AN ANT BASED SIMULATION OPTIMIZATION FOR VEHICLE ROUTING PROBLEM WITH STOCHASTIC DEMANDS. Mukul Tripathi Glenn Kuriger Hung-da Wan

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1 Proeedings of the 29 Winter Simulation Conferene M. D. Rossetti, R. R. Hill, B. Johansson, A. Dunin and R. G. Ingalls, eds. AN ANT BASED SIMULATION OPTIMIZATION FOR VEHICLE ROUTING PROBLEM WITH STOCHASTIC DEMANDS Muul Tripathi Glenn Kuriger Hung-da Wan Center for Advaned Manufaturing and Lean Systems Department of Mehanial Engineering University of Texas at San Antonio San Antonio, TX 78249, USA ABSTRACT The Vehile Routing Problem (VRP) is of onsiderable eonomi signifiane in logisti systems as it manages the distribution of goods to mae an effiient transportation system. Considering a pratial appliation, this paper solves a vehile routing problem with stohasti demand (VRPSD) in whih the ustomer demand has been modeled as a stohasti variable as opposed to onventional VRP. To deal with the additional omputational omplexity, this paper uses a simulation optimization approah to solve the VRPSD. To enhane the algorithm performane, a neighborhood-searh embedded Adaptive Ant Algorithm (ns-aaa), an improved Ant Colony Optimization approah, is proposed. The performane of the proposed methodology is benhmared against a set of test instanes generated using Design of Experiment (DOE) tehniques. The results verified the robustness of the proposed algorithm against Ant Colony Optimization and Geneti Algorithm, over whih it always demonstrated better results, thereby proving its supremay on the onerned problem. INTRODUCTION In a reent manufaturing senario, firms need to deploy effiient methodologies to effetively shedule a fleet of delivery vehiles with an aim to minimize the total osts involved. Sine design of optimal delivery and its allied servies are the major ost promoters, the ey issue for distribution ompanies involves determining best routes to a set of geographially sattered stops with respet to vehile apaity, time windows (Lau, Sim, and Teo 23) and drivers maximum woring time (Bianhi et al. 26) et., while satisfying the ustomer demands. Therefore, the problem of effiient routing and sheduling of vehiles have beome ey onerns for the deision maers of related firms. In literature, suh problems are lassified as Vehile Routing Problems (VRP) and arise as a wide range of pratial deision maing problems suh as shool bus routing, mail, newspaper delivery, fuel oil delivery and muniipal waste olletion (Tan, Lee, and Ou 2). Moreover, strutural similarity between VRP has been established with Traveling Salesman Problem (TSP), Bin Paing Problem (BPP) (Ralphs et al. 2) and Generalized Assignment Problem (GAP) (Fisher and Jaiumar 98). One of the simplest instanes of VRP an be framed as finding a feasible optimum solution for TSP whih has been already shown as NP-omplete in nature (Tan, Lee, and Ou 2). Therefore, determining the optimal solution in VRP is even more ompliated as it involves serving several ustomers using vehiles. To illustrate, Lau, Sim, and Teo (23) explored a variant of VRP with time windows with a limited number of vehiles and proposed a Tabu searh based approah mared by a holding list and a mehanism to enfore dense paing within a route. However, in many real-world appliations, one or more parameters of VRP tend to be stohasti. These parameters an be the set of ustomers to be visited, ustomer demand, travel times or a ombination of these. In general, stohasti variables of nown probability distributions are utilized to model these parameters and the objetive funtion thus formulated is expressed as the expeted ost of the planned routes. These problems are referred to as the Stohasti Vehile Routing Problem (SVRP). One ommon feature of SVRP is that they all have at least one deterministi VRP ounterpart, whih an be modified by mathematially modeling the deterministi variables to obtain the stohasti elements in the problem. Thus, with the inlusion of stohasti modeling the objetive funtion formulated for SVRP tends to be omputationally more expensive than onventional vehile routing problems. Tillman (969) solved the multiple terminal delivery problem with probabilisti demands where as a omprehensive modeling approah towards stohasti vehile routing is presented by Stewart and Golden (983). An intensive survey of models related to SVRP has been arried out by Gendreau, Laporte, and Seguin (996). Cheong et al. (26) onsidered a VRP with Sto /9/$ IEEE 2476

2 hasti Demand (VRPSD) where ustomer demands were assumed to be stohasti and all other data nown in priori. In this paper, we onsider a VRPSD in whih the ustomer demand has been modeled as a stohasti variable independently distributed with nown distributions, suh that its exat value is nown only upon the arrival of the vehiles at the ustomer loation. The model assumes that at some ustomer loation, when the minimum vehile load (load threshold) is attained, the vehile must return to the depot for preventive restoing. However, if the residual load is greater than or equal to a load threshold, the vehile proeeds to the next ustomer in usual fashion. It has been empirially shown, that in the ombinatorial optimization problems of the type defined above, optimality an be guaranteed only in relatively smaller problem instanes through deterministi approahes. Tillman (969) proposed a solution methodology for VRPSD based on Clare and Wright s savings algorithm. However, to deal with the omplexities inherent in ombinatorial optimization problems of real dimensions, several stohasti searh metaheuristis have been proposed. Tan, Lee, and Ou (2) explored different hybridizations of artifiial intelligene based tehniques inluding Simulated Annealing, Tabu searh and Geneti Algorithm for better performane in VRP with time window onstraints. Gendreau, Laporte, and Seguin (996) surveyed the development in the VRPSD with an emphasis on the insights gained and on the algorithms proposed. Cheong et al. (26) presented a multi objetive evolutionary algorithm that inorporates a speifi heuristi for loal exploitation and a route simulation method to evaluate the fitness of solution for VRPSD. Dong and Xiang (26) proposed a novel hybrid algorithm that ombines the Ant inspired meta-heuristis with loal optimization heuristis. This researh follows a strutured approah to determine optimal input parameter values for the VRPSD where optimality is measured by utilizing a simulation model to generate output performane variables. In general, the methodology ombines simulation and an optimization algorithm to solve omputationally omplex problems and is widely now as simulation optimization in the literature. In existing VRPSD models, Ant Colony Optimization (ACO) (Dorigo and Gambardella 997) has empirially shown superior results in resolution of omputational omplexity. Inspired by the suessful appliations of ACO, and with the inreasing hunt for improvement in solution quality, this researh proposes a new metaheuristi having its roots in traditional ant algorithms to develop a simulation optimization framewor for optimizing the problem at hand. The proposed omprehensive modeling enhanes the quality of the solution and dereases the omputational time involved. To improve the searh apability of the proposed algorithm, a neighborhood searh based tehnique is utilized whih attempts to enhane the quality of the solutions by using ompound moves. In order to validate the effiay and effiieny of the proposed tehnique simulations are performed over a set of benhmar instanes. For the generation of the test bed, seven fators are seleted, varied at two different levels, and their impat on the objetive funtion values are investigated using Design of Experiments (DOE) (Taguhi 987). The analysis also reveals the diret relationship between the objetive funtion value and the parameters seleted. The rest of the paper has been organized as follows: Setion 2 formulates the mathematial model of the VRPSD problem followed by setion 3 where the proposed ant algorithm based simulation optimization solution methodology is detailed. Setion 4 reports the experimental design and results showing the effiieny of the proposed algorithm. Finally, setion 5 provides the onlusive remars and states sope for future study. 2 MATHEMATICAL MODELING OF VRPSD AND RELATED CONSTRAINTS Given an undireted graph (,, ) A {( i, j) :, i j V, i j} G V AC where V = {,,..., n} is a set of nodes i.e., ustomers with node denoting the depot. = is the set of ars joining the nodes, having a travel ost (distanes) C= { ij :, ij Vi, j} between the nodes (ustomers). The ost matrix C is symmetri and satisfies the triangular inequality. A fleet of homogeneous vehiles (HNV) eah with vehile apaity ( V ) has to deliver goods to the (N ) ustomers aording to their stohasti demands, minimizing the total expeted distane traveled. We mae the following assumptions before disussing the mathematial model of VRPSD. Customer s demands are stohasti variablesθ i, i = {,... n} whih are independently distributed with nown distributions. The atual demand of eah ustomer is only nown when the vehile arrives at the ustomer loation. It is also assumed that the demand θ i does not exeed the vehile apaityv, and follows a disrete distribution and probability mass funtion πid = prob( θi = d), d =,, 2,..., D V. All the vehiles have to visit the ustomers in the order th speified by the priori tour. Let us assume that vehile has to traverse n ustomers in the order defined by priori tour. Aording to the ustomers atual demand, the vehile deides whether to proeed to the next ustomer or to go to the depot for restoing. Sometimes the hoie of restoing is the better one, even if the vehile is not empty, or if its apaity is bigger than the expeted demand of the next sheduled ustomer. This ation is termed as preventive restoing. The goal of preventive restoing is the batraing of vehiles when they do not have suffiient load to serve the preeding ustomer, thus it performs ba and forth trips to the depot for ompleting the delivery at the ustomer. Let us define some of the variables that are ruial in formulating the onstraints, these are as follows: 2477

3 X i, j, : binary flow variable, equals if path (i, j) is traversed by vehile and otherwise. Y i, j, : amount of ommodity that is traversing ar (i, j) by the means of vehile S i, : amount of ommodity supplied at the th i ustomer by vehile M: a large onstant to transform nonlinear terms to linear ones for some onstraints HNV: total number of vehiles initially available at the depot. The objetive funtion (OBF), i.e. expeted distane overed by the vehile, is omputed as follows. Let R represent the priori tour (defined earlier) traversed by the HNV vehiles and the expeted ost of the priori tour traversed by th vehile f V. Therefore, the total expeted ost for traversing the priori tour by all the HNV vehiles is denoted by variable is ( ) OBF. Now, if the j th ustomer, then, ( ) where L j represents the set of all possible loads that the f j, λ j λ L satisfies equation (). p r ( λ) ( λ), ( λ) th vehile an have after the ompletion of the servie at ( ) f = Min f f () j j j (2) f ( λ) = C + f ( λ d) π + 2 C + f ( λ+ V d) π p j j, j+ j+, j+, d j+, j+, j+, d dd : λ dd : > λ D r j( λ) = j, +, j+ + j+, ( ) π j+, d d = f C C f V d (3) The pseudo ode shown below details the neessary omputation for finding the objetive funtion with the boundary p f λ λ L f λ is the expeted ost orresponding to the hoie of ondition n ( ) = n,, n. In equation (2) and (3) above, j ( ) r proeeding diretly to the next ustomer. Whereas, f ( ) j λ is the expeted ost in ase of preventive restoing by the th vehile. As detailed by Yang, Mathur, and Ballou (2), given a priori tour, for eah j th ustomer, there is a load threshold LT j for vehile suh that, if the residual load after serving j th ustomer is greater than or equal to LT j then it is better to proeed to the next planned ustomer. Otherwise, it is better to go ba to the depot for preventive restoing. Minimize {OBF= f ( V ) To alulate the OBF, equation (4), we use following formulation: for ( =,2,3 HNV) do for ( λ = V, V,, ) do end for return (OBF) f n (λ) = n, for (j= n -, n -2, n -3,, ) do // n is the last ustomer visited by the th vehile ompute f r using f ( + ) (.) (using equation (3)) end for end for end for j for ( λ = V, V,, ) do ompute f ( V ) OBF = OBF + f ( V ) j p ompute f ( λ ) j (using equation (2)) r p ompare f ( λ) j and f ( ) ompute f ( λ) j using f ( ) ( λ) j HNV o } (4) = λ for finding the threshold LT j j+ (using equation ()) 2478

4 Where, OBF an be alulated as illustrated in the setion of ode shown above, subjet to following onstraints: V j= V j= X X HNV X, i, HNV i V =, j, j,, = = (5) {,,2... HNV} (6) {,,2... HNV} (7) Constraint equations (5) to () haraterize the vehile flows on the path. Equation (5) illustrates that the number of vehiles to servie must not exeed available vehiles ready at the depot. Equations (6) and (7) represent the onstraint that eah vehile flows from and ba only to the supply node (denoted by ). V HNV X i, j, j= V HNV X j, i, j= V V X i, j, = X j, i, j= j= = = i V and i (8) i V and i (9) i V, {,, 2... HNV} () V V S i, Y i, j, Y j, i, i V {,,2... HNV} j= j= X V Y i, j V i, j, i, j, () {,, 2... HNV} (2) Equations (8) and (9) state that eah demand node must be visited exatly one. Equation () infers that all vehiles that flow into the ustomer node must also flow out of it. Equation () provides the balaned ommodity flow requirement of the depot. Equation (2) allows the flow of the ommodities as long as there is suffiient vehile apaity. V V V Xi, j, V M Yi, j, + Yji,, i V, i, {,, 2,..., HNV} (3) j= j= j= X i, i, =, Y i, i, (4) HNV n = N (5) = Equation (3) states that whenever a vehile visits a demand node, it should arry some ommodities either from or to that node. These aforementioned onstraints establish the onnetion between the ommodity flow and vehile flow.equation (4) states the non-negativity properties of the deision variables. Finally, onstraint equation (5) illustrates that the sum of ustomers traversed by HNV vehiles must be equal to N. 3 SOLUTION METHODOLOGY The VRPSD is a ombinatorial optimization problem and omputationally omplex in nature. Determining optimality in suh problems requires all possible permutation of the sequenes in whih the vehiles have to be routed. Due to this omplexity, finding optimal solutions (via deterministi approahes) even in smaller problem instanes of real dimensions beomes almost impratial. In literature, ACO has empirially shown superior results in resolution of omplexity for several NP Hard problems (Tripathi et al. 29) and has found suessful appliations in many variants of vehile routing problems (Dong and Xiang 26). In this respet, this researh follows a strutured approah to determine optimal input parameter values for the VRPSD where optimality is measured by utilizing a simulation model to generate output performane variables. In general, the methodology ombines simulation and the proposed algorithm to solve omputationally omplex VRPSD problem. The optimization problem has been defined by the pair (O, CI), where O denotes the objetive funtion to be evaluated and CI is the onstrained input or the group of admissible deisions. If the value of O is provided in analytial form, then the problem redues to a simple optimization problem. However, in various real-world ases, the value of O an only be evaluated as an output of a simulation experiment. In ase of stohasti systems, only estimates for the objetive funtion value an be given. Therefore, multiple repliations are arried out to obtain an authenti result from a simulation experiment. However, just the alulation and repliation of a simulation experiment does not solve the optimization problem. Therefore, 2479

5 an optimization algorithm is integrated with the simulation model to provide the required near optimal results for the problem at hand. In simulation optimization we try to obtain the best input variable values (produing near optimal results) without expliitly evaluating eah possibility. A generi simulation optimization framewor for our problem is illustrated in Figure. The whole system framewor is subdivided into two parts:. The Optimization Module 2. The Simulation Module Initialization Configuration of the simulation model for VRPSD ns-aaa Optimization Algorithm New Configuration Simulation runs/ repliations The Simulation Module Simulation Results Performane Value Stopping Criteria met? No Yes Output Results The Optimization Module Figure : The Simulation Optimization Methodology The system struture learly reveals that the proedure of simulation and optimization are sequential in nature. The optimization algorithm is basially utilized to optimize the onfiguration parameters within the simulation model. It an also be seen that the simulation module is driven by the optimization algorithm whih determines the new onfigurations with eah iteration. The results of the simulation experiment provide the performane values to the optimization algorithm, whih are utilized to eep the adaptive nature of the searh strategy utilized. In this researh, the authors have utilized a new searh algorithm oined as ns-aaa. The simulation optimization embedded ns-aaa has been referred to as ns-aaa SO in the rest of the paper. The overview of the optimization strategy is disussed below. 3. Neighborhood Searh Embedded Adaptive Ant Algorithm In order to enhane the algorithm performane, we introdue a neighborhood searh embedded Adaptive Ant Algorithm as an improvement to the existing Ant Colony Optimization (Dorigo and Gambardella 997). This algorithm provides a resoureful metaheuristi approah having its roots in traditional ant algorithms. The proposed ns-aaa is onstruted so as to deal with the problems of real dimensions suh that the algorithm orients the searh progressively in the diretions favoring the global optimal solution. A fundamental aspet in the performane of this searh strategy followed by ants is to identify an effetive neighborhood for defining moves from one solution to another. The basi onept in this searh metaheuristi is to enhane the quality of the solution by using ompound moves. The basi moves onsist of a simple insertion or exhange of verties /ars on the graph of the problem and ompound moves are usually obtained by ombination of these moves. Osman (993) uses a ombination of insertion moves and exhange moves, vertex shifts from one route to another and exhanges verties between routes based on the 2-opt proess. Based upon this, the distinguishing features of the ns-aaa are detailed below:. A omplete iteration is defined by two tours: the first tour (soial tour) orresponds to the probabilisti movement of ants where as the seond tour (neighborhood tour) is assoiated with the neighborhood generation. 2. Two pheromone update rules are utilized. A loal pheromone update rule just after the soial tour ompletion and a global pheromone update rule at the ompletion of both the tours. 248

6 3. Eah ant generates a omplete tour by hoosing the transition to the next ity aording to a probabilisti state transition rule, alled the adaptive-pseudo random proportional rule. Throughout the run of the algorithm this rule ats as an adaptive and dynami balane between the probabilisti exploration and the exploitation. The progress of the modified algorithm proposed for the optimization problem is as follows: 3.. The Soial Tour The algorithm is initiated by plaing the ants randomly over the feasible ities. Here the number of ants is set equal to the number of ities. The number of ities is made equal to N + HNV. To fore the ant to mae a legal tour, transition to already visited ities are disallowed by forbidding the ant to move to those ities within the same tour. This onstraint is enfored by assoiating a data struture alled tabu list with eah ant. Soial tour starts when transitions are made to the unvisited feasible J r using a state transition rule by allowing the ants to move freely, where y represents the urrent iteration and r ( ) ities y ( ) denotes the urrent ity. The set ( ) ( y ) J r is speifi for eah ant, and the ities that are in the tabu list are exluded from it. In this effet, ants are survived till their tour ompletion by avoiding the possibility of entrapment into any infeasible path. Now for the onstrution of the tour, the adaptive-pseudo-random proportional rule is utilized. This state transition rule provides an adaptive balane between the exploration and exploitation riteria during the run time and is depited below, β arg max ( ) ( ) ( ){ y r,u r,u } if q q s τ η = u Jy r (6) S otherwise β ( r,v) η( r,v) τ ( r,u) η( r,u) τ y if v J y ( r β ) py ( r,v) = y (7) u Jy ( r) otherwise By equation (6), y is the iteration number, r is the ity on whih the ant is urrently positioned and it hooses to move to a feasible ity s. Here, q is the random real number uniformly distributed over the range [, ]. By equation (6) the parameter q ontrols the exploration and exploitation riteria with the ondition that q q for exploitation and q > q for exploration. τ ru, is the long term information obtained from the pheromone level on the path between the ities r and u for The term y ( ) the y th iteration and the term [, ] η ru β is the short term or myopi information obtained from the visibility matrix onstruted using the information based on the problem. Further, the parameter β determines relative importane between the two fators viz, visibility and pheromone information. The adaptive-pseudo random proportional rule fores the ant to move only on the best available path to meet the exploitation riteria. If it fails i.e., q > q the ant swithes over to the exploration riterion given by ondition S in equation (6) in whih any ant stationed at ity r, probabilistially selets ity v for transition, with py ( r,v) transition probability. The ondition v Jy ( r) guarantees that only the ities within the given set Jy ( r ) get seleted. Besides, all the infeasible ities have been assigned zero probability of seletion. In equation (6) the ondition q q forms a balane between exploitation and biased exploration. To avoid entrapment in the loal optima, greater sope for exploration is preferable in the first few stages of the algorithm progress. Towards the final iterations of the algorithm progress, large exploitation is preferable. Due to the fat that the proposed metaheuristi is an intrinsially dynami and adaptive searh tehnique, an adaptive parameter handling strategy is suggested (Eiben, Hinterding, and Mihalewiz 999). The parameter q is adaptively handled by the algorithm during the ourse of its progress aording to following equation log ( y) ( ) log ( y) ( ). if <. log y _ max q =.9 if >.9 (8) log y _ max log ( y ) otherwise log ( y _ max) where y is the urrent iteration of the algorithm and (y_max) is the maximum number of iterations. Equation (8) states that 248

7 the parameter q is allowed neither to fall below a minimum limit nor to surpass a maximum limit in order to provide limited opportunity for exploitation in the initial stages as well as some exploration in the final stages. Eah suh tour orresponds to the soial tour by the ants The Neighborhood Tour After the soial tour ompletion, a set of ants ommuniate important information regarding neighborhood generation to the existing ants in the olony using three neighborhood generation strategies. For this purpose we selet 3w number of ants randomly from the urrent tours and group them in three different sets: Multiple Exhange Set, Multiple Insertion Set and Multiple Reversal Set. w ants are plaed in eah of these sets. Eah ant within a set is allowed to find an enhaned solution to the problem by utilizing a sequential ordering of the neighborhoods by using Multiple Exhange Sheme, Multiple Insertion Sheme and Multiple Reversal Sheme (Rego 2) as shown in Figure g g g 2 g g p g + q (a) (b) () Figure 2: Three neighborhood generation tehniques. (a) Multiple Exhanging Sheme, (b) Multiple Insertion Sheme, and () Multiple Reversal Sheme In Multiple Exhange Sheme, several triplets of the node are taen from the ant s solution string formulated as yli TSP and then multiple exhanging of the nodes (inluding ustomer and dummy nodes) are done by following the proedure: Let there be a triplets, seleted from the solution string. The th triplet is defined by + ( g, g, g ) that inlude two ars ( g, g ) and ( g, g + ). Set T is defined by: a, + {(, )} = T g g g Here T represents the set ontaining a triplets. The proess of multiple exhange is done by transforming T into T as shown, a ' + + a = T {( g, g, g )} {( g, g, g )} The transformation of T, i.e., T T', shows a replaement of the entral node of every triplet. Figure 2(a) shematially illustrates the multiple exhanging shemes detailed above. In Multiple Insertion sheme, set ( gp, gq) is hosen exluding the nodes that are present in set T. Then this set is replaed by another set in T. By performing this move triplet T is transformed into T, where T is defined as: a T {( g, g, g )} { g, g, g )} = + + p a q This type of move has been illustrated in Figure 2(b). In Multiple Reversal Sheme, a type of neighborhood generation sheme, a part of the solution string is taen out and it is again added to the solution string by reversing the node order. The illustration of this sheme is given in Figure 2(). These new paths generated orrespond to the neighborhood tours. For eah 3w ants seleted for these neighborhood 2482

8 tours, the fitness value of eah ant s new tour (the tour length for the orresponding TSP model of the problem or the objetive funtion value of the solution generated) is ompared with the original fitness value of the ant s tour. If the new fitness value is found better, the orresponding ant s tour is replaed by the new tour generated by it. This information pertaining to neighborhood tours is stored on the paths in usual fashion using the global pheromone update rules The Pheromone Update Rules As disussed earlier, we employ two pheromone update rules while the ants onstrut the solution paths. These are the loal pheromone update rule, applied while the ants onstrut the solution during the tour, and the global pheromone update rule, applied at the tour ompletion by ants. The loal pheromone update rule is applied immediately after the ants omplete the soial and just before the neighborhood tour starts. Mathematially we represent the loal pheromone update rule as, τ y+ ( r,v) = ( ρ ) τ y( r,v) + ρ τ ( r,v) τ + rv is the pheromone level after ( y + ) th iteration. The parameter ρ is a real number suh that ρ (,). where ( ) y, τ N + HNV ( r,v) τ ( r,v) = = τ (,) rv = τ τ is defined as a onstant quantity having value equal to the initial pheromone level of the transition paths between the ities. However, it is notieable that the loal pheromone update rule is employed to reward only the paths generated by soial tours. The global pheromone update rule is applied after all ants have ompleted onstruting a solution. The purpose of the global pheromone update rule is to enourage ants to searh for paths in the viinity of the best tour found so far and also to reward the paths generated by neighborhood tours. The global best ant is defined as the ant whih orresponds to the best tour generated till the urrent iteration. It is worth mentioning here that global pheromone update is applied only after the neighborhood tours have been onstruted by the ants designated for the purpose. These neighborhood tours are always taen into aount when onsidering the riteria for seleting the global best ant. The pheromone dropped by the global best ant is defined as, gb τ y+ ( rv, ) = ( α) τ y( rv, ) + α. τ ( rv, ) where τ y + ( rv, ) pheromone level after ( y + ) th iteration. α is a parameter governing pheromone deay suh that < α < and gb Q τ (,) rv = Lgb where L gb determines the quality of the tour onstruted by the globally best ant and Q is the model dependent parameter. By introduing the neighborhood tours in ACO, its searh apability is drastially enhaned; the results onfirm this statement. The minimum level of pheromone trial level is expliitly set over eah path ( τ min ) to mae the proposed algorithm more robust against entrapment into the loal optima and inlude faster onvergene towards the global optimum. Minimum τ is maintained at eah edge by, pheromone level ( ) min τmin if τ y+ < τmin τ y+ ( r,v) = τ y+ ( r,v) otherwise This ensures a finite possibility of exploration of eah feasible path and global optimum is highly liely to be attained in extended runs Visibility of the ants Ants onstrut a solution in an iterative manner by using their short visibility (myopi information or greedy heuristi) with visibility matrix of order n n. Here visibility η [,] rv between any two nodes r and v is the inverse of distane. If its distane is less then its visibility is more, ( r,v) 2483

9 η [,] rv = ( d ) where d rv represents the distane between the nodes. Now the algorithm ends here after ompleting one tour. Immediately the ants expire and fresh ants are randomly positioned at feasible nodes for next iteration. This ourse of ation is ontinued until ( y _ max) yles are ompleted or the algorithm meets some other user speified riteria. 4 EXPERIMENTAL DESIGN AND RESULTS This setion details the experimental design utilized and results obtained by the appliation of ns-aaa SO methodology on the VRPSD. In order to study the effet of the variation of the parameters on the objetive funtion value, we onduted a series of experiments over a arefully generated test bed. In this researh, the objetive funtion involves a number of design parameters out of whih seven were identified to have greater impat on the objetive funtion value. The effet of variation of these fators on the objetive funtion value is studied with the help of Design of Experiments (DOE) (Taguhi 987) by varying them at two different levels. Various permutations of these fators result in 2 7 different experiments. In order to avoid suh an oversized test bed, we utilized Taguhi s L8 orthogonal array (Phade 989), whih ensures a balaned ombination of the levels of the fators and their interations. The resulting test bed onsists of a frational set of 8 experiments but optimally represents the entire test domain of experiments. The first eight olumns of Table represent the Taguhi s L8 orthogonal array where denotes low level and 2 high level. These levels are detailed as below. Table: Experimental design and Results of ns-aaa SO Prob. Objetive funtion values No. N HNV D i S j V r Th j Run Run2 Run3 Run4 Run5 Average D min (%) D max (%) The fators onsidered in this study and the two levels at whih they are varied are disussed below:. Number of Customers (N ): This fator determines number of nodes in the orresponding TSP model stated above and is diretly related to the instane size and omplexity. It was varied at a low level and a high level by hoosing the value of N as 5 and 3 respetively. 2. Number of Vehiles (HNV): The total number of vehiles onsidered in this study were ept at (low level) and 2 (high level). 3. Average demand (D j ): In the VRPSD the ustomer demand is a stohasti variable determined aording to two parameters; D j and Spread (S j ). We vary D i at a higher level of 6 and lower level of Spread (S j ): Apart from the mean values, the stohasti ustomer demands also depend upon the variane S j. Varying demand from minimum to maximum with vehile apaity influenes the problem instanes. Variation of S was at rv 5 for lower level and 5 for higher level. V : Vehile apaity orresponds to the net maximum load that a vehile an bear. Lower level and higher level of apaity vary at and 2 respetively. r, θ : Determines the ustomer loation with polar oordinates r and θ with the distane varying in 5. Capaity of vehile ( ) 6. Distane ( ) terms of r, lower level being (4, 6) and higher level being (8, ). 7. Threshold( Th j ): Given the a priori tour for eah ustomer j there is a load threshold ( Th j ) suh that if the residual load after serving j is greater than or equal to h j, the better option is move forward ation.variation of threshold being taen at lower level as and higher level as 2. j 2484

10 4. Parameter Settings The simulation optimization framewor is oded in MATLAB 7. and run on a worstation with 2 GHz Pentium D proessor and GB of RAM. We initialized the algorithm with τ and η matries as desribed in the previous setion and the minimum pheromone ount τ min onneting the nodes on eah path is set equal to.5. Numerous preliminary experimental runs were arried out before reporting the tuned values of the parameters for the best results obtained. The value of ρ governing the pheromone evaporation rate is set to.55 and the parameter value of α is set equal to.45 and β = 3. As stated earlier, the value of q hanges adaptively during the ourse of the algorithm and varies between. to.9 as shown in equation 8. The model dependent parameter Q is set equal to 5 for N = and, for N = 2. In this researh, maximum number of iterations was hosen as the termination riteria. In our ase we stopped the algorithm after 2 iterations. There were 25 repliations being done for eah iteration within the simulation optimization framewor. For evaluation of performane, eeping the same simulation optimization framewor, we have ompared the results obtained from ns-aaa with the results from traditional Ant Colony Optimization (Dorigo and Gambardella 997) and Geneti Algorithm (GA) (Thangiah 995). Initially, we hose similar parameter setting for ACO as were for ns-aaa. But for tuned performane, we obtained the value of β = 2 andα =.5. Moreover, initial pheromone level on eah path was set equal toτ =.5 and q was taen to be.4. In ase of GA, we utilized a population size of 5. A two point rossover operator with probability.65 and bitwise mutation operator with probability. gave the best results. The terminating riteria for eah of these two algorithms was also set as the same as that for ns-aaa. 4.2 Performane Comparison 7 Using Taguhi s Orthogonal Arrays ( L (2 ) 8 ) as shown in Table, eight problem instanes are designed by varying 7 fators at two levels to obtain maximum insights into the woring of the model by performing limited number of experiments. The Table also lists the average results (for five runs) obtained after appliation of ns-aaa SO over VRPSD with different initial random seeds. From the same Table, we observe the signifiantly lower perentage values of the minimum and maximum deviation from the average objetive funtion value in eah experimental run. Among all the eight experiments designed using orthogonal arrays, the maximum values of D min and D max were found to be.9935% and.3876% respetively, whih learly shows the onsisteny in results obtained and thus the effiay of the proposed ns-aaa SO framewor. The harateristi and the searh apability of ns-aaa SO have been ompared with ACO and GA as shown in Table 2 for problem instane of the test bed. The results learly reveal the supremay of the proposed metaheuristi over the other two approahes ompared, where ns-aaa SO was seen to signifiantly outperform them. The suess of ns-aaa SO lies in the fat that the proposed metaheuristi adapts itself to maintain an adequate balane between exploitation and exploration throughout the run of the algorithm. Further, to prevent the searh from entrapment into loal optima, the minimum quantity of pheromone on any edge, τ min, is always maintained. This ensured the finite possibility of exploration of eah feasible path and thereby inreases the probability of attaining the global optimum. Table 2: Comparison of average results obtained from GA, ACO and ns-aaa SO Problem instanes GA ACO ns-aaa SO Further to present detailed analysis and to estimate the onvergene trends of the algorithm the best fitness values obtained from ns-aaa, ACO and GA were plotted against total number of fitness evaluations in Figure 3. The distinguishing feature of ns-aaa SO is that it adaptively allows for higher initial exploration and promotes exploitation at later stages. This 2485

11 is evident in Figure 3 where ns-aaa SO goes down steeply in the starting phase of the algorithm (maximum exploration) and thereafter searhes in the viinity of the best results found so far (by inreasing the exploitation riteria gradually) as shown by the relatively smooth trends in the algorithm behavior towards its end. The relative importane of visibility over the trail weight is a ritial fator governing the quality of results obtained by the ant algorithms. The effet of variations in the β on the objetive funtion value has been plotted as shown in Figure 4. To start with, equal preferene was given to visibility and pheromone information by onsidering β =. We observed that giving more preferene to pheromone information produed better results. However, when β was inreased to high values, the objetive funtion value started dereasing. In this researh, an optimum balane was found to exist around β = 3..7 x 4.7 x 4 Objetion funtion value GA ACO ns-aaa SO Objetive funtion value beta=2 beta=4 beta= No. of Iterations Figure 3: Convergene trends of ns-aaa SO, ACO and GA No. of iterations Figure 4: Effet of β (beta) values on ns-aaa SO performane 5 CONCLUSION In this paper, a vehile routing problem with stohasti demands (VRPSD) is undertaen for study, and a new algorithm entitled Neighborhood Searh embedded Adaptive Ant Algorithm (ns-aaa) is utilized with simulation optimization. This researh follows a strutured approah to determine optimal input parameter values for the VRPSD where optimality is measured by utilizing a simulation model to generate output performane variables. In order to study the effet of the variation of the parameters on the objetive funtion value, we ondut a series of experiments over a arefully generated test bed. The harateristi and the searh apability of ns-aaa SO have been ompared with ACO and GA. Furthermore, effets of variation of various parameters of ns-aaa SO have also been disussed. The results show that ns-aaa signifiantly outperforms the two algorithms ompared, therefore, stating its effiay. As a sope for future wor, this researh will be expanded to investigate the VRPSD with stohasti ustomer loations and other potential extensions using the ns-aaa SO methodology. ACKNOWLEDGEMENTS The wor presented in this paper has been supported, in part, by the Air Fore Researh Laboratory, ontrat no. FA C REFERENCES Bianhi, L., M. Birattari, M. Chiarandini, M. Manfrin, M. Mastrolilli, L. Paquete, O. Rossi-Doria, and T. Shiavinotto. 26. Hybrid metaheuristis for the vehile routing problem with stohasti demands. Journal of Mathematial Modelling and Algorithms 5():9-. Cheong, C. Y., K. C. Tan, D. K. Liu, and J. X. Xu. 26. A multiobjetive evolutionary algorithm for solving vehile routing problem with stohasti demand. Proeedings of the 26 IEEE World Congress on Evolutionary Computation, July 6-2, 26, Vanouver, BC, Canada,

12 Dong, L. W., and C. T. Xiang. 26. Ant olony optimization for VRP and mail delivery problems. IEEE International Conferene on Industrial Informatis, Dorigo, M., and L. M. Gambardella Ant olony system: A ooperative learning approah to the traveling salesman problem. IEEE Transation on Evolutionary Computation (): Eiben, A. E., R. Hinterding, and Z. Mihalewiz Parameter ontrol in evolutionary algorithms, IEEE Transations on evolutionary Computation 3(2):24-4. Fisher, M. L., and R. Jaiumar. 98. A generalized assignment heuristi for vehile routing. Networs :9-24. Gambardella, L. M., E. D. Taillard, and M. Dorigo Ant olonies for the QAP. Journal of the Operational Researh Soiety 5(2): Gendreau, M., G. Laporte, and R. Seguin Stohasti vehile routing. European Journal of Operational Researh 88:3-2. Lau, H. C., M. Sim, and K. M. Teo. 23. Vehile Routing Problem with time windows and a limited number of vehiles. European Journal of Operational Researh 48: Osman, I. H Metastrategy simulated annealing and tabu searh algorithms for the vehile routing problem. Annals of Operations Researh 4: Phade, M. S., 989. Quality Engineering Using Robust Design. Prentie Hall, Englewood Cliffs, New Jersey. Ralphs, T. K., L. Kopman, W. R. Pulleyblan, and L. E. Trotter. 2. On the Capaitated Vehile Routing Problem. Mathematial Programming Ser. B 94: Rego, C. 2. Node Ejetion Chains for the Vehile Routing Problems: Sequential and Parallel Algorithms. Parallel Computing 27(3):2-222 Stewart, W., and B. Golden Stohasti Vehile Routing: A Comprehensive Approah. European Journal of Operations Researh 4: Taguhi, G., 987. System of experimental design, Vols. and 2. White Plains, New Yor: UNIPUB/Kraus International Publiations. Tan, K. C., L. H. Lee, and K. Ou. 2. Artifiial intelligene heuristis in solving vehile routing problems with time window onstraints. Engineering Appliations of Artifiial Intelligene 4: Thangiah, S. R Vehile routing problems with time windows using geneti algorithms. In Pratial Handboo of Geneti Algorithms: New Frontiers, ed. L. Chambers, Vol. II: Boa Raton, FL: CRC press. Tripathi, M., S. Agrawal, M. K. Pandey, R. Shanar, and M. K. Tiwari. 29. Real world disassembly modeling and sequening problem: optimization by algorithm of self guided ants (ASGA). Robotis and Computer Integrated Manufaturing 25(3): Tillman, F The multiple terminal delivery problem with probabilisti demands. Transportation Siene 3: Yang, W., K. Mathur, and R. H. Ballou. 2. Stohasti vehile routing problem with restoing. Transportation Siene 34():99-2. AUTHOR BIOGRAPHIES MUKUL TRIPATHI is a Graduate Researh Assistant at Flexible Manufaturing and Lean Systems Laboratory at University of Texas at San Antonio (UTSA). He reeived his Bahelors in Tehnology Degree from National Institute of Foundry and Forge Tehnology, India in 28. His researh interests inlude engineering management, worfore foreasting, artifiial intelligene tehniques and their variants, appliation of self-healing mehanism on manufaturing systems, modeling real world disassembly sequening problems, multi-agent system design and appliation for multi-station assembly proesses, produt platform planning and optimization, and reliability optimization. His is <muul.tripathi@utsa.edu>. GLENN KURIGER is a Researh Assistant Professor at the Center for Advaned Manufaturing and Lean Systems (CAMLS) and the Department of Mehanial Engineering at the University of Texas at San Antonio (UTSA). He reeived his Ph.D. degree in Industrial Engineering from the University of Olahoma. His urrent researh interests inlude: simulation, operations researh, simulation optimization, multi-riteria optimization, and lean manufaturing and lean onepts. His address is <glenn.uriger@utsa.edu>. HUNG-DA WAN is an assistant professor in Mehanial Engineering and the o-diretor of Flexible Manufaturing and Lean Systems Laboratory at the University of Texas at San Antonio (UTSA). He reeived his PhD in Industrial and Systems Engineering from Virginia Teh. His researh interests inlude lean manufaturing, web-based manufaturing systems, and omputer integrated manufaturing. He is among the ore faulty of the Center for Advaned Manufaturing and Lean Systems (CAMLS) at UTSA. His is <hungda.wan@utsa.edu>. 2487