Computation Section
Subunit Inventory Analysis

Stochastic Models

Stochastic Inventory System

Of course real inventory systems are not deterministic as in the models considered previously. Although the models that neglect uncertainly may yield results that are useful in many contexts, it is possible to describe and analyze models that explicitly include the uncertainty associated with some aspects. There are many aspects that might be uncertain including the lead time, the quantity that is actually received given the amount ordered, or the amount demanded in any time interval. It is impossible to provide analytical results for the most general case. Here we consider only uncertainty in demand.

One possible stochastic inventory situation is illustrated in the figure above. This is called the (q, r) system and the add-in provides analysis and optimization tools for this system. As for deterministic systems, the inventory level is affected by demands and replenishments, but here we show the demand process as variable, sometimes the demand rate is high causing quickly declining inventory levels, while at other times the demand rate is low with the inventory declining more slowly. For this system, we have some inventory level r at which we place an order for an inventory replenishment. This is called the reorder point. The amount ordered is q, the order quantity. After we place the order, we must wait for some interval, L, the lead time, before the inventory is replenished. Added to the models for the stochastic systems are the risks and costs of shortages, shown as the red areas. If the demand is high during the lead time (greater than r), some customers will not be satisfied. We can set the reorder point high to make shortages unlikely, but that will increase the inventory levels, shown as the blue areas. The order quantity affects both shortages and inventories as well as the cost of replenishment. Our solutions will set system variables to balance the costs of inventory, shortage and replenishment.In the following we describe some of the options available for analysis.

The add-in allows stochastic inventory models to be defined. The models allow the demand during the lead time to be governed by a probability distribution. All results are based on mathematical formulas that are described in theoretical textbooks on inventory theory.

To create a model choose Add Inventory from the menu. When shortages are allowed, several options are presented for demand. The Deterministic option creates the models considered previously. The other three options define the probability distribution for demand during the lead time. We describe the options on the dialog below.

Infinite or Finite Replenishment Rate

Infinite Rate

For the finite replenishment rate, orders arrive in whole lots. The figure illustrates the case where shortages are backordered. The inventory level rises from its lowest level to an amount q greater than the lowest level. The lead time (L) and reorder point (r) are important for the (q, r) system as the uncertainty of demand during the lead time determines the cost of the system. If the lead time were zero, the minimum inventory level would be zero in each cycle and the maximum inventory would be q.

Finite Rate

The same pattern of demand is shown at the left for a finite rate system. This has the same shortages as the infinite case as indicated by the equal time intervals during which shortages occur. The average inventory and backorder levels are reduced because replenishment amounts arrive at a finite rate rather than an infinite rate. Notice that the time intervals in which production is actually taking place are equal. With a production rate P, the interval is q/P. Since the cycle times are not constant in the stochastic system, the value of P must be great enough so that there is a high probability that the complete lot is produced before the next cycle starts. This is another penalty associated with uncertainty. The production capacity must have some excess to allow for the uncertainty in demand. Our add-in requires that P >= D, but does not enforce excess capacity.

Because the low points in inventory occur a little earlier for the finite case than the infinite case, the reorder point must be a little higher for the finite case than the infinite case (r' > r). Because the approximations used by the add-ins, this feature is not reflected in the add-in results.

Demand Probability Distribution

Poisson Distribution

The uncertainty modeled is the demand during the lead time. When the average demand per time interval is D and the lead time is L, the average demand during the lead time is DL. The Poisson distribution is appropriate when demand events occur independently and in single units. The parameter of the Poisson is its mean value (DL). The standard deviation of the Poisson is the square root of the mean. The figure shows the Poisson distribution with a mean value of 10. Excel has built-in functions to evaluate both the individual probabilities and the cumulative distribution.

When the mean demand is greater than 30 it is difficult to evaluate the Poisson probabilities accurately, so we use the Normal distribution as an approximation.

Normal Distribution

The normal distribution is a good approximation of the Poisson for larger values of the mean. This option uses a Normal distribution with mean DL. The standard deviation is an input parameter. For some problems it is interesting to vary the standard deviation to observe the effects of uncertainty. If the Normal is to approximate the Poisson, the standard deviation should set equal to the square root of the mean.

It is convenient to have the parameters of the distribution depend directly on the lead time. Then we can experiment with different lead times without redefining the parameters.

Excel has built-in functions to evaluate the density function and cumulative distribution of the Normal distribution.

Normal-LT For this option, the distribution of the demand during the lead time does not explicitly depend on L. Rather, both the mean and standard deviation are input parameters. This is convenient for some problems, but in reality the distribution of demand does depend on the lead time.

Backorders or Lost Sales

We have illustrated the backorder case in the previous figures. In the figure we illustrate the lost sales case. Here whenever a shortage occurs, the customer is not served. When a replenishment arrives after a shortage interval, the inventory rises to the level q because there is no need to deliver backordered items.

Shortage Cost

Shortages Backordered



The figure shows an inventory cycle assuming shortages are backordered. The cycle shows the relevant expected values. Of course a sample cycle will be different than shown because the demand during the lead time may be less than r and no shortage will be observed. In fact, the theoretical development of stochastic inventories assume that shortages are rare events. Some notation in the figure is defined below.

For the backorder case, we have three alternatives for the cost experienced when a shortage occurs.

Cost per shortage event: Here there is an expense in every cycle where a shortage occurs. The expected shortage cost in a cycle is:

Cost per unit short: Here there is an expense for every unit that is demanded when the inventory position is negative. The expected shortage cost in a cycle is:

Cost per unit-time short: Here there is an expense for every unit that is demanded when the inventory position is negative. The expense is proportional to the time spent in a backordered position. The expected shortage cost in a cycle is:

Lost Sales

Cost per lost sales: Here there is an expense for every unit that is demanded when on-hand inventory is zero. A cycle of inventory appears in the figure with expected values. The expected shortage cost in a cycle is:

With stochastic models, additional parameters must be defined as illustrated in the Parameters dialog below. The parameters available depend on the options selected.

The parameters include the stochastic parameters that define the probability distribution. For the Poisson distribution the demand mean and standard deviation are entirely determined by the expected demand during the lead time. For the Normal distribution, the mean is the expected demand during the lead time, but we specify the standard deviation. For the Normal LT option, we specify the mean and standard deviation of the demand during the lead time. The dialog above results if the Normal LT options is chosen for on the inventory definition dialog.

One additional parameter is the Maximum Probability Short. For stochastic models, the reorder point determines the risk that the inventory will be exhausted during a cycle. Designers often want to limit this to some small value. We provide this option with this parameter. Optimum solutions are limited to assure that this maximum is not exceeded.

The number of display options is also increased for a stochastic model as shown below. The two options on the bottom are only relevant when demand is random.


An Example with a Poisson Distribution for Demand

We show three examples with different probability distributions for demand. We do not show the instance results, but they are available when the user wants to experiment with non-optimum results. The examples show the optimum results. The optimum selects the lot size and fill rate that maximizes total expected profit.

The example describes a situation when the mean demand rate is 20 per week and the lead time is one week. The optimum lot size and fill rate are shown in the display at the left. The reorder point that determines this result is 28.

Note that the expected demand during the week is 20. The extra 8 units guard against the occurrence of shortages. It is optimum for the costs indicated.

All the results that appear below the optimum solution are expected values since they all depend on the random demand. We have changed the format of some of the cells to see the results more accurately

One interesting result is the probability of a shortage (second from the bottom). This is the probability that a shortage will occur in an inventory cycle.

The safety stock measures the increased inventory level that is added due to protect against uncertainty. It is the difference between the reorder point and the expected demand.


An Example with a Normal Distribution for Demand


When demand during the lead time is relatively large, it is more reasonable to assume that the demand is Normally distributed. In the example case, the mean is the expected demand during the lead time (100 for the example). The standard deviation per week is a parameter. We often assume the standard deviation is the square root of the mean.

For the example the mean demand during the one week lead time is 100 and the standard deviation is 10. The optimum solution has a safety stock of about 13 units and an order quantity of 245. The risk of stockout during a cycle is 0.103.

When the Poisson distribution is chosen and the mean demand is greater than 30, the Normal distribution is used as an approximation.



An Example with a Normal Distribution for Demand during the Lead Time


For some situations the lead time is not given, but we know the distribution of the demand during the lead time. The Normal LT is designed for this situation. The data includes the lead time, but its value does not affect the reorder point. Rather the data provides both the mean and standard deviation of demand during the lead time.

This example uses the same data as the previous example, but the standard deviation is 30 rather than 10. The safety stock is increased to 52 units.

The lead time as an impact in all the models in that the order quantity must be at least as great as the expected demand during the lead time plus three times the standard deviation. This makes it very improbable that the demand during a cycle will exceed the order quantity.

  The models can all be adapted for a finite production rate.
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Operations Management / Industrial Engineering
by Paul A. Jensen
Copyright 2004 - All rights reserved