1. INTRODUCTION
A supply chain comprises all activities related to the flow of goods as they are converted from raw materials to final products and delivered to end consumers, as well as all information flows related to these processes. Supply chains are dynamic entities with three types of flow: the product flow, the information flow, and the finances flow. In other words, the term supply chain refers to the flow of materials, products, information, and finances from suppliers to manufacturers, from manufacturers to wholesalers and then retailers, and from them to customers, and vice versa. Supply chain management is the process of integrating supply chain activities and related information flows in order to enhance and coordinate production and distribution processes.
One of the most important decisions in supply chain management, which should take into account a wide range of factors from raw material procurement to product delivery, is supplier selection. Being the first step of supply chain management, supplier selection has a critical impact on all the subsequent steps (Kilic, 2013). Supplier selection involves determining the suppliers from which raw materials should be obtained and the amounts in which each type of raw material should be procured from each selected supplier. Supplier selection is one of the most important decisions in any supply chain, which, when properly done, can significantly reduce the costs of material procurement and production. The first step of supplier selection is to evaluate the performance of potential suppliers, which can be done in terms of several criteria. One of the main goals of this supplier evaluation is to determine the quota that can be assigned to each of them. There have been many studies on the subject of supplier selection. Provided in the following is a brief review of studies carried out in this area (Baghernejad and Fiuzat, 2021;Abdollahbeigi and Asgari Bajgirani, 2020;Barmasi, 2020).
In a study by Seifbarghy and Esfandiari, these researchers modeled and solved a multiobjective supplier quota allocation problem with transaction costs. This model has five objective functions: minimization of purchase costs, rejected units, late delivery, and transaction costs related to purchases from suppliers, and maximization of the evaluation score of selected suppliers. A simple weighting method was used to convert the multiobjective model into a singleobjective model and then two metaheuristic algorithms based on genetic algorithm and simulated annealing were used to solve the model (Seifbarghy and Esfandiari, 2013). Memon et al. (2015) used a hybrid method based on grey systems theory and uncertainty theory to model the group multicriteria supplier selection problem with the objective of minimizing purchasing risks. They developed a goalprogramming model for determining the optimal size of orders assigned to each supplier under uncertainty. In this study, qualitative criteria were modeled as grey parameters and quantitative criteria were considered as uncertainty parameters (Memon et al., 2015). Sodenkamp et al. (2016) studied the problem of multicriteria supplier selection and order allocation. The model presented in this study uses a combination of multicriteria decision analysis and linear programming. This model was developed to investigate the effects of supplier collaboration performance in a hierarchical decisionmaking structure (Sodenkamp et al., 2016).
Facility location is one of the most common spatial decisions in supply chain management. The purpose of this process is to find a set of spatial options suitable for a particular application. In industrial and service applications of facility location, one of the most important questions is how to determine the optimal number of facilities and allocate suppliers to the selected facilities (Martseniuk et al., 2020). Facility location is the process of selecting a location for one or more centers while taking into account the location of other centers and existing constraints such that a specific goal is optimized. The goal of the facility locationallocation problem is to identify the best locations among the nodes of the supply chain and allocate the demands to them (Arabzad et al., 2015). Facility location decisions play a key role in the strategic design of supply chain networks. In general, a supply chain network design project begins with identifying potential sites for the required facilities and capacities (Melo et al., 2009). In the following, studies related to this topic are reviewed.
In a study by Bagherinejad and Dehghani, they introduced a robust multiobjective optimization model for locationallocation decisions in a twostage supply chain network (customer allocation to demand points under uncertainty) and then solved it by a nondominated sorting ant colony optimization algorithm. To check the accuracy of the proposed method, they compared the computational results with the results of a multiobjective genetic algorithm (Bagherinejad and Dehghani, 2015). Ranjbar Tezenji et al. (2016) introduced an integrated model for supplier locationselection and order allocation with capacity constraints under uncertainty. In this study, a biobjective model was used to optimize the mean and variance of costs, and a mixedinteger nonlinear program with two metaheuristic methods based on genetic algorithm and simulated annealing was used to solve the model (Ranjbar Tezenji et al., 2016). In a study by Mehrabad et al. (2017), an evolutionary multiobjective method was used to develop a locationallocation model for a multilevel supply chain. In this study, the total cost of the supply chain and the customer satisfaction level were optimized simultaneously. Ultimately, a hybrid particle swarm optimization algorithm was used to obtain the optimal solutions of the developed model (Mehrabad et al., 2017). Rohaninezhad et al. (2017) proposed a mathematical model for the location of production facilities, in which investors compete for locations and customers. To formulate this problem, these researchers developed a multiobjective model based on game theory with the objective of maximizing investor profitability (Rohaninezhad et al., 2017). Yu et al. (2015) developed a model for integrated locationproductiondistribution planning in a multiproduct supply chain, which determines the suitable locations for the construction of factories and distribution centers during the process of production and distribution decisionmaking. In this study, the mathematical model was developed with the goal of minimizing the total cost of the supply chain and then solved with a hybrid multiobjective particle swarm optimization algorithm (Yu et al., 2015). In 2017, Amin and Baki optimized a facility location model for closedloop supply chains. In this study, which was implemented in Canada, a fuzzy programming approach was used to solve the model (Amin and Baki, 2017). In 2018, Sadeghi Rad et al. (2018) presented a mathematical model for integrated facility location and supplier selection decisionmaking in a closedloop supply chain. In this study, the objective was to minimize the total cost of the chain and maximize the revenue (discounts) from supplier selection (Sadeghi Rad et al., 2018). In 2019, Hu and Dong modeled and optimized a supplier selection model for delivering relief after a disaster. In this study, supplier disruption risks were considered as a key element in supplier selection (Hu and Dong, 2019).
In 2020, Hemmati and Pasandideh (2020) presented a biobjective mathematical model for supplier selection, order allocation, and facility location based on different scenarios. The first objective of this model was to reduce the total cost of the process and its second goal was to reduce the total amount of carbon emitted. The researchers used a multichoice goal programming method to solve this mathematical model (Hemmati and Pasandideh. 2020).
One of the latest researches in this area is a 2021 study by Afify et al. (2021), where they introduced a linearization approach for solving capacitated facility location problems with supply risks taken into account. In this study, a mathematical programming approach was used to create a tradeoff between cost levels and risk levels (Afify et al., 2021).
In the present study, we formulate a multiperiod, multiproduct integrated facility (production center) location, supplier selection, and order allocation problem with an economic objective, which is to minimize total cost while also minimizing product delivery delay. The purpose of this study is to provide a decisionmaking method for optimal facility location, supplier selection, and quota allocation in a typical supply chain so that the costs of the entire chain are minimized. The model can be used by companies that have several factories buying materials from several different suppliers. The solution of this model gives the optimal location of production facilities and the optimal size of orders allocated from each factory to each supplier.
In view of past studies carried out in this field, the present study contributes to the research literature by: (1) maximizing the reliability of the product delivery network in addition to minimizing the total cost; (2) considering the possibility of change in the cost of transactions with suppliers by making the model multiperiod; (3) considering capacity constraints for supply and production levels; (4) performing location in multiple periods. To evaluate the proposed mathematical model, it is implemented using the LPmetric method (with p = 1, 2 and ∞) in the optimization software GAMS.
In the rest of this paper, section 2 provides the problem statement, describes the assumptions, and presents the proposed mathematical model, section 3 explains the method used to solve the model, section 4 discusses the validly and computational complexity of the model, and section 5 presents the conclusions.
2. PROBLEM STATEMENT AND MATHEMATICAL MODEL
This section presents the proposed mathematical formulation for facility location, supplier selection, and quota allocation model with the objective of minimizing the total cost and maximizing the reliability of product delivery. The proposed model can be used in almost all companies in which some materials (parts) are supplied by a few select suppliers in order to optimize this process (e.g., car assembly companies, health products companies, etc.). Provided in the following is a description of the assumptions, indices, parameters, and decision variables of the model.
2.1 Model Assumptions
The model has been developed based on the following assumptions:

•The location of facilities and suppliers is known

•The size of demand in each period is known

•There is an upper bound for late delivery by each supplier
2.2 Model Indices
In this and the following subsections, we first describe the indices, parameters, and decision variables used in the model formulation and then present the proposed biobjective mathematical model.

•s : set of suppliers {1, 2, …, S}

•i : set of candidate sites for production facility {1, 2, …, I}

•p : set of products {1, 2, …, P}

•h : set of periods {1, 2, …, H}
2.3 Model Parameters

• Pi_{sph} : Bid price of product P for supplier s in period h

• g_{spi} : The cost of transferring one unit of product P from supplier s to candidate site i

• t_{sp} : Percentage of units of product p that are delivered late by supplier s

• T'_{p} : Maximum permissible percentage of units of product p that could be delivered late

• D_{ph} : Demand for product P in period h

• C_{sph} : Maximum production capacity of supplier s for product P in period h

• H_{ih} : The cost of establishing a production facility at candidate site i in period h

• B_{sph} : A binary parameter that is equal to one when supplier s can supply product P in period h, and is zero otherwise.

• Max C_{ih} : Maximum production capacity of facility i in period h

•M : A very large number

•n : Maximum number of production facilities that can be established in each period.

• MH_{ih} : Maximum available budget for establishing production facilities in period h
2.4 Decision Variables

X_{spih} : Size of order for product p that is sent from supplier s to facility i in period h

•Y_{spih} : A binary variable that is equal to one if product p is sent from supplier s to facility i in period h, and is zero otherwise.

•Z_{ih} : A binary variable that is equal to one if a facility is established in candidate site i in period h, and is zero otherwise.
2.5 Mathematical Model
The proposed biobjective mathematical model is presented in the following. This model has two objective functions. The first objective function minimizes the total cost of purchasing materials and transferring them to facilities and deploying facilities at candidate sites. The second objective function minimizes the total number of units with late delivery.
s.t.
Equation (3) ensures that the chain satisfies the demand for every product. Equation (4) states that the number of units with late delivery must remain below the maximum allowable level. Equation (5) states that the size of the order for each product from each supplier must be smaller than the production capacity of that supplier. Equation (6) shows the logical relationship between the decision variables and causes the corresponding dependent variable to take the correct value. Equation (7) sets the requirements for the construction of facilities at candidate sites and ensures that products can only be sent to active facilities. In Equation (8), the upper bound for the number of production facilities is set. Equation (9) determines the capacity of production facilities (each facility has a specific capacity in each period). Equation (10) ensures that a product can only be ordered from suppliers that produce this product. Equation (11) determines the maximum budget available for the construction of production facilities in each period (it is assumed that the company has a budget constraint). Finally, Equation (12) defines the decision variables (the range of allowed values for decision variables).
3. SOLUTION METHOD
Suppose the optimal value for the objective i = 1, 2,.., n is ${f}_{i}^{*}.$ In real multiobjective decisionmaking (MODM) problems where the objectives are conflicting, there is usually no single solution like x^{*} ∈ X to optimize all objectives $\nexists {x}^{*}\in X:{f}_{i}^{*}={f}_{i}({x}^{*}).$ Therefore, if a solution method like A produces the solution x^{A}, then A is said to be more efficient the closer ${f}_{i}({x}^{A})$ is to ${f}_{i}^{*}$. To put in terms of deviation from ideal, if the ideal solution is ${F}^{*}=({f}_{1}^{*},\hspace{0.17em}{f}_{2}^{*},\hspace{0.17em}\dots ,\hspace{0.17em}{f}_{n}^{*})$, then a lower deviation of the solution ${F}^{A}=({f}_{1}({x}^{A}),\hspace{0.17em}{f}_{2}({x}^{A}),\hspace{0.17em}\dots ,\hspace{0.17em}{f}_{n}({x}^{A}))$ from F^{*} will indicate the better performance of the method A. However, how this distance or deviation is defined could also affect the result. The LP metric method has been developed to consider this. In the LP metric method, P is the norm of distance measurement
when objectives have different degrees of importance (relative to each other), distance is defined with the norm p as shown below. The lower the ${F}^{*}{F}^{A}{}_{p}$ is, the better is the method A. Equation (13) is then rewritten as follows:
where w_{i} denotes the relative weight (or importance) of each objective (usually specified by the decisionmaker). In the LP metric method, using certain P values results in wellknown models. If P = 1, then the resulting model (the linear version) is called the absolute model or the weighted sum model.
If P = 2, then the model is called the Euclidean norm/distance model. In this case, formulation uses the square of the distance:
This is a convex quadratic model, which has a globally optimal solution. It should be noted that P = 2 has more emphasis on deviation from the optimum of each objective than P =1. Another wellknown choice is P = ∞, which results in the following model:
This shows the maximum deviation from the optimum. In this mode of the method, which is also called the minimax model, we minimize the maximum deviation from the optimum. It should be noted that while this method gives the lowest deviation for objectives, the sum of deviations in this method is usually higher than in the previous two methods. In this study, the defined MODM problem is solved using the LP metric method with P = 1 and P = 2.
4. RESULTS
This section presents and discusses the results obtained by solving the formulated problem for a hypothetical problem instance. After examining previous studies in this area, it was decided to apply it to a test problem designed with the specifications described below. The solution process was carried out using the LPmetric method in the GAMS software
The considered numerical example has five suppliers, each with a certain score, known capacity, and known late delivery rate. In this example, production planning and product distribution are done in four periods. In each period, there are 10 candidate sites for establishing production centers, only some of which will be activated. In this chain, four products are being produced and distributed, and the demand for each product in each period is known. In the following, the specifications of the considered numerical example are given in Tables 1 to 14. It should be noted that the parameters of this numerical example were defined such that problem constraints are not violated.
In this numerical example, the maximum budget available for developing the network in different periods was considered 7750 units. To solve this problem, the software codes of the LP metric method were executed in the CPLEX Solver tool of GAMS version 24.9.1 running on a personal computer with a 5core CPU and 6 GB of RAM.
4.1 Solving the Numerical Example using the LP Metric Method with Norm 1
To solve MODM problems, it is first necessary to obtain the payoff matrix. This matrix helps detect the conflict between objectives and determine the minimum, maximum, and amplitude of changes in each objective, and can be used to normalize the objective function values. For the numerical example considered for the proposed biobjective model (with the objectives of minimizing costs and minimizing late deliveries), the optimal value and the payoff matrix of the objectives were obtained with the CPLEX solver (in GAMS). The results are presented in Table 11.
In Table 11, it can be seen that the lowest cost is 215143 units, but at this cost, the other objective is far from its optimal state. The lowest delay is 960.85 units of total demand, but the solution that has this level of delay also has the highest cost.
The solutions in the payoff matrix of Table 11 are nondominated (because they are optimized for at least one objective). However, there is a degree of deviation from optimum in each of these solutions. In the LP metric method with the norm P = 1, the goal is to minimize the sum of these deviations. If the weight of the objectives is known in advance, only one solution will be obtained. Using the LP metric method with P = 1 (weighted sum of objectives) and with N = 10 changes in weights (N was considered as the input of the LP metric method), Pareto solutions were obtained as shown in Table 12. Since the LP metric method with P = 1 prioritizes minimizing the sum of deviations, in this case, Pareto solution No. 6 is the most efficient.
As shown in Table 12, in the LP metric method with P = 1, Pareto solution No. 6 is the best solution as it has the lowest total relative deviation. In the best solution of this method, the values obtained for the first to third objectives have a relative deviation of 6.5%, 53%, and 3.7% from their optimal values respectively (the optimal values are given in the payoff matrix of Table 11).
4.2 Solving the numerical example using the LP metric method with norm 1
In MODM problems, if the goal is to reduce the deviation of each objective from its optimum as well as the sum of deviations, it may be better to use the LP metric method with the norm P = 2. In this method, which is also known as the quadratic model (or minimization of the sum of squared deviation), the sum of deviations is replaced with the sum of squares of deviation to make the results more sensitive to larger deviations.
Using the LP metric method with P = 2 (quadratic or squared deviation model) and with N = 10 changes in weights, the 10 Pareto solutions given in Table 13 were obtained. Since this method gives higher priority to minimizing the sum of squares of deviations, it identified Pareto solution No. 5 as the best solution.
According to the results presented in Table 13, the LP metric method with P = 2 has identified the Pareto solution No. 5 as the best solution as it has the lowest sum of squares of relative deviations from optimums. In this solution, the values obtained for the first to third objectives have a relative deviation of 8.6%, 36%, and 8.23% from their optimal values, respectively. It can be seen that the maximum deviation in the LP metric method with P = 2 is 36%, which is significantly less than that in the LP metric method with P = 1 (53%).
Figure 1 is drawn to make a comparison between the Pareto fronts obtained with the mentioned norms. It should again be noted that the LP metric method was applied with N = 10 different weights, resulting in 10 Pareto solutions for each norm.
Using the LP metric method with P = 1 results in having the lowest sum of deviations, but it also leads to a high level of maximum deviation from the optimum. Increasing the norm from P = 1 to P = 2, i.e. using the squares of deviations, significantly reduced the maximum deviation, but slightly increased the sum of deviations. Therefore, if the decision is very sensitive to every objective, it is recommended to use the LP metric method with p = 2, which tries to minimize the deviation of each objective from its respective optimum. However, if the sum of deviations is more important than individual deviations, it is recommended to use p = 1.
5. CONCLUSION
In the present study, we formulated a model for locating facilities and selecting suppliers and allocating suppliers to established facilities in a multiperiod, multiproduct supply chain. Optimization was defined as minimizing the total cost (as the economic objective) and minimizing the rate of late delivery of products (as the objective representing chain reliability and customer satisfaction) in the form of a mixed linear biobjective model. To evaluate the performance of the proposed model, a test problem was designed and solved with the LP metric method in GAMS optimization software. The results showed that it is better to solve the model using the LP metric method with p = 2 if the decision is very sensitive to the optimality of every objective, but it is recommended to apply this method with p = 1 if the sum of deviations is more important than individual deviations.
To improve and expand this research, one can consider the uncertainty of the parameters in the model with the help of approaches such as robust, fuzzy, or stochastic optimization. It may also be possible to use metaheuristic algorithms and exact methods including decompositionbased methods to solve largescale problems of this type. Other potential avenues for expanding the model include considering the contingency reserves at production centers and estimating future demands to reduce the risk of shortages, using the evaluation score of suppliers to maximize purchases from top suppliers, and considering the possibility of having defective units during delivery.