Multi-Objective Location-Allocation-Routing Problem of Perishable Multi-Product Supply Chain With Direct Shipment and Open Routing Possibilities Under Sustainability

In this study a multi-objective formulation is proposed for designing a supply chain of perishable products including suppliers, plants, distributors, and customers under sustainable development. In addition to the studies of the literature, direct shipment between producers and customers and also alternative products possibility are allowed. In this problem the objectives like facilities establishment costs, transportation costs, negative environmental impacts, and social impact (fixed and variable employment rates) are optimized simultaneously. As in real situations, most of the transportation activities of such supply chain are performed by hiring transportation devices, the open routing logic is applied to form the travelling path of each hired transportation device. Furthermore, the possibility of direct shipment from the plants to the customers is considered in order to increase profitability of the plants. Because of the NP-hard nature of the supply chain design problems, some meta-heuristic solution approaches of the literature are modified to multi-objective form and applied to solve the problem. Several test problems from small to large sizes are generated randomly to evaluate the meta-heuristic algorithms. As a result, among the proposed algorithms, the multi-objective grey wolf optimizer (MGWO) perform better than others by considering four well-known evaluation metrics. At the end, a case study from perishable products supply chain of Iran is solved and analyzed to show the applicability of the proposed problem.


Introduction
In general, supply chain management uses some effective strategies in order to integrate the flow of material and information among suppliers, producers, inventories, and shops for timely production and distribution of products (Simchi-Levi et al., 2008). In traditional supply chain networks, the products are assumed to be of endless life (Dai et al., 2018). But in reality, some of the products are perishable and lose whole or a part of their value after a while (Levina et al., 2010). Meat, vegetables, dairy products, human's blood, drugs, flowers, etc. are some examples for perishable products (Mirzaei and Seyfi, 2015). One of the most important parts of supply chain management is to design the supply chain network of perishable products. Due to short life cycle of these products (for example foods), the activities such as ordering, pricing, storing, etc. in the supply chain become very complex. Another difficulty of managing the supply chain of perishable products is that keeping high volume of them in inventories has three major negative effects e.g.
(1) the cost of supply chain is increased if the products are perished, (2) the waste of perishable products have negative effects on environment, and (3) the quality of these products start to be decreased immediately after production.
Supply chain costs are approximately 30% of the total value of perishable products for a company. Therefore, managing such costs can increase competitive ability of the company . On the other hand, other criteria such as environmental negative effects and social effects such as employment can improve sustainability of supply chain (Yakavenka et al., 2019). Therefore, these criteria should be considered in supply chain decision making. Generally, decision making in supply chain can be done in strategic, tactical, and operational levels (Shen and Qi, 2007). The strategic decisions are taken mainly about number and locations of facilities e.g. plants, distribution centers, etc. (Ravindran and Warsing Jr, 2016). The tactical decisions are about material flow, information flow, amount of material and product in the facilities, etc. Finally, the operational decisions are taken about transportation activities and optimal transportation routing (Berman et al., 2012). Operations research (OR) is an optimization tool with many applications in supply chain decision making (Snyder et al., 2016). As of the most important applications of OR, design of distribution network in a multi-echelon network with location, allocation and routing decisions can be mentioned (Eiselt and Marianov, 2015). In addition, optimization algorithms (Rabbani et al., 2018) and multi-criteria decision making (Soltani et al., 2015) also are of such applications.
Some important studies of perishable products supply chain is reviewed here. Mohammed 2018) studied a supply chain of perishable products with objectives such as minimizing cost, CO 2 , due date, and accident rate. Hamdan and Diabat (2019) presented a two-level stochastic programming for blood delivery supply chain. The details of the researches performed in this study area are represented by Table 1. These studies are detailed from problem type, structure, solution method, decision type, and sustainability points of view. In this study a supply chain network design problem of perishable products is considered. This supply chain consists of suppliers, producers, distribution centers, and customers. In addition to the studies of the literature, direct shipment between producers and customers and also alternative products possibility are allowed. In order to cover more aspects of the problem, it is formulated as a multi-objective mathematical model. The objective functions of the model are (1) minimization of total costs, including establishment costs of the plants and distribution centers, all transportation costs among the suppliers, facilities and customers, and hiring cost of the transportation facilities, (2) minimization of the negative environmental impact of transportation activities, and (3) maximization of fixed and variable employment rate and minimization of the lost days. The constraints of the model include facilities location constraints, material flow constraints, open transportation routing from plants to customers and distribution centers to customers, and determining alternative products for customers according to their need. As the proposed formulation is generally from NP-hard class of optimization problems, we need to apply meta-heuristic algorithms to solve the medium and large sized instances of the problem. Therefore, some meta-heuristic solution approaches of the literature are modified to multi-objective form and applied to solve the problem. Several test problems from small to large sizes are generated randomly to evaluate the meta-heuristic algorithms. As a result, among the proposed algorithms, the multi-objective grey wolf optimizer (MGWO), perform better than others by considering four well-known evaluation metrics. At the end, a case study from perishable products supply chain of Iran is solved and analyzed to show the applicability of the proposed problem.
The remainder of this paper is organized in some other sections. Section 2 describes the problem of the study, presents its mathematical formulation, and discusses its computational complexity. Section 3 develops some classical and hybrid meta-heuristic algorithms to solve the proposed mathematical model. Section 4 performs a full computational study on the proposed meta-heuristic algorithms. Section 5 gives some conclusions for the study.

Problem description and formulation
In this study a problem of locating and open routing in a supply chain with four echelons such as suppliers, plants, distribution centers, and customers is modeled as a multi-objective problem with sustainability. The network of this problem is depicted by Figure 1. In this problem direct transportation is considered between all echelons, while the routing is done between the established distribution centers (DCs) and the customers and also between the established plants and the customers. The flow of raw material is determined according to the amount of each raw material in final product. The transportation of final products is done in two ways by applying an open routing technique, first sending them to the established DCs and then transporting them to the customers, and second, direct transportation from the plants to the customers. Open routing is applied in this study as in real-world cases, many companies do not own transportation devices. Therefore, they hire some devices to send the products to the destinations. After that the device does not need to come back to the plant and leaves the system. Mathematical model proposed for the problem consists of three objective functions. First objective function minimizes total cost of the network including establishment costs, transportation costs, hiring cost of transportation devices, raw material costs, and production costs. Second objective function minimizes negative environmental impacts of facilities establishment, production, and transportation activities. Third objective function maximizes positive social impacts including fixed and variable employment rates by establishing plants and DCs. In order to consider the concept of open routing in the proposed formulation, we use dummy nodes (customers) in the network of the problem (as shown by Figure 1). In the tour of each transportation device, a dummy customer is considered. The device after passing all customers enters the dummy customer. The distance and cost of this dummy customer to the other customers is zero, therefore, entering to this dummy customer has no cost for the network.
The assumptions used to model the problem is as follow, There are a set of potential plants, a set of potential DCs, and a set of potential suppliers. Some or all of them are to be established or used in the network. The number and place of the customers are known. There are enough available transportation devices with different capacities and costs.

6
The problem is multi-product with a single planning period. Price of raw material k bought from supplier i Cost of producing a unit of product g at plant j by technology e Cost of processing a unit of product g at DC r with capacity l Holding cost of product g at plant j for direct shipment purpose Hiring cost of device v Amount of product g produced by technology e at plant j Excess/shortage production amount comparing to demand Sub-tour elimination variable Mathematical formulation: Objective function (1)  Constraints of the formulation (1)-(39) are explained here. Constraint (4) respects to the capacity of the suppliers. Constraint (5) calculates the amount of raw material for production activities. Constraint (6) applies the capacity limits of raw material given by the suppliers. Constraint (7) ensures that only the established plants should have output. Constraint (8) ensures that in each plant for each product type, only one technology is used. Constraint (9) a plant is established when the production of its products is possible. Constraints (10)-(11) calculate the production amount of the products. Constraints (12)-(13) guarantee that the product flow is between the established plants and DCs. Constraint (14) ensures that the products are delivered to the customers directly or indirectly. Constraint (15)

Meta-heuristic solution approaches
Generally supply chain network design problems are of NP-hard class of optimization problems (Akbari et al., 2020). Therefore, in order to obtain high quality solutions for such problems, use of meta-heuristic solution approaches are unavoidable. The problem introduced in previous section is a typical supply chain network design problem, so, in this section some meta-heuristic algorithms are introduced and applied to solve it effectively. Among the wide range of meta-heuristic algorithms of the literature, some populationbased meta-heuristic solution approaches such as Grey Wolf Optimizer algorithm (GWOA) (proposed by Mirjalili et al., 2014), Ant Lion Optimizer algorithm (ALOA) (Mirjalili et al., 2017), and Dragonfly algorithm (DA) (Mirjalili, 2016) are selected and modified to be used for multi-objective formulation (1)- (39). The efficiency of these algorithms are proved in the literature of combinatorial optimization problems. Furthermore, the results obtained by these algorithms are compared by the NSGAII and SPEAII as two well-known and popular multi-objective meta-heuristic solution approaches of the literature. In the rest of this section, first a solution representation is developed to be used in all of the proposed algorithms, then the multi-objective form of the GWOA, ALOA, and DA approaches are explained in details, and then the comparison metrics used to evaluate the obtained multi-objective solutions are explained.

Solution representation
A matrix-like solution representation is proposed to generate any initial and neighborhood solutions in the meta-heuristic approaches of this study. All values in this solution representation are generated randomly. An example of this representation is given by Figure 2 and its description is given in the following of this sub-section. The length of each row in the matrix is written on the left side of the row in the figure.

Row 2.
This row is used to assign the value of raw material between the suppliers and the plants. The assignment is done by similar logic used in Row 1.

Row 3.
This row is used to assign the value of product between the plants and the DCs. The assignment is done by similar logic used in Row 1.

Row 4.
The device used for transportation between the suppliers and the plants and then to the DCs are determined in this row randomly.

Row 5.
The amount of raw material and products transported be the devices assigned in Raw 4 is determined here. The logic of Raw 1 is used here too.

Row 6.
The capacity level of each DC for each product is determined in this row.  Each solution generated by the proposed approach is evaluated easily using the objective functions (1)-(3) and the parameters of the problem.

Multi-objective grey wolf optimizer algorithm (MGWOA)
The grey wolf optimizer algorithm first was introduced by Mirjalili In these equations, is iteration number, and and are the multipliers determined by the following equations, The values of vector are linearly decreased from 2 to 0. The vectors 1 ⃗⃗⃗ and 2 ⃗⃗⃗ are randomly generated from the interval [0, 1]. In order to simulate the behavior of grey wolf, it is supposed that searching factors , , and have more information about the place of bait. Therefore, the best three solutions are always saved and other searching factors (like ) will update their place according to the best place obtained by other factors. For this reason, the following equations are proposed.
(t + 1) = 1 + 2 + 3 3 Other operators of this algorithm are detailed in the study of Mirjalili et al. (2016). The pseudo code of this algorithm is given by Figure 5.  Figure 5. The pseudo code of the MGWOA.

Multi-objective dragonfly algorithm (MDA)
In order to apply the MDA, an archive should be considered in the classical dragonfly algorithm in order to save the obtained Pareto solutions, but the food source is selected from the archive. The operators in the MDA is the same as the dragonfly algorithm. In order to find an extensive Pareto frontier, the food source is selected from the low density parts of the Pareto frontier. This is the same as the multi-objective PSO algorithm. The low density part of the Pareto frontier is obtained by dividing it into several sub-frontiers. This issue is done by considering the best and the worst Pareto solution, defining a hyper sphere for covering all of the Pareto solutions, and dividing the hyper sphere into some parts in each iteration. Then selecting the low density part of the Pareto frontier is done by a roulette wheel mechanism. The probability of each part is obtained by the following formula, where, is a constant being greater than 1 and is number of parts in -th Pareto frontier. According to this equation, the algorithm selects the food source from the low density parts of the Pareto frontier. Therefore, the dragonfly moves on the low density parts and distributes the solutions in the Pareto frontier uniformly. For being far from the enemies equation (47) is changed to the following equation.
= (48) In each iteration the archive is updated. In order to manage the Pareto frontier, the obtained non-dominated solutions are used to form it. If the archive is full, and still a new solution is generated, a solution from the high density parts is removed and the new solution is replaced. According to the study of Mirjalili et al. (2016), the overall structure of the MDA is depicted by the pseudo code of Figure 6.

Multi-objective ant lion optimizer algorithm (MALOA)
In order to apply the MALOA, an archive should be considered in the classical ant lion optimizer algorithm in order to save the obtained non-dominated Pareto solutions. The convergence of the MALOA is similar to the classical ant lion optimizer algorithm. In order to modify the ant lion optimizer to obtain the MALOA, the structure of multi-objective PSO is used. In order to distribute the solutions of the archive adequately, two operators of leader selection strategy and archive controller are used. The dispersion of the solutions in the archive is measured by considering a radius around any of the solutions. The number of solutions in the area of the radius of any solution is considered as the dispersion criterion of the solution. Similar to the multi-objective PSO and the MDA, a solution from the low density and high density parts of the archive are selected by the following probabilities respectively.
According to the study of Mirjalili et al. (2017), the overall structure of the MALOA is depicted by the pseudo code of Figure 7.

Comparison metrics
As the multi-objective meta-heuristic solution approaches instead of considering a unique solution, consider a set of non-dominated solutions as a Pareto frontier, therefore, their comparison and evaluation become more difficult than single objective meta-heuristics. For this aim, according to the study of Datta and Figueira (2012), the following comparison metrics are considered to compare the solution approaches proposed in previous sub-section.

Number of Pareto solutions (NPS):
It is the number of Pareto solutions found by a meta-heuristic algorithm. Its higher value shows better performance for an algorithm. Spacing: This measure shows that how the Pareto solutions are spaced uniformly. The following formula is used for this measure. The less value of this criterion is favored.

Mean ideal distance (MID):
In this measure, the distance from optimal Pareto is calculated by the following formula, where its lower values are preferred.
where, * is the ideal objective function value for -th objective function.

Computational study
In order to evaluate the proposed meta-heuristic algorithms, those are coded in MATLAB. A case study and some randomly generated test problems are solved by the cods on a PC with 3.2 GHz processor and 16 GB RAM. The details of the computational study are explained at the rest of this section.

Test problems
Three categories of small, medium, and large sizes are considered for test problems of this section. Totally 12 test problems are generated randomly. The parameters of these test problems are generated by Normal distribution. The characteristics of these test problems are detailed by Table 2. Table 2. The characteristics of the generated test problems.

Parameter tuning
In any meta-heuristic algorithm, parameter tuning is done to obtain the best value for each of its controllable parameters for obtaining better performance according the given evaluation criteria. In order to tune the parameters of the proposed meta-heuristics of this study, a typical trial and error method is applied here. For this aim, the values of Table 3 is considered for the potential levels of the parameters of the proposed meta-heuristic algorithms.

Final experiments on the test problems
In order to perform final experiments on the test problems generated in Section 4.1, the parameters of the proposed meta-heuristic algorithms are set to their best values obtained by parameter tuning stage explained by Section 4.2. Then each test problem is solved for 10 times by each meta-heuristic algorithm in order to obtain more reliable results. The obtained results are measured by the comparison metrics of Section 3.3 and the best obtained results for each algorithm is shown by Table 5, 6, and 7. The results of these tables are shown by Figures 8-13.

Case study
In order to show the applicability of this research a case study is presented and solved in this section. This case study considers the Somayyeh industrial group in Iran which produces food related products via establishing a chain consisting of suppliers, plants, distributors, and final customers. This chain has 4 suppliers, 10 potential plants, 31 potential distributors, and 378 customer zones. The raw material are carried from the suppliers to the plants by three types of transportation devices. The products are transported from the plants to the distributors by three types of transportation devices, while five types of device are used to transport the products from the distributors to the customers. All demand values for 15 types of perishable products are available, where, the cost and environmental parameters are valued according to the study of Govindan et al. (2014). The suppliers, potential plants, potential distributors, and customers are shown by Figure 13 and  The MGWO algorithm is used to solve the problem of this case study as this algorithm has shown the best performance comparing to other algorithms in experiments of Section 4.3. The Pareto frontier obtained by this approach is shown by Figure 15. Totally 30 Pareto solution is obtained for the case study. Each of them can be used by managers of the company to be implemented. Selecting any of them requires to consider the criteria which is important for the managers.
24 Figure 15. The Pareto frontier obtained for the case study by the MGWO algorithm.
For more analysis, one of the Pareto solutions is considered. In this solution 14 distributors are established in order to fulfill the demand received by the customers. In this solution a nice customer to distributor assignment structure is obtained which is shown by Figure 16. The flow between the suppliers, plants, and distributors are depicted by Figure 17, where the open routing concept can be seen in the obtained structure. Figure 17. The obtained supply chain and its routing for the case study in a sample Pareto solution.

Concluding remarks
In this study a multi-objective formulation was proposed for designing a supply chain of perishable products including suppliers, plants, distributors, and customers under sustainable development. In this problem the objectives like facilities establishment costs, transportation costs, negative environmental impacts, and social impact (fixed and variable employment rates) should be optimized. As in real situations, most of the transportation activities of such supply chains are performed by hiring transportation devices, the open routing logic is applied to form the travelling path of each hired transportation device. Furthermore, the possibility of direct shipment from the plants to the customers is considered in order to increase profitability of the plants. Because of the NP-hard nature of the supply chain design problems, some meta-heuristic solution approaches of the literature were modified to multi-objective form and applied to solve the problem. Several test problems from small to large sizes were generated randomly to evaluate the metaheuristic algorithms. As a result, among the proposed algorithms, the multi-objective grey wolf optimizer (MGWO) performed better than others by considering four well-known evaluation metrics. At the end, a case study from perishable products supply chain of Iran was solved and analyzed to show the applicability of the proposed problem.

Compliance with Ethical Standards
 Funding: This study was not funded by any organization.

 Conflict of Interest:
-Author Behzad Aghaei Fishani declares that he has no conflict of interest.
-Author Ali Mahmoodirad declares that he has no conflict of interest.
-Author Sadegh Niroomand declares that he has no conflict of interest.
-Author Mohammad Fallah declares that he has no conflict of interest.
 Ethical approval: This article does not contain any studies with human participants or animals performed by any of the authors. Figure 1 Network of the proposed problem. The solution representation scheme used in the proposed meta-heuristic algorithms.  The routing strategy for each device.

Figure 5
The pseudo code of the MGWOA.

Figure 6
The pseudo code of the MDA. The pseudo code of the MALOA. The obtained NPS measure values by the meta-heuristic approaches.

Figure 9
The obtained MID measure values by the meta-heuristic approaches.

Figure 10
The obtained Diversity measure values by the meta-heuristic approaches.

Figure 11
The obtained Spacing measure values by the meta-heuristic approaches.

Figure 12
The CPU running times of the meta-heuristic approaches.

Figure 13
The Pareto frontier obtained for Prob9.

Figure 14
Geographical position of the potential places of the case study. Note: The designations employed and the presentation of the material on this map do not imply the expression of any opinion whatsoever on the part of Research Square concerning the legal status of any country, territory, city or area or of its authorities, or concerning the delimitation of its frontiers or boundaries. This map has been provided by the authors.

Figure 15
Geographical position of the customers of the case study. Note: The designations employed and the presentation of the material on this map do not imply the expression of any opinion whatsoever on the part of Research Square concerning the legal status of any country, territory, city or area or of its authorities, or concerning the delimitation of its frontiers or boundaries. This map has been provided by the authors.

Figure 16
The Pareto frontier obtained for the case study by the MGWO algorithm.

Figure 17
Customer to distributor assignment structure obtained for the case study in a sample Pareto solution.

Figure 18
The obtained supply chain and its routing for the case study in a sample Pareto solution. Note: The designations employed and the presentation of the material on this map do not imply the expression of any opinion whatsoever on the part of Research Square concerning the legal status of any country, territory, city or area or of its authorities, or concerning the delimitation of its frontiers or boundaries. This map has been provided by the authors.