Piecewise linear approximations of the standard normal first order loss function

The first order loss function and its complementary function are extensively used in practical settings. When the random variable of interest is normally distributed, the first order loss function can be easily expressed in terms of the standard normal cumulative distribution and probability density function. However, the standard normal cumulative distribution does not admit a closed form solution and cannot be easily linearised. Several works in the literature discuss approximations for either the standard normal cumulative distribution or the first order loss function and their inverse. However, a comprehensive study on piecewise linear upper and lower bounds for the first order loss function is still missing. In this work, we initially summarise a number of distribution independent results for the first order loss function and its complementary function. We then extend this discussion by focusing first on random variable featuring a symmetric distribution, and then on normally distributed random variables. For the latter, we develop effective piecewise linear upper and lower bounds that can be immediately embedded in MILP models. These linearisations rely on constant parameters that are independent of the mean and standard deviation of the normal distribution of interest. We finally discuss how to compute optimal linearisation parameters that minimise the maximum approximation error.Comment: 22 pages, 7 figures, working draf

Naval Wholesale Inventory Optimization

The article of record as published may be found at https://doi.org/10.1007/978-3-030-28565-4_15The U.S. Naval Supply Systems Command (NAVSUP), Weapon Systems Support, manages an inventory of approximately 400,000 maritime and aviation line items valued at over $20 billion. This work describes NAVSUP’s Wholesale Inventory Optimization Model (WIOM), which helps NAVSUP’s planners establish inventory levels. Under certain assumptions, WIOM determines optimal reorder points (ROPs) to minimize expected shortfalls from fill rate targets and deviations from legacy solutions. Each item’s demand is modeled probabilistically, and negative expected deviations from target fill rates are penalized with nonlinear terms (conveniently approximated by piecewise linear functions). WIOM’s solution obeys a budget constraint. The optimal ROPs and related expected safety stock levels are used by NAVSUP’s Enterprise Resource Planning system to trigger requisitions for procurement and/or repair of items based on forecasted demand. WIOM solves cases with up to 20,000 simultaneous items using both a direct method and Lagrangian relaxation. In particular, this proves to be more efficient in certain cases that would otherwise take many hours to produce a solution