Adapting inventory models for handling various payment structures using net present value equivalence analysis

Abstract

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.

Revisiting foundations in lot sizing—Connections between Harris, Crowther, Monahan, and Clark

“…The insight that Monahan's main model, and all models building on it, should have included Crowther's term from the NPV-derived viewpoint, was first formulated in Boyaci and Gallego (2002). Beullens and Janssens (2011) show that this indeed holds for finite production rates in the model of Banerjee (1986), under conventional payments, and Beullens and Janssens (2013) presented similar findings for the general model of Joglekar (1988) under both conventional and other payment structures.…”

“…Assume that this supplier sells to a buyer with the normal EOQ problem as in (30) but replacing α b w by h b . One can prove from the annuity stream profit functions, following similar procedures as reported in Boyaci and Gallego (2002) and Beullens and Janssens (2013), that in all these cases the buyer and seller can use these profit functions provided that we redefine h b , h and h r as given in Table 6. Table 6: Parameter values in the buyer-vendor supply chain…”

“…When payment structures differ, this equivalence result is no longer guaranteed, see Beullens and Janssens (2013). The expression (1), while being based on new notation, follows closely the original manuscript.…”

“…Abad (1994). The differences between independent firms versus an integrated firm have to be rather sought in the fact that the opportunity cost of capital of individual firms may be very different, see Boyaci and Gallego (2002), or that the payment structures are no longer symmetric, see Beullens and Janssens (2013). This requires a closer investigation into the implications of corporate finance theory on models of logistics in supply chains, and multi-echelon lot sizing models in particular.…”

“…Throughout its presentation, this paper assumes that payments for set-ups and production or procurement occur in full whenever a batch arrives at the firm, unless otherwise stated. This is called a conventional payment structure in Beullens and Janssens (2013). In Section 7.6 we extend this to a class of other payment structures, and show that by using a suitable substitution of parameters, the functional forms of the terms obtained under conventional payment structures are preserved.…”