Solving the Vehicle Routing Problem with Stochastic Demands via Hybrid Genetic Algorithm-Tabu Search

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1 Journal o Mathematics and Statistics: 4(3): , 2008 ISSN: Science Publications Solving the Vehicle Routing Problem with Stochastic Demands via Hybrid Genetic Algorithm-Tabu Search Z. Ismail and Irhamah Department o Mathematics, Faculty o Science University o Technology Malaysia, Sudai, Johor, Malaysia Abstract: This study considers a version o the stochastic vehicle routing problem where customer demands are random variables with nown probability distribution. A new scheme based on a hybrid GA and Tabu Search heuristic is proposed or this problem under a priori approach with preventive restocing. The relative perormance o the proposed HGATS is compared to each GA and TS alone, on a set o randomly generated problems ollowing some discrete probability distributions. The problem data are inspired by real case o VRPSD in waste collection. Results rom the experiment show the advantages o the proposed algorithm that are its robustness and better solution qualities resulted. Key words: Vehicle routing problem with stochastic demands, genetic algorithm, tabu search, hybrid algorithm INTRODUCTION The classical Vehicle Routing Problem (VRP) lies at the center o distribution management and has been extensively studied by operations researchers. Formally, the problem can be described as ollows. V = {0, 1,, n} is a set o nodes with node 0 denotes the depot and nodes 1, 2,, n correspond to the customers and A = {(i, ): i, V, i } is the set o arcs oining the nodes with each arc is associated a distance or a cost c i. With each customer is associated a nonnegative demands ξ i to be collected by a vehicle. It is assumed that the cost matrix C is symmetric and satisies the triangular inequality. The CVRP consists o designing a route that minimize the total cost, starts and ends at the depot, such that each node is visited exactly once and the total demand does not exceed Q. There exist several practical contexts where one or some components o the problem (e.g., demands, customer presence, time, etc.,) are not nown or certainty. These are nown as Stochastic VRP. In this study, we consider a particular SVRP that is VRP with Stochastic Demands, denoted by the abbreviation VRPSD, where the demand is stochastic. Dror et al. [8] mentioned that there are two solution ramewors or Stochastic VRP (including VRPSD), namely stochastic programming and Marov decision process. Further in this study, we ocus the discussion on the stochastic programming. The problems in stochastic programming are usually modeled in two ways: so-called chance Constrained Stochastic Programming (CCP) and Stochastic Programming with Recourse (SPR). In general, the SPR models are more diicult to solve than CCP models but their obective unctions are meaningul [10]. Stewart and Golden [16] provide a comparison o the two mentioned ramewors or solving a multiple VRPSD. They concluded that i the route ailure penalty cost is nown, SPR models produce lower costs than CCP. The theoretical wor in [1] extends the Stewart and Golden ormulations. They revise the SPR model with a ixed penalty or every unit o unsatisied demand to include the expected cost o transportation instead o the transportation cost or a complete route. The recourse action could be in a priori approach [2] or re-optimization. For the rest o study, we conine the discussion on stochastic programming with recourse under a priori approach. The irst two recourse versions were provided by [16]. The irst version is to charge a ixed penalty λ every time a route ails. The second version is to charge a penalty or every unit o demand in excess o the vehicle capacity Q on route. Both versions assumed that all customers must be visited on the route and thereore the obective unction includes the deterministic total cost o the route plus the penalty term. Dror and Trudeau [7] replaced the ixed linear penalty cost in Stewart and Golden by a non-linear Corresponding Author: Zuhaimy Ismail, Department o Mathematics, Faculty o Science, University o Technology Malaysia, UTM Sudai, Johor, Malaysia Tel: Fax:

2 recourse that considers ailure location. Their recourse assumed that ater a ailure all the remaining customers are served through individual deliveries. Assuming this non-optimistic recourse, the deterministic cost in the obective unction is replaced by the expected cost. Dror et al. [8] introduce theoretically a new recourse deined by a return to the depot whenever the vehicle is unable to satisy the demand (the capacity becomes exceeded) and resume the service at the customer on the planned route where route ailure had occurred. Savelsbergh and Goetschalcx [15] simpliied the recourse to achieve computational eiciency assuming at most one ailure in a route. All subsequent publications (e.g., by [10,13] ) have maintained the simple recourse policy deined in [8] with the exception o [3,4,19] that using preventive restocing and the recourse o [5] that is to terminate the route and impose a penalty such as lost revenue and/or emergency deliveries. Recent studies in development o metaheuristics or VRPSD were done by [4]. They considered basic implementation o ive metaheuristics or single vehicle: Iterated Local Search, Tabu Search (TS), Simulated Annealing, Ant Colony Optimization and Evolutionary Algorithm (Genetic Algorithm (GA)) that ound better solution quality in respect to cyclic heuristic. Instead o the wor o [4,10,20], the wor on the application o GA and TS or VRPSD are lacing in the literature, although it is widely nown that GA has been proven eective and successul in a wide variety o combinatorial optimization problems, including certain types o VRP; and as nown also that TS, the approach that dominates the list o successul algorithms, is a robust, eicient and eective approach to the general VRP amily o problem [12,14] and oten outperorms other heuristic techniques in terms o computational speed and solution quality [14]. Although pure GA perorms well, mostly it does not equal TS in terms o solution quality and sometimes pure GA perorm ineicient on practical combinatorial optimization. To improve pure GA perormance, some algorithms are combined with the simple GA, yielding a hybrid algorithm. The statement about GA hybridization is noted by [6] that hybrid algorithms, which combine a GA with more traditional algorithms, have been hinted as a highly powerul combination or J. Math. & Stat., 4(3): , 2008 or academic optimization problems are obtained by hybrid algorithms [17,19]. Based on previous researches on algorithm developed or VRPSD and the nowledge o the basic structure o GA and TS, in this study, we propose a hybrid GA with TS as an alternative o existing algorithms. Our reviews also ound that hybrid GA-TS has not been used to solve VRPSD. It is hope that this hybrid could combine the advantage o GA as population-based method and the strength o TS as traectory method. As nown, population-based methods are better in identiying promising areas in the search space, whereas traectory methods are better in exploring promising areas in search space. MATERIALS AND METHODS Deinitions o some o the requently used notations or the VRPSD are given as ollows: Customers and depot: V = {0, 1,..., n} is a set o nodes with node 0 denotes the depot and nodes 1, 2,, n correspond to the customers to be visited. We assume that all nodes, including the depot, are ully interconnected. Demands: Customers have stochastic demands ξ i, i = 1,..., n which ollows discrete uniorm probability distributions p i = Prob (ξ i = ), = 0, 1, 2,, K. Assume urther that customers demands are independent. Actual demand o each customer is only nown when the vehicle arrives at the customer location. Vehicles and capacity constraint: The vehicle has a capacity limit Q. I the total demand o customer exceeds the vehicle capacity, route ailure is said to be occurred. Route: A route must start at the depot, visit a number o customers and return to the depot. A easible solution to the VRPSD is a permutation o the customers s = (s(1), s(2),, s(n)) starting and ending at the depot (that is, s(1) = s(n) = 0) and it is called a priori tour. Route ailure and recourse action: Route ailure is solving practical problem, also by [11] that it is well said to be occurred i the total demand exceeds the nown that a standard GA must be hybridized with vehicle capacity and the preventive restocing another search procedure to be able to compete with policy [4,19] is employed. metaheuristics lie TS. The approach is also inspired by the emerging interest in hybrid metaheuristics that has Cost and VRPSD obective unction: A = {(i, ): i, risen considerably among researchers in combinatorial V, i } is the set o arcs oining the nodes and a optimization. The best results ound or many practical non-negative matrix C = {c i : i, V, i } denotes the 162

3 travel costs (distances) between node i and. The cost matrix C is symmetric and satisies the triangular inequality. The cost matrix is a unction o Euclidean distance; where the Euclidean distance can be calculated using the ollowing equation: d = (x x ) + (y y ) 2 2 i i i Given a vehicle based at the depot, with capacity Q, VRPSD under restocing policy requires inding vehicle routes and a restocing policy at each node to determine whether or not to return to the depot or restocing beore visiting the next customer to minimize total expected cost. The costs under consideration are: Cost o traveling rom one customer to another as planned Restocing cost: the cost o traveling bac to the depot or restocing The cost o returning to depot or restocing caused by the remaining stoc in the vehicle being insuicient to satisy demand upon arrival at a customer location. This route-ailure cost is a ixed nonnegative cost b plus a cost o traveling to the depot and bac to the route [19] Let n be a particular vehicle route. Upon the service completion at customer, suppose the vehicle has a remaining load q (or the residual capacity o the vehicle ater having serviced customer ) and let (q) denote the total expected cost rom node onward. I S represents the set o all possible loads that a vehicle can have ater service completion at customer, then, (q) or q S satisies: J. Math. & Stat., 4(3): , 2008 In Eq. 2-4, n (q) = c n,0,q Sn (4) p (q) represents the expected cost o r going directly to the next node, whereas (q) represents the expected cost o the restocing action. These equations are used to recursively determine the obective value o the planned vehicle route and the optimal sequence o decisions ater customers are served [4]. Data: We consider VRPSD application in the solid waste collection problem o residential area in Malaysia. The problem o optimization o waste collection can be described as ollows. The solid waste is located in box or containers along the streets and they must be all collected by a leet o vehicles whose capacity can not be exceeded. Each vehicle can service several such sites beore going to dumpsite to unload. Each vehicle starts rom the depot, visits a number o stops and ends at the depot. When a vehicle is ull, it needs to go to the closest available dumpsite to empty its load and then resume their visit. Each vehicle can mae multiple disposal trips per day. Based on experiments reported in [9], three actors seem to impact the diiculty o a given VRP instances: number o customers n, number o vehicles m and illing coeicient. In a stochastic environment, the illing coeicient can be deined as: E( ξ ) n i = (5) i= 1 mq where, E(ξ i ) is the expected demand o customer i, m = 1 or single vehicle and Q denotes the vehicle p (q) capacity. This is the measure o the total amount o (q) = min imum (1) r (q) expected demand relative to vehicle capacity and can be approximately interpreted as the expected number o loads per vehicle needed to serve all customers. In this where experiment, the value o is 1.1. p Several sets o data are randomly generated to (q) = c, (q ξ )p + 1, : ξ q simulate this problem o waste collection. n nodes are (2) + [b + 2c + 1, (q + Q ξ )]p generated in the [0,100] 2 square according to a + 1, : ξ > q continuous uniorm distribution. Nodes are irst assigned to a speciic three demand range in equal and probabilities. Three ranges o demand present low, medium and high solid waste weight. The value o each K demand is then generated in the appropriate range r (q) = c,0 + c0, (Q ξ )p + 1, (3) according to a discrete uniorm distribution. Twenty = 1 instances are generated or each 10, 20 and 50 with the boundary condition: customers with demand range (1,5), (6,10), (11,15). 163

4 The proposed algorithm: The proposed hybrid algorithm o GA and TS consists o several properties such as the order or permutation representation; small population in which all individuals are distinct; inclusion o good simple heuristic solutions in the initial population; Roulette Wheel Selection with elitism; Order Crossover and the Tabu Search as a mutation operator. The algorithm is implemented in the Visual C++ language under the operating windows XP. Step 0: [Deine] Deine operator settings o GA suitable with the problem which is VRPSD. Step 1: [Initialization] Create an initial population P o N chromosomes that consists o constructive heuristics solutions and randomly mutation o it where all individuals are distinct or clones are orbidden. In this study, N was set to be 30. Step 2: [Fitness] Evaluate the itness (C i ) o each chromosome C i in the population. The itness is the unction o VRPSD obective unction. The lower the VRPSD obective unction, the higher itness it is liely to be. Step 4: [Selection] Apply Roulette Wheel Selection. This gives the set o Mating Population M with size N. Step 5: [Crossover] Pair all the chromosomes in M at random orming N/2 pairs. Apply OX crossover with probability pc to each pair and orm N chromosomes o ospring, i random number pc then ospring is the exact copy o parents. Step 6: [Mutation] Dierent with ordinary GA where mutation is applied with probability o pm, in this Hybrid GA-TA, Tabu Search is applied to each ospring. Step 7: [Replace] Evaluate itness o parents and new ospring. Choose the best N chromosomes. Replace the old population with newly generated population. randomized Nearest Neighbour (NN) and randomized Farthest Insertion (FI). The best among them will be chosen as the starting solution to the main body o the search itsel. Neighbourhood structure: The neighbourhood o a solution contains all solutions that can be reached by applying the 2-opt local search. 2-opt proceeds by checing pairs o non adacent edges in a given tour and computing the improvement in the tour length ater rearranging these pairs by exchanging the terminal nodes o the two edges in each pair as shown in the Fig. 1. I no pair o edges can be rearranged to improve the current tour, the algorithm is terminated. Otherwise, the pair o edges that improves the perormance maximum (minimum tour length) is rearranged and the algorithm is executed again. Tabu moves: The aim o tabu moves is to avoid going bac to a solution that has been visited in the last ew iterations. The concept o tabu move is: It is tabu to add an edge which is on the deleted-list It is tabu to delete an edge which is on the addedlist Tabu list: Tabu list is a list o moves that are currently tabu. For the irst iteration, set tabu tenure or all edges is 0. I an edge is moved at iteration v, its addition or deletion is tabu until iteration v + θ. Dynamic tabu list size was applied. I an edge is moved at iteration v, its addition or deletion is tabu until iteration v + θ. θ is randomly selected in the interval N N N N. The, idea o using random tabu list size was introduced by [18] and also used by [10]. Aspiration criteria: As a rule, the search process moves at each iteration rom the current solution to the best nontabu solution in neighbourhood solutions. Step 8: [Test] I the stopping criterion is met then stop and return to the best solution in current population, else go to Step 2. The detail o Tabu Search algorithm implemented is shown below. Initial solution: It is necessary to generate a number o solutions to initialize the main search process. Here two methods are used to construct initial solution: 164 Fig. 1: 2-opt local search

5 Table 1: Standard deviation values o solutions rom GA and HGATS, each or 5 runs Problem instance number N Algorithm GA HGATS GA HGATS GA HGATS N Algorithm GA HGATS GA HGATS GA HGATS However, a tabu solution may be selected i the value o obective unction decreased upon the best value o a solution so ar encountered. Stopping criteria: In our implementation, the number o iteration is equal to the maximum number o iterations. RESULTS AND DISCUSSION Twenty instances were generated on each o the problem size 10, 20 and 50. Each algorithm was tested on each instance or 100 iterations and executed 5 times. Since the result o TS was robust, means that the results ust remain constant rom one run to another run, we only reported the dispersion measure based on standard deviation o GA and HGATS or the need o robustness comparison. Table 1 shown the standard deviation values rom GA and HGATS or each problem instance and each problem size. It can be seen that HGATS was more robust than GA, since the standard deviation values o solution yielded by HGATS were smaller than one produced by GA or most o all problem instances especially or large number o nodes (N = 50) where the value o solutions produced by GA still vary rom one run to another run, but in HGATS the solution values remain constant over the runs. Figure 2 shows the results obtained by the TS, GA and the HGATS or each problem size; each point is the average o best itness unction ound at the end o each run. As it can be observed rom the box plots, the relative perormance o the algorithms was similar on problem size equal to 10 and 20, while the pattern was dierent or problem size equal to 50. To investigate whether the dierences o algorithm perormance are statistically dierent, we conduct the paired samples test. 165 Table 2: Summary o p-values o Kolmogorov-Smirnov normality test or dierences among algorithm perormances Problem size Di GA-TS Di GA-HGATS Di TS-HGATS Table 3: The results o Wilcoxon signed-rans test or problem size = 10 Asymp. Sig. Test statistics b Z (2-tailed) TS_10-GA_ a HGATS_10-GA_ a HGATS_10-TS_ a a : Based on negative rans, b : Wilcoxon signed rans test Firstly we tested the normality assumption o the dierences among the algorithms or each problem sizes using sotware SPSS. The p-values o Kolmogorov-Smirnov test were presented in Table 2. As shown in Table 2, all the dierences on problem size equal to 10 have p-value less than α (= 0.05). It means that we should reect null hypothesis that the data ollows normal distribution. Since the normality assumptions o the dierences are violated, we can not conduct paired t-test and alternatively non parametric Wilcoxon signed rans test should be used. On the other hand or problem size 20 and 50, all the p-values o dierences were greater than 0.05, thus we can not reect the null hypothesis and urthermore we can conduct paired t-test or problem size 20 and 50. The results o Wilcoxon signed rans test or problem size 10 were shown in Table 3 while the results o paired t-test or problem size 20 and 50 were shown in Table 4. From Table 3, we can conclude that the dierence o TS and GA perormances was statistically signiicant or problem size 10 since the p-value is less than α = 0.05, TS and HGATS were also dierent but the

6 Table 4: The result o paired t-test or dierences o GA, TS and HGATS perormance Paired dierences % conidence interval Std. o the dierence Sig. error (2- Mean SD mean Lower Upper t d tailed) Paired samples test or GA and TS, N = 20 GA_20-TS_ Paired samples test or GA and HGATS, N = 20 GA_20-HGATS_ Paired samples test or TS and HGATS, N = 20 TS_20-HGATS_ Paired samples test or GA and TS, N = 50 GA_20-TS_ Paired samples test or GA and HGATS, N = 50 GA_50-HGATS_ Paired samples test or TS and HGATS, N = 50 TS_50-HGATS_ Fig. 2: Box plots o solutions resulted by GA, TS and HGATS perormance o GA and HGATS are similar since the p- value o dierences between TS and HGATS is greater than The same conclusion can be drawn rom Table 4 or problem size 20. The detail conclusion or problem size 10 and 20 in terms o solution quality was given as ollows: the HGATS and GA were better than TS while HGATS and GA perormances were relatively the same. For problem size 50, the perormances o GA and TS were signiicantly dierent (shown by p-value less than α = 0.05), GA and HGATS were dierent while TS and HGATS were similar. Further rom the value o mean dierence, we can draw a conclusion that HGATS and TS were better than GA. 166 CONCLUSION The hybrid GA and TS or solving a priori VRPSD under preventive restocing is designed. We also report the results o comparative study between the proposed HGATS with GA and TS alone or solving single VRPSD. For small problem sizes, the HGATS and GA show superiority compared to TS while the perormances o HGATS and GA are similar. Although the relative perormances o HGATS and GA are similar; the HGATS is more robust than GA. The HGATS also shows its advantage or larger problem size, since it yields better solution quality than GA although the perormances o HGATS and TS are similar. Future wors can be done by implementing more powerul local search move to improve the HGATS perormance. ACKNOWLEDGEMENT The research is supported by IRPA Grant Vote No rom Ministry o Science, Technology and Innovation (MOSTI), Malaysia. The authors would lie to than MOSTI, Research Management Centre (RMC) and Universiti Tenologi Malaysia. The authors also wish to than lecturers and riends or many helpul ideas and discussion. We also wish to than Perniagaan Zawiyah Sdn. Bhd. Johor Bahru, Malaysia or providing necessary data in this study. REFERENCES 1. Bastian, C. and A.H. RinnooyKan, The stochastic vehicle routing problem revisited. Eur. J. Oper. Res., 56:

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