In this paper, a multiobjective sustainable biomass supply chain network under uncertainty is designed by neutrosophic programming method. In this method, for each objective function of the problem, three functions of truth membership, nondetermination and falsehood are considered. Neutrosophic programming method in this paper simultaneously seeks to optimize the total costs of the supply chain network, the amount of greenhouse gas emissions, the number of potential people hired and the time of product transfer along the supply chain network. To achieve the stated objective functions, strategic decisions such as locating potential facilities and tactical decisions such as optimal product flow allocation and vehicle routing must be made. The results of the implementation of neutrosophic programming method show the high efficiency of this method in achieving the optimal values of each objective function. Also, by examining the rate of uncertainty, it was observed that with increasing this rate, the total costs of supply chain network design, greenhouse gas emissions and product transfer times have increased, and in contrast, the potential employment rate of individuals has decreased.
In today’s competitive economics world, focusing on reducing costs or increasing revenue is essential to the longterm competitive advantage and financial profitability of organizations. In addition to economic challenges, the consequences of human activities in the environment and society (such as acid rain, climate change, deforestation, migration of workers to external borders, and reduced job opportunities) have been highlighted. Increasing social awareness of individuals about the social effects of the supply chain cannot be the only indicator of the sustainability of the supply chain economic considerations. In addition, the energy crisis and environmental issues have prompted researchers to move towards the development of renewable energy sources [
Plant fuels are a type of renewable energy that can be a good alternative to fossil fuels. Bioethanol is a type of plant fuel that is widely used today in the field of transportation as a substitute for diesel. Firstgeneration bioethanols are crops such as corn and sugarcane, which have raised concerns such as food security and rising food prices due to the allocation of arable land to biomass crops and environmental sustainability [
Thus, in recent years, different countries of the world have widely used new energy and nonfossil fuels, especially biomass energy and biofuels. Thus, the use of biomass energies is growing worldwide, and the debate between proponents and opponents of the use of such energies is expanding. Due to the increasing desire of people to use such energies, it is predicted that by the middle of the 21st century, the demand for biomass energy and biofuels will double; As a result, biofuels and biomass energy are expected to become an important part of the global energy mix and contribute greatly to energy demand. The structured and gradual replacement of fossil resources with biomass and the calculated construction of interconnected refineries (which in turn requires the expansion of agriculture and the production of strategic foodstuffs) and the production, consumption and export of refined biological products including biofuels are not only a choice but a definite strategic necessity. One of the important reasons for this necessity is that human beings are responsible for their health and environment, employment and food security for themselves and the people, as well as sustainable development [
In a general definition, residues and materials derived from living organisms are called biomass. In fact, all forms of organic matter are considered biomass; Such as plant and animal residues and wastes, municipal and factory wastes. Important sources of biomass include oilseeds (such as sunflower and soybeans), sugar and starch, lignocellulose (such as agricultural waste, forest residues, municipal solid waste, and perennial grasses), and algae with different conversion technologies, including biochemical processes (biodegradation, air fermentation). And alcohol, thermochemical (steam explosion, fireburn), biochemical (carboxylate pathway) and direct combustion, all kinds of energy are produced, including biodiesel, ethanol, biogas, methane, cethanol, hydrocarbonbiooil and bioelectricity [
In different sectors of the industry, the biomass supply chain includes growth (production), harvesting, transportation, collection, storage and conversion of biomass into the final product. Productivity and technology in each sector affect other sectors. The most important issues in the supply chain are the existence and communication with the biomass market and biomass supply. The choice of biomass, stability and profitability in the production of raw materials as well as the scale of production and conversion of biomass into valuable materials on a large scale are among the important issues in this field [
If biomass production is seasonal, these concerns increase. Onsite storage changes biomass characteristics and reduces the productivity of later stages. Storage in places between the place of production and the place of use also carries transportation costs. A more logical solution is to keep the biomass close to where it is used. Given the importance of biomass energy and biofuels in this paper, the aim is to provide a mathematical model of a sustainable biomass supply chain network. Global economic conditions and environmental importance lead to excessive attention of governments, private and governmental organizations and environmental companies to the development of mathematical models of supply chain network [
In this paper, by presenting a mathematical model, an attempt has been made to design a comprehensive network of supply, production, refining and supply of biomass products to customers in which social, economic and environmental issues have been observed.
The structure of the article is as follows, in the second part, the background of research related to the biomass supply chain is discussed and the research gap of the problem is shown. In the third part, an uncertain model of a stable biomass supply chain network is presented and the fuzzy planning and neutrosophic programming methods for controlling and solving the model are described. The fourth section presents the test results and its analysis. The fifth section concludes and presents future research proposals.
The main obstacle to the commercialization of biofuel is its complex production processes. This complexity is due to various forms of various parameters. Uncertainty is one of the common forms of variability in biofuel supply chain parameters. Biomass supply, biofuel production and demand, price and logistics are common parameters that are associated with uncertainty [
Ghaderi et al. [
Their main goal in this paper was to maximize economic, environmental and social performance. They used the Banders analysis method to solve the model and conducted a logistics case study in the United States to prove the effectiveness of their proposed mathematical model. Ghaderi et al. [
Fattahi et al. [
In this paper, labor force, biomass demand and biomass price are considered unknown, and Monte Carlo simulations and evolutionary algorithms (genetic algorithm and differential evolution) are used to find the optimal solution. Akbarian et al. [
In this research, two strategic decisions of facility location and a tactical decision of allocating flow between facilities have been considered. Cao et al. [
Kamal et al. designed a multiobjective selective maintenance allocation problem with fuzzy parameters under neutrosophic environment. They used a new defuzzification technique based on beta distribution to convert fuzzy parameters into crisp values [
Ahmad et al. designed a closed loop supply chain network under fuzzy number. They developed a fuzzy programming model to control the demand and transportation costs, anf used neutrosophic optimization model to optimization the model [
RizkAllah et al. proposed a new algorithm for multiobjective transportation problem, which is inspired by Zimmermann's fuzzy programming and the neutrosophic set terminology. In this paper, a neutrosophic compromise programming model is constructed with the aim to find best compromise solution and optimization the all objectives is developed [
Pratihar et al. presented neutrosophic set to solve the Transportation problem under uncertainty. They considered supply, demand, and cell cost as uncertainty parameters, and used neutrosophic set to obtaint the optimum objective. neutrosophic set is a generalization of crisp sets, fuzzy set, and intuitionistic fuzzy set, which is handle the uncertain, unpredictable, and insufficient information in reallife problem [
In their research, four layers of interconnected decision are considered: biomass supply, conversion processes, storage and markets. This model is formulated as MILP to maximize net present operating value.
According to the literature review and the gaps in the studied papers, the innovations of the model are as follows:
Use of vehicle routing in the distribution of products from retail centers to demand centers.
Using fuzzy programming method in controlling model parameters and neutrosophic programming to solve the model.
Considering the limitation of increasing the quality of purchasing raw materials from cultivation centers with discounts.
Author  Year  Sustainable  Facility 
Allocation problem  Routing problem  Deterministic/ uncertainty  Solution 
Discount  Neutrosophic 

Kim  2011  ∗  ∗  ∗  –  Stochastic  Cplex  –  – 
Awudu et al. [  2013  –  ∗  ∗  –  Stochastic  Cplex  –  – 
Balaman et al. [  2015  –  ∗  ∗  –  Deterministic  Cplex  –  – 
Ghaderi et al. [  2016  –  ∗  ∗  –  Fuzzy  Cplex  –  – 
Osmani  2017  ∗  ∗  ∗  –  Stochastic  Cplex  –  – 
Petridis  2018  ∗  –  ∗  –  Deterministic  GP  –  – 
Fattahi  2018  ∗  ∗  ∗  –  Stochastic  Cplex  –  – 
Khishtandar  2019  –  ∗  ∗  –  Fuzzy  GA – DA  –  – 
Pamucar et al. [  2020  –  ∗  –  –  Fuzzy  TOPSIS  –  ∗ 
Ahmad  2020  –  ∗  ∗  –  Fuzzy  Cplex  –  ∗ 
Akbarian  2020  ∗  ∗  ∗  –  Deterministic  BWM  –  – 
Durmaz et al. [  2020  ∗  ∗  ∗  –  Deterministic  Cplex  –  – 
Mohammadi et al. [  2020  –  ∗  ∗  ∗  Robust  Cplex  –  ∗ 
Theozzo et al. [  2021  –  –  ∗  –  Deterministic  Cplex  –  – 
Cao et al. [  2021  –  ∗  –  ∗  Deterministic  TS  –  – 
Ahmad  2021  –  ∗  ∗  –  Fuzzy  Cplex  –  ∗ 
This Paper  2021  ∗  ∗  ∗  ∗  Fuzzy  Cplex  ∗  ∗ 
The importance of environmental problems and chain network design as well as the use of environmental fuel has led to the design of a sustainable biomass supply chain network in this paper. According to
The assumptions of the sustainable biomass supply chain network model are as follows:
The optimal number and location of facilities such as cultivation centers, preprocessing centers, refineries, major distributors and retailers are unknown.
The capacity of retail and wholesale distributors as well as the elastic parameters of demand, transportation cost, transportation time are indefinite.
The capacity of vehicles in the optimal routing of processed products is known and
The purchase price of raw materials from cultivation centers is known.
Potential set of cultivation centers  
Potential preprocessing set  
Potential set of refineries  
Potential set of major distribution centers  
Potential set of retail distribution centers  
Fixed customer set  
The whole set of potential retail distribution centers and regular customers  
Period set  
Set of raw materials  
Final product set  
Set of vehicle  
Set of discount levels 
In the following equations, the arc
Capacity of vehicles 

Uncertain demand from customer 

Maximum number of vehicles available at the potential retail distribution center 

Distribution capacity for product 

Distribution capacity for product 

Production capacity for product 

Processing capacity for raw materials 

Supply capacity for raw materials 

Fixed cost of using the vehicle 

Fixed cost of establishing a retail distribution center in a potential location 

Fixed cost of establishing a major distribution center in a potential location 

Fixed cost of building a refinery at potential location 

Fixed cost of establishing a processing center in a potential location 

Fixed cost of establishing a cultivation center in a potential location 

Time of transfer of a product unit between two nodes 

The cost of transporting a product unit between two nodes 

Emission of 

Social impact factor of employment of individuals  
Number of potential people hired by establishing a retail distribution center in a potential location 

Number of potential recruits due to use of vehicle 

Number of potential recruits due to the establishment of a major distribution center in the potential location 

Number of potential recruits due to the establishment of a refinery at the potential location 

Number of potential recruits due to the establishment of a processing center at the potential location 

Number of potential recruits due to the establishment of a cultivation center in a potential location 

The cost of maintaining a product unit in the warehouse of a potential retailer center 

The cost of maintaining a product unit in the warehouse of a potential wholesale center 

Number of raw materials 

Purchase price of raw material 

The lower limit of the raw material discount period 

Negative large number 
If vehicle 

If the path between nodes 

If the set of 

If the path between nodes 

If the path between nodes 

If the path between nodes 

If the route between nodes 

The total amount of product 

Total purchase of raw material 

quantity of product 

Product quantity 

The amount of raw material 

The amount of raw material 

The amount of inventory of product 

The amount of inventory of product 

If the retail distributor center is established at potential location 

If a major distributor center is established in a potential location 

If the refinery is located at potential location 

If the processing center is established at potential location 

If the cultivation center is established in potential location 

If the discount level 

Auxiliary variable for the removal of the subtour 
s.t:
Due to the uncertainty of the model parameters in the form of triangular fuzzy execution, in this section, the control of the uncertainty parameters of the model is discussed. Therefore, consider the following linear mathematical programming model with fuzzy parameters:
where
In the above relation
The possible distribution of the fuzzy parameter is denoted by
As mentioned, the demand and cost parameters of delivery are considered as a triangular fuzzy number. Therefore, triangular fuzzy programming method has been used to control the uncertain parameters. As a result, the controlled model of the sustainable biomass supply chain network problem is as follows:
Multiobjective decision models are the most common type of mathematical models that have conflicting goals. In such cases the goal is to achieve the optimal value of all conflicting objective functions simultaneously. In such problems, the decision maker expresses the importance of his preferences by providing an optimal weight
In the above equation,
Therefore, the upper and lower bounds of the neutrosophic membership function can be calculated as follows for truth, nondetermination, and falsehood, respectively:
In the above relation
Therefore, the controlled model of the sustainable biomass supply chain network problem with neutrosophic programming method based on the above relations is as follows:
s.t:
In this section, a small sample problem is analyzed using neutrosophic programming method. The problem of the small sample is designed according to the dimensions presented in
3  5  2  
3  2  3  
3  3  3  
3  1  2 
Due to the fact that the mathematical model has several objective functions, the neutrosophic programming method has been used to simultaneously achieve nearoptimal solutions. Therefore, in
According to
According to
According to the results of
172343.56  117629.97  117630.17  117630.17  172343.56  
1437.69  9412.14  1437.69  1437.89  1437.89  9412.14  
39960  58360  39960  39960.20  39960.20  58360  
4595.17  26918.65  4595.17  5695.37  4595.37  26918.65 
After forming the truth membership function, nondetermination membership function and false membership function according to relation
According to
The above analyzes in this section have been performed assuming an uncertainty rate of 0.5. Therefore, in order to analyze the sensitivity of the uncertainty rate on the values of the objective functions, the value of the uncertainty rate is changed from 0.1 to 0.9 and the values of the objective functions corresponding to each uncertainty rate with the neutrosophic programming method are shown in
According to the results of
0.1  127516.38  846.59  59320  4348.64 
0.2  129465.46  986.25  56460  4394.26 
0.3  132953.15  1064.86  52230  4488.64 
0.4  137591.25  1244.68  50760  4597.44 
0.5  139384.30  1467.83  49440  4631.72 
0.6  141344.65  1577.86  47130  4714.26 
0.7  143976.85  1637.91  45850  4843.50 
0.8  146795.36  1867.79  43660  4996.19 
0.9  148620.30  1976.30  41990  5044.18 
Given the importance of biomass energy and biofuels in this paper, a mathematical model of a sustainable biomass supply chain network under conditions of demand uncertainty, potential facility capacity, transportation costs and transmission time is presented. Sustainability in the proposed biomass supply chain network includes minimizing the total costs of the supply chain network, maximizing the employment rate and minimizing the amount of greenhouse gas emissions along with reducing the transmission time of biomass products. Different types of facility tracking decisions, optimal flow allocation and vehicle routing are considered in the proposed model. Also in this paper, cultivation centers provide the general raw materials needed to produce biomass products by offering a general discount. Triangular fuzzy programming method has been used to control the uncertain parameters of the problem and neutrosophic programming method has been used to solve the multiobjective model. In this method, each of the objective functions is considered in the three membership functions of truth, nondetermination and falsehood. By implementing the neutrosophic programming method, the value of the objective functions obtained is very close to the optimal value of the objective function of the problem by the individual optimization method. Also, by examining the changes in uncertainty rates on the values of the objective functions of the problem, it was observed that with increasing uncertainty rates due to reduced capacity and increased demand, the total cost of network design has increased and vehicle traffic and therefore greenhouse gas emissions and transportation time has increased. Also, in these circumstances, the employment rate has decreased with increasing uncertainty. Since in the present paper, an attempt has been made to provide a comprehensive model of the biomass supply chain network, so more favorable results can be obtained by using the fuzzy stable method in controlling uncertain parameters. It is also proposed to consider the objective function of minimizing the maximum unmet demand as future research.