The incorporation of target performance measures and constrained optimisation in the newsboy problem
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The incorporation of target performance measures and constrained optimisation in the newsboy problem MA Pearson Napier University, Edinburgh The solution of the classical Newsboy problem requires some knowledge of the costs of wastage as well as the costs of lost opportunity. Such knowledge is often hypothetical and generally unavailable. More readily available knowledge, particularly in complex organisations, often relates to targets for wastage and availability, especially where perishable goods are involved. These targets can be incorporated effectively into a generalised forecasting context with a broad distribution of demand patterns by the use of a formulated statistic that measures the suitability of the data for the achievement of such targets. The paper proposes a model and algorithm for optimising supply levels using these two performance measures and constrained optimisation of a target gain function. Keywords: Newsboy; allocation; inventory; forecasting; distribution-free; optimisation
Introduction
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When dealing with inventory where the product has a short shelf life there are usually two con¯icting requirements. Firstly the achievement of high availability of the product measured by the failure of the product to sell out which tends to encourage overstocking. Conversely the expense involved in unsold products (often wasted) tends to make management conservative in stocking levels. The Newsboy problem deals with these limitations by formulating optimality criteria that minimise expected costs or maximise expected pro®ts. These costs are usually the cost Co per unit of overestimating demand, that is the per unit loss incurred when the quantity stocked exceeds demand and the cost Cu per unit of underestimating demand, that is the pro®t per item lost when demand exceeds the quantity stocked. The optimal order size Q* occurs in the simplest problem when
Scarf1 uses the simple costs c, the unit cost of inventory, and r, the unit revenue, in the above formula, (1), where c r Cu and c Co. He then chooses the stocking policy which maximises the minimum pro®t that would occur, considering all demand distributions with the given mean and standard deviation. He therefore chooses the value of Q which maximises his pro®t seen as a function of the mean and standard deviation of the demand and the ratio c=r. The coef®cient of variation, s=m, plays an important role in the formulation of Scarf's solution, indicating that this solution depends strongly on this measure of the variability of the demand distribution. In an important sense the solution is distribution-free, since it only depends on the ®rst two moments m and s of the demand distribution. Scarf also incorporates unit salvage values to calculate an optimal order value by maximising expected pro®t. Some papers since Scarf have concentrated on re®ning his method while maintaining the distribution-free context. Moon and Choi2 i
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