TY - JOUR
AU - Liu, Xiangdong
AB - Abstract In order to further reduce the carbon emission of manufacturing process in flexible job shop, a multi-objective integrated optimization model of flexible job-shop scheduling (FJSP) is proposed. A mathematics model is built in this paper to minimize makespan, total workload of machines and carbon emissions of machines and to optimize process method of each machine characteristic, process sequence and machine allocation. Considering many parameters are interactional and to be optimized in the proposed model, a quantum bacterial foraging optimization is designed to code the related parameters. On the basis of Kacem example through experimental simulation, the performance of the proposed method in the paper was analysed with ANOVA, and by comparing with the algorithms of current separated optimization method of process planning and scheduling, the effect of proposed integrated optimization model on reducing carbon emission in practical requirements of FJSP is verified. 1. Introduction It is known that flexible job-shop scheduling (FJSP) is a NP-hard problem, and the carbon emission generated in the process sequence and machine allocation has a serious effect on the environment. According to the survey, the manufacturing sector accounts for nearly one-third of the total global energy consumption [17]. With the enhancement of energy conservation and emission reduction awareness, the low-carbon scheduling problem of flexible job shop (FJP) has attracted the attention of domestic and foreign scholars. Due to the high complexity of the problem, most researchers use intelligent optimization algorithm to solve the problem, such as shuffled frog-leaping algorithm [5], drosophila algorithm [23], particle swarm optimization [24], ant colony optimization [4], etc. These optimization algorithms have been applied to low-carbon scheduling. In response to the problem of low-carbon scheduling FJSP, domestic scholars have established high-dimensional and low-carbon scheduling models and optimized NSGA-III algorithm with multidimensional economic indicators and green indicators, making up the shortage of insufficient pressure in many-objective space selection [12]. Zhang et al. [28], aiming at dual-resource constrained FJSP of minimize makespan, proposed a knowledge-based fruit fly optimization algorithm. Li and Lei [9] proposed a novel imperialist competitive algorithm (ICA) to solve low-carbon FJSP problem with four targets and produce high-quality Pareto solutions. Although multi-objective low-carbon FJSP is still in its infancy in China, it has become a hot and difficult point in scheduling research. From optimizing green shop scheduling, Wang L. et al. summarized the research progress on such problems in China [21]. Li and Lei proposed a novel ICA to optimize the total energy consumption of key and non-key objectives from considering FJSP low-carbon scheduling problem depending on sequence release time. They analysed the relationship between deterioration of total energy consumption and improvement of key objectives [10]. Xu et al. established a shop scheduling model to minimize machine energy consumption [6, 25]. Compared with the minimize completion times scheduling, this model effectively reduces the energy consumption of the machine idling. Shan et al. considered the carbon emission factors from each energy consumption in shop scheduling problem to reduce the overall carbon emission of the manufacturing process in shop scheduling problem [2, 11]. Literature [12, 14, 15] takes the minimum completion times and minimum total energy consumption in the production process as the optimization goal. Considering the energy consumption during material cutting and normal operation of the machine, the authors optimize the environmental factors within peak power range to achieve the energy saving goal of FJSP. Although relevant scholars have effectively studied the green FJPS problem, the processing method, processing route and cutting parameters of workpieces in the study are determined. By changing the processing order of workpieces on each machine, the adjustment time and waiting time of machines are reduced, thus reducing the energy consumption of machine idling. The single optimizing shop scheduling has limited effect on reducing carbon emissions in manufacturing process [20]. How to achieve efficient energy saving, emission reduction and reduce environmental pollution is one of the urgent problems in the manufacturing industry. Studying low-carbon emission FJPS and realizing low-carbon green manufacturing are not only important directions of FJSP but also inevitable requirements for the benign development of the manufacturing industry. Multi-objective low-carbon FJSP model with minimum makespan, minimum total workload of machines and minimum carbon emissions of machines is established in this paper. The improved quantum genetic algorithm is used to solve the problems, and the validity and efficiency of carbon emission model are verified through case analysis and comparison. 2. FJSP model of low-carbon emission 2.1. Description of problem The low-carbon emission FJSP problem is an extension of the FJSP problem. This kind of problem not only needs to solve the problem of processing machine selection and process sequencing in the production workshop but also needs to achieve the goal of minimizing the carbon emissions of machine processing [13]. The low-carbon emission FJSP problem can be described as follows: there are N processing workpieces and M machines in the shop. Each job i(i|$\in$|{1,2,……,N}) has ni(ni|$\ge$|1) processes, and each process can be produced on any machine with processing capability. Rij represent j(j|$\in$|{1,2,……, ni}) process of workpiece i. Mij(Mij|$\subseteq$|{1,2,……, M}) are defined as the set of machines that can process the j operation of job i. Process Rij can be processed in any machine mk(k = 1,2,…,Mij}) in Mij. Machine m can process multiple processes for different workpieces [19]. Due to difference machine performance, the processing time and energy consumption of the process Rij on the machine m are also different. Allocating a suitable machine m for each process Rij is the optimization goal of low-carbon scheduling. Meanwhile, the processing sequence of the workpieces on machine m is arranged and the start processing time is determined, so as to achieve a synergistic optimization of the efficiency indicators and the low-carbon emission indicators. In this paper, the following assumptions are made when solving the FJSP problem under low-carbon constraints: (1) The same machine can only process one workpiece at a certain time; (2) The same process can only be processed on one machine; (3) No interruptions are allowed while the process is being processed on the machine; (4) If there is no fault in the machine during the processing, it cannot be shut down once it is started up; (5) There is no priority among the processes of different workpieces, and there is a sequence relationship among the processes of the same workpiece; (6) All workpieces can be processed at time t = 0, with the same priority; (7) Carbon emissions are also generated when the machine is idle after starting, and the carbon emissions generated when the machine processes the workpiece are independent of the type of workpiece. 2.2. Equation of carbon emission equation and related parameters tijk, pijk and cijk represent the processing time, processing power and carbon emission of Rij on machine mk, respectively. The carbon emission of job shop mainly comes from the power consumption. Taking electricity consumption as carbon emission index, the coefficient matrix of carbon emission in this paper can be expressed as follows: $$\begin{equation} C={\left\{{c}_{ijk}\right\}}_{m\times n} \end{equation}$$(1) The balance equation of power p can be expressed as follows: $$\begin{equation}p=p^{e}+p^{c}+p^{w}\end{equation}$$(2) Among them, pe is the no-load power to maintain its own running, pc is the cutting power for workpiece processing and pw is the load loss power to bear the processing load. In the actual machining process, pc is proportional to pw, if the ratio is expressed as load loss power coefficient of |$\lambda$|⁠, then Eq.(2) can be simplified as p = pe + (1+|$\lambda$|⁠)pc. Set the processing time to tijk; the no-load power of mk is |${p}_{ijk}^e$|⁠, the cutting power is |${p}_{ijk}^c$| and the carbon emission for Rij on mk can be expressed as: $$\begin{equation} {c}_{ijk}={\int}_0^{t_{ijk}}{p}_{ijk}^e(t)\mathrm{d}t+\left(1+\lambda \right){\int}_0^{t_{ijk}}{p}_{ijk}^c(t)\mathrm{d}t \end{equation}$$(3) Studies have shown that the cutting energy of the same workpiece on the same kind of machine is approximately constant [18] and the change of load loss power factor can be ignored. Therefore, the carbon emission difference in the machine process mainly comes from the no-load power, and the carbon emission coefficient matrix of Rij on mk can be simplified as |${c}_{ijk}\approx{\int}_0^{t_{ijk}}{p}_{ijk}^e(t)\mathrm{d}t$|⁠. The no-load power of the machine mainly depends on the processing speed of the spindle, and the carbon emission calculation can be further expressed as follows: $$\begin{equation} {c}_{ijk}\approx{p}_{ijk}^e(s)\times{t}_{ijk} \end{equation}$$(4) Eq.(4) s are the parameters related to the processing speed. 2.3. Objective function The goal of low-carbon FJSP scheduling is to select the appropriate machine for each process of the workpiece and determine the optimal processing sequence for each process of each machine, so as to minimize the carbon emissions of the processing machines in the workshop. Minimizing the average completion time of the workpiece, minimizing the total cost and minimizing carbon emissions were established in this paper. Considering the differences between multiple targets, the key to solving the problem is to find a satisfactory balanced solution among the multiple targets. Establishing the objective function as follows: Minimize makespan $$\begin{equation} f1=\min (F)=\min \left[\left(\sum \limits_{k=1}^M{F}_{m_k}\right)/M\right] \end{equation}$$(5) $$\begin{equation} {F}_{m_k}=\sum \limits_{i=1}^N\sum \limits_{j=1}^{n_i}\left({S}_{ijk}{b}_{ijk}+{S}_{ijk}{t}_{ijk}\right) \end{equation}$$(6) Minimize carbon emissions $$\begin{equation} f2=\min (C)=\min \left(\sum \limits_{i=1}^N\sum \limits_{j=1}^{n_i}\sum \limits_{k=1}^M{t}_{ijk}{p}_{ijk}{s}_{ijk}\right) \end{equation}$$(7) Minimize total cost $$\begin{equation} f3=\min (A)=\min \left[\sum \limits_{i=1}^N\left({A}_i+\sum \limits_{j=1}^{n_i}\sum \limits_{k=1}^M{A}_{ijk}{S}_{ijk}\right)\right] \end{equation}$$(8) $$\begin{equation} {A}_{ijk}=\left({\mu}_{ijk}+{\nu}_{ijk}\right) \end{equation}$$(9) In Eq.(5), F is the completion time of all machines, which is an important indicator for measuring the workload of the machine; |${F}_{m_k}$| represents the total completion time of machine mk, bijk represents the starting processing time of process Rij on machine mk, tijk represents the processing time of process Rij on machine mk and Sijk represents the processing of process Rij on machine mk in Eq.(6), which means $$ {S}_{i\,j k}=\left\{\begin{array}{l}1,{R}_{i\,j}\ is\ cut\ on\ {m}_k;\\{}0,{R}_{i\, j}\ is\ not\ cut\ on\ {m}_k.\end{array}\right. $$ In Eq. (7), C and pijk represent the total carbon emissions and processing power of the process Rij processed on the machine mk, respectively. In Eq. (8), A represents the total processing cost of workpiece i, Ai represents the raw material cost of workpiece i and Aijk represents the cost of processing Rij on machine mk. In Eq. (9), |${\mu}_{ijk}$| and |${\nu}_{ijk}$| represent the labor cost and machine cost of processing Rij on machine mk, respectively. 2.4. Constraints conditions (1) Operation constraints. Constraints between the processes of the same workpiece are required, and the jth process of the workpiece i must be started after the (j-1)th process is completed. $$\begin{equation} \sum \limits_{k=1}^M{b}_{ijk}{S}_{ijk}\ge \sum \limits_{k=1}^M\left[\left({b}_{i\left(j-1\right)k}{t}_{i\left(j-1\right)k}\right)\right]{S}_{i\left(j-1\right)k} \end{equation}$$(10) In Eq. (10), Sijk = Si(j-1)k = 1. (2) Machine constraints. The same machine can only processed one process at the same time. When process Rij at time t(t > 0), if |$\exists{S}_{ijk}=1$|⁠, then Sxyk = 1 must not be established (when |$i=x,j\ne y$|⁠). (3) Continuity constraints. The process Rij cannot be interrupted during processing: $$\begin{equation} {f}_{ijk}=\left\{\begin{array}{l}\max \left\{{f}_{i\left(j-1\right)k},{b}_{ijk}\right\}+{t}_{ijk},\kern0.5em j>1;\\{}{b}_{ijk}+{t}_{ijk},\kern3em j=1.\end{array}\right. \end{equation}$$(11) In Eq. (11), fijk indicates the completion time of the process Rij. 3. Hybrid method for low-carbon emission The common bacteria foraging optimization algorithm (BFO) is a global optimal algorithm simulating the foraging behavior of the bacteria proposed by Passion and Breg [27]. Chatzis (2011) pointed out that for the bacteria, the convergent speed and precision without communication is better than that with communication. For the multi-dimensional and complex optimization problem, it is difficult to obtain the global optimal solution in case of premature convergence. According to the theorem of NFL [7], the researchers have improved the BFO combining differential evolution, estimation of delivery and immune algorithm [14]; (Khataie et al., 2013). However, some shortcomings may appear in the optimization process of multidimensional peak function, such as the low-optimization precision and success rate. Zhang proposed a quantum algorithm based on bacteria chemotaxis behavior [3]. Khataie (2013) tried to improve the performance of BFO through making the individual perceive the optimal position of the population on the basis of the particle swarm optimization. Saber [16] used a discrete method for the quantum individual in the particle solution space to solve the integer programming. Chatzis (2011) improved the BFO from the view of accelerating the convergence rate, improving the local search precision and reducing the control parameters. 3.1. Common BFOA The common bacterial foraging optimization algorithm (BFOA) is a kind of algorithm based on global random search, and its main operations include chemotaxis, reproduction and elimination. The chemotaxis is the core of BFOA, including tumbling and swimming. When the individual bacterium is searching for nutrients, it will turn to a better fitness value until it reaches the maximum number of chemotaxis or meets worse solutions. According to the fitness value of all individuals in a cycle, the half of an individual with weak ability of foraging will be eliminated. Meanwhile, the other half will be replicated to maintain bacterial populations. Elimination is to create new bacteria individuals to maintain the total number of bacteria. Some individuals may die during the search process, which is conductive for chemotaxis to jump out of the local optimization. If the elimination reaches enough iteration, the algorithm will end (Xu, 2013). 3.2. Improved quantum bacterial foraging optimization It is well known that the short step leads to the lack of effective information sharing among individuals during the evolution process, which is not conductive to the diversity of the population. If the step is too short, the possibility of premature convergence will be great; otherwise, the convergence will reduce too much and the optimal solution cannot be obtained [27]. Therefore, it is difficult to control the optimal solution effectively by using the fixed step size in practical application. Considering the quantum theory can make the individual appear in any position of the whole feasible search space with strong global search performance and population randomness [14], the quantum bacterial foraging optimization (QBFO) is proposed in this paper. The probability density function of bacterial population in the quantum space is constructed by the sharing information of bacterial population in the reproductive stage. Table 1 Standard testing function. Testing function . Function . Search scope . Type . Rosenbrock |${f}_1(x)=\sum \limits_{i=1}^{n-1}\Big(100{\Big({x}_{i+1}-{x}_i^2\Big)}^2{\Big({x}_1-1\Big)}^2\Big)$| [−15,30] Single peak Rastrigrin |${f}_2(x)=\sum \limits_{i=1}^n\Big({x}_i^2-10\cos \Big(2\pi{x}_1\Big)+10\Big)$| [−5.12,5.12] Multi peak Griewank |${f}_3(x)=\frac{1}{4000}\sum \limits_{i=1}^n{x}_i^2-\underset{i=1}{\overset{n}{\Pi}}\cos \Big[\frac{x_1}{\sqrt{i}}\Big]+1$| [−600,600] Multi peak Ackley |${f}_4(x)=20+e-20e\sqrt[-\frac{1}{5}]{\frac{1}{n}\sum \limits_{i=1}^n{x_i^2}_{-e}}-\frac{1}{n}\sum \limits_{i=1}^n\cos \Big(2\pi{x}_1\Big)$| [−32,32] Multi peak Schwefel |${f}_5(x)=418.9829\ n-\sum \limits_{i=1}^n{x}_i\sin \sqrt{\mid{x}_i\mid }$| [−100,500] Multi peak Testing function . Function . Search scope . Type . Rosenbrock |${f}_1(x)=\sum \limits_{i=1}^{n-1}\Big(100{\Big({x}_{i+1}-{x}_i^2\Big)}^2{\Big({x}_1-1\Big)}^2\Big)$| [−15,30] Single peak Rastrigrin |${f}_2(x)=\sum \limits_{i=1}^n\Big({x}_i^2-10\cos \Big(2\pi{x}_1\Big)+10\Big)$| [−5.12,5.12] Multi peak Griewank |${f}_3(x)=\frac{1}{4000}\sum \limits_{i=1}^n{x}_i^2-\underset{i=1}{\overset{n}{\Pi}}\cos \Big[\frac{x_1}{\sqrt{i}}\Big]+1$| [−600,600] Multi peak Ackley |${f}_4(x)=20+e-20e\sqrt[-\frac{1}{5}]{\frac{1}{n}\sum \limits_{i=1}^n{x_i^2}_{-e}}-\frac{1}{n}\sum \limits_{i=1}^n\cos \Big(2\pi{x}_1\Big)$| [−32,32] Multi peak Schwefel |${f}_5(x)=418.9829\ n-\sum \limits_{i=1}^n{x}_i\sin \sqrt{\mid{x}_i\mid }$| [−100,500] Multi peak Open in new tab Table 1 Standard testing function. Testing function . Function . Search scope . Type . Rosenbrock |${f}_1(x)=\sum \limits_{i=1}^{n-1}\Big(100{\Big({x}_{i+1}-{x}_i^2\Big)}^2{\Big({x}_1-1\Big)}^2\Big)$| [−15,30] Single peak Rastrigrin |${f}_2(x)=\sum \limits_{i=1}^n\Big({x}_i^2-10\cos \Big(2\pi{x}_1\Big)+10\Big)$| [−5.12,5.12] Multi peak Griewank |${f}_3(x)=\frac{1}{4000}\sum \limits_{i=1}^n{x}_i^2-\underset{i=1}{\overset{n}{\Pi}}\cos \Big[\frac{x_1}{\sqrt{i}}\Big]+1$| [−600,600] Multi peak Ackley |${f}_4(x)=20+e-20e\sqrt[-\frac{1}{5}]{\frac{1}{n}\sum \limits_{i=1}^n{x_i^2}_{-e}}-\frac{1}{n}\sum \limits_{i=1}^n\cos \Big(2\pi{x}_1\Big)$| [−32,32] Multi peak Schwefel |${f}_5(x)=418.9829\ n-\sum \limits_{i=1}^n{x}_i\sin \sqrt{\mid{x}_i\mid }$| [−100,500] Multi peak Testing function . Function . Search scope . Type . Rosenbrock |${f}_1(x)=\sum \limits_{i=1}^{n-1}\Big(100{\Big({x}_{i+1}-{x}_i^2\Big)}^2{\Big({x}_1-1\Big)}^2\Big)$| [−15,30] Single peak Rastrigrin |${f}_2(x)=\sum \limits_{i=1}^n\Big({x}_i^2-10\cos \Big(2\pi{x}_1\Big)+10\Big)$| [−5.12,5.12] Multi peak Griewank |${f}_3(x)=\frac{1}{4000}\sum \limits_{i=1}^n{x}_i^2-\underset{i=1}{\overset{n}{\Pi}}\cos \Big[\frac{x_1}{\sqrt{i}}\Big]+1$| [−600,600] Multi peak Ackley |${f}_4(x)=20+e-20e\sqrt[-\frac{1}{5}]{\frac{1}{n}\sum \limits_{i=1}^n{x_i^2}_{-e}}-\frac{1}{n}\sum \limits_{i=1}^n\cos \Big(2\pi{x}_1\Big)$| [−32,32] Multi peak Schwefel |${f}_5(x)=418.9829\ n-\sum \limits_{i=1}^n{x}_i\sin \sqrt{\mid{x}_i\mid }$| [−100,500] Multi peak Open in new tab Table 2 The search success ratio and average iteration times after running 20 times. Function . Termination condition . Search success ratio (%) . Average iteration times . Average termination time (s) . QBFO . QPSO . IBFO . QBFO . QPSO . IBFO . QBFO . QPSO . IBFO . f1 1.0 100 0 0 668 / / 1.598 / / f2 1.0 100 0 0 606 / / 1.437 / / f3 1.0e-5 100 41 37 478 886 791 0.990 1.930 1.752 f4 1.0e-5 100 100 10 561 675 945 1.161 1.515 2.102 f5 1.0 100 0 0 722 / / 1.623 / / Function . Termination condition . Search success ratio (%) . Average iteration times . Average termination time (s) . QBFO . QPSO . IBFO . QBFO . QPSO . IBFO . QBFO . QPSO . IBFO . f1 1.0 100 0 0 668 / / 1.598 / / f2 1.0 100 0 0 606 / / 1.437 / / f3 1.0e-5 100 41 37 478 886 791 0.990 1.930 1.752 f4 1.0e-5 100 100 10 561 675 945 1.161 1.515 2.102 f5 1.0 100 0 0 722 / / 1.623 / / Open in new tab Table 2 The search success ratio and average iteration times after running 20 times. Function . Termination condition . Search success ratio (%) . Average iteration times . Average termination time (s) . QBFO . QPSO . IBFO . QBFO . QPSO . IBFO . QBFO . QPSO . IBFO . f1 1.0 100 0 0 668 / / 1.598 / / f2 1.0 100 0 0 606 / / 1.437 / / f3 1.0e-5 100 41 37 478 886 791 0.990 1.930 1.752 f4 1.0e-5 100 100 10 561 675 945 1.161 1.515 2.102 f5 1.0 100 0 0 722 / / 1.623 / / Function . Termination condition . Search success ratio (%) . Average iteration times . Average termination time (s) . QBFO . QPSO . IBFO . QBFO . QPSO . IBFO . QBFO . QPSO . IBFO . f1 1.0 100 0 0 668 / / 1.598 / / f2 1.0 100 0 0 606 / / 1.437 / / f3 1.0e-5 100 41 37 478 886 791 0.990 1.930 1.752 f4 1.0e-5 100 100 10 561 675 945 1.161 1.515 2.102 f5 1.0 100 0 0 722 / / 1.623 / / Open in new tab The state and the position of the individual bacteria are uncertain in the quantum space, and they are determined by wave function of |$\psi (Y,t)$|⁠. The probability density function of the individual position is represented as |${|\psi|}^2$|⁠. The attractive potential based on the delta potential well model is established on each dimension of the attractor. The potential energy function can be expressed as follows: $$ V(Y)=-\gamma \delta (Y) $$ where |$Y={x}_{id}-{P}_d$| is the distance between the individual position of |${x}_{id}$| and the attractor. It is introduced into Schrodinger (Eq.12) to obtain |$\psi (Y,t)$| (Eq.13) and |$Q(Y)$| (Eq.14) in each dimension, in which L represents the feature length of delta. $$\begin{equation} jh\frac{\partial }{\partial t}\psi \left(Y,t\right)=\left[-\frac{h^2}{2m}{\nabla}^2+V(Y)\right]\psi \left(Y,t\right) \end{equation}$$(12) $$\begin{equation} \psi (Y)=\frac{1}{\sqrt{L}}{e}^{-\mid Y\mid /L} \end{equation}$$(13) $$\begin{equation} Q(Y)={\left|\psi (Y)\right|}^2={\left|\psi \left({x}_{id}-{P}_d\right)\right|}^2=\frac{1}{L}{e}^{-2\mid{x}_{id}-{P}_d\mid /L} \end{equation}$$(14) The motion of the individual in the potential field follows the above |${|\psi |}^2$|⁠, and its position is uncertain. But in the practical application, the individual bacteria must have a certain position in any time, so the individual motion equation will be got through the Monte Carlo simulation in this paper. Take the random number of u in (0, 1), and |$u={e}^{-2\mid{x}_{id}-{P}_d\mid /L}$|⁠, then the position update equation of the d dimension variable of individual i is shown in Eq.15: $$\begin{equation} {x}_{id}={P}_d\pm \frac{L}{2}\ln \left(1/u\right),u\sim U\left(0,1\right) \end{equation}$$(15) $$\begin{equation} L=2\beta \cdot \mid mbest-{x}_{id}(t)\mid \end{equation}$$(16) In Eq. 16, |$\beta$| is contraction expansion coefficient, and |$\beta$|<1.782 is to ensure the convergence of the algorithm [22]. The method is to change |${\beta}_1$| into |${\beta}_2$| linearly following the evolutionary algebra (Eq.17), and mbest is the average value of the best position vector in the population (shown in Eq.18). $$\begin{equation} \beta =\frac{\left({\beta}_1-{\beta}_2\right)\times \left( MAXIER-t\right)}{MAXIER}+{\beta}_2 \end{equation}$$(17) $$\begin{equation} mbest=\sum \limits_{i=1}^M{P}_i/M={\left[\sum \limits_{i=1}^M{P}_{i1}/M,\sum \limits_{i=1}^M{P}_{i2}/M,\dots, \sum \limits_{i=1}^M{P}_{iD}/M\right]}^T \end{equation}$$(18) Considering the disadvantages of the fixed step, a dynamic indentation control strategy is proposed here to expand the search space under the premise of the convergence. The implementation steps of QBFO are as follows: (1) Initialize the parameters. Including the number of individual bacteria of s, migration times of Ned, reproductive times of Nre, chemotaxis times of Nc, swimming times of Ns and migration probability of Ped. (2) Initialize population. Generate s individual bacterial vector of xi randomly in the solution space. (3) Calculate the fitness function J of each individual bacteria. (4) Start to cycle. Transfer cycle l = 1:Ned; reproductive cycle k = 1:Nre; chemotaxis cycle j = 1:Nc. (5) Start chemotaxis. The following operation is performed for each bacteria i: (a) Turning: generate a random vector of |$\varDelta \in{R}^n$| to adjust the direction, and each element in vector |$\varDelta$| is a random number in [−1, 1]. Update the position of the individual bacteria |${x}_{id}(d=1,2,\dots, D)$| according to Eq. 10, and the rest D-1variables remain unchanged. $$\begin{equation} {x}_{id}\left(j+1,k,l\right)={x}_{id}\left(j,k,l\right)+C\left(i,j\right)\phi (i) \end{equation}$$(19) $$\begin{equation} C\left(i,0\right)=x\_{\max}_d-x\_{\min}_d \end{equation}$$(20) $$\begin{equation} \phi (i)=\frac{\varDelta (i)}{\sqrt{\varDelta^T(i)\varDelta (i)}} \end{equation}$$(21) In the formula, C(i,j) represents the step size of the forward swimming, |$\phi (i)$| represents the random direction after rotation. (b) Swimming: evaluate the fitness of |${x}_i(j+1,k,l)$|⁠, if the fitness is superior to |${x}_i(j,k,l)$|⁠, |${x}_i(j,k,l)$| will be replaced with |${x}_i(j+1,k,l)$| and swim in accordance with the direction of the move until the fitness value is no longer improved or reaches the maximum number of steps. (c) Make d = d + 1, if d = D, then go to step 5(d), otherwise, go to step 5(a) to continue to operate the next variable. (d) Make j = j + 1, and change the swimming step of individual bacteria according to the dynamic indentation strategy in Eq.22, where A is dynamic indentation coefficient. $$\begin{equation} C\left(i,j+1\right)=A\cdot C\left(i,j\right) \end{equation}$$(22) (6) Reproduction based on quantum behavior. After a complete chemotaxis cycle, the current individual best position and the global optimal position are to update. Calculate the average best position mbest of the population according to Eq. 19 and update its position according to Eq. 15. (7) Migration. All the bacteria are sorted according to the energy, and the bacteria (s, Ped) with low fitness is migrated according to random initialization. (8) Judge whether the cycle is complete. 4. Validation and analysis of experiment 4.1. The design of initial value Five classical functions (shown in Table 1) are introduced to test the performance of the QBFO (Solis F J 1981). The dimensions of the five testing functions are set to 30 in this paper. The number of population is 40, the coefficient of contraction-expansion |${\beta}_1$| =1, |${\beta}_2$| =0.5, the migration probability is 0.25, the number of migration |${N}_{ed}$| =3, the number of reproduction |${N}_{re}$| =5, the number of chemotaxis |${N}_c$| =20, the number of swimming |${N}_s$| =4 and the coefficient of dynamic indentation A = 0.7. Then the total number of chemotaxis iteration MAXIER= |${N}_{ed}\cdot{N}_{re}\cdot{N}_c$| =300. Table 3 Comparison of Kacem of different algorithms. n |$\times$| m . Obj . BFO . IBFO . QPSO . QBFO . rs1 . rs2 . rs1 . rs2 . rs1 . rs2 . rs3 . rs1 . rs2 . rs3 . rs4 . 4 |$\times$| 5 Tl 15 10 10 11 12 10 10 11 10 At 33 31 30 30 32 30 29 30 31 Ct 45.2 45.2 45.2 40.6 38.4 45.2 40.6 40.6 40.6 8 |$\times$| 8 Tl 14 15 14 14 13 14 15 13 14 14 13 At 78 74 76 74 75 74 72 74 74 72 74 Ct 62.5 62.5 60.7 60.8 60.8 60.7 62.5 60.4 60.2 62.5 59.8 10 |$\times$| 7 Tl 11 10 11 11 10 10 At 59 59 58 58 60 58 Ct 55.1 57.9 57.9 54.8 50.3 57.9 10 |$\times$| 10 Tl 6 6 7 5 6 6 5 6 5 At 44 42 40 41 40 40 39 40 40 Ct 25.6 28.2 28.2 25.4 25.4 28.2 25.5 25.4 28.2 15 |$\times$| 10 Tl 22 10 10 10 9 9 At 94 92 89 90 89 90 Ct 52.3 52.1 51.8 51.8 51.3 48.8 n |$\times$| m . Obj . BFO . IBFO . QPSO . QBFO . rs1 . rs2 . rs1 . rs2 . rs1 . rs2 . rs3 . rs1 . rs2 . rs3 . rs4 . 4 |$\times$| 5 Tl 15 10 10 11 12 10 10 11 10 At 33 31 30 30 32 30 29 30 31 Ct 45.2 45.2 45.2 40.6 38.4 45.2 40.6 40.6 40.6 8 |$\times$| 8 Tl 14 15 14 14 13 14 15 13 14 14 13 At 78 74 76 74 75 74 72 74 74 72 74 Ct 62.5 62.5 60.7 60.8 60.8 60.7 62.5 60.4 60.2 62.5 59.8 10 |$\times$| 7 Tl 11 10 11 11 10 10 At 59 59 58 58 60 58 Ct 55.1 57.9 57.9 54.8 50.3 57.9 10 |$\times$| 10 Tl 6 6 7 5 6 6 5 6 5 At 44 42 40 41 40 40 39 40 40 Ct 25.6 28.2 28.2 25.4 25.4 28.2 25.5 25.4 28.2 15 |$\times$| 10 Tl 22 10 10 10 9 9 At 94 92 89 90 89 90 Ct 52.3 52.1 51.8 51.8 51.3 48.8 Open in new tab Table 3 Comparison of Kacem of different algorithms. n |$\times$| m . Obj . BFO . IBFO . QPSO . QBFO . rs1 . rs2 . rs1 . rs2 . rs1 . rs2 . rs3 . rs1 . rs2 . rs3 . rs4 . 4 |$\times$| 5 Tl 15 10 10 11 12 10 10 11 10 At 33 31 30 30 32 30 29 30 31 Ct 45.2 45.2 45.2 40.6 38.4 45.2 40.6 40.6 40.6 8 |$\times$| 8 Tl 14 15 14 14 13 14 15 13 14 14 13 At 78 74 76 74 75 74 72 74 74 72 74 Ct 62.5 62.5 60.7 60.8 60.8 60.7 62.5 60.4 60.2 62.5 59.8 10 |$\times$| 7 Tl 11 10 11 11 10 10 At 59 59 58 58 60 58 Ct 55.1 57.9 57.9 54.8 50.3 57.9 10 |$\times$| 10 Tl 6 6 7 5 6 6 5 6 5 At 44 42 40 41 40 40 39 40 40 Ct 25.6 28.2 28.2 25.4 25.4 28.2 25.5 25.4 28.2 15 |$\times$| 10 Tl 22 10 10 10 9 9 At 94 92 89 90 89 90 Ct 52.3 52.1 51.8 51.8 51.3 48.8 n |$\times$| m . Obj . BFO . IBFO . QPSO . QBFO . rs1 . rs2 . rs1 . rs2 . rs1 . rs2 . rs3 . rs1 . rs2 . rs3 . rs4 . 4 |$\times$| 5 Tl 15 10 10 11 12 10 10 11 10 At 33 31 30 30 32 30 29 30 31 Ct 45.2 45.2 45.2 40.6 38.4 45.2 40.6 40.6 40.6 8 |$\times$| 8 Tl 14 15 14 14 13 14 15 13 14 14 13 At 78 74 76 74 75 74 72 74 74 72 74 Ct 62.5 62.5 60.7 60.8 60.8 60.7 62.5 60.4 60.2 62.5 59.8 10 |$\times$| 7 Tl 11 10 11 11 10 10 At 59 59 58 58 60 58 Ct 55.1 57.9 57.9 54.8 50.3 57.9 10 |$\times$| 10 Tl 6 6 7 5 6 6 5 6 5 At 44 42 40 41 40 40 39 40 40 Ct 25.6 28.2 28.2 25.4 25.4 28.2 25.5 25.4 28.2 15 |$\times$| 10 Tl 22 10 10 10 9 9 At 94 92 89 90 89 90 Ct 52.3 52.1 51.8 51.8 51.3 48.8 Open in new tab Table 4 Data of workpiece. No. . Parent workpiece . Name . Code . No. of processing . Mission . Work hours (m) . Department . 1 - Cardan shaft TF022000–87 1 Assemble 50 Assembly one 2 1 Balance block assembly TF022000–87 1 Assemble 20 Assembly one 3 1 Flange fork assembly TF022013/012–87 1 Assemble 20 Assembly one 4 1 Sliding fork assembly TF022008/012–87 1 Assemble 20 Assembly one 5 1 Spline shaft fork assembly TF022011/012–87 1 Assemble 20 Assembly one 6 1 End cover TF022007–87 1 Assemble 25 Assembly two 7 1 Anti-off nut TF022009–87 1 Assemble 15 Assembly one 8 1 Lining tile TF022010–87 1 Roughcast 0 Processing second class 2 Rough car 8 Processing second class 3 Milling 18 Processing second class 4 Finishing car 20 Processing class 5 Cleanup. Lapping. Assembly welding 10 Small Link Class 9 4 Sliding fork TF021002–87 1 Roughcast 0 Processing second class 2 Rough car 40 Processing second class 3 Mark(1) 8 Processing class 4 Boring and milling inside 30 Processing second class 5 Drill 2 Processing class 6 Boring 12 Processing second class 7 Quenching and tempering 35 Heat treatment class 8 Finishing car 50 Processing class 9 Mark(2) 6 Processing class 10 Tensile spline 55 Processing second class 11 Grinding process surface 26 Processing second class No. . Parent workpiece . Name . Code . No. of processing . Mission . Work hours (m) . Department . 1 - Cardan shaft TF022000–87 1 Assemble 50 Assembly one 2 1 Balance block assembly TF022000–87 1 Assemble 20 Assembly one 3 1 Flange fork assembly TF022013/012–87 1 Assemble 20 Assembly one 4 1 Sliding fork assembly TF022008/012–87 1 Assemble 20 Assembly one 5 1 Spline shaft fork assembly TF022011/012–87 1 Assemble 20 Assembly one 6 1 End cover TF022007–87 1 Assemble 25 Assembly two 7 1 Anti-off nut TF022009–87 1 Assemble 15 Assembly one 8 1 Lining tile TF022010–87 1 Roughcast 0 Processing second class 2 Rough car 8 Processing second class 3 Milling 18 Processing second class 4 Finishing car 20 Processing class 5 Cleanup. Lapping. Assembly welding 10 Small Link Class 9 4 Sliding fork TF021002–87 1 Roughcast 0 Processing second class 2 Rough car 40 Processing second class 3 Mark(1) 8 Processing class 4 Boring and milling inside 30 Processing second class 5 Drill 2 Processing class 6 Boring 12 Processing second class 7 Quenching and tempering 35 Heat treatment class 8 Finishing car 50 Processing class 9 Mark(2) 6 Processing class 10 Tensile spline 55 Processing second class 11 Grinding process surface 26 Processing second class Open in new tab Table 4 Data of workpiece. No. . Parent workpiece . Name . Code . No. of processing . Mission . Work hours (m) . Department . 1 - Cardan shaft TF022000–87 1 Assemble 50 Assembly one 2 1 Balance block assembly TF022000–87 1 Assemble 20 Assembly one 3 1 Flange fork assembly TF022013/012–87 1 Assemble 20 Assembly one 4 1 Sliding fork assembly TF022008/012–87 1 Assemble 20 Assembly one 5 1 Spline shaft fork assembly TF022011/012–87 1 Assemble 20 Assembly one 6 1 End cover TF022007–87 1 Assemble 25 Assembly two 7 1 Anti-off nut TF022009–87 1 Assemble 15 Assembly one 8 1 Lining tile TF022010–87 1 Roughcast 0 Processing second class 2 Rough car 8 Processing second class 3 Milling 18 Processing second class 4 Finishing car 20 Processing class 5 Cleanup. Lapping. Assembly welding 10 Small Link Class 9 4 Sliding fork TF021002–87 1 Roughcast 0 Processing second class 2 Rough car 40 Processing second class 3 Mark(1) 8 Processing class 4 Boring and milling inside 30 Processing second class 5 Drill 2 Processing class 6 Boring 12 Processing second class 7 Quenching and tempering 35 Heat treatment class 8 Finishing car 50 Processing class 9 Mark(2) 6 Processing class 10 Tensile spline 55 Processing second class 11 Grinding process surface 26 Processing second class No. . Parent workpiece . Name . Code . No. of processing . Mission . Work hours (m) . Department . 1 - Cardan shaft TF022000–87 1 Assemble 50 Assembly one 2 1 Balance block assembly TF022000–87 1 Assemble 20 Assembly one 3 1 Flange fork assembly TF022013/012–87 1 Assemble 20 Assembly one 4 1 Sliding fork assembly TF022008/012–87 1 Assemble 20 Assembly one 5 1 Spline shaft fork assembly TF022011/012–87 1 Assemble 20 Assembly one 6 1 End cover TF022007–87 1 Assemble 25 Assembly two 7 1 Anti-off nut TF022009–87 1 Assemble 15 Assembly one 8 1 Lining tile TF022010–87 1 Roughcast 0 Processing second class 2 Rough car 8 Processing second class 3 Milling 18 Processing second class 4 Finishing car 20 Processing class 5 Cleanup. Lapping. Assembly welding 10 Small Link Class 9 4 Sliding fork TF021002–87 1 Roughcast 0 Processing second class 2 Rough car 40 Processing second class 3 Mark(1) 8 Processing class 4 Boring and milling inside 30 Processing second class 5 Drill 2 Processing class 6 Boring 12 Processing second class 7 Quenching and tempering 35 Heat treatment class 8 Finishing car 50 Processing class 9 Mark(2) 6 Processing class 10 Tensile spline 55 Processing second class 11 Grinding process surface 26 Processing second class Open in new tab 4.2. Performance evaluation In order to evaluate the time complexity of QBFO, the performance was evaluated in three aspects: the convergence process of the optimal solution under fixed evolutionary algebra, the average evolution algebra and the success rate (shown in Table 2). The former reflects the convergence speed of the algorithm to a certain extent, the average algebra reflects the evolution speed and the success rate reflects the reliability of the algorithm. The number of the bacteria swimming is recorded as 4, and it is considered that the algorithm has been trapped in local optimal solution if the given value has not been obtained after the iteration number being 1000. QBFO is compared with some existing classical algorithms including the improved bacterial foraging optimization (IBFO) [26] and quantum particle swarm optimization (QPSO) [11, 26, 27]. It can be seen from Table 2 that the success rate of using QPSO and IBFO is low, while the success rate of using QBFO can reach 100%. For f3 and f4, it can achieve a certain success rate using QPSO and IBFO, but the evolution algebra is significantly increased at the same time and the reliability of the two algorithms is poor. For f1 and f2, because there is steep region near the optimal value, it is easy to lie in premature convergence. The complexity of the f5 depends on the local optimal solution far from the global optimal solution, so it is difficult for QPSO and IBFO to search the global extremum and achieve the function optimization. Therefore, it is considered that QBFO has high applicability and can obtain the global optimal solution for multi peak and high dimension problem. 4.3. Case study Aiming at validating the performance of the proposed QBFO and minimizing the average completion time f1, carbon emissions f2 and the total cost f3, the authors took the classic Kacem as an example and designed the population size as 70, the maximum iteration number as 100. Table 5 The mean value and standard deviation of each scheduling target. Algorithm . Objective . Average value . SD . IBFO Makespan (m) 318.071 12.633 Carbon emission (kg) 227.401 2.686 Maximum load (m) 244.623 6.1 Weighted target (%) 67.799 0.011 QPSO Makespan (m) 298.297 5.784 Carbon emission (kg) 222.116 1.235 Maximum load (m) 232.291 2.901 Weighted target (%) 65.099 0.005 QBFO Makespan (m) 292.952 5.596 Carbon emission (kg) 220.263 0.572 Maximum load (m) 232.062 3.092 Weighted target (%) 61.789 0.005 Algorithm . Objective . Average value . SD . IBFO Makespan (m) 318.071 12.633 Carbon emission (kg) 227.401 2.686 Maximum load (m) 244.623 6.1 Weighted target (%) 67.799 0.011 QPSO Makespan (m) 298.297 5.784 Carbon emission (kg) 222.116 1.235 Maximum load (m) 232.291 2.901 Weighted target (%) 65.099 0.005 QBFO Makespan (m) 292.952 5.596 Carbon emission (kg) 220.263 0.572 Maximum load (m) 232.062 3.092 Weighted target (%) 61.789 0.005 Open in new tab Table 5 The mean value and standard deviation of each scheduling target. Algorithm . Objective . Average value . SD . IBFO Makespan (m) 318.071 12.633 Carbon emission (kg) 227.401 2.686 Maximum load (m) 244.623 6.1 Weighted target (%) 67.799 0.011 QPSO Makespan (m) 298.297 5.784 Carbon emission (kg) 222.116 1.235 Maximum load (m) 232.291 2.901 Weighted target (%) 65.099 0.005 QBFO Makespan (m) 292.952 5.596 Carbon emission (kg) 220.263 0.572 Maximum load (m) 232.062 3.092 Weighted target (%) 61.789 0.005 Algorithm . Objective . Average value . SD . IBFO Makespan (m) 318.071 12.633 Carbon emission (kg) 227.401 2.686 Maximum load (m) 244.623 6.1 Weighted target (%) 67.799 0.011 QPSO Makespan (m) 298.297 5.784 Carbon emission (kg) 222.116 1.235 Maximum load (m) 232.291 2.901 Weighted target (%) 65.099 0.005 QBFO Makespan (m) 292.952 5.596 Carbon emission (kg) 220.263 0.572 Maximum load (m) 232.062 3.092 Weighted target (%) 61.789 0.005 Open in new tab Table 6 ANOVA analysis of maximum makespan under different algorithms. . Sum of squares . Expectation function . Mean square . F-test . Sig . Between groups 560.780 2 279.800 8.059 0.001 Within groups 3018.300 87 34.650 Total 3579.080 89 . Sum of squares . Expectation function . Mean square . F-test . Sig . Between groups 560.780 2 279.800 8.059 0.001 Within groups 3018.300 87 34.650 Total 3579.080 89 Open in new tab Table 6 ANOVA analysis of maximum makespan under different algorithms. . Sum of squares . Expectation function . Mean square . F-test . Sig . Between groups 560.780 2 279.800 8.059 0.001 Within groups 3018.300 87 34.650 Total 3579.080 89 . Sum of squares . Expectation function . Mean square . F-test . Sig . Between groups 560.780 2 279.800 8.059 0.001 Within groups 3018.300 87 34.650 Total 3579.080 89 Open in new tab Table 7 The ANOVA analysis result of carbon emission under different algorithms. . Sum of squares . Expectation function . Mean square . F-test . Sig . Between groups 816.509 2 407.638 117.338 0.000 Within groups 301.682 87 3.479 Total 1118.191 89 . Sum of squares . Expectation function . Mean square . F-test . Sig . Between groups 816.509 2 407.638 117.338 0.000 Within groups 301.682 87 3.479 Total 1118.191 89 Open in new tab Table 7 The ANOVA analysis result of carbon emission under different algorithms. . Sum of squares . Expectation function . Mean square . F-test . Sig . Between groups 816.509 2 407.638 117.338 0.000 Within groups 301.682 87 3.479 Total 1118.191 89 . Sum of squares . Expectation function . Mean square . F-test . Sig . Between groups 816.509 2 407.638 117.338 0.000 Within groups 301.682 87 3.479 Total 1118.191 89 Open in new tab Table 8 The ANOVA analysis result of maximum load of machine under different algorithms. . Sum of squares . Expectation function . Mean square . F-test . Sig . Between groups 4278.253 2 2139.204 107.273 0.000 Within groups 1733.938 87 19.838 Total 6012.191 89 . Sum of squares . Expectation function . Mean square . F-test . Sig . Between groups 4278.253 2 2139.204 107.273 0.000 Within groups 1733.938 87 19.838 Total 6012.191 89 Open in new tab Table 8 The ANOVA analysis result of maximum load of machine under different algorithms. . Sum of squares . Expectation function . Mean square . F-test . Sig . Between groups 4278.253 2 2139.204 107.273 0.000 Within groups 1733.938 87 19.838 Total 6012.191 89 . Sum of squares . Expectation function . Mean square . F-test . Sig . Between groups 4278.253 2 2139.204 107.273 0.000 Within groups 1733.938 87 19.838 Total 6012.191 89 Open in new tab Table 9 The ANOVA analysis result of weighted targets under different algorithms. . Sum of squares . Expectation function . Mean square . F-test . Sig . Between groups 0.023 2 0.013 196.891 0.000 Within groups 0.006 87 0.000 Total 0.029 89 . Sum of squares . Expectation function . Mean square . F-test . Sig . Between groups 0.023 2 0.013 196.891 0.000 Within groups 0.006 87 0.000 Total 0.029 89 Open in new tab Table 9 The ANOVA analysis result of weighted targets under different algorithms. . Sum of squares . Expectation function . Mean square . F-test . Sig . Between groups 0.023 2 0.013 196.891 0.000 Within groups 0.006 87 0.000 Total 0.029 89 . Sum of squares . Expectation function . Mean square . F-test . Sig . Between groups 0.023 2 0.013 196.891 0.000 Within groups 0.006 87 0.000 Total 0.029 89 Open in new tab Five kinds of scale standard problems of Kacem (4 |$\times$| 5, 8 |$\times$| 8, 10 |$\times$| 7, 10 |$\times$| 10, 15 |$\times$| 10) is solved with QBFO comparing with the existing algorithms including BFO [1], IBFO and QPSO, the data of comparison result are shown in Table 3. In Table 3, n represents the number of jobs; m represents the number of machines; rsx (x = 1, 2, 3, 4) represents different solutions obtained by the algorithm; Tl represents the maximum completion time of the machine (unit: min); At represents the cost of the job (unit: CNY); Ct represents the total carbon emission of the machine (unit: kg). It can be seen from the results with different algorithms in Table 3 that the QBFO proposed in this paper can obtain more non-dominated solutions. Taking the 10 |$\times$| 7 problem as an example, although the QPSO and QBFO algorithms both get three non-dominated solutions: (11,59,55.1), (10,59,57.9), (11,58,57.9) and (11,58, 54.8), (10,60,50.3), (10,58,57.9), but the solution of QPSO algorithm (11,59,55.1) is dominated by the solution (11,58,54.8) of QBFO algorithm, the solutions (11,58,57.9) of QPSO algorithm are dominated by the solutions (11,58,54.8) and (10,58,57.9) of QBFO algorithm. Figure 1 Open in new tabDownload slide Comparison of different algorithms about carbon emission. Figure 1 Open in new tabDownload slide Comparison of different algorithms about carbon emission. 4.4. Comparison test of algorithm In order to test the performance of the improved QBFO in solving the low-carbon FJSP, MATLAB R2010b was used in the experiment, and the platform has 2.8 GHz Intel Core I6 CPU and 4 GB RAM. The operation termination condition is up to 200 seconds. In order to reflect the influence of parameter optimization on carbon emission and maximum completion time, the parameters in literature [8] are used in the experiment as shown in Table 4. For the above example, IBFO, QPSO and QBFO were tested and run 50 times, respectively. The average value and standard deviation of the obtained results are shown in Table 5. It can be seen from Table 5 that in 50 experiments, the standard deviation of the maximum load of QBFO is slightly higher than the standard deviation of QPSO, but in comparison, the average makespan and standard deviation of QBFO are 7.9% and 55.7% better than IBFO. Compared with QPSO, it is optimized by 1.8% and 3.3%. The average carbon emissions and standard deviation of QBFO are 3.1% and 78.7% better than IBFO. Compared with QPSO, it is optimized by 0.8% and 53.7%. The average maximum load and standard deviation of QBFO are 5.1% and 49.3% better than IBFO. The average weighted target and standard deviation of the QBFO are optimized by 8.9% and 54.5% compared to IBFO, and the mean is optimized by 5.1% compared to QPSO, but the standard deviations of the three are the same. Therefore, for the three algorithms, QBFO has the best solution. The algorithm not only has better targets but also has the smallest standard deviation. It shows that the efficient and improved QBFO algorithm designed in the study has the best convergence and stability in solving FJSP for low-carbon optimization, and the comparison result is shown in Figure 1. The results of the analysis of variance for each target obtained by the three algorithms are shown in Tables 6, 7, 8 and 9: As can be seen from Tables 6, 7, 8 and 9, there are four goals to consider: maximum completion time, carbon emissions, maximum machine load and weighted goals. The solution performance of different algorithms is statistically significant, that is, Sig <0.05. The comparison of the three algorithms shows that the solution performance and robustness of the QBFO algorithm are better than IBFO and QPSO. 5. Conclusion Aiming at the low-carbon management of FJSP, carbon emission model and operational research knowledge are combined in this paper. (1) An integrated optimization model of shop scheduling with the objective of minimizing carbon emissions and minimizing makespan in manufacturing process is proposed. According to the feature of multiple optimization parameters in the integrated model and the carbon emission equations, the optimized quantum bacterial foraging method is designed to sort the population and obtain good individuals. (2) The possibility of improved quantum bacterial foraging and shop scheduling integration optimization to further reduce carbon emissions in the manufacturing process is explored. During formulating the scheduling scheme, only performance indicators such as machine processing time are involved, and other environmental factors such as peak power of machine processing are not considered. (3) Though example verification and comparison with the three commonly used algorithms, each target value is obtained, and then the objective parameters is analysed by variance. The effectiveness and advance of the proposed algorithm is verified through comparing with the existing classical algorithms. The next step of research work should lie in considering how to reduce the electricity consumption of the self-serving cabinets at the time of decreasing the cost and improving the efficiency of the manufacturing, which is to achieve the aim of green manufacturing. Declaration of competing interest. None. This research did not receive any specific grant from funding agencies in the public, commercial or not-for-profit sectors. Acknowledgments Foundation items: project supported by the Liaoning Provincial Natural Science Foundation (20180550499, 2019-ZD-0109) and the Education Department Project of Liaoning Province (JDL2019022). References 1 Mishra A , Shrivastava D. A TLBO and a Jaya heuristics for permutation flow shop scheduling to minimize the sum of inventory holding and batch delay costs . Comput Ind Eng 2018 ; 124 : 509 – 22 . Google Scholar Crossref Search ADS WorldCat 2 Shan G-l , Xu G-g, Qiao C-l. A non-myopic scheduling method of radar sensors for maneuvering target tracking and radiation control . Def Technol 2020 ; 16 : 242 – 50 . Google Scholar Crossref Search ADS WorldCat 3 Vijaychakaravarthy G , Marimuthu S, Naveen Sait A. Comparison of improved sheep flock heredity algorithm and artificial bee colony algorithm for lot streaming in m-machine flow shop scheduling . Arabian J Sci Eng 2014 ; 39 : 4285 – 300 . Google Scholar Crossref Search ADS WorldCat 4 Piroozfard H , Wong KY, Wong WP. Minimizing total carbon footprint and total late work criterion in flexible job shop scheduling by using an improved multi-objective genetic algorithm . Resour Conserv Recycl 2018 ; 128 : 267 – 83 . Google Scholar Crossref Search ADS WorldCat 5 Jiang T . Low-carbon workshop scheduling problem based on grey wolf optimization . Comput Integr Manuf Syst 2018 ; 24 : 2428 – 35 . Google Scholar OpenURL Placeholder Text WorldCat 6 Li J-q , Sang H-y, Han Y-y et al. Efficient multi-objective optimization algorithm for hybrid flow shop scheduling problems with setup energy consumptions . J Clean Prod 2018 ; 181 : 584 – 98 . Google Scholar Crossref Search ADS WorldCat 7 Keller RA , Mehran N, Austin W et al. Athletic performance at the NFL scouting combine after anterior cruciate ligament reconstruction . Am J Sports Med 2015 ; 43 : 3022 – 6 . Google Scholar Crossref Search ADS PubMed WorldCat 8 Sheremetov L , Martínez-Muñoz J, Chi-Chim M. Two-stage genetic algorithm for parallel machines scheduling problem: cyclic steam stimulation of high viscosity oil reservoirs . Appl Soft Comput 2018 ; 64 : 317 – 30 . Google Scholar Crossref Search ADS WorldCat 9 Li M , Lei D. Novel imperialist competitive algorithm for many-objective flexible job shop scheduling . Control Theory Appl 2019a ; 36 : 893 – 901 . Google Scholar OpenURL Placeholder Text WorldCat 10 Li M , Lei D. Research on flexible job shop low carbon scheduling with setup times and key objectives . J Mech Eng 2019b ; 21 : 139 – 49 . Google Scholar OpenURL Placeholder Text WorldCat 11 Liu C , Xu Z, Zhang Q et al. Green scheduling of flexible job shops based on NSGA-II under TOU power Price . China Mech Eng 2020 ; 31 : 576 – 85 . Google Scholar OpenURL Placeholder Text WorldCat 12 Ning T , Wang XP, Hu XP. Disruption management decision model for vehicle scheduling under disruption delay . Syst Eng Theory Pract 2019 ; 39 : 1236 – 45 . Google Scholar OpenURL Placeholder Text WorldCat 13 Ning T , Wang XP. Study on disruption management strategy of job-shop scheduling problem based on prospect theory . J Clean Prod 2018 ; 194 : 174 – 8 . Google Scholar Crossref Search ADS WorldCat 14 Ning T , Huang M, Liang X et al. A novel dynamic scheduling strategy for solving flexible job-shop problems . J Ambient Intell Humaniz Comput 2016 ; 25 : 721 – 9 . Google Scholar Crossref Search ADS WorldCat 15 Ning T , Hua J, Xudong S et al. An improved quantum genetic algorithm based on MAGTD for dynamic FJSP . J Ambient Intell Humaniz Comput 2018a ; 9 : 931 – 40 . Google Scholar Crossref Search ADS WorldCat 16 Saber AY . Economic dispatch using particle swarm optimization bacterial foraging effect . Electr Power Energy Syst 2012 ; 34 : 38 – 46 . Google Scholar Crossref Search ADS WorldCat 17 Wang S , Wang X, Yu J et al. Bi-objective identical parallel machine scheduling to minimize total energy consumption and makespan . J Clean Prod 2018a ; 193 : 424 – 40 . Google Scholar Crossref Search ADS WorldCat 18 Kim S , Meng C, Son Y-J. Simulation-based machine shop operations scheduling system for energy cost reduction . Simul Model Pract Theory 2017 ; 77 : 68 – 83 . Google Scholar Crossref Search ADS WorldCat 19 Ning T , Wang XP, Hu X et al. Disruption management optimal scheduling for logistics distribution based on Prospect theory . Control Decis 2018b ; 33 : 2064 – 8 . Google Scholar OpenURL Placeholder Text WorldCat 20 Ning T , Wang Z, Zhang P et al. Integrated optimization of disruption management and scheduling for reducing carbon emission in manufacturing . J Clean Prod 2020 ; 263 : 1 – 8 . Google Scholar Crossref Search ADS WorldCat 21 Wang L , Wang J-j, Wu C-g. Advances in green shop scheduling and optimization . Control Decis 2018b ; 33 : 385 – 91 . Google Scholar OpenURL Placeholder Text WorldCat 22 Wang W . Research on logistics delivery networks of online retailers . J Syst Manag Sci 2013 ; 3 : 51 – 9 . Google Scholar OpenURL Placeholder Text WorldCat 23 Wu X , Sun Y. Flexible job shop green scheduling problem with multi-speed machine . Comput Integr Manuf Syst 2018a ; 24 : 862 – 75 . Google Scholar OpenURL Placeholder Text WorldCat 24 Wu X , Sun Y. A green scheduling algorithm for flexible job shop with energy-saving measures . J Clean Prod 2018b ; 172 : 3249 – 64 . Google Scholar Crossref Search ADS WorldCat 25 Xu G , De Pessemier , Martens L et al. Energy- and labor-aware flexible job shop scheduling under dynamic electricity pricing: a many-objective optimization investigation . J Clean Prod 2019 ; 209 : 1078 – 94 . Google Scholar Crossref Search ADS WorldCat 26 Yang X-l , Hu R, Qian B et al. Enhanced estimation of distribution algorithm for low carbon scheduling of distributed flow shop problem . Control Theory Appl 2019 ; 36 : 803 – 15 . Google Scholar OpenURL Placeholder Text WorldCat 27 Zhang G-y , Wu Y-g, Tan Y-x. Bacterial foraging optimization algorithm with quantum behavior . J Electron Inf Technol 2013 ; 35 : 614 – 21 . Google Scholar Crossref Search ADS WorldCat 28 Zheng X , Ling W. A knowledge-guided fruit fly optimization algorithm for dual resource constrained flexible job-shop scheduling problem . Int J Prod Res 2018 ; 54 : 1 – 13 . Google Scholar OpenURL Placeholder Text WorldCat © The Author(s) 2021. Published by Oxford University Press. This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/4.0/), which permits non-commercial re-use, distribution, and reproduction in any medium, provided the original work is properly cited. For commercial re-use, please contact journals.permissions@oup.com
TI - Research on flexible job shop scheduling with low-carbon technology based on quantum bacterial foraging optimization
JF - International Journal of Low-Carbon Technologies
DO - 10.1093/ijlct/ctab005
DA - 2021-01-29
UR - https://www.deepdyve.com/lp/oxford-university-press/research-on-flexible-job-shop-scheduling-with-low-carbon-technology-zrnMyRJZKm
SP - 1
EP - 1
VL - Advance Article
IS - 
DP - DeepDyve
ER -