H X2,H+1 <X2,H+1

f2(z2) 0 0 q21 . . . q2H 1.0 1.0

( )g gf Z

gZ

0.8

0.5

0gXg1XgHX, 1g HX +

1

0...

Fig. A1. Piecewise membership function fg(zg).

T.-S. Su / Computers & Industrial Engineering 72 (2014) 72–83 81

models that consider economic and environmental issues are stillneeded. Moreover, the proposed FMOLP model provides a system-atic procedure that production managers can use for decision mak-ing by interactively modifying the membership functions of theobjective until a satisfactory degree is acquired.

Analysis results indicate that the satisfaction degree rises from0.90 to 0.98. This finding suggests that a decision maker focuses onCO2 emissions more than production costs when the weight for to-tal CO2 emissions is increased from 0.5 to 0.9. Therefore, in recov-erable remanufacturing planning, the most appropriate newmaterials and recycled materials must be determined appropri-ately to select the lot size and incorporate them in the most appro-priate machines for production. Therefore, low carbon and low costare achievable.

The case study of a Taiwan toner cartridge manufacturerdemonstrates the feasibility of applying the proposed model.Implementation results confirm that the lot-sizing decisionmodel can greatly facilitate decision makers during productionplanning in fuzzy environments and satisfy practical managerialrequirements.

We recommend that future research extend the proposedmodel to evaluate remanufacturing decisions that involvevarious demand uncertainty and other cost components.Additionally, although this work implements a systematic deci-sion-making procedure and obtains feasible solutions, additionaldata analysis is necessary to implement the proposed FMOLPmodel.

Acknowledgement

The author would like to thank the National Science Council ofthe Republic of China, Taiwan for financially/partially supportingthis research under Contract No. NSC 100-2218-E-020-004. TedKnoy is appreciated for his editorial assistance.

Appendix A

The complete equivalent ordinary LP model for solving therecoverable remanufacturing lot-sizing decision-making problemwith multiple fuzzy objectives is derived (Hannan, 1981) asfollows:

Assuming X10, X11, . . . , X1,H+1, q11, . . . , q1H represent each objec-tive values and fg(zg) scales, then,

Step 1: Derive a membership function fg(zg) for each of the objec-tive functions zg (g = 1, 2). Table A1 represents the piece-wise linear membership functions, f1(z1) and f2(z2).where 0 6 qgh 6 1.0, and qgh 6 qg,h+1 for g = 1, 2; h =1, 2, . . . , H.

Step 2: Connect the points in the discrete membership functionwith line segments (zg, fg(zg)) for g = 1, 2 (see Fig. A1).

Step 3: Express each piecewise linear membership functionfg(zg) in the following form:

fgðzgÞ ¼XH

h¼1

aghjzg � Xghj þ bgzg þ cg ; g ¼ 1;2; . . . ;G;

h ¼ 1;2; . . . ;H ðA:1Þ

where

agh ¼ �tg;hþ1 � tgh

2; bg ¼

tg;Hþ1 þ tg1

2; cg ¼

sg;Hþ1 þ sg1

2

Assume that fg(zg) = tgrzg + Sgr for each segment Xg,r-1 6 zg 6 Xgr

where tgr denotes the slope and Sgr represents the y-intercept of

the line segment at [Xg,r�1, Xgr] in the piecewise linear membershipfunction. Thus, we have

f1ðz1Þ ¼ �t12 � t11

2

� �jz1 � X11j �

t13 � t12

2

� �jz1 � X12j

� � � � � t1;Hþ1 � t1H

2

� �jz1 � X1Hj

þ t1;Hþ1 þ t11

2

� �z1 þ

S1;Hþ1 þ S11

2;

t1;hþ1 � t1h

2

� �–0;

h ¼ 1;2; . . . ;H ðA:2Þ

where

t11 ¼q11 � 0

X11 � X10

� �; t12 ¼

q12 � q11

X12 � X11

� �; . . . ; t1;Hþ1

¼ 1:0� q1H

X1;Hþ1 � X1H

� �;

and S1,H+1 delineates the intercept between the line segment of X1H

and X1,H+1 to the vertical line derived from f1(z1) = t1rz1 + s1r .

f2ðz2Þ ¼ �t22 � t21

2

� �jz2 � X21j �

t23 � t22

2

� �jz2 � X22j

� . . .� t2;Hþ1 � t2H

2

� �jz2 � X2Hj

þ t2;Hþ1 þ t21

2

� �z2 þ

S2;Hþ1 þ S21

2;

t2;hþ1 � t2h

2

� �–0;

h ¼ 1;2; . . . ;H ðA:3Þ

82 T.-S. Su / Computers & Industrial Engineering 72 (2014) 72–83

where

t21 ¼q21 � 0

X21 � X20

� �; t22 ¼

q22 � q21

X22 � X21

� �; . . . ; t2;Hþ1

¼ 1:0� q2H

X2;Hþ1 � X2H

� �

and S2,H+1 delineates the intercept between the line segment of X2H

and X2,H+1 to the vertical line derived from f2(z2) = t2rz2 + s2r .Step 4: Introduce the nonnegative deviational variables dþgh and

d�gh

XI

i¼1

XJ

j¼1

að1Þij W ð1Þj

� �þ bð1Þij nð1Þij

� �þ rð1Þij nð1Þij

� �h iþ �að1Þij W ð1Þ

j

� �hn

þ �bð1Þij�nð1Þij

� �þ �rð1Þij

�nð1Þij

� �iþ nqð1Þj

� �1� hð1Þj

� �Nð1Þj

� �h io

þXK

k¼1

að2Þk W ð2Þk þ bð2Þk nð2Þk

� �h iþ nqð2Þk

� �1� hð2Þk

� �nð2Þk

� �h in oþ d�1h � dþ1h ¼ X1h; h ¼ 1;2; . . . ;H ðA:4Þ

XI

i¼1

XJ

j¼1

cqð1Þij nð1Þij

� �þ �c�qð1Þij

�nð1Þij

� �h iþXK

k¼1

cqð2Þk nð2Þk

� �þ d�2h � dþ2h

¼ X2h; h ¼ 1;2; . . . ;H ðA:5Þ

where d�gh and dþgh denote the negative and positive deviational vari-ables, respectively, at the jth point and where Xgh represents thevalues of the gth objective function at the hth point.

Step 5: Substitute Eqs. (A.4) and (A.5) into (A.2) and (A.3),respectively. Here, the substitution yields

f1ðZ1Þ ¼ �t12 � t11

2

� �d�11 � dþ11

� �� t13 � t12

2

� �d�12 � dþ12

� �� . . .� t1;Hþ1 � t1H

2

� �d�1H � dþ1H

� �þ t1;Hþ1 þ t11

2

� �XI

i¼1

XJ

j¼1

að1Þij W ð1Þj

� �þ bð1Þij nð1Þij

� �þ rð1Þij nð1Þij

� �h in(

þ �að1Þij W ð1Þj

� �þ �bð1Þij

�nð1Þij

� �þ �rð1Þij

�nð1Þij

� �h iþ nqð1Þj

� �1� hð1Þj

� �Nð1Þj

� �h io

þXK

k¼1

að2Þk W ð2Þk þ bð2Þk nð2Þk

� �h in

þ nqð2Þk

� �1� hð2Þk

� �nð2Þk

� �h iooþ S1;Hþ1 þ S11

2ðA:6Þ

f2ðZ2Þ ¼ �t22 � t21

2

� �d�21 � dþ21

� �� t23 � t22

2

� �d�22 � dþ22

� �� . . .� t2;Hþ1 � t2H

2

� �d�2H � dþ2H

� �

þ t2;Hþ1 þ t21

2

� � XI

i¼1

XJ

j¼1

cqð1Þij nð1Þij

� �þ �c�qð1Þij

�nð1Þij

� �h i(

þXK

k¼1

cqð2Þk nð2Þk

� �)þ S2;Hþ1 þ S21

2ðA:7Þ

Step 6: Introduce the auxiliary variable L (0 6 L 6 1), and trans-form the original FMOLP problem into an equivalentcrisp LP form by using the minimum operator to aggre-gate fuzzy sets. The equivalent crisp LP form for solvingthe fuzzy multi-objective recoverable remanufacturinglot-sizing decision-making problem can be formulatedas

max L

s:t: L6� t12� t11

2

� �d�11� dþ11

� �� t13� t12

2

� �d�12� dþ12

� �� . . .� t1;Hþ1� t1H

2

� �d�1H � dþ1H

� �þ t1;Hþ1þ t11

2

� �XI

i¼1

XJ

j¼1

að1Þij W ð1Þj

� �þ bð1Þij nð1Þij

� �þ rð1Þij nð1Þij

� �h in(

þ �að1Þij W ð1Þj

� �þ �bð1Þij

�nð1Þij

� �þ �rð1Þij

�nð1Þij

� �h iþ nqð1Þj

� �1� hð1Þj

� �Nð1Þj

� �h io

þXK

k¼1

að2Þk W ð2Þk þ bð2Þk nð2Þk

� �h iþ nqð2Þk

� �1� hð2Þk

� �nð2Þk

� �h in o)

þ S1;Hþ1þ S11

2

L6� t22� t21

2

� �d�21� dþ21

� �� t23� t22

2

� �d�22� dþ22

� �� . . .� t2;Hþ1� t2H

2

� �d�2H � dþ2H

� �þ t2;Hþ1þ t21

2

� �XI

i¼1

XJ

j¼1

ðcqð1Þij nð1Þij Þþ �c�qð1Þij�nð1Þij

� �h iþXK

k¼1

ðcqð2Þk nð2Þk Þ( )

þ S2;Hþ1þ S21

2XI

i¼1

XJ

j¼1

að1Þij W ð1Þj

� �þ bð1Þij nð1Þij

� �þ rð1Þij nð1Þij

� �h in

þ �að1Þij W ð1Þj

� �þ �bð1Þij

�nð1Þij

� �þ �rð1Þij

�nð1Þij

� �h iþ nqð1Þj

� �1� hð1Þj

� �Nð1Þj

� �h io

þXK

k¼1

að2Þk W ð2Þk þ bð2Þk nð2Þk

� �h iþ nqð2Þk

� �1� hð2Þk

� �nð2Þk

� �h in o

þ d�1h� dþ1h ¼ X1h; h¼ 1;2; . . . ;HXI

i¼1

XJ

j¼1

cqð1Þij nð1Þij

� �h

þ �c�qð1Þij�nð1Þij

� �iþXK

k¼1

cqð2Þk nð2Þk

� �þ d�2h� dþ2h ¼ X2h;

h¼ 1;2; . . . ;H

Eqs. (3)–(15)

L;d�1h;dþ1h; d

�2h;d

þ2h;n

ð1Þij ; �n

ð1Þij ;n

ð2Þk P 0 8h; 8i; 8j; 8k

Step 7: Repeat, implement, and modify the above process untilthe solution is satisfactory.

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