2.1. Food supply chain network design

Due to the growing importance of food, there has been an increasing number of studies on the subject of FSCND in recent years. Mohammed and Wang [24] proposed a multi-objective model to design and plan a meat SC. The researchers used three different methods to solve the multi-objective model and, in the end, opted to use the Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS) to obtain the optimal solution from the Pareto frontier. Mohammed et al. [25] expanded the aforementioned approach by presenting a multi-objective model that used stochastic programming to deal with uncertainty. The authors employed the monitoring system known as Radio-Frequency Identification (RFID) to satisfy the objectives of their model. Mohammed and Wang [26] presented another mathematical model to improve the distribution system of their earlier meat SC model. Aras and Bilge [27] presented a mixed-integer linear programming model to design a FSC in Turkey. The authors utilized the minimax regret approach to deal with uncertain demand. Mohebalizadehgashti et al. [28] formulated a meat SC with the objective of minimizing the total cost and CO2 emission, as well as maximizing the capacity utilization rate of the facilities in the network. Patidar and Agrawal [29] introduced a mixed-integer non-linear programming formulation to optimize the total cost of an agricultural SC. The results indicated that the expenses incurred by farm-to-market transportation were responsible for 85% of the total SC cost. To mitigate this defect, they proposed that the farmers be classified into clusters such that their crops would be aggregated in cluster centers and transported to the market from there. Gholami-Zanjani et al. [30] developed a two-stage scenario-based model with uncertain demand and disruptions caused by an epidemic. The researchers incorporated a number of resilience strategies into the core model to design a resilient FSC. Arabsheybani and Arshadi Khasmeh [31] proposed a robust bi-objective model to design a resilient FSC with the objectives being to maximize the total profit and total resilience score. Relying on four criteria to enhance the resilience of the SC (robustness, agility, leanness, and flexibility). Gholami-Zanjani et al. [13] developed a new framework to configure a resilient green meat SC. The authors implemented two resilience strategies to hedge the network against disruptions and solved the model by applying the sample average approximation and lexicographic weighted Tchebycheff methods.

2.2. Sustainable SCND

The concept of sustainability has gained popularity in the last few years since it demonstrates that a SC’s competitive advantages cannot only be achieved through optimizing its economic aspects [32]. Sustainability is more emphasized in some industries, such as the food industry, due to the specific characteristics of their products. In this subsection, we survey some of the studies on sustainable SCND problems.

Allaoui et al. [2] came up with a hybrid two-stage methodology to optimize a sustainable SC in which sustainable suppliers are selected using the Analytic Hierarchy Process (AHP). Yakavenka et al. [33] presented a distribution network design problem for perishable food products considering the three dimensions of sustainability. The proposed model’s objectives were to minimize the total cost, total emission of transportation modes, and transportation times. Mohammadi et al. [34] developed a sustainable multi-objective model for a processed food company. The decisions made by proposed model included facility location, aggregating the flow of materials between facilities, and selecting the type of product delivery (either directly from plants or indirectly from distribution centers). Jouzdani and Govindan [35] introduced a non-linear mathematical model to design a sustainable perishable FSC. In this study, the perishability of the products followed the Weibull distribution and was affected by the use of vehicle refrigerator. Goodarzian et al. [36] formulated a bi-objective mixed-integer non-linear programming model to design a citrus fruit supply chain network. The contribution of this paper is the development of a mathematical model for designing a three-echelon network that takes into account coefficient water, CO2 emissions, and time window concurrently. Kalantari and Hosseininezhad [12] developed a sustainable model considering four risk factors: delays, fluctuation of exchange rates, quality of raw materials and production. The authors used the cross-entropy method to solve the model for large instances using the optimization software GAMS. Momenitabar et al. [37] introduced a sustainable closed-loop supply chain network that minimizes total and environmental costs, including energy consumption and pollution emissions, and maximizes employment opportunities. To reduce the environmental impacts, the circular economy concept has been addressed in this study. Momenitabar et al. [38] developed a mathematical model for designing a sustainable bioethanol supply chain network. In order to determine the best method to predict the actual amount of demand, they compared three popular machine-learning methods. In addition, the authors considered two meta-heuristic algorithms to find pareto optimum solutions for the large-scale problem. Goodarzian et al. [39] investigated a novel multi-objective mixed-integer linear programming model to establish a sustainable citrus closed-loop supply chain network. The authors included the circularity strategy and the triple bottom lines of sustainability into their model. Mohammadi et al. [40] presented a mixed-integer non-linear programming model to design a sustainable FSC for perishable products. To show the validity of proposed model, they conducted several sensitivity analyses on crucial parameters and applied the model in a food company. Lastly, Momenitabar et al. [41] integrated machine-learning algorithms with quantitative optimization modeling to design a sustainable bioethanol supply chain. In this work, three machine-learning approaches have been applied to forecast the bioethanol demand as model inputs.

2.3. Risk-averse SCND

A few studies have considered risk-aversion in SCND and their mathematical models, some of which are presented in this subsection. Noyan [42] used two-stage stochastic programming (TSSP) to design a disaster management SC under disruption and operational risks. In order to address the variability of non-deterministic parameters, the author incorporated the CVaR measure into the stochastic programming framework. Ahmadi-Javid and Seddighi [43] developed a location-routing optimization model under disruption risks to design a SC consisting of manufacturers and distributors in which a single type of commodity is distributed to several consumers. The problem was formulated as a mixed-integer linear programming model under three risk measurement policies: optimistic, cautious, and pessimistic. The authors utilized the expectation, CVaR, and worst-case risk measures to calculate the risk under each policy. Soleimani and Govindan [44] used a risk-averse TSSP method to plan and design a reverse SC in which the CVaR was used to evaluate the risk associated with the uncertainty in prices and demand for returned products. Govindan and Fattahi [45] proposed a multi-period optimization model under demand uncertainty with the goal of minimizing the total SC cost. The problem was formulated by incorporating TSSP and the CVaR into the model to address risk-aversion. Rahimi and Ghezavati [46] developed a mixed-integer linear programming model to design and plan a sustainable reverse logistics network under uncertainty. The authors applied a risk-averse TSSP approach, utilized the CVaR to cope with the deviation of demand and investment rate from their expected values, and solved a numerical example to obtain potential managerial insights. Rahimi et al. [47] investigated risk-averse SCND by proposing a mixed-integer non-linear programming model to configure a sustainable SC in which the CVaR risk measure was assimilated into the model. This paper contributes mainly by offering various discount policies for selling products to customers. Lotfi et al. [48] developed a model to design an antifragile, sustainable, and agile network that integrates these concepts taking into account several factors such as resilience, risk, robustness, and environmental issues. The authors incorporated the Entropic Value at Risk (EVaR) measure into the stochastic programming framework to address risk-aversion. The cost function of this model encompasses the expected value, maximum, and EVaR. Lotfi et al. [49] proposed a novel hybrid method of robust optimization which employs a stochastic approach to address uncertainty and risk management. The objective function of the model given involves maximizing the coefficients of mean and minimum profits under multiple scenarios, while considering an energy-aware constraint. Lotfi et al. [50] introduced a novel viable SC in which robust stochastic optimization is combined with CVaR as risk criteria to ensure an optimal approach. The objective function of their model minimizes the expected value, maximum, and CVaR of cost. Lotfi et al. [51] presented a new data-driven robust optimization in viable supply chain network design by utilizing Open Innovation (OI) and Blockchain Technology (BCT). The authors employed a robust stochastic optimization technique and assimilated the CVaR measure into the model to effectively tackle uncertainty as well as risks.

2.4. Research gap

As our review of the literature reveals, several studies have investigated sustainable FSCND, but no comprehensive study has been conducted on risk-averse SCND while developing a sustainable FSC at the same time. To address this gap, we present a sustainable risk-averse FSC by incorporating the CVaR into our model to develop a robust network in an uncertain environment. Moreover, a few papers have considered the specific characteristics of food products, such as perishability, in their proposed FSCs (e.g. [13, 30, 34]). Therefore, we also incorporate perishability and its implications into our model by considering the shelf-life of foodstuff to prevent the decay of finished products along the SC. We discovered that making inventory and transportation decisions based on integration of production and delivery times may bridge a significant gap in the literature on FSCND. Furthermore, in the majority of the research papers on sustainability, the only instance of social impact addressed by the authors appeared to be job creation, and other factors such as safety had been largely neglected. Our research methodology outlining the steps undertaken is summarized by the flowchart presented as Fig 1, acting as a guide for the research process. In light of the gap analysis presented in this subsection, and based on Table 1, the present study may contribute to the literature in the following ways:

• Utilizing the CVaR measure for risk management to develop a robust, sustainable, risk-averse FSC;
• Considering perishability in inventory decisions based on the production times, delivery times, and shelf-life of the products;
• Investigating product transportation decisions in terms of both economic and environmental implications in FSC
• Considering job creation and safety together as the social impacts in the FSC;
• Proposing PFGP as a novel extension of goal programming to solve the presented multi-objective model;
• Verifying the applicability and validity of the proposed model and solution approach through a case study from the processed food industry.

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Fig 1. Methodology of the present study.

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3.1. Assumptions

We formulated the model based on the following basic assumptions:

1. The potential locations to establish manufacturers and DCs are predetermined.
2. The capacity of suppliers, manufacturers, and DCs is limited.
3. Retailers can fulfill their demand only through DCs.
4. The demand of retailers is considered as a non-deterministic parameter.
5. Shortage is allowed in the model.
6. The transportation modes are heterogeneous with different capacities and carbon footprints.
7. The products are perishable and have a fixed shelf-life.
8. The decision-makers are only concerned with the costs in the second stage of stochastic programming.

3.2. Mathematical formulation

In this section we introduced the mathematical formulation of the described problem which is as follows (Table 2):

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Table 2. Notation list.

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Mathematical model:

1. Objective functions

(1)

• Economic objective function

We apply a TSSP approach to formulate the problem. In this type of framework, decisions are made over two sequential stages. In the first stage, decision-makers should make scenario-independent decisions that typically determine the number, location, and capacity of facilities. Scenario-independent decisions are made before the realization of uncertain parameters (e.g. demand, supply, transportation costs, etc.) and generate the first-stage (a.k.a. design stage) costs. In the second stage, scenario-dependent decisions (e.g. material flow, inventory level, production rate, etc.) are made regarding the second stage (a.k.a. planning stage) costs after the values of uncertain parameters have been realized [52].

Eq (1) gives the first objective function, i.e. economic sustainability, which aims to minimize total cost of the SC, consisting of first-stage, second-stage, and CVaR costs. The components of Eq (1) are formulated as follows: (2) (3) (4)

Eq (2) obtains the fixed costs of establishing manufacturers and DCs. Eq (3) calculates the second-stage costs, consisting of the costs of purchasing raw materials, production, distribution, inventory holding in manufacturers and DCs, transportation, and lost demand in retailers, with respect to the set of scenarios. The last component of Eq (1) relates to the costs of the CVaR, which is defined as Eq (4) [22].

• Environmental objective function

(5)

Eq (5) states the second objective which represents environmental sustainability and minimizes the total environmental impacts of the proposed SC, comprising the carbon emitted in the processes of establishing the facilities, production, handling, and transportation of the products. The terms of Eq (5) are presented as follows: (6) (7) (8) (9)

Eq (6) computes the amount of carbon emitted in the process of establishing the manufacturers and DCs. Eqs (7) and (8) compute the environmental impacts of production and handling processes in the manufacturers and DCs, respectively. Eq (9) determines the amount of carbon emitted from the transportation of products. The first term of Eq (9) states the environmental impacts of transporting raw materials from suppliers to manufacturers, while the second term represents the environmental impacts of transporting the finished products from manufacturers to DCs. The last term of Eq (9) calculates the environmental impacts of transporting the products from DCs to retailers.

• Social objective function

(10)

Eq (10) concerns social sustainability and aims to maximize the total social impacts of the proposed SC which consists of the number of job opportunities created and workers’ lost days multiplied by their weight. The components of Eq (10) are defined as follows: (11) (12)

The first component of Eq (10) states the number of jobs created by establishing manufacturers and DCs, presented as Eq (11). The second component, calculated by Eq (12), pertains to the number of lost days caused by work-related damages during the process of establishing manufacturers and DCs.

1. Constraints

(13)

(14)

(15)

(16)

(17)

(18)

(19)

(20)

(21)

(22)

(23)

(24)

(25)

(26)

(27)

(28)

(29)

(30)

(31)

(32)

(33)

(34)

(35)

(36)

(37)

(38)

(39)

(40)

(41)

(42)

(43)

Eqs (13) and (14) ensure that each manufacturer and DC can be established only at one capacity level. Eq (15) indicates the amount of raw material purchased from suppliers. Due to the fact that food products are inherently perishable, inventory management and transportation decisions between network nodes require applying more constraints compared to other products. Therefore, in this paper, by considering the shelf life, production and delivery times of products individually, we intend to prevent the perishability of products along the network. In fact, the level of products inventory and the amount of products flow between network nodes are associated with the optimal production and delivery times as well as shelf life of the products. To address this issue, Eqs (16) to (29) have been defined in the proposed model. Eqs (16) to (18) calculate the inventory level of products according to their shelf-life for each manufacturer at different periods. Eqs (16) and (17) show the level of on-hand inventory at time periods shorter than the shelf-life of the product. However it is worth mentioning that in the first period, no inventory is added from the previous period. Therefore, the inventory levels of the products in each manufacturer for t = 1 < sl_p and 1 < t < sl_p are calculated separately using Eqs (16) and (17), respectively. Eq (18) computes the inventory level at time periods longer than the shelf-life of the product. Eqs (19) and (20) compute the flow of each product from a manufacturer to a DC considering its production period. As mentioned in Eqs (16) to (18), the product flows between the relevant nodes are formulated differently for t < sl_p and t ≥ sl_p. Eqs (21) and (22) present the inventory balance of products in each manufacturer based on their production period. Similar to Eqs (16) to (18), the inventory levels of products in the DCs for different intervals of t are calculated by Eqs (23) to (25). Eqs (26) and (27) define the flow of each product from a DC to a retailer based on its production time. Eqs (28) and (29) determine the inventory levels of DCs based on their delivery periods. Eq (28) indicates when the delivery period is equal to the time period, the inventory level of DCs is equal to the quantity of products delivered from manufacturers to DCs subtracted by the flow of products to retailers. Eq (29) states that when t − e < sl_p, the current inventory level is equal to the inventory level of the previous period subtracted by the flow of products to the retailers. Eq (30) determines the quantity of products shipped to retailers as well as the quantity of shortage. Eq (31) defines the raw material procurement capacity of suppliers. Eq (32) presents the production capacity of each manufacturer at each time period. Eqs (34) and (35) express the maximum throughput capacity of each manufacturer and DC, respectively. Eqs (35) to (37) show the capacity constraint of the transportation modes. Eqs (38) to (40) state the decision variables whose values should be zero. Eq (41) calculates the deviational value that exceeds the value-at-risk. Finally, Eqs (42) and (43) express the types of the decision variables.

3.3. Solution approach

In the process of solving real-world problems, there is often more than one objective function, each offering a specific solution if optimized separately. Such circumstances require that a trade-off be established among the objective functions and multi-objective optimization approaches were originally conceived precisely to solve such problems. One of the most commonly-used optimization approaches, introduced by Charnes et al. [53], is known as goal programming. In this approach, decision-makers define an expectation level for each objective function as its goal and try to minimize the deviation from these expectation levels [54]. Ostensibly, decision-makers should accurately determine the expectation level of each objective; however, in reality, the uncertainty of the available information makes it nearly impossible to assign perfectly accurate levels to the objectives. Fuzzy sets theory, introduced by Zadeh [55], is regarded as an effective approach to decision-making under uncertainty. Arikan and Güngör [56] then combined fuzzy set theory and goal programming to create the fuzzy goal programming (FGP) approach. Recently, a novel extension of FGP, known as preemptive fuzzy goal programming (PFGP), was proposed by. The model was developed with different priorities and had the advantage of largely eliminating the limitations of the original approach. One such limitation was that decision-makers were required to set a precise lower bound for the satisfaction degree of the objectives. Moreover, it was often the case that in order to achieve the minimum satisfaction degree for one of the objectives, a better total satisfaction degree for all objectives had to be eliminated. The new model identified and mitigated this issue, accepting solutions that have a higher satisfaction degree while taking into account the priority of the objectives in question. In this approach, the goals are prioritized and the satisfaction degree of the first priority should be greater than the others. By adding certain equations to constraints that illustrate the importance of each goal, this priority structure is taken into account. PFGP is applicable to multi-objective problems that involve uncertainty and objectives with different importance degrees [57].

In the present study, we adopt the PFGP method as it eliminates the requirement that a fixed value be determined for both the objective function and the satisfaction degree. It also takes uncertainty into account when determining the goal of each objective. Using this method, decision-makers can easily adjust the priority of the objective functions and observe the results of the optimal solution [58]. The notations of PFGP are as follows (Table 3):

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Table 3. Notations of PFGP model.

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Using the notations presented above, the mathematical model based on PFGP can be expressed as follows: (44) s.t.

(45)

(46)

(47)

(48)

(49)

(50)

(51)

Eq (44) expresses the goal of the PFGP approach which is maximizing the total satisfaction degree. In Eq (45), h_c(X) represents the constraints of the main model. Eqs (46) and (47) determine the satisfaction degrees of the goals for minimization objectives, where ∀i = (1, 2,…, ii) is the set of minimization objectives. Eqs (48) and (49) calculate the satisfaction degrees of the goals for maximization objectives, where ∀i = (ii + 1,…, i) is the set of maximization objectives. Eq (50) relates to the priority of the goals, stating that the most important objective should achieve a satisfaction degree greater than or equal to the other objectives. Finally, Eq (51) defines the type of the decision variables of the PFGP model.

4.1. Case description

Due to the constantly evolving lifestyles of consumers, the processed food industry plays a significant role in providing the population’s needs by producing and distributing a variety of products, also appearing to be on a growing trajectory. In this study, we investigate a processed food company as a real-life case study to measure the effectiveness of our model. The manufacturing plant is located in Tehran province, Iran, and the products are sold on domestic and international markets. The manufacturer procures its required raw materials, mainly chicken and beef, from multiple suppliers. In the proposed model, we consider only two types of finished products: hamburger 80% beef, and sausage 70% chicken. Because of their perishability, the products are transported by refrigerated trucks. The network size of the case study detailed in Table 4.

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Table 4. Size of the SC.

https://doi.org/10.1371/journal.pone.0308332.t004

In this study, we focus on three three modes of transportation: light truck, medium truck, and heavy truck. The data used in the present research on the capacities of transportation modes and the quantities of CO₂ emissions have been retrieved from Mohammed et al. [25] and Nayeri et al. [59] (see Table 5). Three levels of capacity: low, medium, and high, with different costs, carbon emissions, and job opportunities have been considered for establishing facilities in potential locations. Tables 6 and 7 show the values of the parameters related to each manufacturer and DC with different capacity levels. Also, we examine three prominent demand scenarios for each retailer in our case study: the most pessimistic, the most probable, and the most optimistic. Numerous studies in the literature on SCND problems have repeatedly implemented the same three demand levels to generate their scenarios and have subsequently attested to the reliability and efficiency of the method (e.g. [22, 60, 61]). According to the historical data obtained from the case study as well as the experts interviewed for the purposes of this paper, the occurrence probability of scenarios are estimated at 0.25, 0.5, and 0.25, respectively. Tables 8 to 11 detail the costs of the model, including the fixed cost of establishing facilities, operational cost, raw material purchasing cost, and holding cost of finished products. Since the finished products discussed in this study have different raw material consumption rates, weights, shelf-lives, and production processes, the production capacity and costs associated with each type of product vary, as well (see Table 12). In addition, Table 13 provides the data on the corresponding values of each retailer’s demand under each scenario. The environmental parameters are presented in Tables 14 to 16. We have gathered the data on environmental parameters from several sources, including the food carbon footprint calculator, an online platform provided by MyEmissions.green, and also from academic articles (e.g. [62]). The values of the social parameters can be found in Tables 17 and 18.

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Table 5. Data of the transportation modes operating in the SC.

https://doi.org/10.1371/journal.pone.0308332.t005

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Table 6. Maximum capacity of manufacturers with different capacity levels (kg) for finished products.

https://doi.org/10.1371/journal.pone.0308332.t006

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Table 7. Maximum capacity of DCs with different capacity levels (kg) for finished products.

https://doi.org/10.1371/journal.pone.0308332.t007

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Table 8. Fixed cost of establishing manufacturers and DCs with different capacity levels (×10⁹ Rials).

https://doi.org/10.1371/journal.pone.0308332.t008

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Table 9. Operational cost of manufacturers and DCs (×10⁴ Rials).

https://doi.org/10.1371/journal.pone.0308332.t009

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Table 10. Data of raw material suppliers.

https://doi.org/10.1371/journal.pone.0308332.t010

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Table 11. Holding cost of finished products in manufacturers and DCs (×10⁴ Rials).

https://doi.org/10.1371/journal.pone.0308332.t011

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Table 12. Production data of finished products by manufacturers.

https://doi.org/10.1371/journal.pone.0308332.t012

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Table 13. Demand for finished products (kg).

https://doi.org/10.1371/journal.pone.0308332.t013

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Table 14. Carbon emissions from establishing manufacturers and DCs with different capacity levels (kg).

https://doi.org/10.1371/journal.pone.0308332.t014

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Table 15. Carbon emissions from production of finished products in manufacturers (kg).

https://doi.org/10.1371/journal.pone.0308332.t015

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Table 16. Carbon emissions from handling finished products in DCs (kg).

https://doi.org/10.1371/journal.pone.0308332.t016

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Table 17. Number of job opportunities created in manufacturers and DCs with different capacity levels.

https://doi.org/10.1371/journal.pone.0308332.t017

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Table 18. Number of lost days due to work-related damages in manufacturers and DCs with different capacity levels.

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4.2. Computational results

The proposed mathematical model is solved by the optimization software GAMS using the case study parameters. Furthermore, λ₁ > λ₂ > λ₃ is the fuzzy satisfaction degree reflecting the importance degrees of the objective functions according to the experts’ opinions. Table 19 details the output generated from solving the model. In addition, the numerical results of some analyses performed on the model are presented in the following subsections.

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Table 19. Model output details.

https://doi.org/10.1371/journal.pone.0308332.t019

4.2.1. Comparing model.

The majority of stochastic programming models addressing food supply chain (FSC) under uncertainty are usually designed to be risk-neutral. In this paper, a risk-averse two-stage stochastic programming is aiming to trade off the expected total cost and risk cost, based on a predetermined level of responsive risk aversion employing CvaR as a coherent risk measure. The purpose of this subsection is to compare the main risk-averse problem (considering CvaR) and risk-neutral problem (without considering CvaR). The comparison of the economic objective function of the proposed model between risk-averse and risk-neutral problems is illustrated in Fig 3. As can be seen, the value of economic objective function in risk-averse problem is 8.65% more than risk-neutral one. Since it overcomes the variability of the uncertain parameters, provides more robust solutions compared to a risk-neutral problem in presence of uncertainty, and assist decision makers when faced decisions with uncertain outcomes. It is worth noting that decision-makers can achieve higher levels of risk aversion by increasing the value of network costs, which will cause the model to behave more conservatively and generate better solutions in worst-case scenarios.

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Fig 3. Comparison of risk-averse and risk-neutral problems.

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4.2.2. Analysis of lost demands.

In this subsection, we intend to examine the service level of the SC in terms of lost demands. In the long run, unmet demands can result in customer dissatisfaction which, in turn, could negatively affect the firm’s reputation. Consequently, we can conclude that the more the level of service provided by the SC, the less lost demand will occur. In simple terms, lost demand is a consequence of increasing the total cost with the purpose of establishing balance between economic, social, and environmental sustainability. Table 20 details our analysis of the total value and percentage of lost demands for finished products under each scenario. The finished products in each scenario have varying amounts of unfulfilled demands in which the abovementioned reason is mainly responsible for them. Another noteworthy point inferred from the table is that when the demand for the finished products rises by 50% in the in probable scenario, the quantities of lost demand for the two products increase by 32.1% and 35.7%, respectively. It may thus be concluded that insufficient production capacity is the most likely culprit for the resulting unmet demand.

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Table 20. Details of lost demands.

https://doi.org/10.1371/journal.pone.0308332.t020

4.2.3. Analysis of sustainability dimensions.

This section is dedicated to sensitivity analyses on the sustainability dimensions. For this purpose, we solve the proposed model in three different types: economic, green, and sustainable, and then proceed to discuss the results. In the economic type, only the economic objective function is addressed in the model; in the green type, both the economic and environmental objective functions are addressed simultaneously; and in the sustainable type, the problem is solved by considering the economic, environmental, and social objective functions, i.e. all three dimensions of sustainability. According to Table 21 and Fig 4, solving the problem in the economic type leads to the economic and environmental objective functions to take their best and worst values, respectively. However, focusing solely the economic dimension obtains the best value for the economic objective function. In the green type, by adding an environmental dimension to the problem, the opposite of the results of the economic type is obtained. That is, the economic and environmental objective functions take their worst and best values within the specified cases, respectively. Based on this observation, we may conclude that the green type increases the value of the economic objective in order to ensure a better value for the environmental objective. In the sustainable type, in order to establish a balance between the three dimensions of sustainability, economic and environmental objectives both gain values that lie between the values they were assigned in the economic and environmental types. Moreover, Table 19 indicates that the sustainable type results in a 9% increase in the total cost, as well as a 28.3% decrease in the environmental impacts, and a 53.6% increase in the social impacts compared to the economic type.

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Fig 4. Economic and environmental objective functions sensitivity analysis on sustainability dimensions.

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Table 21. Results of sensitivity analyses on sustainability dimensions.

https://doi.org/10.1371/journal.pone.0308332.t021

4.2.4. Analysis of the PFGP approach.

In the PFGP approach, the influential factor is the priority of objective functions, which is defined by their fuzzy satisfaction degrees (λ_i). Therefore, we solve the model in six cases to assess the impact of changes in the objective functions’ priority on the problem. Table 20 shows the results of solving the model in the presented cases. Based on Table 20 and Fig 5, the conflicts between the economic and environmental goals are evident. In case 1, when the economic objective function takes the highest priority, it is assigned the best value, while the environmental objective function is assigned the worst. In Case 2, in which the priority of the second objective function is higher than the other two, the environmental impact is given the best value, with the economic objective function getting the worst. In case 3, where both the environmental and economic objectives are equally important, the value of both objective functions is squarely between their worst and best values. Table 22 and Fig 6 illustrate the conflict between the economic and social objective functions. In case 4, when the economic objective has the highest priority, it is assigned the best possible value. Conversely, the social objective function receives the worst value among all the cases. Based on case 5, when the social objective function becomes the top priority, it gets the best value, while the economic objective function has the worst value. Finally, in Case 6, in which the social and economic objectives share the top priority, the value of both objective functions is between their lowest and highest.

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Fig 5. Conflict between cost and environmental impacts.

https://doi.org/10.1371/journal.pone.0308332.g005

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Fig 6. Conflict between cost and social impacts.

https://doi.org/10.1371/journal.pone.0308332.g006

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Table 22. Results of sensitivity analyses on the satisfaction degrees of the objective functions.

https://doi.org/10.1371/journal.pone.0308332.t022

4.2.5. Analysis of the risk-averse approach.

In this section, we assess the risk-aversion parameter in order to clarify how it may affect the objective functions of the proposed model and the obtained solutions. Consider an experiment with four different β values where the value of θ = 0.5 remains fixed to better comprehend how β impacts the model. Table 23 and Fig 7 show that as β grows larger, the value of the economic objective function increases, as well.

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Fig 7. CVaR and total cost vs. β.

https://doi.org/10.1371/journal.pone.0308332.g007

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Table 23. Sensitivity of objective functions to the confidence level.

https://doi.org/10.1371/journal.pone.0308332.t023

Since CVaR measures the mean value of the worst (1 − β) % of the total expected cost, when the value of β increases, the risk aversion behavior of the proposed model improves. Hence, the model behaves more conservatively and the CVaR concentrates on the greater realization of total cost. Total costs are comprised of the first-stage costs, CVaR costs, and second-stage costs. It is also important to note that, the first- stage costs and consequently the number of established facilities do not change in response to varying risk parameters since they are not a function of CVaR. In other words, CVaR costs affect the second-stage costs. In general, we may draw the conclusion that a high risk-aversion strategy would result in higher levels of the second-stage costs and the total costs. To better comprehend this mechanism, we need to examine how β influences the environmental and social objective functions. As previously mentioned, by raising the value of β and giving the model a risk-averse attitude, the value of the economic objective function increases and the model should become more conservative, resulting in an upward trend in the value of the environmental impact and a downward trend in social impacts. This conflict can be clearly observed in Table 23 and Fig 8.

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Fig 8. Environmental and social impacts vs. β.

https://doi.org/10.1371/journal.pone.0308332.g008

4.2.6. Analysis of demand values.

To examine the impact of changes in demand on the value of the objective functions, we solve the model in five demand modes: -30%, -15%, 0%, +15%, and +30%). The results of the sensitivity analyses are shown in Fig 9. As expected, rising demand values increase the total SC costs. This may be a consequence of establishing new high-capacity facilities, increased raw material procurement and production of finished products, as well as transportation between the network nodes to satisfy the demand. Furthermore, as more high-capacity facilities are activated and transportation between facilities is raised, which leads to the use of higher capacity transportation modes, the adverse environmental impacts of the network increase. In addition, as the demand grows, the model tends to create more jobs by establishing more facilities and activating higher capacity levels to meet the retailers’ demand. In case of a significant increase in demand, the proposed model tries to keep the total SC costs at a minimum by not establishing new facilities. This is because the costs of opening new facilities to fully meet the demand outweigh the costs incurred by lost demands. Therefore, the model prefers to avoid establishing new facilities, activates lower capacity levels, and only partially satisfies the demand. This results in a less steep upward trend in total cost and social impact. As Fig 9 indicates, the reaction of the economic, environmental, and social objective functions in the proposed model to changes in demand under different modes complies with the above-mentioned explanations, indicating the proper logic of the model.

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Fig 9. Behavior of the objective functions toward changes in demand.

https://doi.org/10.1371/journal.pone.0308332.g009

4.2.7. Analysis of comparison between current and proposed SC conditions.

Taking into account the current situation of the case study, which consists of one medium-capacity manufacturer and two high-capacity DCs, the values of the economic, environmental, and social objective functions are 1999168 × 10⁷, 79453400, and 529.2, respectively. In accordance with the opinions of the case study experts, the potential locations for construction of new facilities were selected based on the company’s revision plans regarding its supply chain to gain a larger market share. The results of solving the proposed model for the economic, environmental, and social objective functions are 1952306 × 10⁷, 75179800, and 491.4, respectively. Fig 10 compares the proposed and current conditions of the case study and displays the values of the model’s objective functions. As can be observed, compared to the current situation, the model proposed in this research causes a 2.3% drop in costs, a 5.4% drop in adverse environmental impacts, and a 7.1% drop in social impacts. This is achieved by constructing a medium-capacity manufacturers along with one medium-capacity and two low-capacity DCs. Thus, it can be said that the proposed model, whose candidate locations for the company’s future plans were selected by the experts, improves the current situation in terms of costs and environmental impacts. In addition, considering the high priority of the total SC costs, the downward trend of the social objective function in favor of the economic objective function seems logical.

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Fig 10. Comparison of current and proposed conditions of the case study.

https://doi.org/10.1371/journal.pone.0308332.g010

4.3. Discussion

In this study, we developed a sustainable risk-averse model to design a SC for processed food products under uncertain conditions. To this end, a multi-period multi-objective model was presented to make decisions on facility location, optimal sourcing, transportation mode selection, capacity planning, and inventory management. The resulting model addresses three dimensions of sustainability as economic, environmental, and social objective functions, thereby minimizing the total cost and carbon emissions and maximizing the number of job opportunities created and safety of the FSC. The main contribution of this research to the literature may lie in our decision to incorporate a risk measure into the design of a sustainable FSC; an innovation that has never been attempted by previous researchers. Furthermore, we applied a scenario-based TSSP approach to cope with the uncertainty of demand for finished products and utilized a novel PFGP multi-objective method to solve the mathematical model. Also, we compared our proposed model with a situation in which the CvaR risk measure was not taken into account. Finally, results show that the value of cost function in risk-averse problem is 8.65% more than risk-neutral one. Because it behaves more conservatively and offers more resilient solutions in the presence of uncertainty compared to a risk-neutral problem.