A Novel Method for Solving the Fully Fuzzy Bilevel Linear Programming Problem

We address a fully fuzzy bilevel linear programming problem in which all the coefficients and variables of both objective functions and constraints are expressed as fuzzy numbers. This paper is to develop a new method to deal with the fully fuzzy bilevel linear programming problem by applying interval programming method. To this end, we first discretize membership grade of fuzzy coefficients and fuzzy decision variables of the problem into a finite number of α-level sets. By using α-level sets of fuzzy numbers, the fully fuzzy bilevel linear programming problem is transformed into an interval bilevel linear programming problem for each α-level set. The main idea to solve the obtained interval bilevel linear programming problem is to convert the problem into two deterministic subproblems which correspond to the lower and upper bounds of the upper level objective function. Based on the Kth-best algorithm, the two subproblems can be solved sequentially. Based on a series of α-level sets, we introduce a linear piecewise trapezoidal fuzzy number to approximate the optimal value of the upper level objective function of the fully fuzzy bilevel linear programming problem. Finally, a numerical example is provided to demonstrate the feasibility of the proposed approach

A compromise-based particle swarm optimization algorithm for solving Bi-level programming problems with fuzzy parameters

© 2015 IEEE. Bi-level programming has arisen to handle decentralized decision-making problems that feature interactive decision entities distributed throughout a bi-level hierarchy. Fuzzy parameters often appear in such a problem in applications and this is called a fuzzy bi-level programming problem. Since the existing approaches lack universality in solving such problems, this study aims to develop a particle swarm optimization (PSO) algorithm to solve fuzzy bi-level programming problems in the linear and nonlinear versions. In this paper, we first present a general fuzzy bi-level programming problem and discuss related theoretical properties based on a fuzzy number ranking method commonly used. A PSO algorithm is then developed to solve the fuzzy bi-level programming problem based on different compromised selections by decision entities on the feasible degree for constraint conditions under fuzziness. Lastly, an illustrative numerical example and two benchmark examples are adopted to state the effectiveness of the compromise-based PSO algorithm

A two-stage stochastic programming with recourse model for determining robust planting plans in horticulture

A two-stage stochastic programming with recourse model for the problem of determining optimal planting plans for a vegetable crop is presented in this paper. Uncertainty caused by factors such as weather on yields is a major influence on many systems arising in horticulture. Traditional linear programming models are generally unsatisfactory in dealing with the uncertainty and produce solutions that are considered to involve an unacceptable level of risk. The first stage of the model relates to finding a planting plan which is common to all scenarios and the second stage is concerned with deriving a harvesting schedule for each scenario. Solutions are obtained for a range of risk aversion factors that not only result in greater expected profit compared to the corresponding deterministic model, but also are more robust

Robotic swarm control from spatio-temporal specifications

In this paper, we study the problem of controlling a two-dimensional robotic swarm with the purpose of achieving high level and complex spatio-temporal patterns. We use a rich spatio-temporal logic that is capable of describing a wide range of time varying and complex spatial configurations, and develop a method to encode such formal specifications as a set of mixed integer linear constraints, which are incorporated into a mixed integer linear programming problem. We plan trajectories for each individual robot such that the whole swarm satisfies the spatio-temporal requirements, while optimizing total robot movement and/or a metric that shows how strongly the swarm trajectory resembles given spatio-temporal behaviors. An illustrative case study is included.This work was partially supported by the National Science Foundation under grants NRI-1426907 and CMMI-1400167. (NRI-1426907 - National Science Foundation; CMMI-1400167 - National Science Foundation

The Weighted Independent Domination Problem: ILP Model and Algorithmic Approaches

This work deals with the so-called weighted independent domination problem, which is an $NP$-hard combinatorial optimization problem in graphs. In contrast to previous work, this paper considers the problem from a non-theoretical perspective. The first contribution consists in the development of three integer linear programming models. Second, two greedy heuristics are proposed. Finally, the last contribution is a population-based iterated greedy metaheuristic which is applied in two different ways: (1) the metaheuristic is applied directly to each problem instance, and (2) the metaheuristic is applied at each iteration of a higher-level framework---known as construct, merge, solve \& adapt---to sub-instances of the tackled problem instances. The results of the considered algorithmic approaches show that integer linear programming approaches can only compete with the developed metaheuristics in the context of graphs with up to 100 nodes. When larger graphs are concerned, the application of the populated-based iterated greedy algorithm within the higher-level framework works generally best. The experimental evaluation considers graphs of different types, sizes, densities, and ways of generating the node and edge weights

Linear Programming with Random Requirements

Linear programming was first developed by George B. Dantzig, Marshall Wood, and associates of the U.S. Air Force, in 1947. At that time, the Air Force organized a research group under the title of project SCOOP (Scientific Computation of Optimum Programs). This project contributed to the developing of a general interindustry model based on the Leontief input-output model, the Air Force programming and budgeting problem, and the problems which involved the relationship between two-person zero sum games and linear programming. The result was the formal development and application of the linear programming model. This project also developed the simplex computational method for finding the optimum feasible program. Early applications of linear programming were made in the military, in economics, and in the theory of games. During the last decade, however, linear programming applications have been extended to such other fields as management, engineering, and agriculture. As the application of linear programming has extended to many other fields, Dantzig (1955), Tinter (1955), Beale (1955), Madansky (1960), and others have been responsible for the formulation and development of stochastic linear programming. The stochastic linear programming problem occurs when some of the coefficients, in the objective function and/or in the constraint system of the linear programming model, are subject to random variation. In the literature, several methods are indicated for formulating the linear programming problem with random requirements to arrive at a solution. The intention of this study is to review some of these methods, and to compare one wit another in terms of the optimum value of the objective function which results from each method. There are three methods that will be considered. The first method is to replace the random element with its expected value and solve the resulting linear programming problem (Hadley, 1964). The second method is Dantzig’s two-stage linear programming problem with a random requirement (Dantzig, 1955). Suppose the following linear programming problem is considered: Min. (or max.) C’X X ≥ 0 Subject to: AX ≤ b, Where C and X are n by 1 vectors, b an m by 1 vector, and A an n matrix, and C’ is C transpose. If vector b is random and matrix A is known, then in the first stage, a decision is made on X, the random vector b is observed, and AX is compared with b. In the second stage, inaccuracies in the first decision are compensated for by a new decision variable Y with some penalty cost F. The problem then becomes, E min. (or max.) C’X + F’Y, X ≥ 0, Y ≥ 0, Subject to : AX + BY = b, Where B is an m by 2n matrix with elements ones, minus ones, and zeroes, and Y is an 2n by 1 vector with elements yi and y-i. E denotes an expectation. In the third method, the constraints with random requirements are to satisfy a given probability level. The problem then is to find values of the decision variables which optimize the expected objective function without violating the given probability measure (Charnes and Cooper, 1962). This report surveys the literature on basic linear programming and the simplex method of solution, describes random requirements, and illustrates three methods of solution. Finally, the optimal value of the objective function of each method is compared with the others