# 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## References

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