Minimization of squared deviation of completion times about a common due date

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Abstract

We discuss a non-preemptive single-machine job sequencing problem where the objective is to minimize the sum of squared deviation of completion times of jobs from a common due date. There are three versions of the problem—tightly restricted, restricted and unrestricted. Separate dynamic programming formulations have already been suggested for each of these versions, but no unified approach is available. We have proposed a pseudo-polynomial DP solution and a polynomial heuristic for general instance. Computational results show that tightly restricted instances of up to 600 jobs can be solved in less than 6s. General instances of up to 80 jobs take less than 2s.

Statement of scope and purpose

In this paper, we have considered an NP-complete single-machine scheduling problem arising in JIT environment, a field of great importance in manufacturing industry. The objective of the problem is to schedule a set of given jobs to minimize the sum of squared deviation of their completion times from a common due date. This paper presents a number of precedence rules, a polynomial heuristic and more importantly a unified pseudo-polynomial dynamic programming formulation. Empirical results show that the dynamic programming formulation performs better than the existing approaches.

Introduction

In this paper, we have considered a non-preemptive single-machine job sequencing problem, where the objective is to minimize the sum of squared deviation of completion times of jobs from a common due date (d). In scheduling literature, this problem is known as mean squared deviation problem (MSDP). MSDP arises in a simplified just-in-time (JIT) production environment where jobs are to be processed non-preemptively and one at a time on a machine. Any job with completion times different from its due date incurs penalty. The objective is to minimize the total penalty of processing the set of given jobs. Depending on the form of penalty functions and due dates, different classes of problems are possible. For MSDP, all the jobs have a common given due date, and the penalty of each job is squared deviation of its completion time from the due date.

MSDP was first formulated by Bagchi et al. [1], [2]. Different versions of this problem have been studied in the literature. The solution strategy for the problem depends on the version of the problem, and version of the problem depends on due date [3]. Hence, due date is expected to play an important role in the solution strategy of MSDP. If the objective function of MSDP is plotted against d for different sequences, the minimum value of d where the global minimum occurs is the critical due date d. As expected, the minimum value of MSDP remains constant for all values of d⩾d. When the due date is sufficiently large (greater than or equal to a critical value d), MSDP is called unrestricted (unconstrained). The unrestricted version of the problem has been shown to be equivalent to completion time variance (CTV) problem. Significant work has been carried out on CTV problem [3], [4], [5], [6], [7], [8], [9], [10], [11].

The graph of objective function of any schedule plotted against different d's has its minimum at a value of d equal to the average completion time of the schedule. The minimum of these local minimum points of all the schedules is referred to as d∗∗. MSDP instances with d⩽d∗∗ are called tightly restricted. The only other possible instances (i.e., those with d∗∗<d<d) are called restricted. Kubiak [7] has shown MSDP to be NP-complete. Even though dynamic programming (DP) formulations [3], [12] have been suggested for different versions of MSDP, no unified approach is available for general MSDP. The objective of this paper is to look for a unified and efficient method for solving large general instances of MSDP.

The paper is organized as follows. In Section 2, we describe MSDP and its properties. A unified DP formulation is introduced in Section 3. A heuristic is suggested in Section 4. Computational results are reported in Section 5. Finally, concluding remarks are given in Section 6.

Section snippets

Problem description and characteristics

For MSDP, N jobs are to be scheduled on single-machine non-preemptively around a given common due date d. Let pj be the processing time of job j, and Cj be the completion time of job j. The objective of MSDP is to minimize F=j(Cj−d)2. Without loss of generality, we assume that the processing times, due date are integers, and jobs are ordered such that p1p2⋯⩽pN. We also assume p1>0.

For presenting the characteristics of optimal schedules, the following notations and terminologies have been used.

Dynamic programming formulation

De et al. [5] have proposed a pseudo-polynomial DP formulation for CTV problem which can be extended for solving MSDP. De et al. [19] have proposed another DP formulation for MSDP. Weng and Ventura [3] have recently proposed a pseudo-polynomial DP formulation for tightly restricted MSDP. A common problem with all of the above formulations is that these either solve a particular category (tightly restricted or unrestricted) of MSDP or consist of number of subroutines. For the latter case, there

Heuristic MUB

De et al. [14] have suggested a heuristic UB for MSDP similar to the procedure suggested by Sundararaghavan and Ahmed [18] for mean absolute deviation (MAD) problem. In UB, jobs are sequenced in non-increasing order of processing times from left or right in the unscheduled interval [0,MS]. Let L and R be the length of the unscheduled intervals before the due date and after the due date, respectively. The job with largest processing time of all the unscheduled job is scheduled on the left if L>R

Implementation and computational results

We have implemented the DP formulation using node representation and pruning rules as discussed in 2 Problem description and characteristics, 3 Dynamic programming formulation. Algorithms were coded in C language and runs were taken on DEC Alphastation 250 4/266. Processing times were generated randomly (using the standard generator drand48 with seed 123456789) from a uniform distribution in the ranges 1–99 [3], [14].

Results are reported for different sizes of problem instances. For each set of

Concluding remarks

In this paper, we have proposed several dominance properties and a heuristic for MSDP. A unified DP formulation of pseudo-polynomial complexity has also been suggested. Performance of DP is considerably better than the existing results.

DP formulation and the heuristic can readily be extended to the weighted version of MSDP with objective function F=∑j∈Eα(Cj−d)2+∑j∈Tβ(Cj−d)2 for non-negative value of α and β. This is true as the optimal schedule in this case also is V-shaped and there is no

Acknowledgements

The author is grateful to Anup K. Sen, Ramkumar Ramaswamy and anonymous referees for their invaluable comments and suggestions. In particular, specific points from one of the referees improved the article considerably.

Sakib A. Mondal is a doctoral Fellow of Indian Institute of Management Calcutta, India. He received B.E. degree in Electronics and Telecommunication Engineering from Bengal Engineering College, India. Presently, he is working with Sofware Concept Laboratory at Infosys Technologies Ltd. His research interests include artificial intelligence search methods, job sequencing and scheduling, metaheuristics, performance engineering.

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Sakib A. Mondal is a doctoral Fellow of Indian Institute of Management Calcutta, India. He received B.E. degree in Electronics and Telecommunication Engineering from Bengal Engineering College, India. Presently, he is working with Sofware Concept Laboratory at Infosys Technologies Ltd. His research interests include artificial intelligence search methods, job sequencing and scheduling, metaheuristics, performance engineering.

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