TOPSIS Based Algorithm for Solving Multi-objective 2 Multi-level Programming Problem with Fuzzy 3 Parameters 4

The paper proposes TOPSIS method for solving multi-objective multi-level programming 13 problem (MO-MLPP) with fuzzy parameters via fuzzy goal programming (FGP). At first, λ cut 14 method is used to transform the fuzzily described MO-MLPP into deterministic MO-MLPP. Then, 15 for specific λ , we construct the membership functions of distance functions from positive ideal 16 solution (PIS) and negative ideal solution (NIS) of all level decision makers (DMs). Thereafter, FGP 17 based multi-objective decision model is established for each level DM for obtaining individual 18 optimal solution. A possible relaxation on decisions for all DMs is taken into account for 19 satisfactory solution. Subsequently, two FGP models are developed and compromise optimal 20 solutions are found by minimizing the sum of negative deviational variables. To recognize the 21 better compromise optimal solution, the concept of distance functions is utilized. Finally, a novel 22 algorithm for MO-MLPP involving fuzzy parameters is provided and an illustrative example is 23 solved to verify the proposed procedure. 24


Introduction
Multi-level programming (MLP) technique is a powerful analytical device for describing decentralized planning problems involving several decision makers (DMs) in a hierarchical organization.MLP has diverse practical applications in such fields as agricultural economics [1], conflict resolution [2], network design [3], pollution control policies [4], warfare [5], and so on.In a multi-level programming problem (MLPP), there exists a single and independent DM at each level and each level DM attempts to optimize its objective function over a common feasible region but the decision of each level DM is exaggerated by the actions and reactions of the other DMs.
Consequently, decision deadlock may occur in the decision making circumstances.However, it has been observed that each level DM should have a motivation to cooperate with each other, and a minimal level of satisfaction of all level DMs must be considered for overall profit of the hierarchical structure.
Using the idea of tolerance membership function of fuzzy set theory [6] to MLPPs for satisfactory decisions, Lai [7] incorporated an efficient fuzzy approach at first in 1996.Shih et al. [8] formulation of MO-MLPP with fuzzy parameters is exhibited.Section 4 provides deterministic formulation of MO-MLPP with fuzzy parameters using λ -cut technique.Some basic concepts relating to distance measures are briefly stated in section 5. TOPSIS based FGP approach for solving MO-MLPP with fuzzy parameters is developed in the next section.Distance functions for obtaining compromise optimal solution are discussed in section 7.In section 8, TOPSIS based algorithm for solving MO-MLPP with fuzzy parameters through FGP method is provided.A MO-MLPP with fuzzy parameters is solved to demonstrate the validity and efficiency of the proposed approach in section 9. Finally the last section concludes the paper with some future scope of research.

Preliminaries
In this section, some basic definitions regarding fuzzy set theory are provided.

Definition 2.2 Normal fuzzy set [31]
 is said to be a normal fuzzy set if there exists a point x in U such that ) ( μ ~x  = 1.

Definition 2.3 Convex fuzzy set [31]
 is called a convex fuzzy set if and only if for any x1, x2 Definition 2.4 λ -cut [31] The λ -cut of a fuzzy set  of U is a non-fuzzy set denoted by  λ defined by a subset of all elements xU such that their membership functions exceed or identical to a real number λ  [0, 1], i.e.
Definition 2.5 Triangular fuzzy number [31] Triangular fuzzy number is both convex and normal fuzzy set in U which is defined by T = ) ( μ T x x, where Generally, a triangular fuzzy number is represented as (r, s, t) (see Fig. 1).
Figure 1.Triangular fuzzy number

Formulation of MO-MLPP with fuzzy parameters
Consider a MO-MLPP where the objective functions at each level are maximization type with fuzzy parameters and common constraints are linear functions with fuzzy parameters.Let, DMi denotes the DM at the i-th level (i = 1, 2, …, p) which controls the variable i x = (x 1 i , x 2 i , …, ∈ Ω , (i = 1, 2, …, p) where x = (x1, x2, ..., xp) and N = N1 + N2 + … + Np and further suppose  , (i = 1, 2, …, p) are the vector of objective functions of the DMi, (i = 1, 2, …, p) at the i-th level.Mathematically, a p-level MO-MLPP with fuzzy parameters is presented as follows: [Second Level]: .

Deterministic formulation of MO-MLPP with fuzzy parameters
At first, we convert the fuzzily described objectives and constraints to deterministic objectives and constraints for a specific value of λ.Now, for specific value of λ, maximization-type objective function ) ( Y ij ~x , (i = 1, 2, …, p), (j = 1, 2, …, Mi) can be replaced by the upper bound of its λ -cut i.e.,

 
Similarly, minimization-type objective function The inequality constraints can be modified by the following constraints: The fuzzy equality constraints can be replaced by two equivalent inequality constraints as given below.
Then, for a prescribed value of λ , the MO-MLPP reduces to the following problem as given below. (4.10)
However, if , (j = 1, 2, ..., M) is not expressed in commensurable unit, then we can employ the modified metric as follows: In order to find the compromise solution of the multi-objective decision making (MODM) problem we solve the following problem: (5.3) According to Lai et al. [33], the above problem (5.3) is transformed into the following auxiliary problem as given below. (5.4) The parameter 'k' is known as the 'balancing factor' between the group utility and maximal individual regret.It is to be noted that if the value of 'k' increases, the group utility i.e.Bk decreases [33].

TOPSIS based FGP approach for MO-MLPP with fuzzy parameters
For specific value of of λ, consider the deterministic MODM problem at i-th level is expressed as follows: TOPSIS model for i-th level DM can be formulated as follows: Here, x , (i = 1, 2, …, p) are the PIS and

The membership functions for
)) x (see Fig. 2) can be constructed as follows: Convert the non-linear membership functions x , (i = 1, 2, …, p) respectively using first order Taylor polynomial series approximation as given below.α < 0, then we consider * PIS i α = 0, (i = 1, 2, ..., p) because the value of the membership function cannot be less than zero [30] (6.9) According to Pramanik and Dey [22], the flexible membership goals of with aspiration level unity can be expressed as follows: where (1, 2, …, p) are the negative deviational variables corresponding to PIS and NIS respectively.The following MODM model is solved based on FGP method to achieve the optimal decision of each level DM as follows: MODM Model: Solving the above Eq.(6.12), let x ) be the optimal solution of i-th level DM.
To avoid any unwanted circumstance i.e decision deadlock, the level DMs should offer some relaxation on decision by assigning preference upper and lower bounds on the decision variables under their control [18,22,30,35,36,37] for ovallall benefit and smooth functioning of the organization and these preference bounds are included in the constraints set.
Consider i i  and i i  , (i = 1, 2, ..., p) be the lower and upper tolerance values on the decision vector considered by i-th level DM such that Therefore, the new hybrid models of FGP and TOPSIS for MO-MLPP for a specific λ can be formulated as follows:

Model (II):
(6.15) The i-th level DM can take the normalized weight i.e.

Selection of compromise optimal solution of MO-MLPP
For selecting compromise optimal solution, we consider a termination criteria based on distance functios.The family of distance functions defined by Zeleny [38] is expressed as given below.
where q  , (q = 1, 2, ..., Q) represents the measure of closeness of the preferred compromise solution to the optimal compromise solution vector regarding q -th objective function.Here, ω = ( 1 ω , 2 ω , ..., Q ω ) denotes the vector of attribute level and  (1 We consider  = 2, then the distance function becomes For maximization type of problem q  = (the preferred compromise solution/ the individual best solution).The solution for which L2 ( ω , q) will be minimal would be the compromise optimal solution for each level DM.

TOPSIS based algorithm to MO-MLPP with fuzzy parameters
The proposed TOPSIS based algorithm (see Fig 3) for MO-MLPP with fuzzy parameters is provided below.
Step 1: For specified value of λ , the upper and lower bounds of the fuzzily described objective functions and constraints are defined at first.Step 2: Calculate the maximum and minimum values for the upper and lower λ -cuts of the objective functions for all level DMs separately subject to the common constraints.
Step 3: Compute PIS and NIS for i-th level DM and formulate distance functions for PIS and )) ( (g i NIS k λ x , (i = 1,2, ...,p) respectively for i-th level DM.
Step 11: L2 ( ω , q) is employed to identify better compromise optimal solution of the problem.
Step 12: If the compromise optimal solution is acceptable to all level DMs then stop.Otherwise, adjust the lower and upper tolerance values of all level DMs and go to Step 8.

Start
Each level DM provides his/her fuzzily described objective functions A flowchart of the proposed algorithms

Numerical example
The following MO-MLPP with fuzzy parameters is considered to demonstrate the proposed procedure.
[First Level] Here, we consider all the fuzzy numbers to be triangular fuzzy numbers and they are given by Replacing the fuzzy coefficient by specified λ , the MO-MLPP can be represented as given below.

__________________________________________________________________________________________________
To obtain the individual worst (minimal) solutions, substitute the fuzzy coefficient by their λ -cuts as follows: For λ = 0.5, the above problem (9.4) reduces to the problem as given below.
The individual worst (minimal) solution , (i = 1, 2, 3; j = 1, 2) of the objective functions of level DMs are demonstrated in the Table 2.
First-level MODM problem: The membership functions of 1 PIS 2 g (x) and 1 NIS 2 g (x) can be formulated as follows: x Solve the following MODM Model to obtain the satisfactory solution of First-level DM: Min α Subject to ((1 + (x1 -10.509)  0.096 + (x2 -0.491)  0.096 + (x3 -0)  (-0.002) -0.004)/ (1-0.004))+ -             Finally, the FGP models due to Dey et al. [30] for solving MO-MLPP involving fuzzy parameters based on TOPSIS method are formulated as follows: The optimal solution of the Model (I) for MO-MLPP is shown in the Table 3.Here, we consider the normalized weights associated with negative deviational variables.The optimal solution of Model (II) is shown in the Table 4. Finally, the comparison of the optimal solutions obtained from the proposed models is presented in the Table 5.On comparing L2 (see the Table 5), we notice that proposed Model (II) provides better compromise optimal solution than the solution obtained by proposed Model (I).Therefore, the better compromise optimal solution of the problem is obtained as x1 = 4.5, x2 = 2.727, x3 = 3.773.

Note:
The models are solved by Lingo ver.11.0.

Conclusions
The paper proposes a new solution methodology for dealing with MO-MLPP involving fuzzy parameters.We examine how the hybrid approach of FGP and TOPSIS can be efficiently used to solve MO-MLPP with fuzzy parameters.TOPSIS based FGP models are developed in the paper to obtain satisfactory solutions of the problem and distance functions are used to identify the better compromise optimal solution.Our proposed hybrid approach is straightforward and effortless to apply in the practical decision making circumstances where each level DM has autonomy to control some preassigned decision variables to obtain minimum level of satisfaction of compromise decision.Also the computational burden of the proposed approach is obviously less because we do not require any positive deviational variables.We hope that the proposed method can be effective in dealing with practical decision making problems such as agriculture planning problems, conflict resolutions, economic systems, managements, network designs, logistics, and other real world problems with fuzzily described different parameters.The proposed approach can be extended to solve decentralized MO-MLPPs, chance constrained MO-MLPPs, etc involving fuzzy parameters.

Definition 2 . 1 :
Fuzzy set [6] A fuzzy set in U is defined by  U  [0, 1] is called the membership function of  and ) ( μ ~xis the degree of membership to which x ψ .

1 Ỹ (x), 2 Ỹ
is the set of decision vector, N = N1 + N2 +…+ Np = total number of decision variables in the system and M is the total number of system constraints.Here, (x), …, p Ỹ (x) are linear and bounded with fuzzy coefficients and let us represent the system constraints (3.4) & (3.5) as J Due to Stanojević[29], we normalize ) , (i = 1, 2, ..., p) since the value of the membership function cannot be superior than one.Also if * PIS i

Step 5 :Step 6 :
Compute the maximum and minimum values of )Transform the non-linear membership functions )i = 1, 2, ..., p) respectively by using suitable transformation technique and then normalize equivalent linear membership functions.

Fuzzily
described constraints are given Upper and lower bounds of the fuzzily described objective functions and constraints are defined for specific value of λ Calculate optimal values of the objective functions Preprints (www.preprints.org)| NOT PEER-REVIEWED | Posted: 15 February 2018 doi:10.20944/preprints201802.0105.v1

(9. 6 )F
The satisfactory solution of the First-level MODM problem is obtained as *

3 NIS 2 g
(x) can be presented as given below.

Table 3 .
The optimal solution of Model (I)

Table 4 .
The optimal solution of Model (II)

Table 5 .
The comparison of the optimal solutions based on distance functions