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Classic inventory models use average cost functions. It is generally accepted that these models should account for the time value of money. They do so not by considering the timing of cash-flows, but by including opportunity costs. The Net Present Value (NPV) framework has long been used to compare these models with. We formalise NPV Equivalence Analysis (NPVEA) under various payment structures, and apply it to a few classic inventory models. While taking the linear approximation is typically part of the process to find equivalence, the essence is to disregard the parameters of a classic inventory model but instead start from cash-flow structures between firms. It is demonstrated how this leads to different plausible interpretations of, or variations to, classic inventory models, in particular for payment structures that differ from conventional assumptions. We identify situations with negative holding costs, which indicates that more features from the real world must be added into the decision model. We illustrate that in addition to capital costs, firms can enjoy capital rewards. These rewards may not always affect the firm's inventory decisions, but are in general useful for finding the impact of changes to various parameters on the firm's future profits.

The foundation that inventory theory is to account for the time value of money goes back to Harris (1913), who was careful to explain that his Economic Order Quantity (EOQ) model is largely about the trade-off with the opportunity cost of capital. His is the archetypal model of classic inventory theory, in which one minimises the average costs per unit of time. The holding cost is commonly found from the integration over a relevant cycle time T : equation(1) View the MathML source1T∫0Th(t)I(t)dt, Turn MathJax on where I (t ) is the inventory level at time t , and h (t )=h the unit holding cost, typically taken to be time-invariant. Costs are not discounted according to their time of occurrence, but the time value of money is implicitly modelled by the inclusion into h of the financial opportunity cost from stock investment. Typically, h is taken to be of the form ( Silver et al., 1998): equation(2) h=αv+f,h=αv+f, Turn MathJax on where v is the money invested per unit of product held in stock, f the unit ‘out-of-pocket’ holding cost, and αα the firm's continuous capital rate. A value α =0.20 (time measured in years) is often used, and the putative view is that the financial holding cost dominates: αv>>fαv>>f. The opportunity cost is also the foundation for the Net Present Value (NPV), which quite generally can be viewed to be the Laplace transform of a cash-flow function a(t) in which the Laplace frequency is taken to be α ( Grubbström, 1967): equation(3) View the MathML source∫0∞a(t)e−αtdt. Turn MathJax on As the time value of money is modelled explicitly, it would be incorrect to include into a(t) the financial holding cost as used in classic models. Instead, it is retrieved in the linearised Annuity Stream (AS) function ( Grubbström, 1980). The AS is the constant payment stream having the same NPV as a given stream of payments; for an infinite horizon, AS =α NPV. The comparison with NPV has been used at least as early as Hadley and Whitin (1963) and Hadley (1964), who demonstrate that Harris' model retains the lot-size relevant terms of the linear AS function. Grubbström, 1980 and Grubbström, 2007 shows how capital costs can be determined for inventories and work-in-process at several stages in more complex systems of production and inventory. See also Gurnani (1983). Porteus (1985) clarifies how the timing of expenditures and revenues relative to the cycle time of a regenerative process affects the valuation of capital costs, and illustrates using Harris' EOQ model. Teunter et al. (2000), Van der Laan and Teunter (2002), Teunter and van der Laan (2002), and Çorbacioğlu and van der Laan (2007) use the linear AS function for setting the unit holding costs at the different stages in systems of remanufacturing. They demonstrate that the mapping of the classic parameters to the AS function is not necessarily injective. Beullens and Janssens (2011) introduce the Anchor Point (AP) in NPV models, and show that its position in the supply chain can affect the valuation of capital costs at the different stages in the system. In this paper we use NPV to study the impact of payment structures on the unit holding cost and other parameters in classic models. Grubbström (1980) was perhaps first to introduce the term. Most studies that use NPV to retrieve the capital costs, except for Porteus (1985) and Beullens et al. (submitted for publication), adopt ‘conventional’ assumptions. Loosely speaking, this means that costs or revenues are assumed to occur either the moment that some (physical) process in the system initiates or terminates, such as the transfer of a batch of materials, or continuously at the rate of process transformation, such as a production rate. As the timing of payments is not described in classic inventory models, this first guess often leads to satisfactory results. However, trends in inventory management, including the use of consignment stocks and credit delays, show that a wider variety of payment structures are adopted in practise. The question is then under which variety of payment structures the classic models, and their solutions, can still be used, or how they should be adapted. The study of equivalence is formalised as NPVEA in Section 2, and a basic idea from propositional logic introduced. We belief that this makes it easier to present results, and their logical consequences, succintly. In Section 3 payment structures are defined and common examples presented. In 4 and 7, the approach is applied to a few well-known inventory models. Next to finding out the strengths and weaknesses of these models, it leads to a number of simple variations which do not appear in our literature, but should be of relevance in the context of various practical applications. As a general conclusion, we find that the study of equivalence under various payment structures can be of great help to better understand inventory theory, and increase and extend its applicability. The paper considers infinite horizon models with constant demand, but NPV can also be used for dynamic lot-sizing, see e.g. Grubbström in press. NPV is not the only possible framework to help making financial decisions about the future, see e.g. Xu et al. (2012) for a comparison with real options.

Classic inventory theory has its roots in an analytical modelling approach which has often been shown to be remarkably accurate and of practical value in a wide range of applications. To help its further development into a theory that is compatible in at least first order with the Net Present Value paradigm, we have formalised an approach that has been in the making at least since 1963, illustrated how this can deal with various payment structures to produce a wide range of possible reference models, and how this leads to different interpretations of, or variations to, classic inventory models. The examples presented illustrate that the method, which we call NPVEA, is a useful modelling and theory development tool in that it can indicate which parts of an existing inventory theory are worth preserving and where perhaps some modifications are in order. The most important idea is not the linear approximation, but is due to the information gap which gives rise to an interpretation problem. Therefore, we cannot rely on the parameters that have become of integral importance in classic inventory theory, when formulating the reference model. The most important practical aspect is thus to start from a different view of the world in which the inventory model resides – a description in terms of a collection of firms that exchange cash-flows with each other in order to execute the activity, where the label ‘firms’ is to be interpreted very broadly (it can also include employees, consultancy firms, financial institutions, etc). As there are many plausible NPV reference models, there are many plausible valid interpretations of a classic model and its underlying theory. Refinements to this theory can be uncovered, including the need to consider revenue rewards and negative holding costs, as has been illustrated in this paper. The further application of NPVEA must be useful in the context of examining the ubiquitous but illustrious unit backorder cost and unit lost sales cost in inventory models in which backorders or lost sales (are allowed to) occur. The method is not restricted to inventory models, but in principle applicable to any model in which monetary flows over time are of importance but in which the construction of cash-flow functions was not the starting point of model development.