Fuzzy solution of fully fuzzy multi-objective linear fractional programming problems

In this study, we present a novel method for solving fully fuzzy multi-objective linear fractional programming problems without transforming to equivalent crisp problems. First, we calculate the fuzzy optimal value for each fractional objective function and then we convert the fully fuzzy multi-objective linear fractional programming problem to a single objective fuzzy linear fractional programming problem and ﬁnd its fuzzy optimal solution which inturn yields a fuzzy Pareto optimal solution for the given fully fuzzy multi-objective linear fractional programming problem. To demonstrate the proposed strategy, a numerical example is provided.


Introduction
Multi-objective linear fractional programming problems(MOLFPP) occurs when there are several ratios of quantities to be optimized simultaneously. It has a variety of important uses, including calculating malting efficiency in DNA and health care, in corporate factory planning (supply/demond ratios, manufacturing/worker), in transport planning (seat/vehicle, speed/distance), in election commission planning (people/areas, booth/ officer) and in educational planning (school/teacher, result output/teacher). Bellman and Zadeh (Zadeh, 1965) introduced the method of decision making in fuzzy environment. Guzel and Sivri (N. Guzel and M.Sivri, 2020) proposed multiple efficient solution to the MOLFPP. Nuran Guzel (Guzel, 2013) introduced a new solution procedure to solve MOLFPP by reducing it to linear programming problem(LPP). Farhana Akond Pramy (Pramy, 2020) introduced graded mean integration method to solve the fuzzy multi objective linear fractional programming problem (FMOLFPP). Surapati Pramanik et.al (Surapati Pramanik and Mandal, 2018) convert each objective function with fuzzy parameters into crisp objective functions using αcut of the fuzzy numbers. Duran Toksari (Toksari, 2008) solving MOLFPP using Taylor's series method. Loganathan and Ganesan (Loganathan and Ganesan, 2019) presented a solution for fully fuzzy linear fractional programming problem(FFLFPP) with a new approach. Suvasis Nayak et.al (Nayak and Ojha, 2019) proposed a solution approach to MPLFPP using parametric functions. El-Saeed Ammar et.al (Ammar and Muamer, 2016) developed an approach for solving MOLFPP with fuzzy rough coefficients. Adem (Cevikel, 2008) tackling FMOLFPP, using Q-Taylor series approach(FMOLFPP). Srikumar Acharya et.al (Srikumar Acharya and Mishra, 2019) solving a multi-probabilistic fractional programming problem Fuzzy solution of fully fuzzy multi-objective linear fractional programming problems with two parameters using the Cauchy distribution. Hamiden et.al (Khalifa and Ammar, 2020)investigated a MOLFPP that involves probabilistic parameters on the right-hand side of constraints. Moslem Ganji et.al (Ganji and Saraj, 2019) solved MOLFPP under uncertainty via robust optimization approach. Stanojevi et.al (B. Stanojevi, 2020)proposed a pair of piece-wise linear membership functions to solve FMOLFPP. In this paper, we consider FFMOLFPP that included triangular fuzzy numbers and proposed a new method for its solution. The proposed method facilitate the decision maker to solve FFMOLFPP without transforming it to an equivalent crisp problem. The remainder of this study is laid out as follows: In Section 2, we recall the notion of fuzzy numbers, ranking and arithmetic operations on the parametric form of fuzzy numbers. In section 3, the general FFMOLFPP, its mathematical formulation and relevant theorems are presented. Also a new algorithm is proposed for the solution of FFMOLFPP without transforming to equivalent crisp problem. In Section 4, a numerical example is given to illustrate the efficacy of the proposed method. Section 5 provides the conclusion part.

Preliminaries
" Definition 2.1. A fuzzy setã defined on the set of real numbers R is said to be a fuzzy number, if its membership functionã : R → [0, 1] has the following characteristics: •ã is normal, (i.e.) there exists an x ∈ R such thatã(x) = 1.
We use F (R) to denote the set of all fuzzy numbers defined on R.
" Definition 2.1. " A triangular fuzzy numberã = (a 1 , a 2 , a 3 ) ∈ F (R) can also be represented as a pairã = (a, a) of functions a(h), a(h) for 0 ≤ h ≤ 1, which satisfies the following requirements: • a(h) is a bounded monotonic increasing left continuous function.
• a(h) is a bounded monotonic decreasing left continuous function.
An arbitrary triangular fuzzy numberã = (a 1 , a 2 , a 3 ) can also be written asã = (a 0 , a * , a * ), where a 0 = a(1) + a(1) 2 is the location index number, a * = (a 0 − a) and a * = (a − a 0 ) are the left and the right fuzziness index functions respectively.

Ranking of Triangular Fuzzy Numbers
Several approaches for the ranking of fuzzy numbers have been proposed in the literature. An efficient approach for comparing the fuzzy numbers is by the use of a ranking function based on their graded means. We define the magnitude of the triangular fuzzy numberã by R(ã) = a * + 4a 0 − a * 4 = a + a + a 0 4 .

Ming Ma et. al. (Ming Ma and Kandel, 2017) have proposed a new fuzzy arithmetic based upon both location index
and fuzziness index functions. The location index number is taken in the ordinary arithmetic, whereas the fuzziness index functions are considered to follow the lattice rule which is least upper bound in the lattice L. i.e. for u, v ∈ L, We define u ∨ v = max{u, v} and u ∧ v = min{u, v}. " The arithmetic operations on any two triangular fuzzy Note: Division is possible only when the location index number of the denominator fuzzy number is non-zero. " 3. Fully Fuzzy Multi-Objective Linear Fractional Programming Problem (FFMOLFPP)

Fully fuzzy linear fractional programming (FFLFPP)
A general FFLFPP is given by " In matrix form, the general FFLFPP is expressed as

Fully Fuzzy Multi-Objective Linear Fractional Programming Problem (FFMOLFPP)
A general FFMOLFPP is given by Based on Guzel's (N.Guzel and M.Sivri, 2020) procedure, the FFMOLFPP (2) is transformed into an equivalent non-fractional multi objective FFLPP as Also the FFMOLFPP (2) is transformed into an equivalent non-fractional single objective FFLPP as Fuzzy solution of fully fuzzy multi-objective linear fractional programming problems whereỹ * i , z * i are maximum point and the value of each objective function, that isz * Theorem 3.2. Letỹ * be a fuzzy pareto optimal solution of P I if and only ifỹ * is also a fuzzy pareto optimal solution of P II .
Assume thatỹ * is a fuzzy pareto optimal solution of P I . Suppose thatỹ * is not a fuzzy pareto optimal solution to P II . Then by definition, there exists añ y ∈ S such that F i (ỹ) F i (ỹ * ) for all i = 1, 2, ..., k and F i (ỹ) F i (ỹ * ) for at least one j = 1, 2, ..., k.
This contradicts the fact thatỹ * is a fuzzy pareto optimal solution of P I . Henceỹ * is also a fuzzy pareto optimal solution of P II . Conversely, letỹ * is a fuzzy pareto optimal solution of P II . Suppose thatỹ * is not a fuzzy pareto optimal for P I . Then there exists anỹ ∈ S such that G i (ỹ) G i (ỹ * ) for all i = 1, 2, ..., k and G i (ỹ) G i (ỹ * ) for at least one j = 1, 2, ..., k.
Fuzzy solution of fully fuzzy multi-objective linear fractional programming problems Proof: Assume thatỹ * is a fuzzy optimal solution of P III . Suppose thatỹ * is not a fuzzy pareto optimal solution of P II . Then there exists anỹ ∈ S such that F i (ỹ) F i (ỹ * ) for all i and F i (ỹ) F i (ỹ * ) for at least one j = 1, 2, ..., k.
By summing over k, we get for at least one j = 1, 2, ..., k, which contradicts our assumption. Henceỹ * is also a fuzzy pareto optimal solution of P II . " Theorem 3.4. Ifỹ * is a fuzzy optimal solution of P III , thenỹ * is also a fuzzy pareto optimal solution of P I .
Proof: Letỹ * be a fuzzy optimal solution of P III . Suppose thatỹ * is not a fuzzy pareto optimal solution of P I . Then there exists aỹ ∈ S such that G i (ỹ) G i (ỹ * ) for all i = 1, 2, ..., k and G i (ỹ) G i (ỹ * ) for at least one j = 1, 2, ..., k.
which is a contradiction to our assumption. There foreỹ * is also a fuzzy pareto optimal solution of P I .
Fuzzy solution of fully fuzzy multi-objective linear fractional programming problems

Algorithm
Step 1: Given a FFMOLFPP with n objective functions.
Step 2: Consider each FFLFPP separately subject to the same set of constraints of the FFMOLFPP.
Step 4: Solve each FFLPP separately by expressing all its fuzzy decision parameters in their parametric forms using fuzzy version of Simplex algorithm.
Step 5: Obtain the fuzzy optimal value of the objective function of each FFLFPP from the optimal solution of the respective FFLPP.
Step 6: Reduce the given FFMOLFPP with n objective functions to an equivalent single objective FFLPP using the optimal values obtaine in Step 5.
Step 7: Using the Simplex approach, solve the single-objective FFLFPP obtained in step 6.
Step 8: Calculate the fuzzy Pareto optimal solution for the given FFMOLFPP using the fuzzy optimal solution for the single objective FFLFPP.

Numerical Example
We consider a FFMOLFPP involving triangular fuzzy numbers.  . By applying the proposed algorithm, we have two FFLFPP. Each FFLFPP is solved separately as follows.
Problem I: Consider the first FFLFPP subject to the constraints of the given FFMOLFPP (5).
subject to2ỹ 1 +2ỹ 2 1 40 The parametric form of this problem is Fuzzy solution of fully fuzzy multi-objective linear fractional programming problems Applying the fuzzy version of simplex method, the fuzzy optimal solution of the FFLPP (7)  problem II: Consider the second FFLFPP subject to the constraints of the given FFMOLFPP (5).
For different values of h ∈ [0, 1], the proposed method gives various fuzzy Pareto optimal solutions for the given FFMOLFPP.

Concluding Remarks
We have proposed a new method for tackling fully fuzzy multi-objective linear fractional programming problems with triangular fuzzy numbers. The proposed method enables the decision maker to solve a fully fuzzy multi-objective linear fractional programming problem (FFMOLFPP) without having to convert a crisp version. It also allows the decision maker to select his or her preferred solution. It can be seen that the proposed method produces less ambiguous results when compared to other existing methods.