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Grey Assignment Problems
Abstract
This paper introduces a new assignment problem called the Grey Assignment Problem, in which the cost of assigning every worker
to every job is a grey number rather than a real number. We establish the mathematical model of the grey assignment problems
and concerned theory, and also give some methods for solving the grey assignment problems.
... A traditional assignment problem concentrates on the usual assignment problems and bottleneck assignment problems. Many new assignment problems, such as "one task -multi persons" (or one person -multi tasks) assignment problem [1,2] , B-assignment problems [3] (optimal efficiency and optimal time assignment problem), C-assignment problems [4,5,6] (the number of the assignments is less than or equal to the number of human resources and the number of the tasks), and gray assignment problems [7,8] (efficiency is indeterminate) have been discussed recently. Each of these assignment problems considers only the efficiency with constrains and provides ways to obtain the best efficiency. ...
This paper establishes a new assignment problem called the time limit assignment problem (TLAP) which pursues the best overall efficiency only if all tasks have been completed in time or very close to the deadline. The time limit assignment problem contains one discrete goal constraint, and therefore it is a special goal programming problem. This paper provides a mathematical model for such a time limit assignment problem and introduces a method to find its optimal solutions and the deviation of the optimal solution to the warning time. We also provide an example that illustrates how to solve the time limit assignment problem.
In some real situations, such as transporting emergency goods when a natural disaster occurs or transporting military supplies during the war, the transport network may be destroyed, the transportation cost (time or mileage) from sources to destinations may not be deterministic, but uncertain grey number. This paper investigated a new bottleneck transportation problem called the grey bottleneck transportation problem, in which the transportation time (or mileage) from sources to destinations may be uncertain, and introduces its mathematical model and algorithms.
In ordinary transportation problems, it is always supposed that the mileage from every source to every destination is a definite number. But in special conditions, such as transporting emergency materials when natural calamity occurs or transporting military supplies during wartime, carrying network may be destroyed, mileage from some sources to some destinations are no longer definite. It is uncertain, a grey number. In these conditions, transportation capacity is often poor; the problems of optimization become even more important. Obviously, traditional methods are not suitable to solve these problems. In this paper, some methods of optimization for the special transportation problems are given.
Based on the demands of constructing a saving society, protecting natural environment and resources, G. Z. Bai introduced a special two-objective programming problem, and gave its mathematical model. This paper will give the definitions of premise optimal solution and satisfying solution, and a new method called the Quasi-Goal Programming Method for solving the problem. Also provide an example that illustrates how to solve the special two-objective programming problem by the method introduced in this paper.
This article introduces the (just-in-time) JIT-transportation problem, which requires that all demanded goods be shipped to their destinations on schedule, at a zero or minimal destination-storage cost. The JIT-transportation problem is a special goal programming problem with discrete constraints. This article provides a mathematical model for such a transportation problem and introduces the JIT solution, the deviation solution, the JIT deviation, etc. By introducing the B(λ)-problem, this article establishes the equivalence between the optimal solutions of the B(λ)-problem and the optimal solutions of the JIT-transportation problem, and then provides an algorithm for the JIT-transportation problems. This algorithm is proven mathematically and is also illustrated by an example.
There are finite zero-sum, two-person games with payoff functions having values given by rational grey number because they cannot be obtained exactly. Such a finite zero-sum, two-person game is called a grey payoff matrix game. In this paper, the grey saddle point, grey game value, quasi-solution and quasi-optimal strategy of grey payoff matrix game is defined. Finally, some methods for solving the grey payoff matrix game are given.
0 Introduction.- I: Linear Programming.- 1 Geometric Linear Programming.- 0. Introduction.- 1. Two Examples: Profit Maximization and Cost Minimization.- 2. Canonical Forms for Linear Programming Problems.- 3. Polyhedral Convex Sets.- 4. The Two Examples Revisited.- 5. A Geometric Method for Linear Programming.- 6. Concluding Remarks.- Exercises.- 2 The Simplex Algorithm.- 0. Introduction.- 1. Canonical Slack Forms for Linear Programming Problems Tucker Tableaus.- 2. An Example: Profit Maximization.- 3. The Pivot Transformation.- 4. An Example: Cost Minimization.- 5. The Simplex Algorithm for Maximum Basic Feasible Tableaus.- 6. The Simplex Algorithm for Maximum Tableaus.- 7. Negative Transposition The Simplex Algorithm for Minimum Tableaus.- 8. Cycling.- 9. Concluding Remarks.- Exercises.- 3 Noncanonical Linear Programming Problems.- 0. Introduction.- 1. Unconstrained Variables.- 2. Equations of Constraint.- 3. Concluding Remarks.- Exercises.- 4 Duality Theory.- 0. Introduction.- 1. Duality in Canonical Tableaus.- 2. The Dual Simplex Algorithm.- 3. Matrix Formulation of Canonical Tableaus.- 4. The Duality Equation.- 5. The Duality Theorem.- 6. Duality in Noncanonical Tableaus.- 7. Concluding Remarks.- Exercises.- II: Applications.- 5 Matrix Games.- 0. Introduction.- 1. An Example Two-Person Zero-Sum Matrix Games.- 2. Linear Programming Formulation of Matrix Games.- 3. The Von Neumann Minimax Theorem.- 4. The Example Revisited.- 5. Two More Examples.- 6. Concluding Remarks.- Exercises.- 6 Transportation and Assignment Problems.- 0. Introduction.- 1. An Example The Balanced Transportation Problem.- 2. The Vogel Advanced-Start Method (VAM).- 3. The Transportation Algorithm.- 4. Another Example.- 5. Unbalanced Transportation Problems.- 6. The Assignment Problem.- 7. Concluding Remarks.- Exercises.- 7 Network-Flow Problems.- 0. Introduction.- 1. Graph-Theoretic Preliminaries.- 2. The Maximal-Flow Network Problem.- 3. The Max-Flow Min-Cut Theorem The Maximal-Flow Algorithm.- 4. The Shortest-Path Network Problem.- 5. The Minimal-Cost-Flow Network Problem.- 6. Transportation and Assignment Problems Revisited.- 7. Concluding Remarks.- Exercises.- APPENDIX A Matrix Algebra.- APPENDIX B Probability.- Answers to Selected Exercises.
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The Tutorial to Grey System Theory
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