Complexity

Volume 2018, Article ID 1973604, 18 pages

https://doi.org/10.1155/2018/1973604

## Solutions to No-Wait Flow Shop Scheduling Problem Using the Flower Pollination Algorithm Based on the Hormone Modulation Mechanism

^{1}School of Information Engineering, Baise University, Baise 533000, China^{2}Computer and Electronic Information College, Guangxi University, Nanning 530004, China^{3}School of Politics and Public Affair Management, Baise University, Baise 533000, China

Correspondence should be addressed to Chiwen Qu; moc.361@newihcuq

Received 9 March 2018; Revised 21 June 2018; Accepted 5 July 2018; Published 6 August 2018

Academic Editor: Irene Otero-Muras

Copyright © 2018 Chiwen Qu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

A flower pollination algorithm is proposed based on the hormone modulation mechanism (HMM-FPA) to solve the no-wait flow shop scheduling problem (NWFSP). This algorithm minimizes the maximum accomplished time. Random keys are encoded based on an ascending sequence of components to make the flower pollination algorithm (FPA) suitable for the no-wait flow shop scheduling problem. The hormone modulation factor is introduced to strengthen information sharing among the flowers and improve FPA cross-pollination to enhance the algorithm global search performance. A variable neighborhood search strategy based on dynamic self-adaptive variable work piece blocks is constructed to improve the local search quality. Three common benchmark instances are applied to test the proposed algorithm. The result verifies that this algorithm is effective.

#### 1. Introduction

The flow shop scheduling problem is a simplified model of many manufacturing enterprise processes, belonging to a kind of important combinatorial optimization problem. The no-wait flow shop scheduling problem is a scheduling problem developed on the flow shop scheduling problem. In manufacturing, such as chemical industry manufacturing, smelting, food processing, and pharmaceutical manufacturing, the processing of work pieces cannot be interrupted from start to finish due to manufacturing craft limitations or storage space. Optimization scheduling of this kind of process can be concluded as a solution to the no-wait flow shop scheduling problem. In a more competitive market, optimal production planning and scheduling methods can reduce enterprises’ production costs and improve their competitiveness. However, when the number of work pieces is over 3, the no-wait flow shop scheduling problem has been proven to be an NP-hard problem [1]. This problem has great engineering research value and important theoretical significance. In the past few decades, many scholars have put forward several approaches to solve such problems, which can be divided into three categories: exact solution, heuristic algorithm, and meta-heuristic method. The exact solution method includes the dynamic programming method, branch and bound method [2], enumeration method, and cutting plane method. Due to the difficulty of the no-wait flow shop scheduling problem, the exact solution is only suitable for problems with relatively small scales. With improvement in the problem size, the time complexity of the algorithm increases rapidly. Nagano and Miyata [3] presented a mechanism that describes in detail how to construct a heuristic algorithm to solve the no-wait flow shop scheduling problem. Laha and Chakraborty [4] developed a constructive heuristic algorithm based on the job insertion principle. Framinan et al. [5] tackled the no-wait flow shop problem with a constructive heuristic algorithm based on an analogy with the objective of minimizing makespan. In order to obtain good approximate solutions, a new constructive heuristic named QUARTS is proposed by Nagano et al*.* [6]. With the total flow time as the criterion, many heuristics were examined by Bonney and Gundry [7], King and Spachis [8], Nawaz et al. [9], Rajendran [10], Gangadharan and Rajendran [11], and Ronconi and Armentano [12]. Grabowski and Pempera [13] used different local search algorithms to solve the no-wait flow shop problem for minimizing makespan. A heuristic algorithm is able to acquire the solutions within short time, but the solutions are usually worse in quality.

The meta-heuristic algorithm based on swarm optimization algorithm can acquire the optimal solution or approximate optimal solution to the no-wait flow shop scheduling problem within feasible time and space complexity. Considering the no-wait flow shop scheduling problem with makespan minimization, Aldowaisan and Allahverdi [14] investigated six heuristics based on simulated annealing (SA) and genetic algorithms (GA). Pan et al*.*, respectively, used the discrete particle swarm optimization algorithm [15], differential evolution algorithm [16], hybrid discrete particle swarm algorithm [17], artificial bee colony algorithm [18], and harmony search algorithm [19] to solve the no-wait flow shop scheduling problem. The goal was to minimize the maximum accomplishment time. For the same criterion, Fink and Voß [20] employed the simulated annealing algorithm and Tabu search algorithm, and Liu et al*.* [21] introduced a hybrid particle swarm optimization algorithm based on local search and self-adaptive learning mechanism. An ant colony optimization algorithm that improved the ant colony algorithm based on local search was applied separately to solutions for NWFSP [22, 23]. Nagano et al. [24, 25] designed an algorithm based on the hybrid meta-heuristic evolutionary clustering search (ECS_NSL). By testing the standard instances and comparison with algorithms provided by other literature, ECS_NSL performance is better. Qi et al. [26] presented a fast local neighborhood search algorithm (FLNS) with makespan criterion. The experimental results show that FLNS outperformed IHA, IBHLS, GA-VNS, and DHS in the solution quality and robustness. Deng et al. [27] proposed a cooperative evolutionary quantum genetic algorithm based on the competition mechanism to solve NWFSP. By maintaining population diversity and a competition mechanism to balance the development and exploit algorithm abilities, good solution results were obtained. Davendra et al. [28] introduced a new discrete self-organizing migrating algorithm. Through standard testing instances, it was shown in the statistical results that the algorithm is better than the heuristic algorithm in the literature.

FPA is a swarm intelligent bionic algorithm [29, 30] proposed by Yang et al. in 2012. The algorithm is highly regarded by scholars due to its simple structure, few parameter settings, relatively stronger global search performance, and easy implementation. Many scholars put forward an improvement method based on the FPA and popularized into industrial and agricultural production. Based on the basic FPA, Wang and Zhou [31] added a neighborhood search strategy and dimension-by-dimension greedy search method to improve cross-pollination. In comparison with other intelligent algorithms, the proposed algorithm is better. Abdel-Raouf and Abdel-Baset [32] proposed a hybrid FPA based on a combination particle swarm optimization algorithm and FPA to solve the constrained optimization problem. FPA is applied to solve the economic load dispatch problem [33–35]. Bekdaş et al. [36] used FPA to optimize the sizing truss structure. Abdelaziz et al. [37, 38] used FPA to obtain the optimal capacitor placement and sizing in distribution systems. They obtained the solution using combined economic and emission dispatch. In order to improve the accuracy and stability of cluster analysis, Wang et al. [39] proposed an FPA cluster analysis method with a bee pollinator. As shown in the experimental statistical results, the algorithm is better than DE, CS, ABC, PSO, FPA, and k-means algorithms in convergence, cluster accuracy, and stability performance. In order to solve the image segmentation problem via multilevel thresholding, Ouadfel and Taleb-Ahmed [40] presented an algorithm that combined social spider optimization (SSO) and FPA. The results showed that the proposed algorithm outperforms the PSO and BAT algorithms. Flower pollen gamete cross-pollination relies on animals like bees and butterflies; however, this is ignored in information sharing among the flowers in the basic FPA. Thus, the algorithm is inclined to engage in local optimum.

Aiming at minimizing the NWFAP accomplishment time, the hormone modulation factor is introduced based on the FPA to realize information sharing and improve the global search performance. A variable neighborhood search strategy based on dynamic self-adaptive variable work piece blocks is constructed to improve the local search quality. The computational results based on benchmark instances show the effectiveness of the proposed algorithm.

#### 2. No-Wait Flow Shop Scheduling Problem Model and Description

Based on the traditional flow shop scheduling problem, the no-wait flow shop scheduling problem is described as follows: Assuming that work pieces need to be processed on machines in the same sequences (without any preemption and interruption). There is a continuous process through machines without interruption when a work piece is started on the first machine. Here, the processing time of work piece on machine is provided. At any time, one work piece can be processed on only one machine and one machine can process only one work piece. To satisfy the no-wait constraints, on a given machine, the work piece completion time should be equal to the work piece start time on the next machine. For the purpose of minimizing the maximum accomplishment time, NWFSP can be described by the mathematics model below: where is a scheduling sequence of work pieces, is the processing time required by the job on the machine , is the accomplishment time of the job on the machine , and is the scheduling sequence aiming at the maximum accomplishment time minimum.

#### 3. Flower Pollination Algorithm

FPA is a new swarm optimization algorithm based on flower pollination behavior. The basic thought is as follows: (i) Each flower is mapped as an individual in the population. (ii) Cross-pollination is conducted by the switch probability. (iii) Self-pollination is conducted at the probability ().
(1)Cross-pollination. Cross-pollination refers to animals like bees and butterflies pollinate among different kinds of flowers using the *levy* flight mode. The flower update mode is shown aswhere is the scaling factor, is the flower individual at generation , is the current optimal individual found among all flower individuals at generation , and is the random number subordinated to *Levy* distribution.

For calculation convenience, Yang et al. [30] used the method proposed by Mantegna to calculate the step size . The calculation method is shown below: where and is a standard gamma function. (2)Self-pollination. Self-pollination simulates close-distance pollination among the same species of flowers. The pollination method is shown below:where is the flower individual at generation , is the random number subordinated to uniform distribution in [0,1], and and are the different flower individuals.

#### 4. FPA Based on the Hormone Modulation Mechanism

##### 4.1. Coding of the Algorithm

A job-permutation-based encoding scheme has been widely used in solving the no-wait flow shop scheduling problem. The job scheduling problem is discrete; it is impossible to use the standard FPA encoding scheme to directly express the sequence of work pieces. A random key coding principle based on the work piece ascending order is applied in this paper to realize the reflecting relationship between individual flowers and the scheduling sequence. Let the solution vector be , where there are two meanings for each vector including its own value and the sequence number representing the value of the new generating sequence. The process from a real vector of flower individual to the work pieces’ sequence is as follows: Each dimension vector value of is sequenced according to the ascending sequence principle; thus, . The sequence of work piece can be acquired through (6). A case on transformation from flower individual to sequence of work piece is provided in Table 1.