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Appendix D: Stochastic Models of Uncertainty and Mathematical Optimization Under Uncertainty
Pages 137-150

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From page 137...
... , referred to as P&R, including recruiting, training, retention, optimal mix, readiness, staffing, career shaping, and resource allocation. Although there are unique considerations for the Department of Defense (DoD)
From page 138...
... The second example concerns trade-offs among the evolution of capabilities and readiness of personnel resources over time given uncertainty around time-varying supply-side dynamics, which relate to P&R missions associated with recruiting, training, retention, optimal mix, readiness, staffing, career shaping, and resource allocation. This material is based on the work of Cao et al.
From page 139...
... , and + denotes the set of non-negative integers. Assuming independent Poisson arrival processes with rate vector v = (vk)
From page 140...
... . Example: Time Inhomogeneous Markov Processes Another example is based on the use of discrete-time stochastic processes to model the evolution of capabilities and readiness of personnel resources over time, given uncertainty around the time-varying supply-side dynamics with personnel acquiring skills, gaining experience, changing roles, and so on; some attrition of personnel; and new personnel being introduced.
From page 141...
... p (t ) , j′ j ′j j′ or, in matrix form (using row vector notation)
From page 142...
... is the vector of decision variables, y := (y1, …, ym) is a vector of dependent variables, f : Rn → R is the objective functional, gi : Rn → R is a constraint functional, i = 1, …, k, A is a matrix, b is a ­scalar, and S is the constraint region.
From page 143...
... To illustrate aspects of the points noted above and described in Chapter 4, a few examples of mathematical optimization under uncertainty involved in decision-making problems related to those of P&R are presented. The first example concerns trade-offs among skill capacities and readiness of resources given uncertainty around the demand for such resources, which relate to P&R missions associated with optimal mix, readiness, staffing, and resource allocation.
From page 144...
... The third example concerns trade-offs among the dynamic allocation of capacity for different types of resources in order to best serve uncertain demand, which relate to P&R missions associated with recruiting, training, retention, optimal mix, readiness, and resource allocation. This material is based on the work of Gao et al.
From page 145...
... Example: Stochastic Dynamic Program Another example is based on optimization of discrete-time stochastic ­ decision processes of the evolution of capabilities and readiness of personnel resources over time given uncertainty around the time-varying supply-side dynamics. One can start with the stochastic dynamic program formulated with respect to a time inhomogeneous Markov process whose dynamics are governed by (A.6)
From page 146...
... that has the lower net benefit. A control policy defines at every time t ∈ R the level of primary resource allocation, denoted by P(t)
From page 147...
... : s ≤ t}. Any adjustments to the primary resource allocation capacity have associated costs, where Ip and Dp denote the per-unit cost for increasing and decreasing the allocation of primary resource capacity P(t)
From page 148...
... 2013. Stochastic optimal control for a general class of dynamic resource allocation problems.
From page 149...
... 2007. Optimal capacity planning in stochastic loss networks.


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