Energy‐efficient and sustainable supply chain in the manufacturing industry

This study aims at reducing energy consumption in supply chain networks by providing optimal integrated production and transportation scheduling. The considered supply chain consists of one main manufacturing center, multiple production units (i.e., suppliers), and multiple heterogeneous vehicles as the transportation fleet. To schedule this complex supply chain network in an energy‐efficient way, several decisions should be made concerning the assignment of orders to suppliers and determining their production sequence, splitting orders, assigning orders to vehicles, and assigning delivery priority to orders. To cope with the problem, a mixed‐integer linear programming model is presented. Due to the complexity of the problem, a novel development of the genetic algorithm named the Multiple Reference Group Genetic Algorithm (MRGGA) is also proposed. Four objectives are considered to be optimized to meet both suitability and energy‐efficiency aspects in the supply chain network. These optimization objectives are to minimize the total orders' delivery times to the manufacturing center, fuel consumption by the vehicles, energy consumption at supplies, and maximize orders' quality. To analyze the performance of the proposed algorithm, a real case and a set of generated instances are solved. The results obtained by the proposed algorithm are compared with an existing genetic algorithm in the literature. Moreover, the results are also compared with the optimal solutions obtained from the mathematical model for small‐size problems. The results of the comparisons show the efficiency of the proposed MRGGA in finding energy‐efficient solutions for the considered supply chain network.


| INTRODUCTION
The supply chain is one of the essential components of the value chain in production and distribution. 1 Energy consumption and air pollution emissions resulting from supply chain activities have been spotlighted by researchers, politicians, and practitioners for the past decades (Saghaei et al., 2020). 2 Energy efficiency in supply chain operations contributes to environmental sustainability and significantly affects economic growth. 3 Moreover, greenhouse gas emission in the transportation operations in the supply chain is considered one of the main environmental concerns worldwide. 4,5 In this regard, the green supply chain is becoming an important strategy for most industries and enterprises. The green supply chain promotes low energy consumption as well as low environmental and air pollution. 6 The scope of the supply chain is broad and covers all relationships between suppliers, manufacturers, distributors, and consumers. In a supply chain, transportation and production can be deemed the most influential operations on energy usage and pollution emissions. Thus, this study seeks to achieve an energy-efficient and sustainable supply chain by optimally scheduling integrated production and transportation operations. The supply chain addressed in this study comprises (1) multiple production units (i.e., suppliers), with different production technologies and energy usage, dispersed in different geographic regions, (2) multiple heterogeneous vehicles with different loading capacities and fuel consumption, and (3) one manufacturing center that produces final products after collecting all the parts/ materials from the suppliers. This type of supply chain network is prevalent in big industries like the automotive industry, where parts are ordered from several suppliers by the automotive manufacturer and the final assembly is made at the main manufacturing plant. It is also common for suppliers to use a shared transport fleet, and each vehicle can service multiple suppliers on a single trip.
Finding an optimal setting of the considered supply chain is challenging and requires the integration of production and transportation scheduling for the following reasons.
• There is more than one supplier for each part type, but not all the suppliers can produce (supply) all the orders. • Each supplier has a different energy consumption for producing an order due to different production technology or machine type. • The distance between each supplier and the main manufacturing center is not the same.
• The vehicles have different fuel consumption.
• The quality of the products received from different suppliers is not the same. • The orders can be split into smaller batches so that different suppliers can fulfill a portion of the order.
Considering the described challenges, this study aims at providing a solution for an energy-efficient and sustainable supply chain by optimally assigning orders to suppliers, determining the production sequence for the suppliers, assigning orders to vehicles, determining the order split, and prioritizing order delivery. The optimization objectives considered are to minimize total orders' delivery times to the manufacturer, total fuel consumption by vehicles, total energy consumption by suppliers (i.e., production units), and maximize the total quality of produced orders.
To tackle this supply chain problem, a mixed-integer linear mathematical (MILP) model is developed. Due to the complexity of the problem, a modified genetic algorithm (GA), named Multiple Reference Group Genetic Algorithm (MRGGA), is also proposed to handle the large size problems. The validity of the MILP model is verified, and the efficiency of the proposed MRGGA is tested by solving a real case and a set of test problems.
The rest of the paper is organized as follows. Section 2 reviews the integrated production and transportation scheduling in the supply chain context. The problem description and specifications are presented in Section 3. The research methodology and research steps are presented in Section 4. The mathematical formulation of the problem is given in Section 5. The details of the proposed MRGGA are described in Section 6. Section 7 contains the computational results and comparisons. Finally, conclusions and future research directions are presented in Section 8.

| LITERATURE REVIEW
A literature review reveals the existence of a rich collection of studies focusing on supply chain optimization. However, this section only provides an overview of the most relevant studies where integrated production and transportation scheduling in the supply chain networks has been addressed. Interested readers are referred to the review study by Becerra and Sanchis 7 and Emenike and Falcone 8 for a comprehensive overview of the green and energy-efficient supply chain using optimization techniques.
Averbakh 9 addressed the integrated production and transportation scheduling problem in a supply chain containing one factory and several customers. The study aimed to minimize the total weighted flow time and delivery cost while assuming that orders were shipped in batches. The authors proposed some heuristic algorithms to tackle the problem. Delavar et al. 10 developed a GA to integrate the production and scheduling problems considering air transportation. After presenting a mathematical model of the problem, two versions of GA were proposed to minimize the transportation cost of flights, departure time earliness penalties, and total delivery earliness tardiness penalties. Scholz-Reiter et al. 11 examined the integration of production and transportation in a public supply chain. They proposed a mathematical model to minimize the sum of tardiness costs, processing costs, order maintenance costs, as well as fixed and variable shipping costs.
Mehravaran and Logendran 12 studied integrated production and transportation scheduling problem in a supply chain considering a flow shop environment with sequence-dependent setup times to minimize work in process and maximize service levels. They presented a linear mathematical model in which the sequence of orders can be different at each stage and proposed a Tabu search algorithm to solve the problem.
Kabra et al. 13 discussed a multiperiod scheduling problem in a multistage multiproduct supply chain in the biopharmaceutical industry. They proposed a mathematical model to maximize net profit (the difference between the total revenue and manufacturing costs, holding, changeovers, waste disposal, and tardiness-related penalties. Ullrich 14 addressed the integration of the machine scheduling and vehicle routing problems with time windows in a two-stage supply chain and proposed a GA to minimize total tardiness. The first stage involved a parallel-machine environment with sequence-dependent setup times. The second stage included a fleet of vehicles with different capacities. Thomas et al. 15 investigated the integration of production and transportation scheduling problems in a coal supply chain with several independent activities that are interrelated by resource constraints. The transportation system was assumed to be composed of various trains with different classes. The problem was divided into two sub-problems: planning and scheduling. The authors presented a mixed-integer mathematical model for the problem. They used the column generation technique to minimize the cost of holding inventory at the mines, the overstock cost at the terminal, the demurrage cost, and the total cost of requesting trains. Sawik 16 integrated supplier selection and supply chain scheduling problems considering disruption risks. Two types of suppliers are considered in the supply chain: domestic suppliers (relatively reliable but more expensive) and foreign suppliers (relatively competitive prices but have more unexpected disruptions). A mathematical model was presented to minimize total cost or maximize customer service level in single and dual-sourcing cases. Han et al. 17 discussed minimizing the makespan and delivery cost in supply chain scheduling problems considering the online environment. They considered ten cases for the problem and proposed a lower bound for all of these cases. Moreover, they proposed optimal algorithms for four of the ten cases and approximate algorithms for other cases. Pei et al. 18 investigated production and transportation planning integration in a supply chain, considering linear order processing times. The objective function of the problem was minimizing the makespan. After presenting the mathematical model of the problem, they proposed an optimum algorithm for the case that there is a buffer for storing the processed batches before transportation. For the case that buffer is not available, a heuristic was proposed to solve the problem.
Beheshtinia et al. 19 presented a developed version of the GA (called RGGA) to solve the integrated production and scheduling problem in a supply chain considering multisite manufacturing to minimize total orders' delivery time. Frazzon et al. 20 investigated the integrated production and scheduling problem in a supply chain. The authors proposed a hybrid approach by merging a mathematical model, discrete event simulation, and a GA to minimize the sum of transportation, production, and storage costs. Borumand and Beheshtinia 21 discussed the integration of production and transportation scheduling in a supply chain considering a manufacturer and its suppliers. They considered five objective functions minimizing total delivery tardiness, production cost, and the emission by suppliers and vehicles and maximizing the production quality. After presenting the mathematical model of the problem, they proposed a new GA using VIKOR as a multicriteria decision-making method in the selection operator of GA. Sarvestani et al. 22 integrated supplier selection and supply chain scheduling problems considering order acceptance and delivery date determination. After presenting the mathematical model of the problem, they proposed a heuristic algorithm to maximize total net profit obtained by the difference between total revenue and total costs tardiness related penalties, distribution, and purchase of raw materials.
Beheshtinia et al. 23 integrated production and routing problems in a pharmaceutical supply chain to minimize total costs of production, transportation, inventory holding, and expired drug treatment. After presenting the mathematical model of the problem, they proposed a possibilistic model and a robust possibilistic model to consider the uncertainty in the problem. Bank et al. 24 integrated production and distribution scheduling problems with lot-sizing decisions to minimize total sequence-dependent setup costs, holding costs, delivery costs, and delay penalties. After presenting two mathematical models for the problem, they proposed a hybrid GA to solve the problem. Purnomo et al. 25 discussed a collaborative supply chain in the furniture industry and tried to minimize material, processing, transportation, and holding costs. They developed a mathematical model to formulate the considered objective functions and proposed a GA to solve the problem. Aminipour et al. 1 studied the cyclic production scheduling problem in a two-stage closed-loop supply chain. The authors assumed that no demand shortage is acceptable, and two objective functions of minimizing the sum of holding and setup costs are considered. After proposing a MILP model of the problem, a heuristic algorithm is proposed to solve large-scale problems. LaRoche-Boisvert et al. 26 investigated the integration of mine-to-port transportation into production planning and proposed a long-term stochastic integer programming model to optimize multiple objective functions, including production scheduling costs at the mines, stockpile reclamation costs, the equipment fixed cost, mining and mine-to-port transportation costs, and the risks related to meeting product demand at the port. Beheshtinia et al. 27 studied integrated production and scheduling problem in a supply chain considering distributed manufacturing system. They developed a new GA named GA-TOPKOR to minimize the total orders' delivery time, transportation and production costs, and pollution level, and maximize the quality of completed orders.
Beheshtinia et al. 28 investigated a supply chain scheduling problem to minimize total delivery time and total fuel consumption simultaneously. They proposed a Social Genetic Algorithm (SGA) to solve the problem. Their considered supply chain comprises a set of suppliers with different production speeds, vehicles with different capacities and speeds, and a manufacturer. Moghimi and Beheshtinia 29 addressed the supply chain scheduling problem and proposed a novel multisociety genetic algorithm to minimize total delay times and environmental pollution produced by vehicles. They assumed multiple suppliers with identical production speed that produces some orders and deliver them to multiple manufacturers with multiple heterogeneous vehicles. Albrecht 30 discussed continuous-time production, distribution, and recycling in a multistage closedloop supply chain with recycling trade-offs. The author developed a mathematical model to maximize the profit of the supply chain. The profit was considered the revenue from sailing products minus the costs of marketing, tardiness, operation, recycling, and forward and reverse logistics. Melchiori et al. 31 proposed a mixed integer linear programming model for logging trucks' simultaneous routing and scheduling in the forest supply chain. The objective function of the model was to minimize total transportation costs. They considered the time as a discrete variable and used an arc-based formulation for routing. Mahmud et al. 32 investigated the supply chain scheduling problem considering a flexible job shop production environment. They presented a mathematical model of the problem and proposed a particle swarm optimization (PSO) algorithm that utilizes the Tabu search algorithm in the search mechanism. They considered objective functions were minimizing costs such as operation, ordering, procurement, and freight costs as well as minimizing the earliness and tardiness of orders. Mahmud et al. 33 discussed a multiobjective optimization approach in a supply chain scheduling problem integrating the supply, production, and batching decision. They proposed a self-adaptive hyper-heuristic using reinforcement learning to choose the right heuristic intelligently. They considered objective functions were minimizing costs such as operation, ordering, procurement, freight, earliness, and tardiness costs and maximizing a collective environmental sustainability reward.
To better highlight the novelty of this study, the current study is compared with the reviewed literature in terms of the objectives and main problem specifications. A summary of the comparison is reported in Table 1. Table 1 also shows that a mathematical model has always been presented in previous studies, while GA is the most dominant algorithm to cope with problem complexity. Moreover, most studies considered the supply time a continuous parameter to get closer to reality. It is also evident from Table 1 that most of the studies considered the availability of multiple suppliers. However, the different production rate of suppliers has not been considered in some of them. In terms of vehicle type, both heterogeneous and homogeneous fleets of vehicles are considered in the former studies.
According to Table 1, the objective functions of minimizing the energy consumption of suppliers are not considered in any of the previous studies when addressing the integrated production and transportation scheduling in the supply chain networks. Moreover, the possibility of splitting orders among suppliers has not been considered before. In this research, these attributes are considered in the problem. Moreover, a novel GA named MRGGA based on reference group theory proposed by socialist Robert K. Merton 19 is introduced to solve the problem. According to the reference group theory, the individuals within a society imitate some reference group persons such as TV superstars, athletes, influencers, and so forth. In MRGGA, these reference groups are created among chromosomes and the chromosomes try to be similar to the good reference groups and different from the bad reference groups. In summary, the main contributions of this study can be summarized as follows.
• Considering order splitting attributes in the integrated production and transportation scheduling in the supply chain network while assuming the time as a continuous parameter. • Considering four optimization objectives aiming at improving the product and service quality of products as well as the suitability aspects. • Presenting a mathematical optimization model of the problem in the form of a mixed-integer programming model. • Developing a modified GA inspired by the reference group theory in sociology, named MRGGA.

| PROBLEM DEFINITION
This study addresses the production and transportation scheduling problems in a supply chain. The considered supply chain consists of multiple suppliers, multiple vehicles as the transportation fleet, and a central manufacturing center that produces the final products.
Other assumptions are as follows: • There are a limited number of orders and suppliers denoted as No and Ns, respectively. • Each order must be assigned to one supplier. After being processed by the supplier, it is transferred to the manufacturing center by a shared transportation fleet consisting of Nv vehicles. • Each supplier is able to process only a certain set of orders. In other words, one supplier may not be able to process all orders. This should be taken into account when allocating orders to suppliers. • The size of each order and the capacity of each vehicle are predetermined. • Each vehicle may be reused multiple times during the planning horizon. Each vehicle in each trip (cargo) collects the orders from several suppliers and delivers them to the manufacturer. • The total size of orders assigned to each cargo of each vehicle shall not exceed the capacity of the vehicle. • The size of some orders may be larger than the capacity of the vehicle. Hence, orders should be split into smaller parts, namely packages. This means that different parts of an order may be transported in different cargo and by different vehicles.
• Some suppliers may have higher production rates than others due to more and better equipment and machinery and may produce orders faster than others. If the processing time required for completing an order is P and the supplier production rate is R, then the order's real process time by the supplier is P/R. • The vehicles may also have different velocities, which are assumed to be constant throughout the route. If the vehicle's velocity is indicated by V, then the real transportation time to travel distance D is D/V. • If an order is split into several packages, the delivery time of the order is the delivery time of its last delivered package to the manufacturer. • The energy consumption for the production of orders by each supplier and the quality of each order processed by each supplier are predetermined. • The fuel consumption and CO 2 emission of vehicles are different and known. This can be justified to be related to the vehicle's load capacity and type. • The distance between suppliers and the main manufacturing center is given.
The purpose of the problem is to assign the orders to suppliers, determine the production sequence of orders by the suppliers, split orders into packages, assign packages to vehicles, and determine the transportation priority of packages. The considered optimization objectives are: (1) minimizing the total delivery time of all orders times to the manufacturer, (2) minimizing the total fuel consumption needed for delivery or orders from suppliers to the main manufacturing center, (3) minimizing the total energy consumption for producing orders by suppliers, and (4) maximizing the total quality of completed orders. Figure 1 shows an overview of the problem investigated in this study. In this example, four suppliers should deliver six orders to the manufacturing center. The presented solution is as follows. Order 6 is assigned to supplier 1, orders 2 and 3 are assigned to supplier 2, order 1 is assigned to supplier 3, and orders 4 and 5 are assigned to supplier 4. After processing the orders by the suppliers, the assignment of the orders to the vehicles and their priorities for conveying to the manufacturers should be determined. Based on the orders' size and vehicles' capacities, the routing of vehicles could be obtained. As shown in Figure 1, vehicle 1 pickups orders 5, and 4 from supplier 4, and order 1 from supplier 3, respectively. Moreover, vehicle 2 pickups order 6 from supplier 1 and orders 2 and 3 from supplier 2, respectively. BEHESHTINIA AND FATHI | 363

| RESEARCH METHOD
This research aims to reduce energy consumption in supply chain networks by integrating and optimally scheduling production and transportation. For this purpose, this study benefits from quantitative solution methods such as mathematical optimization and computerized algorithms.
The main research question of the study can be expressed as follows.
How can integrated production and transportation be scheduled to optimize the total orders' delivery time, total fuel consumption by vehicles, total energy consumption by suppliers, and the total quality of produced orders?
Considering the complexity of the problem, the main research question can be divided into several subquestions, as presented below.
• What is the assignment of orders to suppliers?
• What is the production sequence of the assigned orders to each supplier? • What is order spiriting to packages?
• What is the assignment of packages to vehicles?
• What is the transportation priority of assigned packages to each vehicle?
The main steps followed to reach the aim of the study and answer the research questions are presented in the next section.

| Research steps
The three following research steps are considered to fulfill the aim of the study and address the research questions. The research steps are illustrated in Figure 2.
Step 1: Present a mathematical formulation of the problem that satisfies all the assumptions and constraints. This step is crucial to understand the relationship between variables and constraints.
Step 2: Develop a computerized algorithm to cope with the problem's complexity and enable the experts to handle the problem within a reasonable time. This study proposes a novel GA named MRGGA based on role model theory to solve the problem.
Step 3: Performing computational experiments to evaluate the performance of MRGGA. In this study, the algorithm's performance is tested by conducting three different experiences.
Step 3-1: Implementing MRGGA in a real-life case study and comparing its results with the results of the current decision-making process.
Step 3-2: Comparing the results obtained by MRGGA with an existing algorithm in the literature.

| Theoretical underpinning of the study
This study uses various principles, such as the integration of production and transportation scheduling, multiobjective optimization, and order splitting, to define the problem. Moreover, the proposed solving method is built on underpinning theories such as references group and escaping from the local optimum traps. Since deciding on production scheduling affects decisions in transportation scheduling and vice versa, considering production and transportation scheduling separately does not lead to a global optimum. 34 This research considers integrating production and transportation scheduling problems to make more efficient decisions.
On the other hand, in scheduling problems, the focus is on time-related objective functions, such as total delivery time, tardiness, makespan, etc. But, decisions in a scheduling problem affect other aspects of the supply chain, such as the quality of products and used energy consumption in the supply chain. In this regard, this research considers four objective functions minimizing the total orders' delivery time to the manufacturing center, fuel consumption by the vehicles, energy consumption by suppliers, and maximizing orders' quality.
Because of the probability of falling into an optimal local trap, there is no guarantee that a final solution provided by a computerized algorithm like MRGGA is optimum. 35 Using proper crossover, mutation, and selection operators could reduce this probability but not necessarily eliminate it. 29,36 The proposed MRGGA tries to reduce this probability by the concept of reference group theory proposed by the sociologist Robert K. Merton. 19 The MRGGA simulates social evolution based on Merton's theory. According to this theory, the individuals within a society imitate some reference groups such as TV stars, heroes, sports athletes, and influencers named role models. In MRGGA, based on the objective function's value of chromosomes, some good and bad chromosomes are considered good and bad role models, respectively. The individuals in society try to be similar to good role models and differ from bad role models. Two sets of good and bad role models are created for each of the considered objective functions to increase the diversity of chromosomes in the proposed MRGGA algorithm. This causes the chromosomes to imitate various good role models and differ from the bad ones, consequently guiding the algorithm toward good solutions.

| MATHEMATICAL MODEL
The mathematical formulation of the problem is presented in this section. The presented model is a mixed integer linear programming (MILP) model. The notations used in the model are presented below.      Equation (1) demonstrates the objective functions for minimizing total orders' delivery time (Obj 1 ), minimizing total consumed energy for production by suppliers (Obj 2 ), minimizing total fuel consumption of vehicles (Obj 3 ), and maximizing orders' quality (Obj 4 ). Constraint set 2 indicates that each order should only be allocated to one supplier. Constraint 3 prevents orders from being assigned to unacceptable suppliers. Constraint 4 calculates the completion time of each order in the supplier stage. Constraint 5 indicates that a supplier cannot process more than one order at a time. Constraint 6 removes some redundant variables. Constraint set 7 determines the processing start time for each order. Constraint set 8 assigns a supplier to each package based on the supplier assigned to its corresponding order. Constraint set 9 indicates that a package of an order cannot be assigned to more than one place in a cargo. Constraint set 10 indicates that each place in a vehicle's cargo cannot be occupied by two packages. Constraint set 11 confirms that the total space occupied by the assigned packages in a cargo does not exceed that vehicle's capacity. Constraint 12 confirms that if a package is not allocated to the pth priority of the bth cargo of the vehicle m, then no package is allocated to the p + 1th priority of that cargo. Constraint set 13 ensures that if a package is not allocated to the bth cargo of vehicle m, then no package is allocated to the b + 1th cargo of that vehicle. Constraint sets 14 and 15 indicate that the loading time of each package equals the maximum completion time of the package by the supplier plus the time the vehicle is ready to be loaded (availability time). Constraint set 16 determines the availability time of a vehicle for transporting the package that is assigned to the first priority of its first cargo, while constraint set 17 determines it for transporting the package that is assigned to the first priority of other cargoes. Other availability times are determined by constraint 18, considering the loading time of the previous package and the travel time between the corresponding suppliers. Constraint set 19 calculates the cargo's arrival time, based on the loading time of packages assigned to it. Constraint set 20 determines the delivery time of a package, according to the arrival of the corresponding cargo to the manufacturer. Constraint 21 determines the delivery time of an order based on the delivery time of its packages. Constraint 22 determines the routes traveled by each vehicle in each cargo.
It has already been proven in the literature that a special case of the problem presented in this study, in which order splitting is not permitted, is NP-hard. 29 Therefore, the problem in this study is also NP-hard. In this regard, resorting to heuristic or meta-heuristic algorithms to tackle the problem is inevitable. An overview of the algorithm structure and its main elements are presented in Section 6.

| MRGGA
The GA is a well-known meta-heuristic algorithm widely used to solve NP-hard problems. 37 The GA is inspired by the evolution theory and follows the process of natural selection. This algorithm is used to find good solutions to optimization problems by applying biologically inspired operators, for example, crossover and mutation. 38 In GA, each solution is represented in the form of a chromosome. After creating a population of randomly created chromosomes, the population is increased by creating new chromosomes using crossover and mutation operators. The crossover operator merges two existing chromosomes based on a mechanism and creates new chromosome(s), while the mutation operator tries to create a new chromosome by changing an existing chromosome. After increasing the population, a selection operator is employed, and some chromosomes are selected for the next generation based on their fitness value. After applying the selection operator, a new population of chromosomes is obtained with the same size as the initial population. This new population is considered the current population, and this procedure continues until the termination criterion is met. There are various crossover, mutation, and selection operators. Figure 3 shows the pseudo-code of the GA. The parameters popsize, cross_rate, and mut_rate are population size, crossover rate, and mutation rate, respectively. The cross_rate and mut_rate are coefficients between 0 and 1 that control the diversity of the chromosomes in the algorithm.
Preserving the main concept and features of GA, a modified GA named the MRGGA is developed in this study to address the explained complex supply chain problem. The MRGGA simulates social evolution based on the reference group theory proposed by socialist Robert K. Merton. 19 According to the reference group theory, the individuals within a society imitate some reference group, named role models. In MRGGA, some good and some bad chromosomes (based on their values for objective functions) are considered good and bad role models, respectively. The individuals in society try to be similar to good role models and differ from bad role models. In MRGGA, two sets of good and bad role models are created for each of the considered objective functions in the problem. Following this concept, the main steps of the proposed MRGGA can be explained as follows.
Step I. Create a random initial population: The size of the population is one of the algorithm parameters and is denoted as popsize.
Step Step VI. Check the termination criterion: If the termination condition is not fulfilled, go back to step III; otherwise, terminate the algorithm. In this research, if the best chromosome in each generation is not improved after a certain number of consecutive iterations (denoted as ter_num), the algorithm will be terminated.
Step VII. Selection operators: After performing the crossover and mutation operators, the size of the population is increased. Therefore, some chromosomes (equal to the popsize) should be selected and transferred to the next generation. In this research, the elitism strategy is used to select chromosomes for the next generation.
To better illustrate the solution process, the main steps of the algorithm and its pseudo code are presented in Figures 4 and 5, respectively.

| Chromosome representation
In GA, each solution is represented by a chromosome. The structure of the chromosomes in the proposed MRGGA is two-dimensional. The vertical dimension represents the suppliers and vehicles, and the horizontal dimension indicates the orders assigned to the suppliers and packages assigned to the vehicles. To illustrate the structure of the chromosome and its decoding procedure, suppose a problem in which two suppliers process the orders and two vehicles transport these orders to the manufacturer. Also, assume that all suppliers' production rates and vehicles' velocities equal 1. Other problem information is given in Figure 6. Figure 7 shows a random chromosome for the problem, in which supplier 1 processes order 3. Supplier 2 processes other orders and the orders' transportation priority is order 4, order 2, and order 1, respectively. In this case, vehicle 1 should transport order 4 and order 2, respectively. Vehicle 2 transports order 3 and then order 1 to the manufacturer. The Gantt chart associated with the generated chromosome is presented in Figure 8.
As indicated in Figure 6, the order sizes are greater than the capacity of the vehicles. Therefore, the orders should be split into smaller packages. In this case, the related vehicle delivers the order by various missions. Assume M mb is bth mission of vehicle m. As shown in Figure 7, orders 4 and 2 should be conveyed by vehicle 1. Due to the vehicle's load capacity limitation, order 4 is divided into two packages, and vehicle 1 delivers it by two different missions. As presented in Figure 7, the first package was delivered at 300 (M 11 ) and the second package was delivered at 600 (M 12 ). Order 2 is also divided into two packages; the first package being delivered at 900 (M 13 ) and the second one at 1200 (M 14 ). On the other hand, orders 3 and 1 should be conveyed by vehicle 2. Due to the vehicle's load capacity limitation, order 3 is divided into four packages. The first and second packages are delivered at 200 (M 21 ), and the third and fourth packages are delivered at 400 (M 22 ). Order 1 is also divided into fourth packages. The first and second packages are delivered at 700 (M 23 ), and the third and fourth packages are delivered at 1000 (M 24 ).

| Fitness function
The fitness function of each chromosome is calculated by Equation (23).
where W k , obj k , obj k Max , and obj k Min are the weight of the kth objective function, the value of the kth objective function, and the maximum and minimum value of the kth objective function (during the problem-solving process), respectively.

| Algorithm operators
Two operators, namely crossover and mutation, are used in the proposed MRGGA to guide the algorithm toward a promising solution.
To simulate the crossover and mutation operators, the imitation and differentiation procedures are first explained. In both of these procedures, there are two chromosomes, one defined as the influencer and the other considered as the follower.

| Imitation
An order is randomly selected and the allocation of the order to suppliers and vehicles in the follower chromosome is examined. If these allocations in the follower chromosome are similar to those in the influencer chromosome, no change is needed. Otherwise, the allocation of the order to the supplier (or the vehicle) in the follower chromosome should be similar to the influencer chromosome. In this case, the order priority in the newly assigned supplier (or vehicle) is selected randomly.

| Differentiation
An order is randomly selected, and the allocation of the order to suppliers and vehicles in the follower chromosome is examined. If these allocations in the follower chromosome differ from those in the influencer chromosome, no change is needed. Otherwise, another supplier (or vehicle) should be selected randomly, and the order should be allocated to the new supplier (or vehicle). Figures 9 and 10 show a sample of imitation and differentiation procedures, respectively.

| Crossover operator
The crossover operator simulates the influenced behavior that the society members have toward each other. In this case, two chromosomes are randomly selected from the current generation. One of the chromosomes is considered the influencer and the other is considered the follower. Then, the imitation process is applied to them. Consequently, the influencer and follower chromosomes are swapped and the imitation process is repeated.

| Mutation operator
The mutation operator simulates the influenced behavior that the society members have toward the reference groups. As mentioned previously, before performing the mutation operation, the good and bad role model sets should be created. In the mutation operation, the following steps are performed: Step 1. Select a random chromosome from the current generation and consider it the follower (Set k = 0). Step 2. Let k = k + 1. Step 3. Select a random chromosome from the kth good role model set and consider it the influencer chromosome.
Step 4. Perform the imitation process between the follower chromosome and the influencer chromosome.
Step 5. Select a random chromosome from the kth bad role model set and consider it the influencer chromosome.
Step 6. Perform the differentiation process between the follower chromosome and the influencer chromosome. Step 7. If k equals the number of objective functions, terminate the mutation operation. Otherwise, go to Step 2. Figure 11 shows an overview of performing the mutation and crossover operations.

| Parameter setting
Parameter setting plays a significant role in the preference of the metaheuristic algorithm. 39 Therefore, the parameters of the MRGGA algorithm are carefully determined after conducting the Taguchi method. Details of the Taguchi method and its step by step process can be found in Nourmohammadi et al. 40 Table 2 demonstrates the considered levels for each parameter, and Figure 12 shows the obtained signal-to-noise (SN) ratios by performing the Taguchi method. As a result of the conducted experiments, the decided values for the parameters in this research are 0.7 for cross_rate, 0.3 for mut_rate, 100 for popsize, 10 for Num_Good and Num_Bad, and 20 for ter_num.

| COMPUTATIONAL RESULTS
To evaluate the performance of the proposed MRGGA, its results are compared with the Reference Group Genetic Algorithm (RGGA) proposed by Beheshtinia et al. 19 Both algorithms were coded by Visual Basic 6.0 and run on an Intel Core i3 computer with a 1.70 GHz CPU and 4 GB of RAM. Furthermore, the results of MRGGA are compared with the optimal solutions obtained from the mathematical model for small-size problems. Finally, the results of applying the MRGGA to a real-life problem at a paint production company are compared with the result of the existing decision-making system in the company.

| Comparison with RGGA
To compare the proposed MRGGA with the RGGA, some test problems are generated randomly, considering the information shown in Table 3. All the test problems are solved by both algorithms, and their results are compared for all the objectives. The weight of all the objective functions is assumed to be identical and equal to 0.25.
By combining the scenarios considered in Table 3, 27 test problems are obtained, as shown in Table 4. To compare the performance of the algorithms, a hypothesis test is used with the following hypotheses.  Each problem is solved 30 times by each algorithm, and the hypothesis test is applied to each problem. The results of the comparison are shown in Table 4.
When comparing the solutions presented in Table 4, one solution may be better in one or more objective functions than the other but worse for objective functions. For this reason, an index for comparing the solutions is calculated using the following equation:  Total objectiv function w obj obj where obj i and obj i ideal are the ith objective function and the ideal value of the ith objective function, respectively. The ideal value for each objective function is obtained by running the algorithm with a large population size (in this research, 1000) and a weight equal to 1 for the related objective function and 0 for the others. It is worth noting that a smaller value for "Total objectiv function" shows a higher quality of the solution.
The results show that the p Value is less than 5% in all cases. Thus, the null hypothesis is rejected. Results also show that MRGGA requires more CPU time than RGGA. To further evaluate the performance of the MRGGA as compared to RGGA, two indices of relative improvement percentage (RIP) and RIP per second Process time U [40,60] Distances U [40,60] Vehicle transportation speed U [1,3] Supplier production speed U [1,3] Occupied space by orders U [20,60] Vehicle capacity U [5,20] Quality of production U [1,5] Energy consumption by suppliers U [1,5] Fuel consumption of vehicles per distance unit U [1,5] T RIPS indicates the relative improvement percentage in the total objective function per each extra CPU time unit (second). Reviewing the obtained values for RIPS reported in Table 4 shows that the higher CPU time of MRGGA causes an acceptable improvement in results.

| Comparison with the optimal solution
The proposed algorithm has also been compared with the optimal solution obtained from the mathematical model for some small-size problems. The results of the comparison are presented in Table 5. The optimal solutions are obtained by solving the mathematical model of the problem using LINGO 9.0 optimization software. According to Table 5, MRGGA yields the optimal solution for most problems solved. In cases where the MRGGA does not find the optimal solution, the difference between the obtained solution by MRGGA and the optimal solution is insignificant. Moreover, the CPU time in MRGGA is significantly less than the time required to obtain the optimal solution in all cases.

| Case study
To evaluate the efficiency of the proposed algorithm further, it is applied to a real-life paint manufacturer. The studied supply chain includes a paint manufacturer and its suppliers. Ten suppliers are responsible for producing the raw materials for this paint manufacturer. The raw materials produced by these suppliers are delivered to the paint manufacturer by a fleet of 10 vehicles with different loading capacities and energy consumption. The data was collected from November 1, 2020 to December 1, 2020. Due to the COVID-19 pandemic, the collected data might not be a good representative of an ordinary period.
The pandemic affected strategic decisions in supply chains, such as a tendency to automation (reduces the dependency on human resources), digitalization and changing the communication system (reduces physical contacts of staff), and using local suppliers (decreases disruptions in the transportation of materials). These attributes only affect the input data of the proposed model and algorithm, such as the number of orders, the production rate of suppliers, and transportation time. Therefore, the problem specifications and the proposed solution methods are expected to be unchanged. The proposed solution method is designed to be generic and usable for different values of input data. Moreover, the comparison of results is fair because the input data for MRGGA and the current decision-making system are identical.
The case study information is summarized in Tables 6 and 7. As shown in Table 6, there are 118 orders with 11 different types of raw materials that should be assigned to the suppliers to produce. Moreover, the size of each order is different from the others and usually exceeds the vehicle capacity. It is worth noting that the raw materials are produced by 10 suppliers. The type and number of vehicles as well as their fuel consumption, are presented in Table 7.
By comparing the results of the MRGGA with the current decision-making system reported in Table 8, it was found that MRGGA caused a 42% improvement in the objective function of total order delivery time, a 10% improvement in energy consumption by suppliers, and a 34% improvement in total fuel consumption by vehicles. The objective function value for the total order quality was identical in both cases. Although minimizing CO 2 emission is not one of the optimization objectives in this study, there is a significant correlation between fuel consumption and CO 2 emission by vehicles. In other words, minimizing the total fuel consumption of vehicles in a supply chain results in less CO 2 emission.

| CONCLUSIONS AND FUTURE RESEARCH DIRECTIONS
This study addressed a supply chain scheduling problem while assuming that a shared transportation fleet is responsible for delivering raw materials from suppliers to the manufacturer. The purpose of the study was to determine the allocation of orders to suppliers and determine their production sequence, split the orders into packages, allocate the packages to vehicles, and determine the transportation priority of the packages. Several objectives were considered for optimization, namely total order delivery time, total fuel consumption by vehicles, total energy consumption for production, and the total orders' quality.
To solve the problem, a new GA inspired by the theory of reference groups proposed by American sociologist Robert Merton, named MRGGA, was developed. In this theory, people are influenced by groups such as heroes and celebrities. In MRGGA, a good role model and a bad role model set are considered for each objective function. In MRGGA, the chromosomes replicate themselves similarly to the good role models and differentiate themselves from the bad ones. The imitation procedure causes improvement in the chromosomes' fitness function by inserting good properties into the population.
On the other hand, replicating the structure of chromosomes similar to the structure of good role models lead the algorithm to convergence. Differentiation of chromosomes from bad role models eliminates some bad properties from the population and increases the diversity of chromosomes. Increasing the population diversity delays the algorithm's convergence.
Using multiple good role model sets will also delay the algorithm's convergence because the chromosomes are replicated according to a diverse spectrum of role models. Moreover, employing multiple bad role model sets in the differentiation procedure increases the diversity of the chromosomes' structure more than in the case that only a bad role model set is used. This delays the algorithm's convergence and improves the final results. Considering the promising performance of the proposed algorithm in this study, the algorithm can be adopted by researchers and practitioners for production and transportation scheduling in the supply chain.
This research helps managers to make efficient decisions in their supply chain about the assignment of orders to suppliers, production sequence, splitting orders for transportation, assignment of orders to the vehicles, and routing of the vehicles. The integration of these decisions increases the performance of the supply chain. In addition to the scheduling aspects, considering the quality aspect helps the managers improve the quality of the products in the supply chain network.
Moreover, the sustainability aspect of the supply chain, which is one of the main concerns of managers, is also addressed in this study. The objective functions of minimizing fuel consumption by the vehicles and energy consumption at supplies help the managers to reduce the used energy in their supply chain. Consequently, this causes reduced CO 2 emissions that directly contribute to the environmental aspect of sustainability.
As for future research, considering other objective functions, such as reducing production and transport pollution, might be an extension of the problem. Adding features such as simultaneous pickup and delivery to the problem may also be investigated in future studies. Combining the MRGGA with other meta-heuristic algorithms could be another area for future research. Moreover, the proposed MRGGA can also be adapted to tackle similar problems, such as vehicle routing and machine scheduling.