Biodiesel Supply Chain Network Design Under Hybrid Uncertainties: A Novel Multi-objective Robust Fuzzy Stochastic Approach

In the biofuel supply chain, there may be various and hybrid uncertainties that, if ignored, can lead to inefficient network design . In this study, a multi-objective robust fuzzy stochastic programming (MORFSP) model is proposed for designing biodiesel supply chain network (BSCN) under different and hybrid uncertainties. This model simultaneously minimizes total cost of the BSCN and total environmental impacts of activities of the network. Fixed costs and environmental impact of opening facilities are described as fuzzy variables. Demands, supplies, other costs and environmental impacts are considered as fuzzy scenario based variables. The proposed MORFSP model considers different risks, including possibilistic variability and scenario variability related to economic and environmental objective functions, and unsatisfied demand costs. This model is applied in a real case study to design a BSCN in Iran. Waste cooking oil (WCO), and some non-edible plants like Salvia lerifolia (SL) and Jatropha Curcas L. (JCL) are considered as sources of producing biodiesel. The proposed approach used for designing a four-echelon, multi-period, and multi-product, of BSCN. The results show the effectiveness of the proposed model for designing the BSCN under hybrid uncertainties.

Sustainable development (SD) comprises a thorough and integrated way to the economic, social and environmentally functions (Karakosta and Askounis 2010). An SD approach seeks to provide services which meet basic human needs, in a cleaner and more efficient way that can be sustained in the future (Winkler 2007;Suganthi 2018). Nowadays, energy is undoubtedly crucial for sustainable development and the prosperity of a community (Chaharsooghi and Rezaei 2016;Tan et al. 2018). Increase of energy consumption causes some problems like depleting energy sources and creating pollution (Shakouri and Kazemi 2017). Increment in the energy efficiency of processes applying sustainable resources can very help in achieving sustainable development (Hepbasli 2010;Ulonska et al. 2018). There is a need to developing new energy resources to replace or reduce the use of nonrenewable energy resources (like natural gas, petroleum, coal, etc.) (Li et al. 2009;Ruiz et al. 2016;Rezaei et al. 2020a).
Renewable energies (REs) are considered as one of the potential solutions for climate change, energy security and sustainable growth (Huang et al. 2011;Swain and Karimu 2020). REs are produced applying harmless techniques that have less harmful impact on the environment in contrast to other kinds of energy (Chaharsooghi et al. 2015). So, renewable energy sources seem to be an effective solution for achieving sustainable development (Rezaei et al. 2013). Nowadays, developing RE is becoming a hot topic since it is crucial in dealing with energy supply issue and climate change issue (Cai et al. 2012;Salamanca et al. 2012).
Biomass is becoming among the most typically applied RE sources in the recent years. Biofuels are fuels produced from biomass sources that are commonly used for transportation (Mousavi Ahranjani et al. 2018). Bioethanol and biodiesel that include the most utilized liquid biofuels, are fine substitution for petrol and diesel, respectively (Babazadeh et al. 2017b). Biofuels can be used in existing engines with major modifications (Devarajan et al. 2017;Xu and Xu 2018).
While the biomass is fairly cheap, the logistic costs play main role in the price of the biomass delivered to the biorefinery (Steiner et al. 2012;Atashbar et al. 2016). Since numerous producers of biomass are involved in this supply chain, quantitative models can be beneficial to assess and optimize the related costs, the required resources, and the consumptions of energy (Ba et al. 2016).
Because of restriction of fossil fuel sources and their negative influences on environment, in recent years, there have been a lot of researches on designing biofuel supply chain. In designing the BSCN, the strategic decisions for instance the size, number, capacity and location of facilities, and the tactical decisions, like production quantity, mode of transportation, inventory, and transported product among various facilities are determined Santibañez-Aguilar et al. 2016a).
In this article, multi-objective robust fuzzy stochastic programming model is presented for designing a BSCN under hybrid uncertainties. The main novelties of this study are summarized as follows:  In this study, the different and hybrid uncertainties that exist in the real world in the biofuel supply chains are considered. So that some uncertain variables have fuzzy nature and some have fuzzy stochastic nature.
 According to the literature, fuzzy stochastic approach has not been used to model the biofuel supply chain. Due to the specific characteristics of the supply chain of these fuels and excising different and hybrid uncertainties related to this field, in this study, a robust fuzzy stochastic programming model is proposed for designing the supply chain.
 A multi-objective robust fuzzy stochastic programming model is presented for designing a BSCN under different and hybrid uncertainties that simultaneously minimizes the total cost of the BSCN and the total environmental impacts of the activities of the supply chain.  This model considers the different risks in the network, such as possibilistic variability and scenario variability related to economic and environmental objective functions, and unsatisfied demand costs.
 Possibility of capacity expansion of facilities in different periods is considered.  The presented model is applied in a real case study in Iran.
The Structure of this article is as follows: In Section 2, the researches related to this work are described. In Section 3, the mathematical formulation which groups the main constraints and the objective functions is discussed. In Section 4, the proposed robust fuzzy stochastic programming approach is presented. A case study to examine the proposed approach is discussed in Section 5. Lastly, the final conclusions are stated in Section 6.

2-Literature review
There are different methodologies for designing biofuels supply chain in the literature, like mathematical programming models, simulation, and GIS (Bai et al. 2011). The mathematical programming models were the most applied method (Ghaderi et al. 2016).
In this section, researches most related to our work are reviewed.
In the most studies, deterministic mathematical programming models were applied to design the BSCN. Cambero et al (2015) developed a mixed integer linear programming (MILP) model to design a supply chain of forest residues. This model maximized the net present value of the supply chain over the horizon time. Ahn et al. (2015) presented a deterministic mathematical programming method for designing a BSCN which simultaneously considered resource, and demand constraints. This model helped for determining the location and the amount of feedstock, and the location and the size of refineries. Babazadeh et al. (2017c) proposed a method according to DEA approach and mathematical programming model for designing of BSCN. Firstly, locations for cultivating JCL were evaluated through unified DEA method based on some social and climatic criteria. Afterwards, the areas that had attained the desired scores were selected as candidate areas for cultivating JCL and by a MILP model the BSCN model is designed.
One of the challenges in designing the BSCN is existence of different uncertainties, including, biomass demand, supply, price, etc. (Bairamzadeh et al. 2018a). There are various approaches in the literature to dealing with uncertainty: Some studies have used stochastic planning approach. For example, Giarola et al. (2011) proposed a two-stage stochastic mixed integer programming model to design a biofuel supply chain in the southeastern United States, in which biofuel supply, demand, price and technology are varied in different scenarios. Marufuzzaman et al. (2014) developed a stochastic programming method to design a BSCN under technology development and biomass supply uncertainties. They developed a MOMILP model that optimized costs and emissions of the network.
Fuzzy programming is another widely used approach in dealing with uncertainties. Tong et al. (2014a) presented a linear integer programming model to design a biofuel supply chain. To deal with the uncertainty of the parameters, they have used the fuzzy programming approach. To address the uncertainties, Babazadeh et al. (2017a) proposed a multi-objective possibilistic planning model for the second generation BSCN under uncertainty.
In recent years, the robust optimization approach has paid more attention due to its high capability in the face of risk and uncertainty (Bertsimas et al. 2011;Pishvaee and Khalaf 2016). Kim et al. (2011) developed a scenario-based robust MILP model for designing a biofuel supply chain network under demand uncertainty. Zhang and Jiang (2017) designed a BSCN model, which includes both strategic and tactical levels. They presented a multiobjective robust model under biodiesel price uncertainty. They took Suzhou, a city in China, as a case study for verifying the presented method.
Some of the studies that have used mathematical programming approach under uncertainty are summarized in Table 1.

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According to the literature, in the research related to biofuel supply chain modeling, different and hybrid uncertainties in the real world have not been considered. Therefore, in this study, a novel robust fuzzy stochastic programming model has been presented for this field. In this approach, some variables have fuzzy nature and some have fuzzy scenario nature. This model takes into account the different risks, including possibilistic variability, scenario variability, and unsatisfied demand costs. In addition, a multi-objective robust fuzzy stochastic programming model is presented for designing the BSCN under hybrid uncertainties that simultaneously minimizes the total cost of the network and the total environmental impacts of the activities.

3-Problem description and formulation
The structure of the presented BSCN is shown in Figure 1. It is a four-echelon, multiperiod, and multi-product supply chain network under hybrid uncertainties. As this figure shows, JCL and SL products are shiped from their corresponding cultivation centers to oil extracting facilities. JCL oil and SL oil are obtained in oil extraction facilities and transported to biorefineries. WCOs are collected from suppliers and transported to refineries for pretreating. In biorefineries, biodiesel is produced through JCL oils, SL oils, and waste cooking oils. Afterwards, the produced biodiesel is shipped from refineries to distribution facilities. Lastly, the biodiesel is shipped from distribution facilities to consumption centers.

3-1-Assumptions and notations
The general assumptions used for mathematical modeling are summarized as follows: • The locations of biodiesel consumption centers are known and fixed.
• Potential areas for cultivating of JCL and SL and opening of other facilities are known.
• Transportation is done by 2 modes of road and rail.
• The capacity of JCL and SL farms, established oil extraction facilities, and distribution facilities are not fixed. These capacities are calculated over planning horizon by continuous variables.
• Required capacity for each refinery is calculated through adding capacity alternatives to the initial corresponding capacity. • The opened facilities will be active until the end of the planning horizon.
• Fixed costs and environmental impact of opening facilities are uncertain and are described as fuzzy variables.
• Demands, supply of waste oils, the other costs, and some environmental impacts are uncertain and are considered as fuzzy scenario based variables.
The notations used to formulate the model are described in the following. It should be noted that uncertain parameters are introduced with a tilde on.

Sets
Set of candidate locations for planting JCL f

3-2-Constraints
Constraint (1) guarantees that the quantity of biodiesel shipped from the biorefineries to a city is at least equal to its corresponding demand. Equation (2) indicates the maximum amount of possible shortage. Constraints (3) to (5) ensure that all JCL, SL and WCO will be shipped to their corresponding facilities. (1) Constraints (6) and (7) represent the quantity of JCL and SL yields in the cultivated areas in each period. Equations (8) to (11) indicate the quantity of extracted JCL oil, extracted SL oil, pre-treated WCO, and produced biodiesel, respectively.
Constraint (12) represents that the level of inventory of JCL products at oil extracting facilities in each period of time is equal to the level of inventory in the previous period, in addition to the quantities of JCL shipped to these centers, minus the quantities of JCL converted to JCL oil that transported to biorefineries. Also, Equation (13) shows the level of inventory in oil extraction centers for SL products. Finally, constraints (14) and (15) indicate the level of inventory balance in refineries and distribution centers, respectively. (17) guaranties that when a biorefinery is established, it will be active up to the end of time horizon. Similarly, constraint (18) guaranties that when a capacity option is added to a refinery, it will be active up to the end of time period.

Constraint (16) represents a logical expression. Constraint
Equations (19) and (20) state the maximum and minimum permissible locations for cultivating of JCL and SL. Constraints (21) to (23) represent the maximum and minimum capacities of oil extracting facilities, biorefineries and distribution centers, respectively. Equations (24) and (25) state the minimum number of JCL and SL cultivation centers, respectively. Constraints (26) and (27) show the amount of capacity added to each of the oil extraction facilities and distribution facilities, respectively. Equation (28) calculates total capacity of refineries over the planning horizon.
Constraints (29) to (31) guaranty that the amount of products shipped to each facility does not exceed the capacity of that facility. Constraints (32) to (34) state that the inventory in each facility should not be more than its corresponding capacity.

3-3-Objective functions
The first presented objective function minimizes the total costs, like the fixed costs of establishing the facilities (FC), the variable costs of establishing the facilities (VC), the production costs (PC), the inventory holding costs (IC) and the transportation costs among facilities (TC).
So, according to the above terms, the cost objective function can be formulated as by equation (42): The second presented objective function minimizes the total environmental impact of opening the facilities (EF), production (EP), inventory holding (EI) and transportation (ET).
Similarly, according to the above terms, the Environmental objective function can be formulated by Equation (47):

3-4-Multi-objective approach
Many methods have been presented in the literature for solving multi-objective optimization problems. One of these methods is bounded objective function (BOF) method. In this approach, a problem with several objective functions converts to a single-objective problem. In the BOF method, the minimum and maximum values of objective function F i are determined by the decision maker and are represented by L i and U i , respectively. (48)-(50), in this method, the most important objective function, i.e. F r (X), is minimized and other objective functions are placed in constraints and a high and low limit is considered for them. By changing the values of L i and U i , the Pareto frontiers of the problem can be obtained.

4-The proposed robust fuzzy stochastic programming approach
In this section the presented robust fuzzy stochastic programming (RFSP) model is defined. The presented model is based on the robust scenario-based programming model and fuzzy possibilistic programming model. The proposed RFSP model can consider hybrid uncertainties.
Consider the following mathematical model. Where has fuzzy nature, ̃, , and̃ have fuzzy scenario nature. The general model of fuzzy stochastic programming is as follows: As Equation (56) shows, the objective function is composed of several components, which are as follows:  Finally, the fourth term of Equation (56) represents the unsatisfied demand costs.
In Equation (56), and represents the weight of possibilistic variability and scenario variability, respectively. is a coefficient that adjust the weighted penalty for possible violation of each constraint.

5-Computational experiments and evaluation
The presented MORFSP model is applied in a real case to design the BSCN in Iran. The results of solving the presented model are described in this section. The model is coded in GAMS software and solved using CPLEX solving method.
In this study, a 7 years planning horizon for implementing the proposed model is considered. SimaPro 8 software is applied to obtain the environmental impacts of the processes. Considering the fact that at least 80% of Iran's edible oils are provided by imports, it's not logical using edible oil sources for producing biodiesel. Therefore, in this study, JCL and SL that are non-edible plants and compatible with Iran's climate and require little water are selected as biodiesel resources. WCO is also considered as another source of producing biodiesel. The most appropriate locations for cultivation of these plants are considered as the candidate areas, in which 11 and 7 areas are selected for JCL and SL, respectively. In addition, considering the demand of diesel and air pollution problem of the cities (big and industrial cities), 7 cities are selected as biodiesel consumers.
Therefore, in this model 11 candidate areas for cultivation of JCL, 7 potential areas for cultivating SL, 30 potential WCO suppliers, 30 candidate areas for oil extraction facilities, 30 potential areas for biorefineries, 30 candidate areas for distribution facilities, 7 areas for biodiesel consumers, and 2 transportation modes are considered.
In this research, the scenarios and their probability values are obtained based on Pishvaee et al. (2008) and Rezaei et al. (2020b), in which 3 scenarios are designed. The values of the fuzzy and fuzzy stochastic parameters are obtained based on the experts' opinions and according to the structure of the scenarios.
Bounded objective function method is used to solve the presented multi-objective model. In this research, the cost objective function is minimized as the most important objective function, and the environmental objective function is placed in the constraints for different limits. Figure 2 shows the Pareto frontiers for the objectives.  Figure 2, the cost objective function and the environmental objective function are not compatible. Therefore, according to the preferences of the decision maker a point from the Pareto optimal set is selected. The values of the objective functions for the two points of the Pareto optimal set (points A and B) are compared in Table 2 Also the different components of the objective functions for these points are summarized in Table 3. In the following, the optimal location of each facility and their capacities, which have been obtained by solving the model, are described. It should be noted that these items are related to point A.
The optimal locations for Jatropha cultivation and the capacity of each of these areas are shown in Figure 3.    Finally, the Table 6 shows the established distribution facilities and their corresponding capacities. As shown in this Table, the number of distribution centers is higher than other facilities. The reason for this can be explained by the fact that the model prefers to establish more distribution centers to reduce the transportation costs.  Similarly, Figure 7 and 8 show the importance of the coefficient on the amount of scenario variability of the cost objective function and the total value of cost objective function. As this coefficient increases, the amount of scenario variability decreases and the value of the cost objective function increases.  According to Figure 5 and 7, when the values of these coefficients are zero, there is the greatest risk of decision making. Because at these points, possibilistic variability and scenario variability have their maximum values. In fact, by determining these coefficients, the robustness of the optimal solutions can be controlled according to the preferences of decision makers. Figure 9 shows the importance of the coefficient on the amount of unsatisfied demand cost. According to this figure, by increasing this coefficient, the amount of unsatisfied demand cost decreases. Indeed, by increasing this coefficient, the feasibility

Cost objective function
Weight of robustness will be high. Besides, according to Figure 10, by increasing , the total cost will increase.  Figures 13 and 14. As this coefficient increases, the amount of scenario variability first decreases and then remains constant. However, the total value of the environmental objective function increases with increasing this coefficient.  The results of sensitivity analysis show that the sensitivity of the environmental objective function to risk coefficients is less than the cost objective function. This can be explained by the fact that the fluctuations and uncertainties of the parameters of the environmental objective function are less than the cost objective function ones.

6-Conclusion
Despite the fact that biodiesel has good properties that could be an appropriate alternative to petrodiesel, has not a large share of the energy market due to its high production cost. Effective and efficient design of BSCN can decrease a lot of of these costs. Iran has great

Environmental objective function
Weight of e potential of RE resources especially biomass. While the biomass is fairly cheap, the logistic costs are major obstacles in utilizing these sources.
Biodiesel supply chain can be subject of several uncertain parameters that may greatly affect the optimal configuration. Therefore, in this paper, a fuzzy stochastic optimization model is presented to design a BSCN under hybrid uncertainty. The amount of fixed cost for establishing the facilities, and environmental impact of opening of capacities are described as fuzzy variables. The quantity of demands, supplies and some costs (producing, inventory holding, and shipment) and some environmental impacts (producing, inventory holding, and shipment) are consider as fuzzy scenario based variables.
In this paper, a multi-objective robust fuzzy stochastic programming model is proposed for designing the BSCN under hybrid uncertainty. The presented model aims to simultaneously minimize the total cost and the total environmental impacts of activities of the supply chain network. The results show the effectiveness of the presented approach for designing the BSCN.
The proposed model is implemented in a real case study to design Iran's BSCN. Due to, importing most of edible oils and high cost of producing from these sources, utilizing edible oil sources for producing biodiesel in Iran is irrational. So, in this paper, WCO, JCL and SL are regarded as sources of producing biodiesel. The proposed model determines the optimum location, number, and capacity of the centers of JCL and SL cultivating, oil extraction, distribution and biorefineries.