A multi-objective approach to the parcel express service delivery problem

Transcription

1 JOURNAL OF ADVANCED TRANSPORTATION J. Adv. Transp. 2014; 48: Publshed onlne 7 January 2013 n Wley Onlne Lbrary (wleyonlnelbrary.com) A mult-obectve approach to the parcel express servce delvery problem Aleksandar Cupć* and Dusan Teodorovć Faculty of Transport and Traffc Engneerng, Unversty of Belgrade, Vovode Stepe 305, 11000, Belgrade, Serba ABSTRACT Parcel express servce n many countres assumes door-to-door delvery of parcels and small packages n the fastest possble way. Delvery companes usually organze hub delvery networks, as flows between hubs are characterzed by the economy of scale effect. At hubs, parcels are exchanged across vans, trucks, and planes. To organze parcel delvery n a specfc regon, the parcel delvery company must make approprate decsons about the total number of parcel delvery hubs, ther locatons, and the allocaton of demand for facltes servces to facltes. These ssues are modeled n ths paper as a mult-obectve problem. The model developed s based on compromse programmng and genetc algorthms. We also demonstrate n the paper an nteractve manner n whch a defned problem can be solved. The proposed model could be mplemented n large-scale networks. The paper also shows a case study of parcel delvery servce n Serba. Copyrght 2013 John Wley & Sons, Ltd. KEY WORDS: parcel express servces; hub locatons; compromse programmng; genetc algorthms 1. INTRODUCTION There s ncreasng demand from small-busness owners and other clents for sendng packages wth a tme-senstve deadlne. Parcel express servce n many countres assumes door-to-door delvery of parcels and small packages n the fastest possble way. Every parcel to be delvered s characterzed by ts orgn, destnaton, and delvery deadlne. There are varous optons regardng the parcel delvery servce, such as courer servce, guaranteed next-busness-day delvery, and guaranteed two-busnessdays delvery. As a rule, the parcel express delvery network n the observed country or regon s not fully connected. Delvery companes usually organze a hub delvery network as the flows between hubs are characterzed by economes of scale. At hubs, parcels are exchanged across vans, trucks, and planes. In other words, the maorty of parcels are transported from one node to another wthout a drect servce (Fgure 1). The total number of parcel delvery hubs, ther locatons, and the allocaton of demand for facltes servces to facltes ndcate the way n whch transport facltes and parcel demand are coordnated. When matchng the total number of parcel delvery hubs and ther locatons wth parcel demand, an entre array of problems arses, ncludng fleet sze, vehcle routng on the network, plannng vehcle mantenance, complng a drver rotaton plan, and assgnng drvers to work dutes. When determnng the total number of parcel delvery hubs and ther locatons, the nterests of both the delvery company and the customers must be taken nto account. Because of the conflcts between these nterests, the problem of determnng the total number of parcel delvery hubs and ther locatons s formulated n ths paper as a multcrtera decson-makng problem as opposed to the usual approach under whch *Correspondence to: Dusan Teodorovć, Faculty of Transport and Traffc Engneerng, Unversty of Belgrade, Vovode Stepe 305, Belgrade, Serba. E-mal: dusan@sf.bg.ac.rs Copyrght 2013 John Wley & Sons, Ltd.

2 702 A. CUPIĆ AND D. TEODOROVIĆ H H Collecton Ck k H C kl Transfer W H Hub Node H l Cl Dstrbuton Fgure 1. Hub-and-spoke parcel delvery network. one chosen crteron s optmzed whle dsregardng alternatve ones [1 4]. The delvery company s nterests are expressed n terms of cost mnmzaton and proft earned, and the clents nterests are expressed n terms of the level of servce. Secton 2 defnes these obectve functons. In summary, to organze parcel delvery n a specfc regon, the parcel delvery company should try to fnd the answers to the followng questons: () What should be the total number of parcel delvery hubs? () Where should these facltes be located? () How should demand for the facltes servces be allocated to the facltes? There are numerous papers devoted to locaton analyss. A specal ssue of INFOR on metaheurstcs for locaton and routng problems, edted by Laporte and Semet, was publshed n [5]. A synthess and survey of locaton analyss s gven n revews by ReVelle and Eselt [6] and by Smth et al. [7]. ReVelle et al. [8] publshed a bblography for some fundamental dscrete locaton problems. Current et al. [9] performed mult-obectve analyss of faclty locaton decsons. In the maorty of papers, vehcle routng and locaton problems are formulated as mxed-nteger programmng problems, whch are dffcult to solve. In studyng the confguraton of physcal dstrbuton networks, Daganzo and Newell [10] used a contnuous approach n an attempt to avod some of the dffcultes related to the mxed-nteger programmng formulatons. They modeled customer locatons by a densty surface over the servce area. The authors determned the number of transshpment ponts and the frequency and routng of all the dstrbuton vehcles. The authors also gave the desgn gudelnes. Smlowtz and Daganzo [11] studed the desgn of the complex package dstrbuton systems. They used dealzatons of network geometres, operatng costs, demand and customer dstrbutons, and routng patterns. The consdered desgn problem was reduced to a seres of optmzaton subproblems. The basc goal of ths research was to fnd smple gudelnes to desgn and operate a package dstrbuton network. The total numbers of hubs, hub locatons, and nodes allocated to hubs nfluence the total transportaton costs and the level of servce n the parcel delvery network. The flows between all pars of nodes (orgn destnaton matrx) represent the basc nput data for the hub locaton problem. O Kelly [12,13] was the frst to study the hub locaton problem. Hub locaton problems are known to be notably dffcult ones and are generally consdered to be NP-hard. Varous aspects of ths problem were extensvely studed n the lterature. Locaton analyss surveys that nclude hub locaton problems have been wrtten by, among others, O Kelly and Mller [14], Owen and Daskn [15], ReVelle and Eselt [6], Alumur and Kara [16], and ReVelle at al. [8]. A specal ssue of the Computers & Operatons Research ournal, edted by Hamacher and Meyer and devoted to the new developments on hub locaton, was also recently publshed [17]. In ths paper, we study a sngle allocaton parcel hub-and-spoke network desgn problem. We consder the case n whch there are no capacty constrants at hubs. We also assume that the total number of hubs n the network s not prescrbed n advance. The problem that we consder has a strategc character. In ths paper, we focus on the locaton aspects of the problem. In other words, we do not smultaneously consder vehcle routng and locaton problems. The problem studed n ths paper could be defned n the followng way: for known flows and known dstances between all pars of nodes, determne the total number of hubs and hub locatons, and allocate non-hub nodes to hubs n such a way to maxmze the total proft, and to maxmze level of servce offered to the clents.

3 MULTI-OBJECTIVE APPROACH TO EXPRESS DELIVERY 703 The am of ths paper s to present a mult-obectve approach for solvng parcel delvery hub locaton problems. The consdered parcel delvery hub locaton problem s solved by a combnaton of genetc algorthm and compromse programmng and s also supported by numercal examples. To accept a generated soluton, the decson maker must frequently have other solutons for comparson. The comparson can be performed n an nteractve way wth the actve partcpaton of the decson maker. We also demonstrate n the paper an nteractve manner n whch a defned problem can be solved. The paper s organzed as follows: The problem s stated n Secton 2. Secton 3 descrbes the proposed soluton to the problem. The expermental results are gven n Secton 4. Conclusons and drectons for future research are gven n Secton STATEMENT OF THE PROBLEM We studed the parcel delvery problem for the case of a regon that could be served exclusvely by a fleet of vans and trucks. Thus, planes have not been ncluded n the delvery operatons consdered here. Wthout loss of generalty, we consder the servce opton known as guaranteed one-busnessday delvery. In ths paper, we study a hub locaton problem n the case of a non-orented network, whch s represented by a graph G =(N, A). Ths graph ncludes a set of consecutvely numbered nodes N and a set of consecutvely numbered lnks A. We denote by n the cardnalty of set N. Nodes denoted by H n Fgure 1 are hub nodes, and nodes represented by crcles are spokes. In the maorty of cases, the network of hub nodes s a completely connected graph. We assume that every node s connected to one of the hubs n the network. We denote respectvely by R (R N)andS (S N) the set of parcel orgn nodes and the set of parcel destnaton nodes. We consder the case when the total number of future hubs n the parcel delvery network s not gven n advance. In ths paper, we assume that optmal decsons related to the hub locatons must be made n the presence of trade-offs between two or more conflctng obectves. We attempt to optmze two conflctng obectves subect to specfed constrants. The obectves that we consder are maxmzaton of the total proft and the maxmzaton of level of servce offered to the clents. These obectves conflct wth each other. We propose the mult-obectve formulaton of the parcel delvery hub locaton problem Measurng the total operator s proft We denote by R the total operator s revenue. We also denote by P the fee that operator charges for parcel delvery. The total operator s revenuer s equal to R ¼ X X W ; P (1) where W s the number of unts of flow (parcels) between node and node. Vans are usually used for the collecton and dstrbuton of parcels. Parcels are transported between hubs by trucks, and, consequently, there s a reducton n the transportaton cost of a unt of flow, as the sze of the vehcle ncreases. We consder the case when the operator has two classes of vehcles: small vehcles (vans) and large vehcles (trucks). The operator s fleet usually ncludes a few types of vehcles wthn each class. For example, wthn the van class, an operator can possess a Volkswagen Transporter, Renault Traffc, Ford Transt, and/or Mercedes Sprnter. Because we consder a problem wth a strategc character, we use the average costs of all vehcle types wthn a vehcle class to represent vehcle class costs. When analyzng the parcel delvery system costs, we consder parcel collecton costs from orgns to hubs, transportaton costs from hubs to hubs, dstrbuton costs from hubs to destnatons, the costs of establshng hubs, and the costs of sortng parcels at hubs. We assume that the operator uses small vehcles for collecton and dstrbuton, except n cases of hgh parcel volume between a specfcnodeandahub. Let us denote Q 1 and Q 2 as the capacty of the small and large vehcles, respectvely (Fgure 2). When demand s less than the capacty of a small vehcle, the operator uses a small vehcle (case (a), Fgure 2). When demand s greater than the capacty of a small vehcle and smaller than the capacty of large vehcle (case (b), Fgure 2), the operator uses a large vehcle. Fnally, when demand s greater than the capacty of the large vehcle, the operator uses one, two, three, or more large vehcles, or one, two, three, or more large

4 704 A. CUPIĆ AND D. TEODOROVIĆ a Q 1 Q 2 b c Demand Small vehcle capacty Bg vehcle capacty Fgure 2. Demand values and vehcle capactes. vehcles and one small vehcle. We denote respectvely by y k and z k the total number of small and large vehcles engaged. The total number of small vehcles engaged (number of trps performed by small vehcles) between node and node k equals 8 < y k ¼ 1 f W W k k Q 1 or W k > Q 2 and W k Q 2 Q 1 Q 2 (2) : 0 otherwse wherebxc = max{n 2 Z n x} s the largest nteger not greater than x (n are ntegers, and Z s the set of ntegers). The total number of large vehcles engaged (number of trps performed by large vehcles) between node and node k equals 8 0 f W k Q 1 >< W k z k ¼ Q 2 W k >: Q 2 W k f W k > Q 1 and W k Q 2 Q 1 Q 2 W k þ 1 f W k > Q 1 and Q 1 < W k Q 2 Q 2 Q 2 (3) Let us consder the path! k! m!. Wthout loss of generalty, we assume that all costs are symmetrcal (C = C ) for all vehcle types. The total costs C km of transportng all parcels from orgn through hubs k and m to destnaton equal where C km ¼ C k þ C km þ C m (4) C k ¼ ðy k c 1 þ z k c 2 Þd k (5) C km ¼ ðy km c 1 þ z km c 2 Þd km (6) C m ¼ y m c 1 þ z m c 2 dm (7)

5 MULTI-OBJECTIVE APPROACH TO EXPRESS DELIVERY 705 c 1 s the transportaton cost per unt of dstance for the small vehcle (van), and c 2 s the transportaton cost per unt of dstance for the large vehcle (truck) (the followng numercal values were used n our numercal experments: c 1 = 0.39 /km; c 2 = 0.61 /km), y k s the number of trps made by small vehcles between node and node k, z k s the number of trps made by large vehcles between node and node k, and d k s the dstance from node to node k. Let us ntroduce nto the analyss the followng bnary varables: x k ¼ 1 f node s connected wth hub at k 0 otherwse (8) x ¼ 1 f node s a hub 0 otherwse (9) The total collecton cost C n the whole network equals C ¼ X X x k C k (10) k The total transportaton cost T between hubs equals T ¼ X X X X x k x m C km (11) m k The total dstrbuton costs D throughout the whole network equals D ¼ X X x m C m (12) m We denote by f the cost of establshng the hub n node. The total proft f 1 ð x! Þ of the parcel delvery system equals f 1 ð x! Þ¼ X X W ; P X X x k C k X k X X X x k x m C km X m k X x m C m X m x f (13) The total proft f 1 ð! x Þ of the operator s calculated by takng nto account the planned fee that the operator charges for parcel delvery; planned total collecton, transportaton, and dstrbuton costs; and the total planned costs of establshng the hubs. We take nto account these quanttes because the consdered problem has strategc character. There are other parameters and quanttes that should be ncluded n the analyss f the consdered problem had an operatonal character (e.g., costs of unplanned vehcle falures and breakdowns, costs of crew absenteesm, costs of wrong delveres). In our formulaton, we consder both fxed and varable costs. Costs of establshng the hubs represent the fxed costs. Total collecton, transportaton, and dstrbuton costs are the varable costs, as they depend on the number of unts of flow (parcels) between specfc nodes. In many optons regardng the parcel delvery servce, there are guaranteed tme wndows for collectng parcels. For example, the operator could offer guarantees to clents that all parcels collected between 8:00 AM and 3:00 PM wll be delvered to the destnaton node by 8:00 AM the next mornng (current practce among parcel operators n Serba).

6 706 A. CUPIĆ AND D. TEODOROVIĆ Fgure 3 shows tme wndows for pckng up parcels. The earlest e and the latest l possble tme ponts for acceptng parcels are ndcated n Fgure 3. The th parcel collecton happens at tme pont C. Let us consder the path! k! m!. As n the case of transportaton costs, we also assume that all travel tmes are symmetrcal (t = t ). We denote by t kk the tme necessary to sort a parcel at hub k. The total transportaton tme of the parcel from orgn through hubs k and m to destnaton equals t ¼fmax X k x k t k þ t kk þ X k X x k x m t km þ X x m t m (14) m m We denote by t* the guaranteed tme pont for makng the delvery (e.g., 8:00 AM the next mornng). The latest possble tme pont l for collectng parcels at node must satsfy the followng nequalty: l þ max t t (15) that s, l t max t (16) ( l t max X x k t k þ t kk þ X X x k x m t km þ X ) x m t m k k m m (17) The total number of nodes and/or the total number of clents n the network that can enoy the offered delvery servces represent mportant attrbutes of the level of servce. The total numbers of hubs, hub locatons, and nodes allocated to hubs nfluence the values of these quanttes. For example, the total number of nodes (nhabtants) that can enoy servce could be sgnfcantly ncreased by ntroducng new hubs and/or by relocatng exstng hubs (Fgure 4). There are two possbltes for each node n the network n the parcel delvery system that we study. The node could be outsde of the delvery system, or the node could be connected wth all other nodes n the delvery system. We have not consdered the case n whch a node could have delvery servce wth a lmted number of other nodes n the network. Node s ncluded n the parcel delvery system f the followng s satsfed: l t max t (18) We denote by L the arbtrarly chosen tme pont at the end of the busness day (e.g., 6:45 PM). Let us ntroduce the followng bnary varables: y ¼ 1 when l L 0 otherwse (19) Tme 0 e l t C Tme 0 e l t Fgure 3. Tme wndow for pckng up parcels.

7 707 MULTI-OBJECTIVE APPROACH TO EXPRESS DELIVERY Fgure 4. Increasng the total number of served nodes by ntroducng two new hubs. hubs; nodes outsde the parcel delvery system; nodes that partcpate n the parcel express delvery system; nodes that have the latest possble tme pont for acceptng parcels later than L.! The percentage of clentsf 2 ð y Þ wth the latest possble tme pont for acceptng parcels later than L equals n X n X! w y ¼1 ¼1 n X n X f 2ð y Þ ¼ 100% (20) w ¼1 ¼1! We use f 2 ð y Þ as a measure of the level of servce. The greater the percentage of clents wth the latest possble tme pont for acceptng parcels later than L, the hgher the level of servce. There are other ways to measure the level of servce offered to the clents. For example, f we consdered n the paper the operatonal character of the problem, the tmelness of delveres would be an excellent measure of customer satsfacton. In ths paper, we treat the operator and the customers as the only stakeholders to be consdered. It would be nterestng n future research to nclude n the analyss other stakeholders, f they exst! (e.g., local government, cty councls). We consder the total profit f 1 ð x Þ of the parcel delvery system as the crteron that represents the nterests of the operator. At the same tme, we treat the percentage of! clents f 2 ð y Þ that have the latest possble tme pont for acceptng parcels later than L as the crteron that represents the nterests of the customers. We beleve that the chosen crtera consttute a consstent famly of crtera. The chosen crtera are ndependent, that s, not redundant. Any generated soluton could be easly evaluated accordng to these chosen crtera Mathematcal formulaton of the problem In ths secton, we propose makng optmal or near-optmal decsons related to the parcel delvery hub locaton problem n the presence of trade-offs between two or more conflctng obectves. We attempt to optmze two conflctng obectves smultaneously subect to specfied constrants. The obectves we consder are maxmzaton of the total profit and maxmzaton of the percentage of clents wth the latest possble tme pont for acceptng parcels later than L. These obectves conflct wth each other. We propose the followng mathematcal formulaton of the mult-obectve parcels delvery hub locaton problem: Maxmze XX XXX XXXX! W; P xk C k xk xm C km (21) f 1ð x Þ ¼ X X X m Copyrght 2013 John Wley & Sons, Ltd. xm C m k X k m x f J. Adv. Transp. 2014; 48:

8 708 A. CUPIĆ AND D. TEODOROVIĆ Maxmze subect to f 2 ð y! Þ¼ X n X n ¼1 ¼1 X n X n ¼1 ¼1 w y w 100% (22) ( l t max X x k t k þ t kk þ X X x k x m t km þ X ) x m t m 8 2 N (23) k k m m X X k ¼ N (24) k X kk X k 0 8; k 2 N (25) X k 2 0; 1 8; k 2 N (26) y ¼ 1 when l L 0 otherwse (27) The obectve functon f 1 ðx! Þ represents the total proft of the parcel delvery system, whle the second obectve functon f 2 ðy! Þ represents the percentage of clents wth the latest possble tme pont for acceptng parcels later than L. Constrant (23) defnes the latest possble tme ponts for collectng parcels at nodes. Constrant (24) prescrbes that each node s assgned to one and only one hub. Constrant (25) requres that node s assgned to node k only f k s a hub. The other constrants descrbe varables of the model Multple crtera aspects of the problem. Determnng hub locatons as expressed by relatons (21) (27) s a mult-obectve optmzaton problem (multcrtera optmzaton, vector optmzaton problem). These knds of problems appear n engneerng, computer scence, management, plannng, economy, health, ecology, and other felds. A mult-obectve optmzaton problem s the problem of fndng a vector of decson varables whch satsfes constrants and optmzes a vector functon whose elements represent the obectve functons [18 23]. The general mult-obectve optmzaton problem can be formally defned as follows [20]: Fnd the vector! x ¼½x 1x 2...x nš T (28) that satsfes the m nequalty constrants and the p equalty constrants g ð x! Þ 0 ¼ 1; 2;...; m (29) and optmzes the vector functon h ð x! Þ¼0 ¼ 1; 2;...; p (30)! h! f ð x Þ¼ f 1 ð! x Þ; f 2 ð! x Þ;...; f k ð! T x Þ (31)

9 MULTI-OBJECTIVE APPROACH TO EXPRESS DELIVERY 709 It s mportant to note that by havng several obectve functons rather than one, the concept of optmum changes. In a mult-obectve optmzaton, the ntenton s to dscover good trade-offs nstead of a sngle soluton as n global optmzaton. The chosen obectve functons n our problem conflct wth one another. The total coverage (the total number of nodes served) s ncreased by ntroducng new hubs. On the other hand, addng new hubs ncreases the total costs. Because of the conflctng nature of the gven crtera, there s usually no soluton that smultaneously maxmzes all of the crtera. For ths reason, the soluton to problems (21) (27) usually comprses a Pareto-optmal (effcent, non-nferor) soluton. A pont x!! 2 Ω s Pareto-optmal [20] f, for every x 2 Ω and I = {1,2,...,k}, ether or there s at least one 2 I, such that 8 2I ð f ð!! x Þ¼f ðx ÞÞ (32) f ð!! x Þ > f ðx ÞÞ (33) where Ω s the feasble regon. In other words, the soluton s Pareto-optmal f there exsts no feasble vector! x that would decrease some crtera wth no causng a smultaneous ncrease n at least on other crtera. Fndng the soluton of a mult-obectve optmzaton problem usually nvolves the followng two stages: (1) the optmzaton of the defned obectve functons and (2) artculaton of the decson maker s preferences related to the trade-offs among defned obectve functons. Cohon and Marks [24] classfed all mult-obectve optmzaton methods accordng to the artculaton of the decson maker s preferences. In the case of a pror preference artculaton, the decson maker decdes about trade-offs before searchng for solutons. In contrast, n the case of a posteror preference artculaton, the search for solutons s performed before makng decsons. Progressve preference artculaton technques use three stages. These technques fnd a non-domnated soluton n the frst stage. In the second stage, the technque consders the decson maker s preferences and allows the decson maker to modfy her or hs preferences. These two steps are repeated untl the decson maker s satsfed or untl no addtonal mprovement s possble. The well-known aprorpreference artculaton technques are the Global Crteron Method [18,25 28], Goal Programmng [29,30], the Goal-Attanment Method [31,32], the Lexcographc Method [33,34], Mn Max Optmzaton [35 39], Multattrbute Utlty Theory [40 51], and PROMETHEE [52 55]. The well-known a posteror preference artculaton technques are Lnear Combnaton of Weghts [56] and the e-constrant Method [18,57 59]. Progressve Preference Artculaton technques nclude PROTRADE [60], the STEP Method [61], and SEMOPS [62]. 3. PROPOSED SOLUTION TO THE PROBLEM There are varous approaches and concepts for solvng the consdered mult-obectve parcel delvery hub locaton problem. The chosen approach, compromse programmng, belongs to the group of a pror preference artculaton technques. There are other concepts and other L p metrcs that could be used [18,63,64,26]. We also solve the mult-obectve parcel delvery hub locaton problem n an nteractve manner. It s also possble to approach to the consdered mult-obectve optmzaton problem by mult-obectve evolutonary algorthms [20,65 68] Solvng the mult-obectve parcel delvery hub locaton problem by compromse programmng We use compromse programmng [20] as a tool for solvng the mult-obectve parcel delvery hub locaton problem.

10 710 A. CUPIĆ AND D. TEODOROVIĆ The vector [ f o 1, f o 2,..., f o K ] s called the deal vector where f o ( =1,2,..., K) denotes the optmum of the th obectve functon. The pont that determnes the deal vector s called the deal pont. In real-lfe applcatons, t s rare, f not mpossble, to dscover the deal soluton of the consdered mult-obectve problem. Ducksten [27] proposed the followng measure of possble closeness to deal soluton : L p ¼ " XK w p f ð! x Þ f o p#1 p f ¼1 worst f o (34) where f ð x! Þ s the th obectve functon value as a result of mplementng decson x!, f o s the optmum value of the th obectve functon, f worst s the worst value obtanable for the th obectve functon, K s the total number of obectve functons, w s the th obectve functon s weght, p s the value that shows dstance type: for p = 1, all devatons from optmal solutons are n drect proporton to ther sze, whereas for 2 p 1, larger devatons carry larger weghts n L p metrc. We can generate varous compromse solutons by choosng dfferent parameter values. In ths way, we are able to present several feasble alternatves to the decson maker. The optmal values f o of the defned obectve functons should be dscovered to calculate the closeness to the deal soluton L p.in other words, t s necessary to solve the sngle-obectve parcel delvery hub locaton problem for every defned obectve functon. When solvng sngle-obectve problems, we use an approach based on genetc algorthms Solvng the mult-obectve parcel delvery hub locaton problem n the nteractve manner In real-lfe multcrtera problems, a soluton must be found that s frequently called the mplementaton soluton. For a soluton to be accepted as the best from the users vewpont, the decson maker must have other solutons for comparson. Ths comparson can be acheved n an nteractve fashon wth the actve partcpaton of the decson maker n the process of fndng a soluton to a gven problem. We also solve the mult-obectve parcel delvery hub locaton problem n an nteractve manner by the method of Nakayama and Sawarag [65] to whch certan necessary modfcatons were made for ts applcaton to nteger problems. The goal s to fnd satsfactory solutons x 2 X such that nequaltes f ðþ f x ¼ 1; 2;...; r (35) hold where f (x) s the th crteron and f ¼ ðf 1 ;;f 2 ;...;;f r Þ s the aspraton level of the decson maker. A fnal soluton s found among satsfactory solutons. Herewth follows a short algorthm of the proposed method for solvng the consdered multcrtera problem: Step 1: Assgn an deal pont f ¼ f 1 ; f 2 ;...; f r,wheref s suffcently large, for example, f ¼ maxf f ðþx x 2 Xg. These values reman fxed durng the entre process. There s no need to make too much effort n fndng an deal pont. The values of f should be suffcently large to cover almost all feasble solutons. Step 2: In the kth teraton, the decson maker s asked to gve the aspraton level f k for every crteron f =1,2,..., r. The value of f k s set such that f k < f (36) Step 3: Set w k ¼ 1 f f k ¼ 1; 2;...; r (37) and solve the problem

11 MULTI-OBJECTIVE APPROACH TO EXPRESS DELIVERY 711 mn x max w k f f ðþ x x 2 X and x nteger 1 r (38) Let x k be the soluton of Equaton (38). Step 4: On the bass of the value of f (x k ), =1,2,..., r, the decson maker classfes the crtera nto three groups: (1) the class of crtera that she or he wants to mprove, (2) the class of crtera that she or he may agree to relax, and (3) the class of crtera that she or he accepts as they are. If the decson maker does not ask for mprovement n any crtera, the process s fnshed. A satsfactory soluton x k has been found. Otherwse, the decson maker s new aspraton level s sought for each crteron. In ths step, the new aspraton level s set such that f kþ1 f kþ1 f kþ1 > f ðx k Þ f the decson maker wants to mprove the th crteron, < f ðx k Þ f the decson maker s wllng to worsen the th crteron, and ¼ f ðx k Þ f the decson maker accepts the th crteron as t s. Put k = k + 1 and go to step 3. Contrary to many nteractve methods that make consderable demands of the decson maker and are therefore dffcult to apply n practcal terms, ths method only asks the decson maker to use already obtaned crtera values to assgn her or hs aspraton levels, whch are used to form crtera weghts. In ths way, a new soluton s obtaned by mnmzng the weghted dstance of the crteron from the deal pont. Because of the possblty of assgnng unrealstc aspraton levels, no advance guarantee of reachng them can be gven. Thus, the method allows the decson maker to study the possbltes and contradctons of her or hs problem (that she or he dd not know at the begnnng of the problem-solvng process) and, on ths bass, to determne realstc aspraton levels of crtera that wll lead to the fnal satsfactory soluton. Each problem maxf f ðþx x 2 X and x ntegerg wll be solved usng a genetc algorthm process (the ratonal for usng a genetc algorthm wll be gven n the next secton). Then, an arbtrary upper bound of the largest value found of the th crtera ( =1, 2,..., r) can be taken for the coordnate of the deal pont, f g. For a fxed deal pont and gven aspraton levels n the kth teraton, the genetc algorthm can also be effcently appled to solve mn max problem (38) Genetc algorthm approach to the sngle-obectve parcel express servce delvery problem Delvery companes usually organze a hub delvery network, as flows between hubs are characterzed by economes of scale. Companes behavor s to some extent derved by learnng from dong. Increased experence n parcel delvery operatons has ndcated sgnfcant reductons n unt transportaton cost wth ncreasng sze of the vehcles that operate between hubs. In other words, the cost functons that nclude the concept of economes of scale are non-lnear. In ths way, the economes of scale provde the ratonale to use a genetc algorthm when solvng the parcel express servce delvery problem. There are also other arguments that support the applcaton of a genetc algorthm. When mplementng the descrbed nteractve process, questons reman about assgnng the deal pont and solvng the mn max problem (38). Both can be reduced to problems of non-lnear nteger programmng that are very hard to solve, especally for practcal problems wth large dmensons. Enumeratve schemes for solvng our problem are characterzed by the lack of effcency. Even the very sophstcated enumeratve scheme Dynamc Programmng breaks down on problems of moderate sze. Due to dffcultes n fndng optmal solutons to such large combnatoral problems, greater attenton has been gven lately to heurstc and metaheurstc technques that generate solutons near the optmum.

12 712 A. CUPIĆ AND D. TEODOROVIĆ One possble approach to solvng such problems s to carry out lnearzaton and then use lnear programmng to fnd the soluton, dsregardng the varables nteger aspect. Durng the lnear approxmaton of non-lnear functons, the problem s dmensons sgnfcantly ncrease. The soluton obtaned s usually fractonal and must be rounded off. Ths feature causes consderable problems. In addton, there are no guarantees that the non-nteger soluton obtaned s near to true optmum. Random-search algorthms try to overcome the shortcomngs of the enumeratve schemes. In ths paper, we obtan optmal values of the specfc obectve functons f o by genetc algorthm. Genetc algorthms [70,71] represent search technques used for solvng complex combnatoral optmzaton problems. In the case of genetc algorthms, as opposed to tradtonal search technques, the search s run n parallel from a populaton of solutons. At frst, varous solutons to the consdered maxmzaton (or mnmzaton) problem are generated. In the next step, the evaluaton of these solutons, that s, the estmaton of the obectve (cost) functon, s made. Some of the good solutons yeldng better ftness are further consdered. The remanng solutons are elmnated from consderaton. The chosen solutons undergo phases of reproducton, crossover, and mutaton. Next, a new generaton of solutons s produced, followed by another, and so on. Each new generaton s expected to be better than the prevous one. The producton of new generatons ceases when a prespecfed stoppng condton s satsfed. The fnal soluton of the consdered problem s the best soluton generated durng the search Strng representaton. Our chromosomes contan full nformaton about the solutons they represent. We use bnary strngs to represent solutons. Each bt n the strng offers nformaton about the parcel hub locaton n a specfc node. Fgure 5 shows two examples of the bnary strngs. The frst strng represents a soluton wth three hubs. The hubs are located n node 4, node 5, and node 8. The second strng represents a soluton wth two hubs, located n node 2 and node Intal populaton generaton. We generate the ntal populaton of solutons n a random manner. In the frst step, we randomly generate the number of hubs (number of 1 s) for every bnary strng. The number of hubs H n any strng must be wthn the nterval 0 < H n. In the second step, we randomly generate hub locatons (locatons of 1 s wthn the strng). The probablty that the node wll be selected to be a hub equals where U p ¼ X (39) U U ¼ O þ D O s the total number of parcels orgnatng from node and D s the total number of parcels whose destnaton s node. The quantty U represents the total number of operatons (recevng parcels and shppng parcels) n node. The hgher the total number of operatons n the node, the hgher s the probablty for a node to be selected as a hub. Once the hub locatons are known, the allocaton of non-hub nodes to hubs s performed. The allocaton of non-hub nodes to hubs could be performed n a varety of ways. We assgn every non-hub node to ts nearest hub. Once the allocaton s fnshed, t s possble to calculate the ftness functon value of every generated soluton Fgure 5. Two bnary strngs.

13 MULTI-OBJECTIVE APPROACH TO EXPRESS DELIVERY Selecton. In the frst step, we copy the best chromosomes to a new populaton (usually 10% 20% of the best chromosomes). The remanng chromosomes from the parent populaton are selected n a random manner. The probablty that chromosome wll be selected to be the parent (n the case of a maxmzaton problem) equals p ¼ f X (40) f where f s the ftness functon value of the th chromosome. In other words, we use well-known roulette wheel selecton. The hgher the ftness functon value, the hgher the probablty for a node to be selected as a parent Crossover. New offsprng are created by a crossover operator. The crossover operator selects genes from parents and creates the new offsprng. We used unform crossover n whch bts are coped n a random manner from the frst or from the second parent. The crossover probablty n ths paper equals 90%. (The maorty of our offsprng were made by crossover). Only offsprngs that contan at least one hub were consdered for further analyss Mutaton. Parts of the chromosome can be mutated. In our case, mutaton refers to the change n value from 1 to 0 or vce versa. The probablty of mutaton we used was small (0.75%). The purpose of mutatons s to prevent an rretrevable loss of genetc materal at some ponts along the strng. Once the crossover and mutaton are performed, t s possble to calculate ftness functon value of every generated soluton. 4. EXPERIMENTAL RESULTS Numerous numercal experments were performed. The Serban network that was consdered contans 16 nodes (Fgure 8). All 16 nodes have a correspondng guaranteed tme wndow for collectng parcels. The earlest tme pont for acceptng parcels was 8:00 AM for all 16 nodes. The latest possble tme ponts for acceptng parcels were calculated by usng relaton (17). The L value was 6:45 PM. All accepted parcels are delvered to the destnaton nodes by 8:00 AM the next mornng (current practce among parcel operators n Serba). The tme wndows depend on the total number of hubs, hub locatons, and allocaton of non-hub nodes. In other words, every soluton s characterzed by a correspondng set of tme wndows. Table I shows a set of tme wndows for the case of the network Table I. Nodes and correspondng tme wndows for collectng parcels n the case of the network shown n Fgure 8(b). Node number Rounded tme wndow (L = 6:45 PM) 1 [8:00 AM, 8:00 PM] 2 [8:00 AM, 5:30 PM] 3 [8:00 AM, 5:45 PM] 4 [8:00 AM, 7:30 PM] 5 [8:00 AM, 6:15 PM] 6 [8:00 AM, 6:15 PM] 7 [8:00 AM, 6:45 PM] 8 [8:00 AM, 6:00 PM] 9 [8:00 AM, 5:30 PM] 10 [8:00 AM, 5:00 PM] 11 [8:00 AM, 5:15 PM] 12 [8:00 AM, 4:30 PM] 13 [8:00 AM, 6:45 PM] 14 [8:00 AM, 2:45 PM] 15 [8:00 AM, 7:00 PM] 16 [8:00 AM, 7:00 PM]

14 714 A. CUPIĆ AND D. TEODOROVIĆ shown n Fgure 8(b). (When calculatng tme wndows by usng relaton (17), we rounded the latest possble tme pont for acceptng parcels. For example, the calculated tme pont 5:47 PM for node 3 was rounded to the 5:45 PM.) The code s wrtten n MATLAB (MathWorks, Natck, MA, USA). The nput data related to dstances and travel tmes were taken from Serban operator CtyExpress Expermental results: sngle-obectve case We frst show the expermental results related to the maxmzaton of the total proft of the parcel delvery system. The number of generatons vared between 20 and In most of the cases consdered, the results converged after 100 generatons. In most real-world decson-makng problems, the nput data are not always known precsely, or nformaton s not avalable regardng certan nput parameters that are part of a mathematcal model. We assume that there s uncertanty surroundng the cost of establshng the hub. We multply the estmated cost of establshng the hub by the parameter f, wth values that are wthn the nterval [0, 2]. For example, multplyng the estmated cost by 1.5 means that the cost of establshng the hub could be 50% hgher than estmated. In ths way, we were able to perform senstvty analyss related to the uncertan future costs. There were 80 ndvduals n every generaton. The total number of generatons was equal to 100. The results related to the sngle-obectve problem of proft maxmzaton are gven n Table II. The estmated total proft equals monetary unts (for the case when f = 1). The total number of hubs and of hub locatons depends strongly on the values of parameter f. The hgher the f values, the fewer hubs exst n the network. Fgure 6 shows total proft values through all generatons for the case when f = 0.5. Table II. Results obtaned n the case of the sngle-obectve problem of proft maxmzaton. f Total proft Hub locatons , 3, , , , Total Proft Generatons Fgure 6. Total proft values through all generatons (f = 0.5).

15 MULTI-OBJECTIVE APPROACH TO EXPRESS DELIVERY 715 The full lne represents the best value n every generaton, whle the dashed lne depcts the average ftness functon values. The changes of the ftness functon values through all generatons were notably smlar n all other experments. We also maxmzed the percentage of clents f 2 ð! lþwth the latest possble tme pont for acceptng parcels later than L. The correspondng hub locatons are gven n Table III. Fgure 7 shows changes n the percentage of clents f 2 ð! lþ wth the latest possble tme pont for acceptng parcels later than L through all generatons. As n the prevous case, the full lne represents the best value n every generaton, whle the dashed lne depcts the average ftness functon values. Fgure 8(a) shows hub locatons and allocatons of nodes to hubs n the case of total proft maxmzaton. The correspondng hub locatons and the allocaton of nodes to hubs n the case of percentage of clents maxmzaton are shown n Fgure 8(b) Expermental results: mult-obectves case Solvng the multcrtera problems (21) (27) by compromse programmng frst requres an deal pont. Therefore, we solved two sngle-crteron problems of determnng hub locatons wth the followng obectve functons: maxmzaton of the total proft and maxmzaton of the percentage of clents wth the latest possble tme pont for acceptng parcels later than L. " In the next step, we mnmzed possble closeness to the deal solutonl p ¼ XK w p f ð! x Þ f o p#1 p f ¼1 max f o by the proposed genetc algorthm. We vared the weghts wthn the nterval [0, 1] wth step 0.1. For every par of crtera weghts w 1 and w 2, we solved the defned multcrtera problem 10 tmes. In ths way, we provded numerous solutons for further consderaton. In the next step, we kept for the further analyss only Pareto-optmal (effcent, non-nferor) solutons. In ths manner, every par of crtera weghts w 1 and w 2 had a correspondng set of Pareto-optmal solutons. Pareto-optmal solutons are gven n Table IV. The Pareto fronter s shown n the Fgure 9. None of the ponts shown on the fronter s strctly domnated by any other. These solutons are Pareto-optmal because there are no other solutons that are superor n all obectves. Varyng the Table III. Results obtaned n the case of the sngle-obectve problem (maxmzaton of the percentage of clents! f 2 l wth the latest possble tme pont for acceptng parcels later than L). The percentage of clents wth the latest possble tme pont for acceptng parcels later than 6:45 PM Hub locatons , 3, 4, 8, 11, 14, 16 Percentage of users whch has been offered hgher level of servce Generatons Fgure 7. Changes of the percentage of clents f 2 ð! l Þ wth the latest possble tme pont for acceptng parcels later than L through all generatons.

16 716 A. CUPIĆ AND D. TEODOROVIĆ Fgure 8. Hub locatons and allocatons of nodes to hubs: (a) total proft maxmzaton; (b) maxmzaton of percentage of users wth a hgh level of servce. Table IV. Pareto-optmal solutons n the case of two-obectve functons. w 1 w 2 Total proft f 2 ð! l Þ The percentage of clents wth the latest possble tme pont for acceptng parcels later than Lf 2 ð y! Þ Hub locatons The number of tmes the soluton was obtaned n 10 runs , 13, , 13, , 4, 11, , 13, , 5, 11, 13, , 2, 4, 11, , 5, 11, 13, , 2, 4, 11, , 5, 11, 13, , 2, 4, 11, , 2, 6, 11, 12, 13, , 5, 11, 13, 15, , 2, 6, 11, 12, 13, , 5, 11, 13, 15, , 2, 6, 11, 12, 13, , 5, 11, 13, 15, 16 1 weght values enables the generaton of a large number of solutons that facltate the decson maker s understandng of the problem and the choce of a fnal soluton Expermental results: solvng the problem n the nteractve manner In ths example, the authors took the role of decson maker. The ntal aspraton levels were gven by Desred total proft 1075 Desred percentage of clents 73

17 MULTI-OBJECTIVE APPROACH TO EXPRESS DELIVERY 717 Percentage of users whch has been offered hgher level of servce Total proft Fgure 9. Pareto fronter n the two-obectve case (maxmzaton of the total proft and maxmzaton of the percentage of users wth the latest possble tme pont for acceptng parcels later than L=6.45). The genetc algorthm was used to solve the mn max problem (38) wth the gven deal pont and ntal desred crtera levels. The followng soluton was obtaned: Iteraton 1 Desred proft 1075 Proft earned 1048 Desred percentage of clents 73 Percentage of clents obtaned 59 As the decson maker expresses hs or her preference through acceptng or not acceptng certan crtera values, these values were shown together wth ther gven aspraton levels. In analyzng these results, t could be seen that the ntal aspraton levels were set too hgh to be acheved. Because the soluton was not satsfactory, the decson maker was asked to gve new aspraton levels. The decson maker wanted to study the problem before he or she accepted one of the solutons offered. After the second teraton, the followng soluton was obtaned: Iteraton 2 Desred proft 1050 Proft earned 1020 Desred percentage of clents 70 Percentage of clents obtaned 65 The decson maker reached an acceptable soluton through the greater number of teratons. Table V shows the decson maker s preferences and the results obtaned through nne teratons. Table V. The decson maker s preferences and the results obtaned through nne teratons. Iteraton Aspraton level for crtera 1 Aspraton level for crtera 2 Hub locatons Total proft f 1 ð x! Þ The percentage of clents wth the latest possble tme pont for acceptng parcels later than Lf 2 ð y! Þ , 4, 6, , 5, 11, 13, , 2, 6, 11, 13, , 2, 6, 11, 12, 13, , 5, 11, 13, 15, , 2, 6, 11, 13, , 5, 6, 11, 13, , 2, 4, 11, , 5, 11, 13,

18 718 A. CUPIĆ AND D. TEODOROVIĆ 5. CONCLUSION Ths study examned the parcel delvery hub locaton problem. When determnng hub locatons, the nterests of both the operator and the clent must be taken nto consderaton. Due to the conflcts between these nterests, the task of determnng hub locatons s formulated as a multcrtera decsonmakng problem as opposed to the typcal approach under whch one chosen crteron s optmzed whle dsregardng alternatve ones. We have attempted to make good decsons related to the parcel delvery hub locatons problem n the presence of trade-offs between two or more conflctng obectves. We determned the total number of hubs and hub locatons, and we allocated non-hub nodes to hubs n such a way as to maxmze total proft and maxmze the level of servce offered to the clents smultaneously. The obectves consdered n the paper were maxmzaton of the total proft and maxmzaton of the percentage of clents wth the latest possble tme pont for acceptng parcels later than L. The mathematcal formulaton of the problem consdered s descrbed n the paper. We used a combnaton of compromse programmng and genetc algorthms to smultaneously optmze two conflctng obectves that were subect to specfed constrants. We performed a computatonal study to test the performance of the proposed approach. The crtera weght values were vared, whch allowed us to generate a large number of solutons for the decson makers. We also solved the mult-obectve parcel delvery hub locaton problem n an nteractve manner. The proposed nteractve method for solvng multcrtera problems s easy to apply because the decson maker s only asked to estmate the acceptablty of the crtera values acheved and gve hs or her aspraton levels on the bass of ths acceptablty. The method allows for changes n the decson maker s preferences and a return to earler reected solutons. The proposed model has been mplemented on a relatvely small network n Serba. The model developed could be also easly mplemented on a large-scale network. The basc nput data for the problem consdered are the estmated numbers of parcels and packages between pars of ctes. It s often mpossble to estmate these numbers wth suffcent precson (.e., there s a degree of uncertanty surroundng the numbers of parcels between pars of ctes). Uncertanty s also frequent n the estmaton of the operator s future revenue and costs. A future research drecton would be to develop a model that ncorporates uncertanty n numbers of parcels and n the operator s revenue and costs. ACKNOWLEDGEMENTS Ths research was partally supported by the Mnstry of Scence of Serba. The authors are grateful to the revewers for ther valuable comments and suggestons for mprovng the paper. REFERENCES 1. Topcuoglu H, Corut F, Erms M, Ylmaz G. Solvng the uncapactated hub locaton problem usng genetc algorthms. Computers and Operatons Research 2005; 32: Tan PZ, Kara BY. A hub coverng model for cargo delvery systems. Networks 2007; 4: Zäpfel G, Wasner G. Plannng and optmzaton of hub-and-spoke transportaton networks of cooperatve thrd-party logstcs provders. Internatonal Journal of Producton Economcs 2002; 78: Wasner G, Zäpfel G. An ntegrated mult-depot hub-locaton vehcle routng model for network plannng of parcel servce. Internatonal Journal of Producton Economcs 2004; 90: Laporte G, Semet F. Specal ssue of INFOR on metaheurstcs for locaton and routng problems. Infor 1999; 37(3): ReVelle CS, Eselt HA. Locaton analyss: a synthess and survey, European Journal of Operatonal Research 2005; 165: Smth HK, Laporte G, Harper PR. Locatonal analyss: hghlghts of growth to maturty. Journal of the Operatonal Research Socety 2009; 60: S140 S ReVelle CS, Eselt HA, Daskn MS. A bblography for some fundamental problem categores n dscrete locaton scence, European Journal of Operatonal Research 2008; 184: Current J, Mn H, Schllng D. Multobectve analyss of faclty locaton decsons. European Journal of Operatonal Research 1990; 49(3): Daganzo CF, Newell GF. Confguraton of physcal dstrbuton networks. Networks 1986; 16(2):

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