Group scheduling and job-dependent due window assignment based on a common flow allowance

doi:10.1016/j.cie.2013.11.017

Computers & Industrial Engineering

Volume 68, February 2014, Pages 35-41

https://doi.org/10.1016/j.cie.2013.11.017 Get rights and content

Highlights

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We consider group scheduling and job-dependent due window assignment.
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Each job is assigned an individual due window based on a common flow allowance.
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The goal is to find the optimal sequence for both the groups and jobs.
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Polynomial time optimal algorithm is proposed.

Abstract

We study a single-machine group scheduling and job-dependent due window assignment problem in which each job is assigned an individual due window based on a common flow allowance. In the group technology environment, the jobs are divided into groups in advance according to their processing similarities and all the jobs of the same group are processed consecutively in order to improve production efficiency. A sequence-independent machine setup time precedes the processing of the first job of each group. A job completed earlier (later) than its due window will incur an earliness (tardiness) penalty. Our goal is to find the optimal sequence for both the groups and jobs, together with the optimal due window assignment, to minimize the total cost that comprises the earliness and tardiness penalties, and the due window starting time and due window size costs. We give an O(n log n)time algorithm to solve this problem.

Introduction

We study a single-machine scheduling and job-dependent due window assignment problem in a group technology (GT) environment. In the manufacturing industry, it is well known that firms can increase production efficiency by adopting GT. GT is an approach to manufacturing that seeks to improve efficiency in high-volume production by exploiting the similarities of different products and activities in their production. Through decades of application, people have found many advantages of GT. For instance, changeover between different jobs in the same group is simplified, reducing the costs or time involved; jobs in the same group spend less time waiting, which results in less work-in-process inventory; jobs in the same group tend to move through production in a direct route, reducing the manufacturing lead time (Li, Ng, and Yuan, 2011). As an example of GT in practice, Conway, Maxwell, and Miller (1967) consider paint production on a single machine. Customer orders vary in colors, but they can be divided into major color groups, such as red, blue, and green. Within a color group, e.g., red, colors may range from very light to dark red. The times to set up the machine to produce paint in colors of the same group are small, since it is natural to produce paint from lighter to darker colors, so these setup times are negligible. On the other hand, the time to switch the machine from the production of paint of one color group to another color group is substantial, which may include the times to clean the machine and change the tools, and this setup time is generally color-independent, i.e., the setup time is sequence-independent. In view of these observations, it is clear that dividing the orders into groups according to their processing similarities can significantly increase production efficiency. While GT research was originated by Mitrofanov (1966) and Opitz (1970), scheduling studies taking GT into consideration include Chen et al., 1997, Ng et al., 2005, Logendran et al., 2005, Allahverdi et al., 2008, Cheng et al., 2008, Shabtay et al., 2010, Behnamian et al., 2010, Li et al., 2011, and Bai, Li, and Huang (2012), among others.

While there is a large body of literature on due date scheduling involving the constant due date, i.e., each job is given the same due date, only a few studies focus on the common flow allowance, i.e., each job is given a due date that is equal to the sum of its processing time and a common flow allowance. The common flow allowance that reflects equal waiting time is a decision variable. Based on the common flow allowance, we set a due date for each job, the value of which has different implications. While a small value of the common flow allowance (leading to earlier due dates) makes the supplier more attractive to customers, a large value (i.e. late due dates) increases the flexibility of production but decreases attractiveness to customers. Depending on the value of the common flow allowance, a total cost is incurred, which the system seeks to minimize. The system intends to inform each customer of a time that they have to wait until their job is completed through the optimal value of the common flow allowance. The delivery date announced will not generally be realized. However, the cost to the system will be minimized (Adamopoulos and Pappis, 1996). Due date scheduling research involving the common flow allowance includes Cheng, 1986, Alidaee, 1991, Cheng et al., 1996, Wang, 2006, Shabtay and Steiner, 2007, and Wang and Wang (2010).

In a recent paper, Mosheiov and Oron (2010) extend the flow allowance notion to the due window assignment setting in which each job is assigned an individual due window. Following Liman, Panwalker, and Thongmee (1998), they consider that a job completed earlier (later) than its due window will incur an earliness (tardiness) penalty, while a job completed within its due window is considered as on time that will incur no penalty. In this new due window assignment model, in addition to the standard job scheduling decisions, one has to determine the job-dependent due windows. The due window starting time of a job J_j is $d_{j}^{1}$ , which is equal to the sum of its processing time p_j and a common flow allowance q₁, which is a job-independent constant, i.e., $d_{j}^{1} = p_{j} + q_{1}$ . Similarly, the due window completion time of the job J_j is $d_{j}^{2}$ , which is equal to the sum of its processing time p_j and a larger common flow allowance q₂, i.e., $d_{j}^{2} = p_{j} + q_{2}$ . Mosheiov and Oron (2010) provide an O(n log n) solution for this problem. Mor and Mosheiov (2012) and Chen, Ji, and Ge (2013) extend this problem to the case including a maintenance activity. They consider several versions of this problem, including the duration of the maintenance activity is (i) a constant time, (ii) an increasing function of its starting time, and (iii) position-dependent. They also investigate extensions of the position-dependent processing times. They solve all these problems by converting them into an Assignment Problem.

To the best of our knowledge, there is no research considering GT and due window assignment based on a common flow allowance simultaneously. The published papers in this area either study a job-dependent due window based on a common flow allowance without group technology or a job-dependent due date based on a common flow allowance with group technology. There is no paper studying a job-dependent due window based on a common flow allowance with group technology. Compared with the due date assignment problem based on a common flow allowance, we need to determine two values of the common flow allowance. Compared with the due window assignment problem based on a common flow allowance without group technology, we need to determine where the common flow allowance is, i.e., the common flow allowance may exist in other groups. Therefore our model is more general and covers the results of several papers, such as Li et al. (2011) and Mosheiov and Oron (2010).

Shabtay et al. (2010) provide some practical examples of scheduling multiple groups of similar products with an assignable common due date and resource allocation on a single machine. They point out that GT implementation is especially important when the specifications of products destined to different markets, different market segments, or even different customers vary. This may be due to specific packaging, labelling, or other processes. In this paper we extend the notion of a due date to a job-dependent due window based on a common flow allowance. Generally, in real-life production, a set of independent jobs divided into groups are given, where jobs of the same group must be processed consecutively and groups are not allowed to be interweaved so that GT can be used to take advantage of their processing similarities. Mosheiov and Oron (2010) first introduced job-dependent due windows based on a common flow allowance. In our paper each job is assigned an individual due window, but all the jobs within the same group share a common due window size. We adopt the common flow allowance procedure provided by Adamopoulos and Pappis (1996) to assign the due windows. We seek to determine the optimal group sequence, the optimal job sequence, and the optimal due window assignment to minimize the total cost that comprises the earliness and tardiness penalties, and the due window starting time and due window size costs.

The remainder of this paper is organized as follows: In Section 2 we introduce the notation and formulate the problem. In Section 3 we present some properties of the optimal solution. In Section 4 we provide an O(n log n) algorithm to solve the problem. In Section 5 we conclude the paper and suggest future research topics.

Section snippets

Problem formulation

The scheduling problem under study is formulated as follows: We are given a set of n independent jobs. In order to take advantage of GT in production, all the jobs are divided into m groups {G₁, G₂, …, G_m} in advance according to their processing similarities. All the jobs are available at time zero, and job preemption is not allowed. A single machine is available to process all the jobs. Each group G_i has n_i jobs ${J_{i 1}, J_{i 2}, \dots, J_{{in}_{i}}}$ , where J_ij denotes the job J_j of group G_i and $\sum_{i = 1}^{m} n_{i} = n$ . Jobs of

Preliminary results

In this section, some useful preliminary results are given to solve the scheduling problem. It is clear that an optimal schedule exists that starts at time zero and contains no machine idle time.

Lemma 1

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$If C_{ij} ⩽ d_{ij}^{1} \Rightarrow C_{i (j - 1)} ⩽ d_{i (j - 1)}^{1}, for i = 1, \dots, m; j = 1, \dots, n_{i} .$
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$If C_{ij} ⩾ d_{ij}^{2} \Rightarrow C_{i (j + 1)} ⩾ d_{i (j + 1)}^{2}, for i = 1, \dots, m; j = 1, \dots, n_{i} .$

Proof

For a given schedule π, if $C_{ij} ⩽ d_{ij}^{1}$ , then we have $C_{ij} = C_{i (j - 1)} + p_{ij} ⩽ d_{ij}^{1} = q_{i}^{1} + p_{ij} \Leftrightarrow C_{i (j - 1)} ⩽ q_{i}^{1} \Rightarrow C_{i (j - 1)} ⩽ q_{i}^{1} + p_{i (j - 1)} \Leftrightarrow C_{i (j - 1)} ⩽ d_{i (j - 1)}^{1} .$ If $C_{ij} ⩾ d_{ij}^{2}$ , the we have $C_{ij} = C_{i (j - 1)} + p_{ij} ⩾ d_{ij}^{2} = q_{i}^{2} + p_{ij} \Leftrightarrow C_{i (j - 1)} ⩾ q_{i}^{2} \Rightarrow C_{i (j - 1)} + p_{ij} ⩾ q_{i}^{2} \Leftrightarrow C$

Analysis of case I

Note that the values of k_i and l_i given in Lemma 3 are independent of the job processing times and the job sequence. Therefore, it is optimal for any job sequence within each group of case I.

Without loss of generality, we assume G_i is case I. For group G_i. The optimal k_i and l_i can be observed from Lemma 3, so does the optimal values of $q_{i}^{1}$ and $q_{i}^{2}$ , which $q_{i}^{1} = S_{i} + s_{i} + \sum_{j = 1}^{k_{i} - 1} p_{i [j]}$ and $q_{i}^{2} = S_{i} + s_{i} + \sum_{j = 1}^{l_{i} - 1} p_{i [j]}$ . Recall that S_i denotes the starting time of G_i and s_i is the sequence-independent setup

An optimal solution for the group sequence

By the analysis of Sections 3 Preliminary results, 4 Optimal properties of the problem, the optimal internal job sequence is observed. And the following lemma enables us to determine the processing orders of the groups.

Lemma 10

For a given schedule π, there is an optimal group processing order in which all the groups are scheduled in non-decreasing order of φ_i, $where ϕ_{i} = \{\begin{matrix} (s_{i} + \sum_{j = 1}^{n_{i}} p_{ij}) / n_{i} γ_{i}, & if group G_{i} is case I \\ (s_{i} + \sum_{j = 1}^{n_{i}} p_{ij}) / n_{i} δ_{i}, & if group G_{i} is case II \\ (s_{i} + \sum_{j = 1}^{n_{i}} p_{ij}) / n_{i} ϖ_{i}, & if group G_{i} is case III \end{matrix}$

Proof

We prove the result by

A numerical example

A three-group due window assignment problem based on a common flow allowance is solved here to illustrate the solution procedure. Each group has four jobs. The job processing times, cost parameters, and setup times are given is Table 1.

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Step 1. For group G₁, $k_{1} = ⌈ n_{1} (δ_{1} - γ_{1}) / α_{1} ⌉ = ⌈ 4 (7 - 5) / 5 ⌉ = 2$ $l_{1} = ⌈ n_{1} (β_{1} - δ_{1}) / β_{1} ⌉ = ⌈ 4 (16 - 7) / 16 ⌉ = 3$ . According to the Remarks, G₁ is case I. For group G₂, the given parameters are β₂ < δ₂, so according to Lemma 6, the due window assignment problem reduces to a due date

Conclusions

We consider a single-machine scheduling problem that considers both group technology and job-dependent due window assignment simultaneously. The objective is to find the optimal job and group schedules, and the optimal job-dependent due windows to minimize an objective function that includes the earliness and tardiness penalties, and due window starting time and due window size costs. We show that the problem can be solved in polynomial time. Future research can consider the problem in other

Acknowledgements

We are grateful to three anonymous referees for their helpful comments on an earlier version of our paper. This research was supported in part by the Humanities and Social Sciences Planning Foundation of the Ministry of Education (Grant No. 13YJA630034), the National Basic Research Program of China (973 Program) (Grant No. 2012CB315804), the key innovation team of Science Technology Department of Zhejiang Province (Grant No. 2010R50041), and the Contemporary Business and Trade Research Center

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^☆: This manuscript was processed by Area Editor Subhash C. Sarin.

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Group scheduling and job-dependent due window assignment based on a common flow allowance☆

Highlights

Abstract

Introduction

Section snippets

Problem formulation

Preliminary results

Analysis of case I

An optimal solution for the group sequence

A numerical example

Conclusions

Acknowledgements

Applied Mathematics Letters

European Journal of Operational Research

Applied Mathematical Modelling

International Journal of Production Economics

Engineering Costs and Production Economics

Discrete Optimization

International Journal of Production Economics

Annals of Operations Research

International Journal of Production Economics

International Journal of Production Economics

International Journal of Production Economics

Computers & Operations Research