Local search and suffix tree for car-sequencing problem with colors

doi:10.1016/j.ejor.2007.04.051

European Journal of Operational Research

Volume 191, Issue 3, 16 December 2008, Pages 972-980

https://doi.org/10.1016/j.ejor.2007.04.051 Get rights and content

Abstract

This paper describes an approximation solution method for the car sequencing problem with colors. Firstly, we study the optimality of problems with a single ratio constraint. This study also introduces a data structure for efficient calculation of the penalties related to ratio constraints. We describe the constructive greedy algorithm and variable neighborhood search adjusted for the problem with colors. Tabu metaheuristic is used to improve the results obtained by VNS. We then represent the cars with their constraints as letters over an alphabet and apply the algorithm to spell the motifs in order to improve the number of batch colors without decreasing the costs associated to the set of ratio constraints. The algorithm achieves 19 out of the 64 best results for instance sets A and B. These instances are the reference instances for Challenge ROADEF.

Introduction

We describe here the method applied to approximatively solve the car-sequencing problem which was the subject of the ROADEF Challenge 2005. This type of problem is thoroughly studied from both the theoretical and practical standpoint. Recently, Gent [3] proved that the problem is NP-complete by reducing Hamiltonian path problem to the car-sequencing problem. Thus, approximation methods are mostly used to study and solve the problem [1], [2], [3], [4], [5], [7]. This work develops the known methods further and combines them to solve the car-sequencing problem with colors. We also propose a new approach to improve the solution for the problem where the number of batch colors is second or third objective.

Firstly, we discuss the optimality of the problem with only one ratio constraint. We develop an optimal algorithm for this particular case. Since the approximation method that we present could be divided in three different stages we describe it in three sections. The second section describes the greedy algorithms. The constructive greedy algorithms are initially used to build the starting solution. Among many tested greedy strategies we choose just three to apply on every instance and continue with the best solution. Local search with variable neighborhood, described in the fourth section, mostly uses the variations of the well known approaches cited in literature for solving this type of problem without colors. However, we develop some problem specific neighborhoods able to efficiently improve the solution at hand. Probably the most interesting and innovative part of the method is described in the Section 5. We describe how to use a particular computational biology technique [6] to deal with the instances that have the number of batch colors as the second or third objective. Section 6 gives the most interesting computational results for the instances used either for developing or tuning the method, or for the evaluation of the finalists. Finally, we discuss a way to improve this method and possibly combine it with other approximation algorithms.

Section snippets

One ratio constraint optimality

The first thing we would like to know about an optimization problem is range of the bounds for the optimal solution. The calculation of these bounds often helps us to get better insight into the structure of the problem and ultimately to devise better algorithms for its solutions. The fact that the car-sequencing problem is NP-complete leaves us little hope for achieving the optimal value for the general problem. We will therefore consider optimality for the simplified car-sequencing problems

Constructive greedy algorithms

In the current literature is reported (e.g. [4]) that the greedy methods often provide a good quality solution for this type of problem. We develop several kinds of greedy algorithms to find the initial solution quickly. Mainly, we modify the heuristics from [4], [5] by including the optimization objective and constraints dealing with batches of colors. It is worth mentioning that all these heuristics are improved here by better estimation of the utilization percentage [1] for every ratio

Notations

In order to explain the basics of our algorithms we use the following notation that originated from [4] in order to describe different phases of the method:

•
A sequence, noted $π = 〈 c_{i_{1}}, c_{i_{2}}, \dots, c_{i_{n}} 〉$ is a sequence of n cars and its length is denoted ∣π∣.
•
The concatenation of two sequences π₁ and π₂, noted π₁ · π₂, is a new sequence composed of the cars from π₁ followed by the cars from π₂.

Variable neighborhood search

The neighborhoods used to implement the local search are natural and already applied for the classic type of

Suffix trees and the colors for the car-sequencing problem

In this section we present one efficient method for improving the cost associated with color paint batches without deteriorating the other costs. Thus, in this section we are dealing only with the instances where the set of higher priority level ratio constraints is not empty and the objective associated with it is the most important. Consider the problem where we want to optimize with respect to three objectives without compensation between them. They follow in order of their importance

Implementation and computational results

The method is implemented in C++ using the STL library. There is still room for code optimization especially for the type of computational environment used for evaluation.

Algorithm 3 Approximation algorithm for car-sequencing pb

1: choose the best greedy result
2: while time ⩽ MAX − TIME do
3: local search enhanced with Tabu search
4: end while
5: if the instance is not (RAF_…) type then
6: ImproveColors(s)
7: end if

The method is tested and tuned on the 64 instances (instance sets A and B) provided by Renault. All results are grouped in three different tables

Conclusion

It was expected that these kinds of algorithms show their strength on this particular type of car-sequencing problem as they are “state of the art” algorithms for the classic version of the problem. The particular strength of the method is its robustness and speed. For some instances, the application reaches a solution very close to the best known within a few seconds. It is also important to mention that it produces a relatively big number of good but structurally different solutions. This is

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