مدل موجودی (R، Q) تنزیل شده؛ اکتشاف هوشمندانه حسابداری

The discounted continuous-review (R,Q) inventory model with continuous and stochastic demand is investigated. New optimality conditions are derived, clarifying the difference to the average-cost case, also graphically. Supported by depreciation theory, applied to the value of a setup, the results suggest an insightful and very precise approximation – The Shrewd Accountant's Heuristic – based on a new average-cost model. It deepens and extends the work of Hadley (1964). Three examples are worked out in detail and the model is generalized to Poisson demand and to stochastic lead-times.

We extend understanding of discounting in inventory theory by analysis of a simple version of the (R,Q) inventory model with demand following an idealized stochastic process with continuous sample path. An (R,Q) policy then coincides with an (s,S) policy and the inventory position is raised to R+Q each time it hits R. New optimality conditions are found, generalizing those of the average-cost model. The difference to the average-cost model is clarified and depicted, and a very precise heuristic, based on a new average-cost formulation and the new optimality conditions, is derived by use of depreciation theory. It is little better than the optimal average-cost policy – as is well-known – but the exercise renders insight. Discounting is the economists' well-justified way of valuing dynamic streams of payments. It has been used for valuation of inventory policies since the beginning of inventory theory, e.g. Arrow et al. (1951), Veinott and Wagner (1965). Discounted models are frequently used to demonstrate the optimality of inventory policies, e.g., Clark and Scarf (1960), Scarf (1960). However, in practice discounted-cost inventory models are most often approximated by the more convenient average-cost formulations that charge interest cost of inventory and backlog. Quite surprisingly, comparisons of the two approaches are rare in the literature. Hadley (1964) presented an average-cost approximation of the discounted, deterministic EOQ model. It appears in some textbooks, e.g., Zipkin (2000, ch. 3.7) and Porteus (2002, App. C) and it has been extended and commented in quite a few articles, e.g., Grubbström (1980), Grubbström and Thorstenson (1986), Haneveld and Teunter (1998), Corbacioglu and van der Laan (2007) and Beullens and Janssens (2011). Chao (1992) developed an exact model with Wiener demand. Hadley approximated the discounted annuity by the average cost plus interest on half the setup cost. By applying depreciation theory and the new average-cost model, we improve his estimate. We then use the new cost estimate to derive a heuristic policy through the new optimality conditions. The precision of this simple heuristic turns out to be extremely high. The sequel is organized as follows: Section 2 finds the net present value of an (R,Q) policy and the discount rate, r (an insight); Section 3 presents the new optimality result ( Proposition 2 – a major insight). Section 4 applies depreciation theory to calculate the annuity through a (re-)discovered accounting identity ( Proposition 3 – another insight) and compares the two approaches ( Corollary 3, Fig. 2). Section 5 presents the heuristic, 6, 7 and 8 applies it to three models – the deterministic EOQ model without and with backlogging and to the model with Normal lead-time demand, Section 9 generalizes the optimality result to Poisson demand ( Proposition 4) – this model is close to Johansson and Thorstenson (1996). Section 10 compares the new and the traditional average-cost formulation (an insight!) and discusses stochastic lead-times. Section 11 concludes. Two appendices contain omitted proofs and a third, omitted formulae.

When demand follows an idealized stochastic process, and lead-times may be stochastic, a new optimality condition for the discounted (R,Q) policy was derived ( Proposition 2). It generalizes to Poisson demands ( Proposition 4) and deviates interestingly from the average-cost condition ( Section 3). The Shrewd Accountant's Heuristic ( Section 5) estimates the optimal discounted annuity by the optimal average cost plus interest on the average depreciated value of the setup. This average is approximated through the new average-cost model. Finally a policy is estimated through the new optimality condition. The approximation is extremely precise and may numerically be useful for stochastic models, which can be tedious to optimize ( Section 8). There is little gain in using it, though, compared to the optimal average-cost policy (new or old), as already noted by Hadley (1964) for deterministic EOQ problems. Yet the analysis renders several insights. The models are formulated with interest costs on inventory and backlog included in the expected holding and backlogging costs as is generally done in practice. The interest rate, r, is somewhat tricky to determine, because time has to be translated into number of demanded units ( Section 2). When demand is continuous, r is smaller than the regular continuous discount rate, α, but for Poisson demand there is no difference ( 2 and 9). To allow for discounting by r, costs of inventory and backlog are multiplied by r/α. This makes the difference between the new and the traditional average-cost model. However, for our continuous demand, r and α seem very close. The new average-cost policy simplifies analysis, but is very close to the traditional one: R moves a little closer to optimum and R+Q a little further away ( Section 10). It is superior, though, as a component of our “shrewd” heuristic. The depreciation schedule of a setup begins at inventory position R+Q with its nominal value, K. For an optimal discounted policy, it then depreciates to zero as the inventory position approaches R. For an average cost optimal policy, the value first appreciates above K and then depreciates to zero. For a general policy, there may be increasing phases both at the beginning and the end – in the latter case the value of the setup is negative when finally increasing to zero ( Fig. 1, Appendix 2). Exact discounting lowers the inventory levels, both R and R+Q, compared to the average-cost solution, although this solution includes the interest costs on inventory and backlog. The effect on Q depends on the service level in our examples. At high service levels, Q decreases; at low service levels, it may increase ( Section 7). The average depreciated value of the setup cost, K, is typically higher than Hadley's estimate, K/2. For his simple EOQ model, our heuristic makes the estimate K2/3 ( Section 6). High demand uncertainty makes our average closer to Hadley's estimate, slightly dependent on the character of the shortage cost ( Section 8). A final, possibly trivial insight is the fact that the average-cost problem is just an approximation of the discounted problem. It should include an interest rate chosen with equal care. It offers computational simplification – that's all! Our results can certainly be extended to Renewal demand processes in some way, and possibly further to encompass (s,S) and (R,nQ) policies for Compound Renewal and Gamma demand processes, or for periodic review; but we guess that most of the present simplicity will then be lost, except possibly for asymptotic cases where Q/D→∞ (cf. Roberts 1962).