1. INTRODUCTION
In recent years, research in the field of operations management has been dominated by the subject of supply chain management. A supply chain consists of all facilities, jobs, tasks and activities involved in the production and delivery of a good or service, from manufacturers and their suppliers to vendors and their customers. Supply chain management is the act of planning, scheduling, and managing supply and demand, procurement of materials, production of products or delivery of services, warehousing, inventory control, distribution, delivery, and customer service. Supply chains exist in service businesses as well as manufacturing industries, but the complexity of supply chains can vary greatly from industry to industry and from business to business (Barzinpour and Esmaeili. 2014;Ahmadi et al., 2018). Nowadays, with the intense competition between manufacturers of goods and services in constantly changing markets with diverse customer demands, supply chains play a critical role in the efforts to reduce costs, improve product/service quality, and achieve enhanced efficiency needed for companies to survive. Success and failure of a supply chain is ultimately determined by its customers or end consumers. Giving the customer the right product, at the right price and time is the most important factor of success in competition, thus playing a key role in the survival of a business. Therefore, a good knowledge of determinants of customer satisfaction as well as market conditions is essential for deploying any new supply chain strategy (John, 2018). Only when market constraints are known a business can create, develop, and adopt strategies to meet the needs of the supply chain and its end customers. In today’s competitive environment, businesses are increasingly reviewing their supply chains to find ways to achieve improved performance in terms of cost saving, inventory management, efficiency, and demand satisfaction. Today, shortening the transportation cycles, controlling inventory holding and handling costs and timely delivery of orders are the main concerns of all supply chain managers, as they are crucial for gaining more market share in competitive environments. Supply chain management with an integrated approach to the management of flows of materials, goods, information and finances has the ability to meet these requirements (Boonjing et al., 2017). Supply chain management can coordinate all of these activities in such a way that customers can obtain quality products and reliable services at the lowest cost. Even on its own, supply chain, management can create a competitive advantage for a company. Storing produced goods near consumer markets for timely and appropriate distribution in response to demand is one of the most important issues in any supply chain, as not making the right decisions in this regard can impose heavy costs on the supply chain (Fard and HajaghaeiKeshteli, 2018). To remain competitive in today’s industrialized world, manufacturers absolutely need to minimize their supply chain costs by properly storing the goods demanded by customers and routing vehicles to transport these goods to the selected warehouses or end customers (Maghsoudlou et al., 2016;Tretiak et al., 2019).
The subject of routing vehicles in distribution operations falls in the domain of Vehicle Routing Problem (VRP). VRP refers to a set of problems in which fleets of vehicles need to transport goods from one or more warehouses to multiple customers located in different geographical locations under certain requirements such as minimum cost or maximum coverage. Typically, customers are visited only once and all their demand must be met by only one vehicle. Each vehicle has a certain capacity and all routes start from a specific point (origin of loading) and end at the same point after going through a series of customers (Barzinpour and Esmaeili, 2014). These problems are also known as transportation planning or scheduling problems. VRP is at the heart of distribution management and is an issue that thousands of businesses and organizations companies deal with every day in their delivery and collection operations.
Another important factor in supply chain management is the effective control of physical flows over the chain. Given the great importance of this issue, companies tend to always seek more efficient methods for this purpose to achieve enhanced customer satisfaction and cost saving. Among the various methods available for this purpose, the use of crossdocking systems is a particularly efficient method to reduce inventory costs and improve response to customer needs. Cross docking is a type of distribution strategy that integrates transportation with retail to prevent longterm storage. In these systems, the goods are transferred as directly as possible to delivery vehicles once they are offloaded. In other words, one of the goals of cross docking is to reduce the time of transfer of goods. In these systems, networkscheduling decisions determine the details of shipments, i.e. how many shipments will be sent to a specific transport service. In areas near a crossdocking station, network scheduling may involve vehicle routing to collect and deliver goods from (to) the station. Since transportation and distribution costs can make up a significant portion of supply chain costs, some companies use this system to minimize these costs and increase the speed of distribution.
The problem studied in this paper is the transportation in companies that operate in a large geographical area, especially those with cross docking (Yu et al., 2016;Gaikwad et al. 2021). Gil et al. (2010) have developed a twolevel model consisting of warehouses and retailers for this purpose. Boonjing et al. (2017) have argued that the location and number of warehouses can serve as measures for analyzing supply chain performance. Therefore, the main reasons for opening a new warehouse could be to provide better service to retailers, lower shipping costs, reduce the risk of disruption in one of the warehouses and adhering to the FIFO system.
The classic VRP involves delivering products/services with a fleet of vehicles from one distribution center to a set of customers with known demands. The notion of cross docking has been introduced as an innovative strategy to minimize unnecessary costs, especially from inventory holding, and improve the customer service level. Many companies are working to develop effective strategies to control the physical flow of their supply chains. The most important goals pursued in the development of these new strategies are to minimize the total cost and achieve a high level of agility, flexibility and reliability for different demands. Vahdani et al. (2012) presented a crossdocking strategy for a fleet of vehicles, which allows separate delivery to and collection from different nodes of the network. They optimized this strategy by formulating it as a mixed integerprogramming model with the objective of minimizing shipping costs and solving it with GAMS software (Vahdani et al., 2012). In 2014, Agustina et al. (2014) examined cross docking and vehicle scheduling in a food supply chain where reliability depends on the time of delivery. They formulated this problem as a mixed integer linear program, used the concept of customer zones and hard time window for deliveries to reduce the solution space, and then solved the problem with simplex (Agustina et al., 2014). In 2015, Ahmadizar et al. (2014) presented a model for twolevel vehicle routing with cross docking in a threeechelon supply chain consisting of suppliers, crossdocking stations, and retailers. They considered two levels for net work routing, the first consisting of suppliers and crossdocking stations and the second comprised of sales warehouses and retailers. They also proposed a hybrid genetic algorithm with local search to solve the problem (Ahmadizar et al., 2015). In 2016, Yu et al. (2016) formulated the problem of open vehicle routing with cross docking as a mixed integer linear programming model. They used the simulationannealing algorithm to solve the problem for the distribution of one product through one distribution warehouse with a homogenous fleet of vehicles (Yu et al., 2016;Bagheri Sadr and Bozorgian, 2021;Samimi, 2021). In 2018, Naik and Suresh (2018) studied a location routing problem, in which decisions about hub location, allocation of nodes to hubs, and routing between nodes assigned to the same hub should be made to minimize the total cost of transportation (Naik and Suresh, 2018: Fallahtafti et al., 2019).
Mousavi et al. (2019) optimized a locationinventory problem with periodic demand and demand uncertainty. They developed a (r, Q) inventory model for this purpose and ultimetely solve this problem with a genetic algorithm (GA) and a particle swarm optimization algorithm (PSO). The results of this study showed that GA produced better solution than PSO. In 2020, Aziz and Hu modeled a supply chain with simultaneous pickup and delivery. In this study, the possibility of sending goods directly from the factory to customers was considered and decisions regarding location and distribution were optimized on this basis. The results obtained by solving the model with Benders decomposition method showed the efficiency of this solution method (Azizi and Hu, 2020). In 2021, Karim and Nakade (2021) formulated a model for a spare parts supply chain with disruption risks and environmental pollution taken into account. These researchers developed a mixed integer nonlinear model was developed for this purpose. Numerical analyses showed that low risk of production centers could control the amount of greenhouse gas emission released in the supply chain (Karim and Nakade, 2021). A section initially gives a survey of the establishments of serious area models. It at that point follows resulting improvements through time under extraordinary thought of client conduct (Eiselt et al., 2019).
Since the review of the literature showed that, so far, there has been no study on distribution scheduling and profitability optimization through transportation cost minimization under uncertainty, the present study was conducted to close this gap in the literature.
2. METHODOLOGY
Over the last two decades, intense competition between companies in delivering goods and services has changed how companies develop and expand. Today, companies need to integrate all activities involved in the process of production and distribution from the supply of raw materials to the delivery of final goods to consumers. Distribution and support systems are also important parts of the supply chain process, as they contribute to effective planning, execution and control of management strategies for flow and storage of goods, services and related information from the top of the chain down to consumers. The location of distribution centers also plays a key role in facility planning. In recent years, facility location has become known as one of the key elements of success and survival of manufacturing businesses. The problem of location of distribution centers is focused on how to select the best sites for establishing distribution centers from a set of potential points so that the total cost of distribution is minimized. The model of this study utilizes a multiobjective function, where the first objective is to maximize the profit from sales and the second objective is to optimize the distribution makespan. In the first objective function, the sales profit is deducted from the costs of routing, location and inventory holding (due to lost sales) and the profit from proper location is obtained.
Therefore, the model has been formulated based on following assumptions:

1. Product pickup and delivery is allowed and there is a planning horizon for receiving the products in the crossdocking stations.

2. Vehicles must arrive at the loading docks simultaneously.

3. There is a known demand at each demand point.

4. The scale of products and the amount of product loaded in each loading session is known.

5. All vehicle routes start from and end at crossdocking stations.

6. Pickup and delivery in multiple loads is not allowed (customer must receive the order in one package).

7. Vehicles have limited capacity.

8. There are a limited number of vehicles.

9. Vehicles can carry one or more types of goods.

10. All vehicles are based at crossdocking stations.

11. Each route ends at the same crossdocking station from which it starts.

12. Transport scheduling is subject to uncertainty with a probabilistic distribution function.

13. Incoming vehicles must reach the crossdocking station at the beginning of each period and outgoing vehicles must distribute the shipments during the day.
• Indices

i : Index of retail centers (i = 1,…, N)

j : Index of customers (j = 1,…, N)

r : Index of products (r = 1,…, R_{i})

l : Index of vehicles (l = 1,…, L)

t : Time period
• Parameters

D_{jrt} : Demand of customer j for product r in period t

CV_{l} : Fixed operating cost of vehicle l

CC_{lij} : Cost of transporting product by vehicle l from retail center i to customer j

CCCC_{ljj1} : Cost of transporting product by vehicle l from customer j to customer j1

FC_{it} : Fixed cost of establishing retail center i in period t

Te_{ij} : Travel time between retail center i and customer j

CT_{ij} : Cost of travel between retail center i and customer j

TT_{jj1} : Time of travel between customer i and customer j1

TIME_{j} : Unloading time at customer j

CCT_{jj1} : Cost of travel between customer j and customer j1

VP_{r} : Volume of product r (per unit)

VV_{l} : Capacity of vehicke l

HI_{rt} : Cost of lost sales of product r (per unit) at customer j in period t (due to losing sales to competitor)

U : A very large number

s_{JRT} : Revenue from the sale of product r to customer j in period t
• Variables

QP_{rlijt} : Amount of product r that is sent from retail center i to the location of customer j by vehicle l in period t

QPT_{rlijj1t} : Amount of product r that is sent from customer j to customer j1 by vehicle l in period t

SH_{jrt} : Amount of lost sales of product r (per unit) at customer j in period t

Y_{lijt} : A binary variable that equals 1 if vehicle l travels from the retail center i to the location of customer j in period t, and is zero otherwise.

YY_{lijj1t} : A binary variable that equals 1 if vehicle l travels from the location of customer j to the location of customer j1 in period t, and is zero otherwise

YYY_{ljit} : A binary variable that equals 1 if vehicle l travels from the location of customer j to retail center i in period t, and is zero otherwise

bX_{it} : A binary variable that equals 1 if retail center i is established in period t, and is zero otherwise

s_{j} : Start time of service to customer j

f_{j} : Finish time of service to customer j
This section presents the proposed mixed integer linear programming model for the discussed problem based on the above notations.
The second objective function optimizes the scheduling of product distribution among customers.
s.t
Constraint (3) ensures that the total volume of products loaded for delivery is less than the capacity of vehicles. In other words, it guarantees that cargo does not exceed vehicle capacity.
Constraint (4) ensures that all demand is met either directly from a retail center or indirectly from another customer.
Constraint (5) guarantees that in each period, there will one route open from a retail center either to customer or from one customer to another.
Constraint (6) guarantees that in each period, the vehicle arriving at a customer will either travel to the next customer or return to the retail center.
Constraint (7) ensures that when a vehicle travels from a retail center to a customer, it can return to the retail center if needed.
Constraint (8) guarantees that a vehicle can travel from a retail center to a customer only once in each period.
Constraint (9) ensures that in each period where products are to be shipped from a retail center to a customer, there will a route open between that retail center and that customer.
Constraint (10) ensures that if a product is to be moved between two customers, there will a route open between these customers.
Constraint (11) ensures that no loop is created between customers.
Constraint (12) guarantees that in each period, a product can be shipped from a retail center to a customer only if that retail center is established.
Constraint (13) balances the incoming and outgoing vehicles of each customer.
Constraint (14) ensures that the schedule of product distribution among customers is respected.
Constraint (15) states that the finish time of the distribution activity will be after transportation and unloading operations are competed.
3. RESULTS AND DISCUSSION
In this section, we solve the proposed mathematical model for the discussed supply chain. Given the inputs needed for the exact solution, the travel time between the retailer and customer locations is considered probabilistic.
In the goal programming, since the feedback data received from customers suggests that delivery time of products should be less than 180 hours, the ideal distribution make span was considered 180 hours or less.
NSGAII is one of the most widely used and powerful algorithms available for solving multiobjective optimization problems. The efficiency of this algorithm in solving various problems has been proven in countless studies. NSGA was developed in 1994 by Srinivas and Deb (1994) for solving multiobjective optimization problems. The most important features of this optimization method are as follows:

•The solutions are ranked in terms of how many times they are dominated by other solutions. The solution that is dominated by no other solution is the most desirable.

•Fitness of a solution is determined based on its ranking in terms of being nondominated by other solutions.

•A fitnesssharing scheme is used for close solutions to make sure that solutions are evenly distributed over the search space.
Given the high sensitivity of the performance of NSGA to the fitness sharing parameters as well as other parameters, in 2000, Deb et al. (2000) introduced the second version of NSGA called NSGA‐II. Besides having a multitude of functions, NSGA‐II has been used as a model for the development of many other multiobjective optimization algorithms. This algorithm and its unique approach to dealing with multiobjective optimization problems have been used repeatedly to create newer multi objective optimization algorithms.
The parameter values selected for NSGAII using the design of experiments method are given in Table 1.
As mentioned in the introduction, to analyze the small instances of the problem, eight test problems were designed and their exact solutions were obtained. The results of this process are presented in Table 2.
As the table above shows, there is a great degree of similarity between the exact values obtained from GP (exact solution) and those obtained from NSGAII, which means the performance of the algorithm is largely acceptable. Figure 1 makes a comparison between the solution times of the two methods.
As can be seen, the solution time of NSGAII has increased at much lower rate with the problem dimension. This is while GP has had longer solution times with a faster increase with the problem dimension. Therefore, NSGAII can be considered more efficient than GP.
Given the potential importance of settings used for the basic parameters of the model, it is necessary to examine the sensitivity of the model output to the input pa rameters so that solutions can be judged accordingly. Obviously, revenue and cost parameters have a direct impact on the total profit of the system and distribution make span, which is why companies should always formulate and follow a comprehensive longterm plan to reduce costs and increase revenue. Therefore, in this section, sensitivity analysis is performed on parameters that do not have a direct impact on the objective function but should be considered by businesses and organizations.
As stated earlier, one of the factors that influence the objectives and can play an active role in the analysis is customer demand. Thus, this section examines the sensitivity of outputs to changes in this parameter.
The analysis of sensitivity to increased demand showed that the higher the demand, the higher the company’s profit. However, a 25% increase in demand caused the profit to increase by only 3%, which indicates that the impact of increased demand on the company’s distribution costs almost matches the impact on the profit. Meanwhile, the increased demand also led to extreme prolongation of transport make span. This is due to the multitude of trips needed to cover the extra demand, which causes the transport make span to increase by 27%.
The analysis also showed that while decreasing the capacity of transport vehicles will result in a matching decrease in the sales profit (in the first objective function), increasing this capacity has no impact on the sales profit. In addition, decreasing the capacity of vehicles caused the distribution make span to decrease by 2% (in the second objective function) and increasing this capacity led to a 17% increase in the distribution make span.
4. CONCLUSIONS
Given the prominent role of distribution processes in manufacturing businesses, transportation planning, routing, and inventory and warehousing optimization are extremely important subjects of discussion in this field. The distribution of goods and services from their point of origin to the point of consumption is also one of the important components of GDP, reflecting the wealth generation capability of a country. Thus, logistics capabilities can be considered an asset for a country. Transportation is one of the main aspects of logistics with a significant impact on the cost of goods and services. This paper presented a model for vehicle routing with time windows for a multiproduct distribution system. The objective of the modelwas to determine the best route and the optimal number of vehicles that should be used in the distribution network to achieve the shortest distribution makespan with maximum operating profit
First, the sets of problem assumptions, objectives and constraint were formulated in an integrated way. After formulating the final model, it was solved with GAMS software to check its validity. Considering the complexity of this problem and the fact that realworld problems are usually large and cannot be easily solved with exact solution methods, it was necessary to develop a heuristic or metaheuristic algorithm for solving the problem. Therefore, the metaheuristic algorithm known as NSGAII was adopted for solving largescale variants of the problem. The code of the model was created in MATLAB to allow problem instances of various sizes from small to large to be produced at random.
In view of the obtained results, future researches are recommended to develop other metaheuristic algorithms for solving largescale instances of the problem and compare their performance with that of the proposed algorithm. It is also recommended to solve the model with the exact solution techniques such as Epsilon constraint method and Pareto and to formulate the model with fuzzy parameters.