A Firework Algorithm for Solving Capacitated Vehicle Routing Problem

Transcription

1 A Firework Algorithm for Solving Capacitated Vehicle Routing Problem 1 Noora Hani Abdulmajeed and 2* Masri Ayob 1,2 Data Mining and Optimization Research Group, Center for Artificial Intelligence, Faculty of Information Science and Technology, Universiti Kebangsaan Malaysia, UKM, Bangi, Selangor, Malaysia, 1 noora.hani88@gmail.com; *2 masri@ftsm.ukm.my; Abstract The capacitated vehicle routing problem (CVRP) is a combinatorial optimization problem that is very hard to solve for optimality. In CVRP, a set of vehicles of the same capacity located at a central depot need to be routed to serve a set of customers with known demands. This work presents a firework algorithm for solving CVRP. Firework algorithm mimics the explosion phenomenon in searching nearby the space of the solutions by generating sparks around the solutions. We tested the proposed algorithm on14 Christofides benchmark instances and compared it with other heuristics in the literature. Computational results showed that the proposed firework algorithm is competitive in terms of quality of the solutions compared to other methods. Keywords: Vehicle Routing Problem, Meta-Heuristic, Swarm Intelligence 1. Introduction The capacitated vehicle routing problem (CVRP) is an important problem in the industry sector with several applications in the field of transportation, distribution and logistics and thus it is a very interesting problem to study for computer scientists. CVRP can be defined as a complete undirected graph G=(V,E), where V={0,1 n} is the set of vertices and E={(i,j)}: i,j ϵ V,i<j} is the set of edge. Vertices 1 n represent customers, each customer i with known demand and a service time, vertex v 0 represents the depot, while the other vertices represent the customers. CVRP seeks to generate a set of routes where each route starts and ends at the depot. Each customer is visited exactly once and by only one vehicle. The total demand of any route cannot exceed the capacity of vehicle and the total duration of any route is no longer than a given bound. The use of exact methods to solve NP-Hard problem can guarantee the optimal solutions. However, when the problem size is large, the required computational time will grow exponentially [12-13]. Some examples of exact methods that were used to solve CVRP are: branch and bound and dynamic programming [8], and mathematical programming approach [9]. Since CVRP is a NP-hard problem, many researchers proposed various kinds of heuristics methods to solve the problem. For example, [1] presented an optimized crossover genetic algorithm for CVRP. [2] Applied a hybrid meta-heuristics for CVRP. [3] Presented an adaptation of the Active-guided evolution strategies for large-scale CVRP. [4] Applied Iterated variable neighborhood descent algorithm for the CVRP. [5] Proposed the Multiple Phase Neighborhood Search-GRASPS for the CVRP. [6] Applied a hybrid approach to CVRP using ant colony optimization and simulated annealing. Heuristic approaches can be divided into the constructive and improvement approaches [2]. The constructive approaches can generate a feasible solution quickly, but the qualities of the solutions are usually not very good. These approaches can obtain feasible solutions within a reasonable computation time. On the other hand, the improvement approach could obtain near optimal solutions. Therefore, improvement approaches such as Tabu search (TS) and simulated annealing (SA) are often applied to solve many NP-hard problems [12-14]. Many algorithms have been proposed and successfully applied to solve CVRP, but there is still a room for improvement. Therefore this motivates us to move forwards and investigate more algorithms that have not been studies yet for solving CVRP. This work applies a new algorithm named fireworks algorithm (FA) for the CVRP. FA is originally proposed by Tan and Zhu [7] for optimization problem, but it has not been studied yet to solve CVRP. Therefore, this work investigates the capability of FA in solving CVRP and then enhanced the FA performance by hybridizing it with a mutation operator as a diversification mechanism. We select FA in this paper due to its ability in dealing with many International Journal of Advancements in Computing Technology(IJACT) Volume 6, Number 1, January

2 difficult optimization problems such as mathematical function [7-10], engineering [7-10] and design [7-10] problems. 2. Problem definition of CVRP The aim of CVRP is to minimize the traveling cost of vehicles while respecting some constraints. The objective function of CVRP can be formulated as in equation (1) [2]: Minimize (1) Subject to 1 k K, (2) 1 k K, (3) For i=0, (4) Where, C ij is the travelling time from customer i to customer j, K is the number of vehicles, N is the number of customers, S i is the service time at customer i, Q k is the loading capacity of vehicle k, T k is the maximal traveling (route) time of vehicle k, d i is the demand at customer i, X k 0,1 where i j; i, ij j ϵ {0, 1,..., N}. Equation (1) is the objective function of the problem that aims to minimize the traveling cost of vehicles. Equation (2) is the constraint of loading capacity, where X k 1 if vehicle k ij travels from customer i to customer j directly, and 0 otherwise. Equation (3) is the constraint of maximum traveling time. Equation (4) specifies that there are maximum K routes going out of the delivery depot. 3. Solution representation and initial solution generation FA algorithm uses a direct representation scheme. The solution is indicated with onedimensional vector where each integer number represents a customer, and the length of each solution represents the total number of customers. The number of routes in the solution is counted based on the total number of 0 minus one. A given solution is divided into a set of routes which is delimitated by 0. Figure 1 shows an example of solution representation. In this example, there are three routes, first one serve two customer 4 and 1, the second route serve three customers 2, 7 and 5, third route serve two customers 3 and Figure 1. Solution representation The initial solution is generated by randomly select customers to the current route without violating the capacity of the vehicle. If violation occurs, create a new route and repeat this procedure until all customers are routed. 80

3 4. Neighborhood operators A neighbourhood operator is used to obtain a new solution x from the current solution x in the FA heuristic. Two common neighbourhood operators are chosen for this work. The selected neighbourhood operators are: a. Random swap (NS1) This operator randomly selects positions (in the solution vector) i and j where i j and swaps the customers located in positions i and j. See figure. 2, where i=3 and j=7. Only a feasible swap is accepted. Before: Swap point Swap point After: Figure 2. Random swap b. Random move (NS2) This operator randomly selects positions i and j where i j and relocating the customer from position i to position j. See Figure 3, where customer 5 is relocated from position 7 to position 3. Where number 1 and 5 represent the customer id. Only feasible move is accepted. Before: Move Position Move Point After: Figure 3. Random move 5. Firework algorithm (FA) The FA is a swarm intelligence algorithm that is based on simulation of the fireworks explosion process [7]. In the FA, there are two explosion (search) processes being considered and the mechanisms are also designed to maintain the diversity of sparks. Fireworks as well as the newly generated sparks represent potential solutions in the search space. Similar to other optimization algorithms, the goal is to find a good quality solution [15]. In analogy with real fireworks exploding and illuminating the night sky, FA uses these procedures to explore the potential search space of the given problem. For each firework, an explosion process is initiated and a shower of sparks fills the local space around it. The framework of the FA is described as follows. As the firework is exploded, the local space close to the firework is filled with a shower of sparks. Thus, firework explosion process can be indicated as a search in the closet area near a certain point. To find a point, we continually set off fireworks in possible space until one spark targets or is fairly close to the point. Table 1 shows the analogy between the real firework and FA. Table 1. The analogy between real real fireworks and Firework Algorithm (FA) Real Firework Firework Algorithm Firework One solution Sparks neighbor solutions Firework location Solution space Quality of firework explosion Quality of solution. 81

4 In each generation of explosion in the FA requires selecting n locations (where location represent a solution space in the optimization problem), since n fireworks are exploded in n different location. The quality of sparks are then evaluated (using the objective function Equation (2) is used for the CVRP) after each explosion. FA will stop when the optimal spark is found or the stopping condition is met. Otherwise n other locations will be chosen from the current fireworks and sparks locations for the next explosion generation Design of fireworks explosion The number of sparks generated by each firework is defined as follows. S y x f ( ) i m. max i n ( y f ( x )) max i i 1 (5) is a parameter which controls the total number of sparks gained from the n fireworks, the size of population, which considered the maximum (worst) value of the objective function among the n fireworks, and, ξ indicates the smallest constant in the computer, is utilized to avoid zero-divisionerror. From equation (5) if the gap between the quality of solution and the worse one is big, this mean that this solution is good in term of quality and many sparks will be generated from this solution. From equation (5) we will get a real number. Therefore, equations (6) will convert the real number obtained from equation (5) to integer number. In order to convert to integer number, show equation (6) (6) Where and are constant parameters, that confine the range of the generated sparks Selection of locations Each explosion generation should start with n locations which are assigned for the fireworks explosion. The n solutions are selected based on the distance among them. The definition of distance between a location with other locations are calculate using equation 7. R ( ) = d (, ) = ǁ - ǁ, (7) Where is the set off all current locations of both fireworks and sparks. Then the selection probability of a location is defined as follows. P ( ) = (8) Note: - refer to norm in mathematics, which mean the distance formula: 82

5 Figure 4 shows the FA pseudo-code. The algorithm starts by generating a set of fireworks and set the stopping condition (line 1). Next the main loop of the algorithm begins (line 2). First select n fireworks for the explosion process (line 3). At each iteration two types of sparks are generated using two kinds of explosions. The first explosion takes place as follows (line 4- line7): for each firework calculates the number of sparks ( ) to be generated from this firework (line 5) using Equations 5 and 6. Next based on the calculated ( ) generate the sparks from each firework using move operator (NS2) where each move generate one spark (solution) and calculate the solutions quality using Equation 2 (line 6). Subsequently, the second explosion is performed as follows (line 8 line 11): first set the number of firework ( ) that to be used for the second sparks generation (line 9). Randomly select a firework that is not belong the n set (line 10), generate a spark from the selected firework using swap operator (NS1) and evaluate the quality of the generated spark using Equation 2 (line 10). Select the best sparks from both explosions to represent the population of the fireworks for the next generation (line 12). Select a set of n fireworks for the next generation (line 13). Check the stopping condition (line 14). If satisfied, then stop. Otherwise go to line 2. (1) Generates a set of fireworks; set the stopping condition. (2) While stopping criteria = false do (3) Set off a set of n fireworks ; //explosion 1 (4) For each firework n do (5) Calculate the number of sparks to be generated from the firework x i ; using Eq. 5 and 6. (6) Generate sparks from and evaluate their quality using Eq.2; (7) End for (8) For K=1: do, where is the number of iterations per generation //explosion 2 (9) Randomly selects a firework that are not in n set; (10) Generate a spark from the selected firework and evaluate the quality using Eq. 2; (11) End for (12) Select the best sparks from the first and second explosions and keep them for the next explosion generation; (13) Randomly select n-1 sparks from both explosions using based on Eq.8; (14) End while; (15) Return the best firework. Figure 4. The pseudo-code of the proposed algorithm As discussed in Algorithm 1, there are two kinds of sparks which are generated each explosion. In the first kind, the explosion and the number of sparks rely on the quality of the firework where we generate a number of solution using move operators (NS2). While, the second kind is generated by applying a swap operator (NS1) for the purpose of conducting a search within a local search space close to a current firework. The swap operator is applied at each selected solution. Moreover, after identifying the locations for the two kinds of sparks, the n locations will be assigned for the next generation explosion. It should be noted that in both types of spark generation only improving solutions are accepted and replaced with the current one Strength of FA FA deals with a population of solutions that can help in dealing with a problem with huge search space. The main strength of FA is in its ability of focusing the search process on different area by using two types of exploitation that use different way in exploring the search space. By using different exploitation process that algorithm can jump from the current local optima. 83

6 5.4. Limitation of FA The main limitation of FA is that the employed explosions process only accepts improving solution. Furthermore, only good quality solutions are added into the population. This makes FA more exploitative that explorative. Thus, FA may face the problem of fast convergence. 6. Results and discussions The proposed algorithm is programmed by using the Java creator running on the Intel Pentium computer of GHz using Windows 7. We use 14 benchmark datasets from Christofides [11]. FA parameters are set based on preliminary experiments which are based on the recommendation of the original paper. See Table 2. Table 2. FA parameters Parameter Value 1 N 5 2 M 50 3 A b That is used in all comparing experiments. For every dataset, the algorithm was run 31 times. Every run was stopped when the fitness cannot be enhanced after 100 generations. In addition, we do comparisons between our suggested algorithm and the Best-Known-Result (BKR) findings that are reported in previous literature regarding instances of Christofides [11]. Some algorithms have been selected namely: optimized crossover genetic algorithm [1]. Appling hybrid meta-heuristics [2]. Active-guided evolution strategies [3]. Multiple Phase Neighborhood Search-GRASP [5]. An enhanced ant colony optimization (EACO) [6]. The best results are presented in Table 3. Table 3. The computational results of FA compared to the algorithms from the literature and the bestknown- results Instanc OCGA AHMH AGES MPNS- EACO FA BKV e GRASP C C C C C C C C C C C C C C In the first, we compare the results of FA against the algorithms from the literature which are tested at the same benchmark instances of Christofides [11]. In addition, we also report the percentage deviation from the best know results (BKR), PD= ((FA-BKR)/BKR)*100, where FA is the best results of FA over 31 runs and BKR is the best known results. It is clear from Table 3 that FA produced the 84

7 best quality solution of the most instances. Where the values in bold indicate the best results obtained by the algorithm. In particular, we can make the following conclusion: - FA is obtained the same results as OCGA for 6 (C1, C2, C6, C7, C12, C14), out of 14 instances. - FA is obtained the same results as AHMH for 6 (C1, C2, C6, C7, C12, C14), and better than AHMH in one (C13). - FA is obtained the same results as AGES for 3 (C1, C2, C12), out of 14 instances. - FA is obtained the same results as MPNS-GRASP for 3 (C1, C6, C7), and better than MPNS- GRASP for 3 (C2, C12, C14). - FA is obtained the same results as EACO for (C1, C2, C6, C7, C12, C14), and better than EACO for (C13). - FA is obtained the same results as BKR for 6 (C1, C2, C6, C7, C12, C14), out of 14 instances. It should be noted that although FA did not managed to obtained the best results for the other 8 instances, the quality of the these solutions (C3, C4, C5, C8, C9, C10, C11, C13) are very competitive and the percentage deviation for the best known results are very small (between 6.90 to 0.007) which indicate that our results are very good. We also would like to indicate that none of the compared methods managed to produce the best results for all the 14 instances. Second, in Table 4, we compare the best time of FA gained OCGA [1], AGES [3] and AHMH [2], by take the best results of the run. It shows that the best time for FA in instance C3, C4, C5, C7, C8, C9, C10, C13, C14 are better than the compared algorithms. Table 4. The best times in FA compared to algorithms from the litterateur (time in second) Instance FA (S) OCGA(S) AGES(S) AHMH(S) C C C C C C C C C C C C C C Conclusion In this paper, we have proposed a fireworks algorithm for solving the capacitated vehicle routing problem. Fireworks algorithm is a new population based algorithm that mimics the explosion of the fireworks. Fireworks algorithm generates solutions by generating a set of sparks around the current fireworks. The proposed algorithm is tested on 14 instances of Christofides benchmarks. The computational experiments showed that the proposed algorithm is competitive in terms of the quality of the solutions when compared to other methods in the literature. 8. Acknowledgments The authors wish to thank Ministry of Higher Education for supporting this work under the ERGS Research Grant Scheme (ERGS/1/2013/ICT02/UKM/02/3). 85

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