## 2 Literature Review  
### Bi-level Programming  
BLP methods are established for solving decentralized decision-making problems at two-levels, the first level is known as leaders or FLDM, and the second one is followers or SLDM. BLP performs the hierarchical structure of decision-making problem, where FLDM solved their problem in separation and then after, delivers the values of their controlling factors. SLDM tries to find their solutions under the guidelines considered by the FLDM in the form of their predominant factors. During the last decade, many researchers studied BLPP deeply, and many methodologies have been developed to solve BLPP. First, [1] used the geometric characterizations and algorithms to get the solution of a particular case of the two-level linear programming problem. Reference [2] proposed a two-person bi-level multi-objective decision-making model and suggested an interactive algorithm to solve the decision-making problem. In this algorithm, they simplified the problem by converting it into separate multi-objective decision-making problems at each level, after that they achieved an optimal solution. In addition, [3] considered the transportation planning and assignment problem as a decentralized two-level integer programming problem and applied an interactive fuzzy programming approach to get a satisfactory solution to the problem. Reference [4] considered bi-level multi-objective programming problems and used the Interactive balance space approach to get an optimal solution.  
Some notable work includes [5] which considered a BLPP under integer restrictions and used the concept of tolerance membership function and the branch and bound technique to develop a fuzzy decision model and calculate an optimal solution. Reference [6] presented a fuzzy BLPP Reference [7] considered multi-objective bi-level decision problems with fuzzy demands and used branch-and-bound approximation algorithm to solve it. Reference [8] considered a decentralized bi-level multi-objective programming problem and proposed a fuzzy goal programming approach to solve it. In this algorithm, they defined the membership functions for all fuzzy goals and objectives at each level, and this algorithm was also extended for solving bi-level multi-objective linear fractional programming problems. Reference [9] proposed \(\alpha\)-cut and goal programming technique for solving fuzzy linear bi-level multi-objective programming problem. Reference [10] suggested a new particle swarm optimization technique with chaos searching technique to get optimum solutions of the non-linear BLPP and KT condition was used to convert a BLPP into a single objective problem. Reference [11] developed a hybrid model of TOPSIS and fuzzy goal programming algorithm to solve the linear fractional bi-level multi-objective decision-making problem. Reference [12] attempted to solve a bi-level integer multi-objective fractional programming problem with an interactive approach. Reference [13] formulated a chance-constrained algorithm to solve the fractional integer bi-levelproblems. In recent years, [14] considered bi-level multi-objective fractional programming problems and used the Jacobian matrix technique and fuzzy goal programming algorithm to get a satisfactory optimum solution. Reference [15] considered BLPP and suggested an approach for solving the problem by eliminating the terminated constraints, which cause the elimination of cycling to get the solution with infinite steps. Reference [16] developed a hybrid technique with the help of machine learning and optimization concept for solving BLPP.