Solving a bi-objective vehicle routing problem under uncertainty by a revised multichoice goal programming approach

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1 Internatonal Journal of Industral Engneerng Computatons 8 (2017) Contents lsts avalable at GrowngScence Internatonal Journal of Industral Engneerng Computatons homepage: Solvng a b-objectve vehcle routng problem under uncertanty by a revsed multchoce goal programmng approach Hossen Yousef a, Reza Tavakkol-Moghaddam b*, Mahyar Taher Bavl Olae b, Mohammad Mohammad a and Al Mozaffar c a Faculty of New Scences and Technologes, Unversty of Tehran, Tehran, Iran b School of Industral Engneerng, College of Engneerng, Unversty of Tehran, Tehran, Iran c Structural Cvl Engneerng, Islamc Azad Unversty Tabrz Branch, Tabrz, Iran C H R O N I C L E A B S T R A C T Artcle hstory: Receved September Receved n Revsed Format October Accepted January Avalable onlne January Keywords: Vehcle routng problem Mult-choce goal programmng Customer prorty Customer satsfacton A vehcle routng problem wth tme wndows (VRPTW) s an mportant problem wth many real applcatons n a transportaton problem. The optmum set of routes wth the mnmum dstance and vehcles used s determned to delver goods from a central depot, usng a vehcle wth capacty constrant. In the real cases, there are other objectve functons that should be consdered. Ths paper consders not only the mnmum dstance and the number of vehcles used as the objectve functon, the customer s satsfacton wth the prorty of customers s also consdered. Addtonally, t presents a new model for a b-objectve VRPTW solved by a revsed mult-choce goal programmng approach, n whch the decson maker determnes optmstc aspraton levels for each objectve functon. Two meta-heurstc methods, namely smulated annealng (SA) and genetc algorthm (GA), are proposed to solve large-szed problems. Moreover, the expermental desgn s used to tune the parameters of the proposed algorthms. The presented model s verfed by a real-world case study and a number of test problems. The computatonal results verfy the effcency of the proposed SA and GA Growng Scence Ltd. All rghts reserved 1. Introducton A vehcle routng problem (VRP) s an essental ssue wth wdely studed optmzaton problems, such as dstrbuton and transportaton logstc problems (Huang & Hu, 2012). Moreover, n the recent years, some ssues (e.g., global warmng, resource scarcty, traffc congeston and polluton) are causng companes and governments to desgn logstc systems effectvely (Archett et al., 2014). The VRP s a complex problem n the feld of operatons research. Ths problem regards to determne a set of mnmum-cost vehcle routes, whch startng and fnshng from a central depot, usng a vehcle wth a capacty constrant to servce a set of customers. Each customer s servced by only one vehcle by consderng the capacty constrant of vehcles. The VRP also has some varants that nclude some * Correspondng author Tel: ; Fax: E-mal: tavakol@ut.ac.r (R. Tavakkol-Moghaddam) 2017 Growng Scence Ltd. All rghts reserved. do: /j.jec

2 284 addtonal constrants, such as tme wndows. The VRP wth tme wndows (VRPTW) s a varant of the VRP that consders the tme wndows for customers and makes sure that the servces must start n ther tme wndows. In the recent years, t has attracted more and more attenton and studed wdely. For the frst tme, the VRP was proposed by Dantzg and Ramser (1959) and then has been studed wdely (e.g., Montoya-Torres et al., 2015; Pllac et al., 2013). In the VRPTW, the tme wndow concept s consdered due to a customer may need the earlest and latest servce tmes. The earlest and latest tmes represent the lower and upper bounds of a tme wndow, respectvely. The tme wndow condton s useful n many cases, such as goods dstrbuton, school bus routng, and after-sales servce problems. In the mentoned problems, customers must be servced n a determned tme wndow. The tme wndow restrcton s classfed to hard and soft tme wndows. In the hard tme wndow condton, ths constrant must be satsfed strctly. However, the tme wndow condton can be volated by consderng the penaltes cost n soft tme wndows. Gehrng and Homberger (2002) descrbed the parallelzaton of a two-phase meta-heurstc algorthm for the VRPTW. Eksoglu et al. (2009) presented the comprehensve detals of used heurstc and metaheurstc methods n the VRP. A large neghborhood and varable neghborhood search (VNS) methods are presented n Rncon-Garca et al. (2017) to solve the VRPTW by consderng hard tme wndows. In ther proposed model, the soluton procedure ncludes two steps. In the frst step, the requred number of vehcles s mnmzed by a large neghborhood search method. Fnally, the total travel dstance s mnmzed n a feasble search space. The real applcaton problems are mult-objectve problems (Melán-Batsta et al., 2014; Rath et al., 2015). For nstance, the ant colony algorthm (ACA) s presented n Santa Chávez et al. (2016) to mnmze the total travel dstance, travelng tmes and energy consumpton smultaneously. In the more of real cases, on-tme delvery s consdered as a key to measurng the performance of servcng (Forslund & Jonsson, 2010). Transportaton companes seek to satsfy customers tme wndows. However, full satsfacton of customers' requrements s not able n some condtons due to resource restrctons or cost reducton. In fact, companes try to reduce the total system cost and also on the other hand, companes would lke to satsfy the customers' requrements n order to enhance the customers' satsfacton. In the VRPTW, the servce delay s consdered as a customer wated tme when a vehcle arrves at a customer after ts earlest servce tme. In real-lfe problems, the earlest servce tme can be consdered as the most favorable tme for customers. In the compettve condton, the customer's satsfacton degree has earned more consderaton and companes try to enhance the customer's satsfacton by mnmzng the customer watng tme. In the realty, customers are nterested to be servced as soon as possble; moreover, each customer has ts own prorty for the companes. In the recent years, because of ncreasng competton for effcent servce and rgd necesstes of customers, a physcal dstrbuton turns nto more complcated. In applcable problems n the VRP, there are several objectves, such as the number of used vehcles, the total watng tme, total traveled dstance, makespan (.e., the longest route), total delay tme and so on (Castro-Guterrez et al., 2011). A few studes can be found that deal wth mult-objectve functons, especally the ones that consder another aspect of the objectve functons (e.g., customer satsfacton) and consder the opnon of the decson makers (DMs) n the decson process. Zografos and Androutsopoulos (2004) presented a mathematcal model n the VRPTW for transportaton hazardous goods by consderng prorty n order to mnmze the total travel tme and the total rsk. Afshar-Bakeshloo et al. (2016) presented the mxednteger lnear program (MILP) model n the VRPTW and consdered the customer satsfacton and polluton n addton total dstance travel and the total requred number of vehcles. Ghannadpour et al. (2014) presented a mathematcal model n VRPTW by consderng the uncertanty n a customers request. The man am of the presented model s to mnmze the total dstance travel and the requred number of vehcles and maxmze the customers satsfacton smultaneously. A three-phase

3 H. Yousef et al. / Internatonal Journal of Industral Engneerng Computatons 8 (2017) 285 tabu search algorthm s presented n Taş et al. (2013) to mnmze the total cost and the customers' expected earlness and lateness. Lee et al. (2012) presented a robust optmzaton method n the VRP to mnmze the total dstance travel and the requred number of vehcles n the VRPTW by consderng travel and demand as an uncertan parameter and customers deadlnes. Barkaou et al. (2015) presented a mathematcal model n a dynamc VRPTW and ntegratng antcpated future vst requests durng plan generaton to mprove the customer s satsfacton. Svaramkumar et al. (2015) proposed a mathematcal model n the VRPTW by consderng the customer s satsfacton n order to mnmze the total dstance cost and the total requred number of vehcles. They consdered customer satsfacton by mnmzng the sum of the total gap between ready and arrvng tmes. A new hybrd varable neghborhood-tabu search heurstc n the VRPTW has been presented n Belhaza et al. (2014) to mnmze a backward tme slack n the VRP wth multple tme wndows. As can be seen from the presented lterature revew of ths paper, a mult-objectve optmzaton problem by consderng the prorty and customers' satsfacton has been consdered rarely. Goal programmng (GP) s one of the common methods to solve the mult-crtera decson-makng (MCDM) problems and attempts to optmze a number of objectves smultaneously. In the recent decade, t has earned more consderaton. In GP, specfyng the aspraton level for each objectve s essental. The man am of ths method s to reduce the devatons from aspraton levels. Usually, n realty, the aspraton level of objectves s mprecse for the DMs. The mathematcal model n the VRP wth soft tme wndows (VRPSTW) usng a GP method s presented n Calvete et al. (2007). The proposed soluton procedure conssts of two steps. In the frst step, the feasble routng has been determned and then n the second; the set of best ones has been selected as the optmum soluton. The computatonal results demonstrated that the mentoned soluton procedure has an effcent performance for medum-szed problems. Hong and Park (1999) proposed a b-objectve model n the VRPTW and used a GP method to solve the problem as well as developed a heurstc algorthm to reduce the computatonal tme for medum and large-szed problems. Ghoser and Ghannadpour (2010) proposed a novel mathematcal model and soluton procedure n the VRPTW usng GP and genetc algorthm (GA) to mnmze the total dstance cost and vehcles used smultaneously. Most of the researchers assume that the nput parameters are determnstc and certan. However, n the realty, many parameters (e.g., customer demand, travelng tme and vehcle capacty) are mprecse (Delage et al., 2010). HO (1989) addressed the categorzaton of uncertanty nto two groups; envronmental uncertanty and system uncertanty. In the context of the VRP, the uncertanty of the vehcle capacty s ncluded n the system uncertanty, and envronmental uncertanty conssts of the uncertantes n customers demand and the travelng tme. There are three reasons for consderng the uncertanty: (1) for the consderable tme gap between desgn and mplementaton, (2) because of hgh cost for obtanng the parameters of problems exactly, and (3) the lack of nformaton. In real cases, adequate nformaton s not always accessble for predctng uncertan parameters. The fuzzy approach s a suffcent method to demonstrate uncertantes over the experts or decson makers knowledge. Customer demand s the sgnfcant parameter that s mprecse because of nadequacy and/or unavalablty of requred nformaton and unpredctable customer s behavor. The fuzzy travel tme n the VRP s proposed n La et al. (2003), who employed the fuzzy programmng wth a possblty measurement and they used the GA to solve the proposed model. Moghadam and Seyedhossen (2010) consdered the customers demand as an uncertan parameter wth the unknown locaton. Zheng and Lu (2006) addressed the VRP by fuzzy travel tme and proposed a chance-constraned programmng (CCP) model wth credblty measurement. They ncorporated a fuzzy smulaton and GA as a hybrd ntellgent algorthm to solve the proposed model. Hashmoto et al. (2006) proposed local search n the VRPTW wth flexble tme wndows and travel tmes. The mathematcal model n VRPTW has been proposed n Goncalves et al. (2009) by consderng customer s demand as a fuzzy random varable. They ntegrated a stochastc smulaton and GA as a hybrd algorthm to solve

4 286 the proposed model. In ths paper, the demand of customers s consdered as a fuzzy parameter. Based on the revew study presented n ths paper, there are few efforts to consder the customer satsfacton n the VRPTW. As the best of our knowledge, ths s the frst study that employs the revsed mult-choce goal programmng (RMCGP) n the VRPTW. Hence, n ths paper, a new b-objectve possblstc programmng model n the VRPTW by consderng the prorty and customers satsfacton s proposed. The man am of the proposed model s to solve a new mathematcal model n order to nvestgate the trade-off between the cost of the desgned system and the customers satsfacton. The frst objectve s to mnmze the sum of a fxed cost assocated wth the number of vehcles used and the total travel dstance. The second objectve s to mnmze the gap tme between the arrval tme and the ready tme by consderng the prorty of the customers. The rest of ths paper s organzed as follows. The formal descrpton of the proposed mathematcal model s presented n Secton 2. Secton 3 presents the method to convert the fuzzy model nto ts crsp equvalent model, revsed mult-choce goal programmng, and the procedure of meta-heurstc algorthms, namely smulated annealng (SA) and genetc algorthm. Secton 4 llustrates the expermental results and the comparson between SA and GA. Fnally, Secton 5 s dedcated to the concluson. 2. Problem defnton The VRPTW s descrbed by a set of customer n to be served; a specal node (named a depot) and a possble network connectng between the nodes. The vehcles leave the depot, serve the customers and must return to the depot. The customers to be servced are denoted by nodes 1, 2,, n. The nodes 0 and n+1 represent the same node (central depot). Therefore, the network connecton of the problem conssts of n+2 nodes. In ths study, each customer has the predefned prorty to be servced. A route s defned as startng from a depot, travelng through the arc to serve the customers and returnng to the depot. The travel tme tj s dentfed by each arc of the network. The demand of each customer s denoted by d as mprecse data. The customers must be served only once by one of the vehcles by consderng vehcles lmted capacty. Therefore, the capacty of the vehcles must be greater than or equal to the sum of ts allocated customer. On the other hand, each customer must be served by vehcles durng a predefned tme wndow [a, b], that a, and b denote the earlest and latest arrval tmes, respectvely. Each vehcle arrves to customer later than b s penalzed. Moreover, f t arrves earler than a, t should be watng untl openng tme. In ths paper, a b-objectve mathematcal model for the VRPTW s proposed. The frst objectve functon mnmzes the total travel dstance and vehcles used costs. The second objectve functon mnmzes the sum of gap tme between the arrval tme and the ready tme of customers by consderng the prorty of them. The second objectve functon tres to servce the customers as soon as possble by consderng the prorty of them. Ths objectve functon s appled as a customer satsfacton term. The detals of the proposed model are as follows: 1. Each vehcle starts from node 0, to servce customers then returns to node n The travellng tme between nodes 0 and n 1 s zero. 2. Customers are servced once by only one vehcle. 3. The demand of customers represented as a fuzzy parameter. Moreover, the demand of nodes 0 and n 1 s zero. 4. The vehcle cannot leave the node untl the servce s completed. 5. The customer wth a hgher prorty s shown by a hgher number. For nstance, f the problem conssts of 10 customers, 10 shows the frst prorty and nne shows the second one. In the presented model, the prorty of customers s determned by the populaton of the customers (ctes).

5 H. Yousef et al. / Internatonal Journal of Industral Engneerng Computatons 8 (2017) 287 Parameters: n Number of locatons K Number of vehcles t Travel tme between customers and j ( 0 n,1 j n 1, j ) j d Demand of customer 1 ( n) S Servce tme of customer 1 C a b ( n) Capacty of the vehcle Lower bound of tme wndow of customer Upper bound of tme wndow of customer F Fxed cost per vehcle n use M A suffcently large number 0,1 1, j P Prorty of customer 1 ( n) ( n j n j ) Decson Varables: x 1 f vehcle k vsts mmedately before vstng customer j; 0, otherwse jk w k Tme when vehcle k starts to servce node K n K n n 1 Mn Z F x x t 1 0jk jk j k 1 j 1 k 1 0 j 1 (1) Mn Z s.t. K n 1 k 1 j 1 n K 2 1 k 1 Pg k xjk 1 1 n (3) n x0 jk 1 1 j 1 n n 1 jk 0 1 (2) k K (4) x x jk 0 1 j n,1 k K (5) w s t w 1 x M 0 n,1 j n 1,1 k K (6) k j jk jk j n 1 a x w 1 n,1 k K (7) jk k j 1 n 1 w b x 1 n,1 k K (8) k jk j 1 n n 1 d x C 1 k K (9) jk 0 j 1 g k wk a f a wk 0 otherwse 1 n,1 k K (10) xjk {0,1}, 0 n, 1 j n 1, 0 k K (11)

6 288 In ths model, the frst objectve functon (Eq. (1)) mnmzes the sum of the total requred number of vehcles and the total travel dstance. The second objectve functon (Eq. (2)) mnmzes the gap tme between the arrval tme and ready tme by consderng the prorty of the customers. Eq. (3) nsures that each customer served once by only one vehcle. Eq. (4) guarantes each vehcle starts to servce the customers from the depot (0) and fnally return to the depot (n+1), t should be noted nodes (n+1) and (0) demonstrate the same depot. Eq. (5) guarantees that each vehcle must leave for other customers once t servces at a customer node. Eqs. (6) to (8) ensure the tme wndow requrement s observed. If the customer s not served by vehcle k, the value of w k wll be zero by consderng Eqs. (7) and (8). Eq. (9) guarantees that the sum of the customers loads n each route does not exceed the vehcle capacty lmtaton. The relatve gap tme of customers s determned by Eq. (10). For each customer the gap tme determned by takng the dfference between the arrval tme of vehcle to the customers and the lower bound of the tme wndows. Eq. (11) ndcates the logcal bnary requrement of the decson varables. Eq. (10) can be lnearzed by replacng Constrant (12); that s a bnary varable and M s a sgnfcant k large number. The constrant set s defned by:, k (12a) g w a M k k k g w a M k k k g M(1- ) k k g M (1 - ) k 3. Soluton procedure k, k, k, k (12b) (12c) (12d) The uncertanty can be categorzed as uncertanty n nput parameters and flexble programmng (Dubos & Perny, 2016). Pshvaee and Razm (2012) categorzed the uncertanty of the nput parameters as: (1) randomness, the random nature of the nformaton can result n randomness; typcally, researchers used stochastc programmng methods to confront wth t; (2) epstemc uncertanty; ths knd of uncertanty deals wth obscure coeffcents n objectves and constrants. In the proposed model, a possblstc programmng method s used to deal wth the uncertan data. To solve a proposed model, n the frst step, the model s converted nto an equvalent auxlary crsp model. Then n the second step, a revsed multchoce goal programmng (RMCGP) method s used to fnd a preferred compromse soluton. Some researchers (e.g., da Slva et al., 2013; Lao & Kao, 2010) used ths method to solve ther proposed model. The proposed model s coded n LINGO software. The valdty of the model s tested on some randomly generated data sets. Due to the proposed model belongs to NP-hard problems (Ghannadpour et al., 2014); the requred tme to solve large-szed problems s too much (more than 6 hours). Thus, smulated annealng (SA) and genetc algorthm (GA) are used to obtan a near-optmal soluton n a reasonable computatonal tme Step 1: An auxlary crsp model Some methods have been presented to fnd compromse solutons to confront wth possblstc programmng models (Jménez et al., 2007; La & Hwang, 1992). In ths paper, we employ an extended verson of the method proposed by La and Hwang (1992) to transform the proposed model nto an auxlary crsp model. To deal wth the mprecse demand of customers on the left-hand sde of Eq. (9), the weghted average method has been employed n La and Hwang (1992) to defuzzfy and convert ths parameter to a crsp number. In ths paper, a trangular fuzzy number s utlzed to consder customer demand as a fuzzy parameter (Torab & Hassn, 2008). Fg. 1 demonstrates the trangular possblty

7 dstrbuton of a fuzzy number A Ap, Am, Ao H. Yousef et al. / Internatonal Journal of Industral Engneerng Computatons 8 (2017) 289, where A, A, and p m A are the most pessmstc value, the o most possble value and the most optmstc value, respectvely, whch determned by the DMs. A p A m A Fg. 1. Trangular possblty dstrbuton of the fuzzy parameter. p A A The equvalent auxlary crsp constrants can be demonstrated by: n n 1 p m o (w1 d, w2 d, w 3 d, ) xjk Ck 0 j 1 1 k K (13) where, w1 w2 w3 1 and w 1, w 2, and w 3 denote the weghts of the most pessmstc, the most possble, and the most optmstc value of the fuzzy demand of customers, respectvely. Moreover, β s the mnmum acceptable possblty. The optmum values of weghts and β usually are determned by the knowledge of the DMs. However, based on the concept of the most lkely values proposed by La and Hwang (1992), these parameters are consdered as: 4 1 w 1, w 1 w3 and β = Step 2: RMCGP Most of the prevous studes are formulated as a sngle-objectve programmng model. However, n realty, researchers and managers are confronted wth mult-crtera problems ( Jones & Tamz, 2016). GP s a method to solve mult-objectve problems by achevng a set of compromsng solutons. For the frst tme, ths method has been ntroduced n Charnes and Cooper (1957). Ths method mnmzes the devatons between aspraton levels and achevements of each goal. The aspraton level of each goal s determned by the DMs. The comprehensve revew of the GP method can be found n Aoun et al. (2009). The lnear and non-lnear GP can be solved usng well-developed software, such as LINGO software or meta-heurstcs methods (e.g. SA and GA) (Jones et al., 2002). GP s more straghtforward and adaptable to employ varous scenaros by accommodatng ether target values or weghts. The prmary ntenton n GP s to present extra auxlary varables, whch named as devatons to formulate the model. These devatons demonstrate the dfference between aspraton levels of each goal and the optmum soluton. The negatve devaton ( d ) represents under-achevement of the goal and postve devaton ( d ) presents over achevement of the goal. In the formulaton of the GP, goals are declared as a lnear equaton wth both postve and negatve devaton. The man am of ths method s to mnmze the undesrable devatons between the aspraton level of goals and the optmum soluton. The GP model conssts of two types of constrants; goal and system constrants. System constrants are the man constrants of the model that usually are formulated as lnear programmng. The goal constrants are auxlary constrants, whch used only n a GP method. The goal constrants regulate a soluton by consderng desred goals. The model of GP demonstrated as follows:

8 290 n mn W( d d ) (14a) s.t. 1 h ( X ) ( or )0 k 1, 2,..., q k (14b) f ( X) d d b 1, 2,..., n (14c) d, d 0 1, 2,..., n (14d) where, hk ( X ) s system constrant k, f( X ) represents goal constrant, b represents the aspraton level of goal, d and d are postve and negatve devatons from the target value of goal, respectvely. b f( x) f f( x) b d (15) 0 otherwse f ( x) b f f ( x) b d (16) 0 otherwse The standard GP method emphaszes to fnd the best soluton that s near to the aspraton level of objectves, and also attempts to mnmze devatons from aspraton levels. However, n realty, consderng exactly one level for aspraton level s not useful and DMs lke to determne cautous prmary aspraton levels based on avalable nformaton. The mult-choce goal programmng helps to the DMs n makng decsons by preventng understatement of the decsons. Chang (2007) presented a new method to solve the mult-choce goal programng (MCGP) for mult-objectve decson problems by multple aspraton levels. The MCGP s formulated as follows: n mn WfX ( ) g 1 org2 or... gm (17a) s.t. 1 h ( ) ( )0 k x or k 1, 2,..., q (17b) where, g j (=1,2,,n and j=1,2,,m) s the j-th aspraton level of the -th goal gj 1 gj gj 1. The MCGP method can be formulated by: n mn W( d d ) (18a) s.t. 1 h ( X ) ( or )0 k 1, 2,..., q k m k j j j 1 f ( X ) d d g S (B) 1, 2,..., n d, d 0 (18b) (18c) 1, 2,..., n (18d) S (B) R ( X ) 1, 2,..., n j (18e) where, Sj B represents a functon of bnary seral numbers. Chang (2008) presented the revsed approach for MCGP, whch named RMCGP to solve the mult-objectve decson problems wthout employng bnary varables, used n the MCGP achevement functon. The MCGP-achevement can reformulate n two condtons; the more the better and the less the better. The frst condton s formulated by: n mn Wd ( d ) ( e e ) (19a) s.t. 1

9 H. Yousef et al. / Internatonal Journal of Industral Engneerng Computatons 8 (2017) 291 h ( X ) ( or )0 k 1, 2,..., q k f ( X ) d d y 1, 2,..., n k y e e g.max (19b) (19c) 1, 2,..., n (19d) g y g 1, 2,...,.mn.max d, d, e, e 0 1, 2,..., n The second condton s formulated by: n 1 n (19e) mn W( d d ) ( e e ) (20a) s.t. h ( X ) ( or )0 k 1, 2,..., q k (19f) (20b) f ( X) d d y 1, 2,..., n (20c) k y e e g.mn 1, 2,..., n (20d) g y g 1, 2,..., n.mn.max (20e) d, d, e, e 0 1, 2,..., n (20f) where, g max. s the upper bound of the -th aspraton level, gmn. s the lower bound of the -th aspraton level, s the contnuous varable wth a range of gmn. y gmax., d and d are postve and negatve devatons from f( X) y and w s the weght of the -th goal. For the frst case: e and e are postve and negatve devatons from y g.max and s the weght of the sum of devatons of y g.max. In the second case: e and e are postve and negatve devatons from y g.mn and s the weght of the sum of devatons of y g.mn. Regardng to the explaned method n Sectons 3.1 and 3.2, the proposed model s presented by: mn W ( d d ) ( e e ) W ( d d ) ( e e ) (21) s.t. Constrants (3)-(8),(11)-(13) Z d d y (22) y e e g (23) mn g1.mn y1 g1.max (24) Z d d y (25) y e e g (26) mn g2.mn y1 g2.max (27) d, d, e, e, d, d, e, e 0 (28) where, d 1 s the postve devaton varable of Z 1; d 1 s the negatve devaton varable of Z 1; d 2 s the postve devaton varable of Z 2; d 2 s the negatve devaton varable of Z 2; e 1 s the postve devatons

10 292 from y1 g1.mn ; e 1 s the negatve devatons from y 1 g 1.mn ; e 2 s the postve devatons from y2 g2.mn ; e 2 s the negatve devatons from y 2 g 2.mn. The value of g mn. and g max. can determned below: g 1.mn : can be determned by Mnmzng Z 1 g 1.max : can be determned by Maxmzng Z 1 g 2.mn : can be determned by Mnmzng Z 2 g 2.max : can be determned by Maxmzng Z 2 The DMs can determne the g mn. and g max. by consderng the solutons of the mentoned sub-problems and by consultng wth experts Smulated annealng SA s derved from the analogy to annealng n solds and the solvng method of combnatoral optmzaton problems. For the frst tme, the man dea proposed by Metropols et al. (1953). For the frst tme, SA s used to solve the optmzaton problems by Krkpatrck (1984). Usually, optmzaton problems have some local optmal ponts. Smple optmzaton algorthms search the local optmum ponts by selectng a random ntal soluton and obtanng the neghbor from solutons. Usually, the smple algorthms stopped the searchng due to convergence to local optmum ponts. However, SA prevents to stay n the local optmum pont through acceptng cost ncreasng neghbors wth some probablty. The parameters settng of SA algorthms such as the ntal temperature (T 0), the number of neghborhoods, and a temperature reducton factor (α) have a sgnfcant mpact on the performance of SA. In ths algorthm to fnd the optmum soluton, n the frst step, the ntal soluton generated randomly and then n the around of the ntal soluton neghbor s searched. SA has some advantages and dsadvantages compared to other meta-heurstcs, such as GA and partcle swarm soluton (PSO). The advantages of ths algorthm (e.g., easer mplementaton, convergence attrbutes and utlzng hll clmbng) make t popular from meta-heurstc algorthms n the recent decade (Subramanan et al., 2013) Genetc algorthm As mentoned n the lterature revew secton, the GA s one of the most popular algorthms used to solve VRPTW problems. Ths algorthm s a heurstc search algorthm that mmcs evoluton through natural selecton. In ths algorthm, the procedure of searchng begns wth a set of chromosomes mentoned as the ntal populaton. The ntal soluton can be generated randomly or created by heurstc methods. The new populaton s created based on the crossover operator and then mutaton operator s utlzed. The ftness functon s used to select the best soluton. Fnally, a maxmum number of generatons or other stoppng crtera such as a computatonal tme lmt can stop the algorthm Soluton representaton The soluton structure for two proposed meta-heurstc algorthms s same. Fg. 2 demonstrates the possble soluton of nstance conssts four vehcles and 10 customers. Fg. 2. Instance of a soluton representaton.

11 H. Yousef et al. / Internatonal Journal of Industral Engneerng Computatons 8 (2017) 293 As can be seen from Fg.2, two dfferent routes separated wth each other by ndex 0; each number demonstrates the customer that s served by vehcles. If the number of routes be less than vehcles number, t means to serve the customers, there s no need to use all of the vehcles. The proposed soluton structure permts to mnmze the number of vehcles used and the requred number of vehcles smultaneously Best cost-best route crossover (BCBRC) In ths paper, the ntal soluton s generated through an approprate proposed structure that presented n the prevous secton. A crossover operator exchanges the nformaton between two chromosomes. In the feld of the VRPTW, an approprate crossover operaton should not damage the best soluton. Moreover, t should mprove the well-known soluton to create a better soluton. However, some unsutable crossover operators may create nfeasble solutons for VRPTW problems (.e., classcal one-pont crossover; due to the excludng or falng of vertces after reproducton) (Ghoser & Ghannadpour, 2010). Ombuk et al. (2006) presented a best cost-route crossover (BCRC) to mnmze the requred number of vehcles and total dstance cost by consderng the feasblty of constrants. The other verson of the BCRC operator proposed by Ombuk-Berman and Hanshar (2009), named best cost-best route crossover (BCBRC). In ths operator, the best route s selected by consderng the average objectve functon of nodes. Fg. 3 demonstrates the structure of the BCBRC operator Sequence-based mutaton (SBM) The man am of the mutaton operaton s to prepare algorthms to gan local random research ablty. The procedure of the proposed SBM operator ncludes two steps. Two produced soluton from the crossover operator s selected. In the frst step, from each soluton, a lnk s selected randomly to break a route and then make a change n the routes before and after the break ponts to make new solutons. Fg. 4 demonstrates the procedure of the SBM operator to create a new soluton. As can be nferred from ths fgure, ths operator chooses break pont 1 from soluton 1 and break pont 2 from soluton 2, randomly. Then a connecton between the route of customers served before the break pont 1 and the route of customer servced after the break pont 2 s created. By applyng ths procedure, the new soluton s created. As can be seen from Fg. 4, n the generated soluton, a new route s created that namely New Route1 from removng Route1 and preservng Route2. Lkewse, the second new soluton can be made by creatng a connecton between the route of customers served before the break pont 2 and the route of customer servced after the break pont 1. Fg. 3. Example of the BCBRC operator. To create feasble solutons, customers wth early tme wndows and hgher prorty are served at the begnnng of a route. Moreover, the set of customers that have a lower prorty and late tme wndows are

12 294 served at the end of the route. Due to some customers are duplcated or removed for servcng, a repar operator s utlzed to repar ths soluton to feasble soluton. For nstance, n Fg. 4, two customers n new soluton 1 are located on the both of the routes (customers g and h), and the customers d and e are not served. The repar operator reforms the unfeasble soluton to feasble one n the followng ways: If a customer located n both of the routes, the customers removed from one of two routes randomly. If a customer s removed from routes, then by consderng the tme wndow and capacty constrants, the sets of unserved customers nserted nto one of two routes. Obvously, there s no guarantee to create a feasble soluton by applyng ths way. Therefore, f the new soluton s not reformed to feasble one, the new soluton s rejected, and the old soluton s restored. Fg. 5 llustrates the procedure of the repar operator to reform the unfeasble soluton to a feasble one. Soluton1 Soluton 2 New soluton1 Fg. 4. Procedure of the SBM operator. Fg. 5. Procedure of a repar operator.

13 4. Computatonal experments H. Yousef et al. / Internatonal Journal of Industral Engneerng Computatons 8 (2017) 295 The proposed model s valdated by solvng some test problems. The detal of the parameters dstrbuton functons s lsted n Table 1. To solve large-szed test problems, SA and GA are used. The performance of proposed meta-heurstc algorthms s compared wth each other and LINGO software. Both of SA and GA algorthms are compled n MATLAB 7.1 on the personal computer ncludng, Intel Core 2 Duo 2.6 GHz processors and 2 GB RAM. In ths secton, the performance of the proposed SA and GA n terms of soluton qualty and computatonal tme s evaluated n some randomly generated problems. Each algorthm runs for fve tmes and the best result s reported. The parameter settng of meta-heurstc algorthms have consderable effect on ther performance. In ths paper, the parameter settngs of the proposed meta-heurstcs s tuned by the response surface methodology (RSM). The optmum parameter settngs of SA and GA algorthms are tabulated n Tables 2 and 3, respectvely. The solutons of GA and SA are compared wth the optmal solutons obtaned from LINGO software n small-szed test problems. Moreover, for large-szed problems that LINGO software cannot reach the optmum soluton at the reasonable tme. The comparson result between the proposed meta-heurstcs n small-szed problems s reported n Table 4. In small-szed problems, a gap between SA and GA wth LINGO software s reported through the percentage of relatve gap measure that calculated based on [100 (G LINGO G Meta)/G Meta], where G LINGO and G Meta are the objectve functon value (Eq. (21)) of LINGO software and meta-heurstc algorthm, respectvely. Moreover, for large-szed problems, the gap between GA and SA s presented based on [100 (G best-meta G Meta)/G Meta] n whch, G Meta s the objectve functon of the obtaned soluton over meta-heurstc methods and G best-meta s the objectve functon value of meta-heurstc methods that have better performance. Each meta-heurstc method runs for fve tmes and the average gap s reported n Table 5. As can be nferred from Table 5, SA outperforms GA n terms of the soluton qualty n the all of the test problems. Based on the RMCGP method, the soluton procedure of meta-heurstcs algorthms ncludes two steps. Frst, the values of g 1.mn, g 1.max, g 2.mn, and g2.max are obtaned n each replcaton for fve tmes and the best one s selected. Then, based on the obtaned results from the prevous step, the value of the objectve functon (Eq. (21)) s calculated. The requred computatonal tmes for the proposed meta-heurstcs are reported n Table 5. The requred computatonal tme ncreased sharply as the test problem sze becomes larger. The results demonstrate that GA and SA can obtan a near-optmum soluton n a reasonable tme especally n large-szed problems. As can be seen from Table 5, the requred computatonal tme of GA s more than SA. Ths dfference s derved from the fact that GA has some addtonal mechansms (e.g., selecton mechansm and crossover), whch s tme consumng. Moreover, a pared t-test was carred out to compare the runtme of SA and GA. The result of the t-test s llustrated n Table 6. As can be nferred from Table 6, the sgnfcant dfference s not shown from the result of the t-test. Table 1 Sources of random generaton of the parameters. Parameter Value Parameter Value t j C k d S U(0.5,5.25) U(95,115) a b U(3.5,6) U(5,9.5) U(11,25) F 2000 U(0.2,1)

14 296 Table 2 Parameter settngs of SA. Parameter Intal temperature Temperature reducton rate No. of neghborhood value Table 3 Parameter settngs of GA. Parameter Populaton sze Crossover rate Mutaton rate value Table 4 Average relatve gaps and CPU tme for small-szed test problems. SA GA Data set n K Replcatons Gap (%) Tme(s) Replcatons Gap (%) Tme(s) Table 5 Average relatve gaps and CPU Tme for large-szed problems. Data set SA GA n K Tme(s) Gap (%) Tme(s) Table 6 Result of the t-test for SA and GA computatonal tme. Meta-heurstcs SA GA Mean Varance Observatons Hypotheszed Mean Dfference 0 df 35 t Stat P(T<=t) one-tal t Crtcal one-tal P(T<=t) two-tal t Crtcal two-tal

15 H. Yousef et al. / Internatonal Journal of Industral Engneerng Computatons 8 (2017) 297 Fg. 6. Typcal output of test problem 8 Fg. 6 shows a typcal output of test problem 8. In order to determne the effect of the fxed cost of vehcles on the second objectve functon, the senstvty analyss s carred out. Ths experment s carred out for test problem 7 and all trends can be generalzed for other test problems. Fg. 7 demonstrates the senstvty of the second objectve functon upon the fxed cost of vehcles ncrease, n whch the second objectve functon s gettng constant or worse regardng by ncreasng the constant cost of vehcles. The value of second objectve functon Percentage of ncresng the fxed cost of vehcles Fg. 7. Impact of ncreasng the cost of vehcles on the second objectve functon. As can be nferred from Fg. 7, the value of the second objectve functon wll be ncreased by ncreasng the constant cost of vehcles; however, f the fxed cost of vehcles cost s ncreased from 300 to 1200 percent, the value of the second objectve functon remans constant Case study In ths secton, a real case of goods transportaton s studed to valdate the performance of the proposed model. The case study s carred out for an nternatonal transportaton company n Iran. Tranome Tabrz Company (TTC) founded n Ths company transports dfferent type of products to all parts of Iran and some Iran s neghbor countres (e.g., Azerbajan, Turkey, and Iraq). Ths company has contracted wth the Mohandesan Company (MC) for transportng gas capsule to 25 ctes by fve avalable vehcles. The requred nformaton gathered from both of the companes. As mentoned n the prevous sectons, accordng to a dynamc nature of the envronment, the customers demand s consdered as a fuzzy parameter. The proposed model for TTC s solved n LINGO software. To solve the proposed model, n the frst step, the proposed model s converted to an equvalent crsp model. Then, the RMCGP method s used. Table 7 demonstrates the results. As can be seen from Table 7, snce the postve and negatve devatons of the frst goal are zero, the frst goal s fully satsfed. However, n the second goal, snce the

16 298 postve devaton s bgger than zero, the second goal has postve devaton. The soluton does not satsfy the aspraton level 198 y2 279 and acheves 93.33% of the second goal. Fg. 8 demonstrates the soluton of the case study, n whch 25 ctes have servced by fve vehcles. Table 7 Result of the sub-problems by fve vehcles From solvng Z 1 Value From solvng Z 2 Value g 1.mn g 2.mn 198 g 1.max g 2.max 279 d 1 0 d 2 15 d 1 0 d 2 0 e 1 85 e 2 27 e e 0 Z Z y y Fg. 8. Soluton of the case study by fve vehcles Fg. 9. Soluton of the case study by sx vehcles If the DMs decded to ncrease the number of the avalable vehcles from fve to sx vehcles, the proposed model s resolved. Table 8 demonstrates the result of the problem by consderng the sx avalable vehcles. Table 8 Result of the sub-problems by sx vehcles From solvng Z 1 Value From solvng Z 2 Value g 1.mn g 2.mn 181 g 1.max g 2.max 264 d d 2 0 d 1 0 d 2 0 e e 2 22 e 1 0 e 2 0 Z Z y y 2 203

17 H. Yousef et al. / Internatonal Journal of Industral Engneerng Computatons 8 (2017) 299 As can be seen from Table 8, postve and negatve devatons of the second goal are zero; therefore, the second goal s satsfed fully. However, n the frst goal, snce the postve devaton s bgger than zero, the frst goal has a postve devaton. The soluton does not satsfy the aspraton level y and acheves 99% of the frst goal. Fg. 9 demonstrates the optmum vehcle routng problem by sx avalable vehcles. As a senstvty analyss by accordng to Table 9, t can be comprehended that the value of each objectve functon ncreased by ncreasng the value of W correspondng objectve functon. Therefore, the DMs can adjust the value of objectve functons by changng the correspondng W. Moreover, the Pareto fronter of the problems s llustrated n Fgs. (10-11). As can be nferred from these fgures, due to the both of objectve functons are mnmzed, thus the Pareto fronters for the problem s concave. Table 9 Result of a senstvty analyss. Number of vehcles W 1 W 2 Z 1 Z Objectve functon 2 Objectve functon Objectve functon 1 Fg. 10. Pareto fronter of the problem by fve vehcles Objectve functon 1 Fg. 11. Pareto fronter of the problem by sx vehcles.

18 Concluson In ths paper, a b-objectve possblstc programmng model was developed to formulate the VRPTW by consderng the customers satsfacton. In the proposed mathematcal problem, the frst objectve functon was to mnmze the sum of a fxed cost related to the number of vehcles and travel dstance, and the second objectve functon consdered the customers satsfacton by mnmzng the gap tme between the arrval tme and ready tme by consderng the customers prorty. In ths paper, for solvng the proposed problem, two steps were consdered; n the frst step, the possblstc proposed model was converted nto an equvalent auxlary crsp model and n the second step, an RMCGP method was employed to attan an approved adjustment soluton. The proposed model provdes useful awareness to help the DMs n dentfyng effectve parameters and creates an optmal decson closer to realty. The proposed model provdes a sutable way to solve b-objectves decson-makng problems, whch ncludes mult-choce of aspraton levels. The customers demand was consdered as fuzzy numbers. The proposed model was valdated by LINGO software. In order to solve the model n large-szed problems, two meta-heurstc algorthms (.e., smulated annealng (SA) and genetc algorthm (GA)) s used. The results demonstrated that SA outperforms GA n both objectve functon values and computatonal tmes. Fnally, to demonstrate the valdaton of the proposed model, an ndustral case study related to the TTC was nvestgated. References Afshar-Bakeshloo, M, Mehrab, A, Safar, H, Malek, M, & Jola, F. (2016). A green vehcle routng problem wth customer satsfacton crtera. Journal of Industral Engneerng Internatonal, 12(4), Aoun, B., Martel, J. M., & Hassane, A. (2009). Fuzzy goal programmng model: an overvew of the current state of the art. Journal of Mult Crtera Decson Analyss, 16(5 6), Archett, C., Speranza, M. G., & Vgo, D. (2014). Vehcle routng problems wth profts. Vehcle Routng: Problems, Methods, and Applcatons, 18, 273. Barkaou, M., Berger, J., & Boukhtouta, A. (2015). Customer satsfacton n dynamc vehcle routng problem wth tme wndows. Appled Soft Computng, 35, Belhaza, S., Hansen, P., & Laporte, G. (2014). A hybrd varable neghborhood tabu search heurstc for the vehcle routng problem wth multple tme wndows. Computers & Operatons Research, 52, Calvete, H. I., Galé, C., Olveros, M. J., & Sánchez-Valverde, B. (2007). A goal programmng approach to vehcle routng problems wth soft tme wndows. European Journal of Operatonal Research, 177(3), Castro-Guterrez, J., Landa-Slva, D., & Pérez, J. M. (2011, October). Nature of real-world multobjectve vehcle routng wth evolutonary algorthms. In Systems, Man, and Cybernetcs (SMC), 2011 IEEE Internatonal Conference on (pp ). IEEE. Chang, C. T. (2007). Mult-choce goal programmng. Omega, 35(4), Chang, C. T. (2008). Revsed mult-choce goal programmng. Appled Mathematcal Modellng, 32(12), Charnes, A., & Cooper, W. W. (1957). Management models and ndustral applcatons of lnear programmng. Management Scence, 4(1), da Slva, A. F., Marns, F. A. S., & Montevech, J. A. B. (2013). Mult-choce mxed nteger goal programmng optmzaton for real problems n a sugar and ethanol mllng company. Appled Mathematcal Modellng, 37(9), Dantzg, G. B., & Ramser, J. H. (1959). The truck dspatchng problem. Management scence, 6(1), Delage, E., Bostel, N., Dejax, P., & Gendreau, M. (2010). Re-optmzaton of techncan tours n dynamc envronments wth stochastc servce tme. Rapport de stage du Master ORO.

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