A ranking-based differential evolution algorithm for hybrid ow shop sustainable scheduling

7 With the increasing of environmental pressure, the sustainable production for hybrid flow shops 8 (HFS) has attracted more attention due to its broad industrial applications. In implementing the 9 sustainable production of HFS, the selection of parallel machines for various jobs is a vital step. In light 10 of this, a multi-objective mathematical model for minimization of makespan and energy consumption 11 of HFSP was formulated. The sustainability of parallel machines were evaluated and ranked according 12 to fuzzy TOPSIS method. To solve the multi-objective model of HFSP, an improved differential 13 evolution algorithm was presented to assign the job with the ranked parallel machines, which can 14 narrow the search scope and accelerate the convergence speed. Finally, a case study was presented to 15 evaluate the effectiveness of the proposed ranking-based algorithm and to prove the feasibility of the 16 model. The results showed that the proposed improved algorithm outperforms NSGA-II and PSO in 17 searching for non-dominated solutions, which can effectively solve the sustainable production of HFSP.


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According to the characteristic of parallel machines, the classic HFSP can be divided into three

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Hence, to implement sustainable production, reasonable evaluation of sustainability for the parallel 39 machines become an urgent demand for scheduling.

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The sustainability of parallel machines can be evaluated from three aspects, i.e., the environmental 41 impact, the technical performance and the production cost. Energy consumption of the parallel

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To this end, this work presented a sustainable model of HFSP for minimization of makespan and 70 energy consumption. In addition, an improved differential evolution algorithm integrated with the 71 ranking sustainability of parallel machines was developed to support the HSFP sustainable scheduling.

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The innovations of the approach are summarized below:

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• A multi-objective mathematical model for minimization of makespan and energy consumption of 74 HFSP was formulated.

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• The sustainability of parallel machines were evaluated and ranked according to fuzzy TOPSIS 76 method.
set of n jobs has to be processed at w stages in series. Each job i consists of a pre-determined sequence 87 of operations. Each operation requires one machine selected from a set of available parallel machines, 88 which is denoted as Mij. The formal mathematical definition of the problem will be described in detail 89 in the following sections.

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Hybrid flow shop scheduling has been extensively examined and the main objective has been to 93 improve production efficiency. However, limited attention has been paid to the consideration of energy 94 consumption with the advent of green manufacturing. In order to reduce resource and energy 95 consumption and achieve sustainable production, this paper set the assignment of machines and the 96 sequence of operations on all the machines as variables in HFSP to minimize the makespan T and 97 energy consumption E to realize sustainable production.

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Hypotheses considered in this paper are summarized as follows:

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(1) Jobs are independent, and have equal priority.

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(2) After a job is processed on a machine, it is transported to the next machine immediately.

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(3) All jobs and machines are available at time zero, turning off the idle machines is not allowed and 102 machine failure is not considered.

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(4) The machine cannot be turned off completely until it has finished all operations assigned to it.

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(5) The order of operations for each job is predefined and cannot be modified.

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(6) Pre-emption is not allowed, that is, no task can be interrupted before the completion of its current 106 operation.

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The notations used throughout the study are listed in Table 1.

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The total energy consumption was composed of the energy consumption for the processing 121 stage PEtotal, the waiting stage WEtotal and the transportation stage TEtotal. The processing energy is 122 determined by processing time and the processing power per unit time of machine tools. The waiting energy of machine tools is the energy consumed by machine tools when they are not machining, that is, 124 waiting to process next jobs. The transportation energy consumption TEtotal is determined by the 125 transportation process of the work-pieces between different stages, which can be calculated as Eq. (5).

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Therefore, the second objective of this problem is simplified below.
Mathematically, an integer linear programming model of the HFSP was formulated as the 128 following, which will be used throughout the paper.  148 149 Fig. 2 The Framework of sustainable evaluation for machine tools 150

Sustainable evaluation system of machines 151
To rank the sustainability of parallel machines, the hierarchical structure of this research decision 152 problem is shown in Fig. 2. The sustainability of machines was evaluated by three aspects described as 153 following.
• Technical performance: the technical performance refers to the maximum machining quality and 155 efficiency that the machines can achieve, which were indicated by the maximum spindle speed 156 ( N1), feedrate ( N2 ), the accuracy ( N3) and reliability ( N4 ).

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• Economic performance: the economic performance of machines is indicated by the cost to 158 processing the corresponding jobs. Generally, the cost can be calculated by the summary of the 159 machining cost ( N5 ), human cost ( N6 ) and maintenance cost ( N7 ).   genetic operation, i.e., differential mutation, crossover and selection. It is a parallel search evolution strategy that is fairly fast and reasonably robust, which make differential evolution the versatile tool

Ranking-based Decoding 185
Decoding method is the key factor to decode the iterative chromosome sequence into a reasonable 186 production scheduling scheme, which has a great impact on the efficiency of the solution. The heuristic 187 rule proposed in this paper mainly refers to the directional selection of machine tools according to the 188 sustainability of the parallel machines. The parallel machines with the highest sustainability are more 189 suitable for the jobs. In this work, the jobs is encoded by the number and the sequence as shown in        Table 3.

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In addition, the sustainability of each machine were evaluated and ranked with the fuzzy TOPSIS 226 method listed in Table 4. The higher rank would be selected in the decoding processing of the RBDE  Table 5.

Effectiveness of RBDE 235
To test the effectiveness of RBDE, we expanded the instance size by increasing the times of 236 the data of workpiece/process/machines in Table 2. For the simplicity of presentation, an instance 237 with n jobs, s processes and m machines were denoted as an n*s*m. For each instance, RBDE 238 algorithm was run for 50 times independently and took the average value of the solutions. The running 239 time and two extreme results of the Pareto front set were listed in Table 6. 240

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Different ranking value of parallel machine tools for each process and jobs can be obtained according 251 to section 2, as shown in Table 6. It can be seen that the utilization rates of M1, M8, M9, M10, M13,

Comparisons with other algorithms 260
In order to test the performance of the proposed algorithms, the heuristic rule was implanted in the 261 NSGA-II and PSO named as HMNSGA-II and HMPSO. The 'Max.', 'Avg.', and 'Min.' column 262 represent the maximum, average, and minimum number of non-dominated solutions, respectively. It 263 can be clearly found from Table 7 Table 8.  Fig. 8.

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Besides, it is noted that the RBDE algorithm provides solutions that are relatively closer to the Pareto 318 fronts, whereas the RBDE and HMNSGA-II algorithms provide solutions that are more diversified.

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This result well explains the above-mentioned performance anomaly between the three algorithms.

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Overall, the proposed RBDE algorithm is capable of providing better solutions than HMNSGA-II

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(2)The sustainability of parallel machines were defined by a hierarchy criteria, which was ranked 333 by fuzzy TOPSIS method.

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(3)A ranking-based DE was developed to produce feasible scheduling sequences, in which 335 realized active selection of parallel machines with high sustainability. The effectiveness of the proposed 336 methods was verified.

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Although the efficiency of the proposed method has been verified, this paper has some limitations.