An Application Model for Linear Programming with an Evolutionary Ranking Function

One of the most important methodologies in operations research (OR) is the linear programming problem (LPP). Many real-world problems can be turned into linear programming models (LPM), making this model an essential tool for today's financial, hotel, and industrial applications, among others. Fuzzy linear programming (FLP) issues are important in fuzzy modeling because they can express uncertainty in the real world. There are several ways to tackle fuzzy linear programming problems now available. An efficient method for FLP has been proposed in this research to find the best answer. This method is simple in structure and is based on crisp linear programming. To solve the fuzzy linear programming problem (FLPP), a new ranking function (RF) with the trapezoidal fuzzy number (TFN) is devised in this study. The fuzzy quantities are de-fuzzified by applying the proposed ranking function (RF) transformation to crisp value linear programming problems (LPP) in the objective function (OF). Then the simplex method (SM) is used to determine the best solution (BS). To demonstrate our findings, we provide a numerical example (NE).


Introduction
Many approaches exist to address mathematical issues in operation research, and these methods help reduce the complexity. LP is a powerful decision-making tool. This is commonly used in real-world problems using the applied operational research approach. The theory of fuzzy sets (FS) has been used in various fields, including industrial applications, systems theory, organizational theory, and mathematical modeling. Ranking fuzzy numbers (RFN) is a critical decision-making method in a fuzzy environment. [1], the idea of a fuzzy set was originally introduced as a way of dealing with uncertainty caused by imprecision rather than randomness. [2] were the first to establish the concept of FLP. In linguistic multi-criteria decision-making situations, fuzzy number ranking is extremely important. Many researchers [3][4][5][6][7][8] applied different kinds of fuzzy linear programming problems with ranking functions. One of the important types of fuzzy numbers is trapezoidal fuzzy. It used many articles with different methods.
[9] introduced the Trapezoidal Fuzzy Number Linear Programming. [10], the ranking function methods were used to solve fuzzy fractional linear programming problems. [11], using symmetric trapezoidal fuzzy numbers to solve a type of FLP problem has been presented. [12] used LPP with some Multi-Choice Fuzzy Parameters (FP). [13] suggested A New Approach for Solving the Type-2-Fuzzy Transportation Problem. [14] studied analytical preparations' level-dependent weighted average value of discrete trapezoidal fuzzy numbers. [15] presented a new method for determining the tolerance level and aspiration level based on Zimmermans approach. In [16], a Generalized Model for Fuzzy Linear Programs with TFN was discussed in this paper.The objective of this paper is to explain a novel RF for solving the fuzzy linear programming problems by simplex method by comparing it to other ranking functions. This paper is separated into (6) sections. Section one contains the introduction. In section two, we recall the basic definitions. Section three introduces trapezoidal fuzzy numbers (TFN). In section four, we concept RF. Section five suggests the evolutionary Maleki ranking function. In section six, the proposed case study, finally, the paper arrives at a conclusion.

Basic Definitions
The background and foundations for fuzzy set theory are given in this section.  ii. Every ɕ ∈ [0, 1] must have a closed interval in ̃ .
iii. The support of ̃ must be constrained.

Definition 2.4 Fuzzy Linear Programming Problem (FLPP) [20]:
The following is the description of a fuzzy number linear programming (FLP) problem:

Evolutionary Maleki Ranking Function
In this section, we propose a new ranking function that depends on the idea of The Maleki's ranking function with a new weight ( 15 16 ) . Assume ̃= ( , , , ) is a trapezoidal fuzzy number, and use the following formula to find the ranking function of ̃.

Case Study
To apply for trapezoidal fuzzy number in linear programming problem, we have data from monthly production of the yogurt section in the abo ghreeb factory in the general company for the food products for the problem below and expressing the types of yogurt product (yogurt, diet yogurt, shenena, and diet shenena).
The objective function represents the types of yogurt products.
The constraints represent the basic materials (milk powder, skim milk powder, salt, starter, and water).
The following steps can be used to find the fuzzy optimal solution (FOS) to the chosen FLPP: Step1. The practical LPP is: Step 3. In this problem, we take two cases in the following trapezoidal fuzzy number : Step 6. To determine the optimal value (OV) of the above problem, we take transform TFN in the objective function (OF) to LPP by the Evolutionary Maleki ranking function (MRF) Step 11. Finally, the OS of CLPP is obtained. Thus, 1 = 17.857 , 2 = 166.667 , 3 = 33.333 , 4 = 0 , = 36.314060.

Conclusion
The simplex method is an important method for solving linear programming problems. This paper proposes an evolutionary ranking function for solving the FLPP by SM. The proposed method may transform the FLPP into its equivalent crisp linear fractional programming problems (CLFPP). A practical problem was used to demonstrate the efficacy of our proposed strategy.