# Double and Triple Optimization

• Michael J. Vidalis
• Michael E. J. O’Kelly
• Diomidis Spinellis
Chapter
Part of the Springer Optimization and Its Applications book series (SOIA, volume 31)

## 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:

$$\max X({\bf w}) =\max X({w}_{1},{w}_{2}, \ldots ,{w}_{K})$$
subject to:
$$\sum _{i=1}^{K}{w}_{ i} = 1,\ \ \ \mbox{ for {w}_{i} > 0}$$
for normalized total work-load equal to unity and fixed allocation of servers and fixed buffer allocation.
The server allocation problem, SAP:
$$\max X({\bf s}) =\max X({S}_{1},{S}_{2}, \ldots ,{S}_{K}))$$
subject to:
$$\sum _{i=1}^{K}{S}_{ i} = S,\ \ \ \mbox{ for {S}_{i} \geq 1 and integer}$$
for fixed allocation of work to each station and fixed buffer allocation.
The buffer allocation problem, BAP:
$$\max X({\bf n}) = X({N}_{2}, \ldots ,{N}_{K})$$
subject to:
$$\sum _{i=2}^{K}{N}_{ i} = N,\ \ \ \mbox{ for {N}_{i} \geq 0 and integer}$$
for fixed allocation of work to each station and fixed allocation of servers.

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:
$$\max X({\bf w},{\bf s})$$
subject to:
$$\sum _{i=1}^{K}{w}_{ i} = 1,\ \ \ \mbox{ for {w}_{i} > 0 and normalized work-load}$$
and
$$\sum _{i=1}^{K}{S}_{ i} = S,\ \ \ \mbox{ for {S}_{i} \geq 1 and integer}$$
and for fixed buffer allocation.

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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## Authors and Affiliations

• 1
• Michael J. Vidalis
• 2
• Michael E. J. O’Kelly
• 3
• Diomidis Spinellis
• 4
1. 1.Department of EconomicsAristotle University of ThessalonikiThessalonikiGreece