1. INTRODUCTION
Energy and power plants are one of the important parts of life in the 21^{st} century and are one of vital resources for production. Therefore, given our access to energy, energy consumption is of paramount importance to the people. Overall, focusing on energy consumption during planning is a way to increase energy efficiency in production (Amit and Song, 2021;Tan et al., 2021). In this regard, planning is the process of creating a balance between supply and demand. In addition, the concept of scheduling defines the order and time allocated to the activities of a task. In other words, scheduling is the structure of a detailed timeline that shows the date and time of initiation and cessation of tasks (Baybalinova et al., 2021;Slack, 1999). In a shop manufacturing schedule method, each product has its own specific sequence, and scheduling of resources must be carried out in a way that the desired goals could be achieved. Some of the objectives of a shop scheduling problem include reducing the delivery time of the last work, decreasing the average running time, and reducing the average earliness or tardiness of tasks (Fang et al., 2011;Seow and Rahimifard, 2011). In general, a shop scheduling problem is defined as finding an optimal sequence and performing various operations related to each machine while considering maximum constraints and different variables and permutations (Liu et al., 2013).
In recent decades, researchers’ attention to the decrease of energy consumption has increased, for which they have more focused on production planning. In a research, Dai et al. (2013) evaluated energyefficient scheduling for a flexible flow shop while taking two objectives of completion time and energy consumption into account. They proposed a mathematical model for the problem of a flexible shop flow environment. The problem was solved based on an efficient energy mechanism, and simulated annealing was applied to assess the variables of completion time and energy consumption. In the end, a case study of a production scheduling problem for a metalworking workshop in a plant was simulated. In another study, Fang et al. (Mansouri et al., 2015;Zhang et al., 2021) proposed a multiobjective integer linear programming formulation for optimization of scheduling operations in a shop flow environment while considering productivity (completion time) and energy (peak power load and associated carbon footprint). The model was assessed by a case study in a flow shop where two machines were employed to produce a variety of parts. Gilmanova et al. (2021) considered the energy consumption minimization in permutation flow shop (PFS) scheduling problem. According to these scholars, the energy consumption of each machine is decomposed into two parts: the useful part which completes the operation at the current stage and the wasted part which is consumed during the idle period. In the foregoing study, the objective was to minimize the total wasted energy consumption which is a weighted summation of the idle time of each machine. Ultimately, a branchandbound algorithm was developed to solve the problem. In another study, Mansouri et al. (2015) introduced a twomachine sequencedependent PFS to find the Pareto frontier comprised of makespan and total energy consumption. The mathematical model proposed in the mentioned study is comprised of two service and energy consumption levels. Using variable speeds and preliminary startup times, the model aimed to reduce inherently conflicting completion times and energy consumption. In addition, lower bounds were used for the model, which was outdated and still usable for all problems. Ultimately, the researchers used the exploratory algorithm, which is applied for big problems. Moreover, Tang et al. (2015) presented an article entitled “energyefficient dynamic scheduling for a flexible flow shop using an improved particle swarm optimization”. The primary goal was to address the dynamic scheduling problem to reduce energy consumption and makespan for a flexible flow shop scheduling. Since the problem was strongly NPhard, a novel algorithm based on an improved particle swarm optimization was adopted to search for the Pareto optimal solution in dynamic flexible flow shop scheduling problems. Finally, numerical experiments were performed to evaluate the performance and efficiency of the proposed approach.
2. MATHEMATICAL MODELING AND STATEMENT OF THE PROBLEM
In most operating workshops and industries, tasks with multiple operations are performed on several different machines. The shop flow model is created when the path of all tasks is specified (e.g., all tasks are performed on machines with a certain order) (Song, 2021). In the present study, the mathematical modeling technique is used to express the optimization problem. The mathematical model is a threeobjective model; the first objective is to reduce the time of tardiness, whereas the second and third goals are to decrease the completion time and energy consumption, respectively. In this study, we evaluate the multimachine PFS problem, where machines have variable speeds, based on the relationship between tardiness and energy consumption. Our primary objective is to present a mixedinteger programming model to eliminate the manufacturing scheduling problem with the attitude of energy consumption reduction. The model premises are, as follows (Table 1):
Work time in the environment is definite. The machines have variable speeds. There is a processing order for tasks. Work is performed at variable speeds on dedicated machines. Processing of works is done in a flow shop environment.
The model presented in the current study is based on the model developed by Setiawan (2021), who evaluated the relationship between completion time and energy consumption by proposing a twomachine shop flow problem, where machines had variable speeds. According to these scholars, machining operations with variable speeds caused energy consumption to vary at different levels, and a specific speed was considered for processing tasks. Indexes and parameters are as follows:

i Index of machines (i1,…,m)

j,k Index of works (j,k1,…,n)

l Index of processing speed

n Number of works

m Number of machines

v_{l} Processing speed factor (12, 1, 3, processing at low, moderate and high speeds, respectively)

a_{l} The conversion factor for the lth processing speed

_{i}β The conversion factor for the idle time of the ith machine

M A very big number

p_{ij} The processing time of the jth work on the ith machine

c_{ij} The completion time of the jth work on the ith machine

θ_{i} The idle time of the ith machine

C_{max} Completion time of the work

T_{max} Time of tardiness

T_{i} Tardiness

L_{j} Delay in the jth work

D_{j} Time of delivery of the jth work

c_{j} Completion time of the jth work

TEC Total energy consumption (kilowatt/hour)
Indexes and parameters:

y_{jk} 1, if the jth work is immediately scheduled before the kth work; otherwise, 0.

x_{ijt} 1, if the jth work is processed on the ith machine; otherwise, 0.

Ω_{j} 1, if the jth work is the first work; otherwise, 0.
The model is described after defining the symbols used in the model.
Objective Function
Constraints
Constraints (4) show that a speed factor is selected for each work. Constraints (5) demonstrate the completion time of work on the first machine. Constraints (6) show the completion time of the last work. Constraints (7) guarantee that the completion time of consecutive works is increasing (based on the completion time of the previous work). Moreover, constraints (8) indicate that the total completion time must be higher than the completion time of the last work on the last machine. Constraints (9) show the machine idle time, and constraints (10) mark the total energy consumption level (kilowatt/hour). Constraints (11) show the completion time of the jth work. Based on constraints (12), delay indicates the time it takes for the work to be completed before the due date. According to constraints (13), tardiness shows the amount of delay in the completion of the work, compared to the delivery time. Constraints (14) show the total tardiness while constraints (15) guarantee that there is only one first work. Constraints (16) and (17) show the observance of the sequence of works, and constraints (18) show the binary and nonnegative variables. Note: Ω_{j} determines the first work and all completion time computations are done based on this factor. Finally, constraints (7) are necessary for consecutive works, and the last work is linked to the first work in the model.
3. NUMERICAL EXAMPLE SOLUTION AND ANALYSIS OF RESULTS
A numerical example is presented to assess the model. Given the fact that the model is of multiobjective type, it is solved using the weighted summation method (giving weight to each of the objective functions). The information related to these examples is shown in Table 1. In total, there are four machines and six works. In addition, the objective functions are obtained in four scenarios.
Start of works from zero time
Factor of conversion for processing speed ${\alpha}_{l}=1.5,\hspace{0.17em}1,\hspace{0.17em}0.6$
Factor of conversion for machine idle time ${\beta}_{l}=0.2,\hspace{0.17em}0.5,\hspace{0.17em}0.3,0.6$
The factor of processing speed (for processing with low, normal and high speeds) ${v}_{l}=1.2,\hspace{0.17em}1,\hspace{0.17em}0.8$
Delivery date ${d}_{l}=9,8,7,6,5$
The GAMS software is used to solve the model, the results of which are presented below.
In the first scenario, the first objective T_{max} is weighted one, and the rest of the objective functions are weighted zero. The sequence of works obtained in the present study is shown in the Figure 1.
In the second scenario, the second objective function C_{max} is weighted one, and the rest of the objective functions are weighted zero. The sequence of works obtained is shown in the Figure 2.
In the third scenario, the third objective function TEC is weighted one and the rest of the objective functions are weighted zero. The sequence of works obtained is presented in the Figure 3.
In the fourth scenario, each objective function is given an equal weight of 0.3333. The sequence of works obtained is presented in the Figure 4.
Table 2 presents information obtained from the optimal solution of the problem while considering each of the objective functions.
According to the solutions obtained from the GAMS software, work 4 is the first processed task in all scenarios. Scheduling difference in the first and second scenarios: given the zero weight of TEC, energy consumption is not important and is mostly used to minimize the completion time and delay time of works. The machines work at a high speed and energy consumption increases. Scheduling difference in the second and third scenarios: given the zero weight of completion time, energy consumption decreases and time of tardiness and completion time of works increase.
4. CONCLUSION
In this study, we proposed a mathematical model related to the desired optimization (i.e., manufacturing scheduling with the attitude of energy consumption decrease in a shop flow environment). The model was unique in terms of innovation since no similar study has been carried out on the simultaneous decrease of tardiness completion time and energy consumption. The model was confirmed by the GAMS software, which is one of the most robust optimization software. Given the type of the proposed model (i.e., mixed integer programming), CPLEX in GAMS was applied to solve the example. The model primarily aimed to minimize the tardiness completion time and energy consumption. According to the results, delays in works decreased by an increase in the tardiness time. Meanwhile, energy consumption increased in the example. Given the variable processing speed of the machines, the objective functions improved by an increase in the processing speed. In addition, increased idle time of the machines led to a decrease in their energy consumption. Given the complexity of the problem at realworld sizes, it is recommended that more attention is heeded to the use of heuristic and metaheuristic techniques in future studies.