Optimistic Stackelberg solutions to bilevel linear programming with fuzzy random variable coefficients

In this paper, we propose an alternate modified solution procedure based on fuzzy goal programming (FGP) approach for solving bi-level multiobjective linear fractional programming problems (BL-MOLFPPs) with some major modifications in the algorithm suggested by Abo-Sinna and Baky (2010). The proposed modified technique is simpler and requires less computational works than that of earlier technique. In the formulation of FGP models, each objective function (at both the levels - leader's level and follower's level) as well as control variables (of leader's level) are transformed into fuzzy goals with modified membership functions. Then FGP approach is used for achieving the highest degree of each of membership goals. To avoid decision deadlock situations and difficulties associated with bi-level programming problems, solution preferences by decision makers (at leader's and follower's levels) are not taken into account in proposed algorithm. A comparative analysis is also carried out with numerical example in order to show efficiency of proposed algorithm over earlier technique.

This article considers bilevel linear programming problems where the coefficients of the objective functions and the constraints in the problem are given as fuzzy parameters. Stackelberg problems and Stackelberg solutions under fuzziness have not been fully discussed so far. New Stackelberg solution concepts under fuzziness, called possibilistic Stackelberg solutions , are proposed by incorporating possibility theory into Stackelberg solutions in non-fuzzy environments. This paper shows some theoretical properties and theorems on the formulated fuzzy bilevel programming problems. It is shown that the proposed possibilistic Stackelberg solutions are obtained by using linear or nonlinear programming techniques. The effectiveness of the proposed models is demonstrated by numerical experiments using transportation problems where the upper-level decision maker is a buyer and the lower-level one is a supplier.

Multi-parametric programming based algorithms for the global solution of bi-level mixed-integer linear and quadratic programming problems

Multi-objective linear bi-level multi-follower programming problem (MOBLMFPP) is a special case of two level hierarchical programming problems in which the second level hierarchy includes multiple followers. This paper presents an alternate method based on fuzzy goal programming approach for the solution of multi-objective linear bi-level multi-follower programming (MOLBMFP) problem in which there is no sharing of information among followers. In the proposed FGP model formulation, each of objective functions of each level (leader and follower's level of MOLBMFPP) as well as decision variables at each level are characterised into fuzzy goals. Suitable linear membership functions are defined for each objective function and decision variables. Then minimising the sum of the negative deviational variables of both levels, the highest membership value of each of fuzzy goals is obtained. Existence of compromise optimal solution for MOLBMFPP is established in context of proposed approach. A numerical example is illustrated in support of proposed methodology. A comparative analysis is also carried out in order to show efficiency of proposed method over earlier method for the solution of MOLBMFPP.

This paper considers bilevel linear programming with fuzzy random variables. Through the introduction of a novel optimisation model that is to minimise the absolute deviation of degree of possibility and necessity that each fuzzy goal is fulfilled, the original bilevel programming problems with fuzzy random variables can be transformed into deterministic bilevel fractional programming problems. It is shown that the extended Stackelberg solutions can be obtained by solving the transformed bilevel linear programming problem through the combined use of the variable transformation method and the Kth best algorithm. A numerical example of bilevel purchase problem for food retailing is provided to illustrate the usefulness of the proposed methods.

A global optimization algorithm for solving the bi-level linear fractional programming problem

This paper represents the bi-level linear programming problem in an imprecise and uncertain mixed environment. The aim of this paper is to introduce coefficients of leader and the follower’s objective function and the constraints as the fuzzy random variable. To determine the optimal value of the leader and follower’s objectives a new methodology is developed for bi-level linear programming model in the presence of fuzzy random variable. A numerical example is solved to demonstrate the methodology.

This paper investigates a kind of bilevel programming with fuzzy random variable coefficients in both objective functions and the right hand side of constraints. On the basis of the notion of Er-expected value of fuzzy random variable, the upper and lower level objective functions can be replaced with their corresponding Er-expected values. In terms of probability over defuzzified operator, fuzzy stochastic constraints can be converted into the equivalent forms. Based on these, the fuzzy random bilevel programming problem can be transformed into its deterministic one. Then we suggest differential evolution algorithm to solve the final crisp problem. Finally, a numerical example is given to illustrate the proposed method.

This paper presents an efficient method for solving a multi objective bilevel programming problem with multiple number of objective functions in both first level as well as in second level. The uncertain parameters are present in second level in form of fuzzy random variables. In this case the fuzzy random variables are assumed to be fuzzy normal random variables. First the fuzziness is removed by using alpha cut technique and randomness is removed by chance constrained method. Then multi objective bilevel programming problem is transformed into a single objective non-linear mathematical model by using fuzzy programming technique. This non-linear mathematical model is solved by existing methodology or software. A numerical example is presented to illustrate the efficiency and feasibility of the proposed method.

Multilevel optimization problems often arise in various applications (in economics, ecology, power engineering and other areas) when modeling complex systems with a hierarchical structure associated with independent actions of subsystems. The difficulty of analyzing such complex systems requires first of all the study of bilevel models, the management of which would be an integral part of the analysis of more complex systems. In solving bilevel programming problems, an important role is played by the property of partial calmness, the presence of which allows us to reduce the bilevel problem to the classical nonlinear programming problem with a nonsmooth objective function. It is known that linear bilevel programming problems are partially stable. The proof of this property for more complex problems meets difficulties. In particular, our article shows the inaccuracy of some results in this area. The goal of the paper is to obtain some new results in the partial calmness of bilevel programming. In particular, new sufficient conditions for bilevel problems are proved. The results are obtained on the base of Lipschitz-like properties for multivalued mappings. In the paper we propose new sufficient conditions for partial calmness which are based on some modification of the known constraint qualification RCPLD which have been proposed by the researches Andreani, Haeser, Schuverdt and Silva.

Structural reliability analysis with fuzzy random variables using error principle

Raw-materials base (hereinafter RMB) is one of the largest industries for financial investments in Russia. There are various mathematical descriptions for the development of regions with resource-based economy. Earlier in [1] the researchers considered the model based on bilevel integer stochastic programming problems with Boolean variables. This paper proposes a new approach to public-private partnership modelling, including a bilevel linear stochastic programming problem. This model assumes that budget constraints of the state and the investor can vary in a random manner with a specific probability distribution. We put forward two methods to solve this problem: problem’s reduction to the deterministic bilevel one and formulation of deterministic problems sequence with help of Monte Carlo methods. In order to solve deterministic problems of integer programming, we suggest two approaches: direct enumeration and heuristic “Game” approach. The numerical experiments for proposed algorithms validation are conducted on the basis of actual data of Zabaykalsky Krai development. Multiple input parameters of the model vary in these experiments. Finally, we present a brief analysis of the obtained solutions to the stochastic linear programming problems with Boolean variables.

The bilevel programming problem (BLPP) is a sequence of two optimization problems where the constraint region of the upper level problem is determined implicitly by the solution set to the lower level problem. To obtain optimality conditions, we reformulate BLPP as a single level mathematical programming problem (SLPP) which involves the value function of the lower level problem. For this mathematical programming problem, it is shown that in general the usual constraint qualifications do not hold and the right constraint qualification is the calmness condition. It is also shown that the linear bilevel programming problem and the minmax problem satisfy the calmness condition automatically. A sufficient condition for the calmness for the bilevel programming problem with quadratic lower level problem and nondegenerate linear complementar¬ity lower level problem are given. First order necessary optimality condition are given using nonsmooth analysis. Second order sufficient optimality conditions are also give...

A new evaluation of mean value for fuzzy numbers and its application to American put option under uncertainty

In this paper, we propose a linear programming based interactive method for multiobjective linear programming problems, in which fuzzy coefficients and random variable coefficients are involved in the objective functions simultaneously. In the proposed method, it is assumed that the decision maker has a fuzzy goal for each objective function, and such a fuzzy goal can be quantified by eliciting the membership function. Through the possibility measure and a fractile optimization model, the original problem is transformed to the well-defined multiobjective programming problem. Then, a generalized Pareto optimal concept is defined, and a linear programming based interactive algorithm is proposed to obtain a satisfactory solution from among a generalized Pareto optimal solution set.