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 [9]. Examining complex linear integer linear programming models, many studies can be observed that are related to biomass supply chain network optimization models under uncertainty. For example, Dal-Mas et al. [10] designed a dynamic randomized integer linear programming model to design a biofuel supply chain under market uncertaint. Giarola et al. [11] proposed a multi-period multi-level integer linear integer programming model to optimize the bioethanol supply chain network design taking into account biomass and carbon market uncertainties.

Ghaderi et al. [12] designed a mixed integer mathematical model for a biomass supply chain network in which demand was considered indefinitely. They conducted their study on a real issue in the state of North Dakota in the United States. Osmani et al. [13] applied stochastic optimization for a chain network to produce stable second-generation ethanol.

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. [14] presented a robust multi-objective feasibility optimization model in designing a sustainable biomass supply chain network that simultaneously considered economic, social and environmental goals. Rabbani et al. [15] designed a network of sustainable biomass supply chains with conflicting economic, social and environmental goals. They used the Epsilon constraint method to solve their model and showed that in order to achieve a desirable level of social and environmental performance, the share of economic function should be increased by 15%. Petridis et al. [16] designed a sustainable biomass supply chain network in a paper. They provided an integrated mathematical framework for the transportation and storage of biomass at different nodes of the supply chain, using the ideal planning method to minimize the economic, social, and environmental goals of the problem.

Fattahi et al. [17] designed a biomass supply chain network under supply and demand uncertainty. In their model, they used the stochastic planning method and examined different supply and demand values in different scenarios. In their model, objective functions included environmental, economic and social goals. Foo et al. [18] presented a simple linear model for the optimal allocation of palm biomass in a multilevel supply chain network. Their main purpose in this paper was to maximize the benefits of grid design along with minimizing the cost of carbon dioxide emissions. Bragado et al. [19] developed a mathematical model for optimizing the flow system of biomass supply chain products. In this paper, they aimed to increase the profits from the design of the entire network and obtain the most optimal source of biomass energy storage and product distribution. khishtandar [20] designed a mathematical programming model in conditions of uncertainty for the biogas supply chain network.

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. [21] proposed a mixed integer programming model for the sustainable biomass supply chain and used the BWM and TH methods to solve the problem. The results showed that adapting the appropriate ethanol distribution policy to a planning horizon could lead to a 33% reduction in total costs. Durmaz et al. [22] designed and planned the optimal biomass supply chain network, which includes the flow from poultry farms to biogas facilities.

In this research, two strategic decisions of facility location and a tactical decision of allocating flow between facilities have been considered. Cao et al. [23] proposed a routing location model in the biomass supply chain network and solved the model using a hierarchical method based on prohibited search. Thezozzo et al. [24] provided an optimization framework for the design of biorefinery. Ghahremani Nahr et al. designed a green closed loop supply chain under uncertainty. They developed a net robust possibilistic optimization method to control the uncertainty parameters. At the end, they used five meta-heuristic algorithms to solve the designed model [25].

Kamal et al. designed a multi-objective 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 [26]. Junaid et al. [27] studied a supply chain with aims at identifying and assessing supply chain risks and developing criteria for managing these risks. They presented a neutrosophic (N) theory is merged with the analytic hierarchy process (AHP) and technique for order of preference by similarity to ideal solution (TOPSIS) to deal with complexity, uncertainty, and vagueness. Abdel-Basset et al. desiged a framework to handle scheduling problems using neutrosophic activity duration times. They considered important aspects of time-cost tradeoffs while taking into account the real and uncertain situation surrounded by the projects [28].

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 [29]. Edalatpanah designed a new model of DEA that the information on decision-making units is triangular neutrosophic numbers. In this paper, after comparative the conventional DEA model and the triangular neutrosophic numbers, an algorithm for solving the new model has been used [30]. Abdel-Baset et al. developed a green closed loop supply chain under uncertainty. In this paper the robust ranking technique with neutrosophic set to handle practices and performances in the developed model is used. The results display that “reverse logistics”, “supplier environmental collaboration”, “carbon management” are the significant in GSCM practices [31]. Edalatpanah proposed a new algorithm to solve the neutrosophic linear programming. In this paper, the variables andright-hand side represented with triangular neutrosophic numbers. The results show that the new algorithm is useful to design of a wide range of neutrosophic optimization [32].

Rizk-Allah et al. proposed a new algorithm for multi-objective 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 [33].

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 real-life problem [34]. Touqeer et al. solved a linear programming network problems with constraints concerning interval-valued neutrosophic numbers. For this, they used score function and upper and lower membership functions of interval-valued neutrosophic numbers [35].

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.

Tab. 1 compares some of the most important papers on the subject under study.

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.

In the following equations, the arc (ι,ϱ)=(i,j)∪(j,k)∪(k,l)∪(l,m)∪(mn,mn′) is considered for the transfer of products.

Parameters:

Decision variable:

Eq. (1) shows the first objective function of the problem and includes minimizing the costs of the entire biomass supply chain network (transportation costs, fixed construction costs, maintenance costs, and the cost of purchasing discounted raw materials). In Eq. (2), the second objective function illustrates the problem and involves minimizing the amount of greenhouse gas emissions from vehicle traffic on the routes allocated to them. Eq. (3) shows the social objective function of the problem and includes maximizing the employment rate and employing people to create new centers. Eq. (4) is related to the fourth objective function of the problem and is to minimize the time of transfer of products and raw materials by freight vehicles on the routes allocated to them. Eq. (5) states that the optimal route of transport by a vehicle can be serviced if that vehicle has already been selected. Eq. (6) ensures that each optimal route of transport should only be traveled by a maximum of one selected vehicle. Eqs. (7) and (8) show that all customer products can only be delivered from the route for which a vehicle is intended. Eq. (9) ensures that vehicles must visit each node at most once. Eq. (10) guarantees that if a vehicle enters a node, it will inevitably leave that node. Eq. (11) ensures that the amount of product carried by each vehicle does not exceed the capacity intended for that product. Eq. (12) ensures that the maximum number of available vehicles can be used for each selected retail distribution center. Eq. (13) ensures that each retailer has the ability to distribute products and provide services to the maximum of its product capacity. Eqs. (14) and (15) are auxiliary constraints for sub-net removal. Eq. (16) shows the total amount of products distributed in each retail distribution center. Eqs. (17) and (18) show the balance of product flow in the retail and wholesale center, respectively, along with the amount of product inventory in these centers. Eq. (19) shows the amount of raw materials needed to convert products in each refinery. Eq. (20) shows the amount of raw materials transferred from each cultivation center to the pre-processing center. Relationships (21)–(24) calculate the optimal route of vehicles between each level of the supply chain network. Relationships (25)–(28) are related to the capacity constraints of each potential center, and if those centers are constructed, they can benefit from the services of the centers to the maximum of their capacity. Eqs. (29)–(31) show the constraints related to the general discount, so that in Eq. (29) shows the total amount of raw material purchased from each cultivation center, Eq. (30) expresses the optimal level of purchase discount from each cultivation center and Eq. (31) shows the guarantee of using a discount level for each crop in each cultivation center. Eqs. (32) and (33) show the type and nature of decision variables.

(1) MinW1=∑v=1FixVv.ZVv,t+∑m=1FixMm.ZMm+∑j=1FixJj.ZJj+∑l=1FixLl.ZLl+∑i=1FixIi.ZIi+∑i=1∑j=1∑v=1∑t=1Tr~i,j.Gi,j,v,t′+∑j=1∑k=1∑v=1∑t=1Tr~j,k.Sj,k,v,t′+∑k=1∑l=1∑v=1∑t=1Tr~k,l.Rk,l,v,t′+∑mn=1∑mn′=1∑v=1∑t=1Tr~mn,mn′.Xm,n,v,t′+∑l=1∑m=1∑v=1∑t=1Tr~l,m.Ul,m,v,t′+∑m=1∑p=1∑t=1Hm,t.IPm,p,t+∑l=1∑p=1∑t=1Hl,t.IMl,p,t+∑i∑b∑c∑tpri,b,cMi,b,c,tΘi,c,t

(2) MinW2=∑mn=1∑mn′=1∑v=1∑t=1Co2mn,mn′,v.Xm,n,v,t′+∑l=1∑m=1∑v=1∑t=1Co2l,m,v.Ul,m,v,t′+∑k=1∑l=1∑v=1∑t=1Co2k,l,v.Rk,l,v,t′+∑j=1∑k=1∑v=1∑t=1Co2j,k,v.Sj,k,v,t′+∑i=1∑j=1∑v=1∑t=1Co2i,j,v.Gi,j,v,t′

(3) MaxW3=∑i=1φFiZIi+∑j=1φEjZJj+∑k=1φDkZKk+∑l=1φClZLl+∑m=1φAmZMm+∑v=1∑t=1φBvZVv,t

(4) MinW4=∑mn=1∑mn′=1∑v=1∑t=1Ti~mn,mn′.Xm,n,v,t′+∑l=1∑m=1∑v=1∑t=1Ti~l,m.Ul,m,v,t′+∑k=1∑l=1∑v=1∑t=1Ti~k,l.Rk,l,v,t′+∑j=1∑k=1∑v=1∑t=1Ti~j,k.Sj,k,v,t′+∑i=1∑j=1∑v=1∑t=1Ti~i,j.Gi,j,v,t′

s.t:

(5) ∑m=1∑n=1Xm,n,v,t′≤ZVv,t,∀v,t

(6) ∑v=1Yn,p,v,t′=1,∀n,p,t

(7) ∑p=1Yn,p,v,t′≤M.∑mn=1Xmn,n,v,t′,∀n,v,t

(8) ∑mn=1Xmn,n,v,t′≤∑p=1Yn,p,v,t′,∀n,v,t

(9) ∑mn=1Xmn,n,v,t′≤1,∀n,v,t

(10) ∑mn=1Xmn,mn′,v,t′=∑mn=1Xmn′,mn,v,t′,∀mn′,v,t

(11) ∑n=1Dem~n,p,t.Yn,p,v,t′≤CapVvZVv,t,∀p,v,t

(12) ∑n=1∑v=1Xm,n,v,t′≤DQm.ZMm,∀m,t

(13) ∑n=1∑v=1Dem~n,p,t.Xm,n,v,t′≤CapM~m,p.ZMm,∀m,p,t

(14) ∑m=1∑v=1STm,v=0

(15) STmn,v+1≤STn,v+M(1−Xmn,n,v,t′),∀mn,n,v,t

(16) ∑n=1∑v=1Dem~n,p,t.Xm,n,v,t′=Qm,p,t,∀m,p,t

(17) ∑l=1∑v=1Ul,m,p,v,t+IPm,p,t−1−IPm,p,t=Qm,p,t,∀m,p,t

(18) ∑k=1∑v=1Rk,l,p,v,t+IMl,p,t−1−IMl,p,t=∑k=1∑v=1Ul,m,p,v,t,∀l,p,t

(19) ∑j=1∑v=1Sj,k,c,v,t=∑l=1∑v=1∑p=1Rk,l,p,v,tOc,p,∀c,k,t

(20) ∑i=1∑v=1Gi,j,c,v,t=∑k=1∑v=1Sj,k,c,v,t,∀j,c,t

(21) ∑p=1Ul,m,p,v,t≤M.Ul,m,v,t′,∀l,m,v,t

(22) ∑p=1Rk,l,p,v,t≤M.Rk,l,v,t′,∀k,l,v,t

(23) ∑c=1Sj,k,c,v,t≤M.Sj,k,v,t′,∀j,k,v,t

(24) ∑c=1Gi,j,c,v,t≤M.Gi,j,v,t′,∀i,j,v,t

(25) ∑m=1∑v=1Ul,m,p,v,t≤CapL~l,p.ZLl,∀l,p,t

(26) ∑l=1∑v=1Rk,l,p,v,t≤CapKk,p.ZKk,∀k,p,t

(27) ∑k=1∑v=1Sj,k,c,v,t≤CapJj,c.ZJj,∀j,c,t

(28) ∑j=1∑v=1Gi,j,c,v,t≤CapIi,c.ZIi,∀i,c,t

(29) Θi,c,t=∑j=1∑v=1Gi,j,c,v,t,∀i,c,t

(30) Mi,b,c,tΨi,b,c,t≤Θi,c,t,∀i,b,c,t

(31) ∑b=1Mi,b,c,t=ZIi,∀i,c,t

(32) ZVv,t,Xmm,mn′,v,t′,Yn,p,v,t′,Ul,m,v,t′,Rk,l,v,t′,Sj,k,v,t′,Gi,j,v,t′,ZMm,ZLl,ZKk,ZJj,ZIi,Mi,b,c,t∈{0,1},∀i,j,k,l,m,mn,mn′,v,t,b,c

(33) Qm,p,t,Θi,p,t,Ul,m,p,v,t,Rk,l,p,v,t,Sj,k,c,v,t,Gi,j,c,v,t,IPm,p,t,IMl,p,t,STmn,v≥0,∀m,p,t,i,l,v,k,j,n,c,mn

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:

(34) Min Z=c~txs.t.:x∈N(A~,B~)={x∈Rn|a~ix≥b~i,i=1,…,mx≥0}x∈N(A~,D~)={x∈Rn|a~ix≤d~i,i=1,…,mx≥0}

where c~=(c~1,c~2,…,c~n),A=[a~ij]m×n,b~=(b~1,b~2,…,b~n)t,d~=(d~1,d~2,…,d~n)t, (fuzzy parameters used in the objective function of the problem) is the vector coefficient and the parameter to the right is the constraint. The probabilistic distribution function of fuzzy parameters is assumed based on the properties of fuzzy numbers. Finally, x=(x1,x2,…,xn) shows the decision vector. For the feasibility and optimization of the problem presented in the above model, it is necessary to control the uncertain parameters presented in the objective and constraint function. Therefore, assuming the α parameter as the minimum degree of constraint, the controlled model is as follows:

(35) Min Z=EV(c~)xs.t.:[(1−α)E2ai+αE1ai]x≥(1−α)E1bi+αE2bi,i=1,…,mx≥0,α∈[0,1][(1−α)E1ai+αE2ai]x≤(1−α)E2di+αE1di,i=1,…,mx≥0,α∈[0,1]

In the above relation EV(c~) is the expected value of the fuzzy number used in the objective function of the model, which is calculated as follows:

(36) EV(c~)=E1c+E2c2

The possible distribution of the fuzzy parameter is denoted by C~=(Cp,Cm,Co). Cp and Cm, Co represent the optimistic, probable, pessimistic values of the fuzzy number C, which are determined by the decision maker. Therefore, the expected value of the fuzzy parameter of the objective function can be calculated as follows:

(37) EI(c~)=[E1c,E2c]=[cp+cm2,cm+co2]

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:

(38) Min W1=∑v=1FixVv.ZVv,t+∑m=1FixMm.ZMm+∑j=1FixJj.ZJj+∑l=1FixLl.ZLl+∑i=1FixIi.ZIi+∑i=1∑j=1∑v=1∑t=1(Tri,jp+2Tri,jm+Tri,jo4).Gi,j,v,t′+∑j=1∑k=1∑v=1∑t=1(Trj,kp+2Trj,km+Trj,ko4).Sj,k,v,t′+∑i∑b∑c∑tpri,b,cMi,b,c,tΘi,c,t+∑k=1∑l=1∑v=1∑t=1(Trk,lp+2Trk,lm+Trk,lo4).Rk,l,v,t′+∑l=1∑p=1∑t=1Hl,t.IMl,p,t+∑mn=1∑mn′=1∑v=1∑t=1(Trmn,mn′p+2Trmn,mn′m+Trmn,mn′o4).Xm,n,v,t′+∑l=1∑m=1∑v=1∑t=1(Trl,mp+2Trl,mm+Trl,mo4).Ul,m,v,t′+∑m=1∑p=1∑t=1Hm,t.IPm,p,t

(39) Min W4=∑mn=1∑mn′=1∑v=1∑t=1(Timn,mn′p+2Timn,mn′m+Timn,mn′o4).Xm,n,v,t′+∑l=1∑m=1∑v=1∑t=1(Til,mp+2Til,mm+Til,mo4).Ul,m,v,t′+∑k=1∑l=1∑v=1∑t=1(Tik,lp+2Tik,lm+Tik,lo4).Rk,l,v,t′+∑j=1∑k=1∑v=1∑t=1(Tij,kp+2Tij,km+Tij,ko4).Sj,k,v,t′+∑i=1∑j=1∑v=1∑t=1(Tii,jp+2Tii,jm+Tii,jo4).Gi,j,v,t′

s.t.:

(40) ∑n=1(αDemn,p,t(m)+Demn,p,t(o)2+(1−α)Demn,p,t(p)+Demn,p,t(m)2).Yn,p,v,t′≤CapVvZVv,t,∀p,v,t

(41) ∑n=1∑v=1(αDemn,p,t(m)+Demn,p,t(o)2+(1−α)Demn,p,t(p)+Demn,p,t(m)2).Xm,n,v,t′≤(αCapMm,p(p)+CapMm,p(m)2+(1−α)CapMm,p(m)+CapMm,p(o)2).ZMm,∀m,p,t

(42) ∑n=1∑v=1(αDemn,p,t(m)+Demn,p,t(o)2+(1−α)Demn,p,t(p)+Demn,p,t(m)2).Xm,n,v,t′=Qm,p,t,∀m,p,t

(43) ∑m=1∑v=1Ul,m,p,v,t≤(αCapLl,p(p)+CapLl,p(m)2+(1−α)CapLl,p(m)+CapLl,p(o)2).ZLl,∀l,p,t

(44) Eqs. (2), (3); Eq. (5)--Eqs. (10)--(12); Eqs. (14), (15); Eqs. (17)--(24); Eqs. (26)--(33)

Multi-objective 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 β∈[0,1] to each objective function, So that with a high value of weight β in an objective function, decision maker preferences are higher in that objective function. Zimmermann [41] by introducing a multi-objective fuzzy programming method, maximized the decision maker's preferences in achieving the objective functions simultaneously. In the following years, Faizulkhakov [42] and Angelov [43] were able to solve a variety of mathematical problems with the development of Russian multi-objective fuzzy programming called intuitive fuzzy programming. In these planning methods, flexibility in element membership functions was also possible. This method has been extensively studied in various real life issues and problems. In recent years it has been observed that living conditions may have neutral thoughts about an element in the collection. Neutral or uncertain thoughts about the elements fall between a degree of falsehood and truth. Thus, with the development of the intuitive fuzzy programming method, Smarandache [44] examined the neutrosophic programming method, which has three membership sets: truth (degree of belonging), uncertainty (degree of belonging to some extent) and falsehood (degree of non-belonging). According to the developed method of neutrosophic programming, in this paper, the model of sustainable biomass supply chain network model with the objectives of four opposite objective functions is discussed. Hence, each objective function has three functions: truth membership, non-determination and falsehood. Therefore, the neutrosophic programming method plays an important role in optimizing multi-objective problems by considering neutral thoughts. Consider a multi-objective model with contrasting objective functions in which (D) represents a set of fuzzy decisions, (G) a set of fuzzy objective functions, and (C) fuzzy constraints. Therefore, the set of fuzzy decisions is represented as D=G∩C. Fuzzy neutrosophic Decision Set (Dn) with the set of neutrosophic fuzzy target functions (Go) and the set of neutrosophic constraints (Cm) [45]:

(45) Dn=(⋂o=1OGo)(⋂m=1MCm)=(w,PD(w),QD(w),RD(w))s.t.:PD(w)={minPGo(w),∀o∈Os.t.PCm(w),∀m∈M}QD(w)={maxQGo(w),∀o∈Os.t.QCm(w),∀m∈M}

RD(w)={maxRGo(w),∀o∈Os.t.RCm(w),∀m∈M}

In the above equation, PD(w) is a membership function of truth; QD(w) is the membership function of non-determination and RD(w) is the membership function of false under neutrosophic fuzzy decisions Dn. Each of the above membership functions has upper and lower bound, which is obtained as the following relation for all membership functions:

(46) Uo=max(Zo(X))

Lo=min(Zo(X))

Therefore, the upper and lower bounds of the neutrosophic membership function can be calculated as follows for truth, non-determination, and falsehood, respectively:

(47) UoP=Uo,LoP=Lo

UoQ=LoP+ao,LoQ=Lo

UoR=UoP,LoR=LoP+bo

In the above relation ao and bo is a predefined value between 0 and 1. According to the above materials, the linear membership function for a neutrosophic fuzzy framework is as follows:

(48) Dn=(⋂o=1OGo)(⋂m=1MCm)=(w,PD(w),QD(w),RD(w))s.t.:Po(Zo(X))={1ifZo(X)<LoPUoP−Zo(X)UoP−LoPifLoP≤Zo(X)≤UoP0if Zo(X)>UoP}Qo(Zo(X))={1ifZo(X)<LoQUoQ−Zo(X)UoQ−LoQifLoQ≤Zo(X)≤UoQ0ifZo(X)>UoQ}Ro(Zo(X))={1ifZo(X)>UoRZo(X)−LoRUoR−LoRifLoR≤Zo(X)≤UoR0ifZo(X)<LoR}

Therefore, the controlled model of the sustainable biomass supply chain network problem with neutrosophic programming method based on the above relations is as follows:

(49) max∑o=1(μo+ϑo−δo)