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The resource allocation problem addressed in this book is an optimization problem with a single simple constraint. Given a fixed amount of the resource (this is the constraint), one is asked to determine its allocation to n activities in such a way that the objective function under consideration is optimized. The amount of resource allocated to each activity is treated as a continuous or discrete (integer) variable, depending upon the cases. This is a special case of the nonlinear programming problem or the nonlinear integer programming problem.
Due to its simple structure, this problem is encountered in a variety of application areas, including, among many others, load distribution, production planning, computer scheduling, portfolio selection, and apportionment. It also appears as subproblems of more complex problems. Since B.O. Koopman first wrote a paper on the optimal distribution of effort in 1953, a significant number of papers have been published on these problems. Efficient algorithms are known, depending upon the form of objective function – e.g., separable, convex, minimax, maximin, fair, fractional, or general – and the form of constraint-continuous or discrete. Problems of considerably large sizes can be effectively solved by these algorithms in reasonable computation time. Algorithm development seems to have reached its maturity.
It is also observed that similar algorithms have been recurrently proposed in the literature, perhaps because of their simplicity of structure and the diversity of their applications. In view of this, it appears timely to summarize the past thirty years of achievement. This motivated us to write this book.
This is believed to be the first book to discuss comprehensively the algorithmic aspects of the resource allocation problem and its variants. Included are the Lagrangean multiplier method, dynamic programming, greedy algorithms, and their generalizations. The analysis of the computational complexity, exploiting modern data structures, accompanies each algorithm description in the book. Extensions to more general problems involving more than one constraint and/or more than one resource type are also treated.
Resource Allocation with Continuous Variables
Resource Allocation with Integer Variables
Minimizing a Separable Convex Function
Minimax and Maximin Resource Allocation Problems
The Fair Resource Allocation Problem
The Apportionment Problem
Fundamentals of Submodular Systems
Resource Allocation Problems under Submodular Constraints
Further Topics on Resource Allocation Problems