| Zusammenfassung: | Consider a stochastic allocation problem where a total resource of R units are to be allocated among m competing facilities in a system. An allocation of ri units to facility i results in a random response $X_{i}(r_{i}),i=1,\ldots ,m$ . The system response is then defined by the random variable $Y({\bf r})=h(X_{1}(r_{1}),\ldots ,X_{m}(r_{m}))$ where $h\colon \text{IR}^{m}\rightarrow \text{IR}$ is the system performance function. Let ${\cal S}\subset \text{IR}_{+}^{m}$ be the set of all feasible allocations. We are then interested in the stochastic allocation problem $\text{min}\{Eg(Y({\bf r}))\colon \sum_{i=1}^{m}r_{i}=R,{\bf r}\in {\cal S}\}$ for some utility function g. The aim of the paper is to obtain a partial or a full characterization of the optimal solution to this problem with minimal restriction on g. For this we introduce notions of stochastic Schur convexity and stochastic transposition increasingness and identify sufficient conditions on $X_{i}(r_{i}),i=1,\ldots ,m$ and h under which Y(r) will be either stochastically Schur convex or transposition increasing with respect to r. Then under appropriate condition on g it can be shown that the stochastic Schur convexity of Y(r) will imply the optimality of balanced resource allocation and the transposition increasingness will imply a partial characterization of the optimal solution thus reducing the computational effort needed to find the optimal solution. Several examples in the telecommunication, manufacturing and reliability/performability systems are presented to illustrate the main results of this paper.
|
|---|