Linear bilevel programming with interval coefficients

This paper considers a particular case of linear bilevel programming problems with one leader and multiple followers. In this model, the followers are independent, meaning that the objective function and the set of constraints of each follower only include the leader's variables and his own variables. We prove that this problem can be reformulated into a linear bilevel problem with one leader and one follower by defining an adequate second level objective function and constraint region. In the second part of the paper we show that the results on the optimality of the linear bilevel problem with multiple independent followers presented in Shi et al. [The kth-best approach for linear bilevel multi-follower programming, J. Global Optim. 33, 563---578 (2005)] are based on a misconstruction of the inducible region.

Linear bilevel programs with multiple objectives at the upper level

In this paper, a linear fractional programming problem with interval coefficients in both objective function and constraints is solved. The uncertain objective function is transformed into two deterministic objective functions. The two objective functions are converted into a single objective problem through the linear combination method. Through a modified possibility degree, the uncertain inequality and equality constraints are transformed into deterministic inequality constraints. The existing results are concerning the qualitative and quantitative analysis of basic notions in parametric linear fractional programming problem. These notions are the set of feasible parameters, the solvability set and the stability set of the first kind. A numerical example is investigated to demonstrate the effectiveness of the present approach.

An extended branch and bound algorithm for linear bilevel programming

A penalty function method based on Kuhn–Tucker condition for solving linear bilevel programming

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

In this paper, we study general linear programs in which right hand sides are interval numbers. This model is relevant when uncertain and inaccurate factors make difficult the assignment of a single value to each right hand side. When objective function coefficients are interval numbers in a linear program, classical criteria coming from decision theory (like the worst case criterion) are usually applied to determine robust solutions. When the set of feasible solutions is uncertain, we identify a class of linear programs for which these classical approaches are no longer relevant. However, it is possible to compute the worst optimum solution. We study the complexity of this optimization problem when each right hand side is an interval number. Then, we exhibit some duality relationships between the worst optimum solution problem and the best optimum solution to the dual problem.

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.

A nonlinear interval number programming method for uncertain optimization problems

In recent years, many researchers in the field of fuzzy mathematics and operations research in the world are interested to solve the interval type-2 fuzzy linear programming (IT2FLP) problems. In IT2FLP problems all parameters, including coefficients in objective function, coefficients in the constraints and right hand sides of the constraints are interval type-2 fuzzy numbers (IT2FN). Hence, the IT2FLP model is one of the most important and the most appropriate models to interpret and determine the optimal decision. In this paper, a new ranking function method is proposed for solving these problems. Moreover, an example is used to study and describe the proposed method.

Linear program under changes in the system matrix coefficients has proved to be more complex than changes of the coefficients in objective functions and right hand sides. The most of the previous studies deals with problems where only one coefficient, a row (column), or few rows (columns) are linear functions of a parameter. This work considers a more general case, where all the coefficients are polynomial (in the particular case linear) functions of the parameter tT??R. For such problems, assuming that some non-singularity conditions hold and an optimal base matrix is known for some particular value t of the parameter, corresponding explicit optimal basic solution in the neighborhood of t is determined by solving an augmented LP problem with real system matrix coefficients. Parametric LP can be utilized for example to model the production problem where, technology, resources, costs and similar categories vary with time. .

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.

In this paper, a linear programming (LP) problem is considered where some or all of its coefficients in the objective function and/or constraints are rough intervals. In order to solve this problem, we will construct two LP problems with interval coefficients. One of these problems is an LP where all of its coefficients are upper approximations of rough intervals and the other is an LP where all of its coefficients are lower approximations of rough intervals. Via these two LPs, two newly solutions (completely and rather satisfactory) are defined. Some examples are given to demonstrate the results.

The traditional linear programming model is deterministic. The way that uncertainty is handled is to compute the range of optimality. After the optimal solution is obtained, typically by the simplex method, one considers the effect of varying each objective function coefficient, one at a time. This yields the range of optimality within which the decision variables remain constant. This sensitivity analysis is useful for helping the analyst get a sense for the problem. However, it is unrealistic because objective function coefficients tend not to stand still. They are typically profit contributions from products sold and are subject to randomly varying selling prices. In this paper, a realistic linear program is created for simultaneously randomizing the coefficients from any probability distribution. Furthermore, we present a novel approach for designing a copula of random objective function coefficients according to a specified rank correlation. The corresponding distribution of objective function values is created. This distribution is examined directly for central tendency, spread, skewness and extreme values for the purpose of risk analysis. This enables risk analysis and business analytics, emerging topics in education and preparation for the knowledge economy.

PurposeThe purpose of this paper is to propose a method for solving multi-objective linear programming (MOLP) with interval coefficients using positioned programming and interactive fuzzy programming approaches.Design/methodology/approachIn the proposed algorithm, first, lower and upper bounds of each objective function in its feasible region will be determined. Afterwards using fuzzy approach, considering a membership function for each objective function and finally using grey linear programming, the solution for this problem will be obtained.FindingsAccording to the presented example, in this paper, the proposed method is both simple in use and suitable for solving different problems. In the numerical example mentioned in this paper, the proposed method provides an acceptable solution for such problems.Practical implicationsAs in most real-world situations, the coefficients of decision models are not known and exact. In this paper, the authors consider the model of MOLP with interval data, since one of the solutions to cover uncertainty is using interval theory.Originality/valueBased on using grey theory and interactive fuzzy programming approaches, an appropriate method has been presented for solving MOLP problems with interval coefficients. The proposed method, against the complex methods, has less effort and offers acceptable solutions.