Linear Integer Programming: Theory, Applications, Recent Developments
- Length: 200 pages
- Edition: 1
- Language: English
- Publisher: de Gruyter
- Publication Date: 2021-12-06
- ISBN-10: 3110702924
- ISBN-13: 9783110702927
- Sales Rank: #0 (See Top 100 Books)
This book presents the state-of-the-art methods in Linear Integer Programming, including some new algorithms and heuristic methods developed by the authors in recent years. Topics as Characteristic equation (CE), application of CE to bi-objective and multi-objective problems, Binary integer problems, Mixed-integer models, Knapsack models, Complexity reduction, Feasible-space reduction, Random search, Connected graph are also treated.
Without mathematics no science would survive. This especially applies to the engineering sciences which highly depend on the applications of mathematics and mathematical tools such as optimization techniques, finite element methods, differential equations, fluid dynamics, mathematical modelling, and simulation. Neither optimization in engineering, nor the performance of safety-critical system and system security; nor high assurance software architecture and design would be possible without the development of mathematical applications.
De Gruyter Series on the Applications of Mathematics in Engineering and Information Sciences (AMEIS) focusses on the latest applications of engineering and information technology that are possible only with the use of mathematical methods. By identifying the gaps in knowledge of engineering applications the AMEIS series fosters the international interchange between the sciences and keeps the reader informed about the latest developments.
Title Page Copyright Contents Acknowledgements Preface About the authors Chapter 1 Segment search approach for the general linear integer model 1.0 Introduction 1.1 The Linear Integer Programming (LIP) Model and some preliminaries 1.2 The concept of segments 1.2.1 Selection of the segment interval 1.2.2 Decreasing the objective value 1.2.3 Original variable sum limit 1.2.4 Determination of h0 value 1.3 Segment-search approach 1.3.1 General integer model 1.4 Mixed integer model 1.5 Numerical illustration 1.5.1 General linear integer model: Example 1.1 1.5.2 Solution by the proposed segment-search algorithm 1.5.3 Mixed integer model: Example 1.2 1.6 Conclusions References Chapter 2 Improved solution method for the 0-1 GAP model 2.0 Introduction 2.1 Generalized assignment problem 2.2 Relaxation process 2.3 GAP model in relaxed form 2.4 The relaxed transportation model 2.5 GAP transportation branch and bound algorithm 2.5.1 Optimality 2.5.2 Numerical illustration 2.6 Improved solution method for GAP 2.6.1 Proposed algorithm 2.6.2 Strength of proposed algorithm 2.6.3 Reconsider the same numerical example 2.7 Conclusions References Chapter 3 A search for an optimal integer solution over the integer polyhedron – Two iterative approaches 3.0 Introduction 3.1 Background information 3.1.1 Geometry of integer-points in a convex space defined by the linear constraints 3.1.2 Optimality of the solution 3.2 Young’s primal integer programming approach (1965) 3.2.1 Numerical Illustration of Young’s primal approach 3.3 The Integer Polyhedron Search Algorithm (IPSA) 3.3.1 Integer Polyhedron Search Algorithm (IPSA) by Munapo, Kumar and Khan (2010) 3.4 More numerical illustrations of IPSA Example 3.4.1 Example 3.4.2 3.5 Concluding remarks References Chapter 4 Use of variable sum limits to solve the knapsack problem 4.0 Introduction 4.1 The knapsack model 4.2 Development of the variable range for a knapsack problem 4.2.1 Variable range 4.2.2 Objective value upper bound 4.2.3 Objective value lower bound 4.2.4 How to overcome this challenge? 4.2.5 Variable sum bounds and subsets 4.2.6 Subsets of variable sum bound 4.3 Variable sum bounding algorithm 4.3.1 Algorithm 4.4 Numerical illustration Direct branch and bound algorithm Solution by the proposed method - Variable sum bounding algorithm 4.5 Conclusions References Chapter 5 The characteristic equation for linear integer programs 5.0 Introduction 5.1 Development of a characteristic equation for a pure linear integer program 5.1.1 Analysis of a trivial example 5.1.2 The characteristic equation 5.1.3 Some interesting properties of the CE 5.1.4 An algorithm to find the kth best optimal solutions, k≥1 using the CE approach 5.1.5 Features of the CE 5.1.6 A numerical illustration 5.1.7 An ill conditioned integer programming problem 5.1.8 Analogies of the characteristic equation with other systems and models 5.2 The ordered tree method for an integer solution of a given CE by Munapo et al. (2009) 5.2.1 A Numerical illustration of the ordered tree search technique 5.3 The CE for the binary integer program 5.3.1 Numerical illustration of a binary program 5.4 CE applied to a bi-objective integer programming model 5.4.1 Numerical illustration for bi-objective model 5.5 Characteristics equation for mixed integer program 5.5.1 Mathematical developments 5.5.2 A characteristic equation approach to solve a mixed-integer program 5.5.3 Numerical illustration – MIP 1 5.6 Concluding remarks References Chapter 6 Random search method for integer programming 6.0 Introduction 6.1 The random search method for an integer programming model 6.1.1 Integer linear program, notation, and definitions 6.1.2 The random search method for integer programming 6.1.3 Reduction in the region for search 6.1.4 The algorithm 6.1.5 Numerical illustrations for an integer program 6.2 Random search method for mixed-integer programming 6.2.1 The mixed-integer programming problem, notation and definitions 6.2.2 Steps of the random search algorithm 6.2.3 Numerical illustration 6.3 An extreme point mathematical programming problem 6.3.1 Mathematical formulation of an extreme point mathematical programming model 6.3.2 Problems that can be reformulated as an extreme point mathematical programming model: Some applications 6.4 Development of the random search method for the EPMP model 6.4.1 Randomly generated solution 6.4.2 The number of search points 6.4.3 Reduction in search region or a successful solution 6.4.4 A feasible pivot for an EPMP 6.4.5 Algorithmic steps 6.4.6 Illustrative example 6.4 6.5 Conclusion References Chapter 7 Some special linear integer models and related problems 7.0 Introduction 7.1 The assignment problem 7.1.1 Features of the assignment model 7.1.2 Kuhn-Tucker conditions 7.1.3 Transportation simplex method 7.1.4 Hungarian approach 7.1.5 Tsoro and Hungarian hybrid approach 7.2 See-Saw rule and its application to an assignment model Pairing of columns 7.2.1 Starting solutions 7.2.2 See-Saw algorithm 7.2.3 Numerical Illustration 1 7.2.4 Proof of optimality 7.3 The transportation problem 7.3.1 Existing methods to find a starting solution for the transportation problem 7.3.2 Transportation simplex method 7.3.3 Network simplex method 7.3.4 The method of subtractions for an initial starting solution 7.4 The See-Saw algorithm for a general transportation problem 7.4.1 A General transportation model 7.4.2 The assignment-transportation model relationship 7.4.3 See-Saw rule for the transportation model 7.4.4 Initial position before the See-Saw move 7.4.5 See-Saw algorithm for the general transportation model 7.4.6 Numerical illustration of transportation model 7.5 Determination of kth (k ≥ 2) ranked optimal solution 7.5.1 Murthy’s (1968) approach 7.5.2 Minimal cost assignment approach for the ranked solution 7.6 Concluding remarks References Chapter 8 The travelling salesman problem: Sub-tour elimination approaches and algorithms 8.1 Introduction 8.2 Binary formulation of the TSP 8.2.1 Sub-tour elimination constraints 8.2.2 Some conceptual ideas and typical structure of the TSP model 8.2.3 Changing model (8.5) from linear integer to quadratic convex program (QP) 8.2.4 Convexity of f(Xˉ) 8.2.5 Complexity of convex quadratic programming 8.2.6 Other considerations 8.3 Construction of sub-tour elimination cuts 8.4 Proposed algorithm 8.4.1 Numerical illustration 8.4.2 Conclusion 8.5 The transshipment approach to the travelling salesman problem 8.5.1 Conventional formulation 8.5.2 Some important properties of a totally unimodular matrix 8.5.3 Breaking a TSP into transshipment sub-problems 8.5.4 General case – transshipment sub-problem 8.5.5 Standard constraints 8.5.6 Infeasibility 8.6 The transshipment TSP linear integer model 8.6.1 Numerical illustration 8.6.2 The formulated transshipment TSP linear integer model 8.7 Conclusions References Index De Gruyter Series on the Applications of Mathematics in Engineering and Information Sciences Already published in the series
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