Job-shop scheduling problem considering power cost and makespan under time-of-use electricity price

Abstract

In order to solve the problem of minimizing power cost and makespan with time-of-use electricity. A genetic algorithm based on individual concentration and similarity vector distance strategy is proposed. The proposed genetic algorithm overcomes premature convergence problem by keeping the fittest individual through computing individual concentration and similarity vector distance. Production power cost reduction is achieved by using right-shift local search algorithm. The effectiveness of the proposed algorithm is illustrated by comparing the proposed algorithm with other scheduling algorithms. The comparative experiments indicate the proposed algorithm has better performance on minimizing power cost as well as makespan.

Keywords

Job-shop scheduling machine power cost time-of-use electricity genetic algorithm

1. Introduction

Most of manufacturing systems deal with job-shop scheduling problem. They are extensions of classical job-shop scheduling problems where a set of jobs need to be processed in a group of machines following different routes. Jobs have to be processed by one of the machines at each stage, and objective function is commonly to minimize the makespan. However, the manager should not only consider the makespan, but also the power costs.

Luo et al. use hybrid ant colony algorithm to solve hybrid flow-shop scheduling problem considering time-of-use electricity [1]. Tan et al. integrated response in power demand side with generation in power generation side to established joint optimization model based on global optimization for power cost [2]. Xie et al. used ant colony optimization algorithm for solving the short-term optimization scheduling problem of the cascaded hydroelectricity stations [3]. Besides, there are some researches about Time-of-Use (TOU) price to adjust process time between peak and off-peak period [4, 5, 6, 7]. Until now, there are a very few studies on job-shop scheduling considering TOU electricity price can be found.

Most of the manufacturing process is a multi-stage and there are multiple parallel machines, which belongs to job-shop scheduling problem. As early as 2004, Pfund et al. would prove that the job-shop scheduling problem about maximizing the number of completed product belongs to NP-hard problem [8]. Liu et al., with the minimum of power cost as objective function, established a mixed-integer nonlinear programming model, and a hybrid genetic algorithm (Memetic) is presented [9]. Jiang et al. propose a multi-objective flexible job-shop scheduling problem (FJSP) optimization model where power cost, makespan, processing cost and cost-weighted processing quality were considered [10]. References about energy consumption forecasting model for job-shop scheduling problem are proposed [11, 12, 13, 14, 15]. In summary, on one hand, only little literatures mention about energy consumption model of job-shop problem. On the other hand, traditional genetic algorithm is used to solve scheduling problems, but the traditional genetic algorithm has the problem of early convergence.

2. Problem definition

The proposed problem is a multi-objective job shop problem which can be stated as followed:

(1)
There are a set of jobs $J=\{J_{1},J_{2},\ldots,J_{n}\}$ to be processed at $m$ stages in a series. Each job $j$ should be processed by a set of machines $M=\{M,M_{2},\ldots,M_{m}\}$ with a set of process sequence $O=\{O_{11},O_{12},\ldots,O_{1m};\ldots;O_{n1},O_{n2},\ldots,O_{nm}\}$ .
(2)
Within each job there is a set of operations which need to be processed in a specific order (known as Precedence constraints). Each operation can be processed on any machine and only one operation in a job can be processed at a given time.

In real life manufacturing management, idle machine will not shut down until all products have been processed. It will lead to a great power cost especially in power peak period under Time-of-Use (TOU) electricity price. TOU price means it charge difference electricity price depend on time. It may lead to different production cost. Figure 1 shows different price of electricity in each period of time. 9 pm to 7 am is the non-peak period (the price of electricity is 5.1 yuan per kilowatt hour). 7 am to 11 am and 5 pm to 9 pm is peak electricity (the price of electricity is 9.9 yuan per kilowatt hour). 11 am to 5 pm is the normal part (the price of electricity is 8.1 yuan per kilowatt hour). TOU price can be formulated as follows, where t represents time.

$\displaystyle f(t)=\left\{\begin{array}[]{ll}5&0\leqslant t\%24\leqslant 7,\>2% 1\leqslant t\%24\\ 8&7\leqslant t\%24\leqslant 11,\>17\leqslant t\%21\\ 10&11\leqslant t\%24\leqslant 17\end{array}\right.$ (1)

Figure 1.
A distribution map of TOU price.

3. Mathematical model

The mathematical model of scheduling problem is described as follows:

Sets:

$J=\{{J_{1}},{J_{2}},\ldots,{J_{n}}\}$ : Job set

$O=\{{O_{11}},{O_{12}},\ldots,{O_{1m}};\ldots;{O_{n1}},{O_{n2}},\ldots,{O_{nm}}\}$ : Operation set

$M=\{M,{M_{2}},\ldots,{M_{m}}\}$ : Machine set

Parameters:

${T_{ij}}$ : Processing time of job $i,i=1,2,\ldots,m$ in stage $j,j=1,2,\ldots,n$ ; Unit of time is hour.

${r_{ij}}$ : Release time of job $i$ at stage $j$ ;

${C_{ij}}$ : Completion time of job $i$ at stage $j$ ;

$R$ : A sufficiently large integer;

${E_{jh}}$ : Power of processing machine $h$ at stage $j$ in one time unit, $h=1,2,\cdots,{m_{j}}$ .

Variables: ${x_{is}}$ : A Boolean variable, if job $i$ is in the position $s$ , then ${x_{is}}=$ 1, otherwise ${x_{is}}=$ 0.

${y_{ijh}}$ : A Boolean variable, if job $i$ is on machine $h$ at stage $j$ , then ${y_{ijh}}=$ 1, otherwise ${y_{ijh}}=$ 0.

$z_{h}^{t}$ : A Boolean variable, if machine $h$ is working at time $t$ , then $z_{h}^{t}=$ 1, otherwise $z_{h}^{t}=$ 0.

Objective function:

The objective function is divided into two sub-objective functions: makespan and power cost (PC).

The first objective is to minimize power cost (PC)

$\displaystyle\textit{PC}=\sum\limits_{t=1}^{c_{\max}}f(t)\sum\limits_{j=1}^{k}% \sum\limits_{h=1}^{m_{j}}E_{jh}z_{h}^{t}.$

The second objective is to minimize the makespan

$\displaystyle C_{\max}=\min\mathop{\max}\limits_{1\leqslant j\leqslant k}\{% \mathop{\max}\limits_{1\leqslant i\leqslant n}c_{ij}\},$

which is the completion time of the last job leaving the system.

All relevant constraints are as follows:

(1)
All machines and jobs are simultaneously available at time zero, and there is no machine breakdown.
(2)
Machines at each stage cannot process operations corresponding to any other stages.
(3)
The storage or buffer capacity between any two successive stages is unlimited.
(4)
Each machine can process at most one job at a time.
(5)
Preemption of jobs is not allowed, i.e., any commenced operation must be completed without interruptions.
(6)
Each operation should start immediately after the previous operation is finished if at least one machine at the stage is available.

4. Genetic algorithm with right-shift (GA-RS)

The GA-RS algorithm solves the scheduling problem by two steps instead of using multi-object genetic algorithm. The first step is to minimize the makespan to find the solution, and the second step is to minimize process energy costs (EP) by using right-shift strategy.

4.1 Encoding and initialization population

A chromosome that represent GA-RS solution is an array with $n\times m$ length cells, where $n$ is the number of jobs and $m$ is the number of job stages. The representation can be viewed as a kind of enlarged job-list. Since each job is assumed to be processed in each machine only one time, so each job appears in the job-list exactly $m$ times.

For example, a chromosome [1,2,1,3,2,1,3,2,3] represents 3 jobs processing list and the processing sequence is job1 in stage1, job2 in stage1, job1 in stage2, …, job3 in stage3. The machine processing matrix $m_{i,j}$ [2,1,3; 1,2,3; 3,1,2] represents jobs processed by machines at each stage. The index of $m_{i,j}$ denotes $i$ -th job in $j$ -th stage, and the value of $m_{i,j}$ denotes processing machine.

4.2 Fitness function

$N$ chromosomes are evaluated using the fitness function. ${f_{s}}$ , $s=1,2,\ldots,N$ , represents fitness value of each chromosome. Fitness function is as Eq. (2).

$\displaystyle{f_{s}}=\frac{1}{(C_{\max}^{s}/\mathop{\max}\limits_{s}C_{\max}^{% s})}$ (2)

$C_{\max}^{s}$ denote the maximum completion time of $s$ -th chromosome. $\mathop{\max}\limits_{s}C_{\max}^{s}$ denotes the maximum completion time in $N$ chromosomes.

4.3 Selection

In this case, we select the excellent individuals by calculating the probability of individual selection by using strategies based on concentration ${C_{t}}$ and similarity vector distance $\rho(x)$ . Given $n$ selected chromosomes, $H$ denotes length of chromosome and $x_{i}^{k}$ denotes $k$ -th chromosome in $i$ -th individuals and $f(x_{i})$ denotes fitness value of individual ${x_{i}}$ . The representation of concentration and similarity vector distance of individual ${x_{i}}$ is shown as follows:

$\displaystyle C(x_{i})=\frac{1}{n*H}\sum\limits_{j}^{n}\sum\limits_{k=1}^{H}% \textit{pos}(x_{i}^{k},x_{j}^{k})$ (3) $\displaystyle\textit{pos}(x_{i}^{k},x_{j}^{k})=\left\{\begin{array}[]{ll}1,&% \text{when }x_{i}^{k}=x_{j}^{k}\\ 0,&\text{other}\end{array}\right.$ (4) $\displaystyle\rho(x_{i})=\frac{\sum\limits_{j=1}^{n}|f(x_{i})-f(x_{j})|}{\sum% \limits_{i=1}^{n}\sum\limits_{j=1}^{n}|f(x_{i})-f(x_{j})|}$ (5)

The last probability based on concentration ${C_{t}}$ and similarity vector distance $\rho(x)$ is denoted by the Eq. (6).

$\displaystyle p(x_{i})=\alpha*C(x_{i})*\left(1-\frac{f(x_{i})}{\max f(x_{i})}% \right)+(1-\alpha)*\rho(x_{i})$ (6)

On Eq. (6), $\max f(x_{i})$ denotes the largest value of the fitness in individuals. $\alpha$ is an arbitrary number of [0, 1], which denotes a constant to adjust for the weight.

4.4 Crossover

We utilize the following crossover strategy on two parent chromosomes to obtain two newly child chromosomes. The procedure of crossover is introduced as follows.

Step1: Radomly selecct two genes for two parents. The selected gene positions are called mapping section.

Step2: Two selected genes exchange. The mapping relationship is map each gene in one parent chromosome with the corresponding gene in another parent chromosome which is on the relative position.

Step3: Repair the offspring by replacing the reduplicate gene outside the mapping section according the mapping relationship.

Suppose two parents are (1, 2, 2, 3, 3, 1, 2, 3, 1) and (2, 1, 2, 3, 2, 1, 1, 3, 3), an example is given in Fig. 2 to show the main step of crossover operation.

Figure 2.

Example of crossover operation.

4.5 Mutation

This operator randomly chooses two gene from the individual and then exchanges their positions. Meanwhile a shift-left operation will be added. More solution space could be explored and the change of dropping in the local optimum could be reduced. Figure 3 illustrates the process of mutation.

Figure 3.

Example of mutation operation.

4.6 Termination condition

The maximum number of generations is set to $N_{\max}$ as termination condition.

4.7 Right-shift procedure

Some intelligence algorithms use local search methods to optimize solution results, which help to find good solutions more quickly by increasing search solution space. In this paper, we apply a right-shift procedure instead of local search methods to further improve the quality of solutions obtained by proposed genetic algorithm. The right-shift procedure is used to reduce power cost while remaining the same makespan. It can be observed that some operation can be delayed to power off-peak periods without affecting the makespan. This procedure can reduce the power cost by taking advantage of difference of TOU electricity price. The detail of right-shift procedure is described as followed.

Input: A complete schedule decoded from a job sequence.

Output: A better schedule with less electric power cost.

For loop stage

j=k:2

Sort all jobs in the step

j

in decreasing order according to the job’s sequence. The result is represented by

{\sigma_{j}}

and

{\sigma_{j}}(i)

denotes

i

-th job in sorting result.

For job

i=1:n

Change the start time to the start time of

i

-th job at

j

-th step

{T_{\min}}={r_{ij}}

Change the start time to the latest start time of

i

-th job at (

j+1

)-th step

{T_{\max}}={r_{ij+1}}-{t_{ij}}

Change the start time of

i

-th job at (

j+1

)-th step to

\mathop{\arg\min}\limits_{T_{\min}\leqslant T\leqslant T_{\max}}\{\sum% \nolimits_{q=T}^{q+t_{ij}}f(q)E_{ijh}\}

End For

Output current scheduling with less electric power cost

5. Computational experiments and results

5.1 Experimental design

Impact factors affecting the performance of the algorithm are machines $m$ , jobs $n$ , stage $O$ , processing times $t_{ij}$ , the unit time of machine setup power consumption ${E_{jh}}$ and TOU price. Each factor is set as followed.

The processing time is uniformly distributed, recorded as ${t_{ij}}=$ $U$ [1, 10] hours The per unit time of machine setup power consumption ${E_{jh}}$ takes from a random integer 1 through 10 randi(1,10) per $k w h$ . TOU electricity price $f(t)$ is shown as Eq. (1).

Table 1
Factors and levels of GA parameter tuning experiments

Factors	Indexs of levels	Levels
Machine (m)	1	8
	2	10
Stage (O)	1	8
	2	10
Job (n)	1	8
	2	10
Pop	1	50
	2	100
	3	200
Cr	1	0.9
	2	0.5
	3	0.2
Mr	1	0.05
	2	0.1
	3	0.2

Table 2

ANOVA result in $m$ 10_ $n$ 10_ $O$ 10 dataset

Source	Type III sum of squares	df	Mean square	F	Sig.
Pop	630.463	2	315.232	125.417	0.000
Cr	1884.276	2	942.138	374.835	0.000
Mr	41.54	2	20.77	8.263	0.002
Error	50.27	20	2.513
Total	1202657.235	27

Table 3

Multiple comparisons of Pop in $m$ 10_ $n$ 10_ $O$ 10 dataset

(I) Pop	(J) Pop	Mean difference (I-J)	Standard error
1	2	6.2822*	0.74736
	3	11.8289*	0.74736
2	1	$-$ 6.2822*	0.74736
	3	5.5467*	0.74736
3	1	$-$ 11.8289*	0.74736
	2	$-$ 5.5467*	0.74736

Table 4

Multiple comparisons of Cr in $m$ 10_ $n$ 10_ $O$ 10 dataset

(I) Cr	(J) Cr	Mean difference (I-J)	Standard error
0.2	0.5	8.2311*	0.74736
	0.9	20.3400*	0.74736
0.5	0.2	$-$ 8.2311*	0.74736
	0.9	12.1089*	0.74736
0.9	0.2	$-$ 20.3400*	0.74736
	0.5	$-$ 12.1089*	0.74736

Table 5

Multiple comparisons of Mr in $m$ 10_ $n$ 10_ $O$ 10 dataset

(I) mr	(J) mr	Mean difference (I-J)	Standard error	Sig.
0.05	0.1	1.6289*	0.74736	0.041
	0.2	3.0356*	0.74736	0.001
0.1	0.05	$-$ 1.6289*	0.74736	0.041
	0.2	1.4067	0.74736	0.074
0.2	0.05	$-$ 3.0356*	0.74736	0.001
	0.1	$-$ 1.4067	0.74736	0.074

Table 6

Results of GA-RS algorithm parameter tuning

Dataset	GA-RS algorithm parameters
	Pop	Cr	Mr
$m$ 8_ $n$ 8_ $O$ 8	200	0.9	0.2
$m$ 8_ $n$ 8_ $O$ 10	200	0.9	0.2
$m$ 8_ $n$ 10_ $O$ 8	200	0.9	0.2
$m$ 8_ $n$ 10_ $O$ 10	200	0.9	0.2
$m$ 10_ $n$ 8_ $O$ 8	200	0.9	0.1
$m$ 10_ $n$ 8_ $O$ 10	200	0.9	0.2
$m$ 10_ $n$ 10_ $O$ 8	200	0.5	0.2
$m$ 10_ $n$ 10_ $O$ 10	200	0.9	0.2

Figure 4.

Estimated marginal means plot.

5.2 Parameters of GA-RS

The performance of GA algorithm depends on the control parameters, namely, population size (Pop), crossover rate (Cr) and mutation rate (Mr). The factors and levels are listed in Table 1. The GA parameter tuning experiments are applied to test all 81 possible combinations of these factors by using the approach proposed by Ruiz and Maroto in 2006.

The experimental results are analysed by means of a multifactor analysis of variance (ANOVA) by using. statistic software SPSS 22.0. In the univariate model, makespan is the dependent variable. Pop, Cr and Mr are set as fixed factor. All the experiments were carried out at 95% confidence level.

Since all the parameters tunning experimental results are alike, one of experimental results carried on dataset 10 machines and 10 jobs for 10 stages ( $m$ 10_ $n$ 10_ $O$ 10) is shown in Table 2. If the factor Sig $<$ 0.05 at the 95% confidence level, it means that this factor has a significant impact on the dependent variable. As shown in Table 2, Parameters Pop, Cr and Mr have much significant impact on the makespan. F-ratio represents the variance explained by a factor and the unexplained variance. The parameter Cr has greatest F-ratio value, which means that the experimental results have significant difference between different Cr level. The multiple comparisons among the levels of all parameters are shown in Tables 3–5. As we can seen in Table 3, Pop $=$ 200 is significant different from with 50 and 100. The mean difference shows that the algorithm obtain best performance when Pop $=$ 200. Cr is set to 0.9 and Mr is set to 0.2 according to the Tables 4 and 5. The best level of each factor for the rest of 7 datasets is listed in Table 6.

Figure 5.

makespan convergence pattern of proposed GA-RS algorithm.

Figure 6.

Gantt charts of solution without right-shift.

An estimated marginal means is used to select to the best level of the factor for the proposed GA-RS algorithm. It is clearly shown in Fig. 7 that Cr and Pop have a significant impact on Makespan, and Pop $=$ 200, and Cr $=$ 0.9 result in a much more effective genetic algorithm. There is no significant difference with Mr in three levels, Mr $=$ 0.2 has a slender advantage than others.

Figure 7.

Power cost convergence pattern of proposed GA-RS algorithm.

Figure 8.

Gantt charts of solution after right-shift.

5.3 Results and discussion

In this section, we compare the relative effectiveness of the proposed GA-RS algorithm with traditional genetic algorithm. Firstly, we introduce the strategy of this computational evaluation.

Since we attempt to convert the multi-objective optimization problem into single-objective optimization problem, the GA-RS algorithm is divided into two steps. The first step is to optimize the makespan of JSP problem, and the second step is to optimize the power cost of machine by using right-shift strategy. The experimental is carried on $m$ 10_ $n$ 10_ $O$ 10 dataset.

In the frist step evaluation, Fig. 5 demonstrates the makespan convergence pattern of proposed GA-RS algorithm. With the increasing of generation, the makespan is steadily on the decrease. The convergence stops at generation 48 and the makespan is reduced from 256 to 194. Figure 6 denotes the Gantt charts of solution without right-shift.

In the second step evaluation, Fig. 7 represents the power cost convergence pattern of right-shift strategy. It can be observed that right-shift procedure just delayed the release time of jobs without changing the completion time, in which reducing power cost by adjusting process time in power off-peak period. Figure 8 denotes the Gantt charts of solution after right-shift.

In this section, the performance of proposed GA-RS algorithm is evaluated we compare our proposed GA-RS algorithm with traditional genetic algorithm. Figures 6 and 8 demonstrate the difference of Gantt charts before right-shift procedure and after right-shift procedure. It can be observed that right-shift procedure just delayed the release time of jobs without changing the completion time, in which reducing electric energy consumption by adjusting process time in power off-peak period.

In order to further demonstrate the validity of proposed GA-RS algorithm, we make a comprehensive assessment of job-shop scheduling problem with different experimental data as practical application experiments. The proposed GA-RS algorithm compares with traditional multi-objective genetic algorithm. The traditional multi-objective genetic algorithm converts multi-object optimization problem into single-objective optimization problem, by using relative weights ${v_{1}}$ and ${v_{2}}$ . ${v_{1}}$ represents the weight of makespan, while ${v_{2}}$ represents the weight of power cost. In our experiment, ${v_{1}}$ is set to 0.5 and ${v_{2}}$ is set to 0.5. The parameter of multi-objective genetic algorithm is set to Pop $=$ 200, Cr $=$ 0.9 and Mr $=$ 0.2, generation $=$ 1000, which is the same as proposed GA-RS algorithm.

Table 7
Comparison of two algorithm with different instances

Dataset (m/n/O)	GA-RS algorithm		Traditional GA
	Make-span (hour)	Power cost (kwh)	Make-span (hour)	Power cost (kwh)
8/8/8	151	5042	151	5079
8/8/10	179	6421	179	6775
8/10/8	167	6392	167	6489
8/10/10	202	7811	202	8231
10/8/8	142	5080	142	5456
10/8/10	176	6480	176	6765
10/10/8	146	6405	146	6590
10/10/10	192	7899	192	8291

All the algorithms are coded in Matlab R2012b. An intel Core i5 2.30 GHz CPU computer with 4 GB RAM is used to russn the experiments. The performance of proposed algorithm was evaluated by makespan and power cost, and the experimental results are shown in Table 7, some conclusions can be obtained from Table 7.

When the data scale is small, there are little difference between multi-objective genetic algorithm and GA-RS algorithm. However, with the increasing of data scale, our proposed algorithm outperform than the multi-objective genetic algorithm. It can be explained that when the data scale increase, the idle time of jobs will also increase, which provide the opportunity for right-shift procedure to adjust the process time in power off-peak period to reduce electricity energy cost.

6. Conclusion

In this paper, we begin the research of minimizing power cost and makespan on job-shop scheduling problem. The problem is formulated by a multi-object optimization of power cost and makespan. For solving this problem, we develop an a genetic algorithm based on individual concentration and similarity vector distance strategy in which a right-shift procedure is applied to adjust off-peak period. The results are as follows.

(1)

In terms of job and machine combination, under the same power cost proportion, the power cost can be reduce by increasing machine numbers.

(2)

In terms of power cost proportion, under the same job and machine combination, the power cost can be reduce by increasing power cost proportion which means more efficient machine are used.

In the future research, there is a need to investigate in several important directions. Firstly, the solutions of the proposed GA-RS approach are encoded only job sequence and machine assignment in a small portion of the search space. Therefore, different solution encoding strategy can be considered for further expand searching space. Secondly, the multi-objective scheduling algorithm may be further improved. The study deals with two objective functions with coefficient. How to make a tradeoff between power cost and makespan is still an open question.

Footnotes

Acknowledgments

This work was supported by Natural Science Foundation of Guangdong Province (no. 2015A030310 340), Science Technology Planning Project of Guangdong Province (no. 2016B010126006, no. 2017A040405058), and the New PearlRiver Star Program of Guangzhou City (no. 201610010035).

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Job-shop scheduling problem considering power cost and makespan under time-of-use electricity price

Abstract

Keywords

1. Introduction

2. Problem definition

4.1 Encoding and initialization population

4.2 Fitness function

4.7 Right-shift procedure

5. Computational experiments and results

5.1 Experimental design

Table 1 Factors and levels of GA parameter tuning experiments

Table 7 Comparison of two algorithm with different instances

Footnotes

Acknowledgments

References

Table 1
Factors and levels of GA parameter tuning experiments

Table 7
Comparison of two algorithm with different instances