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This report constitutes the Doctoral Dissertation for Anh Ninh and consists of three topics: log-concavity of compound Poisson and general compound distributions, discrete moment problems with fractional moments, and the recruitment stocking problems. In the first topic, we find the conditions for the compound Poisson and general compound distributions to be log-concave (log-convex). This problem is very important not only from the stochastic optimization perspective but also from the theory of maximum entropy in probability. Some interesting connection to Tur{'a}n-type inequality will also be mentioned. In the second topic, we formulate a linear programming problem to find the minimum and/or maximum of the expectation of a function of a discrete random variable, given the knowledge of fractional moments. Using a determinant theorem we fully characterize the dual feasible basis for this discrete fractional moment problem. With the dual feasible basis structure, Pr{'e}kopa dual method can be applied for its solution. Numerical examples show that by the use of fractional moments, we obtain tighter bounds for the objective. In the third topic, we introduce a new class of inventory control model - the recruitment stocking problems. In particular, we analyze a general class of inventory control problem, in which we need to recruit a target number of individuals through designated outlets. As soon as the recruits of all outlets add up to the target number, the recruitment is done and no more individuals will be admitted. The arrivals of individuals at each outlet are random. To recruit an individual upon its arrival, we must provide a pack of materials. We order the packs of materials in advance and hold them in the outlets. Outlets can neither transfer recruits nor cross-ship materials among themselves. If an outlet runs out of stock, any futher recruit at the outlet will be lost. We propose both exact and approximation methods to measure key performance metrics for the system: Type I and II service levels and recruitment time. Extensive numerical study shows the effectiveness of our proposed framework. |
