Exact algorithms for a scheduling problem with unrelated parallel machines and sequence and machine-dependent setup times

doi:10.1016/j.cor.2006.07.015

Computers & Operations Research

Volume 35, Issue 4, April 2008, Pages 1250-1264

https://doi.org/10.1016/j.cor.2006.07.015 Get rights and content

Abstract

A scheduling problem with unrelated parallel machines, sequence and machine-dependent setup times, due dates and weighted jobs is considered in this work. A branch-and-bound algorithm (B&B) is developed and a solution provided by the metaheuristic GRASP is used as an upper bound. We also propose a set of instances for this type of problem. The results are compared to the solutions provided by two mixed integer programming models (MIP) with the solver CPLEX 9.0. We carry out computational experiments and the algorithm performs extremely well on instances with up to 30 jobs.

Introduction

In the industrial context, scheduling problems are related to manufacturing resource planning (MRP). Programming is made according to the planning horizon. Long term decisions have strategic characteristics, and therefore are taken by high administration. The short term is known as the tactical level. In this level, the objective is to order the production sequence in order to minimize the manufacturing costs. The demand and due dates are, usually, already defined. The main objective is to define the processing sequence in order to meet both demands and due dates. Moreover, the algorithms developed here seek to minimize the makespan and the sum of the weighted tardiness of each job.

The study of scheduling problems with sequence-dependent setup times has attracted considerable attention in recent years [1]. They are also some of the most complex scheduling problems. Considering one machine and one stage, the problem is equivalent to the traveling salesman problem [2]. Garey and Johnson [3] showed that minimizing the makespan on two identical machines is a NP hard problem. Certainly, a problem with unrelated machines and due dates is also NP hard.

There is much research work considering parallel machines, but few unrelated parallel machines or sequence-dependent setups. Luh et al. [4] apply dynamic programming for a problem also with sequence-dependent setup times. Liu and Liao [5] consider several stages, flexibility and sequence-dependent setup times. They use a lagrangean relaxation on a model considering flow constraints, but also identical machines at each stage. Acero and Delgado [6] use a heuristic based on tabu search considering a flow shop problem with unrelated parallel machines at each stage, but they do not consider sequence and machine-dependent setup times. Meyr [7] shows an interesting approach to a lot-sizing problem, considering a flow shop in a single production line with sequence-dependent setup times. In posterior work, Meyr [8] adds unrelated parallel machines to the original problem. Hans-Joaquim and John [9] explore a problem of job shop with no sequence-dependent setup times using constraint programming. Koulamas and Kyparisis [10] consider a makespan minimization problem on uniform parallel machines with release times. Pinedo [11] suggests several heuristics for problems with similar characteristics. However, we were unable to find other papers considering a problem with these characteristics. We also refer to Błażewicz et al. [2], Pinedo [11], Lee and Pinedo [12] and Brucker [13].

Algorithms based on Branch-and-Bound (B&B) and GRASP are not uncommon when dealing with scheduling problems, as well as other similar problems [14]. Rabadi et al. [15] propose a B&B for an early/tardy scheduling problem considering one machine with sequence-dependent setup times. An adaptation of the metaheuristic GRASP (greedy random adaptive search procedure) [16] is used to find an upper bound (UB) for our problem. We have also referred to Feo et al. [17], [18], where applications of GRASP for one machine scheduling problems are shown and Resende et al. [19] who show an interesting application of the metaheuristic for job shop problems. Much work has been done about the application of GRASP to the considered problem [20].

This paper is organized as follows: Section 2 shows the MIP models used to be compared to our B&B. Section 3 gives the definition of a B&B algorithm and the customization for our problem. In Section 4, the instances for the problem are defined. Tests are detailed and results are shown in Section 5. Section 6 gives conclusions and future research directions.

Section snippets

MIP (mixed integer programming) model

The problem considered in this work is a scheduling problem with unrelated parallel machines, due dates, and setup times that depend on both job sequence and machine. It consists of programming several jobs to be processed by several machines. Each job should be scheduled to a specific machine, and the order each machine will process its jobs should be decided. It takes each job a different amount of time to be processed by each machine. When a machine finishes processing a job, it has a setup

The branch-and-bound algorithm

Algorithm 1. A generic Branch & Bound
1:	procedure BRANCH(node)
2:		if IsACompleteSolution(node) then
3:			$UB \leftarrow$ SolutionValue(node);
4:			return
5:		end if
6:		for all $n \in$ ChildrenOf(node) do
7:			if $LB (n) < UB$ then
8:				Branch(n);
9:			end if
10:		end for
11:	end procedure
12:	$UB = InitialSolution ()$ ;
13:	Branch(firstNode);
14	return UB;

A B&B is a specific enumeration tree strategy. In a B&B, there are three main procedures: initialization, branching and bounding. During the initialization, a fast heuristic is used to find a good initial

Instances

To analyze the algorithms and models developed for this problem, several classes of instances are defined. In each class there is a change in one of the inputs. All values are randomly generated from an uniform distribution. Their maximal and minimal values are listed in Table 1, where $U (x, y)$ is a value generated from an uniform distribution between x and y.

To generate the due dates, several variations of formulas found in other work [28], [29] were tested. To ensure that the ratio $weighted$

Tests and results

We present results comparing the solutions found by the developed B&B to the solutions found by both models when solved by CPLEX 9.0² in Section 5.1. Tests with the B&B solving the various classes described in Section 4 are shown in the following sections.

We use an interquartile mean for all tests, discarding 50% of the values found. Since there are 20

Conclusions

In this paper, we considered a scheduling problem with unrelated parallel machines, due dates and setup times that depend both on the machine and the sequence. A B&B algorithm using GRASP as an initialization procedure, and a MIP model were proposed and tested. Also as a main contribution, we generated a set of instances with different values of due dates, setups and processing times.

Concerning the performance of the algorithms and models, we solved instances with up to 25 jobs. We showed that

Acknowledgement

The authors gratefully thanks the anonymous reviewers for their helpful comments. This research has been partially supported by NSF, US Air Force and CNPq grants.

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Exact algorithms for a scheduling problem with unrelated parallel machines and sequence and machine-dependent setup times

Abstract

Introduction

Section snippets

MIP (mixed integer programming) model

The branch-and-bound algorithm

Instances

Tests and results

Conclusions

Acknowledgement

European Journal of Operational Research

European Journal of Operations Research

European Journal of Operational Research

Computers & Operations Research

Computers & Operations Research

Computers & Operations Research

European Journal of Operational Research

Impact of sequence-dependent setup time on job shop scheduling performance

International Journal of Production Research

Scheduling computer and manufacturing processes

Computers and intractability: a guide to the theory of NP-completeness

Job shop scheduling with group-dependent setups, finite buffers, and long time horizon

Annals of Operations Research

Scheduling flexible flowshops with sequence-dependent setup effects

IEEE Transactions on Robotics and Automation

Methods for solving pratical problems of job-shop scheduling modelled in CLPFD