Performance Analyses on ODV-EV Technique for Capacitated Vehicle Routing Problem using Genetic Algorithm

Performance Analyses on ODV-EV Technique for Capacitated Vehicle Routing Problem using Genetic Algorithm A. Ramalingam 1 and K. Vivekanandan 2 1 Department of MCA, Sri Manakula Vinayagar Engineering College, Puducherry, India. 2 Department of CSE, Pondicherry Engineering College, Puducherry, India. 1 a.ramalingam@smvec.ac.in, 2 k.vivekanandan@pec.edu Abstract The Capacitated Vehicle Routing Problem (CVRP) is an extension of classical VRP. The objective of the CVRP is to minimize the distance with an additional constraint that every vehicle has to sustain the capacity of the vehicle. Although several techniques have been proposed to solve CVRP and it is rare to find a specific technique to adopt for this problem, in addition to that the models are complex. As CVRP is NP-complete problem, optimization methods may be difficult to solve these problems, furthermore, it is observed that the results also inadequate to explore for real world enlargements. Therefore solution to the CVRP is obtained through Genetic Algorithm (GA) which is based on heuristic approach. The traditional GA with Random and Nearest Neighbour population seeding techniques having poor fitness individuals causes long time to converge to the optimal solutions. In this paper, we analyse the performance of Random and Nearest Neighbour population seeding techniques in terms of convergence rate, error rate and convergence diversity with proposed Equi-begin with Variablediversity (EV) technique for Capacitated Vehicle Routing Problem (CVRP) using Genetic Algorithm. Experimental results are carried out using the CVRP benchmark instances obtained from the VRPLIB were experimented using MATLAB software. Keyword Genetic Algorithms, Capacitated Vehicle Routing Problem, EV Population Seeding Technique, Performance Analysis, VRPLIB, MATLAB. I. INTRODUCTION A typical Vehicle Routing Problem (VRP) can be desired as the problem of designing minimum cost routes from one depot to a set of geographically scattered points (cities, stores, warehouses, schools, customers and so on). The routes must be designed in such a way that each point is visited only once by exactly one vehicle, all routes start and end at the depot, and the total demands of all points on one route must not exceed the capacity of the vehicle [1]. It is quite close to one of the most famous combinatorial optimization problems, Travelling Salesperson Problem (TSP), where only one person has to visit all the customers [2-5]. A Capacitated Vehicle Routing Problem (CVRP) is like VRP with an additional constraint that every vehicle must have uniform capacity of a single commodity. The Objective is to minimize the vehicle fleet, the sum of travel time, and the total demand of commodities for each route may not exceed the capacity of the vehicle which serves that route. A solution is feasible if the total quantity assigned to each route does not exceed the capacity of the vehicle which services the route. In GA, the population consists of a set of solutions or individuals instead of chromosomes. A Crossover operator plays the role of reproduction and a mutation operator is assigned to make random changes in the solutions [4,8,9]. Genetic Algorithms (GA) is based on a parallel search mechanism, which makes it more efficient than other classical optimization techniques such as branch and bound, Tabu search method and simulated annealing etc. The basic idea of GA is to maintain a population of candidate solutions that evolves under selective pressure [4, 5, 15, 16]. In this paper we compare other population seeding technique such as Random population seeding technique and Nearest Neighbour with the novel Ordered Distance Vector A. Ramalingam & K. Vivekanandan ijesird, Vol. I (V) November 2014/ 173

C2 Route 2 C3 C4 5 8 C1 Depot (C0) Route 3 C5 3 Route 1 C7 C6 2 2 Route 1 Route 2 Route 3 C1, C2, C3, C4, C5, C6, C7 Customers and corresponding demands δ= Capacity of the Vehicle Fig 1. A solution for CVRP with 7 customers and demands (ODV) based EV Technique, the comparison is done to find the best initial population technique which plays an imperative role in finding optimal solution [4,12, 15, 17]. II. CVRP - MODEL AND EXPLANATION In this model, the standard CVRP is considered that the set of vehicles with identical capacity has to visit a set of customers in order to minimize the total travelled distance. At any point the maximum capacity of the vehicle should not exceed the demands of the customer in the current route [15,16]. The CVRP is a variant of VRP, let be a complete undirected graph, where is the vertex set, and are expressed as is an edge set. They and Let be used as the set of customers, represent the depot. The Demand / quantity of goods requested by each customer, is associated with the corresponding vertex [15,16]. Let is non-negative distance, travel time or cost matrix between customers and. Furthermore, a set of homogeneous vehicles originate from a single depot, let signifies the capacity of a vehicle. Consider a problem size with 7 customers and corresponding demands as shown in the Fig 1. The objective of CVRP is that the vehicles have to start from the depot and visit the customer one by one in order to minimize the total distance [4,15,16]. Suppose total demand of all the customers in the route exceeds the total capacity of the vehicle then the route has to be aborted, the vehicle requisite to return the depot [4,13,15]. From the depot the vehicle has starts a different route to visit the unvisited customers and this progression is continued to visit all the customers exactly once. In the above example show, the vehicle has to start the route from the depot and has to visit the next customer which is, before visiting the customer verify the condition that the total demand of the customers has exceed the vehicle the total capacity (i.e.). A. Ramalingam & K. Vivekanandan ijesird, Vol. I (V) November 2014/ 174

At this moment, if the condition has satisfied the vehicle has to abort the route else add the demand of current customer in total demand and visit the next customer. Likewise, vehicle has to visit the next customer which is from the current customer, before that verify the condition that the vehicle has exceed the total capacity(i.e.). Succeeded by the previous stage, the vehicle has to visit the next customer, however while checking the condition has violated. Hence it is obvious that the vehicle has to abort the route and departure to the depot. The current route has moved to the individual, same procedure is repeated until the possible routes has been generated for number of customers and in this example have got three routes that are all relocated to the individual. It has not ensured that the individuals in the population yield the optimal solution to the problem. Based on the genetic operations the individuals in the populations are improved, for that different process should be provided. III. ALGORITHM DESIGN AND DEVELOPMENT A Genetic Algorithm (GA) has been proposed to solve the CVRP. However, different genetic operators applied to improve the efficiency of the CVRP of the population in each generation. In genetic algorithm, various population seeding techniques are applied to generate the initial population. Furthermore, the various population seeding techniques are altered based on the problem and applied with the enhanced GA. A. Selection Even though, the optimal solutions have been determined for each generation, due to the genetic operations there is a possibility of destroying that optimal solution in next generation [4,6,]. Consequently, the preservation of optimal solutions, has compulsory, thus the total cost of each individual in the population is determined w.r.t to the objective function [11,14,18]. The individuals those having least distance are selected as an elitist individual based on the elitist rate [4,5]. The elitist individuals are passed succeeding generation without any modification of additional genetic operations. B. Crossover The crossover process is completed through greedy crossover [4,5]. In crossover, choose any two random Parent individuals from the whole population. The leading customer in the parent individual is moved and Assign to the offspring individual current customer[4,5,]. Now estimate the position of the current customer in both the parent individuals and then detect the right side and left side customer to the current customer in both the parent individuals. Note that, if the position of current customer is beginning customer then the position of left customer to the current customer is last customer in the individual and if the position of current customer is last customer then the position of Right customer to the current customer is last customer. C. Mutation Ultimately, it is valuable to have a Mutation process, Generate two random locations of the customer in the individual, which is having same demand and then swap the customers in the offspring individual in their respective positions and vice versa. Succeeded by mutation process offspring individual to have been displaced to the population set, execute this process till the population makes the maximum population size. Evaluate the total cost for all the individuals from the whole population, it is mandatory to add the depot at the beginning and the end of the each route in the individuals while calculating the cost[4,5,16]. A. Ramalingam & K. Vivekanandan ijesird, Vol. I (V) November 2014/ 175

A generalized equation for calculating the cost of the routes present in all the individuals. However, the objective of CVRP is that to minimize the cost of the total cost of the routes present in the individuals with a single constraint the total demand of the route should not exceed the limit of the capacity of the vehicle[3,4,6]. The objective function has derived from the generalized equation and it is described below: The cost of all the routes are estimating, The solution cost of the problem / individuals are Estimating, and the objective function of the problem is,. Using these equations the fitness of each individual is calculated and the individual having least total cost is selected as best individual then analyzed with various performance factors. IV. POPULATION SEEDING TECHNIQUES In this section, a brief description on other population seeding techniques such as Random and Nearest Neighbor has been studied and both the techniques are compared with Ordered Distance Vector (ODV) based EV technique. A. Random Initialization Technique In this technique, randomly a finite set of individuals which are generated by choosing random adjacent cities is called a population. To improve the search space exploration an uniform random number generator has been used. A variety of random number generation techniques have been proposed such as quasi random, sobol random and uniform random sequence [6,7,8].The time taken to generate the initial population is less in random population seeding technique. B. Nearest Neighbor Technique Nearest neighbor (NN) tour construction heuristic is a common choice[4], in alternative for random population initialization, to construct the initial population. In NN technique, individuals in the population seeding are constructed that the gene y can be selected as adjacent gene for the gene x such that it would be the nearest unallocated gene of the individual of gene x. C. ODV based Equi-begin with Variable diversity (EV) Technique An effective population initialization technique are also used to initialize the initial population, since the starting customer is fixed in our proposal, subsequently The EV (Equi-begin with Variable diversity) based ODV (Ordered Distance Vector) population seeding technique based on the ODV matrix is used. In ODV the customers are organized based on the distance that is calculated by the permutation of customers [4,16]. The ODV of a is, Then, For each customer, the ODV generates corresponding least distance customers in sorted order and rank the customers based on the distance, then it will moved to the ODM (Ordered Division Matrix) that is given by matrix, A. Ramalingam & K. Vivekanandan ijesird, Vol. I (V) November 2014/ 176

BEST CONVERGENCE RATE International Journal of Engineering Science Invention Research & Development; Vol. I Issue V November 2014 a) Equi-begin ( ) The starting customer of the each individual is always same (i.e.) is fixed for all the individuals in the population. In our application, the starting customer of the individuals is fixed so we are applying this method.[15,16] b) Variable diversity ( ) The Next customer in the individual is added based on the value, is an integer that selected within the range of best adjacent ( value. The customer in the position of bax value is moved to the next customer location of the individual.[15,16] V. EXPERIMENTATION AND RESULT ANALYSIS All the implementations are carried out using MATLAB with CVRP benchmark datasets obtained from VRPLIB (http://neo.lcc.uma.es / vrp/ vrpinstances / capacitated vrp instances/). The CVPR instances that have been chosen for the experimentation are A-n39-k5, B-n50-k7, A-n60-k9, P-n70- k and P-n76-k5. For each technique, the executions are carried out for 25 times and the average of each case has been considered for experimental analyses. A. Convergence rate The convergence rate is one of the important performance analysis factor in GA, Convergence Rate of an individual refers to the supremacy of the individual with respect to the known optimal solution for the problem The Fig.2 shows that the Best Convergence Rate of the all the population seeding techniques. The EV technique has showed extraordinary performance for all the instances when compared to other population seeding techniques. The A-n39-k5 is exposed the higher convergence rate for ODV-EV Technique and P-n76-k5 has the lowest convergence rate in NN technique. The performance of random technique is unpredictable; it varies for all the instances which is shown in Table I. EV NN RANDOM 120 0 80 60 40 20 0 INSTANCES Fig 2. Best Convergence Solutions Generated by EV A. Ramalingam & K. Vivekanandan ijesird, Vol. I (V) November 2014/ 177

BEST ERROR RATE International Journal of Engineering Science Invention Research & Development; Vol. I Issue V November 2014 B. Error rate Best error rate is one of the main performance analysis factors in transportation, and logistics problems, which indirectly shows the quality of the individual in the population. The Error rate is indirectly proportional to convergence rate, Fig 3. shows the error rate of the best individuals in the population. The best individuals in random technique has showed lower performance in convergence rate, this reviles that the random technique has the highest error rate for the entire instance when compared with other techniques which is showed in Fig 3. NN technique has the next highest error rate then the random technique and the error rate increase with increase in number of customer for the corresponding instances. The EV technique has the lowest error rate for the entire instance which is shown in Table III C. Convergence Diversity The maintenance of a diverse solution in the population is required to ensure that the solution space is effectively searched, especially in the earlier stages of the optimization process. Population Diversity is considered as the primary reason for premature convergence. Fig. 4 shows the Convergence diversity of ODV-EV techniques reviled good result for most of the instances and the random technique exposed poor performance with respective to convergence diversity except for the instance B-n50-k7. The performance of NN technique is better when compared to the Random technique is shown in Table II. D. Computation time Fig.5 significantly proves that the computation time increases as increase with the problem instances, each technique has its own computation time for every problem instances. From the Fig. 5, it is clearly observable that the instance A-n39- k5 has the lowest computation time and P- n76-k5 has the highest computation time. The results of ODV-EV Technique with different CVRP instances is illustrates in Table III, the overall computation time of all the instances ranges from 39.812 to 78.473. The performance of ODV-EV with respective to computation time is better for most of the instances when compared to other two techniques and then the Nearest Neighbor technique showed decent computation time. 30 EV NN RANDOM 25 20 15 5 0 Fig 3. Best Error Rate Solutions Generated by EV INSTANCES A. Ramalingam & K. Vivekanandan ijesird, Vol. I (V) November 2014/ 178

TIME DIVERSITY International Journal of Engineering Science Invention Research & Development; Vol. I Issue V November 2014 0 EV NN RANDOM 90 80 70 60 50 40 30 20 0 A-n39-k5 B-n50-k7 A-n60-k9 P-n70-k P-n76-k5 INSTANCES Fig 4. Convergence Diversity Generated by EV 90 80 70 60 50 40 30 20 0 EV NN RANDOM A-n39-k5 B-n50-k7 A-n60-k9 P-n70-k P-n76-k5 INSTANCES Fig 5. Computation Time Generated by EV A. Ramalingam & K. Vivekanandan ijesird, Vol. I (V) November 2014/ 179

TABLE I. Result Analysis of CVRP using Random Technique Sl Instance Optimal Value Capacity Route Time (sec) Quality Solution Convergence Rate (%) Error Rate (%) Best Average Worst Best Average Worst Best Average Worst Diversity 1 A-n39-k5 822 0 5 39.812 864.000 11.031 1541.000 94.891 64.960 12.530 5.9 35.040 87.470 82.360 2 B-n50-k7 741 0 7 50.281 805.000 971.182 1448.000 91.363 68.936 4.588 8.637 31.064 95.4116 86.774 3 A-n60-k9 1354 0 9 62.729 1518.869 1664.535 2324.473 87.824 77.065 28.325 12.176 22.935 71.675 59.498 4 P-n70-k 827 135 72.628 909.836 1141.239 1546.759 89.984 62.003 12.968.016 37.997 87.032 77.016 5 P-n76-k5 627 280 5 78.473 783.817 858.841 1150.075 74.989 63.024 16.574 25.011 36.976 83.425 58.414 TABLE II.Result Analysis of CVRP using NN Technique Sl Instance Optimal Value Capacity Route Time (sec) Quality Solution Convergence Rate (%) Error Rate (%) Best Average Worst Best Average Worst Best Average Worst Diversity 1 A-n39-k5 822 0 5 39.795 861.935 1123.317 1467.531 95.142 63.343 21.468 4.858 36.657 78.532 73.674 p2 B-n50-k7 741 0 7 51.728 794.845 996.899 1476.287 92.733 65.466 0.771 7.267 34.534 99.229 91.963 3 A-n60-k9 1354 0 9 62.889 1446.878 1686.125 2461.748 93.140 75.471 18.187 6.860 24.529 81.813 74.953 4 P-n70-k 827 135 72.436 904.5 1118.560 1454.403 90.677 64.745 24.135 9.323 35.255 75.865 66.541 5 P-n76-k5 627 280 5 79.949 791.417 849.259 1176.169 73.777 64.552 12.413 26.223 35.448 87.586 61.363 TABLE III. Result Analysis of CVRP using ODV-EV Technique Sl Instance Optimal Value Capacity Rou te Time (sec) Quality Solution Convergence Rate (%) Error Rate (%) Best Average Worst Best Average worst Best Average Worst Diversity 1 A-n39-k5 822 0 5 39.850 849.3 1130.583 1532.444 96.703 62.459 13.571 3.297 37.541 86.429 83.132 2 B-n50-k7 741 0 7 51.198 764.923 08.546 1416.342 96.772 63.894 8.860 3.228 36.6 91.139 87.91 3 A-n60-k9 1354 0 9 63.083 1392.19 9 1671.927 2455.313 97.179 76.519 18.662 2.821 23.481 81.338 78.517 4 P-n70-k 827 135 72.784 849.332 11.660 1501.377 97.300 65.700 18.455 2.700 34.300 81.545 78.845 5 P-n76-k5 627 280 5 78.650 690.819 823.797 1216.637 89.822 68.613 5.959.178 31.387 94.041 83.863 A. Ramalingam & K. Vivekanandan ijesird, Vol. I (V) November 2014/ 180

VI. CONCLUSION In this paper, the Ordered Distance Vector (ODV) based Equi-begin with Variable diversity (EV) population seeding technique is compared with the other population seeding techniques such as Nearest neighbour and Random seeding technique. The CVRP has been chosen as the test bed and the experiments are performed on different sized CVRP benchmark datasets obtained from VRPLIB. The experimental results obtained using EV technique shows that the proposed EV technique gives the best convergence values obtained from various instances is found to be best for the most of the instances when compared with other techniques. The error rate is also reduced for the most of the instances in EV than the other two techniques with minimum computation time when compared to Random population seeding and Nearest Neighbour technique. We are extending the work also for other instances of VRP to compare and evaluate the performance of the proposed technique. REFERENCES [1] M. 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