# The Probabilistic Model of Inventory Control Explained

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Regardless of the industry, it is necessary for manufacturers and retailers to maintain the right quantities of inventory stock to ensure the smooth running of production operations and sales activities.

Holding certain levels of inventory stock helps companies to avoid lost sales, will reduce ordering costs, help to maintain efficient production runs and reduce customer service dissatisfaction. However, in addition to these benefits, there are two broad costs associated with holding inventory stock: order processing costs and carrying costs.

To mitigate some of the costs associated with ordering and carrying inventory stock inventory models have been developed to help companies determine the optimal inventory stock levels to maintain relative to their organisation.

These inventory control models are classified into two major types the Deterministic Models, built on the assumption there is no uncertainty in the demand and replenishment of inventory stock and Probabilistic Models which acknowledge a degree of uncertainty in the demand pattern and lead time of inventories.

### Probabilistic model of inventory control

The Probabilistic inventory model is closely aligned to the manufacturing and retail reality that from time to time, demand will vary. Demand variations cause shortages, particularly during lead time if a retailer only has a limited amount of inventory stock to cover the demand during the lead time when replenishment stock has not arrived.

The probabilistic inventory model incorporates demand variation and lead time uncertainty based on three possibilities.

The first is when lead time demand is constant but the lead time itself varies and the second is when lead time is constant but demand fluctuates during lead time. The third possibility is when both lead time and demand during lead time vary.

Employing known economic, geological and production data the probabilistic inventory model creates a collection of approximate inventory stock quantities and their related probabilities. The advantage of a probabilistic approach is that by using values within a bandwidth, modelled by a defined distribution density, you achieved greater reliability than when using deterministic figures.

### Probabilistic inventory methods

Probabilistic inventory models consisting of probabilistic supply and demand are more suitable in most circumstances. Two methods are used based on the frequency of order placement for procuring inventory stock, these are single period and multi-period inventory systems.

• The term single period term refers to the situation where the inventory stock is perishable, and orders are typically only made once. Generally, for one time ordering of seasonal products or where demand exists only for the period in which it is ordered. For example, a newspaper sold today will not be sold at the same price tomorrow nor will summer clothing items be likely to sell during the winter season
• An incremental analysis is used to determine the optimal order quantity for a single period inventory with probabilistic demand. Assessing how much to order by comparing the cost or loss of ordering one additional unit with the cost or loss of not ordering that one additional unit

With the multi-period method orders are placed multiple times over an entire production cycle and are further classified as continuous review or periodic review inventory.

• Continuous review inventory is reviewed constantly and when inventory stock drops to a certain predetermined par or reorder level, a fixed quantity is ordered. Continuous review is commonly used for high volume, valuable or important stock items
• Periodic review inventory is examined at periodic intervals in predetermined timeframes, irrespective of the levels to which inventory levels drop. At this time an order is then placed to bring inventory up to the maximum level, the method is largely used for moderate volume items

In plain terms, the probabilistic model of inventory control is based on or adapted to a theory of probability which involves or is subject to chance variation. Multiple possible outcomes exist, each having varying degrees of certainty or uncertainty of its occurrence.

## Digging deeper: the Newsvendor model

One of the models used in probabilistic inventory control is the Newsvendor or Newsboy model. This is a single-period inventory model used for seasonal or perishable items with a discrete demand pattern. In fact, the name derives from the Newsvendor’s dilemma of how many newspapers to order to meet the demand for today, knowing that tomorrow, the product no longer has the same value.

We can delve into the involved algebra of this inventory model, or we can consider things more simply to better understand its value.

Take for example a business that is selling sun umbrellas. The question they are asking is how many sun umbrellas they need to satisfy demand over the summer; this unknown quantity is Q. We must also determine the cost of buying the product initially (c), the price we sell the product for or revenue (r) and the discounted price we sell the product for as a salvage value at the end of the season (s).

If the lead time is so long that we can only order the product once in the period, then we are extremely interested in ordering the right amount because this cannot be rectified later.

### Marginal costs

We are also interested in the trade-off between overordering and having to sell the excess for salvage value, and not ordering enough which therefore renders us short and forfeiting possible sales. Defining these (the cost of excess and cost of shortage) are our marginal costs.
So now, let us substitute in some values. Let’s assume the following:

• c = \$300
• r = \$500
• s = \$50

Using these numbers, we can work out the cost of excess and cost of shortage. The cost of excess is calculated as Ce = c-s because it is the amount the product cost us initially minus the amount we managed to salvage at the end of the season. The cost of shortage is calculated as Cs = r-c because it is the amount we would have sold the product for if we had it minus the amount the product would have cost us. Therefore, if we had one item in excess and one item short respectively, these costs are as follows:

• Ce = \$300 – \$50 = \$250
• Cs = \$500 – \$300 = \$200

### Service level

This is the chance we can meet all demand in a single period (the summer season in this case). For simplicity’s sake, if we imagine that the number of products we sell can be anywhere from 1 to 10 where each number of products sold is discrete, and we order a quantity of 5 units at the start of the summer season. We will meet the demand with our outset of 5 units if 5 or fewer units are sold over the season. So, the probability of meeting this demand with the outset is cumulative and is the sum of each of the discreet demand values (for example, the probability of 1 unit, 2 units 3 units, 4 units and 5 units being sold).

The probability of 1 unit sold out of 10 is 0.10. For two units, it is 0.10 x 2 and so on. Therefore, the probability of 5 units being sold is 0.5. This means if we order 5 units at the outset we will provide a service level of 50%.

#### Optimal service level

The optimal service level is given by the following formula:
Cost of shortage ÷ (Cost of shortage + Cost of excess)
So, in the above example, this would be:
200 ÷ (200+250) = 0.44

#### Optimal order quantity

The optimal order quantity is the minimum order size needed to meet the optimal service level. So, we need to meet the optimal service level of 0.44 with our order. We have determined that the probability of demand for each item is 0.10 with a cumulative effect up to a maximum of 10. So, to meet 0.44, we would have to order at least 5 units (0.1 x 5 = 0.5). This is the optimal order amount to avoid shortages and excess.

This is a very simple portrayal of using a probabilistic model for estimating demand and managing your inventory. Of course, as with any part of inventory control, the estimations and subsequent decisions are only as good as the input data. This is derived from accurate and reliable inventory management software which keeps track of every item as it moves through your warehouse. 