General

Deterministic

 No Shortages Infinite Finite Shortages Backordered Infinite Finite Lost Sales Infinite Finite Summary
 Inventory Theory - Deterministic Inventory

In this section we consider an isolated inventory in which external demanders remove items from the inventory and external suppliers replenish the inventory. Rather than demand occurring in a random and uncertain manner, we assume that items are withdrawn from the inventory at a continuous rate. Replenishments to the inventory are of a fixed size q, called the lot size. The time between when a replenishment is requested and when the amount enters the inventory is called the lead-time. We assume that the lead-time is zero or a constant.

Six different models are developed in this section. They vary in whether shortages are allowed or not and whether the replenishment rate is infinite or finite. Two customer responses to shortages are considered: the backorder case when the customer will wait for delivery and the lost sales case when the customer will not wait. For the backorder case, the cost associated with a backorder is either proportional to the waiting time or it is independent of the waiting time. The cases are illustrated by the graphs of inventory position versus time shown below. Links adjacent to the figures and in the navigation bar on the left lead to pages providing detailed development of the optimization and analysis formulas. The formulas are implemented in the Inventory add-in.

Practitioners sometimes criticize of the results of inventory theory because the simple models described in this section do not often mirror reality. We justify their consideration on several grounds. First because of the simplicity we are able to find algebraic formulas that give the optimum solutions for the models. In some cases the computed optimum values may prove useful. The familiar EOQ formula is perhaps one of the most used formulas in practice. Perhaps more valuable are the insights into the effects of the several parameters on inventory cost and optimum operating policies. Further, the results for the deterministic system are used as approximation of the formulas describing stochastic systems. Further, we use the deterministic models for individual stations in models that involve systems of inventories. Finally, the process of creating a cost model and finding its optimum using calculus is a good example for the application of mathematics to the solution of other practical problems.

Care must be taken that the models are not used inappropriately. Just because a formula has been derived does not justify its use without a careful analysis of its validity in a particular situation. We hope that the presentation of inventory theory provided here and in many excellent textbooks provides both the basis and cautionary limitations for application.

Infinite Replenishment Rate and No Shortages

The figure shows the quantity in inventory versus time. The entire lot arrives instantaneously. Continuous demand draws down the inventory.

Finite Replenishment Rate and No Shortages

The figure shows the quantity in inventory versus time. The lot enters the system at a finite rate.

Infinite Replenishment Rate with Shortages Backordered

The figure shows the inventory position versus time. When the position is positive, the on-hand inventory is the same as inventory position and there are no shortages. When the inventory position is negative, the on-hand inventory is zero and the amount of shortage is the negative of the inventory position.

Two cost models are considered in the backordered case: the cost for each customer experiencing a backorder is proportional to the waiting time for delivery, and the cost is independent of the waiting time.

Finite Replenishment Rate with Shortages Backordered

This is the same as the previous case except the replenishment rate is finite.

Infinite Replenishment Rate with Shortages Lost

Here when the on-hand inventory is exhausted, customers who arrive before the next replenishment will not wait. Rather, the sale is lost. The next replenishment delivers the entire lot. There is a cost associated with each lost sale.

Finite Replenishment Rate with Shortages Lost

This is the same as the previous case except replenishment occurs at a finite rate.

Operations Management / Industrial Engineering
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by Paul A. Jensen