Fuzzy Mathematical Model for a Lot-Sizing Problem In Closed-Loop Supply Chain

Transcription

1 ournal of Opimizaion in Indusrial Engineering, Vol. 11, Issue 1,Winer and Spring 2018, DOI /OIE Fuzzy Mahemaical Model for a o-sizing roblem In losed-oop Supply hain Amir Faehi Kivi a,*, Amir Aydin Aashi Abkenar b, Hossin Alipuor c a Young Researchers and Elie lud, KhalkhalBranch Islamic Azad Universiy, Khalkhal, Iran, b Young Researchers and elie lud, KhalkhalBranch Islamic Azad Universiy, Khalkhal, Iran, c Assisan rofessor, Indusrial Eng. Dep., Khalkhal Branch, Islamic Azad Universiy, Khalkhal, Iran Received 6 Sepember 2015; Revised 16 ovember 2016; Acceped 26 December 2016 Absrac he aim of lo sizing problems is o deermine he periods where producion akes place and he quaniies o be produced in order o saisfy he cusomer demand while minimizing he oal cos. Due o is imporance on he efficiency of he producion and invenory sysems, o sizing problems are one of he mos challenging producion planning problems and have been sudied for many years wih differen modeling feaures. In his paper, we propose a fuzzy mahemaical model for he single-iem capaciaed lo-sizing problem in closed-loop supply chain. he possibiliy approach is chosen o conver he fuzzy mahemaical model o crisp mahemaical model. he obained crisp model is in he form of mixed ineger linear programming (MI), which can be solved by exising solver in crisp environmen o find opimal soluion. Due o he complexiy of he problems harmony search (HS) algorihm and geneic algorihm (GA) have been used o solve he model for fifeen problem. o verify he performance of he algorihm, we compuaionally compared he resuls obained by he algorihms wih he resuls of he branch-and-bound mehod. Addiionally, aguchi mehod was used o calibrae he parameers of he mea-heurisic algorihms. he compuaional resuls show ha, he objecive values obained by HS are beer from GA resuls for large dimensions es problems, also U ime obained by HS are beer han GA for arge dimensions. Keywords o-sizing, Harmony search, Reurned producs 1. Inroducion he lo-sizing problem is o find he righ balance beween hese coss so as o minimize he oal coss (Hoesel and Wagelmans, (1991). Wagner and Whiin presened a dynamic programming soluion algorihm for single produc, muli-period invenory lo-sizing problem. he Wagner-Whiin (1958) proposed algorihm for dynamic lo-sizing has ofen been misundersood as requiring inordinae compuaional ime and sorage requiremens. Mongomery e al. (1973) resened several single-echelon, single-iem, saic demand invenory models for siuaions in which, during he sock ou period, a fracion b of he demand is backordered and he remaining fracion 1 - b is los forever. Akbalik and enz (2009) sudied a special case of he single-iem capaciaed lo-sizing problem, where alernaive machines are used for he producion of a single-iem. hey proposed an exac pseudo-polynomial dynamic programming algorihm which makes i -hard in he ordinary sense. Akbalik and oche (2009) presened a new class of valid inequaliies for he single-iem capaciaed lo-sizing problem wih sep-wise producion coss. hey proposed a cuing plane algorihm for differen classes of valid inequaliies inroduced. Akbalik and Rapine (2012) considered he single-iem uncapaciaed lo-sizing problem wih bach delivery, focusing on he general case of ime-dependen bach sizes. hey showed ha he problem is polynomial solvable in ime O ( 3 ), where denoes he number of periods of he horizon. Abad (2001) considered he problem of deermining he opimal price and lo-size for a reseller. He assumed ha demand can be backlogged and ha he selling price is consan wihin he invenory cycle. Aksen e al. (2003) addressed a profi maximizaion version of he well-known Wagner Whiin model for he deerminisic single-iem uncapaciaed lo-sizing problem wih los sales. he auhors proposed an O ( 2 ) forward dynamic programming algorihm o solve he problem.brahimi e al. (2006) presened four differen mahemaical programming formulaions of he Single-iem lo sizing problems. hu e al. (2013) addressed a real-life producion planning problem arising in a manufacurer of luxury goods. his problem can be modeled as a single-iem dynamic losizing model wih backlogging, ousourcing and invenory *orresponding auhor addressesamir.faehi.g@gmail.com 133

2 Amir Faehi Kivi e al./fuzzy Mahemaical Model.. capaciy. hey showed ha his problem can be solved in O ( 4 log) ime where is he number of periods in he planning horizon. ang (2004) provides a brief presenaion of simulaed annealing echniques and heir applicaion in lo-sizing problems. Sadjadi e al. (2009) proposed an improved algorihm of he Wagner and Whiin mehod. In so doing, hey firs assumed ha shorage is no permied and invenory holding and seup coss are fixed, and hen assumed he possibiliy of shorage and variabiliy of seup and holding coss. Golany e al. (2001) sudied a producion planning problem wih remanufacuring. hey proved he problem is -complee and obain an O ( 3 ) algorihm for solve he problem. i e al. (2007) analyzed a version of he capaciaed dynamic lo-sizing problem wih subsiuions and reurn producs. hey firs applied a geneic algorihm o deermine all periods requiring seups for bach manufacuring and bach remanufacuring, and hen developed a dynamic programming approach o provide he opimal soluion o deermine how many new producs are manufacured or reurn producs are remanufacured in each of hese periods. an e al. (2009) addressed he capaciaed dynamic lo- sizing problem arising in closed-loop supply chain where reurned producs are colleced from cusomers. hey assumed ha he capaciies of producion, disposal and remanufacuring are limied, and backlogging is no allowed. Moreover, hey proposed a pseudo-polynomial algorihm for solving he problem wih boh capaciaed disposal and remanufacuring. Zhang e al. (2012) invesigaed he capaciaed lo-sizing problem in closed-loop supply chain considering seup coss, produc reurns, and remanufacuring. hey formulaed he problem as a mixed ineger program and propose a agrangian relaxaionbased soluion approach Fuzzy lo-sizing problem Fuzzy se heory was firs proposed by Zadeh(1965). I is a mahemaical ool o describe imprecision in he fuzzy environmen. Imprecision refers o he sense of vagueness raher han he lack of knowledge abou he value of parameers. he vagueness is due o he unique experiences and judgmens of decision makers. Fuzzy mahemaical programming or fuzzy opimizaion, which was proposed by Zimmermann (1996), is one applicaion of fuzzy se heory. here has been a lo of research which deals wih vagueness in lo-sizing models as one of he fuzzy mahemaical programming models. For example, Yao and ee (1999) invesigaed a group of compuing schemas for he economic order quaniy as fuzzy values of he invenory wih/wihou backorder. Yao and hiang (2003) fuzzified he oal demand and he cos of soring one uni per day o he riangular fuzzy numbers. hey defuzzified by he cenroid and he signed disance mehods. Finally, hey compared he resuls obained by cenroid and signed disance mehods in he case of he oal cos of invenory wihou backorders. hey expressed he fuzzy order quaniy as he normal rapezoid fuzzy number and hen solved he aforemenioned opimizaion problem. ai (2003) applied he fuzzy se heory o solve he capaciaed lo-size problem and by using numerical examples. Wong e al. (2012) proposed a sochasic dynamic lo-sizing problem wih asymmeric deerioraing commodiy, in which he opimal uni cos of maerial and uni holding cos would be deermined. hey used arificial neural nework (A) and modified an colony opimizaion (AO) o solve his sochasic dynamic losizing problem. Guillaume e al. (2003) invesigaed losizing problem wih fuzzy demands. Mandala e al. (2005) invesigaed muli-iem muli-objecive invenory model wih shorages and demand dependen uni cos has been formulaed along wih sorage space, number of orders and producion cos resricions. hey imposed he cos parameers, he objecive funcions and consrains are in fuzzy environmen. hey used geomeric programming mehod o solve he model. hang e al. (2006) presened a fuzzy exension of he economic lo-size scheduling problem (ES) for fuzzy demands. hey used a geneic algorihm o solve he problem. hen and hang (2008) sudied fuzzy economic producion quaniy (FEQ) model wih defecive producions which canno be repaired, fuzzy opporuniy cos. hey used funcion principle as arihmeical operaions of fuzzy oal producion invenory cos, and used he graded mean inegraion represenaion mehod o defuzzify he fuzzy oal producion and invenory cos. Halim e al. (2011) consider a single-uni unreliable producion sysem which produces a singleiem. hey developed wo producion planning models on he basis of fuzzy and sochasic demand paerns and defuzzified by using he graded mean inegraion represenaion mehod. Kesarapong e al. (2011) proposed a single-iem lo-sizing problem wih fuzzy parameers, which is called he fuzzy single-iem lo-sizing problem. hey used he possibiliy approach o conver he fuzzy model o he equivalen crisp single-iem lo- sizing problem (E-SIS). Sahebjamnia and orabi (2011) developed a fuzzy sochasic muli-objecive linear programming model for a muli-level, muli-iem capaciaed lo-sizing problem. In his paper, he single-iem capaciaed lo-sizing problem is discussed. An inegraed model wih backlogging, safey socks and ousourcing wih differen producion mehods and limied warehouse space is presened anda fuzzy mahemaical model is proposed. Four meaheurisic algorihms named simulaed annealing (SA), harmony search (HS), vibraion damping opimizaion (VDO) and 134

3 ournal of Opimizaion in Indusrial Engineering, Vol. 11, Issue 1,Winer and Spring 2018, geneic algorihm (GA) have been used o solve he proposed model. he remaining of his paper is organized as follows Secion 2 describes he single-iem capaciaed lo-sizing problem wih fuzzy parameers and equivalen crisp of he single-iem capaciaed lo-sizing problem. he soluion approaches for solving he proposed model inroduced in Secion 3. he aguchi mehod for uning he parameers and compuaional experimens is presened in Secion 4. he conclusions and suggesions for fu ur e sudies are included in Secion roblem Formulaion In his secion, w e presen an MI formulaion of he problem. he single-iem capaciaed lo-sizing problem wih backlogging, safey socks and limied ousourcing in closed-loop supply chain, is a producion planning problem in which here is a ime-varying demand for an iem over periods. In his secion, we presen a formulaion of he problem. Firs, he problem assumpions, parameers, and decision variables have horoughly been inroduced and hen he proposed model has been defined Assumpions Be for e he formulaion is considered, he oh er following assumpions are made on he problem. I. he s h o r a g e, invenory, safey sock defici coss, variable cos, seup cos, ou-sourcing cos and demand are non-deerminisic. II. he amoun of he reurned producs is regarded deerminisic over he planning horizon. III. Shorage is backlogged. IV. Shorage and invenory coss mus be aken ino consideraion a he end. V. he quaniy of invenory and shorage a he beginning of he planning horizon is zero. VI. he quaniy of invenory and shorage a he end of he planning horizon is zero arameers umber of periods in he planning horizon, =1,, umber of producion manner, j=1,, j he producion cos of each uni in he period hrough he manner j 2.4. he proposed model A j h h d he seup cos of he producion in he period hrough he manner j he uni holding cos in he period Uniary safey sock defici cos in period he demand in he period he quaniy of he safey sock of produc i in he period Uniary shorage cos in period Uni ou-sourcing cos a period M A large number d r R he uni holding cos of reurned producs in period he maximum number of reurned producs ha could be disposed in period he maximum number of reurned producs ha could be remanufacured in period he number of reurned producs in period 2.3. Decision Variables j roducion quaniy in he period hrough he manner j f he number of reurned producs ha remanufacured in period he number of reurned producs ha disposed in period s y Bi nar y variable, 1 if he produced in he period j hrough he manner j, oherwise U Ou-sourcing level a period I S y j = 0 he quaniy of shorage in he period he quaniy of oversock defici in he period S he quaniy of safey sock defici in he period f s r MinZ ( ( A y ) I h S h S U F g I ) Subjec o 1 j 1 j j j j i i (1) 135

4 Amir Faehi Kivi e al./fuzzy Mahemaical Model j 1 j 1 S S I I U S S d 1,2,..., S 0 (3) I 0 (4) I I R r r f s 1 j My j 1,2,...,, 1,2,..., j I d 1, 2,..., 1 (6) S 1,2,..., (8) 0 U I 1 S 1 d f s d r 1, 2,..., 1, 2,..., 1, 2,..., y {0,1} j 1,2,..., 1,2,..., (12) j (2) (5) (7) (9) (10) (11), f, s, r I, I, S, S 0 j 1,2,..., 1,2,... j (13) he objecive funcion (1) shows oal cos. onsrains (2) are he invenory flow conservaion equaions hrough he planning horizon. onsrains (3) and (4) define respecively, he demand shorage and he safey sock defici for iem a end period is zero. onsrains (5) are he invenory flow conservaion equaions for reurned producs. consrains (6) impose ha he quaniy produced mus no exceed a maximum producion level M. M is he oal demand requiremen for produc on secion [, ] of he horizon, M is hen equal Eq. (14) o M 1 d ( 14) onsrains (7) and (8) define upper bounds on, respecively, he demand shorage and he safey sock defici in period. onsrains (9) ensure ha ousourcing level U a period is nonnegaive and canno exceed he sum of he demand, safey sock of period and he quaniy backlogged, safey sock defici from previous periods. onsrains (10) and (11) are he capaciy consrains of disposal, remanufacuring.onsrains (12) and (13) characerize y j is a binary variable and he f s r variable's domains,,, I, I, S, S non-negaive for j and. j are 2.5. Equivalen crispsingle-iem capaciaed lo-sizing problem he possibiliy approach in he conex of fuzzy se heory was inroduced by Zadeh (1978) o deal wih nonsochasic imprecision and vagueness. According o he Dubois and rade (1988) and Dubois (2006), he possibiliy approach appropriaely was used o model various kinds of informaion, such as linguisic informaion and uncerain formulae, in logical seings. In his secion, he possibiliy approach will be used o conver he fuzzy model o he equivalen crisp model. In his paper, hance-onsrained rogramming () by harnes and ooper (1959), which is normally used o confron sochasic linear programming (S), is adoped as a way o conver he fuzzy single-iem capaciaed losizing problem o he Equivalen crisp single-iem capaciaed lo-sizing problem. he concep of guaranees ha he probabiliy of sochasic consrains is greaer han or equal o a pre-specified minimum probabiliy. erworasirikul e al. (2003) proved and proposed he below emma e ai for i = 1,..., n be fuzzy variables wih normal and convex membership funcions and b be a crisp variable. he lower and upper bounds of he -level se of ai are 136

5 ournal of Opimizaion in Indusrial Engineering, Vol. 11, Issue 1,Winer and Spring 2018, denoed by i ) α ( a and U ( a i ) α, respecively. hen, for any given possibiliy levels 1, 2 and 3 wih 0 < 1, 2, 3 < 1, (Kesarapong e al, 2011) iff ( a1) α ( an) α b, 1 1 if U U ( a1) α ( an ) α b, 2 2 if ( a ) ( a ) b and U U ( a ) ( a ) b. (i) π( a1 an b) α1 (ii) π( a1 an b) α2 (iii) π( a1 an b) α3 α n α M i n Z Subjec o 1 α n α he single-iem capaciaed lo-sizing problem wih fuzzy parameers is ransformed ino he equivalen crispsingleiem capaciaed lo-sizing problem by he above emma, as follows j 1 j 0 j A j 1 A j 0 y j I j1 ( (( ( ) ( 1 ) ( ) ) ( ( ) ( 1 )( ) ) ) ( ) ( ( h ) ( 1 )( h ) ) S ( ( h ) ( 1 )( h ) ) S ( ( ) ( 1 )( ) ) U f i 1 i 0 i s r ( ( F ) ( 1 ) ( F ) ) ( ( g ) ( 1 )( g ) ) ( ( ) ( 1 )( ) ) I ) U U U j d 1) j S S I I U S S ( ) ( ) ( 1, 2,...., j d 1) j1 (15) (16) (17) S S I I U S S ( ) ( ) ( 1, 2,...., S 0 (18) I 0 (19) r r f s I I R 1 j M y j 1,2,...,, 1,2,..., j U I ( d ) 1, 2,..., 1 (21) (22) S ( ) U 1,2,..., U U I S ( d ) ( ) f s d U ( ) r U ( ) 1, 2,..., 1, 2,..., U y { 0, 1 } j 1,2,..., 1,2,..., j 1, 2,..., (23) (24) (25) (26) (27), f, s, r I, I, S, S 0 j 1,2,..., 1, 2,.... (28) j 3. Soluion Approaches 3.1. Harmony search algorihm H a r m o n y search (HS) is a new heurisic mehod ha mimics he improvisaion of music players. H S w a s 137

6 Amir Faehi Kivi e al./fuzzy Mahemaical Model.. proposed by Geeme al. (2001). Inspiraion was drawn from musical performance processes ha occur when a musician searches for a beer sae of harmony, improvising he insrumen piches owards a beer aesheic oucome. he HS algorihm imposes fewer mahemaical requiremens and does no require specific iniial value seings of he decision variables. Because he HS algorihm is based on sochasic random searches, he derivaive informaion is also no necessary. In he HS algorihm, musicians search for a perfec sae of he harmony which deermined by aesheic esimaion, as he opimizaion algorihms search for a bes sae (i.e., global opimum) deermined by an objecive funcion. Harmony search Objecive funcion f(x i ), i=1 o Define HS parameers HMS, HMR, AR, and BW Generae iniial harmonics (for i=1 o HMS) Evaluae f (x i ) While (unil erminaing condiion) reae a new harmony x i new, i=1 o If(U(0,1) HMR), x i new =x j old, where x j old is a random from {1,,HMS} Else if (U (0, 1) AR), x i new =x (i)+ U(0, 1) [x U (i) - x (i)] Else x i new =x j old + BW[(2 U(0,1))-1], where x j old is a random from {1,,HMS} end if Evaluaion f (x i new ) Accep he new harmonics (soluions) if beer End while Fine he curren bes esimaes 3.2. Geneic algorihm he geneic algorihm (GA) has been proven o be powerful mehod for combinaorial opimizaion problems. I was firs proposed by Holland (1975) o encode he feaures of a problem by chromosomes, in which each gene represens a feaure of he problem. In general, he GA consiss of he following seps Sep 1 Iniialize a populaion of chromosomes. Sep 2 Evaluae he finess of each chromosome. Sep 3 reae new chromosomes by applying geneic operaors (e.g., crossover and muaion) o he curren seleced chromosomes. Sep 4 Evaluae he finess of he new populaion of chromosomes. Sep 5 If he erminaion condiion is saisfied, sop and reurn he bes chromosome; oherwise, go o Sep Represenaion schema In his paper, each chromosome is a producion plan and each chromosome formed as an ineger vecor wih genes as shown in Fig 1, where is he number of periods Selecion he selecion provides he opporuniy o deliver he gene of a good soluion o nex generaion. here are various selecion operaors available ha can be used o selec he parens. In his sudy, he ournamen selecion is employed rossover rossover is a process, in which chromosomes exchange genes hrough he breakage and reunion of wo chromosomes o generae a number of children. rossover s offspring should represen soluions ha combine subsrucures of heir parens. In his sudy, crossover generaes an offspring by combining wo selecive parens as shown in Fig Muaion A muaion scheme ha swaps he value of he (number of periods srongly muaion) random seleced genes of he curren soluion wih each oher. Fig. 3 illusraes his operaion on he=8 and Sm=0.5. In our implemened GA, crossover and muaion operaors are used wih he given probabiliies Finess funcion he finess funcion is he same as he objecive funcion defined in Secion erminaion condiion he search process sops if he sopping crierion (i.e., maximum number of improvisaions) is saisfied, hen compuaion is erminaed. 4. Experimenal Resuls We ry o es he performance of he GA and HS in finding good qualiy soluions in reasonable ime for he problem. For his purpose, 15 problems wih differen sizes are generaed for each algorihms. he producion manners and periods has he mos impac on problem hardness. he proposed model coded wih ingo (ver.8) sofware using for solving he insances, GA and HS is implemened o solve each insance in five imes o obain more reliable daa. he geneic algorihm and harmony search are coded in MAAB R2011a and all ess are conduced on a no 138

7 ournal of Opimizaion in Indusrial Engineering, Vol. 11, Issue 1,Winer and Spring 2018, book a Inel ore i5 rocessor 1.7 GHz and 6 GB of RAM. In his paper, he aguchi mehod applied o calibrae he parameers of he proposed algorihms. he aguchi mehod was developed by aguchi (2000). his mehod is ba sed on maximizing he performance measures, called signal-o-noise (S/) raio, in o r d e r o fin d he opimized levels of he effecive facors in he experimens. he S/ raio refers o he mean-square deviaion of he objecive funcion ha minimizes he mean and variance of qualiy characerisics o make hem closer o he expeced values. For he fa c or s ha have significan impac o n he S / raio, he highes S/ raio provides he opimum level for ha facor. able 1 liss differen levels of he facors for HS and GA. In his paper, according o he levels and he number of he facors, respecively he aguchi mehod 25 A (17000,18000,19000, 20000), A (19000,20000,21000,22000) E is used for he adjusmen of he parameers for he HS and GA. F i g s 4 and 5 shown S / raios. Bes evel of he facor for each algorihm is shown in able 2. ompuaional experimens are conduced o validae and ver i fy he behavior and he performance of he meaheurisic algorihms employed o solve he considered single-iem capaciaed lo-sizing problem. We ry o es he performance of he proposed GA and HS finding good qualiy soluions in a reasonable ime for he problem. For his purpose, 15 problems w i h differen sizes and 0.4 are generaed. For hese problems, w e used he rapezoidal fuzzy seup cos and uni price which would be classified as hree levels cheap, normal and expensive. I is respecively denoed by, A (21000,22000,23000,24000) (60,65,70,75), (70,75,80,85) and (70,75,80,85) he rapezoidal fuzzy demand and safey sock are classified as hree levels, low, medium and high. I is respecively denoed by d (8, 9,10,11), d (10,11,12,13), d H (13,14,15,16), ( 1, 2, 3, 4 ), (3,4,5,6) and (5, 6, 7,8). M H E M, he riangular fuzz y invenory holding cos, s h o r a g e cos, safey sock defici cos, ou-sourcing cos, repair cos, h h u F E h (6,7,8), (7,8,9) (12,13,14), h E (13,14,15), h E (8,9,10) dispose cos and invenory holding cos for reurn producs are classified as hree levels cheap, normal and expensive., (16,17,18), (17,18,19), (18,19,20), u (30000,35000, 40000), u (35000,40000,45000), (40000,45000,50000), g (55,60,65), g (60,65,70), g (65,70,75) Medium ow Medium (20,25,30), F (25,30,35), ow (4,5,6), Medium (5,6,7), (6,7,8). High High E High, h (11,12,13),, F (15,20,25), he number of manners and periods has he mos impac on he problem hardness. he proposed model coded wih ingo (Ver.8) sofware using for solving he insances. he bes resuls are recorded as a measure for he relaed problem. able 3 shows deails of compuaional resuls obained by soluion mehods for all es problems. he resuls of running he proposed GA and H S are compared w i h he opimal soluion of he insances, obained f r o m ingo sofware, F i g 6 depic comparison beween soluion qualiy of he ingo, GA and H S of he insances, and Fig 7 depic comparison beween soluion qualiy of GA and HS. he GA and HS algorihms can solve all he es problems for boh models. he GA and HS can find good qualiy soluions for small dimensions problems. he objecive values obained by HS are beer fr om GA resuls for arge dimensions es problems. Fig 8 shows he U ime of he HS is less han GA. 5. onclusion In his paper, we propose a mahemaical formulaion of a new single-iem capaciaed lo-sizing problem wih backlogging, safey socks and limied ousourcing in closed-loop supply chain. his formulaion akes ino accoun several indusrial consrains such as shorage coss, safey sock defici coss and limied ousourcing. Due o he complexiy of he problem, GA and H S algorihms is used o solve problem insances. Several problems wih differen sizes generaed and solved by HS and ingo sofware. he resuls s h o w ha he H S algorihm able o fi n d good qualiy soluions in 39

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