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A multi-start dynasearch algorithm for the time dependent single-machine total weighted tardiness scheduling problem

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Abstract

We extend the dynasearch technique, recently proposed by Congram et al., in the context of time-dependent combinatorial optimization problems. As an application we consider a general time-dependent (idleness) version of the well known single-machine total weighted tardiness scheduling problem, in which the processing time of a job depends on its starting time of execution. We develop a multi-start local search algorithm and present experimental results on several types of instances showing the superiority of the dynasearch neighborhood over the traditional one.

Introduction

In this paper we evaluate the performance of dynasearch in the context of time-dependent scheduling. Dynasearch is a recently proposed neighborhood search technique [6] that allows a series of moves to be performed at each iteration of a local search algorithm, generating in that way an exponential size neighborhood. In order to efficiently explore such a neighborhood, dynasearch uses dynamic programming. Congram et al. applied dynasearch to the classical single machine total weighted tardiness problem (1∥∑wjTj) [6] and compared the quality of the obtained solutions with traditional multi-start and iterated descent algorithms. The obtained computational results were very encouraging in the case where dynasearch was applied inside an iterated local search algorithm when compared to classical iterated descent algorithms. However, the application of dynasearch in the case of a multi-start algorithm gave marginally better results than the classical multi-start methods. It is then natural to ask if dynasearch is not so appropriate for multi-start local search algorithms. In this work, we show that this is not true. More precisely, we study the time dependent version of the single machine total weighted tardiness problem, and we present computation results showing that multi-start dynasearch clearly dominates the classical multi-start local search algorithms.

In scheduling theory there have been an increasing interest, in the last few years, for scheduling problems with time-dependent processing times [2]. In this paper we consider a general time-dependent version of the well known single-machine total weighted tardiness scheduling problem. The problem can be stated as follows. We are given a set of n jobs, each job j has a due date dj and a positive weight wj. The processing time fj(t) of each job j depends on its starting time of execution t and is given by a function fj. We shall denote fj(t) by pjt. So, if a job j immediately starts after a job i, its duration is pjCi, where Ci represents the completion time of job i. We shall consider only the idleness version (it means that we do not allow idle times between jobs, i.e. each job starts immediately after the previous one has terminated, and therefore the solution we seek is a permutation of the jobs), and consider that all values are integer ones.

Since we make no assumptions on the functions fj, this model captures a wide range of practical applications. A first example of applications is when the availability of the resources (e.g. processing power) vary (e.g. in a monotone or cyclic way) over time; think for example at the load of a computer network. A second example of applications is when any delay in the execution of a job may lead to an increase (resp. decrease) of the difficulty of the job and therefore to a modified duration; think for example to fire fighting (resp. destroying a target which is getting closer).

We denote by Cj the completion time and by Tj=max{Cjdj,0} the tardiness of job j. The objective is to find a schedule which minimizes the total weighted tardiness ∑j=1nwjTj. Adopting the three-field standard notation of Graham et al. [10] we will denote this problem by 1|pjt,idleness|∑jwjTj. This problem is strongly NP-hard since it is a generalization of the single-machine total weighted tardiness problem 1∥∑jwjTj [15]. Indeed, there exists a dynamic programming algorithm with a running time O(n3j=1npj) for the problem 1∥∑jwjTj, but only when weights are agreeable, that is pjpkwjwk for all jobs j and k [15]. There exists a branch and bound algorithm for the 1∥∑jwjTj problem [18], but as it is reported in [6] it cannot be used in practice on instances with more than 50 jobs. Moreover the design of approximation algorithms seems very hard, since the only known results concern only the far less general 1∥∑jTj problem with a FPTAS due to Lawler [16] in O(n7/ϵ) time, and slightly improved by Kovalyov [14] in O(n6logn+n6/ϵ) time. Since the problem we consider is a broad generalization of the 1∥∑jwjTj problem, these results stress the importance of the metaheuristic approach if one wants to practically deal with instances of this problem.

Section snippets

Dynasearch

Local search algorithms, and their generalizations such as simulated annealing and tabu search (also called metaheuristics), are often used to obtain near optimal solutions for a wide range of NP-hard combinatorial optimization problems [4], [5], [19]. In these methods a neighborhood is defined, usually by giving a set of transformations that can be applied to the current solution. In the simplest local search method, at each iteration the algorithm searches the neighborhood of the current

Experimental results

There is certainly a tradeoff between the benefit of using a large neighborhood in terms of the quality of local optima and the induced time increase, relative to a small neighborhood, in order to search it. Our experiments were designed in order to determine if it was worth to spend more time exploring a larger neighborhood. To compare the performance of the standard and the dynasearch swap neighborhoods in a fair way, we have used a simple multi-start local search algorithm.

Conclusion and extensions

Our work is in the continuity of Congram et al. [6] which have introduced the dynasearch swap neighborhood for the 1∥∑jwjTj problem. By introducing the time parameter inside the dynamic programming algorithm we obtain a pseudopolynomial algorithm in time and space, whereas their algorithm needed O(n3) time and O(n) space, but we enlarge considerably the class of problems which can now be treated. We need not consider problems in which the cost change between neighboring solutions depends only

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