A genetic algorithm metaheuristic for Baery Distribution Vehicle Routing Problem with Load Balancing Timur Kesintur Department of Quantitative Methods, School of Business Istanbul University Avcilar Yerlesesi, Bogazici Cad. Istanbul, 34320 TURKEY ttur@istanbul.edu.tr Mehmet Bayram Yildirim Department of Industrial and Manufacturing Engineering Wichita State University 1845 N Fairmount St. Wichita, KS, 67260-0035 USA Bayram.yildirim@wichita.edu Abstract In this paper, we analyze a distribution problem at a baery company in Istanbul, Turey. The baery wants to distribute products with the available fleets of trucs, while maintaining balance in worload between different truc drivers. The resulting problem is very similar to a capacitated vehicle routing problem with load balancing objective with a maximum distribution time and truc capacity constraints. A genetic algorithm meta-heuristic approach is proposed to obtain good quality solutions. The proposed meta-heuristic is utilized on a case study where different trucs serve customers in a large metropolitan area. The distance matrix is generated via a web application using Google Maps. The solutions obtained via the meta-heuristic are compared with the company s own distribution plan and global ant colony optimization algorithm. Keywords-component; Load balancing, Vehicle Routing with capacity constraints, Baery Distribution, metaheuristics, genetic algorithm I. INTRODUCTION In this paper, we propose a mathematical model for a distribution planning problem at a baery in Istanbul Turey with a load balancing objective [1]. The baery owns a fleet of trucs which are used in distribution of fresh perishable baery products. The trucs loaded with fresh baery products may visit several grocery stores, marets and restaurants on a tour to satisfy the customer demand and return bac to the baery three times a day. Currently, the baery has partitioned the demand by regions, and serves each region via a truc. We model this problem as a capacitated vehicle routing problem with load balancing objective and maximum tour length. Our goal is to propose a framewor to improve the current distribution plan by the baery management by minimizing the imbalance between the worloads of drivers. The VRP introduced by Dantzig ve Ramser [2] is a hard combinatorial optimization problem and only relatively small instances can be solved to optimality. Exact algorithms are only capable of solving small instances with the amount of customers typically being less than 50. This is because the lower bounds of the objective value are difficult to derive, so partial enumerations based on exact algorithms have a slow convergence rate. Because exact approaches are generally inadequate and inflexible, heuristics are commonly used. In this paper, to solve the proposed model, we utilize a genetic algorithm with local search to obtain good results in reasonable amount of time. For discussions on VRP, we refer the reader to Toth and Vigo [3] and Ralphs et al. [4]. Worload balancing on similar resources is a well-studied problem. For example, in a manufacturing setting Duman et al. [5] show that a balanced load in parallel resources may improve the flexibility and deficiency of the production line. Assigning tass to worers in an office or shop floor is another viable example. To maintain the morale of the worforce, it is of utmost important to eep the assignment of jobs to employees woring in the same area as equally balanced as possible (i.e., assign approximately equal worload) to minimize tension on the office or shop floor, or in this case, between the truc drivers. In vehicle routing problems, to minimize disparities and achieve fairness between drivers, worload balancing objectives have been introduced. In a vehicle routing problem, worload for a driver on a tour can be defined in terms of the length of the tour, volume transported during a period [6] duration of the tour (which may include loading and unloading operations [7], [8], or as a number of customers whom needs to be visited. The objective may be minimization of the sum of the differences between the worload of each tour and the smallest worload [8] or the difference between the worload of the longest tour and the one of the shortest tour [7], [9], [10]. The Vehicle Routing Problem (VRP) with load balancing consists of designing a set of delivery or collection routes such that: 1) Each route starts and ends at the depot, 2) each service request is visited exactly once by exactly one vehicle, 3) the total demand of each route does not exceed the capacity of a truc; 4) the total duration of each route (including travel and service times) does not exceed a preset limit due to perishability of baery products; and 5) the total worload imbalance is minimized. The organization of the paper is as follows: In section 2, the assumptions, parameters and the model for the VRP problem 978-1-61284-922-5/11/$26.00 2011 IEEE 287
with load balancing are presented. Next, a genetic algorithm meta-heuristic to solve VRPLB is described. After results computational experimentation on a case study, conclusions are presented in section 5. II. PROBLEM STATEMENT To describe the VRPLB model the following notation is utilized: K is the set of vehicles, and K denotes the number of vehicles. Similarly, J is the set of demand points. K j is the subset of vehicles which may serve demand at j, and J is the subset of demand points which can be served by vehicle. S ij is the distance (duration) for vehicle if demand point i precedes j on vehicle s tour. p i is the amount of time required to load/unload the demand at i by if vehicle is utilized. Let C be the total completion time of the tour for vehicle. The imbalance is defined as C C where the minimization max objective will force C max to have the value of maximum completion time of jobs on all vehicles, i.e., C max C. max K The relative imbalance for vehicle is the ratio of imbalance and the maximum completion time on all machines, i.e., C C max relative imbalance =. C max Below is a mathematical program to minimize average relative percentage imbalance for a vehicle routing problem with heterogeneous fleet where the goal is to minimize the average relative percentage of imbalance (ARPI). 1 C C (1) max min 100 K K C max Mathematically, the total completion time of vehicle s tour, C is defined as where and x ij (2) C y p x s K i i ij ij ij ij jj y i 1 if demand i is assigned to vehicle 0 otherwise 1 if i is the immediate predecessor of j on vehicle 's tour 0 otherwise Constraint (3) ensures that the maximum worload is greater than or equivalent to individual worloads. C C K (3) max Constraint (4) ensures that each demand point is assigned to a vehicle. y 1 i J (4) i Ki Constraint (5) guarantees that only two demand points on the same vehicle s tour can precede each other. x y i J, K, j J (5) ij i i Constraint (6)/constraint (7) ensures that a demand point must be before/after another demand point on a vehicle s tour. x y K, j J (6) ij i i J x y K, i J (7) ij i j J Constraint (8) represents sub-tour elimination constraints, which ensure that a demand point cannot be the immediate predecessor or successor of two or more different demand points at the same time on a vehicle s tour. ' ' x J 1 J J (8) ij ' ' ij jj Constraint 9 limits the amount of demand that can be satisfied by vehicle as Q. d y Q, K (9) i i ij where d i is the demand at i. To ensure the daily number of deliveries to a demand point, a limit on the total length of a tour by vehicle (T ) is imposed by the following constraint: C T K (10) Note that any solution with a zero ARPI value is optimal. The VRPLB model has an exponential number of possible solutions and is a very difficult problem to solve. This motivated us to develop a genetic algorithm to determine good solutions in a reasonable amount of time. III. META-HEURISTIC GENETIC ALGORITHM Genetic algorithms (developed by Holland [11]-[12] and a meta-heuristic that mimic the process of evolution in order to solve complex combinatorial problems), have been applied successfully to vehicle routing problems. Genetic Algorithms (GAs) updates a population of solutions via genetic operators such as crossover, mutation and selection to achieve offsprings with better quality until some convergence criteria are met. At each generation, a genetic algorithm is capable of producing and maintaining a set of feasible solutions, maintaining a population of candidate solutions, and evaluating the quality of each candidate solution according to the problem-specific fitness function. The pseudo-code of the genetic algorithm is presented in Fig.1. The components of the proposed genetic algorithm are explained in detail below: Representation (Coding): Each solution to the genetic algorithm is represented as a super chromosome having the size of the number of customers ( J ). This chromosome has subchromosomes that represent the sequence of visits to customers on a tour for each vehicle (see Figure 2 for a chromosome that represents ten costumers and three vehicles). The customers which will be served by the first vehicle are listed in the scheduled order on a tour is listed first on the super 288
chromosome, followed by customers of vehicle 2 and then vehicle 3. In another array of size of the number of vehicles ( K ), information on the number of customers scheduled for each vehicle is ept. Genetic Algorithm STEP 0: Generate an initial population STEP 1: Evaluate the fitness value of the chromosomes STEP 2: Perform selection operation and give those individual that have better fitness values a more chance to survive in the next generation. STEP 3: Perform crossover and mutation operations. Apply Feasibility Chec for each new offspring. STEP 4: Perform local search (optional) STEP 5: Repeat steps 1, 2, 3 and 4 until the GA is run for the predetermined number of generation STEP 5: Select the best chromosome. Figure1. Pseudocode for the genetic algorithm Vehicle 1 Vehicle 2 Vehicle 3 Customers 5 8 1 4 10 3 7 6 9 2 Figure 2. Representation of a chromosome for a 3 vehicle 10 customers example Evaluation of Fitness Function: The fitness function f for a chromosome is the objective function value, (ARPI) of solution while severely penalizing the infeasibilities in the capacity and maximum trip length constraints (equations 9 and 10): (11) min0, Q diyi min0, C T K ij f( ARPI ) ARPI e Selection: The selection model should reflect nature's survival of the fittest. In this paper, the roulette wheel system is used to select parents for the next generation [13]. Crossover: Once parents are selected, the single point crossover operation is applied with a probability of P c to generate two new offspring solutions: Two parent strings are selected randomly from the population. A random number between 2 to J -1 is generated to determine the crossover point. When crossover is finished, the genes before the crossover point in the first chromosome are the first part of the first child chromosome. The second part of the first child chromosome is generated by checing the genes from the second chromosome one by one and adding those genes that are not yet in the child chromosome. Similarly, the 2 nd child chromosome is generated. Note that the number of demand points for each vehicle is ept constant. Mutation: The mutation operator moves a gene randomly to another position on the chromosome, with a probability equal to the mutation probability, P m. If the new position is in the same vehicle, then only the customers order is changed. However, if the gene is moved to another vehicle, the number of customers and order of customers on both vehicles may change. Feasibility Chec: For each new offspring obtained via mutation or crossover operation, the feasibility should be checed. If not feasible, i.e., at least one of the trucs tour violate the total capacity of a vehicle, apply the following one pass heuristic possibly to obtain a feasible solution: if the total demand/duration for vehicle is more than its capacity/max tour length, include all customers on the tour that the vehicle capacity/tour length can handle. Move those who cannot be handled to the next vehicle, +1. Local Search (optional): A 2-exchange heuristic, a first improvement local search, is utilized to switch positions of two customers on a chromosome (i.e., either in the same vehicle or on different vehicles). Apply feasibility chec. If the exchange improves the current solution, it is ept. At each iteration, J 2 number of exchanges is performed. Two variants of the genetic algorithm without (GA) and with local search (GA*) are utilized to solve the problem. IV. COMPUTATIONAL EXPERIMENTATION The baery company has 74 customers and 4 trucs. These vehicles are used to deliver fresh baery products three times a day. Each truc has a capacity of 6000 units, and can deliver to all customer locations. It is assumed that the demand for each customer is nown through historical data, and each customer can be served by any vehicle. Furthermore, it is assumed that it taes the same amount of time to load/unload at each demand point. The locations of each customer are mared on the map in Figure 3 (Google Inc.[14]). The distances and duration times between each customer are calculated using Google Maps. These distances were verified by the baery management team and the drivers of the trucs [1]. The computational experimentation is performed on a Pentium Dual Core Machine with 4 GB of memory and 120 GB of hard drive using MATLAB R2007b as the programming medium to code develop the meta-heuristics. The value of performance measures is obtained by averaging the results over 10 runs of 1000 iterations for 10. Through experimentation, the crossover and mutation probability and a population size is determined as 0.9, 0.01, and 40, respectively. The computational results are presented in Table 1. In Table 1, the ARPI values for the current solution by the company, and also by a global ant colony optimization algorithm (GACO) to solve the vehicle routing problem without any side constraints by Kesintur [1]. The GACO, 289
originally proposed by Kesintür and Soyler [15], is a population based ant colony optimization meta-heuristic, was developed to solve the lot sizing problems [1]. To avoid local minimums, GACO utilizes an operator similar to mutation operators in genetic, by mutating the routes of the ants to obtain new potential solutions. When applied to VRP problems, computational experimentation proved the effectiveness of GACO algorithm over classical metaheuristics such as variants of ant colony optimization algorithms and genetic algorithms [1]. current ARPI is 16%, whereas the GA* has an ARPI value of 8%, so an improvement of 50% is ARPI values. As can been seen in figure 4, when the maximum allowable tour length increases, the ARPI decreases. However, this also results in longer tour lengths and durations. The GACO which provides the solution with the shortest tour length (since its aim is to minimize the total tour length, but not minimization of ARPI, and does not have any capacity constraint as well as total tour length constraint) has the worst ARPI value. With some modifications in the length of the tour, a more balanced GACO algorithm which determines the minimum length distribution plan has a total tour length of 342 minutes while having an ARPI value of 41% By increasing the total distribution time by 25% (i.e., 80 minutes), the ARPI value can be decreased to 9%. Figure 3. Locations of the customers [1] The maximum tour length parameter is varied to 102, 108, 114 and 120 minutes for most of the vehicles. However, it was observed that regardless of the objective function, the trucs serving customers located in the northwest quadrant could not be served in less than 120 minutes when there is meaningful worload for that vehicle, so the minimum tour length for that vehicle was ept constant at 120 minutes. In Table 1, firm represents the company s current schedule and GACO is the ARPI values for VRP problem without capacity and tour lengths obtained with GACO algorithm, unconstrained column represents the ARPI values for the VRP problem with load balancing objective, however, without tour length constraint. For all solutions, the tour length for each vehicle is listed in Table 1. Note that the ARPI values for firm and GACO algorithm are significantly higher than solutions for which load balancing was the objective. If no tour duration contraint is imposed, one can obtain perfect ARPI values with an objective function value of 0.01. The results indicate that the genetic algorithm with local search (GA*) outperforms other solutions. The company s Figure 4. ARPI as a function of tour travel time constraint V. CONCLUSION In this paper, a capacitated vehicle routing problem with load balancing objective and maximum tour length constraint is proposed. A genetic algorithm with local search is developed. The solution framewor is applied on a real life case study from a baery. It was observed that the genetic algorithm with local search outperformed the solutions obtained from the one without local search, and the company s solution. Furthermore, it was observed that when the maximum tour length constraint is relaxed, better ARPI values are obtained. However, the tour lengths increase. As a future research, one can try to develop a solution algorithm for a multi-objective capacitated VRP problem with load balancing and total tour length objectives. ACKNOWLEDGMENT This paper was partially supported by the Istanbul University. Research Foundation Project (BAP) No: 10625. 290
TABLE I. TOTAL TOUR DURATION TIMES AND ARPI GA (minutes) GA* (minutes) FIRM GACO Unconstrained 102 108 114 120 102 108 114 120 Vehicle 1 87.00 52.00 476.70 101.62 107.97 113.94 119.88 101.94 107.03 114.00 119.88 Vehicle 2 55.00 86.00 476.58 101.84 107.84 113.94 119.88 102.06 107.99 114.00 119.88 Vehicle 3 118.00 73.00 476.70 101.89 107.28 113.64 119.88 101.88 107.95 114.00 119.76 Vehicle 4 * 121.00 141.00 476.70 116.29 118.54 118.14 119.88 113.82 117.12 116.64 119.88 ARPI 16.01 41.25 0.01 9.36 6.86 2.71 0.00 7.81 6.06 1.71 0.03 Total Delivery Duration 381.00 342.00 1906.68 421.64 441.63 459.66 479.52 419.70 440.08 458.64 479.40 * Vehicle 4 s constraint was upped to 120 minutes REFERENCES [1] T. Kesintur, Global Ant Colony Optimization for Vehicle Routing Problems, PhD Dissertation, Istanbul Universitesi, Institute of Social Sciences, Published by Economic Research Foundation, Istanbul, Turey, 2010. [2] G.B. Dantzig, and J.H. Ramser, The truc dispatching problem, Management Science, Vol. 6(1), (6), No:1,1959, pp. 80-91. [3] P. Toth ve D. Vigo, The vehicle routing problem, SIAM, Philadelphia, (2002a). [4] T.K. Ralphs, L. Kopman, W.R. Pulleyblan and L.E. Trotter, On the Capacitated Vehicle Routing Problem, Math. Programming, Vol. 94, 2003, pp.343 359. [5] E. Duman, M. B. Yildirim and A. F. Alaya, Scheduling continuous aluminum casting lines, International Journal of Production Research, Vol. 46 (20), 2008, pp. 5701-5718 [6] R. Ribeiro and H.R. Lourenco, A multi-objective model for a multiperiod distribution management problem, in proceedings of Metaheuristic International Conference 2001, pp.91 102, 2001. [7] N. El-Sherbeny, Resolution of a vehicle routing problemwith multiobjective simulated annealing method. Ph.D. thesis, Faculte Polytechnique de Mons, Mons, Belgium, 2001. [8] T.R. Lee and J.H. Ueng, A study of vehicle routing problem with load balancing, International Journal of Physical Distribution and Logistics Management, 29, 1998, pp. 646 648. [9] N. Jozefowiez, Mode lisation et re solution approche es de proble`mes de tourne es multi-objectif. Ph.D. thesis, Laboratoire d Informatique Fondamentale de Lille, Universite des SciencesetTechnologiesde Lille,Villeneuved Ascq, France, December 2004. [10] N. Jozefowiez, F. Semet, E-G. Talbi, Parallel and hybrid models for multi-objective optimization: Application to the vehicle routing problem, in: J.J. Merelo Guervos et al. (Eds.), Parallel Problem Solving from Nature VII, Lecture Notes in Computer Science, vol. 2439, Springer- Verlag, 2002, pp. 271 280. [11] D.E. Goldberg, Genetic Algorithms in Search Optimization and Machine Learning, Addison Wesley Publishing Company, USA, 1989. [12] C.R. Reeves, Modern Heuristic Techniques for Combinatorial Problems, Mcgraw-Hill Boo Company Inc., Europe, 1995. [13] Ip, W.H., Li, Y., Man, K.F., Tang, K.S., Multi-product planning and scheduling using genetic algorithm approach Computers & Industrial Engineering, Vol. 38(2), 2000, pp. 283-296. [14] Google Inc. (2009). Google Maps. (Online): http://maps.google.com/ (10/11/09). [15] Kesintur, T., Soyler H., Global Ant Colony Optimization Journal of The Faculty of Engineering and Architecture of Gazi University,Vol. 21(4), 2006, pp. 689-698, 2006. (In Turish) 291