On characterizing solution for multi-objective fractional two-stage solid transportation problem under fuzzy environment

problem with fuzzy cost coefficients c̃ijk r , fuzzy supply quantities ãi, fuzzy demands b̃j, and/or fuzzy conveyances ẽk. The fuzzy efficient concept is introduced in which the crisp efficient solution is extended. A necessary and sufficient condition for the solution is established. Fuzzy geometric programming approach is applied to solve the crisp problem by defining membership function so as to obtain the optimal compromise solution of a multi-objective two-stage problem. A linear membership function for the objective function is defined. The stability set of the first kind is defined and determined. A numerical example is given for illustration and to check the validity of the proposed approach.


Introduction
Solid transportation problem (STP) is a generalization of the well-known classical transportation problem (TP), where three item properties are taken into account in the constraint set of the STP (namely, supply, demand, and mode of transportation or conveyance) instead of two constraints (source and destination). The STP was first proposed by Shell [1] in his work by introducing the distribution of a product by some properties. Later many researchers discussed the STP in different aspects. Haley [2] introduced a solution procedure for STP as an extension of the modified distribution method. Patel and Tripathy [3] investigated a computationally superior method for an STP with mixed constraints. Bit et al. [4] applied fuzzy programming approach to solve the multi-objective STP with real-life applications. Vejda [5] developed an algorithm for a multi-index TP, which is the extension of the distribution modification method. The zero-point method for finding the optimal solution of TP was introduced by Pandian and Natarajan [6]. Pandian and Anuradha [7] developed an efficient methodology to determine the optimal solution of STP with the help of the principle of zero-point method.
Fuzzy sets theory was first introduced by Zadeh [8]. Dubois and Prade [9] extended the use of algebraic operations on real numbers to fuzzy numbers. Jimenez and Verdegay [10] applied two ways under uncertainty for STP: interval and fuzzy STP. Orlovski [11] formulated general multi-objective non-linear programming problems with fuzzy parameters. Sakawa and Yano [12] introduced the concept of -α pareto optimality of fuzzy parametric programs. Recently, Das et al. [13] introduced an STP with mixed type of constraints under different environment: crisp, fuzzy, and intuitionistic fuzzy. Baidya et al. [14] introduced a new concept safety factor in a TP and also considered an STP with imprecise unit cost, sources, destinations, and capacities of conveyances represented by triangular and trapezoidal fuzzy numbers. Kundu et al. [15] studied multi-objective STP under different uncertain environment, in which the unit transportation costs are represented as fuzzy, random, and hybrid variables, respectively. Numerous researchers presented their work on STP by introducing new method, for example, Sinha et al. [16], Aggarwal and Gupta [17], Sinha et al. [16], etc. addressed a novel concept regarding the TP where they maximized the profit and minimized the transporting time subject to constraints. They considered all the parameters as trapezoidal interval type-2 fuzzy numbers. Aggarwal and Gupta [17] introduced a new ranking system for signed distance of intuitionistic fuzzy numbers and formulated an STP in intuitionistic environment to compute initial basic feasible solution. Acharya et al. [18] applied an interactive fuzzy goal programming approach for solving multi-objective generalized STP. Sobana and Anuradha [19] used the -α cut under imprecise environment, and they proposed a new algorithm to find an optimal solution for STP. Singh et al. [20] formulated a general model of the multiobjective STP with some random parameters and they proposed a solution method by using the chanceconstraint programming technique to solve the model of multi-objective STP. Kumar et al. [21] proposed a new computing procedure for solving fuzzy Pythagorean TP, where they extended the interval basic feasible solution, then existing optimality method to obtain the cost of transportation. Khalifa et al. [22] investigated a neutrosophic programming using lexicographic order to determine the optimal solution. Arqub and Al-Smadi [23] presented the fractional differential equation and solved by using the fuzzy approach.
Fractional programming (FP) is considered as one of the various applications on non-linear programming, and it is applicable in numerous fields such as finance, economic, financial and corporate planning, and health care. Normally, the minimization or maximization of objective functions such as return on investment, return/risk, time/cost, or output/input under a limitation of constraints are some other examples of the applications of FP. Charnes and Cooper [24] introduced the linear fractional programming (LFP). Tantawy [25] investigated an iterative method using the conjugate gradient projection method for solving LFP problems. Stanojevic and Stanojevic [26] applied the efficiency test introduced by Lotfi et al. (2010) to the proposed two procedures for deriving weakly and strongly efficient solutions in multi-objective LFP problems. They started from any feasible solution and introduced its applications in the multi-criteria decisionmaking process. Das and Mandal [27] addressed an efficient approach for solving a class of single-stage constraint LFP problems, based on the transformation of the objective value and the constraints also. Dutta and Kumar [28] presented an application of FP approach to inventory control problem. Simi and Talukder [29] introduced a new method for solving LFP problem. In their work, they first transformed the LFP into linear programming and hence solved this problem algebraically using the duality concept. Rubi and Pitam [30] proposed an iterative fuzzy approach for solving LFP.
In this research article, the cost minimizing fuzzy multi-objective fractional STP is studied under uncertainty. Fuzzy programming approach is applied to solve the corresponding crisp problem and hence the notions of solvability set and the stability set of the first kind are defined and characterized.
The rest of the article is organized as follows: in Section 2, multi-objective two-stage fuzzy STP is formulated. Section 3 proposes a solution procedure for solving the problem. Section 4 provides a numerical example to illustrate the efficiency of the solution procedure. Finally, some concluding remarks are reported in Section 5.

Problem formulation and solution concepts
Let p q andĩ jk r ijk r be the coefficients of the objective functions,  a i be the availability of the product at the source i, b j be the minimum requirement at the destination j, and ẽ k be the conveyance. All of  p q a b ,˜,ĩ is a vector r objective function and the subscript on both Z r , p q ,ĩ jk r ijk r identified the number of objectives ( = … r K 1, 2, , ). Without loss of generality, it is assumed that: The problem can be formulated as: Alternatively, defining the interval of confidence at level α, the triangular fuzzy number is characterized as: q p α p r s α r α , ; f o ra l l 0 , 1 .  for some r.
and with strict inequality holds for at least one r, which is contradiction.
From the continuity and convexity of the membership function, we get 1, 2, , ; 1, 2, , , which is a contradiction. □ (3) is equivalent to the following problem:

By the transformation
The membership function of each objective function can be constructed as: For each = r K 1, , applying Zadeh's min operator [8], problem (3) reduces to the following model (5).
It clear that the constraints in (6) may be reduced into the following form: Model (5) can be rewritten as in the equivalent form as in Model (6):

Solution procedure
The steps of the solution procedure for solving the STP can be summarized as follows: Step 1: Calculate the individual minimum and maximum of each objective function subject to the given constraints so as to determine the lower and upper bounds of the objectives Z r using the variable transformation method.
Step 2: Using the variable transformation method, problem (2) can be converted into problem (3).
Step 3: Determine the membership function as in (4).
Step 4: By introducing an auxiliary variable δ, problem (5) is equivalent to the following classical linear programming (6).
Step 5: Solve problem (6) using any software package (say, MATLAB), to obtain the optimal compromise solution.
Step 6: Combining stage I and stage II to obtain the optimal solution for the two-stage problem.

The solution is
with respect to the given constraints Subject to

Stage II
Step 1: Let us take the following data:   In the same way, solving Subject to

( ) Constraints in 14 .
We obtain the following solution: 1.466667, 0.5000000, and 0.8500000, i.e., The membership function for ( ) ( ) Z x Z x , 1 2 , and ( ) Z x 3 are as follows: Step 2: Let us solve the following problem: Step 5: The solution is Step 6: The stability set S can be determined as      geometric programming approach has been applied to determine the optimal compromise solution for a multi-objective two-stage fuzzy STPs in which sources' availabilities and destination's demands are triangular fuzzy numbers, and membership function for the objective functions has been defined rather than the crisp value provides more information for the decision-maker. MATALB software has been used to find out the optimal compromise solution. This approach provides an easy and simple analyst mathematical programming problem.