A travelling salesman approach to solve the F/no-idle/C_max problem

Abstract

This paper investigates the F/no-idle/C_max problem, where machines work continuously without idle time intervals. The idle characteristic is a very strong constraint and it affects seriously the value of C_max criterion. We treat here only the permutation flow-shop configuration for machine no-idle problems with the objective to minimise the makespan. Based on the idea that this problem can be modelled as a travelling salesman problem, an adaptation of the well-known nearest insertion rule is proposed to solve it. A computational study shows the result quality.

Introduction

In a flow-shop organisation, all jobs follow the same route (machine sequence) and each job has exactly one operation on each machine. A permutation flow-shop is a flow-shop where all machines process the jobs in the same sequence. An important class of these scheduling problems is constrained by the no-idle production environment, where machines work continuously without any interruption from the start of the first job processing to the last job completion. In this paper, we propose a heuristic for the F/no-idle/C_max permutation problem. The obtained order minimises the C_max (or makespan i.e. the last job completion date), without any machine idle intervals. Such no-idle environment is particularly important when a very expensive equipment is used or when its using cost depends on time consumption.

Literature is quite poor for this problem. Adiri and Pohoryles [1] treated the F/no-idle/∑C_i permutation flow-shop which is shown to be NP-hard by Garey and co-workers [8], [17]. A polynomial bounded algorithm is proposed for F/no-wait/∑C_i and F/no-idle/∑C_i with increasing (the lowest processing time on a machine is greater than the largest one on the previous machine) and decreasing (the largest processing time on a machine is lower than the lowest one on the previous machine) series of dominating machines.

The complexity of the F/no-idle/C_max has only been mentioned in [20]. Authors referred to a Russian communication in 1981 [18] that proves the NP-Hardness of the problem. In [9], Giaro studied the complexity of some particular flow-shop or open-shop scheduling with no-idle and no-wait constraint.

Baptiste and Hguny [2] have proved the NP-hardness of F3/no-idle/C_max. Many heuristics have been developed for the general Fm//C_max problem: see, for example, [3], [6], [19], [21].

Baptiste proposed a branch and bound algorithm for the case of M no-idle machines (M>2) based on the evaluation of M−1 F2//C_max sub-problems. Narasimhan and Panwalkar in 1984 [15] and Narasimhan and Mangiameli in 1987 [16] have studied a manufacturing environment which could be found in many industries: a hybrid flow-shop where the production system has a continuous-process machinery on the first stage and a repetitive-batch equipment on the second one. In [15], Narasimhan and Panwalkar have presented the CMD rule (cumulative minimum deviation) as a good rule to minimise the idle-time on the second stage and consequently, the makespan. The CMD was applied in the case of a single machine on the first stage. In [16], Narasimhan and Mangiameli have shown the efficiency of the GCMD rule (generalised cumulative minimum deviation) which generalises the CMD rule to treat the case of multiple parallel machines on the first stage. The results of simulation runs demonstrate that the GCMD rule is better than other rules to minimise the sum of machine idle-times, the sum of job waiting times, the makespan or the sum of machine freeing dates.

In this paper, we propose a heuristic to solve the permutation flow-shop taking into account the machine no-idle constraint and makespan criterion. In Section 2, a model of this problem is presented. Section 3, gives a heuristic based on modelling our problem as a travelling salesman problem in Section 4, experiment results are shown. Our heuristic has been experimented on a wide range of problem test data. The obtained results are compared to the optimal solutions given by an enumerative algorithm for small job size problems, then to results obtained by an integer mixed programming solver using the model presented in Section 2 for larger size problem. A conclusion and some perspectives end this paper.