Minimizing makespan subject to minimum flowtime on two identical parallel machines

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Abstract

We consider the problem of scheduling jobs on two parallel identical machines where an optimal schedule is defined as one that gives the smallest makespan (the completion time of the last job) among the set of schedules with optimal total flowtime (the sum of the completion times of all jobs). We propose an algorithm to determine optimal schedules for the problem, and describe a modified multifit algorithm to find an approximate solution to the problem in polynomial computational time. Results of a computational study to compare the performance of the proposed algorithms with a known heuristic shows that the proposed heuristic and optimization algorithms are quite effective and efficient in solving the problem.

Scope and purpose

Multiple objective optimization problems are quite common in practice. However, while solving scheduling problems, optimization algorithms often consider only a single objective function. Consideration of multiple objectives makes even the simplest multi-machine scheduling problems NP-hard. Therefore, enumerative optimization techniques and heuristic solution procedures are required to solve multi-objective scheduling problems. This paper illustrates the development of an optimization algorithm and polynomially bounded heuristic solution procedures for the scheduling jobs on two identical parallel machines to hierarchically minimize the makespan subject to the optimality of the total flowtime.

Introduction

Consider the following scheduling problem: a set N={1,2,…,n} of n jobs available at time zero is to be processed on m identical parallel machines. Each job iN is to be processed without interruption on one of the m machines with processing time pi. Each machine can process only one job at a time and no job may be processed by more than one machine. Setup time, if any, is included in the processing time. It is desired to minimize the total flow-time as the primary objective and minimize makespan (maximum completion time) as the secondary objective. Thus, it is required to find a schedule for which the maximum completion time (makespan) is minimized, subject to the constraint that no reduction in the total flow-time is possible. Both makespan and flowtime performance measures have significant impact on a schedule's cost, since the former generally represents the amount of resources tied to a set of jobs; while the latter is a useful indicator of the amount of work-in-process inventory [1]. With the current trend to minimize work-in-process inventory, and the desire to maximize production rate, finding a minimum makespan schedule among those that minimize total flowtime is a useful criterion in practice. Parallel-machine scheduling problems arise often in practice as in the scheduling of jobs to a number of computer processors or the scheduling of jobs to a set of identical lathes.

The need to consider multiple criteria in scheduling is widely recognized. Either a simultaneous or a hierarchical approach can be adopted. For simultaneous optimization, there are two approaches. First, all efficient schedules can be generated, where an efficient schedule is one in which any improvement to the performance with respect to one of the criteria causes a deterioration with respect to one of the other criteria. Second, a single objective function can be constructed, for example by forming a linear combination of the various criteria, which is then optimized. Under a hierarchical approach, the criteria are ranked in order of importance; the first criterion is optimized first, the second criterion is then optimized, subject to achieving the optimum with respect to the first criterion, and so on. Surveys of algorithms and complexity results in this area are given by Chen and Bulfin [2], Lee and Vairaktarakis [3] and Nagar et al. [4]. Clearly, the above described problem is one of the hierarchical multi-criteria scheduling.

Following the three-field notation of scheduling problems, we will designate the identical parallel machine problem to minimize makespan subject to minimum total flowtime as a P||Fh(Cmax/∑Ci) problem where P designates the identical parallel machines, Cmax denotes the maximum completion time (makespan), ∑Ci represents the total flowtime, and the functional notation Fh(Cmax/∑Ci) designates that we hierarchically minimize makespan subject to minimum total flowtime. This problem has been shown to be NP-hard by Bruno et al. [5]. Heuristic algorithms for their solution are developed by Coffman and Sethi [6]. Eck and Pinedo [7] improved the results of Coffman and Sethi [6] and proposed a heuristic method for solving the P2||Fh(Cmax/∑Ci) problem that gives a minimum flowtime schedule with makespan that is guaranteed to be no more than 3.7037% above the makespan of the optimal schedule. Leung and Young [8] considered the preemptive case of the problem and developed an algorithm for its solution. To our knowledge, no algorithm is available to optimally solve the P2||Fh(Cmax/∑Ci) problem.

This paper proposes an optimization algorithm to solve the P2||Fh(Cmax/∑Ci) problem which, while exponential in its computational complexity, is quite efficient in solving large-sized problem instances. In view of the NP-hard nature of the problem, we also propose a modification of the multifit algorithm to find an approximate solution to the P2||Fh(Cmax/∑Ci) problem and compare its performance to the LPT-based heuristic developed by Eck and Pinedo [7].

The rest of the paper is organized as follows. 2 Minimizing total flowtime, 3 Minimizing makespan review the available results for minimizing total flowtime and makespan individually and describe the optimization and heuristic algorithms specifically tailored to the two-machine case. Section 4 discusses the transformation of the P2||Fh(Cmax/∑Ci) problem to an equivalent P2||Cmax problem which can be solved using the algorithms in Section 3. The simulation experiment and computational results are given in Section 5. Finally, we conclude the paper in Section 6 and provide some fruitful directions for future research.

Section snippets

Minimizing total flowtime

Conway et al. [9] describe a simple extension of the SPT rule to optimally solve the P||∑Ci problem. An optimal schedule is obtained as follows: at any stage of the assignment of jobs to various machines, m shortest processing time jobs to different machines. At the end, if the number of jobs left, k, is less than m, (i.e. k<m), they are assigned to k different machines. For the P2||∑Ci problem, this procedure simplifies as follows:

Algorithm S shortest processing time procedure

Input: pi for i=1,…,n;σ=(σ(1),…,σ(n)) such that pσ(1)pσ(2)⩽⋯⩽pσ

Minimizing makespan

The P||Cmax problem is known to be NP-hard in the ordinary sense [10]. Hence, most of the research to solve the P||Cmax problem aims at providing a near-optimal solution in a polynomially bounded computational time. In this section, we describe an optimization algorithm and two heuristic algorithms to solve the P2||Cmax problem.

Minimizing makespan subject to minimum flowtime

In this section, we show that the P2||Fh(Cmax/∑Ci) problem can be formulated as an integer program which can be transformed to an equivalent P2||Cmax roblem and hence solved by using the solution approaches described in Section 3 above. To do so, without the loss of generality, we assume that the total number of jobs n is even. If n is odd, an artificial job with zero processing time is added to the set of jobs. Further, let the schedule σ=(σ(1),…,σ(n)) be such that pσ(1)pσ(2)⩾⋯⩾pσ(n). From

Experimental results

A simulation experiment is performed to evaluate the effectiveness of the two heuristic algorithms in finding an optimal solution. In addition, the optimization algorithm HT is evaluated as to its efficiency in solving problems involving large number of jobs.

We consider two factors in this experimental study: the variability of processing time and the number of jobs. Problem hardness is likely to depend on the range and distribution of the processing times. Therefore, the processing times are

Conclusions

This paper considered the two-identical-parallel-machine problem to minimize makespan subject to minimum total flowtime and proposed an optimization algorithm and a heuristic algorithm for its solution. Computational results of a simulation study with randomly generated problems show that the proposed optimization algorithm HT is quite efficient in optimizing large-sized problems. Further, the heuristic algorithm HM is relatively more effective in finding optimal solutions than the existing

Jatinder N.D. Gupta is Professor of Management, Information and Communication Sciences, and Industry and Technology at the Ball State University, Muncie, Indiana, USA. He holds a Ph.D. in Industrial Engineering (with specialization in Production Management and Information Systems) from Texas Tech University. Coauthor of a textbook in Operations Research, Dr. Gupta serves on the editorial boards of several national and international journals. He has published numerous research and technical

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Jatinder N.D. Gupta is Professor of Management, Information and Communication Sciences, and Industry and Technology at the Ball State University, Muncie, Indiana, USA. He holds a Ph.D. in Industrial Engineering (with specialization in Production Management and Information Systems) from Texas Tech University. Coauthor of a textbook in Operations Research, Dr. Gupta serves on the editorial boards of several national and international journals. He has published numerous research and technical papers in such journals as International Journal of Information Management, Journal of Management Information Systems, Operations Research, IIE Transactions, Naval Research Logistics, European Journal of Operational Research, etc. His current research interests include information technology, scheduling, planning and control, organizational learning and effectiveness, systems education, and knowledge management.

Johnny C. Ho is Associate Professor of Operations Management at Columbus State University. He holds a Ph.D. in Management from Georgia Institute of Technology. Dr. Ho has published articles such journals as Naval Research of Logistics, Annals of Operations Research, European Journal of Operational Research, Production Planning and Control, and International Journal of Production Economics. His current research interests include scheduling, planning and control, and technology justification.

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