Dynamic lotsizing with a finite production rate

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.

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“…Grubbström (2013). NPV is not the only possible framework to help making finanical decisions about the future, see e.g.…”

Section: Introductionmentioning

confidence: 99%

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

Beullens

Janssens

2014

International Journal of Production Economics

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“…As shown in a recent paper (Grubbström 2012), when the production rate is assumed to be finite, rather than that production takes place instantaneously, then inner-corners no longer appear. Instead a condition of tangency applies, and cumulative production now built up of a set of ramps either touch an upper corner of the demand staircase, or not, see Figure 2.…”

Section: Figure 1 the Inner-corner Condition As A Necessary Provisiomentioning

confidence: 97%

“…The problem treated in this paper is a follow-up of (Grubbström 2012) allowing for shortages to be possible at the cost of delaying the in-payments for backlogged items. Other penalties for negative consequences from shortages (such as badwill) are not considered here.…”

Section: Figure 1 the Inner-corner Condition As A Necessary Provisiomentioning

confidence: 99%

Cumulative staircase considerations for dynamic lotsizing when backlogging is allowed

Grubbström

2014

International Journal of Production Economics

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The dynamic lotsizing problem concerns the determination of optimal batch quantities, when given required amounts appear at discrete points in time. The standard formulation assumes that no shortages are allowed and that replenishments are made instantaneously.For the case when no shortage is allowed, previously it has been demonstrated that the innercorner condition for an optimal production plan in continuous time reduces the number of possible replenishment times to a finite set of given points at which either a replenishment is made, or not. The problem is thus turned into choosing from a set of zero/one decisions with 1 2 n− alternatives, of which at least one solution must be optimal, where n is the number of requirement events. Recently, the instantaneous replenishment assumption has been replaced by allowing for a finite production rate, which turned the inner-corner condition into a condition of tangency between the cumulative demand staircase and cumulative production.In this paper we investigate relationships between optimal cumulative production and cumulative demand, when backlogging is permitted. The production rate is assumed constant and cumulative production will then be a set of consecutive ramps. Cumulative demand is a given staircase function. The Net Present Value (NPV) principle is applied, assuming a fixed setup cost for each ramp, a unit production cost for each item produced and a unit revenue for each item sold at the time it is delivered.Among other results, it is shown that optimal cumulative production necessarily intersects the demand staircase. Instead of having 1 2 n− production staircases as candidates for optimality, there are 1 2 n− production structures as candidates. These are made up of sequences of batches, of which the set of batches may be optimised individually.Results for the Average Cost approach are obtained from a zeroth/first order approximation of the objective function (NPV).

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“…(For a graphical illustration of the problem, see Fig. 1 in Grubbström (2014) in which the demand curve is now a staircase of uniform steps.) Theorem 1.…”

Section: The Eoq For Uniform Stepsize Cumulative Demandmentioning

confidence: 99%

“…When viewing the system as a dynamic lot sizing problem with an infinite production rate, it is known that from both the NPV and the average cost perspective a so-called inner corner condition must be satisfied, see Grubbström and Molinder (1994) and Grubbström (2014). This means that the firm will only order at moments when a customer order is to be delivered, or in the terminology introduced in Roundy (1985), that the policy must be nested.…”

Section: The Eoq For Uniform Stepsize Cumulative Demandmentioning

confidence: 99%

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

Beullens

2014

International Journal of Production Economics

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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.

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Dynamic lotsizing with a finite production rate

Cited by 11 publications

References 20 publications

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

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

Cumulative staircase considerations for dynamic lotsizing when backlogging is allowed

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

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