The distributed flowshop scheduling problem with delivery dates and cumulative payoffs

doi:10.1016/j.cie.2022.107961

Computers & Industrial Engineering

Volume 165, March 2022, 107961

https://doi.org/10.1016/j.cie.2022.107961 Get rights and content

Highlights

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The DPFSP with delivery dates and cumulative payoffs is addressed.
•
A mathematical model is built to formulate the new problem.
•
The upper and lower bounds of the problem are given.
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The Insert-Pruning and nine heuristic algorithms are proposed.
•
An improved IG algorithm is proposed.

Abstract

In the classic distributed permutation flowshop scheduling problem (DPFSP), there are more studies on the minimization of makespan, total flow time, total tardiness, etc. This paper studies a new problem with a new optimization goal, the DPFSP with delivery dates and cumulative payoffs. It is a variation of the DPFSP with job release dates that maximizes the total payoff with a stepwise job objective function. The main contributions are summarized as follows. (1) A mathematical model is built to formulate the new problem. (2) The characteristics of the problem are explored, and the upper and lower bounds of the problem are given. Based on the problem-specific knowledge, an algorithm named Insert-Pruning is proposed to improve the efficiency of search. (3) Nine heuristic algorithms are proposed, including DRI, DRA, DEI, DEA, DNI, DNA, DII, DIA and DFF. (4) Combined with the characteristics of the problem, some modifications and improvements have been made to the IG algorithm to solve it, including the destruction method, the local search method and the acceptance criterion. (5) The experimental results show that the presented algorithm significantly outperforms the existing algorithms in the literature. In comparison with other competing algorithms in different dimensions, our algorithm has shown better performance, which verifies the effectiveness of this algorithm.

Introduction

In recent years, the widespread application of distributed manufacturing has attracted research on different types of distributed scheduling problems (Wang and Wang, 2019). The distributed permutation flowshop scheduling problem (DPFSP) is a generalization of the traditional permutation flowshop scheduling problem (PFSP) (Lei et al., 2020, Naderi and Ruiz, 2010). There are f identical factories in the DPFSP (Ruiz et al., 2019). Each factory consists of m machines that are arranged as a flow shop. A set of n jobs must be processed in one of these f identical factories.

The Flowshop Scheduling Problem with Delivery Dates and Cumulative Payoffs proposed by Luciana S. Pessoa and Carlos E. Andrade (2018) is a new variant of the PFSP. In this problem, each job is not ready at the beginning, but has a release date. Each job can be processed after its release date. Moreover, a set of K delivery dates $D = {D_{1}, D_{2}, \dots, D_{K}}$ is given, where $0 < D_{1} < D_{2} < \dots < D_{K}$ . The objective value of a given schedule S is equal to $\sum_{j = 1}^{n} F (C_{j} (S))$ , where C_j(S) is the completion time of the job j and F(t) is a non-increasing step function of t which is shown in the equation (1). A real example of this problem can be found in the reference (Seddik et al., 2013) which describes the content of book digitization. Several digitization firms are hired to process many books. There are several delivery dates agreed in advance. And on the delivery date, the digitization firm's payment is calculated. The earlier the digitization of a book is completed, the greater the payoff associated with it. In addition, the calculation of a book payoff obeys a step function, which decrease at the delivery date. For example, the equation (1) can be used to express the step function. $F (C_{j}) = \{\begin{matrix} \begin{matrix} 0 \\ 1 \\ . . . \\ K \end{matrix} & \begin{matrix} if \\ if \\ if \end{matrix} & \begin{matrix} D_{K} < C_{j} \\ D_{K - 1} < C_{j} ⩽ D_{K} \\ 0 < C_{j} ⩽ D_{1} \end{matrix} \end{matrix}$

In this paper, the flowshop with Delivery Dates and Cumulative Payoffs is extended to the distributed permutation flowshop because of the widespread application of distributed manufacturing. First of all, the problem is analyzed and a mathematical model is proposed to formulate it. The upper and lower bounds are given. Based on the characteristics of the problem, an algorithm named Insert-Pruning, is proposed to improve the efficiency of search, which inserts a job from back to front. According to the problem information, such as release dates, processing time, earliest possible completion time, etc., nine constructive heuristic algorithms are proposed, including DRI, DRA, DEI, DEA, DNI, DNA, DII, DIA and DFF. To adapt the Iterated Greedy (IG) algorithm to the problem, some modifications has been done, including the destruction method, the local search method and the acceptance criterion. A detailed design experiment is carried out to determine the best parameter configuration. Finally, a comprehensive computational campaign based on the 540 instances shows that the improved IG algorithm is the best performer among all the algorithms in comparison.

The remainder of this paper is organized as follows. Section 2 presents a literature review of related works. In Section 3, the distributed flowshop scheduling problem with delivery dates and cumulative payoffs is formulated, and several problem characteristics are discussed. Nine heuristic algorithms are introduced in Section 4. Section 5 presents the proposed IG in detail. We report the computational results and comparisons in Section 6 following the parameter setting. Finally, Section 7 provides the concluding remarks and suggests some future work.

Section snippets

Literature review

As early as 2010, Naderi and Ruiz (2010) first began to research on the DPFSP. In their pioneering work, Naderi and Ruiz (2010) proposed six different alternative mixed integer linear programming (MILP) models to characterize the DPFSP problem. Later, a modified iterated greedy (MIG) algorithm for the DPFSP to minimize the maximum completion time was proposed (Lin et al., 2013). The superior performance of the MIG algorithm was shown by the computational results. A bounded-search iterated

Problem description

In this section, we introduce the mathematical model of the DPFSP with Delivery Dates and Cumulative Payoffs. The mathematical model is expanded from the models introduced by Naderi and Ruiz (2010), and Pessoa, L. S. and Andrade, C. E. (2018), who addressed the distributed flow shop scheduling problem and the PFSP with Delivery Dates and Cumulative Payoffs, respectively. First of all, the original maximum problem is transformed into the minimum problem, using Equation (2). This function is

Proposed constructive heuristics algorithms

The quality of the initial solutions has a direct impact on the performance of the whole algorithm. For PFSP, the NEH heuristic algorithm (Rad et al., 2009) is the most effective algorithm. Based on the NEH algorithm, we propose 9 initial solution generation algorithms suitable for the characteristics of the problem in this paper.

The solution representation method mentioned in the literature (Pan et al., 2019) is used for both the heuristic algorithms in this section and the Iterated Greedy

Proposed Iterated greedy algorithm

The Iterated Greedy algorithm (IG) always records two solutions: the best solution found by the algorithm and the current solution used. In each iteration process, IG starts from the current solution, destroys and reconstructs the current solution through iterative greedy operations, and then generates a series of new solutions. If there is a better solution found than the best solution, the best solution is updated. At the same time, an acceptance criterion similar is used to update the

Experimental setting

In response to the problems addressed in this paper, we randomly generated two groups of test instances. The first group contains 54 instances and the second group contains 540. For all the benchmark instances, the processing time of the jobs are generated following the uniform distribution in the U [1, 99] range that is common in scheduling literature. The test problems included n jobs and f shops, in which each had m machines, where f∈{2, 3, 4, 5, 6, 7}, n∈{100, 200, 500} and m∈{5, 10, 20}.

Conclusions

This paper studies a variation of the DPFSP with job release dates that maximizes the total payoff with a stepwise job objective function. The characteristics of this new problem are analyzed, and the IG algorithm is adaptively modified based on the characteristics of the problem. A large number of numerical experiments are used to test the performance of the algorithm. The experimental results show that the new algorithm has better performance than existing algorithms.

In the future, other

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

This research is partially supported by the National Science Foundation of China 61973203, Shanghai Science and Technology Innovation (project 21XD1401000), and Shanghai Key Laboratory of Power Station Automation Technology.

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Distributed scheduling problems have been extensively studied due to their critical roles in industrial applications. Most research focuses on identical distributed factories and the objective of minimizing the makespan, ignoring the widely existing heterogeneous factories and delivery targets in reality. This paper aims to establish a distributed heterogeneous flowshop scheduling scenario with the objective of minimizing total tardiness (DHPFSP-t), and solve it using an improved iterative greedy algorithm. A mixed-integer programming model and several problem-specific properties are proposed for this novel scheduling scenario. Based on the distributed heterogeneous characteristics of the problem, four constructive algorithms are established to quickly generate initial solutions. Among them, the better solution is selected as the initial solution of the iterative greedy algorithm. To enhance the performance of the iterative greedy algorithm, the concept of tabu memory is incorporated and novel destruction and local search processes are devised. Thorough experiments are conducted to verify the effectiveness of the mixed-integer programming model of the scenario, the constructive heuristics and the improved iterative greedy algorithm. The result confirms the validity of the proposed model and the good performance of the proposed method.
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The distributed flowshop scheduling problem with delivery dates and cumulative payoffs

Highlights

Abstract

Introduction

Section snippets

Literature review

Problem description

Proposed constructive heuristics algorithms

Proposed Iterated greedy algorithm

Experimental setting

Conclusions

Declaration of Competing Interest

Acknowledgements

Applied Soft Computing

Computers & Operations Research

Computers & Industrial Engineering

Engineering Applications of Artificial Intelligence

Swarm and Evolutionary Computation

Computers & Industrial Engineering

European Journal of Operational Research

Computers & Operations Research

European Journal of Operational Research

Procedia Computer Science

Expert Systems with Applications

European Journal of Operational Research

Omega

Omega

Knowledge-Based Systems

Expert Systems with Applications

IEEE Transactions on Systems, Man, and Cybernetics: Systems

Computers & Industrial Engineering

Computers & Operations Research