A new bi-objective model for a closed-loop supply chain problem with inventory and transportation times

Transcription

1 Scienia Iranica E (2016) 23(3), 1441{1458 Sharif Universiy of Technology Scienia Iranica Transacions E: Indusrial Engineering A new bi-obecive model for a closed-loop supply chain problem wih invenory and ransporaion imes F. Forouzanfar a, R. Tavakkoli-Moghaddam b;, M. Bashiri c and A. Baboli d a. Deparmen of Indusrial Engineering, Science and Research Branch, Islamic Azad Universiy, Tehran, Iran. b. School of Indusrial Engineering and Cener of Excellence for Inelligence Based Experimenal Mechanics, College of Engineering, Universiy of Tehran, Tehran, Iran. c. Deparmen of Indusrial Engineering, Faculy of Engineering, Shahed Universiy, Tehran, Iran. d. DISP Laboraory, INSA-Lyon, Villeurbanne Cedex, France. Received 12 Ocober 2014; received in revised form 7 February 2105; acceped 25 April 2015 KEYWORDS Closed-loop supply chain; Ineger nonlinear programming; Transporaion; Invenory; Bee algorihm. Absrac. This paper considers a closed-loop supply chain design problem including several producers, disribuors, cusomers, collecing ceners, recycle ceners, revival ceners, raw maerials cusomers considering several periods, exising invenory and shorage in disribuion ceners, and ransporaion cos and ime. This problem is formulaed as a bi-obecive ineger nonlinear programming model. The aim of his model is o deermine numbers and locaions of supply chain elemens, heir capaciy levels, allocaion srucure, mode of ransporaion beween hem, amoun of ranspored producs beween hem, amoun of exising invenory, and shorage in disribuion ceners in each period o minimize he sum of sysem coss and ransporaion ime in he nework. To validae his model and show he applicabiliy of i for small-sized problems, GAMS sofware is used. Because his given problem is NP-hard, a Bee Colony Opimizaion (BCO) algorihm is proposed o solve medium and large-sized problems. Furhermore, o examine he eciency of he proposed BCO algorihm, he associaed resuls are compared wih he resuls obained by he Geneic Algorihm (GA). Finally, he conclusion is provided Sharif Universiy of Technology. All righs reserved. 1. Inroducion Nowadays, rapid economic changes and compeiive pressures in curren global markes force rms o inves and focus on ecien managemen of heir logisics sysem. Reurn policies, environmenal concerns, and emphasis on servicing and reusing pieces lead o improvemen in forwarding radiional supply chain, where Reverse Logisics (RLs) componens have been combined. Precise, on ime, and accurae ransfer of *. Corresponding auhor. addresses: ie foroozanfar@yahoo.com (F. Foroozanfar); avakoli@u.ac.ir (R. Tavakkoli-Moghaddam); bashiri.m@gmail.com (M. Bashiri); armand.baboli@insa-lyon.fr (A. Baboli) useless maerials, iems, and producs from he end poin and ulimae consumer o suiable and relevan uni hrough supply chain has been described by he erm `reverse logisics' [1]. Using reverse logisics no only saves invenory ransporaion coss, wase maerial ransporaion coss, and disposal coss, bu also ensures fuure sale and cusomer saisfacion. Global compeiive condiions, legal obligaions, and especially environmenal concerns oblige organizaions o collec heir reurned producs, so ha hey revive, recycle, and dispose hese producs for he sake of environmenal conservaion. Collecing producs afer consumpion by cusomers and reurning hem ino supply chain or disposing hem have raised he issue of closed loop supply chain, which considers inegraed managemen of forwarding and reverse sreams in his

2 1442 F. Forouzanfar e al./scienia Iranica, Transacions E: Indusrial Engineering 23 (2016) 1441{1458 chain. Real design of supply chain nework srucure leads rms o gain more compeiive advanages. Therefore, closed-loop supply chain designing ha simulaneously considers he forward and reverse chains can be ecien in gaining more advanages. Organizaions mus reduce ime and cos of supplying cusomer demands o survive in global markes. Time for doing he order depends on several facors including a ransporaion sae. Dieren ransporaion saes include a reverse relaion beween ime and cos. Undoubedly, when his value is sligh, i is aken as value added by which one could reach long erm and shor erm compeiive advanages in marke. On he oher hand, decisions regarding he amoun of invenory or shorage in Disribuion Ceners (DC's) in each period, amoun of ranspored producs beween levels in each phase during each period, and cos of esablishing ceners wih a specic capaciy level depend on heir coss. In his paper, we propose a mahemaical model for a closed-loop supply chain including several producers, disribuors, cusomers, collecing ceners, recycle ceners, revival ceners, and raw maerials cusomers. Addiionally, we consider several periods, escalaing facor of cos, exising invenory and shorage in disribuion ceners, several levels of capaciy and mode of ransporaion beween ceners in each period, and ransporaion cos and ime in he modeling as novel innovaions. Furhermore, a soluion approach based on he aricial bee colony is developed. To he bes of our knowledge, his paper is among he rs publicaions ha consider he bee colony opimizaion algorihm o solve a closed-loop supply chain nework problem. The resuls of is soluion are compared wih he resuls obained by he Geneic Algorihm (GA). The res of his paper is organized as follows. In Secion 2, he given problem is dened and a mahemaical model is proposed. In Secion 3, o solve his model, he bee colony opimizaion algorihm is developed and a geneic algorihm is also applied. In Secion 4, some experimens and resuls are discussed. Finally, conclusion is provided in Secion Lieraure review Some sudies relaed o a closed-loop supply chain are menioned in his secion in order o clarify he necessiy of his sudy. Some recen sudies on a closedloop supply chain and characerisics of his paper are illusraed in Table 1. Tavakkoli-Moghaddam e al. [11] presened a hree-level muli-period supply chain model ha minimizes he ime and cos of ransporaion along he chain. They considered exising invenory or shorage in disribuion ceners and also ook ino accoun dieren ransporaion saes beween wo ceners in wo dieren levels. Guide and Van Wassenhove [12] used ve phases o describe he evoluion of he closed-loop supply chain research. Krikke e al. [13] surveyed a wide sudy in he basic closed-loop supply chain abou reurn pracices. They analyzed hese pracices and provided some recommendaions for convering value desrucion ino value creaion. Sind and Sahamie [14] reviewed he research on closed-loop supply chain managemen in a process indusry. They invesigaed he main characerisic of CLSC planning in he process indusry o deermine he evoluion and gaps of his curren research and o explore he opic area and mehodology in his eld. Govindan e al. [15] presened a universal lieraure review of recen papers in a RL/CLSC and suggesed fuure direcions and opporuniies of relaed research. According o he reviewed papers, no paper has considered minimizaion of he cos and ransporaion ime hroughou he closed-loop supply chain, simulaneously, so far. Naure-inspired algorihms are very useful in solving muli-variable opimizaion problems. The Bee Algorihm (BA) [16] is one of he well-known group algorihms, which simulaes he foraging behavior of honey bee colonies. We use he BA for opimizing a closed-loop supply chain nework and compare he resuls wih he Geneic Algorihm (GA). The GA [17] is a specic kind of evoluion algorihms using biological echniques, such as inheriance and muaion. I is an innovaive populaion based algorihm. This paper is among he rs publicaions ha consider he bee colony algorihm o solve a closed-loop supply chain nework problem. Soleimani and Kannan [18] applied a hybrid Paricle Swarm Opimizaion (PSO) and GA for solving a closed-loop supply chain nework design problem in large-scale neworks. Kannan e al. [19] used a GA o solve a closed-loop supply chain model. Min e al. [20] proposed a mixed-ineger nonlinear model and GA o provide a minimum cos soluion for he closed-loop supply chain nework design problem involving he spaial and emporal consolidaions of produc reurns. To he bes of our knowledge, here is no paper ha considers he bee colony opimizaion algorihm o solve a closed-loop supply chain problem. 3. Problem deniion This problem consiss of several producers, disribuors, cusomers, collecing ceners, recycle ceners, revival ceners, and raw maerials cusomers (see Figure 1) considering several periods, exising invenory and shorage in disribuion ceners, and ransporaion cos and ime. Trade-o beween cos and ime creaes a bi-obecive problem. One crierion ries o minimize he xed cos for opening ceners wih a cerain capaciy level, ransporaion and allocaion coss, consrucion, process, resrucuring and sepa-

3 F. Forouzanfar e al./scienia Iranica, Transacions E: Indusrial Engineering 23 (2016) 1441{ Ref. LP MILP Table 1. Some recen sudies on a closed-loop supply chain. Modeling Obecive funcion Period Max Min Cos MINLP RMIP MIP ILP Pro Weighs of suppliers [2] [3] Defec raes Shorage Oher Time [4] [5] [6] [7] [8] [9] [10] This paper Single Muliple LP: Linear Programming; MINLP: Mixed-Ineger Nonlinear Programming; MIP: Mixed-Ineger Programming; MILP: Mixed-Ineger Linear Programming; RMIP: Relaxed Mixed-Ineger Programming; ILP: Ineger Linear Programming. Conribuion Considering uncerainy on producs' demand and prices, dieren decision scenarios in he planning model. Developing a model o deermine he raw maerial level, producion level, disribuion and invenory level, disposal level, and recycling level a dieren faciliies. A case of baery recycling is considered. A robus opimizaion model (for handling he inheren uncerainy of inpu daa). The model capures he rade-os beween various coss wih considering a se of available ransporaion modes. Tha is scenario-based. Considering seup, producion and invenory cos for new producs and remanufacured producs in all periods. Using a Lagrangian relaxaion based approach for he capaciaed lo sizing problem. Considering supplier selecion and using he fuzzy ses heory (o overcome he uncerainy in assessmen of eligible suppliers). Considering uncerainy in demand, faciliy locaion and using sochasic programming. The model is scenario-based. Considering a single-produc and consan demand. The uiliy of he proposed PRISM algorihm. Describing a model which opimizes he sraegic and acical Decisions. Considering escalaing facor of cos and a single-produc. Using one ype of vehicles beween wo nodes and selecing one capaciy for each DCs.

4 1444 F. Forouzanfar e al./scienia Iranica, Transacions E: Indusrial Engineering 23 (2016) 1441{1458 Figure 1. Closed-loop supply chain nework. raion coss, and invenory and shorage coss in he closed-loop supply chain. The oher crierion reduces ransporaion ime along he chain Assumpions The main assumpions of he presened model are as follows: 1. This problem is single produc [21]; 2. Several capaciy levels are considered for ceners and nally one capaciy will be chosen for each cener; 3. There are several available saes for ransporaion beween wo consecuive levels [21]; 4. Only one kind of ransporaion vehicles is used beween wo knos among levels in each period [21]; 5. Faser ransporaion vehicle is cosly [21]; 6. There is a balance beween inpus and oupus of each cener; 7. Several periods are considered along he planning horizon; 8. A he end of he las period, invenory or shorage is supposed o be zero; 9. Ceners allocaion has been done a he beginning of he rs period and unil he end of he planning horizon, i will no change; 10. Shorage in DCs in each period should be given in he nex period; 11. The analysis us considers invenory of DCs Ses I Se of plans; J Se of DCs; K Se of demand zones of cusomers; L Se of collecing ceners; M Se of revival ceners; N Se of recycle ceners; D i D D l D m D n T Se of capaciy levels available o plan i (i 2 I); Se of capaciy levels available o DC ( 2 J); Se of capaciy levels available o collecing cener l (l 2 L); Se of capaciy levels available o revival cener m (m 2 M); Se of capaciy levels available o recycle cener n (n 2 N); Time period Parameers B il1 Time for ransporing any quaniy of produc from plan i o DC using ransporaion mode l 1 in period ; B kl2 Time for ransporing any quaniy of produc from DC o cusomer's demand zone k using ransporaion mode l 2 in period ; B kll3 Time for ransporing any quaniy of produc from cusomer's demand zone k o collecing cener l using ransporaion mode l 3 in period ; B ln l4 Time for ransporing any quaniy of produc from collecing cener l o recycle cener n using ransporaion mode l 4 in period ; B nel5 Time for ransporing any quaniy of produc from recycle cener n o maerial cusomer e using ransporaion mode l 5 in period ; B lml6 Time for ransporing any quaniy of produc from collecing cener l o revival cener m using ransporaion mode l 6 in period ; B ml7 Time for ransporing any quaniy of produc from revival cener m o DC using ransporaion mode l 7 in period ; F d i F d F d m F d n Yearly xed cos for opening and operaing plan i wih capaciy level d (d 2 D i ); (i 2 I); Yearly xed cos for opening and operaing DC wih capaciy level d (d 2 D ); ( 2 J); Yearly xed cos for opening and operaing revival cener m wih capaciy level d (d 2 D m ); (m 2 M); Yearly xed cos for opening and operaing recycle cener n wih capaciy level d (d 2 D n ); (n 2 N);

5 F. Forouzanfar e al./scienia Iranica, Transacions E: Indusrial Engineering 23 (2016) 1441{ A il1 A kl2 A kll3 A ln l4 A nel5 A lml6 A ml7 P i P P l P m P n V Uni cos of ransporaion from plan i o DC using ransporaion mode l 1 in period ; Uni cos of ransporaion from DC o cusomer's demand zone k using ransporaion mode l 2 in period ; Uni cos of ransporaion from cusomer's demand zone k o collecing cener l using ransporaion mode l 3 in period ; Uni cos of ransporaion from collecing cener l o recycle cener n using ransporaion mode l 4 in period ; Uni cos of ransporaion from recycle cener n o maerial cusomer e using ransporaion mode l 5 in period ; Uni cos of ransporaion from collecing cener l o revival cener m using ransporaion mode l 6 in period ; Uni cos of ransporaion from revival cener m o DC using ransporaion mode l 7 in period ; Producion cos of each uni of produc in plan i; Process cos of each uni of produc in DC ; Process cos of each uni of produc in collecing cener l; Resrucuring cos of each uni of produc in revival cener m; Separaion cos of each uni of produc in recycle cener n; Holding cos of each uni of invenory in DC ; Z Cos of each uni of shorage in DC ; e a e b e c e f e g e v Cos increase facor for producion of each uni of produc in plan i; Cos increase facor for process of each uni of produc in DC ; Cos increase facor for process of each uni of produc in collecing cener l; Cos increase facor for resrucuring each uni of produc in revival cener m; Cos increase facor for separaion of each uni of produc in recycle cener n; Cos increase facor for holding each uni of invenory in DC ; e z Cos increase facor for shorage of each uni in DC ; C e Maerial cusomers' demand e in period ; l Un-reviving percenage of colleced producs in each collecion cener l in period ; w i Capaciy of plan i; w Capaciy of DC ; w l Capaciy of collecing cener l; w m Capaciy of revival cener m; w n Capaciy of recycle cener n; Disrepair percenage of producs sen from DC in period ; k Percenage of supplied demand during period which is reurned by cusomer demand of zone k; H k Amoun of cusomer's demand of zone k in period ; L Iniial invenory level in DC Decision variables il1 1 if plan i is allocaed o DC during period via ransporaion mode l 1 ; and 0, oherwise; kl2 1 if DC is allocaed o cusomer's demand zone k during period via ransporaion mode l 2 ; and 0, oherwise; kll3 1 if cusomer's demand zone k is allocaed o collecing cener l during period via ransporaion mode l 3 ; and 0, oherwise; ln l4 1 if collecing cener l is allocaed o recycle cener n during period via ransporaion mode l 4 ; and 0, oherwise; nel5 1 if recycle cener n is allocaed o maerial cusomer e during period via ransporaion mode l 5 ; and 0, oherwise; lml6 1 if collecing cener l is allocaed o revival cener m during period via ransporaion mode l 6 ; and 0, oherwise; ml7 1 if revival cener m is allocaed o DC during period via ransporaion mode l 7 ; and 0, oherwise; U d i U d 1 if plan i is opened wih capaciy level d; and 0, oherwise; 1 if DC is opened wih capaciy level d; and 0, oherwise;

6 1446 F. Forouzanfar e al./scienia Iranica, Transacions E: Indusrial Engineering 23 (2016) 1440{1457 U d l U d m U d n O i O k O kl O lm O ln O m O ne 1 if collecing cener l is opened wih capaciy level d; and 0, oherwise; 1 if revival cener m is opened wih capaciy level d; and 0, oherwise; 1 if recycle cener n is opened wih capaciy level d; and 0, oherwise; Amoun of produc ranspored from plan i o DC during period ; Amoun of produc ranspored from DC o cusomer's demand zone k during period ; Amoun of produc ranspored from cusomer's demand zone k o collecing cener l during period ; Amoun of produc ranspored from collecing cener l o revival cener m during period ; Amoun of produc ranspored from collecing cener l o recycle cener n during period ; Amoun of produc ranspored from revival cener m o DC during period ; Amoun of produc ranspored from recycle cener n o maerial cusomer e during period Q Invenory level of DC in period ; E Shorage of DC in period ; Z k min f 1 = max k; Amoun of supplied demand of cusomer in zone k in period. 8 >< >: max ;l 2 f kl2t kl2g + " P P l 2 kl2 " + max f kll3 T kll3 g + P l;l 3 l 8 " max l # P 3.5. Mahemaical model Minf 1 is calculaed as shown in Box I, and Minf 2 is obained by he following equaion: minf 2 = Fi d Ui d + + d2d i i d2d l L k + L d2d n N + M + L + N + i + + L F d l U d l + F d nu d n + i k l 2 l l 3 M l 6 l 7 N l 4 e l 5 k M max f il1t i;l il1g 1 l 3 kll3 " max f ln l4 T ln l4 g + P P >< N;l 4 N " " >: max f lml6 T lml6 g + P P M;l 6 M # l 4 ln l4 # l 6 lml6 # d2d d2d m M F d U d F d mu d m l 1 A kl2 kl2 A kll3 kll3 A lml6 lml6 A ml7 ml7 A ln l4 lml6 A nel5 nel5 P i O i (1 + e a ) P O k (1 + e b ) P l O lm (1 + e c ) # max f nel5 T nel5 g ; e;l 5 max f ml7 T ml7 g ;l 7 A il1 il1 9 >= 9 >= # >; >; ; Box I

7 F. Forouzanfar e al./scienia Iranica, Transacions E: Indusrial Engineering 23 (2016) 1441{ s..: + L + M + N + L N M N L l 3 N e P l O ln (1 + e c ) P m O m (1 + e f ) P n O ne (1 + e g ) Q V (1+e v ) + E Z (1+e z ) ; l 3 kll3 = 1 8k; ; (1) l 4 ln l4 1 8l; ; (2) l 6 lml6 1 8l; ; (3) l 2 kl2 = 1 8k; ; (4) l 5 O ne Nel5 C e 8e; ; (5) l 3 O kl kll3 k Z k 8k; ; (6) k l 3 k l 7 l 5 e l 1 l O kl kll3 = N (1 l )O kl kll3 = M O m ml7 = L O ne nel5 L O i il1 l 4 O ln ln l4 8l; ; (7) l 6 (8) O lm lml6 8l; ; O lm lml6 8m; ; l 6 (9) l 4 (10) O ln ln l4 8n; ; d2d i U d i w i 8i; ; (11) d2d i U d i 1 8i; (12) d2d U d 1 8; (13) Ul d 1 d2d l 8l; (14) Um d 1 d2d m 8m; (15) Un d 1 d2d n 8n; (16) i l 1 O i il1 + M l 7 O m Ml7 + Q ( 1) d2d U d w 8; ; (17) U d w d2d k l 2 O k kl2 8; ; (18) O kl kll3 Ul d w l k l 3 d2d l 8l; ; (19) O lm lml6 Umw d m L l 6 d2d m 8m; ; (20) O ln ln l4 Unw d n ; L l 4 d2d n 8n; ; (21) ml7 = U d M d2d 8; ; (22) l 7 M l 7 = k k Q ( O m ml7 + i O i il1 + Q ( 1) l 2 O k kl2 + Q ; 8; ; (23) l 2 (1 )O k kl2 E ( 1) + l 2 k 1) + i k( 1) Z k( 1) kl2 8; ; (24) l 1 k l 2 k O i il1 + M l 7 l 2 H k kl2 E ( 1) O m Ml7 k( 1) Z k( 1) kl2 =Q E ; 8; ; (25) Q E = 0 8; ; (26)

8 1448 F. Forouzanfar e al./scienia Iranica, Transacions E: Indusrial Engineering 23 (2016) 1441{1458 Q T = 0 8; ; (27) E T = 0 8; ; (28) Q T0 = L 0 8; ; (29) Q d2d U d w 8; ; (30) l 3 k + N O kl kll3 = M l 6 O lm lml6 l 4 O ln ln l4 8l; (31) il1 1 8i; ; ; (32) l 1 kl2 1 8; k; ; (33) l 2 kll3 1 8k; l; ; (34) l 3 ln l4 1 8l; n; ; (35) l 4 nel5 1 8n; e; ; (36) l 5 lml6 1 8l; m; ; (37) l 6 ml7 1 8m; ; ; (38) l 7 kl2 = Z k 8k; ; (39) Z k H k 8k; ; (40) il1 ; kl2 ; kll3 ; ln l4 ; nel5 ; lml6 ; ml7 ; U d i ; U d ; U d l ; U d m; U d n 2 [0; 1]; (41) O i ; O k ; O lm ; O ln ; O m ; O ne ; Q ; E 0: (42) In his model, he rs obecive funcion ries o nd he minimum ime o ranspor producs along any pah in he closed-loop supply chain. The second obecive funcion minimizes he xed cos for opening ceners wih a cerain capaciy level, ransporaion and allocaion coss, consrucion, process, resrucuring and separaion coss, and invenory and shorage coss in a closed-loop supply chain. Consrain (1) shows ha each cusomer's demand zone is allocaed o a collecing cener. Consrains (2) and (3) sugges ha each collecing cener is allocaed o a recycle cener (in Consrain 2) and a revival cener (in Consrain 3) when a collecing cener is esablished. Consrain (4) emphasizes ha each cusomer's demand zone akes service from one DC. Consrain (5) suggess ha cusomer demand for maerials is supplied during each period. Consrain(6) shows he amoun of reurned produc from cusomer k during period. Consrains (7) and (8) show ha he reurned producs from cusomer k o collecing cener l are sen o revival cener m and recycle cener n. Consrain (9) shows ha inpus and oupus of he revival cener are equal. Consrain (10) indicaes ha recycled produc consiuens are sold o cusomers afer recycle process. Consrain (11) relaes o he plan capaciy. Consrains (12) o (16) show ha if a cener has been esablished, a capaciy level will be allocaed o i. Consrains (17) and (18) relae o capaciy of DC. Consrains (19) o (21) relae o he capaciies of collecing, revival, and recycle ceners, respecively. Consrain (22) shows wheher DC is esablished wih he capaciy level and allocaed o a revival cener. Consrain (23) suggess ha he summaion of period ( 1) invenory in DC and producs received from recycle ceners and plans during period is equal o he summaion of period invenory and producs sen o he cusomers' demand zones. Consrain (24) emphasizes ha shorage in DC and impaired sen producs should be compensaed in he nex phase. Consrains (25) and (26) show ha i is impossible o have invenory and shorage, simulaneously, in DC in each phase. Consrains (27) and (28) show ha in he las period, we do no have invenory and shorage in DC. Consrain (29) suggess ha we consider an iniial invenory level for DCs. Consrain (30) relaes o DC capaciy. Consrain (31) indicaes ha a revivable par of he colleced producs from cusomer k in collecing cener l is allocaed o a revival cener and a recyclable par of i is allocaed o a recycle cener. Consrains (32) o (38) relae o using one ype of ransporaion vehicles beween wo ceners a wo levels. Consrain (39) shows he amoun of supplied demand of cusomer k. Consrain (40) says ha his amoun canno be greaer han he real demand of cusomer k. Finally, Consrains (41) and (42) dene variables. 4. Bee colony opimizaion and geneic algorihms To show applicabiliy and validiy of he presened model, we solve several small-scale problems hrough a branch-and-bound module in GAMS. This sofware is a robus ool for solving and analyzing mahemaical models of linear and nonlinear opimizaion problems.

9 F. Forouzanfar e al./scienia Iranica, Transacions E: Indusrial Engineering 23 (2016) 1441{ To solve large-scale problems, we use he bee algorihm. In order o show is eciency, he associaed resuls are compared wih he resuls obained from he geneic algorihm Bee colony opimizaion algorihm The proposed srucure of he bee colony opimizaion algorihm is as follows (please see Figure 2): Purposed bee colony opimizaion algorihm finiializaion: Iniialize he algorihm parameer. Generae N feasible soluions as he iniial populaion. While crierion is mee Calculae he ness for each soluion in curren populaion. Figure 2. Srucure of bee algorihm.

10 1450 F. Forouzanfar e al./scienia Iranica, Transacions E: Indusrial Engineering 23 (2016) 1441{1458 g Selec he bes bees and heir locaion as p 1 se. Selec he oher bees and heir locaion as p 2 se. Apply neighborhood search operaor o p 1 se. Apply feasibiliy check mehod o he obained soluions. Assign some bees o he obained soluions and calculae heir ness. Apply random neighborhood search operaor on p 2. Apply feasibiliy check mehod on obained soluions. Calculae heir ness. Selec he N bes bees of each locaion. Apply improvemen mehod on seleced soluions and ake oupu of his mehod as populaion of he nex generaion. Updae Pareo archive End while Reurn he Pareo archive Soluion represenaion We use a marix o presen each soluion. Each soluion includes several marices which are designed in accordance wih model oupus. For example, for variable il1, we dene a four-dimensional marix wih dimensions I J l 1 T. In he same way, we dene a marix for oher oupus Soluion iniializaion mehod Given ha he Bee Colony Opimizaion (BCO) algorihm is a populaion-based algorihm, a he beginning of he algorihm, we need a populaion of soluions as he iniial populaion. This paper uses a random process o produce possible iniial soluions. Available soluions indicaed by N in each repea of he BCO process are supposed o be xed during he opimizaion process. To produce N possible iniial soluions, he designed process should be repeaed N imes Applicabiliy of soluions Since during performing he algorihm new soluions are produced, an approach has been designed o check he applicabiliy of soluions. The algorihm make he infeasible soluions o feasible ones where i checks ou all he limiaions of he produced soluions. If one or more limiaions have been violaed in ha soluion, i ries o make he soluion feasible Finess funcion calculaion In his paper, since he proposed model is a muliobecive one, we calculae he ness index of each soluion, which classies soluions and calculaes he crowding disance crierion of soluions [22]. The ness value of individual c, considering is rank and obecive values, can be calculaed by [23]: 1 Finess c = 2 3 ; DP 6 f 4 (c) 7 (43) N rank(c) 5+(rank(c) 1) D =1 P f (i) i=1 where D is he number of obecives, f (i) is he value of he h obecive of he ih individual, and N rank(c) is he number of soluions in rank(c) Local search (bee P 1 group) To solve he sudied problem, a new process is designed based on local search. P 1 group soluions populaion is he inpu of his process. This process bases upon neighborhood search. In oher words, his process receives group of soluions as inpu and ries o reach suiable neighbor soluions by recovering each soluion. In he presen paper, we use a muli-operaor local search process as well as a repeaable local search process wih guided muaion o design he above process [24]. By combining hese wo local search processes, we ry o presen a new process of local search. A general srucure of he proposed operaor is presened below: f Sep 1: Ge inpu soluions (p 1 ). Sep 2: For each soluion in p 1 se, use he muli-operaor search o enhance soluions. Consruc he pool of rial soluions generaed in he sep before. g In his approach, a muli-operaor search process is execued on each available soluion in P 1 group and he resul is a se of local opimized soluions, which are in he neighborhood of he aforesaid soluion. All local opimized soluions obained from execuion of his process are pooled. For recovering soluions, heir neighborhood is searched and analyzed. Neighborhood search operaors used here include an exchange operaor ha will be execued on four locaion-relaed marices. These operaors work as follows: Operaor 1: Two indices i and are consruced randomly during consisen periods f1; ; ng (i.e., n facories) and f1; ; mg (i.e., m disribuion ceners); index is produced during period f1; ; Tg, and if square il1 is equal o zero, i will be considered 1 and if i is 1, i will be considered zero; also, a sream of maerials beween facory i and disribuor during period by vehicle l 1 is changed according o capaciies. Then, a modicaion process is execued on he soluion marices and i modies hem according o model limiaions.

11 F. Forouzanfar e al./scienia Iranica, Transacions E: Indusrial Engineering 23 (2016) 1441{ Operaor 2: Two indices k and l are consruced randomly during consisen periods f1; ; kg (i.e., k cusomer regions) and f1 lg (i.e., l collecing ceners); index is produced during period f1; ; Tg and if square kll3 is equal o zero, i will be considered 1 and if i is 1, i will be considered zero; also a sream of maerials beween cusomer k and collecing cener l during period by vehicle l 3 is changed according o capaciies. Then, a modicaion process is execued on he soluion marices and i modies hem according o model limiaions. Operaor 3: Two indices l and m are consruced randomly during consisen periods f1; ; lg (i.e., l collecing ceners) and f1; ; mg (i.e., m revival ceners); index is produced during period f1; ; Tg, and if square lml6 is equal o zero, i will be considered 1 and if i is 1, i will be considered zero; also sream of maerials beween collecing cener l and revival cener m during period by vehicle l 6 is changed according o capaciies. Then, a modicaion process is execued on he soluion marices and i modies hem according o model limiaions. Operaor 4: Two indices n and l are consruced randomly during consisen periods f1; ; ng (i.e., n recycle ceners) and f1; ; lg (i.e., l collecing ceners), index is produced during period f1; ; Tg, and if square ln l4 is equal o zero, i will be considered 1 and if i is 1, i will be considered zero; also sream of maerials beween ceners l and n during period by vehicle l 4 is changed according o capaciies. Then, a modicaion process is execued on he soluion marices and i modies hem according o model limiaions. Each of hese operaors search a par of neighbor soluions and oher pars may be searched by oher operaors; hus, using one neighborhood search operaor may lead o losing some poenial neighbor soluions. Therefore, his sudy uses a muli-operaor search process, which will be explained laer. In a muli-operaor search process, we use he above four operaors o produce neighbor soluions. A mulioperaor search process algorihm is presened below: fmuli operaor search framework 1. For each inpu soluion x, se p approx = fxg 2. Repea 3. Randomly selec some x in p approx for which nh(x) has no been invesigaed ye 4. For all NR = fnh 1 ; nh 2 ; ; nh r g Generae neighborhood nh i (x) 5. Updae p approx wih all elemens x nh in nh i (x) 6. If x in p approx, hen 7. Mark x as `invesigaed' g 8. End if 9. Unil no elemen as x in p approx wih x sill o be invesigaed 10. Reurn p approx In he dened search process, as can be seen in is algorihm, a se of soluions (P approx ) is consruced, which includes iniial soluions. In each repea of his process, one member ha has no been analyzed so far is chosen randomly and neighbor soluions are produced by he aforemenioned four neighborhood search operaors. Then, approx se P is updaed hrough comparing soluions wih non-dominaed relaions and his cycle will be repeaed unil all members are analyzed. A he end of he algorihm, approx se P ha includes a local opimized soluion in he neighborhood of an iniial soluion is reurned as he algorihm resul Random neighborhood search (P 2 group) To execue random neighborhood search for he second bee group, his sudy uses a parallel neighborhood search operaor. This process uses four operaors menioned in he previous secion simulaneously or in a parallel way. If we indicae hose operaors wih ls 1, ls 2, ls 3, ls 4 symbols, he srucure of parallel process is as follows: ffor inpu soluion s: S 1 = ls 1 (s); S 1 =Apply feasibiliy check mehod on s 1 ; S 2 = ls 2 (s); S 2 =Apply feasibiliy check mehod on s 2 ; S 3 = ls 3 (s); S 3 =Apply feasibiliy check mehod on s 3 ; S 4 = ls 4 (s); S 4 = Apply feasibiliy check mehod on s 4 ; Oupu soluion = accepance(s, s 1, s 2, s 3, s 4 ). g As can be seen in he above srucure, for each exising soluion in P 2, each operaor will be execued separaely on soluion and, ulimaely, among oupus and inpus of he four operaors, one will be chosen, given non-dominaed relaions, and will be repored as process oupu Selecion In each repea, he algorihm needs a populaion of soluions. In his sudy, for choosing he nex repea populaion, he exising soluions in he curren repea populaion and new soluions produced by he algorihm are pooled and afer classicaion and calculaion of he crowding disance crierion for each soluion given is level, hrough Deb's formula [22], N soluions

12 1452 F. Forouzanfar e al./scienia Iranica, Transacions E: Indusrial Engineering 23 (2016) 1441{1458 which have he highes qualiy and highes variance will be chosen as he nex repea populaion of he algorihm Improvemen srucure In he proposed srucure of he bee colony opimizaion algorihm, an improvemen srucure is designed which is execued on he seleced soluions in he previous repea o improve hem. Oupu soluions of he improvemen srucure are chosen as he nex repea populaion of he algorihm. Execuion of he improvemen process bases upon Variable Neighborhood Search (VNS), as will be explained laer. The VNS srucure uses four neighborhood search srucures. These srucures are local search operaors, explained in he previous secion. These four srucures are used in he conex of VNS, whose general srucure (i.e., pseudo-code) is as follows: ffor each inpu soluion K = 1 While sopping crierion is me, do New soluion=apply NSS ype k If new soluion is beer han K = 1 Else, K = k + 1 If k = 5 hen K = 1 End if End if End while g The VNS algorihm receives all he available soluions in a populaion and gives an oupu soluion. Then, a recovery process is execued on oher soluion marices and afer recovery, i replaces he inpu soluion. In fac, a general srucure of he recovery process is as follows: Improvemen mehod ffor each s i in inpu populaion S i =apply VNS procedure on s i ; S i =check feasibiliy mehod. g 4.2. Geneic algorihm The proposed general srucure of he geneic algorihm is as follows: Geneic algorihm finiializaion: Iniialize algorihm parameers; Generae N feasible soluion as he iniial populaion. While sopping crierion is me, do g Calculae ness for soluion in he curren populaion Calculae number of soluions (nc) for crossover operaor. Couner=0; While couner nc do Selec wo parens. Couner=couner+2. Apply crossover operaor on he seleced parens. End while Calculae number of soluions (nm) for muaion operaor Couner=0; While couner nm do Selec one soluion ha has no been seleced so far. Apply muaion operaor on he seleced soluion. Couner=couner+1; End while Apply reproducion operaor on he oher soluions in populaion ha has no been seleced so far. Selec N bes soluions as he populaion of he nex generaion. End while Reurn he bes soluion Esablishing an iniial populaion Iniial soluions are produced randomly Paren selecion mehod This sudy uses a roulee wheel mehod for selecing parens. The probabiliy of parallel selecion wih each chromosome is calculaed in erms of is ness. If f k is he ness value of chromosome k, he probabiliy of parallel survival wih ha chromosome is as follows: P k = f k np i=1 f i : (44) Now, we arrange chromosomes according o P k and q k, which are a cumulaive frequency of P k, calculaed by: q k = k i P i : (45) This mehod simulaes a roulee wheel in order o deermine which members have a chance o reproduce. Each member in erms of is conformiy receives some pars of he rolling wheel. Then, in each phase, one member is seleced and his process is subeced o repea unil enough pairs are seleced for producing he nex generaion Muaion operaor For execuing he muaion operaor used in he geneic algorihm, his sudy uses a parallel neighborhood search srucure explained in he bee algorihm.

13 F. Forouzanfar e al./scienia Iranica, Transacions E: Indusrial Engineering 23 (2016) 1441{ Crossover operaor The crossover operaor designed in his algorihm is a single poin crossover operaor, which is execued on all soluion marices Finess funcion Calculaion of he ness value of each soluion is similar o he bee algorihm Selecing a populaion for he nex repea A he end of each repea, among he soluions of ha repea and he new produced soluions, we selec n soluions wih he highes ness value as a populaion of he nex repea (or generaion). 5. Designing experimen and resuls Given he considered hypoheses and parameers of he proposed mahemaical model, we dene several small, medium, and large-scale problems, randomly. In order o analyze eciency of he proposed algorihm, we execue i in he MATLAB sofware environmen and resuls of he execuion on experimenal problems are compared wih he resuls of GAMS sofware precise calculaions. Comparisons base upon a crierion, which is he gap beween arge funcion and execuion ime of each process Comparison crieria To solve he presened model, we propose and execue he bee colony algorihm and he geneic algorihm. However, his model is solved in he GAMS sofware environmen. Given ha his model is bi-obecive, in order o solve he model wih GAMS sofware, we consider he weigh composiion of arges. To compare he resuls obained by he algorihms and GAMS sofware, we use comparison crierion ha shows he gap beween arge funcions. This crierion is explained laer. Eqs. (46) and (47), represening he disance beween he proposed opimizaion algorihms, are used as eciency crierion. This crierion shows validiy of he developed algorihms. error(%) = (ans.bco ans.gams) ; (46) ans.gams error(%) = (ans.ga ans.gams) : (47) ans.gams This crierion is dened as he gap beween he obecive funcion values (i.e., weigh composiion of model obecives) of he algorihm and GAMS sofware. Eq. (46) calculaes he gap beween he bee colony algorihm and GAMS sofware. In his formula, we have: - ans. BCO: obecive funcion values of he bee colony algorihm; - ans. GAMS: obecive funcion values of solving he model by GMS sofware; - % error: he gap beween wo obecive funcion values of he bee colony algorihm and GAMS sofware. Eq. (47) calculaes he gap beween he obecive funcion values of he geneic algorihm and GAMS sofware. In his formula, we have: - ans. BCO: obecive funcion values of he geneic algorihm; - ans. GAMS: obecive funcion values of solving he model by GMS sofware; - % error: he gap beween wo obecive funcion values of he geneic algorihm and GAMS sofware. In addiion o he above crierion and weigh composiion of obecives, he resuls of hese algorihms are compared based on comparison indices of muli-obecive problems on he basis of Pareo archive. There are many dieren indices for analyzing qualiy and variance of muli-obecive innovaive algorihms. This sudy considers hree comparison crieria presened as follows [25]: Qualiy meric: This meric compares he qualiy of Pareo soluions in each process. Indeed, qualiy meric classies all he obained Pareo soluions in boh algorihms and indicaes wha percenage of high level soluions belongs o each process. Higher percenage shows ha algorihm has a high qualiy. Spacing meric: This meric analyzes he consisency of Pareo soluion disribuion around soluions border. This index is dened by: P N 1 s = i=1 d mean d i : (N 1) d mean In his relaion, d i indicaes he Euclidean disance beween wo adacen non-subdued soluions and d mean shows he average d i values. Diversicaion meric: This meric is used o deermine non-subdued soluion rae on he opimized border and is dened by: D = r N i=1 max( x i y i ): In his relaion, kx i yk i indicaes he Euclidean disance beween wo adacen soluions x i and y i on he opimized border.

14 1454 F. Forouzanfar e al./scienia Iranica, Transacions E: Indusrial Engineering 23 (2016) 1441{1458 References Table 2. Sizes of some exising problems in he lieraure. plans DCs cusomer's ceners collecing ceners revival ceners [26] [27] [28] [29] [30] Size Small Medium Large Insance Table 3. Small, medium and large-scale problems. plans DCs cusomer's ceners collecing ceners revival ceners recycle ceners period Design of experimens This sudy presens dieren experimenal and real problems wih dieren sizes in he area of direc and reverse logisics nework design. In order o design and consruc experimenal problems, we analyze some problems in he lieraure ha are repored in Table 2. Given he exising sizes in he lieraure, we consider hree groups of problems wih small, medium, and large sizes o analyze he eciency of he proposed algorihms ha are presened in Table 3. For designing experimenal problems groups, we ry o dene he problem size according o he exising area in he previous sudies as shown in Table Parameer seing ˆ The populaion size in boh algorihms is se o 150; ˆ Repea number (sopping crierion) in boh algorihms is se o 500; ˆ In GA, crossover rae is considered equal o 0.7 and muaion rae is se o 0.2;

15 F. Forouzanfar e al./scienia Iranica, Transacions E: Indusrial Engineering 23 (2016) 1441{ ˆ Producs ransporaion ime beween all ceners U[1; 100]; ˆ Annual cos of opening ceners U[1; 40]; ˆ Transporaion cos of each uni beween all ceners U[50; 100]; ˆ Coss of producion, process, rebuild, separaion in heir relaed ceners U[1; 20]; ˆ Shorage and invenory holding coss in DCs U[1; 40]; ˆ Cusomers' demand for maerial e U[1; 20]; ˆ Cusomers' demand U[20; 100]; ˆ Iniial invenory level U[1; 500]; ˆ Capaciy of ceners in all levels U[1; 1000] Soluion resuls As can be seen in Table 4, in he rs sample wih a small size, error is low and equal o 0.07 per cen for he GA and 0.02 per cen for he BCO algorihm. Regarding qualiy, as can be seen, he BCO algorihm can produce higher qualiy soluions han he GA and GAMS. Regarding ime in a proposed srucure of he BCO algorihm and using cerain search algorihms, ime of he BCO algorihm is more han ha of he GA and GAMS. As illusraed in Table 4, one can see ha he given problem is more complicaed han he exising problems in he lieraure due o numerous variables. Therefore, GAMS sofware can solve only small-size problems wihin an accepable ime, while in largesized calculaion of he exac soluion during he reasonable ime i is no possible. For medium- and largesized problems, as can be seen in he BCO algorihm, i produces higher qualiy, bu needs much ime for soluions as compared o GA. Given he execuive resuls, we conclude ha for solving he presened model, he proposed BCO algorihm is more ecien and robus han he GA. As i was menioned, he resuls of boh algorihms are compared on he basis of Pareo archive using qualiy, diversicaion, and space Size Small Insance Table 4. Resuls of small, medium and large-scale problems. OFV CPU ime (sec.) Gap (%) BCO GA GAMS BCO GA GAMS BCO GA Medium > > > > > > > > > Large > > > > > > > > >

16 1456 F. Forouzanfar e al./scienia Iranica, Transacions E: Indusrial Engineering 23 (2016) 1441{1458 Size Small Insance Table 5. Resuls of small, medium and large-scale problems. Qualiy meric Diversicaion meric Spacing meric BCO GA BCO GA BCO GA Medium Large comparison merics. The resuls of hese comparisons based on hese hree merics are repored in Table 5. As illusraed in Table 4, in all cases, he proposed BCO algorihm can produce soluions wih higher qualiy and higher diversicaion han he GA. Regarding he space meric, in mos cases, he obained soluions by he geneic algorihm are more consisen han he obained soluions of he proposed BCO, which is he disadvanage of BCO. 6. Conclusion This sudy has presened a new mahemaical model o design a closed-loop supply chain considering several periods, invenory and shorage in disribuion ceners, and ime and cos of ransporaion. This problem has been formulaed as a bi-obecive ineger nonlinear programming model, whose applicabiliy has been analyzed by solving several small-sized problems by GAMS sofware. Because his problem is NPhard, for solving medium- and large-sized problems, a Bee Colony Opimizaion (BCO) algorihm has been proposed and is resuls have been compared wih he resuls obained by he Geneic Algorihm (GA). The compuaional resuls have indicaed ha he proposed BCO algorihm produces higher qualiy soluions han he GA; however, is CPU ime is no less han ha of he GA. Some useful comparison merics (i.e., qualiy, space, and diversiy merics) have been applied o validae he eciency of he proposed BCO algorihm. The resuls have shown ha he proposed BCO algorihm produces soluions wih higher qualiy and diversicaion han he GA; herefore, i is more ecien. The experimenal resuls have indicaed ha our proposed algorihm ouperforms he GA. There are some areas for fuure research in his paper. A deerminisic model has been developed in his research. I is valuable o consider uncerain parameers in he model and examine he eecs of uncerainy on he resuls. Furhermore, i is worhwhile o apply oher muli-obecive soluion approaches in he lieraure o solve he model and compare he resuls.

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