Two due date assignment problems with position-dependent processing time on a single-machine

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Abstract

The focus of this study is to analyze single-machine scheduling and due date assignment problems with position-dependent processing time. Two generally positional deterioration models and two frequent due date assignment methods are investigated. The objective functions include the cost of changing the due dates, the total cost of positional weight earliness, and the total cost of the discarded jobs that cannot be completed by their due dates. We conclude that the problems are polynomial time solvable. Significantly enough, after assessing the special case of each problem, this research found out that they can be optimally solved by lower order algorithms.

Introduction

There are various situations in which the processing time of jobs may be subject to change due to deterioration and/or learning phenomena or additional resources consumption. Machine scheduling problems with deterioration and/or learning effects or resource allocation have been extensively studied in the last two decades in various machine settings and performance measures. For a complete list of studies, the reader may refer to the comprehensive survey by Cheng et al., 2004, Shabtay and Steiner, 2007, Biskup, 2008.

The problems with due date determination have received considerable attention since the last two decades due to the introduction of new methods of inventory management such as just-in-time production system and supply chain management. A survey or review on this line of the scheduling problems could be found in Baker and Scudder, 1990, Cheng and Gupta, 1989, Gordon et al., 2002, Lauff and Werner, 2004. In just-in-time systems, jobs are to be completed neither too early nor too late. Otherwise, they lead to the scheduling problems with both earliness and tardiness costs and assigning due dates. In supply chain systems, managers of manufacturing or service organizations quote the delivery dates (due dates) to clients frequently. Concerning the resources constraint in a company, supplier cannot quote the acceptable delivery dates for all customers. Failing to promise the delivery dates may not be acceptable to the customer and may force a company to offer price discounts in order to retain the business. Gordon and Strusevich (2009) studied an interesting scenario of supply chain management scheduling. They focused on a segment of the supply chain that includes two participants: a manufacturer and a customer. Since most of the manufacturers nowadays wish to market their production, all the accepted orders will be completed in full and in time to satisfy the customer’s demand. The due dates by which the manufacturer has to complete are the subject of negotiation with the customer. In this model, the due dates are assigned according to a certain rule. For the mutually agreed due dates, the manufacturer has an option of discarding some of the orders if they cannot be completed in time. In this case, a penalty for each discarded order will be paid.

Motivated by a scenario of supply chain management scheduling in Gordon and Strusevich (2009), this study investigates the single-machine scheduling and due date assignment problems with position-dependent processing time. The objective functions are slightly extended from what is proposed by Gordon and Strusevich (2009) to a general form. We concluded that the problems remain polynomial time solvable.

Section snippets

Problem formulation

In all models that we consider in this study, the jobs of set N = {1, 2,  , n} have to be processed without preemption on a single-machine. The jobs are simultaneously available at time zero. The machine can handle only one job at a time and is permanently available from beginning. For each job j, where j ϵ N, the value of its basic processing time pj and due date dj is known. Then the actual processing time of job j, if scheduled in position r of a sequence, is given by pj[r]=pjg(r), where g(r) is a

Due date assignment problems

First, a useful lemma is given as follows.

Lemma 1

Hardy, Littlewood, & Polya, 1967

Let there be two sequences of numbers xi and yi. The sum ∑ixiyi of products of the corresponding elements is the least (largest) if the sequences are monotonic in the opposite (same) sense.

The following properties are applicable for the due date assignment method. Its proofs are obvious and omitted.

Property 1

In a feasible schedule, there exists at most one on-time job.

Property 2

There exists an optimal schedule with no idle time between the jobs in NE and the common due

Conclusion

This study investigated the single-machine scheduling and due date assignment problems with position-dependent processing time. We extended part of the objective functions proposed by Gordon and Strusevich (2009) to the positional weighted earliness penalty. We, therefore, concluded that the problems remain polynomial time solvable based on the new objective functions. Significantly enough, after assessing the special case of each problem, this research found out that they can be optimally

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    In chemical and food production, the common due date model applies if some of the involved substances or components have a limited life span (a “best before” time), which imposes a common due date on the whole mixture or the final product. There are many papers that focus on the common due date assignment problem (e.g., Adamopoulos & Pappis, 1995; Birman & Mosheiov, 2004; Biskup & Jahnke, 2001; Cheng, 1984, 1987, 1989; Cheng, Chen, & Shakhlevich, 2002, 2004, 2007; De, Ghosh, & Wells, 1991; Gordon & Strusevich, 2009; Hsu, Yang, & Yang, 2011; Kahlbacher & Cheng, 1993; Li, Ng, & Yuan, 2011; Min & Cheng, 2006; Mosheiov, 2001; Mosheiov & Yovel, 2006; Ng, Cheng, Kovalyov, & Lam, 2003; Panwalkar, Smith, & Seidmann, 1982; Shabtay & Steiner, 2006). For state-of-the-art reviews of scheduling models considering common due date assignment, as well as practical applications of such models, the reader may refer to Cheng and Gupta (1989), Gordon, Proth, and Chu (2002, 2010), and Lauff and Werner (2004).

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