An application of interactive fuzzy optimization model for redesigning supply chain for resilience

Supply chain disruptions compel professionals all over the world to consider alternate strategies for addressing these issues and remaining profitable in the future. In this study, we considered a four-stage global supply chain and designed the network with the objectives of maximizing profit and minimizing disruption risk. We quantified and modeled disruption risk as a function of the geographic diversification of facilities called supply density (evaluated based on the interstage distance between nodes) to mitigate the risk caused by disruptions. Furthermore, we developed a bi-criteria mixed-integer linear programming model for designing the supply chain in order to maximize profit and supply density. We propose an interactive fuzzy optimization algorithm that generates efficient frontiers by systematically taking decision-maker inputs and solves the bi-criteria model problem in the context of a realistic example. We also conducted disruption analysis using a discrete set of disruption scenarios to determine the advantages of the network design from the bi-criteria model over the traditional profit maximization model. Our study demonstrates that the network design from the bi-criteria model has a 2% higher expected profit and a 2.2% lower profit variance under disruption than the traditional profit maximization solution. We envisage that this model will help firms evaluate the trade-offs between mitigation benefits and mitigation costs.

the traditional profit maximization model. Our study demonstrates that the network design propagation and explains how a small shock can cause havoc in systems (Elsinger et al., 2006). Keynes (1937) identified systemic risk during the Great Depression, and it has subsequently been used in fields other than economics, including climate and biology. This theory remains true in the case of complex SCs since a small shock in one part of the SC can propagate across the entire SC, causing havoc. This is further supplemented with the contingency theory (Burns & Stalker, 1961), which is widely used in organizational studies. According to this theory, an organization should optimize its performance by implementing a strategy that aligns its capabilities with environmental requirements (Mintzberg, 1978). These two theoretical foundations provide a lens for studying SC disruption. Since new trends such as lean, JIT, and other efficiency-focused activities are increasing, SCs are always at risk of potential disruptions. Hence, determining the impact of alternate SC designs on disruptions and SC profits is essential.
The SC network considered in this study is a well-known published network structure for real-world problems. The network structure is well-suited for a range of industries, including those related to consumer electronics, automotive, batteries, plastic goods, and glass. Based on the strategic nature of the problem, cost and resiliency are the main factors to consider while designing today's global SCs. An SC disruption can cause a firm to have significant financial losses. This necessitates the search for the best SC design that can allow firms to operate cost-efficiently with the best network structure while remaining resilient to SC disruptions. With this motivation, the purpose of this study is to address the issue of designing a four-stage SC that considers profit and supply density. The problem involves determining the best suppliers, warehouse locations, and their capacity, as well as the distribution flow among the chosen facilities in the SC. Furthermore, the geographical dispersion of suppliers, which affects investment costs, product distribution, and redundancy, was considered in this study. According to our knowledge, there are few studies on SC network design models that explicitly integrate SCD characteristics, making it relevant in the current context and necessitating a separate study. We propose a bi-criteria mixed-integer linear programming (MILP) approach (Pazhani et al., 2018) for an SC network design problem with the dual objective of maximizing profit and SCD.
It is difficult to forecast and plan for demands for a strategic problem with long planning horizons. Firms may have to run the model with different input settings (e.g., different demand profiles) to finalize the best network structure. Furthermore, owing to the intrinsic multicriteria nature of the problem, firms would require an efficient algorithm to run the model and choose the best possible network structure from the available efficient solutions. This necessitates the search for and implementation of such an algorithm. An interactive fuzzy optimization algorithm is widely used to systematically solve the bi-criteria problem using the input of DMs. We adopted this algorithm and defined steps to guide users in this process. The flexibility of this methodology in analyzing and comparing scenarios helps firms to make high-quality, confident decisions. The model and proposed methodology are illustrated using a realistic example. Furthermore, the advantages of designing the SC while considering both cost and supply density are illustrated using disruption analysis. The analysis shows that the mitigation benefit outweighs the mitigation cost in the case of disruptions.
The remainder of this article is organized as follows. Section 2 reviews the literature on SC network design and disruption. Section 3 provides a detailed description of the SC network design problem for a four-stage SC with cost and supply density objectives and proposes a new bi-criteria MILP model for the problem. We also propose an interactive bi-criteria fuzzy optimization model to solve the bi-criteria model. Section 4 presents an example to illustrate the proposed bi-criteria mathematical model. Section 5 discusses the disruption scenario analysis to demonstrate the advantages of considering supply density to improve SC resiliency against disruptions. Section 6 presents the conclusions and future research directions.

Literature review
Since the 1990s, competition in external businesses has encouraged organizations to improve efficiencies, and research on SCs has gained prominence as a result. Although SC network design has been studied extensively (Khalilpourazari & Arshadi Khamseh, 2019;Özceylan & Paksoy, 2013aÖzceylan & Paksoy, , 2013bPervin et al., 2018;Sangaiah et al., 2020), awareness of the impact of disruptions in SC activities has recently increased.
SC disruption as a field of study is not new; it has coexisted with the SC field since its inception (Snyder et al., 2016). However, over the last two decades, the term "disruption" regarding SC has gained traction as follows: (a) Several high-profile events, such as the terrorist attacks of September 11, 2001 (Stecke & Kumar, 2009), Hurricane Katrina in 2005(Wachtendorf et al., 2013, and the more recent COVID-19 pandemic (Ivanov & Dolgui, 2020), have brought studies on SC disruptions to the forefront of public attention. (b) According to Anderson (2007), McGillivray (2000, and Peck (2005), the JIT philosophy tends to intensify SC vulnerability during disruptions. This vulnerability is attributed to the marginal room for error, which is demanded by the inherent nature of a tightly optimized, lean design. (c) As the vertical integration in supplier firms are decreasing, global SCs with suppliers all over the world are increasing (Cohen & Lee, 2020).
Furthermore, literature reviews conducted by contemporary researchers demonstrate the emergence of new dimensions to the SC design theme (Table 1).
The field has evolved from simple decision support models in SC network design (Farahani et al., 2014;Meixell & Gargeya, 2005;Melo et al., 2009) to complex topics such as supply disruptions, sourcing decisions, and facility location (Snyder et al., 2016). For example, Fazli-Khalaf et al. (2017) proposed an effective hybrid robust fuzzy stochastic programming method to control parameter uncertainty and risk-aversion level in the context of a lead-acid battery SC case study. Özceylan and Paksoy (2013a) proposed a mixed-integer programming model for optimizing a general closed-loop SC network model with forward and reverse components. Khalilpourazari et al. (2020) used a neural-learning process to overcome new challenges based on past experiences. They considered three objective functions that minimized total transportation time and cost while minimizing unfulfilled demand in a real-world case in Iran. Goli and Aazami (2018) used an accelerated cuckoo optimization algorithm to optimize vehicle routing in a case study on dairy product distribution. Similarly, research on SC has demonstrated the use of algorithms and mathematical models in diverse scenarios.
Recent research has demonstrated that the field of supplier selection is expanding (Table 2). Some prominent examples are studies on framework development (De Boer et al., 2001), the need to focus on customer-oriented criteria (Ho et al., 2010), and the evaluation of uncertainty as a critical factor in supplier selection (Chai et al., 2013).
The literature reviews outlined in Tables 1 and 2 emphasize the importance of investigating uncertainty in SC designs. According to Sabri and Beamon (2000), uncertainty is one of the most challenging and prominent problems in SC management. The inherent stochastic nature of uncertainties is what makes a system complex, and it has attracted the interest of researchers all over the globe. For example, Goh et al. (2007) developed a stochastic model   Santoso et al. (2005) proposed a stochastic programming model and solution algorithm to solve a realistic SC network design problem. They combined the sample average approximation scheme with an accelerated Benders decomposition algorithm to solve largescale stochastic SC design problems. Additionally, Salehi et al. (2017) adopted a new robust two-stage multiperiod stochastic model to design a blood supply network in Iran, taking into account the possibility of a natural disaster. They concluded that such studies are helpful in developing alternative strategies during SC disruptions.
The severity of SC disruptions is related to the geographical concentration of nodes within an SC, known as SCD, which results from an SC network design decision (Craighead et al., 2007;Falasca et al., 2008). SCD can be measured as the number of nodes divided by the average internode distance. The SCD is said to be high when many nodes are clustered within the SC. Therefore, the severity of SC disruptions is directly related to the SCD. Snyder et al. (2016) emphasized that disruption can be mitigated by sourcing flexibility, which can have an impact on overall SC profits. Thus, we focused on SCD and SC profits in this study.
Other studies have assessed the vulnerability of SCs and evaluated suitable mitigation strategies in response to uncertainty and disruption. Schmitt and Singh (2009) used the Monte Carlo analysis to generate a risk profile and discrete-event simulation to evaluate inventory policies suitable for distribution networks that consider demand uncertainty, supply uncertainty, or both. Klibi and Martel (2012) developed a scenario-based risk model to generate resilient SCs using the Monte Carlo analysis. In both studies, multiperiod risk profiles were generated to cover a specified planning horizon. Harrison et al. (2013) proposed an optimization approach called resiliency enhancement analysis via deletion and insertion (READI) to improve SC network resiliency. READI is used to evaluate network resiliency (when an important SC node or flow is disabled) and mitigation strategies for resilience improvements.
Empirical research has shown the relationship between SC network characteristics and SC disruption (Wagner & Neshat, 2010). SC network characteristics, including network decentralization, geographical dispersion, number of nodes, and number of tiers, appear to be related to the occurrence of SC disruptions (Bode & Wagner, 2015;Kim et al., 2015;Squire, 2010). Conversely, as shown in Table 3, characteristics such as density, complexity, node criticality, node centrality, and lack of redundancy are related to the severity of SC disruptions (Craighead et al., 2007;Falasca et al., 2008;Squire, 2010;Wagner & Bode, 2006). It is noteworthy that most studies on SC design have focused on optimizing a single criterion, particularly profit (see Chan et al., 2016;Cheraghalipour et al., 2018;Darestani & Hemmati, 2019;Jiang et al., 2019;Latha Shankar et al., 2013;Sangaiah et al., 2020). However, DMs in many fields, including industry, engineering, and social sectors, are increasingly required to consider multiple conflicting objectives in their decision processes (Ravindran, 2016). Multicriteria decision-making problems are categorized based on whether the constraints are (i) finite and known or (ii) infinite and unknown (Ravindran, 2016). SC network design studies with multiple criteria have used a variety of solution methodologies, including variants of goal programming (Ravindran et al., 2010), Benders decomposition algorithm (Garcia-Herreros et al., 2014), exact mathematical modeling (Huang & Goetschalckx, 2014; (2015) The number of suppliers in each tier, the number of levels, and the geographical dispersion among members within the network have a positive relationship with the frequency of SC disruptions Occurrence Peng et al., 2011), and network optimization (Mari et al., 2014). Melo et al. (2009) provided a detailed review of location design in SCs. We used multicriteria decision-making modeling (Pazhani et al., 2018;Pinto-Varela et al., 2011) to simulate the problem as a bi-criteria MILP in order to maximize profit and supply density. The interstage distance between SC nodes was used to calculate and maximize supply density, which resulted in a geographically dispersed network design. Subsequently, we propose an interactive fuzzy optimization algorithm that uses the 2-constraint method to solve the problem and generate a Pareto-efficient frontier. The interactive optimization algorithm guides the user/DM in choosing the best network design solution from those available in the Pareto-efficient frontier, based on the firm's objectives. The model is illustrated using a realistic example. We also evaluated the resilience of the SC network solutions under disruptive scenarios and demonstrated the value of incorporating supply density into the network design. This research contributes to the existing literature by incorporating SC characteristics into SC design. Additionally, the proposed interactive optimization algorithm systematically solves the bi-criteria problem. The algorithm reduces the cognitive burden on the DM by accelerating the convergence of the best compromise solution.

Bi-criteria network design model
This section presents the proposed bi-criteria network design model for the four-stage SC network with the objectives of maximizing profit and supply density. Let S {1, 2,…, n S } be the set of suppliers, M {1, 2,…, n M } the set of manufacturing plants, and C {1, 2,…, n C } the set of retailers. Let W {1, 2,…, n W } be the set of potential warehouse sites and L {1,2,…, n L } the set of the warehouse capacity levels. Figure 1 shows the considered SC network. The SC network includes suppliers, manufacturing plants, warehouses, and retailers. The manufacturing plants procure the raw material required for production from the set of selected suppliers. The product flows from the manufacturing plants to the warehouses, which are managed by a single DM (i.e., a centralized control system). The retailers are faced with demands from customers and are supplied by a set of potential warehouses.
The proposed model is developed to (i) select the appropriate suppliers and determine the quantities allocated to the chosen suppliers, (ii) select the appropriate set of warehouses and their capacity levels to distribute products from the manufacturing plants to retailers, and (iii) determine the flow of products in the SC between the selected set of facilities. The model's objective is to maximize SC profit and supply density. The total cost includes purchasing costs, manufacturing costs, warehouse opening costs, and transportation costs between SC stages. The supply density objective was calculated based on the interstage distances between the selected set of suppliers and the manufacturing plants to which they supply raw materials and the intrastage distances between the selected group of suppliers. Hence, a lower supply density value implies that the nodes in the SC are clustered. Thus, we maximized the supply density.
The model parameters, decision variables, and cost function components are provided below:  The following assumptions were considered when developing the proposed model:

Input Parameters
(i) Retailer demands are deterministic. Since the proposed model is a strategic decisionmaking model, this assumption is reasonable. (ii) The suppliers and manufacturing plants have finite production capacity. (iii) The cost of transporting raw materials from the supplier to the manufacturing plant is included in the raw material purchasing cost. (iv) The warehousing facilities in the SC have capacity restrictions.

The proposed bi-criteria MILP model
Considering the purchasing cost, manufacturing cost, transportation cost, fixed cost components, interdistance between the suppliers and manufacturing plants, and intradistance between the suppliers, the problem can be formulated as an MILP with the objectives of maximizing the total profit and the supply density of the SC: where the components of the SC profit objective are: {revenue}-{purchasing cost + production cost + transportation cost from plants to warehouses + transportation cost from warehouses to retailers + fixed cost for opening warehouses + lost sales cost}.

Objective 2: maximizing the supply density based on the interstage distance (Z 2 )
The supply density is calculated based on the interstage distance between the selected suppliers and plants and the intrastage distance between the selected suppliers. We will demonstrate the computation with an example. Consider a two-stage SC where two suppliers (S1 and S3) both supply products to a manufacturer (M1). The interstage distance is the distance between stages. In this example, the interstage distance is the sum of the distance between supplier S1 and manufacturer M1 and the distance between supplier S3 and manufacturer M1 (see Fig. 2). The interstage density of the suppliers in the SC is calculated using this metric. The intrastage distance is the distance within a stage. In this example, the intrastage distance is the distance between suppliers S1 and S3, as shown in Fig. 2. The intrastage density of suppliers within a stage in the SC is measured using this metric. In this study, the supply density objective is the density of the supply entity per unit of demand, which is defined by the interstage and intrastage distances for the supplier stage in the objective function: Each supplier s has a finite supply capacity: cap s . Constraint set (1) ensures that the quantity of raw materials supplied by supplier s to all manufacturing plants is less than or equal to its capacity. Constraint sets (2) and (3) determine the binary variables for the interstage flow between suppliers and manufacturers. Constraint (2) ensures that if there is a shipment between supplier s and manufacturer m, the binary variable Sα sm 1. Constraint (3) ensures that if there is no shipment between stages, the binary variable Sα sm 0. Constraint (3) also ensures minimum shipment if there is a shipment between supplier s and plant m. Constraint sets (4) and (5) determine the binary variables for the intrastage flow for the supplier stage. The right-side term in Constraint (4) represents the product flow from suppliers i and j to manufacturer m. If both (supplier i to manufacturer m and supplier j to manufacturer m) links have product flow, the binary variable Sβ ijm 1 and Sβ' ijm 0. If product flow exists in either the supplier i to manufacturer m link or the supplier j to manufacturer m link, then Sβ' ijm 1 and Sβ ijm 0. If there is no flow in these links, Sβ ijm 0 and Sβ' ijm 0. Constraint (5) ensures that one of the following cases is true: Sβ ijm 1 and Sβ' ijm 0, Sβ ijm 0 and Sβ' ijm 1, or Sβ ijm 0 and Sβ' ijm 0. Constraint (6) is the production capacity constraints at the plants. The left-most term represents the total quantity of products transported to warehouses from plant m, which should be less than or equal to its capacity. Constraint set (7) ensures that the quantity of raw materials flowing into plant m equals the number of products flowing out of the plant to the warehouses. Constraint set (8) ensures that if a warehouse is selected, the number of products flowing into the warehouse, w, does not exceed the storage capacity. The left side represents the total quantity of products flowing into warehouse w. The right side is the capacity of the selected warehouse. If warehouse w is opened, constraint set (9) ensures that only one capacity level is preferred. Constraint set (10) ensures that the quantity of products flowing into warehouse w equals the number of new products flowing out of the warehouse to the retailers. Constraint set (11) represents the demand satisfaction constraints. The total quantity of products flowing into retailer c and the lost sales at retailer c should be equal to the demand at that retailer.
Constraint set (12) ensures that the variable, SUP s , is set to 1 when there is a flow from supplier s. Note that M is a large positive number. Constraint (13) also ensures that the total number of selected suppliers does not exceed the maximum. Finally, Constraints (14) and (15) describe the nonnegativity and binary conditions of the decision variables.

Interactive fuzzy optimization algorithm
Fuzzy programming methods are popular approaches for solving multiobjective programming models because of their ability to explicitly measure and adjust the satisfaction level of each objective function (Pishvaee & Razmi, 2012). Herein, we propose a fuzzy solution method based on the 2-constraint method (Hwang & Masud, 1979). This method provides DMs an appropriate picture of the entire Pareto-optimal set, allowing them to select their preferred solution. The advantage of this method is that after the entire Pareto-optimal set has been defined, the DM can determine the final decision more confidently based on comprehensive available information (Pishvaee & Razmi, 2012; for a detailed description of the 2-constraint method, see Ehrgott, 2005).
We first generate a set of efficient solutions by varying the right-hand side of the 2constraint. The 2 values are first varied in more comprehensive steps to develop the entire Pareto-optimal solution. Next, we present the efficient frontier to the DM, and the DM chooses a range of 2 values that they are interested in. We use the DM's input to generate solutions in the interested range by varying the 2 values in more adequate steps. Subsequently, the DM is presented with this solution to determine the best compromise solution. We used an interactive fuzzy optimization algorithm based on the 2-constraint method for solving the bi-criteria MILP model.
The general form of the bi-criteria mathematical programming model is as follows: where x is an n-dimensional vector of the decision variables, f 1 and f 2 represent the profit and density objectives, respectively, and y i represents Constraint sets (1) to (15). Let S {x|y i (x) ≤ 0 } denote the feasible region. We aimed to find the best compromise solution that maximizes profit and SCD based on the DMs' utility function. The steps of the proposed method are as follows: At each iteration, we minimized μ 1 (x) with constraint on μ 2 (x). The problem can also be solved by minimizing μ 2 (x) with a constraint on μ 1 (x).

Illustrative example
This section presents an analysis of a four-stage SC system as an example. The SC network comprises the following: • Twenty potential suppliers of materials required to manufacture new products, • Five manufacturing plants that produce new products and refurbish returned products, • Twenty-five possible warehousing facilities for distributing new products to retailers, and • Approximately 100 retailers, who face demand from customers.
The input and cost-parameter settings used in this example are more realistic with respect to the location of the facilities and the distances between them. The SC facilities are geographically spread across six different regions around the world (see Table 4). We selected latitude and longitude for each facility in the SC. Thereafter, we used the FINDDIST procedure of Ramkumar et al. (2012) to determine the interstage and intrastage distances between the facilities. See "Appendix" for Tables 18 and 19: Table 18 shows the interstage distance between the suppliers and manufacturers, and Table 19 shows the intrastage distance between the suppliers. Note that the intrastage distance between suppliers i and j, where i j is not feasible, is assigned a significant value of 100,000.
The cost parameters are modeled as a function of the product price. Table 5 shows the cost settings. Note that the purchasing cost of the raw materials includes the distances between the supplier and the manufacturing plants (dis sm ).
Retailer demand was generated from a uniform distribution between 500 and 700 units. The maximum number of suppliers (mas) to be selected was set as 10. Table 20 (see "Appendix") shows the purchasing cost and capacity of the suppliers, as well as the capacity of the manufacturing plants. Table 21 (see "Appendix") shows the capacity of the warehouses and fixed costs.
We then solved and analyzed the proposed bi-criteria model and the interactive solution method using the illustrative example. The example was coded in Microsoft Visual C++ S5, S8, S10, S11, S12, S15, S17, S18  Purchasing cost of raw material p sm~U nif (60%, 65%) * pp + (dis sm / 250) Production cost for a new product pc m~U nif (8%, 12%) * pp Transportation cost per unit between plant and warehouse tr mw~U nif (5.5%, 6.5%) * pp Transportation cost per unit between retailer and warehouse tr wc~U nif (8.5%, 9.5%) * pp 6.0 and solved using ILOG Concert Technology with CPLEX 12.1 on a personal computer with a 2.8 GHz INTEL(R) Core (TM) 2 Duo Processor and 2.0 GB RAM. We first ran the model as a single-objective problem with profit (Z 1 ) and density (Z 2 ) as separate objectives. Thereafter, we ran the model as a bi-criteria model with both profit (Z 1 ) and density (Z 2 ) objectives using the proposed interactive fuzzy optimization algorithm.

Single-objective model solutions
We started by solving the single-objective models to obtain the ideal value of the objective functions. For the profit maximization model, the ideal value of SC profit (v u ) was $13,160,455.48, with a supply density (w l ) of 0.81. For the supply density maximization model, the ideal value of the supply density (w u ) was 30.01, with an SC profit (v l ) of $11,257,836.05. Table 6 summarizes the network design from the single-objective models. Figure 3 presents the geographical dispersion of the SC network solutions. Region 1 S16 -S16 -Region 2 S5, S8, S11, S15 -S5, S8, S10, S12 W15

Bi-criteria model solutions
In this section, we solve the bi-criteria model with profit and density objectives using the proposed interactive fuzzy optimization algorithm based on the 2-constraint method. Pishvaee and Razmi (2012) suggested that the DM could adjust the range of the 2 values throughout the calculation process. In early iterations, the DM starts with more comprehensive steps to quickly generate the whole range of Pareto-optimal solutions. In later iterations, the DM may be interested in selecting the final preferred solution using finer steps such that interesting areas can be investigated more precisely. Initially, we vary the 2 values between 0 and 1 in steps of 0.05. Table 7 shows the revenue, costs, SC profit, and supply density for 2 values ranging from 0 to 1. Figure 4 shows the entire efficient frontier. We present the DM with this efficient frontier solution. We assume that the DM selects an 2 value range of 0.45-0.50. We generate the Pareto-optimal solutions using a finer step value of 0.01. Table 8 shows the corresponding results, and Fig. 5 shows the graphical representation of the efficient frontier.
We assume that the DM selects the efficient solution for 2 0.45. Table 9 presents the selected suppliers and warehouses from the bi-criteria model with an 2 value of 0.45. Suppliers from five regions are selected. Table 10 shows the material flows across the SC stages. Three warehouses (warehouses 8, 12, and 23), each with a size of 3, are selected to distribute the finished products to the retailers. The product flows between the manufacturing plants, and the selected warehouses are presented in Table 11. Figure 6 shows the SC network configuration from the bi-criteria model for 2 0.45.

Disruption analysis
In this section, we present the disruption analysis using a discrete set of scenarios. The objective is to compare the severity of disruption to the SC performance of the different SC network solutions obtained in Sect. 4. We generated six independent disruptive scenarios, each representing a disruption in each region. Table 12 presents the list of suppliers, plants, and warehouses that were affected by a disruption in each scenario.
In the MILP model, we set the binary variables relating to the disrupted entities to zero and re-optimized the model to measure the SC profit and unfulfilled demand. Note that the SC profit is the difference between the revenue and variable costs (purchasing cost, production cost, transportation cost, and lost sales cost). The fixed cost was not included in the profit calculation because the location decision had already been made. Tables 13  and 14 provide the revenue, variable costs, and unfulfilled demand associated with each disruptive scenario. Table 15 compares the SC profit of two SC network design solutions under disruptive scenarios.
Since the occurrence of the disruptive scenarios is different, we referred to the number of disasters, both natural and technological disasters reported by the Centre for Research on the Epidemiology of Disasters (Guha-Sapir et al., n.d.) between 1900 and 2014, to estimate the probability of disruption (see Table 16). The likelihood of occurrence for each scenario is the ratio between the number of disasters reported in a region and the total number of disasters reported. Table 16 shows the expected profit value and variance for each network design solution.
The expected profit of the SC network solution that was obtained from the profit maximization model is $9,959,864.60, with a variance of 8.5 × 10 12 . The expected payoff of the SC network solution that was obtained from the bi-criteria model is $10,162,625.72, with a variance of 8.31 × 10 12 . The high value of anticipated profit and the low value of profit variance both indicate that the SC network design solution obtained from the bi-criteria model is more resilient than the one obtained from the profit maximization model. The bi-criteria solution yielded a higher expected profit value of $202,771 or 2% than the profit maximization solution. The bi-criteria solution had a lower profit variance than the profit maximization solution (about 2.2%). Note that the cost of redundancy is $13,160,455-$12,880,839 $279,615.68,  Fig. 4 The efficient frontier for 2 values between 0 and 1 as calculated from the difference in profit values between the profit maximization and the bi-criteria solutions.

Discussions and managerial implications
In Sect. 5, we compared the SC performance of two SC network designs in a normal situation and under disruption. This section compares the number of selected suppliers of our model to that of Rienkhemaniyom and Pazhani (2015), as shown in Table 17.
The following are some of the inferences drawn from the analysis and the managerial implications from the model: • SC network design, which is highly focused on profit maximization, tends to select suppliers that are clustered in regions that offer low costs. Even though the region has a relatively low possibility of facing random disruptions, our analysis shows that the expected profit loss and variance are high. • The SC network design from the bi-criteria model diversifies risk by locating suppliers in different geographical locations. Even though it increases the probability of facing SC disruption, the expected profit loss, as well as its variance, is relatively low. Hence, the SC network design from the bi-criteria model is more resilient than that from the singleobjective model. other. • By adding a constraint to the model (limiting the maximum number of selected suppliers to 10), the SC profit decreases. In terms of resiliency, the SC network design from the bi-criteria model is more resilient than the one from the profit maximization model in both cases. • Mitigation cost versus mitigation benefit: the mitigation benefit from the bi-criteria model (without the maximum number of supplier constraints) outweighs the mitigation cost. Conversely, the mitigation benefit from the bi-criteria model (with the maximum number of suppliers) is higher than the mitigation benefit from the profit maximization model. SC managers can use this comparison to evaluate the effectiveness of various mitigation strategies.

Fig. 5
The efficient frontier for 2 values between 0.44 and 0.50 Table 9 Selected suppliers and warehouses from the bi-criteria model solution

Conclusions and future research
To manufacture and distribute products efficiently and effectively, network design in SC is one of the most crucial decisions to make. In this study, we attempted to integrate systemic risk theory and contingency theory to investigate the impact of supply density on SC resilience and the design of a four-stage SC network. Furthermore, the consideration of SCD and profitability in network design is novel, and this will open up new avenues for similar studies in the field. We formulated a bi-criteria MILP model to optimally select suppliers, determine the location of facilities, and design a distribution plan between the selected set of facilities in order to maximize SC profit and supply density. We developed an interactive fuzzy solution approach based on the ε-constraint method to solve the proposed bi-criteria MILP model. The solution approach was able to interactively generate a Pareto-efficient frontier that represented a trade-off between SC profit and supply density objectives. A realistic illustrative example was solved to demonstrate the use of the bi-criteria MILP model and the interactive algorithm.
We also evaluated the resiliency of the SC network solutions and compared them based on the expected profit and variance. According to the findings, the SC network design that prioritized profit maximization tended to select facilities that were nearby such that the total cost was minimized. However, this resulted in a high expected severity of random disruptions. Conversely, the bi-criterion SC network design allowed redundancy in the SC by spreading facilities to different regions. Thus, the SC expected the disruptions to be less severe. For the Table 10 Product flows from the selected suppliers to the manufacturing plants  illustrative example, the results indicate that the bi-criterion SC network design yielded a 2% higher expected profit and a 2.2% lower expected profit variance than the profit maximization network design. The model can help companies evaluate the trade-off between mitigation benefit and mitigation cost. However, this study has limitations that allow scope for future work. First, the current model does not discuss the profit and risk-sharing mechanisms that govern the SC. We acknowledge that these have significant implications for sharing benefits and risks along the SC, and we believe that this can be considered as a follow-up study.
Second, the density of plants and warehouses in the network can be considered. In this case, the interstage and intrastage functions must also be extended to the other stages. We can also develop the mathematical model to incorporate other SC network characteristics, such as network complexity and node criticality, to gain a better understanding of SC resilience. The model can also be developed into a multiperiod model, and a risk profile can be formulated to evaluate SC resiliency on a tactical level. Finally, the model does not consider the stochastic nature of the variables that impact SC profitability and resilience. Parameters such as demand and cost factors in the SC directly affect the network design and SC performance indicators. Based on its strategic nature, the problem can be extended by incorporating demand and cost uncertainty. Several researchers (Belen et al., 2009;Savku & Weber, 2017;Yılmaz et al., 2015) have investigated stochastical optimal control problems in diverse topics; such studies can also be conducted in the context of SC design and optimization.   (continued) S11 S12 S13 S14 S15 S16 S17 S18 S19 S20 S1 11,450