EODVGA : An Enhanced ODV Based Genetic Algorithm for Multi-Depot Vehicle Routing Problem

Multi-Depot Vehicle Routing Problem (MDVRP) is a familiar combinative optimization problem that simultaneously determines the direction for different vehicles from over one depot to a collection of consumers. Researchers have suggested variety of meta-heuristic and heuristic algorithms to elucidate MDVRP, but none of the existing technique has improved the fitness of the solution at the time of initial population generation. This motivates to propose an enhanced ODV based population initialization for Genetic Algorithm (GA) to solve MDVRP effectively. The Ordered Distance Vector (ODV) based population seeding method is a current and effective population initialization method for Genetic Algorithm to produce an early population with quality, individual diversity and randomness. In the proposed model, the customers are first grouped based on distance to their nearest depots and then routes are scheduled and optimized using enhanced ODV based GA. The experiments are performed based on different types of instances of Cordeau. From the experimental results, it is very clear that the proposed technique outperforms the existing techniques in terms of convergence rate, error rate and convergence diversity


Related Work
Genetic Algorithm (GA) is one of the methods of Metaheuristic algorithm.Meta-heuristic algorithm finds solutions in less computational time for routing problems.Variable Neighbourhood Search Algorithm for the Multi Depot Heterogeneous Vehicle Routing Problem with Time Windows is one of the variants of heuristic and meta-heuristic methods presented by Yingcheng Xu et al., [2].Multi-Phase Meta-Heuristic for Multi-Depots Vehicle Routing Problem uses extremal optimization (EO) for re-adjusting the solutions.For every cluster it implements the Shuffled Frog Leaping Algorithm (SFLA).It was proposed by Jianping Luo, Xia Li et al., [3].For solving min-max Multi-Depot Vehicle Routing Problem K. Venkata Narasimha et al., [4] proposed Ant colony optimization technique.Modular heuristic algorithm to solve Fleet-sizing for multi-depot and periodic vehicle routing problem was proposed by Alireza Rahimi Vahed., [5].Under capacity and route length constraint for solving multi-depot vehicle routing problem C. Contardo, R. Martinelli et al., [6] proposed variable fixing, column-and-cut generation.For solving large-scale multi-depot vehicle routing problem W. Tu et al., [7] proposed Bi-level Voronoi diagram-based metaheuristic.
There are various methods namely Tabu search [8,9], Simulated annealing [10,11], Genetic approach [12], Ant colony optimization [13,14,20], Particle swarm optimization [15,16].Genetic Algorithm (GA) takes less computation time and results in reasonable and optimal solutions when compared with further meta-heuristic algorithms.Because of these things still Genetic Algorithm (GA) is used for solving NP hard problems [21,22], Vehicle routing problem, Travelling salesman person problem.Still today there exists no solution for determining how Genetic Algorithm influences the process of searching upcoming variants, different aspects setting and operator of GA.Ordered Distance Vector (ODV) [17] is a recent population seeding and effective population initialization method for Genetic Algorithm.

Ordered Distance Vector (ODV) Based Genetic Algorithm
Genetic algorithm (GAs) is a seeking procedure used to discover sufficient results for optimization.It has been ended up being effective at seeking ideal arrangement among a substantial and complex examination space in a versatile way.The need of it is to locate the best arrangement in the substantial search space which is the gathering of every doable arrangement among which the coveted arrangement re-sides.The Ordered Distance Vector (ODV) based population seeding system is a successful population initialization strategy for GA to produce an underlying population with irregularity, quality and individual assorted variety.It has three sorts of population initialization strategies; they are Vari-begin with Variable diversity (VV), Equi-begin with Variable diversity (EV), Vari-begin with Equal diversity (VE).In our proposed work Equi-begin with Variable diversity (EV) is used to solve MDVRP.

ODV based EV technique
The effective population initialization technique is used to initialize the initial population in Genetic Algorithm.
Since the starting city is fixed in our work, we are using EV (Equi-begin with Variable diversity) based ODV (Ordered Distance Vector) population seeding technique which is built on the ODV Matrix (ODVM).
Ordered Distance Vector (ODV): ODV based population seeding system produces an arrangement of change of "n" cities utilizing the Ordered Distance Vector matrix.In every permutation, the categorization of the cities is picked in such a way where the sum of distance among the cities is close least.By the permutation of cities, the distance is computed and based on it the cities are sorted here.The ODV of is as appeared in the Equation and the condition for the distance between the city and the following cities are given.request and rank the cities on the basis of distance, at that point it will be moved to the ODM (Ordered Division Matrix) that is given by n(n − 1) matrix as given.

Equi-begin and Vari-diversity (EV) Equi-begin (E):
The starting city of the each individual is always same here.So remains stable for all the individuals in the population.In VRP the starting city of the individuals is fixed (depot) so here this method is applied.

Variable diversity (V):
The succeeding city in the individual is added based on the value, is a selected integer whose range is within value.The city in the position of value is progressed to the succeeding city location of the individual.
The population is generated using Equi-begin with Variable diversity (EV), the starting city of each individual is fixed and based on the value and the succeeding city of the individual is chosen and added.The individuals in the population have high permutation of cities and the time complexity can be reduced.During initialization the number of maximum individuals in the population is as given, ( Where ( ) total amount of individuals in the population, best adjacent value and total cities.EV method generates the initial population as given, 4 generated, using that best adjacent (ba) value the next customer is visited, these process is done for until the service is done for all the customers.

Experimentation and Result Analysis
The enhanced ODV EV population initialization technique based genetic algorithm is implemented in MATLAB 2011b.The experiments are performed different types of Cordeau"s Instances (p01, p02, p03, p04, po7, p15, p18, pr05, pr06, pr10) obtained from (http://neo.lcc.uma.es/vrp/vrp-instances).The significance of the proposed techniques is examined by the performance factors such as Convergence rate, Error rate and Convergence diversity.Convergence rate: Convergence rate of an individual of a population set is defined as the percentage of fitness value obtained by the individual according to the optimal fitness value given below,

( )
Error rate: Error rate of an individual of a population set is defined as the percentage of difference between fitness obtained by the individual and optimal fitness value given below,

Convergence diversity:
The convergence diversity of the population is the difference between the convergence best rate and convergence error rate.This factor shows that diversity among the individuals in the population.It is calculated by the below shown equation.

Result analysis
The result analysis of the proposed enhanced ODV EV technique is described in this section and comparison of ODV with other state of the art methods.The proposed enhanced ODV-EV based Genetic Algorithm is compared with other state of the art methods [18,5,19].There are three performance factors used to examine the significance of the proposed techniques and they are Convergence rate, Error Rate and Convergence Diversity.

Convergence Rate
Convergence rate is commonly used for technical analysis.It indicates the series of convergence between the various instances.Different types of Cordeau"s Instances (p01, p02, p03, p04, po7, p09, p10, p11, p12, p16, p17, p19, p10, pr10) are taken.For convergence rate (%) consider the Table 1 and Table 2 Convergence Diversity ODV EV technique produces low convergence when compare to other existing techniques.Hence our proposed technique is proven to be efficient.To overcome the premature convergence problem convergence diversity is an important factor which illustrates the different range between the individuals.This shown in Table 6.5 and ODV-EV is efficient as per computational results.

Figure 1 .
Figure 1.Multi-Depot Vehicle Routing Problem ( ) distance between the and Ordered Division Matrix (ODM) : For every city, the ODV creates relating least distance cities in arranged EAI Endorsed Transactions on Scalable Information Systems 03 2019 -06 2019 | Volume 6 | Issue 21 | e8

] 4 .
Proposed System4.1 ODV based EV for MDVRPThe Ordered Distance Vector (ODV) Equi-begin and variable diversity (EV) population seeding technique based on GA is proposed to solve MDVRP.The proposed algorithm consists of two stages first; an Order Distance Matrix (ODM) will be generated from the Distance Matrix (DM) and followed by generation of the initial population based on the ODM.The second stage of the technique can be performed by EV, VE and VV depending on the type of population seeding technique to be chosen.In our proposal we are using Equi-begin with Variable diversity (EV) technique.The algorithm for MDVRP using ODV based EV techniques as shown below.Algorithm for MDVRP using ODV-EVStep1: Initialization of no_cities, gen, pop_size, Q, max_gen, len, ba Step2: Iterating Step1 to 6 until Step3: Assign the distance matrix into temp.Distance matrix Step 4: Iterating Step 5 to 7 until Step 5: Customer with minimum distance from Sth customer is identified and moved to ODM * + Step 6: The customer with min distance has changed as maximum * + * + Step 7: Incrementing Step 8: Completed current customer and iterating for next customer S Step 9: Initialization the initial customer Step10: Iterate step 19 to 20 until Step 11: Iterate through Step 18 until Step 12: Initialize the initial customer and total demand Step 13: Random number is generated ( ), based on best adjacent (ba) value Step 14: Obtaining the next customer from ODM using current customer Step 15: Checking next customer belongs to individuals Step 16: Obtaining the demand of the customer and then added to total demand Step 17: Iterate Step 18 until total demand is less than are equal to Q Else goto Step 11 Step 18: Incrementing the length of individual and initializing the next customers to the individuals Step 19: Move the current individual of the population Step 20: Increment the size Step 21: End The above shown algorithm is ODV-EV technique for MDVRP.The flow of algorithm is, first the Initialization is done for number of cities, generation, capacity, best adjacent and population size.Then distance matrix is assigned to temporary distance matrix then distance vector matrix is created by using temporary distance matrix.Customer with minimum distance from size of customer is identified and moved to ODM.The minimum distance customer is visited and then random ba value is EAI Endorsed Transactions on Scalable Information Systems 03 2019 -06 2019 | Volume 6 | Issue 21 | e8

Table 2 .
. The proposed ODV technique system has high convergence for more number of instances.This shows ODV EV technique is best when comparing to other state of the art methods.The fig.2shows the performance of different technique w.r.t best convergence rate and fig.3shows the performance of different technique w.r.t worst convergence rate.Experimental results w.r.tError RateDifferent types of Cordeau"s Instances are chooses, now the instances are evaluated for the proposed technique and compared with the existing techniques.As per the computational results, the error rate for the ODV-EV technique is less when compare to state of art methods.This is shown in Table3.

Table 4 .
Experiment results w.r.t Performance of different techniques w.r.t Convergence diversity (%)6.Conclusion and Future WorkEnhanced Ordered Distance Vector (ODV) based Genetic Algorithm (GA) is proposed to solve Multi-Depot Vehicle Routing Problem (MDVRP).For generating an initial population with individual diversity, randomness and quality, the ODV based population seeding technique is an active and recent population initialization method for GA and ODV based GA improves the fitness of the solution at the time of population initialization itself.The performance of the proposed technique is compared with state of the art methods and the experimental analysis is done through based performance factors such as convergence rate, error rate and convergence diversity.The values obtain from various instances shows that our proposed ODV EV population seeding technique based genetic algorithm solves MDVRP effectively than other methods.In future these ODV EV work can be extended to solve the others variants of vehicle routing problem in efficient manner.