# Double and Triple Optimization

Chapter

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## Abstract

There are three pure allocation problems, viz., the work-load allocation problem, the server allocation problem and the buffer allocation problem, all concerned with maximizing throughput. Mathematically, these problems may be described as follows:

The work-load allocation problem, WAP: subject to: for normalized total work-load equal to unity and fixed allocation of servers and fixed buffer allocation.

$$\max X({\bf w}) =\max X({w}_{1},{w}_{2}, \ldots ,{w}_{K})$$

$$\sum _{i=1}^{K}{w}_{ i} = 1,\ \ \ \mbox{ for ${w}_{i} > 0$}$$

The server allocation problem, SAP: subject to: for fixed allocation of work to each station and fixed buffer allocation.

$$\max X({\bf s}) =\max X({S}_{1},{S}_{2}, \ldots ,{S}_{K}))$$

$$\sum _{i=1}^{K}{S}_{ i} = S,\ \ \ \mbox{ for ${S}_{i} \geq 1$ and integer}$$

The buffer allocation problem, BAP: subject to: for fixed allocation of work to each station and fixed allocation of servers.

$$\max X({\bf n}) = X({N}_{2}, \ldots ,{N}_{K})$$

$$\sum _{i=2}^{K}{N}_{ i} = N,\ \ \ \mbox{ for ${N}_{i} \geq 0$ and integer}$$

As indicated above, there are three single-variable decision problems. Combining these problems into two-variable problems leads to the following three problems which may be mathematically described as follows:

The combined work-load allocation and server allocation problems, W + S: subject to: and and for fixed buffer allocation.

$$\max X({\bf w},{\bf s})$$

$$\sum _{i=1}^{K}{w}_{ i} = 1,\ \ \ \mbox{ for ${w}_{i} > 0$ and normalized work-load}$$

$$\sum _{i=1}^{K}{S}_{ i} = S,\ \ \ \mbox{ for ${S}_{i} \geq 1$ and integer}$$

The reader may note that this problem has already been discussed in Chapter 4.

## Keywords

Simulated Annealing Allocation Problem Expansion Method Service Time Distribution Complete Enumeration
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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