Grey wolf optimization (GWO) algorithm is a new population-oriented intelligence algorithm, which is originally proposed to solve continuous optimization problems inspired from the social hierarchy and hunting behaviors of grey wolves. It has been proved that GWO can provide competitive results compared with some well-known meta-heuristics. This paper aims to employ the GWO to deal with two combinatorial optimization problems in the manufacturing field: job shop and flexible job shop scheduling cases. The effectiveness of GWO algorithm on the two problems can give an idea about its possible application on solving other scheduling problems. For the discrete characteristics of the scheduling solutions, we developed a kind of discrete GWO algorithm with the objective of minimizing the maximum completion time (makespan). In the proposed algorithm, searching operator is designed based on the crossover operation to maintain the algorithm work directly in a discrete domain. Then an adaptive mutation method is introduced to keep the population diversity and avoid permutation convergence. In addition, a variable neighborhood search method is embedded to further enhance the exploration. To evaluate the effectiveness, the discrete GWO algorithm is compared with other published algorithms in the literature for the two scheduling cases. Experimental results demonstrate that our algorithm outperforms other algorithms for the scheduling problems under study.

Under the current environmental pressure, many manufacturing enterprises are urged or forced to adopt effective energy-saving measures. However, environmental metrics, such as energy consumption and CO₂ emission, are seldom considered in the traditional production scheduling problems. Recently, the energy-related scheduling problem has been paid increasingly more attention by researchers. In this paper, an energy-efficient job shop scheduling problem (EJSP) is investigated with the objective of minimizing the sum of the energy consumption cost and the completion-time cost. As the classical JSP is well known as a non-deterministic polynomial-time hard (NP-hard) problem, an improved whale optimization algorithm (IWOA) is presented to solve the energy-efficient scheduling problem. The improvement is performed using dispatching rules (DR), a nonlinear convergence factor (NCF), and a mutation operation (MO). The DR is used to enhance the initial solution quality and overcome the drawbacks of the random population. The NCF is adopted to balance the abilities of exploration and exploitation of the algorithm. The MO is employed to reduce the possibility of falling into local optimum to avoid the premature convergence. To validate the effectiveness of the proposed algorithm, extensive simulations have been performed in the experiment section. The computational data demonstrate the promising advantages of the proposed IWOA for the energy-efficient job shop scheduling problem.

Flexible job shop scheduling problem (FJSP) is a typical discrete combinatorial optimization problem, which can be viewed as an extended version of the classical job shop scheduling problem. In previous researches, the scheduling problem has historically emphasized the production efficiency. Recently, scheduling problems with green criterion have been paid great attention by researchers. In this paper, the mathematical model of the low-carbon flexible job shop scheduling problem is established with the objective of minimizing the sum of the energy consumption cost and the earliness/tardiness cost. For solving the model, a kind of bi-population based discrete cat swarm optimization algorithm (BDCSO) is presented to obtain the optimal scheduling scheme in the workshop. In the framework of the BDCSO, two sub-populations are used to adjust the machine assignment and operation sequence respectively. At the initialization stage, a two-component discrete encoding mechanism is first employed to represent each individual, and then a heuristic method is adopted to generate the initial solutions with good quality and diversity. By considering the discrete characteristics of the scheduling problem, the modified updating methods are developed for the seeking and tracing modes to ensure the algorithm work directly in a discrete domain. To coordinate the global and local search in each sub-population, six adjustment curves are used to change the number of cats in the seeking and tracing modes, based on which six algorithms are developed, i.e., LBDCSO, SinBDCSO, CosBDCSO, TanBDCSO, LnBDCSO, and SquareBDCSO. In addition, the information exchanging strategy is introduced to implement the cooperation of the two sub-populations. Finally, extensive simulation based on random instances and benchmark instances is carried out. The comparisons results demonstrate the effectiveness of the proposed algorithms in solving the FJSP under study.

Job shop scheduling problem (JSP) is a combinatorial optimization problem, which has been widely studied due to its strong theoretical and background for application. However, in previous researches on the traditional JSP, the optimization objective is mainly relative to time, such as makespan, flow time, tardiness, earliness and workload, and so on. With the advent of green manufacturing, energy consumption should be considered in the JSP. Therefore, a low-carbon job shop scheduling problem is studied in this paper. Due to the NP-hard nature, a meta-heuristic algorithm, bat algorithm (BA), is considered in this study. According to the characteristics of the problem, a kind of bi-population based discrete bat algorithm (BDBA) is proposed to minimize the sum of the energy consumption cost and the completion-time cost. A parallel searching mechanism is first introduced to the algorithm, by which the population is divided into two sub-populations to respectively adjust the job permutation and the processing speed of each machine. Three communication strategies are used to implement the cooperation between the sub-populations. In addition, due that the original BA was developed to deal with the continuous problems, a modified discrete updating approach is proposed to make the BA algorithm directly work in a discrete domain. Finally, extensive simulations have been conducted to test the effectiveness of the proposed BDBA algorithm. The experimental data demonstrate that the proposed BDBA is effective in solving the lowcarbon job shop scheduling problem under study.

In this paper, a novel improved whale optimization algorithm (IWOA), based on the integrated approach, is presented for solving the flexible job shop scheduling problem (FJSP) with the objective of minimizing makespan. First of all, to make the whale optimization algorithm (WOA) adaptive to the FJSP, the conversion method between the whale individual position vector and the scheduling solution is firstly proposed. Secondly, a resultful initialization scheme with certain quality is obtained using chaotic reverse learning (CRL) strategies. Thirdly, a nonlinear convergence factor (NFC) and an adaptive weight (AW) are introduced to balance the abilities of exploitation and exploration of the algorithm. Furthermore, a variable neighborhood search (VNS) operation is performed on the current optimal individual to enhance the accuracy and effectiveness of the local exploration. Experimental results on various benchmark instances show that the proposed IWOA can obtain competitive results compared to the existing algorithms in a short time.