No Shortages

Shortages Backordered

Lost Sales


Inventory Theory
- Deterministic/Infinite Replenishment/No Shortages

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 Rather than demand occurring in a random and uncertain manner, we assume that items are withdrawn from the inventory at a continuous rate.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. The resulting behavior of the inventory is shown in Fig. 1. We use this deterministic model of the system to explain some of the notation associated with inventory. Because of its simplicity, we are able to find an optimal solution to this deterministic model. The solution specifies the optimum lot size. This appears in many texts on operations management with the name economic order quantity or EOQ.


Infinite Replenishment Rate and No Shortages


The first model considered is illustrated by the figure below which shows the variation of the inventory level with time.

Figure 1. Inventory with Infinite Replenishment rate and no shortages

The figure shows time on the horizontal axis and inventory level on the vertical axis. We begin at time 0 with an order arriving. The amount of the order is the lot size, q. The lot is delivered all at one time causing the inventory to shoot from 0 to q instantaneously. Material is withdrawn from inventory at a continuous demand rate, D, measured in units per time interval. We are assuming that the material is withdrawn in a continuous fashion, rather than in discrete units, so we show the inventory level declining as a straight line. After an amount of time q/D, the inventory is depleted. At that time another order of size q arrives and the cycle repeats. The cycle time is .

The inventory pattern shown in the figure is obviously an abstraction of reality in that we expect no real system to operate exactly as shown. The abstraction does provide an estimate of the optimum lot size, called the economic order quantity (EOQ), and related quantities. We consider alternatives to those assumptions on later pages.


Formulas for Instance Results



Click buttons to see the notation

Here we derive the formulas for the results for an instance. An instance is defined by a collection of inventory parameters and a value for the lot size q. Click the buttons on the left for a description of the notation for parameters and results.

These formulas are implemented in the Inventory add-in.


To construct a mathematical model describing the economic costs or profits associated with the inventory system, we show the cash flows below. This figure is a mixed representation of discrete as well as continuous cash flows. The arrows represent amounts paid or received at points in time. The areas represent continuous cash flows given by rates. Amounts appearing above the 0 axis are revenues, while amounts below are expenditures or costs.

The table below shows the various revenue and cost components and their respective cost rates. In all of the following, we assume that the parameters are nonnegative and that q > 0.

Amount during a cycle
Cost or revenue rate

Ordering Cost

In each cycle an order is placed for the quantity q

Product Cost
Holding Cost for Product in Inventory
Linearly varies from Hq to 0 in each cycle
Revenue from product Sales

For this model there is a single decision variable that is the lot size or the replenishment, q. All other quantities are a function of q. The inventory cost is a strictly convex function of q, so there is a unique global minimum with respect to q.

Additional quantities associated with the inventory policy are derived below.

When the lead time is greater than the cycle time, an order for replenishment must be placed more than one cycle before the order is delivered. The add-in performs this more complicated computation.

The measures are computed below for the example using a lot size of 400 units. This is called an instance of the inventory model. The figure shows a single cycle of the inventory pattern.


Optimum Policy


For determination of the optimum lot size q*, the unit revenue and unit cost can be neglected and we write the expression for the cost rate for operating the inventory.

Since the second derivative is positive for positive parameters, the inventory cost function is convex and the solution for q* is a global minimum.

When q* is between the minimum and maximum lot sizes, the inventory measures with the optimum lot size are found by substituting q* into the instance formula.

At the optimum, the holding cost is equal to the setup cost. We see that optimal inventory cost is a concave function of product flow through the inventory, indicating that there is an economy of scale associated with the flow through inventory. The optimal lot size increases with increasing setup cost and flow rate and decreases with increasing holding cost.

The table below shows the parameters and instance results as well as the results for the optimum lot size (q*= 816).


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Operations Management / Industrial Engineering
by Paul A. Jensen
Copyright 2004 - All rights reserved

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