On The Application of the Laplace Transform to Certain Economic Problems

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.

Section: Introductionmentioning

confidence: 99%

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

Beullens

Janssens

2014

“…We are now in a position to evaluate the system according to the Net Present Value principle (NPV), provided that values of economic parameters in vector form concerning unit production prices/costs p and setup costs K and an interest rate  are introduced (and interpreted as elements of the relevant cash flow [Grubbström 1967] we then obtain the NPV of the cash flow involved as…”

Section: A Numerical Examplementioning

confidence: 99%

“…The NPV principle has been a natural way of expressing the economic consequences in MRP Theory, since cash flows written as transforms are immediately evaluated as to their NPV by exchanging the Laplace frequency for the continuous interest rate following the NPV Theorem of the Laplace Transform [Grubbström 1967]. …”

mentioning

confidence: 99%

On the representation of timing for different structures within MRP theory

Bogataj

Grubbström

2012

Self Cite

MRP Theory has been developed during the last 25 years for capturing processes concerning multi-level, multi-stage production-inventory systems in a compact way. Input-output analysis has been used to describe structures, and Laplace transforms to describe the timing relations. This theory has mainly dealt with assembly systems, in which each item has only one successor. The lead times for the assembly of an item have usually been constants and equal for all items entering a given successor. For such systems, the equations describing the flows of components may be written to include the generalised input matrix as the product of an input matrix containing needed amounts, and a diagonal lead time matrix with lead time operators along its main diagonal. less thanbrgreater than less thanbrgreater thanOn occasion, there has been a need to deviate from this representation enabling lead times to vary depending on which input item that is considered. This paper deals with how to represent lead times and similar output delays (in diverging, arborescent systems), when the assumption of equal times is relaxed, in order to retain the basic structure of the fundamental balance equations involved. The intention of this paper is to create a general taxonomy for the representation of timing in algebraic form for a variety of systems covering both assembly systems and arborescent systems (such as extraction, distribution and remanufacturing), as well as systems with mixed properties. For instance, this method may be used directly for the evaluation of investments in capacity or in the location of activities in a production network, or even in a global supply chain

“…Since Harris (1913), it is well-known that lot sizing leads to financial holding costs that depend on the firm's opportunity cost of capital α. Key contributions to the theory that allows a more refined investigation were made in Hadley (1964), Grubbström (1967), Grubbström (1980), Porteus (1985), Haneveld and Teunter (1998), Van der Laan and Teunter (2002) and Boyaci and Gallego (2002). At the time of writing, more than a decade later, this should have generated an important shift in modelling the financial implications of lot-sizing, in particular in the context of price/quantity discounts, joint economic lot-sizing, and supply chain coordination.…”

Section: Introductionmentioning

confidence: 99%

“…NPV modelling has been used at least for fifty years to check whether the lot sizing models we investigate are capable of maximising the NPV of the future profits of the firm(s) involved. The NPV value to the firm of its lot sizing activity is found as the Laplace transform of a cash-flow function (Grubbström, 1967) in which the Laplace frequency is taken to be the firm's opportunity cost of capital rate α. As in Hadley (1964) and Grubbström (1980), we construct from this the Annuity Stream (AS) function.…”

Section: Introductionmentioning

confidence: 99%

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

Beullens

2014

While many review articles exist on (deterministic) lot sizing models used in the context of price and quantity discounts, buyer-vendor coordination, supply chain management, and joint economic lot sizing problems, they do not convey the impact of important findings which date back to at least 2002, or, in hindsight, to 1984. As a result, many recent articles still model the financial implications of lot sizing decisions without having the assurance that these models would help the firm(s) involved in maximising the Net Present Value (NPV). This paper therefore reviews these findings, while adding also its own contributions, as to convey the general importance to lot sizing theory. We show that the underlying principles used in the four key articles that have led to a division in modelling approaches are in fact all in line with NPV, and argue that therefore there should not be these discrepancies that currently persist in the literature. We establish the connections between these four strands of literature using the solution to a simple variation of Harris' EOQ model, deriving thereby results from Boyaci and Gallego (2002) and Beullens and Janssens (2011), but showing their general applicability to any type of supply-chain structure. The breath of implications to deterministic lot sizing theory is illustrated using practical examples. We present a stochastic version of the model of Crowther (1964), which is arguably the least understood and applied model, but on the other hand the most important one in realising how these modelling strands can be unified.