A Bi-Objective Model and Efficient Heuristic for Hazardous Material Inventory Routing Problem

Transcription

1 Inernaional Conference on Compuaional Techniques and Arificial Inelligence (ICCTAI'2012) Penang, Malaysia A Bi-Objecive Model and Efficien Heurisic for Hazardous Maerial Invenory Rouing Problem ooraddin Dabiri and Mohammad J. Tarokh Absrac Two ypes of decisions ha direcly affec he performance of a disribuion sysem is he vehicle rouing and invenory assignmen. These problems have been sudied separaely for many years. The invenory rouing problem (IRP) is inegraion of above menioned disribuion problems. In he oher words, IRP assumes each cusomer mainains a local invenory of a produc and consumes a cerain amoun of ha produc each day. Each day a flee of vehicles is dispached over a se of roues o resupply a subse of he cusomers. Specifically, IRP deermines invenory allocaion and vehicle rous by objecive o minimize he disribuion cos. Therefore, he deerminaion of hazardous maerials disribuion roues can be defined as a bi-objecive invenory rouing problem (BiIRP) while risk minimizaion accompanies he cos minimizaion in he objecive funcion. In his paper we inroduce an efficien biobjecive mixed ineger linear programming model for IRP. For his purpose, we exend a well-known vehicle rouing problem (VRP) formulaion o consider he IRP assumpions. Also, we propose a binary chromosome represenaion o use in Muli-Objecive GA algorihm. Finally, in order o evaluae he efficiency of proposed heurisic we design some novel experimens and solve hem by boh proposed algorihm and Cplex Keywords Geneic algorihm, hazardous maerial disribuion, invenory rouing problem, vehicle rouing. S I. ITRODUCTIO UPPLY chain managemen is a se of approaches uilized o efficienly inegrae suppliers, manufacurers, warehouses and sores, so ha merchandise is produced and disribued a he righ quaniies, o he righ locaion, and a he righ ime, in order o minimize sysem-wide coss while saisfying service level requiremens. This involves a se of managemen aciviies like purchasing, invenory conrol, producion, sales and disribuion. The overall goal of supply chain managemen is o inegrae organizaional unis and coordinae flows of maerial, informaion and money so ha he compeiiveness of he supply chain is improved. For his purpose, implemening he reliable logisic sysem is vial. On he oher hand, o remain compeiive and profiable, producs need o ge o poin of sale faser and more cos effecively ooraddin Dabiri is wih K.. Toosi Universiy of Technology, Tehran, Iran. (Corresponding auhor: phone: ; fax: ; n.dabiri@yahoo.com). Mohammad Jafar Tarokh is wih K.. Toosi Universiy of Technology, Tehran, Iran. ( mjarokh@knu.ac.ir). han hose of he compeiion. Invenory and ransporaion coss are wo imporan cos componens wihin enormous logisic sysems. In order o deal wih hem, he vehicle rouing problem (VRP) and he invenory managemen problem have been sudied separaely for many years. In order o reach more cos effecive logisic sysem we should inegrae VRP and he invenory relaed decisions. An invenory rouing problem (IRP) is a good example of inegraion beween invenory and ransporaion problems which is cos effecively applicable in vendor managed invenory (VMI) environmen. In he VMI model a vendor observes and conrols he invenory levels of is cusomers, as opposed o convenional approaches where cusomers monior heir own invenory and decide he ime and amoun of producs o reorder. In hese ypes of disribuion sysems, he decision of how much invenory o mainain a he supplier and he reailers is affeced by delivery imes and amouns for he reailers, which in urn is affeced by he capaciy of he vehicles used for he deliveries. Thus, simulaneous decision making is imporan in hese sysems o obain significan cos savings [1-2]. In he oher words, The Invenory Rouing Problem (IRP) combines rouing and invenory as follows. Cusomers, based on heir expeced periodically demand, mus be assigned o one or more days, and hen a VRP problem mus be solved for each day o assign vehicles o cusomers and deermine roues for he vehicles, wih a goal of minimizing he oal delivery cos [3-5]. According o our finding, here are five publicaions which focus on lieraure survey of invenory rouing problem. Firs effor was presened as book chaper by [6]. Second publicaion was a survey paper prepared by [7]. They called his problem as Dynamic Rouing and Invenory Problems (DRIP). They define dynamiciy as a siuaion where repeaed decisions have o be made a differen imes wihin a planning horizon, and earlier decisions influence laer decisions. This is a common feaure in mos of he IRP presened in he lieraure. Sarmieno and agi [8] provide hird survey which was a review on inegraed analysis of producion disribuion sysems. Recenly anoher research has been done by [9]. They claim o focus on logisical overview of invenory rouing problems, and very lile researches have been menioned abou he applicaion oriened works wihin he field. Finally, comprehensive review has been done in he 283

2 Inernaional Conference on Compuaional Techniques and Arificial Inelligence (ICCTAI'2012) Penang, Malaysia published paper by [10]. Auhors of las paper srongly emphasize on indusrial aspec of invenory rouing problem. We refer an ineresed reader o described five surveys for references o several of ypes and applicaions of IRP. The hazardous maerials ransporaion covers a large par of he economic aciviies of he indusrialized counries. I has been esimaed ha four billion ons of hazardous maerials are being ranspored annually a a worldwide level. Apar from is significan role in he economy of he indusrialized counries, hazardous maerials ransporaion raises concerns abou he safey of he human and naural environmen [11]. Therefore he consideraion of he hazardous maerials disribuion as a bi-objecive IRP may consequences many benefis. In his paper we inroduce an efficien bi-objecive mixed ineger linear programming (MILP) model for IRP in second secion. In he hird secion we propose a binary marix as chromosome represenaion o use in Muli-Objecive GA (MOGA) algorihm as a soluion approach. In order o evaluae he efficiency of proposed heurisic we design some experimens and solve hem by boh proposed algorihm and Cplex 12.2 in secion four. Finally, we summarize he resuls of our research in he las secion. II. MODEL FORMULATIO A. Problem Saemen The IRP considered in his sudy is he problem of deermining roues of vehicles and delivery quaniies for each reailer over a given planning horizon which composed of several discree ime periods. The objecive is o minimize he ransporaion and invenory holding coss simulaneously. Transporaion coss consis of variable disance relaed cos and fixed cos which considered for each vehicle used in a period. Assumpions made in his sudy are given below: 1) There is a single warehouse supplying muliple reailers wih producs of a muliple ype. 2) There is no limi on he availabiliy of he producs in he warehouse. 3) Vehicles are homogeneous, ha is, vehicles have he same loading capaciy. 4) A vehicle makes a single rip in a period and i may visi several reailers in each rip. 5) The se of reailers served by a vehicle may change hroughou he planning horizon. Sequenially, reailers are o be pariioned ino several groups in each period. 6) Transporaion cos includes wo componens, raveling disance relaed cos and vehicle fixed cos. 7) The ravel coss beween he warehouse and each reailer and beween all pairs of reailers are known, symmeric and consan over planning horizon. 8) The demand quaniy of each reailer in each period is deerminisic and known in advance, bu may vary. 9) There is limied sorage faciliy for invenory a he reailers. 10) Shorages are no allowed. In order o formulae proposed IRP model we exend a basic VRP model ha is firsly proposed by [12]. The noaions and mahemaical modeling for bi-objecive invenory rouing problem is given as follows. Unlike previously proposed formulaions such as [13], in he following formulaion we relax he vehicle indicaor indices. Therefore our model is very useful when we use non-inernal and non-much specified vehicles in he disribuion flee. oaions for parameers and variables are illusraed in Table I and Table II, respecively x ij I i D i U i TC V B. Mahemaical Model The following is a mixed ineger programming formulaion for he problem. minimize TC = h I + Subjec o: T = 1 i= 0 i T T c x + f V τ = 1 i= 0 j= 0 = 1 T i τ = 1 i= 0 j= 0 ij ij (1) Zrisk = r x ij ij (2) I + D = I + d i (3), i, 1 i i i I C i, (4) i j= 0 j i Symbol Symbol i TABLE I OTATIOS FOR MODEL PARAMETERS Descripion number of cusomers T number of periods i, j=0, 1,..., denoe he indices of locaions in which index 0 indicaes supplier s locaion =1, 2,..., T denoes he indices of he muliple period d i demand of cusomer i a period h i holding cos per produc uni a cusomer i f fixed usage cos per vehicle a period c ij raveling disance from locaion i o locaion j q vehicle capaciy r ij denoes he ransporaion risk generaed on he pah from locaion i o locaion j sorage capaciy of cusomer i C i TABLE II OTATIOS FOR MODEL VARIABLES Descripion equals o 1 if a vehicle ravels from locaion i o j in period, and 0 if i does no on-hand invenory a locaion i a he end of period delivery amoun for cusomer i a period arbirary real numbers ha are jus used for model formulaion oal cos of disribuion number of vehicles is used in period xij 1 i, (5) 284

3 Inernaional Conference on Compuaional Techniques and Arificial Inelligence (ICCTAI'2012) Penang, Malaysia x = xij i, (6) ji j= 0 j i j= 0 j i 1 D, i xij M Di i (7) M j= 0 j i U - U + q x q D i, j i j and (8) i j ij i x = V 0 j (9) j = 1 Di Ii xij 0 i, (10) 0 i, (11) { } 0,1 i, j (12) Equaion (1) ha is firs objecive funcion will minimize he disribuion coss including invenory holding coss, variable ransporaion coss and vehicle usage fix coss. Hazardous maerial handling as second objecive funcion is represened by Equaion (2). Consrains (3) ensure invenory balance for each cusomer a he end of each period. For each period, consrains (4) bound he invenory level of he cusomers o heir sorage capaciy. Consrains (5) make sure ha a vehicle will visi a locaion a mos once in a ime period and consrains (6) ensure roue coninuiy. If he delivery for a cusomer i a period is equal o zero (DRiR=0), hen any vehicle should no visi i. This fac has been implemened by consrains (7). Consrains (8) serve for wo purposes. The firs one is o ensure ha he assigned amoun for a vehicle is less han or equal o he vehicle s capaciy, and he second is o ensure he sub-our eliminaion for each vehicle in each period. Consrains (9) will deermine number of vehicles is used in each period. Consrains (10)-(12) are he domain consrains. III. 2BSOLUTIO APPROACH Invenory rouing problem is P-hard [1]. As a consequence exac soluion mehods canno find he opimum soluion even for moderae size problems. Alernaively several heurisic approaches have been developed wihin he las wo decades aiming a idenifying near opimal soluions. Along wih he mos exensively adoped echniques are: simulaed annealing, an colonies, geneic algorihm, neural nework, paricle swarm opimizaion and ec. Here we implemen he geneic algorihm (GA) o solve described problem in he previous secion. GA is a sochasic search echnique ha explores he problem domain by mainaining a populaion of individuals, which represens a se of poenial soluions in he search space. GA aemps o combine he good feaures found in each individual using a srucured ye randomized informaion exchange in order o consruc individuals who are beer suied o heir environmen han he individuals ha hey were creaed from. Through he evoluion of beer and beer individuals, i is anicipaed ha he desired soluion will be found. In his paper we jus inroduce a proper binary chromosome represenaion and oher aspecs of MOGA mehod will be same as ohers in lieraure. More specifically proposed chromosome is based on deciding when a cusomer will be visied among planning periods. In he oher words, proposed chromosome is a binary marix ha indicaes in each period which cusomers should be visied by vehicles. Figure 1 illusraes an example of he chromosome represenaion for a problem having 5 cusomers and 5 periods. For insance cusomer 3 will be visied in he firs and las periods. According o he visiing chromosome we could simply deermine delivery marix (invenory assignmen). The amoun o be delivered depends on wheher here will be visi in he subsequen period or no. The oal delivery o cusomer i in period is he sum of all he demands in period, +1,..., k-1 where he nex visi will be made a period k. Afer delivery scheduling deerminaion we should decide abou vehicle rouing, for his purpose in his paper we use wellknown saving algorihm proposed by [14] in each period. Since he saving algorihm does no require flee size as inpu parameer, so i s compleely compaible wih proposed decision problem in previous secion. Here we ignore he descripion of he MOGA, because i s very famous so for more illusraion we refer researchers o [15]. Cusomers Cusomers Cusomers Chromosome is an (T-1) Array Demand marix (d i ) Delivery marix (D i ) Fig. 1 Chromosome represenaion and corresponding delivery marix 285

4 Inernaional Conference on Compuaional Techniques and Arificial Inelligence (ICCTAI'2012) Penang, Malaysia IV. COMPUTATIOAL AALYSIS In his secion, we aemp o evaluae proposed muliobjecive GA algorihm which discussed in previous secion. According o our design of experimens, cusomers are allocaed in a square of disance unis and heir coordinaes are generaed using a uniform disribuion wihin hese limis. The depo is locaed in he middle of he square. The Euclidean disance beween each pair of locaions was compued o obain a complee graph whose pah weighs are he ravel disance. Traveling coss (c ij ) was calculaed by rounding up corresponding ravel disance and muliplying i by ransporaion cos rae. Associaed risk for each pah (r ij ) is generaed using a uniform disribuion wihin (0, 10). In his paper we suppose ransporaion cos raes are consan and equal o 2. Holding coss are differen for each cusomer and have absolue normal disribuion wih a mean of 0.03 and a sandard deviaion of A consan cos value of 10 for fixed he vehicle usage cos in each period (f ) is assumed. For each combinaion of periods and producs, cusomer demands (d i ) are generaed using a uniform disribuion from 5 o 25. I is assumed ha vehicle flee is homogenous and each vehicle capaciy is 100 iems. Also, each cusomer has a sorage capaciy of 150 iems. We generae hree levels of (10), (20), and (30) and wo levels of T (5) and (7). For each problem seing defined by a combinaion of and T, we randomly generae wo replicaes. The naming convenion used for he es problems sars wih wo digis ha are assigned for he number of cusomers. Afer a hyphen, a digi refers o presening he number of he planning periods. Finally, he replicae number is given a he las digi afer a hyphen. Thus, he problem represens he firs replicae of a es wih 10 cusomers and planning periods of 5. We have implemened he run numerical examples on a personal compuer wih 2.53 GHz Core-2-Due CPU and 3 GB RAM. As saed earlier, Muli-objecive GA oolbox of he Malab sofware is used o solve he proposed model. For his purpose afer many examinaions we apply following seing o oolbox required argumens. We se populaion size 150, and uniform populaion creaion. Also, ournamen operaor is used as selecion funcion. Two poin wih rae 0.8 and uniform operaors wih rae 0.05 are used as crossover and muaion funcions, respecively. Well-known disance crowding funcion wih fracion rae 0.35 is used for conrolling he diversiy of Pareo soluions. Also, we se 1200 seconds as sopping crieria. In he lieraure many algorihms have been developed o find he exac soluion for Pareo se. The ε-consrain mehod was proposed by [16] in 1971, his mehod is one of he bes known approaches for solving muli-objecive problems. In his mehod one objecive is seleced as he main objecive and oher objecive are ransformed ino consrains. The drawback of he convenional ε-consrain mehod lies on efficiency of is Pareo froniers and here is no guaranee for he efficien soluions and i may generae inefficien soluions. The augmened ε-consrain (AUGMECO) mehod is novel version of he convenional ε- consrain mehod ha provides efficien Pareo froniers. We refer an ineresed reader o find he descripion of AUGMECO o see [17]. In his paper, we evaluae he efficiency of he proposed algorihm comparing is resuls wih pay-off able of efficien Pareo soluion obained by AUGMECO mehod. Table III shows deailed compuaional resuls of he experimens. A firs, he AUGMECO mehod for proposed model was implemened in GAMS sofware using Cplex 12.2 o obain efficien wo Pareo soluion. Then middle poin is calculaed by average of above poins. Also, MOGA heurisic was applied o 12 es problems and wo of he obained Pareo soluions froniers were seleced based on being bes of wo objecive funcions. The seleced soluions are used o calculae he improvemen percen according Eq. (13). OF F AUGMECO MOGA improvemen P. = 100 ( OF ) (13) AUGMECO According o he resuls, i can be said ha if we use MOGA algorihm he firs objecive funcion will improve abou 25% in average agains he use of AUGMECO mehod. Bu, second objecive funcion will be worse abou 7% in he resul of he MOGA raher han AUGMECO mehod. This disadvanage in is no considerable so much, because according o Figure (2) improvemen percenage will increase by raising he number of binary variables. Therefore we could say he resuls demonsrae he pracicabiliy of he proposed bi-objecive model as well as he proposed soluion algorihm. 286

5 Inernaional Conference on Compuaional Techniques and Arificial Inelligence (ICCTAI'2012) Penang, Malaysia TABLE III DETAILED COMPUTATIOAL RESULTS OF TEST PROBLEMS Tes O. F. AUGMECO MOGA Improvemen % Z rans Z risk Z rans Z risk Z rans Z risk Z rans Z risk Middle Z rans Z risk Middle Z rans Z risk Middle Z rans Z risk Middle Z rans Z risk Middle Z rans Z risk Middle Z rans Z risk Middle Z rans Z risk Middle Z rans Z risk Middle Z rans Z risk Middle Z rans Z risk Middle Z rans Z risk Middle A V E R A G E Fig. 2 Improvemen % in objecives vs. number of binary variables V. COCLUSIO The Invenory Rouing Problem (IRP) combines rouing and invenory as follows. Cusomers, based on heir expeced periodically demand, mus be assigned o one or more days, and hen a VRP problem mus be solved for each day o assign vehicles o cusomers and deermine roues for he vehicles, wih a goal of minimizing he oal disribuion cos. In his paper we propose an exension of IRP models in order o consider ransporaion risk as well as is cos. For his purpose, we exend a well-known VRP formulaion o saisfy IRP assumpions. Also, we implemen MOGA soluion approach o solve proposed bi-objecive mahemaical model. Compuaional resuls illusrae he pracicabiliy of he proposed bi-objecive model as well as he proposed soluion algorihm. REFERECES [1] Q.-H. Zhao, e al., "A pariion approach o he invenory/rouing problem," European Journal of Operaional Research, vol. 177, pp , [2] S. Çeinkaya, e al., "A comparison of oubound dispach policies for inegraed invenory and ransporaion decisions," European Journal of Operaional Research, vol. 171, pp , [3] B. Golden, e al., "Analysis of a large scale vehicle rouing problem wih an invenory componen," Large Scale Sysems, vol. 7, pp , [4] J. F. Bard, e al., "A decomposiion approach o he invenory rouing problem wih saellie faciliies," TRASPORTATIO SCIECE, vol. 32, pp , [5] P. Jaille, e al., "Delivery cos approximaions for invenory rouing problems in a rolling horizon framework," TRASPORTATIO SCIECE, vol. 36, pp , [6] A. Federgruen and D. Simchi-Levi, "Chaper 4 Analysis of vehicle rouing and invenory-rouing problems," in Handbooks in Operaions Research and Managemen Science. vol. Volume 8, T. L. M. C. L. M. M.O. Ball and G. L. emhauser, Eds., ed: Elsevier, 1995, pp [7] F. Baia, e al., "Dynamic rouing-and-invenory problems: a review," Transporaion Research Par A: Policy and Pracice, vol. 32A, pp , [8] A. M. Sarmieno and R. agi, "A review of inegraed analysis of producion disribuion sysems," IIE Transacions, vol. 31, pp , [9]. H. Moin and S. Salhi, "Invenory rouing problems: A logisical overview," Journal of he Operaional Research Sociey, vol. 58, pp , [10] H. Andersson, e al., "Indusrial aspecs and lieraure survey: Combined invenory managemen and rouing," Compuers & Operaions Research, vol. 37, pp , [11] K. G. Zografos and K.. Androusopoulos, "A heurisic algorihm for solving hazardous maerials disribuion problems," European Journal of Operaional Research, vol. 152, pp , [12] R. V. Kulkarni and P. R. Bhave, "Ineger programming formulaions of vehicle rouing problems," European Journal of Operaional Research, vol. 20, pp , [13] T. F. Abdelmaguid, e al., "Heurisic approaches for he invenoryrouing problem wih backlogging," Compuers & Indusrial Engineering, vol. 56, pp , [14] G. Clarke and J. Wrigh, "Scheduling of vehicles from a cenral depo o a number of delivery poins," Operaions Research, pp , [15] K. Deb, Muli-objecive opimizaion using evoluionary algorihms vol. 16: Wiley, [16] Y. Haimes, e al., "On a bicrierion formulaion of he problems of inegraed sysem idenificaion and sysem opimizaion," IEEE Transacions on Sysems, Man, and Cyberneics, vol. 1, pp , [17] P. Xidonas, e al., "Equiy porfolio consrucion and selecion using muliobjecive mahemaical programming," Journal of Global Opimizaion, vol. 47, pp ,