In this study, we discussed a fuzzy programming approach to bi-level linear programming problems and their application. Bi-level linear programming is characterized as mathematical programming to solve decentralized problems with two decision-makers in the hierarchal organization. They become more important for the contemporary decentralized organization where each unit seeks to optimize its own objective. In addition to this, we have considered Bi-level Linear programming (BLPP) and applied the Fuzzy Mathematical programming (FMP) approach to get the solution of the system. We have suggested the FMP method for the minimization of the objectives in terms of the linear membership functions. FMP is a supervised search procedure (supervised by the upper Decision Maker (DM)). The upper-level decision-maker provides the preferred values of decision variables under his control (to enable the lower level DM to search for his optimum in a wider feasible space) and the bounds of his objective function (to direct the lower level DM to search for his solutions in the right direction).

This paper presents a fuzzy goal programming (FGP) methodology for solving bi-level quadratic programming (BLQP) problems. In the FGP model formulation, firstly the objectives are transformed into fuzzy goals (membership functions) by means of assigning an aspiration level to each of them, and suitable membership function is defined for each objectives, and also the membership functions for vector of fuzzy goals of the decision variables controlled by decision maker at the first level are developed in the model formulation of the problem. To achieve the highest membership value of each of the fuzzy goals, we formulate the problem by minimizing the negative deviational variables and thereby obtaining the most satisfactory solution for all decision makers. A numerical example is given to demonstrate the proposed approach.

In this paper, a Multi-Choice Stochastic Bi-level programming Problem (MCSBLPP) is considered where all the parameters of constraints are followed by normal distribution. The cost coefficients of the objective functions are multi-choice types. At first, all the probabilistic constraints are transformed into deterministic constraints using stochastic programming approach. Further, a general transformation technique with the help of binary variables is used to transform the multi-choice type cost coefficients of the objective functions of Decision Makers (DMs). Then the transformed problem is considered as a deterministic multichoice bi-level programming problem. Finally, a numerical example is presented to illustrate the usefulness of the paper.

In some countries, regional authorities may attempt to rebalance the allocation of national facilities in benefit of their own region which, in turn, may cause disturbances in the central government’s decision-making proces. Regarding the hierarchical nature of these types of decisions, classical optimization models are not effective in decision-making and the use of multi-level programming can increase the efficiency of planning. Our paper aims to address the issue of a bi-level programming model to conduct the location analysis of emergency warehouses. A three-echelon relief supply chain is considered in which the relief network involves national and regional warehouses and demand cities. The upper-level model decides on the location of national warehouses, allocating them to regional warehouses. The lower-level model determines the location of regional warehouses and allocates them to demand points. The structure of both levels is based on the median location-allocation problem. Three solution approaches are presented based on the full enumeration and two types of nested evolutionarymethods (genetic and heuristic local search algorithms). For the model to be used in Iran, the efficiency of algorithms is analyzed for two sizes of problems. The obtained results show the proper functioning of the solution approaches.

Protecting important nodes in a network against natural disasters, security threats, attacks, and so on is one of the main goals of network planners. In this paper, a new model is presented for protecting an important node (NMPN) in a typical network based on a defensive location problem where the threatening agent (t-agent) can reinforce its power at some nodes. The NMPN is a bi-level programming problem. At the upper level, the planner agent (p-agent) try to , nd the best lo-cations for protecting resources in order to protect the important node. The lower level problem is represented as the shortest path problem in the network in which the edges are weighted with positive values and sometimes negative values. Thus, the Bellman-Ford algorithm is applied to solve the lower level problem. The NMPN is an NP-hard problem. In this work, the genetic, ant colony optimization, binary arti, cial bee colony with di, erential evolution, arti, cial bee colony algorithms, and a modi, ed tabu search (MTS) algorithm are used to solve it. A test problem is randomly generated to investigate the performance of the metaheuristic algorithms in this paper. Parameters of the metaheuris-tic algorithms are tuned by the Taguchi method for solving the test problem. Also, the ANOVA test and Tukey's test are used to compare the performance of the metaheuristic algorithms. The best results are obtained by the MTS algorithm.