Two Novel Heuristics Based on a New Density ... - Semantic Scholar
78 International Journal of Operations Research and Information Systems, 6(1), 78-90, January-March 2015
Two Novel Heuristics Based on a New Density Measure for Vehicle Routing Problem Abdesslem Layeb, Department of Information and Applications, University of Constantine II, Constantine, Algeria
ABSTRACT The vehicle routing problem (VRP) is a known optimization problem. The objective is to construct an optimal set of routes serving a number of customers where the demand of each customer is less than the vehicle’ capacity, and each customer is visited exactly once like in TSP problem. The purpose of this paper is to present new deterministic heuristic and its stochastic version for solving the vehicle routing problem. The proposed algorithms are inspired from the density heuristic of knapsack problem. The proposed heuristic is based on four steps. In the first step a density matrix (demand/distance) is constructed by using given equations. In the second step, a giant tour is constructed by using the density matrix; the customer with highest density is firstly visited, the process is repeated until all customers will be visited. In the third phase, the split procedure is applied to this giant tour in order to get feasible routes subject to vehicles capacities. Finally, each route is improved by the application of the nearest neighbor heuristic. The results of the experiment indicate that the proposed heuristic is better than the nearest neighbor heuristic for VRP. Moreover, the proposed algorithm can easily be used to generate initial solutions for a wide variety of VRP algorithms. Keywords:
Constructive Heuristics, Nearest Neighbor, Operational Research, Optimization Problems, Vehicle Routing Problem (VRP)
INTRODUCTION The Vehicle Routing Problem (VRP), proposed by Dantzig and Ramser (1959), is an important research area in operations research given its importance in logistic planning. The use of route planning and scheduling algorithms can help to save time and shipping costs. VRP is an extension of the well-known TSP problem
with several vehicles, and each customer has a demand less than the vehicle capacity. All the vehicles start from one or several depots and finish their travels at the same start depot after visiting a set of customers (Figure 1). The aim is to find the best route for each vehicle in order to minimize the total distance traveled by all the vehicles, subject to a set of constraints like vehicle capacity, time, pickup and delivery, etc.
DOI: 10.4018/ijoris.2015010106 Copyright © 2015, IGI Global. Copying or distributing in print or electronic forms without written permission of IGI Global is prohibited.
International Journal of Operations Research and Information Systems, 6(1), 78-90, January-March 2015 79
Figure 1. A simple VRP example
Although the problem appears to be easy to state, it belongs to NP-hard problems category, and needs a huge time and space to resolve, especially with complex VRP instances (Toth & Vigo, 2002; Samanta & Jha, 2011). There are several variants of VRP. Each one of these variants differs from another by constraints or features that have to be taken into account. The basic variant of VRP problem is the Capacitated Vehicle Routing Problem (CVRP), in which only the capacity constraint is considered (Laporte & Semet 2001; Charikar, Khuller & Raghavachari, 2001). In the CVRP, there is only one depot from which all the vehicles stars. Each vehicle has a limited capacity cap. There are a set of n customers and each customer has a certain demand d < cap. Each customer is served by one vehicle, and the total demand carried by one vehicle is at most cap. Consequently, the solution of CVRP is a set of routes with the lowest cost. In order to solve the CVRP, several methods were proposed that we can classify in tow main classes exact algorithms and approximate algorithms. The exact algorithms are able to find the exact solution of the CVRP. The most popular algorithms of this class are based on the Branch & Bound (Toth & Vigo, 2001), Branch & Cut (Lysgaard, Letchford & Eglese, 2004), column
generation (Baldacci & Mingozzi, 2009), and set partitioning (Baldacci, Christofides, & Mingozzi, 2008). Unfortunately, the exact algorithms have an exponential complexity, and they are impracticable for large CVRP instances (Samanta & Jha, 2011). On the other hand, the approximate algorithms constitute an alternative way to find near optimal solutions in reasonable time compared to the exact algorithms. The approximate algorithms can be divided into three subclasses: constructive heuristics, improvement heuristics and metaheuristics. The constructive heuristics gradually build a feasible solution. The main features of the constructive heuristics are the simplicity and the rapidity to find quite acceptable solutions. The most known and used heuristics for CVRP are the savings heuristic of Clarke and Wright (Clarke & Wright, 1964), the sweep algorithm (Gillett & Miller, 1974), the insertion heuristic (Campbell & Savelsbergh, 2004), the petal algorithm (Ryan, Hjorring & Glover, 1993), and the nearest neighbor algorithm. The main disadvantage of heuristics is the fact that they do not guarantee the optimality, and sometimes the solutions are too far from the best solutions. In order to improve the quality of the heuristic solutions, improvement heuristics are used (Kinderwater & Savelsbergh,
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Two Novel Heuristics Based on a New Density Measure for Vehicle Routing Problem Abdesslem Layeb, Department of Information and Applications, University of Constantine II, Constantine, Algeria
ABSTRACT The vehicle routing problem (VRP) is a known optimization problem. The objective is to construct an optimal set of routes serving a number of customers where the demand of each customer is less than the vehicle’ capacity, and each customer is visited exactly once like in TSP problem. The purpose of this paper is to present new deterministic heuristic and its stochastic version for solving the vehicle routing problem. The proposed algorithms are inspired from the density heuristic of knapsack problem. The proposed heuristic is based on four steps. In the first step a density matrix (demand/distance) is constructed by using given equations. In the second step, a giant tour is constructed by using the density matrix; the customer with highest density is firstly visited, the process is repeated until all customers will be visited. In the third phase, the split procedure is applied to this giant tour in order to get feasible routes subject to vehicles capacities. Finally, each route is improved by the application of the nearest neighbor heuristic. The results of the experiment indicate that the proposed heuristic is better than the nearest neighbor heuristic for VRP. Moreover, the proposed algorithm can easily be used to generate initial solutions for a wide variety of VRP algorithms. Keywords:
Constructive Heuristics, Nearest Neighbor, Operational Research, Optimization Problems, Vehicle Routing Problem (VRP)
INTRODUCTION The Vehicle Routing Problem (VRP), proposed by Dantzig and Ramser (1959), is an important research area in operations research given its importance in logistic planning. The use of route planning and scheduling algorithms can help to save time and shipping costs. VRP is an extension of the well-known TSP problem
with several vehicles, and each customer has a demand less than the vehicle capacity. All the vehicles start from one or several depots and finish their travels at the same start depot after visiting a set of customers (Figure 1). The aim is to find the best route for each vehicle in order to minimize the total distance traveled by all the vehicles, subject to a set of constraints like vehicle capacity, time, pickup and delivery, etc.
DOI: 10.4018/ijoris.2015010106 Copyright © 2015, IGI Global. Copying or distributing in print or electronic forms without written permission of IGI Global is prohibited.
International Journal of Operations Research and Information Systems, 6(1), 78-90, January-March 2015 79
Figure 1. A simple VRP example
Although the problem appears to be easy to state, it belongs to NP-hard problems category, and needs a huge time and space to resolve, especially with complex VRP instances (Toth & Vigo, 2002; Samanta & Jha, 2011). There are several variants of VRP. Each one of these variants differs from another by constraints or features that have to be taken into account. The basic variant of VRP problem is the Capacitated Vehicle Routing Problem (CVRP), in which only the capacity constraint is considered (Laporte & Semet 2001; Charikar, Khuller & Raghavachari, 2001). In the CVRP, there is only one depot from which all the vehicles stars. Each vehicle has a limited capacity cap. There are a set of n customers and each customer has a certain demand d < cap. Each customer is served by one vehicle, and the total demand carried by one vehicle is at most cap. Consequently, the solution of CVRP is a set of routes with the lowest cost. In order to solve the CVRP, several methods were proposed that we can classify in tow main classes exact algorithms and approximate algorithms. The exact algorithms are able to find the exact solution of the CVRP. The most popular algorithms of this class are based on the Branch & Bound (Toth & Vigo, 2001), Branch & Cut (Lysgaard, Letchford & Eglese, 2004), column
generation (Baldacci & Mingozzi, 2009), and set partitioning (Baldacci, Christofides, & Mingozzi, 2008). Unfortunately, the exact algorithms have an exponential complexity, and they are impracticable for large CVRP instances (Samanta & Jha, 2011). On the other hand, the approximate algorithms constitute an alternative way to find near optimal solutions in reasonable time compared to the exact algorithms. The approximate algorithms can be divided into three subclasses: constructive heuristics, improvement heuristics and metaheuristics. The constructive heuristics gradually build a feasible solution. The main features of the constructive heuristics are the simplicity and the rapidity to find quite acceptable solutions. The most known and used heuristics for CVRP are the savings heuristic of Clarke and Wright (Clarke & Wright, 1964), the sweep algorithm (Gillett & Miller, 1974), the insertion heuristic (Campbell & Savelsbergh, 2004), the petal algorithm (Ryan, Hjorring & Glover, 1993), and the nearest neighbor algorithm. The main disadvantage of heuristics is the fact that they do not guarantee the optimality, and sometimes the solutions are too far from the best solutions. In order to improve the quality of the heuristic solutions, improvement heuristics are used (Kinderwater & Savelsbergh,
Copyright © 2015, IGI Global. Copying or distributing in print or electronic forms without written permission of IGI Global is prohibited.
11 more pages are available in the full version of this document, which may be purchased using the "Add to Cart" button on the product's webpage: www.igi-global.com/article/two-novel-heuristics-based-on-anew-density-measure-for-vehicle-routingproblem/124763?camid=4v1
This title is available in InfoSci-Journals, InfoSci-Journal Disciplines Business, Administration, and Management. Recommend this product to your librarian: www.igi-global.com/e-resources/libraryrecommendation/?id=2
Related Content Examining Double Marginalization Effect for Innovative Product Supply Chain Wenjing Shen (2012). International Journal of Operations Research and Information Systems (pp. 37-52).
www.igi-global.com/article/examining-double-marginalization-effectinnovative/62257?camid=4v1a Model Development and Hypotheses (2015). Business Process Standardization: A Multi-Methodological Analysis of Drivers and Consequences (pp. 156-196).
www.igi-global.com/chapter/model-development-andhypotheses/121931?camid=4v1a Sequential Test for Arbitrary Ratio of Mean Times Between Failures Yefim H. Michlin, Dov Ingman and Yoram Dayan (2011). International Journal of Operations Research and Information Systems (pp. 66-81).
www.igi-global.com/article/sequential-test-arbitrary-ratiomean/50561?camid=4v1a
Global Collaborative Business Bhuvan Unhelkar, Abbass Ghanbary and Houman Younessi (2010). Collaborative Business Process Engineering and Global Organizations: Frameworks for Service Integration (pp. 65-97).
www.igi-global.com/chapter/global-collaborativebusiness/36533?camid=4v1a
