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A Bee Algorithm for the Multi-Objective Vehicle Routing Problem

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A Bee Algorithm for the Multi-Objective Vehicle Routing Problem Bee-route: A Bee Algorithm for the Multi-objective Vehicle Routing Problem Jamila Sassi a, Ines Alaya b and Moncef Tagina COSMOS-ENSI, National School of Computer Sciences, University of Manouba 2010, Tunisia Keywords: Combinatorial Multi-objective Optimization, Artificial Bee Colony Algorithm (ABC), Multi-objective Vehicle Routing Problem with Time Window (VRPTW). Abstract: The vehicle routing problem has attracted a lot of interest during many decades because of its wide range of applications in real life problems. This paper aims to test the efficiency and capability of bee colony optimization for this kind of problem. We present a Bee-route algorithm: a multi-objective artificial Bee Colony algorithm for the Vehicle Routing Problem with Time Windows. We have performed our experiments on well known benchmarks in the literature to compare our proposed algorithm results with other state-of-the- art algorithms. 1 INTRODUCTION The VRPTW aims to determine optimal routes for a number of vehicles when serving a set of customers The vehicle routing problem (VRP) is a part of a within predefined time intervals (the time windows). large application domain. It could be applied to var- The main formulation of the VRPTW was proposed in ious fields such as transportation, network connec- 1987 by (Solomon, 1987) where the time constraints tion, health management techniques, recycling meth- must be respected by each vehicle. In fact, it has been ods, etc. The VRP aims at finding an optimal set of shown that the classic VRP problem is NP-hard, and routes for a number of vehicles, initially located at a this result could be extended to the VRPTW. While depot, to serve a given set of customers. Each vehi- it is quite possible to determine an optimal solution cle leaving the depot returns to the initial depart af- for small instances, it quickly becomes unfeasible for ter having completed its tour in a certain time slot. medium or large instances. The cumulative demand of customers visited by a ve- Given the complexity of the VRPTW, many res- hicle must not exceed its capacity. We introduce in olution methods have been developed. In (Lim and this article a multi-objective artificial Bee Colony al- Zhang, 2007), proposed a two-stage algorithm for gorithm for Vehicle Routing Problem with Time Win- the VRPTW. The authors extended the VRPTW algo- dows (VRPTW) called Bee-route algorithm. In the rithm for m-VRPTW. Vehicle routing problem with multi-objective VRPTW, we consider two objectives both time window and limited number of vehicles. to be optimized: the first is to minimize the number of The m-VRPTW is an useful extension of VRPTW vehicles used to deliver the demand of customers and problem in real applications. The algorithm first mini- the second objective is to minimize the total distance mizes the number of vehicles with an ejection pool to of the routes. hold temporarily unserved customers which enables Approaches proposed in recent decades have been the algorithm to go through the infeasible solution characterized by prioritizing customers, and for time space. (Ghoseiri and Ghannadpour, 2010) have pro- optimization, reaching a reasonable time frames. The posed a goal programming approach for the formula- problem of Vehicle Routing Problems with Time tion of the VRPTW and an adapted efficient genetic Windows (VRPTW) constitutes a generalization of algorithm for it in which the decision maker speci- the VRP in which each customer has a window of fies optimistic aspiration levels to the objectives (total time within this time the customer must be served. distance and number of vehicles) and the deviations from those aspirations are minimized. (Tsung-Che a https://orcid.org/0000-0002-9857-6467 and Wei-Huai, 2014) have presented an evolutionary b https://orcid.org/0000-0002-7037-3326 algorithm to find a set of Pareto optimal solutions. 307 Sassi, J., Alaya, I. and Tagina, M. Bee-route: A Bee Algorithm for the Multi-objective Vehicle Routing Problem. DOI: 10.5220/0009831303070318 In Proceedings of the 15th International Conference on Software Technologies (ICSOFT 2020), pages 307-318 ISBN: 978-989-758-443-5 Copyright c 2020 by SCITEPRESS – Science and Technology Publications, Lda. All rights reserved ICSOFT 2020 - 15th International Conference on Software Technologies They incorporate problem specific knowledge into the new algorithm that solves the multi-objective vehi- genetic operators. The objectives are to minimize the cle routing problem with time windows (VRPTW) number of vehicles and the total distance simultane- with the artificial bee colony optimization. Since ously. we consider the VRPTW as a multi-objective prob- In recent years, swarm intelligence has also at- lem, in which we have to minimize the number of tracted the interest of many researches from several vehicles and the total distance, optimizing one ob- scientific areas (Alzaqebah et al., 2018), (Alaya I., jective usually leads to degrading the other objec- 2003), (Zouari W., 2017) : biology, ethology, com- tive. The conventional single-objective approaches puter science , etc. Many researchers have success- for VRPTW, and even some approaches that claims fully proposed Artificial Bee Colony (ABC) algo- to be multi-objective, are unable to explore this con- rithm to quickly find good quality solutions to prob- flicting behaviour of objectives and return a single lems. An ABC has been proposed by (Ghaleini E. N., best solution. Whereas for multi-objective problems, 2018) in the network application, to optimize weight there is a set of optimal solutions and not a single and biases of artificial neural networks to receive best solution, called the non-dominated solutions or higher levels of accuracy and performance prediction. the Pareto optimal solutions. A feasible solution is They combined artificial bee colony and neural net- non-dominated if it does not exist another feasible so- work for the specific purpose of approximating the lution better than the current one in some objective safety factor of retaining walls. (Alzaqebah and Sana, function without worsening other objective function. 2016) investigate the use of bee algorithms (BA) to Our algorithm approximates this set of Pareto opti- address the VRPTW and identify their strengths and mal solutions, since it is based on a metaheuristic ap- weaknesses. The strength of BA is that the algorithm proach, considering both of the objectives at the same has both global exploration performed by Scout bees time. In fact, in the proposed algorithm, we propose and local exploitation performed by recruiter bees. to employ a weighted approach called Bee-route al- Then again the weakness of the algorithm is that it gorithm: a multi-objective artificial Bee Colony algo- is parameter dependent, so each instance may require rithm for Vehicle Routing Problem with Time Win- different parameter values. The proposed work in dows. This approach will solve the problem consid- (Alzaqebah and Sana, 2016) involves the recruitment ering the two objectives simultaneously relatively to of additional Employed bees and applies local search different weight vectors. to a set of elite solutions, which are considered the The remaining parts of the paper are organized as most promising solutions in the search space. (Shao- follows: In section 2, we will describe the formula- qiang and Linjie, 2016) has been presented a real tion of VRPTW. Our approach will be defined with western-style food delivery problem in Dalian city, more details in section 3. In section 4 we will present China, which can be described as a vehicle routing our experiments and results. Finally, we will draw the problem with time windows. An integer linear model conclusion and we will provide further research per- for the problem is developed, and an improved arti- spectives. ficial bee colony algorithm, which possesses a new strategy called an adaptive strategy, a crossover op- eration, and a mutation operation, which are pro- 2 FORMULATION OF THE posed for the problem. (Yao, 2017) propose an im- VRPTW proved artificial bee colony (IABC) algorithm for the VRPTW.The ABC algorithm is improved by a local In our approach, we aim at the vehicle routing prob- optimization based on a crossover operation borrowed lem with time window (VRPTW). This problem is from the genetic algorithm and a scanning strategy. a variant of the well-kown VRP complemented by (Alzaqebah and Sana, 2016) proposed a Modified a time window in which each customer should be ABC algorithm to improve the solution quality of the served. Formally, we present the problem description original ABC. In the Modified ABC a list of aban- of the VRPTW as follows: doned solutions is used by the scout bees to memo- The VRPTW is defined by a directed graph G rise the abandoned solutions. This algorithm using a = (V,E), where V 2 f0;1;:::;ng is the node set and memory by scout bees. They can memorise the aban- E 2 f(i; j) : 0 ≤ i; j ≤ n;i 6= jg is the arc set. Node doned solutions and select one of these solutions to 0 is the depot and N = Vnf0g denotes the set of cus- be replaced by a new generated solution. They re- tomers. For customer i, a vehicle may arrive before placed all the abandoned solutions by randomly gen- the start time of window e and wait until e to start the erated solutions as in the original ABC algorithm. i i service, but it may not arrive after li the end of time. In this paper, we are interested in introducing a At the same time, each customer can only call for 308 Bee-route: A Bee Algorithm for the Multi-objective Vehicle Routing Problem one vehicle but the same vehicle is allowed to serve Constraints (3)restrict the assignment of each cus- more than one customer.
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