Double and Triple Optimization

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