Analyzing Decomposition Procedures in LP and Unraveling for the Two Person Zero Sum Game and Transportation Problems

2020 ◽  
Vol 3 (1) ◽  
pp. 21-41
Author(s):  
Haridas Kumar Das ◽  
Abir Sutra Dhar
Keyword(s):  
Transportation Problem ◽  
Benders Decomposition ◽  
Decomposition Methods ◽  
Decomposition Algorithm ◽  
Linear Programs ◽  
Transportation Problems ◽  
Unbounded Solutions ◽  
Mathematical Programs ◽  
Iteration Number ◽  
Zero Sum

This paper studies some decomposition methods, including Dantzig-Wolfe decomposition (DWD), decomposition-based pricing (DBP), Benders decomposition (BD), and a recently proposed improved decomposition (ID) method for solving linear programs (LPs). The authors then develop a new decomposition algorithm for solving LPs in a general form, allowing authors to combine the concept of Benders decomposition and decomposition-based pricing methods. The authors generate conditions for solving problems that have either infeasible or unbounded solutions. As an illustration, the authors give the corresponding models and numerical results for two standard mathematical programs: the two-person zero-sum game and the transportation problem. The authors compare several procedures and identify which one produces the best solution by giving the authors the smallest iteration number. This study reveals that the algorithm along with Benders decomposition produce the most efficient computational solutions of LPs.

scholarly journals A Heuristic Algorithm for solving Integer Linear Programming Problem and Unveiling the Applications

2020 ◽  
Vol 44 (1) ◽  
pp. 13-31
Author(s):  
Abir Sutra Dhar ◽  
HK Das
Keyword(s):  
Heuristic Algorithm ◽  
Benders Decomposition ◽  
Facility Location Problem ◽  
Decomposition Methods ◽  
Unbounded Solutions ◽  
Mathematical Programs ◽  
Integer Linear Programming Problem ◽  
Integer Programming Problems ◽  
Increasing Demand ◽  
Academy Of Sciences

There is a growing need for integer solutions in industries, production units, etc. Specifically, there is an increasing demand to develop precise methods for solving integer-programming problems (IPPs). In this paper, we propose a new algorithm for solving IPPs in a general form by combining two decomposition techniques: Benders decomposition (BD) and decomposition-based pricing methods (DBP). Moreover, we generate some conditions for solving problems having either infeasible or unbounded solutions. In addition, we present an application and evaluation of a solution method for solving IPPs, while also giving a brief description of the different classical decomposition methods, namely the Dantzig-Wolfe decomposition (DWD), decomposition-based pricing (DBP), Benders decomposition (BD), and recently proposed improved decomposition (ID) methods for solving IPPs.We also discuss the use of the decomposition methods for solving IPPS to develop a heuristic algorithm, describe the limitations of the classical algorithms, and present extensions enabling its application to a broader range of problems. To illustrate the decomposition procedures, we will provide corresponding models and numerical results for two standard mathematical programs: the Fixed Charge Problem (FCP) and the Facility Location Problem (FLP). Our findings from this study suggest that our algorithm produces the most efficient computational solutions of IPPs. Journal of Bangladesh Academy of Sciences, Vol. 44, No. 1, 13-31, 2020

The Star Degree Centrality Problem: A Decomposition Approach

2021 ◽  
Author(s):  
Mustafa C. Camur ◽  
Thomas Sharkey ◽  
Chrysafis Vogiatzis
Keyword(s):  
Integer Programming ◽  
Large Scale ◽  
Benders Decomposition ◽  
Open Neighborhood ◽  
Decomposition Algorithm ◽  
Solution Time ◽  
Set Cover ◽  
Degree Centrality ◽  
Decomposition Approach ◽  
Acceleration Techniques

We consider the problem of identifying the induced star with the largest cardinality open neighborhood in a graph. This problem, also known as the star degree centrality (SDC) problem, is shown to be [Formula: see text]-complete. In this work, we first propose a new integer programming (IP) formulation, which has a smaller number of constraints and nonzero coefficients in them than the existing formulation in the literature. We present classes of networks in which the problem is solvable in polynomial time and offer a new proof of [Formula: see text]-completeness that shows the problem remains [Formula: see text]-complete for both bipartite and split graphs. In addition, we propose a decomposition framework that is suitable for both the existing and our formulations. We implement several acceleration techniques in this framework, motivated by techniques used in Benders decomposition. We test our approaches on networks generated based on the Barabási–Albert, Erdös–Rényi, and Watts–Strogatz models. Our decomposition approach outperforms solving the IP formulations in most of the instances in terms of both solution time and quality; this is especially true for larger and denser graphs. We then test the decomposition algorithm on large-scale protein–protein interaction networks, for which SDC is shown to be an important centrality metric. Summary of Contribution: In this study, we first introduce a new integer programming (NIP) formulation for the star degree centrality (SDC) problem in which the goal is to identify the induced star with the largest open neighborhood. We then show that, although the SDC can be efficiently solved in tree graphs, it remains [Formula: see text]-complete in both split and bipartite graphs via a reduction performed from the set cover problem. In addition, we implement a decomposition algorithm motivated by Benders decomposition together with several acceleration techniques to both the NIP formulation and the existing formulation in the literature. Our experimental results indicate that the decomposition implementation on the NIP is the best solution method in terms of both solution time and quality.

Methods for Solving Fully Fuzzy Transportation Problems Based on Classical Transportation Methods

2013 ◽  
pp. 328-347 ◽  
Author(s):  
Amit Kumar ◽  
Amarpreet Kaur
Keyword(s):  
Transportation Problem ◽  
Fuzzy Numbers ◽  
Real Life ◽  
Optimal Solution ◽  
Fuzzy Linear Programming ◽  
Programming Formulation ◽  
Transportation Problems ◽  
Fuzzy Optimal Solution ◽  
Linear Programming Formulation ◽  
Fuzzy Transportation Problem

There are several methods, in literature, for finding the fuzzy optimal solution of fully fuzzy transportation problems (transportation problems in which all the parameters are represented by fuzzy numbers). In this paper, the shortcomings of some existing methods are pointed out and to overcome these shortcomings, two new methods (based on fuzzy linear programming formulation and classical transportation methods) are proposed to find the fuzzy optimal solution of unbalanced fuzzy transportation problems by representing all the parameters as trapezoidal fuzzy numbers. The advantages of the proposed methods over existing methods are also discussed. To illustrate the proposed methods a fuzzy transportation problem (FTP) is solved by using the proposed methods and the obtained results are discussed. The proposed methods are easy to understand and to apply for finding the fuzzy optimal solution of fuzzy transportation problems occurring in real life situations.

A Novel Ranking Approach to Solving Fully LR-Intuitionistic Fuzzy Transportation Problems

2018 ◽  
Vol 15 (01) ◽  
pp. 95-112 ◽  
Author(s):  
Abhishekh ◽  
A. K. Nishad
Keyword(s):  
Transportation Problem ◽  
Fuzzy Numbers ◽  
Optimal Solution ◽  
Solution Procedure ◽  
Ranking Functions ◽  
Transportation Problems ◽  
Intuitionistic Fuzzy Numbers ◽  
Intuitionistic Fuzzy ◽  
Fuzzy Environment ◽  
Fuzzy Transportation Problem

To the extent of our knowledge, there is no method in fuzzy environment to solving the fully LR-intuitionistic fuzzy transportation problems (LR-IFTPs) in which all the parameters are represented by LR-intuitionistic fuzzy numbers (LR-IFNs). In this paper, a novel ranking function is proposed to finding an optimal solution of fully LR-intuitionistic fuzzy transportation problem by using the distance minimizer of two LR-IFNs. It is shown that the proposed ranking method for LR-intuitionistic fuzzy numbers satisfies the general axioms of ranking functions. Further, we have applied ranking approach to solve an LR-intuitionistic fuzzy transportation problem in which all the parameters (supply, cost and demand) are transformed into LR-intuitionistic fuzzy numbers. The proposed method is illustrated with a numerical example to show the solution procedure and to demonstrate the efficiency of the proposed method by comparison with some existing ranking methods available in the literature.

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