1. INTRODUCTION
In markets where there is fierce competition between multiple manufacturers, to become more competitive, many companies focus on optimizing their supply chain rather than their organization (Zegordi and Nia, 2009;Abdollahbeigi, 2021). The term supply chain refers to the set of suppliers, manufacturers, and distributors that work together toward meeting the needs of customers. Supply chain decisions should be made in such a way as to meet different requirements in terms of ontime delivery, product quality, and product cost. Green supply chains try to consider the environmental implications of decisions in different areas of supply chain management (Srivastava, 2007;Daryakenari and Nasiri, 2021). One of the most important motivations of supply chain management is to take an integrated and coordinated approach to decision making. Production and distribution are the two key operations in the supply chain that play a central role in achieving optimal efficiency in integrated supply chain planning and scheduling. Coordinated planning of these two operations tends to yield better results in terms of both performance and cost, as separate planning of these processes may not be able to consider their mutual needs.
With the rising awareness about the trends of resource consumption and greenhouse gas emissions over the last decade, the international academic community has shown a growing interest in these subjects. Each year, many researchers propose a wide variety of ways to reduce resource consumption. Good examples of these efforts are the works done to develop enhanced materials, equipment, and refining processes. While these innovations could be very valuable, in many cases, they can be very expensive and timeconsuming. Supply chain decision makers can also benefit from the strategies that reduce resource consumption while maintaining an appropriate service level (Hassanzadeh and RastiBarzoki, 2017;Gaikwad, 2021). Given the increasing competition between manufacturers to provide better products and services without contributing to environmental problems, this study presents an integrated production and distribution scheduling model for a twostage supply chain with the objective of reducing emissions and minimizing tardiness in deliveries to retain customer satisfaction. In this model, it is assumed that orders are processed in batches and vehicles can move at different speed levels. In the following section, we review the literature on the subject of production and distribution scheduling in supply chains. In Section 3, we present the mathematical formulation of the considered problem. In Section 4, we solve a numerical example with GAMS and present the results, and in Section 5 we provide the conclusions and offer some suggestions for future research.
2. LITERATURE REVIEW
Over the years, there have been many studies in the field of production and transportation scheduling in supply chains.
In a study by Chen and Vairaktarakis (2005), they developed an integrated production and distributionscheduling model. In this model, a set of customer orders are first processed by the facility and then delivered to customers. This model was designed to create an integrated production and distribution schedule with the optimized service level and distribution costs. Service level was measured by a function of the time the products are delivered to customers. This study examined two types of integrated scheduling problems. In the first type, the service level is measured by the average time of delivery to customers, and in the second type, it is measured by the maximum time of delivery to customers.
Geismar et al. (2008), proposed an integrated production and transportationscheduling model in which products have a short lifespan and therefore there is no inventory holding during the process. When a large quantity of the product is produced, it should be delivered directly to customers during its limited lifespan. The purpose of this assumption was to minimize the time required to produce and deliver the product in order to meet customer demand. To solve this model, these researchers proposed a twostep heuristic method making use of a genetic algorithm in the first stage and the Gilmore Gomory algorithm in the second stage.
Zegordi and Nia (2009) formulated the problem of integrated production and transportation scheduling in a twostage supply chain. In the first stage of this supply chain, a number of suppliers are distributed over different locations and in the second stage, several vehicles with different speeds and capacities transport a number of products from these suppliers to the factory. Also, it is assumed that each product requires a certain type of vehicle and can only be processed by certain suppliers. After developing a mixedinteger programming model, a dynamic genetic algorithm was used to solve this model.
In another study, Zegordi et al. (2010), presented a formulation for production and transportation scheduling in a twostage supply chain, in which the first stage is focused on a number of suppliers with different production speeds and the second stage is concentrated on multiple vehicles with different speeds and capacities. Also, it was assumed that different products occupy a different percentage of each vehicle. This problem was modeled as a mixedinteger program, which was then solved with a genetic algorithm with two chromosomes. This study reported that the gendered genetic algorithm performed better than the genetic algorithm in solving this problem.
In a study by Pei et al. (2015), they formulated a supply chainscheduling problem, where each job has a different volume and processing time. In this study, the supply chain consisted of a manufacturer and a consumer, and its goal was to minimize the maximum processing time. This problem was formulated as a mixedinteger programming model. In the end, these researchers evaluated the proposed model by applying it to a set of random numerical examples of different sizes.
Liao et al. (2015) studied the integration of production and transportation scheduling in a twostage supply chain, where the first stage involves scheduling multiple suppliers with different production speeds and the second stage involves scheduling multiple vehicles with different transport capacities. In this research, the main objective was to minimize the maximum time for completing all jobs. A new heuristic algorithm was developed to search for optimal or nearoptimal solutions. Numerical results showed that because of the electromagnetismlike mechanism used in the developed algorithm, it performed 20.66% better than the gendered genetic while also having a shorter computation time.
In another study by Pei et al. (2016) a threestage supply chainscheduling problem for an aluminum production supply chain was formulated. In this model, the first and third stages are two factories, the extrusion factory of the supplier and the aging factory of the manufacturer, where serial batching machine and parallel batching machine respectively process jobs in different ways. In the second stage, a vehicle transports jobs between the two factories. In this study, setup time and capacity constraints were explicitly considered. A mixedinteger programming model was formulated to minimize the maximum processing time. Because of the computational complexity of the model, two heuristic algorithms were developed for two different cases of this problem. In the end, random problem instances of different sizes were developed to test the performance of the proposed algorithms. Numerical results demonstrated the effectiveness of the proposed algorithms, especially in solving largescale problem instances.
Cheng et al. (2015) developed a model for integrated production and distribution scheduling for manufacturers. In the production part of this model, batchprocessing machines have a fixed capacity and the jobs have arbitrary sizes and processing times. Jobs in a batch can be processed together, as long as the total volume of jobs in each batch does not exceed the capacity of the machine. The processing time of each batch is the longest processing time of all jobs in that batch. In the distribution part of this model, vehicles have the same transport capacity. The objective is to minimize the cost of production and distribution for the manufacturer. Given that the problem is NPhard, an ant colony optimization algorithm was developed to solve its production part and a heuristic method was used for its distribution part. These researchers also obtained a lower bound for the optimal total cost. A large set of random data was generated to test the performance of the proposed approach compared to the lower bound. The results showed that the proposed heuristic method performs outstandingly with less than 5second execution time for 200 jobs.
In a study by Marandi and Zegordi (2017) research they tried to come up with a solution for shortening the time interval between production and distribution to maintain the quality of perishable goods, like dairy products. This study proposed a modified formulation for integrated production and distribution scheduling where products have a short lifespan and therefore there is no inventory holding involved in the process. Once a certain quantity of the product is produced, it should be transported directly to customers during its limited lifespan with the goal of minimizing delivery costs and tardiness during production and distribution in order to satisfy the demand before the deadline. Given the complexity of the mixedinteger programming model developed for the problem, the improved particle swarm optimization (IPSO) algorithm was also developed to solve this model. For small and mediumscale problem instances, the performance of this algorithm was compared with commercial optimization software. For large problems, the algorithm was compared with other genetic algorithms proposed in the literature.
Cheng et al. (2016) studied a number of supply chain scheduling problems, where the manufacturer uses batchprocessing machines with limited capacities and customers place orders of different sizes. Once the products are processed, they are divided into batches so that the size of the batches does not exceed the vehicle capacity. After production, the products are delivered to the customers by the manufacturer’s vehicles. In each type of problem, two machine configurations, including single machines and similar parallel machines, were considered. The researchers examined the computational complexity of the problem and formulated it as a mixedinteger programming model. They then proposed polynomial algorithms based on approximation and dynamic programming for solving the model. In the end, the time complexity and performance of each algorithm were also analyzed.
In a study by Yılmaz and Pardalos (2017) the problem of twostage supply chain scheduling with multiple customers and multiple manufacturers was modeled. In the first stage of this model, manufacturers produce the products and in the second stage, each batch of products is transported from the manufacturer to consumer by several vehicles. These researchers stated that despite the importance of multiple customers and average leadtime, no study had investigated them together in the context of a twostage supply chainscheduling problem. The innovation of this study was the coordination of production and distribution decisions for better scheduling in a twostage supply chain with multiple customers and multiple manufacturers. This study presented a mixedinteger linear optimization model to formulate the problem. Since the problem is NPhard, the researchers also developed a hybrid algorithm consisting of an artificial bee colony algorithm and simulated annealing for solving the problem. They also conducted a series of computational experiments with random problem instances with different capacity levels based on real aluminum production data to evaluate the performance of the proposed algorithm.
According to Frazzon et al. (2018) the emergence of new information and communication technologies can potentially improve information transparency in supply chains. However, for this potential to be realized, it is necessary to develop new scheduling methods capable of processing large amounts of data and coping with dynamic disturbances of manufacturing and transport stages. For this purpose, Frazon et al. (2018) proposed a hybrid method consisting of mixedinteger linear programming, discrete event simulation, and a genetic algorithm for the integrated scheduling of production and transport processes in supply chains. The results of this study showed a significant decrease in the number of tardy orders.
In a study by Salehi et al. (2017) they proposed a mixedinteger nonlinear programming model for the green transportationscheduling problem of certain manufacturing companies. The first goal of this model is to minimize the cost of transportation for the manufacturer and the second goal is to minimize total carbon emissions as a measure of environmental sustainability.
He et al. (2019) formulated a model for integrated production and distribution scheduling in a global MTO supply chain and then used the memetic metaheuristic algorithm to optimize the model. The computational results showed that the proposed method could be very efficient in solving largescale problem instances.
Kumar et al. (2020) reviewed the previous studies on the quantitative approaches to the integration of production and distribution in supply chains. According to this review study, one of the most important gaps in this literature is the consideration of environmental issues.
Goli et al. (2020) developed a model for production and distribution scheduling under financial constraints. In this study, several metaheuristic algorithms were used to solve the mathematical model.
Wang (2021) studied the integration of production, distribution, and inventory holding processes in an ecommerce supply chain. In this chain, semifinished products are provided to manufacturers and then sent to retailers once production is completed. For this problem, they developed a nonlinear mathematical model that minimizes all production, distribution, and inventory costs. They then used a particle swarm optimization algorithm to solve this problem. The results showed that the proposed mathematical model yields significant cost reduction.
Reviewing the previous studies on integrated production and distribution scheduling in supply chains will show that most of them have considered a single objective rather than taking the multiobjective approach. Also, very few studies have been done with the two objectives of reducing tardy deliveries and decreasing greenhouse gas emissions. Also, many studies on integrated production and distribution scheduling have not considered the vehicle speed levels, which can have a great impact on the amount of emission released from vehicles due to fuel consumption. In this study, we develop a mathematical model with the objectives of reducing order tardiness and decreasing emissions due to fuel consumption while considering different speed levels for the vehicles that deliver orders to customers.
3. METHODOLOGY
The present study presents an integrated production and distributionscheduling model in a twostage supply chain. In the first stage of this chain, customer orders are processed by the factory in batches. In the second stage, orders are delivered to customers by vehicles moving at certain speed levels. The first objective is to minimize tardiness in delivering orders to ensure customer satisfaction. The second objective, which concerns the environmental aspect of the problem, tries to minimize the amount of greenhouse gas released during the distribution phase.
3.1 Problem Assumptions
The assumptions considered in this study are as follows:

•Job batches cannot be broken down.

•Each vehicle has a specific weight and fuel emission factor.

•Vehicle has a constant speed while delivering an order.

•There is a leadtime for each batch.
3.2 Mathematical Model
The mixedinteger nonlinear mathematical model has been developed to minimize the total tardiness of orders and the amount of emission released due to fuel consumption based on the above assumptions. The notation and description of sets, parameters, and variables used in this model are given in Table 1.
Based on the described parameters and variables, the mathematical formulation of the problem will be as follows:
s.t.
Equation (1) is the first objective function, which minimizes the total tardiness of all orders. Equation (2), which is the second objective function, minimizes the emission of all vehicles. Equation (3) calculates the delivery time of batches based on the vehicle speed level. Equation (4) computes the tardiness of each order, or in other words, ensures that the tardiness is greater than the difference between the actual delivery time and the delivery date of the batch in which the order is placed. Equations (5) and (6) determine the speed level constraints. Equation (7) calculates the finish time of each batch in the production stage. Equations (8) and (9) are used to prevent overlap between the two batches processed on the factory machine. Since the finish time of each batch cannot be greater than the sum of start time and processing time for that batch, the constant B is calculated by summing these times. Equation (10) shows that the load time of a batch is after its finish time. Equations (11) and (12) ensure that each job is assigned to only one batch. Equation (13) states that the total size of orders allocated to a batch should not exceed the capacity of the factory machine. According to Equation (14), the total size of a batch transported by a vehicle should not exceed the capacity of that vehicle. Equations (15) and (16) ensure that two jobs ordered by different customers are not assigned to one batch. Equations (17) and (18) are related to the assignment of batches to vehicles. Equation (19) determines the actual load of vehicles. Equation (20) calculates the weight of each batch. Equations (21)(23) ensure that each batch is carried by only one vehicle at one speed level. Equation (24) shows the precedence relationship of batches. Equations (25) and (26) define the type of decision variables.
4. RESULTS AND DISCUSSION
To evaluate the performance of the proposed model, we generated 15 instances of the problem, solved them in GAMS software, and then analyzed the results. Since the model has two objectives, it was solved using the Epsilon constraint method, which is known to be a suitable method for solving multiobjective problems. The specifications of the problem instances and the corresponding computational results are given in Table 2.
The quality of Pareto solutions of the multiobjective model was measured in terms of the average distance from the ideal point, the value of which is specified in Table 2. In problem instance 1, vehicle 1 carries order 1 at speed level 2 so that the order can be delivered with the least tardiness and emission. In problem instance 2, where orders are double the size of those in problem instance 1, more emission has been released as the increased weight of the orders has increased the amount of emission produced (fuel consumed) by the vehicle.
For a better analysis of the results, the value of each objective function, the number of Pareto solutions, and the quality of solutions are plotted in diagrams of Figures 1 to 4.
As shown in Figure 1, the total tardiness tends to increase with increasing problem size. This trend is seen in all instances except problems no. 9 and 48. In problems no. 14 and 15, tardiness has increased at a sharper rate than in other problem instances. For the emission rate, the objective function has a strictly increasing trend with problem no. 5 being the only exception.
As Figure 3 shows, the highest number of Pareto solutions has been obtained for problem no. 15 and the lowest for problems no. 1 and 2. In other words, the number of Pareto solutions tends to increase with the problem number. In addition, as shown in Figure 4, there is an increasing trend in the quality of Pareto solutions.
In general, it can be concluded that the designed mathematical model can give decisionmakers a set of highquality Pareto solutions for problems of different dimensions.
5. CONCLUSIONS
In this study, we formulated the problem of integrated production and distribution scheduling in supply chains and presented a mixed integer programming mathematical model for this problem. The objective functions of this model aim to minimize the tardiness of orders as well as greenhouse gas emissions due to fuel consumption. Since vehicle speed can affect tardiness as well as greenhouse gas emission, the model considered different speed levels for vehicles. The instances of the mathematical model were solved using GAMS software. Given its multiobjective nature, the problem was solved with the Epsilon constraint method and the results were analyzed and studied. Since solving the largescale instances of the problem could be timeintensive, future studies may try using metaheuristic algorithms for optimization. Also, one may consider different distances between the factory and its customers or formulate the problem for multiple factories and multiple customers.