Elsevier

Computers & Operations Research

Volume 109, September 2019, Pages 188-199
Computers & Operations Research

A branch and price algorithm for single-machine completion time variance

https://doi.org/10.1016/j.cor.2019.05.007Get rights and content

Highlights

  • The single-machine completion time variance problem 1||CTV is studied from the viewpoint of exact solutions.

  • A new time-indexed quadratic programming is formulated, which is further linearized into a MIP formulation.

  • Based on the formulation, a branch and price algorithm is developed, which combines the Lagrangian relaxation with clusters.

  • Computational results for various instances demonstrate the effectiveness and efficiency of the proposed branch and price algorithm.

Abstract

This paper studies a single machine scheduling problem to minimize the completion time variance (CTV) of all jobs. A new time-indexed quadratic programming formulation is proposed, in which the quadratic terms are further linearized into a mixed integer programming (MIP) with non-negative continuous variables. To solve the problem to optimality, a branch and price (BP) algorithm is designed, which combines the Lagrangian relaxation with clusters. Extensive computational experiments on three sets of benchmark problem instances and randomly generated ones are conducted and the results show that the proposed BP algorithm outperforms state-of-the-art approaches.

Introduction

Scheduling deals with the allocation of resources to tasks over time periods and it is one of the typical decision-making processes in many manufacturing and service industries (Leung, 2004, Pinedo, 2012). A popular assumption in the scheduling literature is that performance measures are nondecreasing functions of job completion times, i.e., regular performance measures, including total weighted completion time, makespan, total weighted tardiness, maximum lateness, and so on. However, there are many important occasions when nonregular performance measures should apply, for instance, scheduling with Just-in-Time (JIT) objectives like total weighted tardiness plus total weighted earliness, completion time variance (CTV) and mean squares of deviation (MSD) of completion times from a given common due date.

This paper studies the case with the objective function of minimization of CTV on a single machine, which is usually denoted by 1||CTV with the well-known three field “α|β|γ” notation (Graham et al., 1979). The problem is to find a sequence of n jobs among all n! possible ones, which minimizes the completion time variance. This problem finds applications in Just-in-Time services (Manna, Prasad, 1999, Nessah, Chu, 2010, Rajkanth, Rajendran, Ziegler, 2017, Srirangacharyulu, 2017, Viswanathkumar, Srinivasan, 2003) and in manufacturing and service industries where customer classes are not differentiable and have similar job properties (Mehta et al., 2012), i.e., with the requirement of approximately the same level of service. In these applications, the manufacturer or service provider must provide a fair treatment in terms of order fulfillments. The objective is to increase customer service rates by minimizing the variance among all order fulfillments.

As a seminal work of 1||CTV, Merten and Muller (1972) introduced this problem as a model for the file organization problem in which it is important to provide uniform response time to users. Kubiak (1993) proved that the 1||CTV problem is NP-hard and no polynomial-time algorithm exists. A pseudo-polynomial algorithm was developed to solve the problem. For the first time, Schrage (1975) proved that the position of the job with the largest processing time should be scheduled first and conjectured the next three largest jobs, which has been further proven true in Vani and Raghavachari (1987). Eilon and Chowdhury (1977) proved the V-shape property, which means that the jobs must be arranged in descending order of processing times if they are placed before the shortest job, but in ascending order of processing times if placed after it. For problem instances with the number of jobs n ≥ 20, they proposed five heuristic procedures to approximately solve the problem.

By far, various heuristic or meta-heuristic methods have been proposed for the problem 1||CTV (see Table 1). A heuristic based on the complementary pair-exchange principle was presented in (Gupta et al., 1990). De et al. (1992) designed a pseudo-polynomial dynamic programming algorithm and a fully polynomial approximation scheme. In addition, they derived an effective lower bound, which can be obtained iteratively in O(n) time. Mittenthal et al. (1993) designed an algorithm of a hybrid greedy approach, followed by a simulated annealing search of the V-shaped sequence solution space. Kubiak (1995) proposed two pseudo-polynomial dynamic programming algorithms. A zero-one quadratic integer programming was also formulated, which is a submodular function with a special cost structure. Ventura and Weng (1995) introduced a Lagrangian relaxation procedure to find a lower bound. They also presented a two-phase heuristic algorithm: the first phase is a variant of the heuristic given by De et al. (1989) and the second phase is to ameliorate the sequence obtained in the first phase through the use of an interchange procedure. Manna and Prasad (1999) developed the bounds on the position of the shortest job in the optimal sequence. Furthermore, they proposed a new heuristic method by using these bounds. Al-Turki et al. (2001) developed a tabu search-based method for the 1||CTV problem. Viswanathkumar and Srinivasan (2003) presented a branch and bound method. For small-scale problems, the exact method was compared with the pseudo-polynomial dynamic programming in (De et al., 1992). And for larger-scale problems, the optimal solutions were compared with the results of simulated annealing in (Mittenthal et al., 1993), heuristics in (Manna and Prasad, 1999) and the tabu search in (Al-Turki et al., 2001). Kubiak et al. (2002) introduced fully polynomial approximation schemes (FPASs) within O(n2/ϵ) time where n is the number of jobs and ϵ > 0. Sharma (2002) developed a heuristic with a running time of low polynomial order. For problem instances with number of jobs n < 10, their heuristic can always obtain an optimal solution. Chaudhry and Drake (2008) proposed a spreadsheet-based genetic algorithm. Nessah and Chu (2010) derived a very tight lower bound and proposed lower bound based heuristic procedures for the weighted 1||CTV problem. The computational results show that solutions obtained by their heuristic methods are very close to optimal ones. Srirangacharyulu and Srinivasan (2010) presented a heuristic method and a genetic algorithm based method. The computational results show that the methods provide better solutions compared to a few existing heuristics when n is large. Wu et al. (2011) presented a V-shaped property based heuristic method. Rajkanth et al. (2017) proposed a heuristic method based on swapping complimentary pairs.

There are also many researches on CTV problems with multiple machines, including parallel machine (Cai, Cheng, 1998, Chen, Li, Sawhney, 2009, Federgruen, Mosheiov, 1996, Li, Chen, Sun, 2010, Rajkanth, Rajendran, Ziegler, 2017, Srirangacharyulu, 2017, Srirangacharyulu, Srinivasan, 2010), flow-shop (Gajpal, Rajendran, 2006, Gowrishankar, Rajendran, Srinivasan, 2001, Mehta, Pandit, Philip, Sharma, 2012) and job-shop (Ganesan et al., 2006). The approaches for CTV problems with multiple machines are summarized in Table 2.

In this paper, we focus on the 1||CTV problem. On the one hand, single machine settings are frequent in several industries and in some computer systems (Pereira and Vásquez, 2017). On the other hand, complex machine environments are often decomposed into single machine problems in many practical settings (Pinedo, 2012, p.35). The study of single machine problems can provide methods for multiple-machine problems, since many problems with multiple machines can be solved with decomposition-based methods like Benders decomposition (Tran et al., 2016) or column generation (Chen and Powell, 1999), in which the subproblem is a single machine problem.

The literature review above shows that: the 1||CTV problem has been attracted extensive concerns in the literature. However, to the best of our knowledge, there is not any attempt to solve the 1||CTV problem optimally with a branch and price algorithm. With this motivation, we develop a branch and price algorithm in this paper. The main contributions are as follows:

  • A new time-indexed quadratic programming is formulated for the 1||CTV problem. The quadratic terms in the formulation are further linearized with non-negative continuous variables, which results in a mixed integer programming (MIP) formulation.

  • A branch and price (BP) algorithm is proposed based on the MIP formulation. The restricted master problem (RMP) and sub-problems are obtained by the Lagrangian relaxation with clusters. In each iteration, all columns with reduced cost less than 0 are inserted into RMP. When all reduced costs are zero or positive and the optimal solution to the RMP is fractional, a branch and bound procedure is applied to the RMP with its current columns.

  • Extensive computational experiments are conducted on benchmark problem instances and randomly generated ones. The computational results show both the effectiveness and efficiency of the proposed branch and price algorithm.

The remainder of this paper is organized as follows. In Section 2, we present the basic formulation for the 1||CTV problem and the equivalent time-indexed formulation which is further linearized. In Section 3, we describe the proposed branch and price algorithm, in which the column generation method based on the Lagrangian relaxation with clusters and the branching strategy are introduced in detail. In Section 4, we conduct computational experiments to test the performance of the proposed BP algorithm on a set of benchmark problem instances and randomly generated ones. Section 5 concludes our study.

Section snippets

The formulation based on completion time variables

Given a set of jobs, N={1,2,,n}, where each job jN has a non-negative processing time pj, and the jobs are to be processed on a single machine, we aim to minimize the completion time variance of all jobs. The assumptions for the 1||CTV problem are described as follows:

  • (1)

    All jobs are available at time zero and the processing times of jobs are known in advance.

  • (2)

    The machine can process only one job at a time.

  • (3)

    Setup times are included in processing times or can be negligible.

  • (4)

    Preemption is not allowed.

A branch and price algorithm

In this section, a branch and price algorithm is developed for the problem. More specifically, Section 3.1 presents the column generation method based on Lagrangian relaxation with clusters, Section 3.2 describes the branching strategy in the BP algorithm and Section 3.3 gives the complete BP algorithm.

Computational experiments

The proposed algorithm was coded in Python 2.7.8 and run in WIN 10 operating system on a PC equipped with an Intel 2.10 GHz i7-4588U Processor and 4 GB of RAM. The Gurobi 7.5.2 was used as the IP solver to solve the models. We conducted two groups of computational experiments.

  • On benchmark problem instances. The performance of the proposed BP algorithm is compared with those of the EC1.2 in (Eilon and Chowdhury, 1977), the GGB in (Gupta et al., 1990), the SS in (Srirangacharyulu and

Conclusions

This paper addresses the completion time variance problem on a single machine. A time-indexed quadratic programming formulation is given, in which the quadratic terms are further linearized with non-negative continuous variables and a MIP formulation is obtained. Based on the formulation, a branch and price algorithm is designed to solve the problem to optimality, which combines the Lagrangian relaxation with clusters. Extensive computational experiments are conducted on both the benchmark

Acknowledgements

The authors thank the editors and anonymous referees for their constructive comments that helped significantly improve the quality of the paper. This work was supported by the National Science Foundation of China (NSFC) under Grants 71571135. The work was also supported by the Fundamental Research Funds for the Central Universities.

References (41)

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