Nondominated Extreme Points Research Articles

Solving a multi-objective solid transportation problem: a comparative study of alternative methods for decision-making

The transportation problem in operations research aims to minimize costs by optimizing the allocation of goods from multiple sources to destinations, considering supply, demand, and transportation constraints. This paper applies the multi-dimensional solid transportation problem approach to a private sector company in Egypt, aiming to determine the ideal allocation of their truck fleet.In order to provide decision-makers with a comprehensive set of options to reduce fuel consumption costs during transportation or minimize total transportation time, a multi-objective approach is employed. The study explores the best compromise solution by leveraging three multi-objective approaches: the Zimmermann Programming Technique, Global Criteria Method, and Minimum Distance Method. Optimal solutions are derived for time and fuel consumption objectives, offering decision-makers a broad range to make informed decisions for the company and the flexibility to adapt them as needed.Lingo codes are developed to facilitate the identification of the best compromise solution using different methods. Furthermore, non-dominated extreme points are established based on the weights assigned to the different objectives. This approach expands the potential ranges for enhancing the transfer problem, yielding more comprehensive solutions.This research contributes to the field by addressing the transportation problem practically and applying a multi-objective approach to support decision-making. The findings provide valuable insights for optimizing the distribution of the truck fleet, reducing fuel consumption costs, and improving overall transportation efficiency.

Using Dynamic Programming Technique for Solving a Class of Bi-Criteria Multistage Transportation Problem

This paper presents a solution algorithm to solve one class of bicriteria multistage transportation problem. This class represents bi-criteria multistage transportation problem without any restrictions on the intermediate stages. A way to find the non-dominated extreme points in the criteria space is developed. This method involves a parametric search in the criteria space. The presented algorithm is based also on fruitful application of the methods for treating transportation problems and the dynamic programming technique to obtain the shortest route for transportation networks and methods for solving bicriteria linear programming. The solution algorithm as well as an illustrative example are also included in this paper.

Using dynamic programming technique for solving a class of bi-criteria multistage transportation problem

This paper presents a solution algorithm to solve one class of bicriteria multistage transportation problem. This class represents bi-criteria multistage transportation problem without any restrictions on the intermediate stages. A way to find the non-dominated extreme points in the criteria space is developed. This method involves a parametric search in the criteria space. The presented algorithm is based also on fruitful application of the methods for treating transportation problems and the dynamic programming technique to obtain the shortest route for transportation networks and methods for solving bicriteria linear programming. The solution algorithm as well as an illustrative example are also included in this paper.

A bi-objective column generation algorithm for the multi-commodity minimum cost flow problem

We present a column generation algorithm for solving the bi-objective multi-commodity minimum cost flow problem. This method is based on the bi-objective simplex method and Dantzig–Wolfe decomposition. The method is initialised by optimising the problem with respect to the first objective, a single objective multi-commodity flow problem, which is solved using Dantzig–Wolfe decomposition. Then, similar to the bi-objective simplex method, our algorithm iteratively moves from one non-dominated extreme point to the next one by finding entering variables with the maximum ratio of improvement of the second objective over deterioration of the first objective. Our method reformulates the problem into a bi-objective master problem over a set of capacity constraints and several single objective linear fractional sub-problems each over a set of network flow conservation constraints. The master problem iteratively updates cost coefficients for the fractional sub-problems. Based on these cost coefficients an optimal solution of each sub-problem is obtained. The solution with the best ratio objective value out of all sub-problems represents the entering variable for the master basis. The algorithm terminates when there is no entering variable which can improve the second objective by deteriorating the first objective. This implies that all non-dominated extreme points of the original problem are obtained. We report on the performance of the algorithm on several directed bi-objective network instances with different characteristics and different numbers of commodities.

A Recursive Algorithm for Finding All Nondominated Extreme Points in the Outcome Set of a Multiobjective Integer Programme

In this paper, we present two versions of an algorithm for the computation of all nondominated extreme points in the outcome set of a multiobjective integer programme. We define adjacency of these points based on weight space decomposition. Thus, our algorithms generalise the well-known dichotomic scheme to compute the set of nondominated extreme points in the outcome set of a biobjective programme. Both algorithms are illustrated with and numerically tested on instances of the assignment and knapsack problems with three objectives.

A two-phase algorithm for the biobjective integer minimum cost flow problem

We present an algorithm to compute a complete set of efficient solutions for the biobjective integer minimum cost flow problem. We use the two phase method, with a parametric network simplex algorithm in phase 1 to compute all non-dominated extreme points. In phase 2, the remaining non-dominated points (non-extreme supported and non-supported) are computed using a k best flow algorithm on single-objective weighted sum problems. We implement the algorithm and report run-times on problem instances generated with a modified version of the NETGEN generator and also for networks with a grid structure.

Detecting Non-Dominated Extreme Points for Multiple Objective Linear Programming

Efficient extreme points in decision space of multiple objective linear programming (MOLP) may not map to non dominated extreme points in objective space under the linear mapping, condition that efficient extreme points have a non dominated extreme is given, the important of this study is that the decision-Maker may depends on extreme points of the set of the objective space than that of the decision space since they have fewer extreme points.

Solving bicriteria solid transportation problem with fuzzy numbers by a genetic algorithm

The proposed Genetic Algorithms (GAs) are incorporated with problem-specific knowledge and the decision maker's degree of optimism, and the criteria space approach for bicriteria linear program conductive to find out the nondominated extreme points in the criteria space.

On degeneracy and collapsing in the construction of the set of objective values in a multiple objective linear program

In this paper an algorithm is developed to generate all nondominated extreme points and edges of the set of objective values of a multiple objective linear program. The approach uses simplex tableaux but avoids generating unnecessary extreme points or bases of extreme points. The procedure is based on, and improves, an algorithm Dauer and Liu developed for this problem. Essential to this approach is the work of Gal and Kruse on the neighborhood problem of determining all extreme points of a convex polytope that are adjacent to a given (degenerate) extreme point of the set. The algorithm will incorporate Gal's degeneracy graph approach to the neighborhood problem with Dauer's objective space analysis of multiple objective linear programs.

VMDP-program for solving optimality problems in vector criterion Markov and semi-Markov decision processes

Optimality problems in infinite horizon discrete time vector criterion Markov and semi-Markov decision processes can be expressed as standard problems of multiple objective linear programming. “VMDP” is a FORTRAN program for enumerating the objective function coefficients and constraint coefficients in these problems and finding all nondominated extreme points, i.e. all nondominated pure stationary strategies, and all nondominated edges. Using VMDP, computational experiments are presented. The characteristics of Markov and semi-Markov decision problems (the VMDP code) with those of general muitiple objective linear programming problems (the Steuer Adbase code) are compared.

Solving multiple objective linear programs in objective space

This paper presents an analysis of the objective space for a multiple objective linear program and develops a procedure for determining the nondominated extreme points and edges in objective space. Since not all extreme points (nor edges) of the constraint space necessarily map to extreme points (edges) of objective space, this technique analyzes a simpler structure than that analyzed by algorithms based on the extreme points of the constraint space.

Multicriteria analysis of water allocation in a river basin: The Tchebycheff Approach

A new interactive multiple objective methodology is applied to the seasonal water allocation problem. Linear optimization and filtering procedures are combined within a multiobjective frame‐work in order to identify nondominated solutions associated with randomly sampled criteria vectors. The procedure samples from the entire nondominated set, not just the set of nondominated extreme points, by computing the nondominated criteria vector that is closest to an ideal criteria vector according to a weighted Tchebycheff metric. Unlike many other multicriteria methods, the Tchebycheff procedure is capable of finding solutions of highest utility which may not correspond to extreme point solutions. The procedure is applied to the problem of water allocation in a river basin where a large number of conflicting objectives, including environmental objectives, compete for the available supply.

Bicriteria Transportation Problem

In a transportation problem, generally, a single criterion of minimizing the total cost is considered. But in certain practical situations two or more objectives are relevant. For example, the objectives may be minimizations of total cost, consumption of certain scarce resources such as energy, total deterioration of goods during transportation, etc. Clearly, this problem can be solved using any of the multiobjective linear programming techniques; but the computational efforts needed would be prohibitive in many cases. The computational complexity in these techniques arises from the fact that each of the methods finds the set of nondominated extreme points in the solution space where such extreme points are, generally, many. Therefore, this paper develops a method of finding the nondominated extreme points in the criteria space. Such extreme points in the criteria space would be generally less and only these are needed while choosing a nondominated solution for implementation. The method involves a parametric search in the criteria space. Although the method is developed with respect to a bicriteria transportation problem, it is applicable to any bicriteria linear program in general. The bottleneck criterion included as a third objective is particularly significant in time bound transportation schedules. A numerical example is included.

Linear Multiparametric Programming by Multicriteria Simplex Method

A recently derived Multicriteria Simplex Method [Yu, P. L., M. Zeleny. 1975. The set of all nondominated solutions in linear cases and a multicriteria simplex method. J. Math. Anal. Appl. 49 (2, February) 430–468.] is used to study some basic properties in the decomposition of parametric space. A new type of parametric space, which arises naturally in its formulation, is used. Two computational methods are discussed. The first one is an indirect algebraic method which locates the corresponding set of all nondominated extreme points. The second one is a direct geometric decomposition method which is similar to that discussed by Gal and Nedoma [Gal, T., J. Nedoma. 1972 Multiparametric linear programming. Management Sci.. 18 406–421.]. The difficulties of such a geometric method are discussed through an example.

The set of all nondominated solutions in linear cases and a multicriteria simplex method

In this note we are interested in the properties of, and methods for locating the set of all nondominated solutions of multiple linear criteria defined over a polyhedron. We first show that the set of all dominated solutions is convex and that the set of all nondominated solutions is a subset of the convex hull of the nondominated extreme points. When the domination cone is polyhedral, we derive a necessary and sufficient condition for a point to be nondominated. The condition is stronger than that of Ref. [1] and enables us to give a simple proof that the set of all nondominated extreme points indeed is connected. In order to locate the entire set of all nondominated extreme points, we derive a generalized version of simplex method—multicriteria simplex method. In addition to some useful results, a necessary and sufficient condition for an extreme point to be nondominated is derived. Examples and computer experience are also given. Finally, we focus on how to generate the entire set of all nondominated solutions through the set of all nondominated extreme points. A decomposition theorem and some necessary and sufficient conditions for a face to be nondominated are derived. We then describe a systematic way to identify the entire set of all nondominated solutions. Through examples, we show that in fact our procedure is quite efficient.