November 18, 2019      < 1 min read

Inventory control is exceedingly difficult if you go into it blind, with no tips and tricks and no supporting software. However, with the right knowledge and methods in place, it can be something you can manage and control with few problems. Part of the art of inventory control lies in the ability to adapt to demand in an efficient way so that supply is maintained appropriately. However, this is difficult when there are inherent levels of uncertainty associated with customer demand, product costs, holding costs and lead times. So, how can businesses cope with inventory uncertainties?

### The probabilistic model of inventory control

This model is based on the assumption that there is inevitable uncertainty associated with factors such as demand and lead times in some retail industries. Taking this into consideration, probabilistic models of the demand are then drafted with minimum reorder quantities or timeframes. These essentially form calculations of ‘best fit’ from which appropriate orders can be made that ensure supply is maintained and resources are optimised.

### 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.

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