# EOQ: How much inventory to buy? – to stock or not to stock that inventory

Retail Managers are responsible to ensure stock availability to customers by replenishing stocks regularly. At the same time, they are also responsible for profitability of their category or division, leading to questions on – whether we should carry that product and how much inventory should we carry on hand? This is where it gets interesting. Because if you want to cater to all customer demand, then theoretically you must carry infinite inventory so that you cater to all customers who may or may not come. And, carrying infinite inventory is a loss-making proposition, leading to unhealthy inventory and write-offs. Let us understand this through an example below.

Let’s say the product you are going to sell costs \$6 and you will sell it at \$10, making a profit of \$4. However, if you cannot sell that product then you will incur a loss of \$6 (product cost).

Therefore, the tradeoff here is between:

a). the profit that you’d earn if the customer walks in and you have it in your inventory

b). the loss that you’d incur if you stock and the customer doesn’t turn up

To make the right decision of whether to hold that extra unit of inventory or not, you need to understand the probability of selling that inventory (say one unit) denoted as Px.

• If the probability of selling that extra unit is 100% (P100), then you should surely stock it and make a profit of \$4.
• If the probability of selling that extra unit is 25% (P25), then the payoff is

25%*\$4 + 75%*(-\$6) = -\$3.5 (negative payoff). Since this is going to make a negative payoff, you shouldn’t stock that unit of inventory.

• If the probability of selling that extra unit is 80%(P80), then the payoff is

80%*\$4 + 20%*(-\$6) = \$2 (positive payoff)

• If the probability of selling that extra unit is 60%(P60), then the payoff is

60%*\$4 + 40%*(-\$6) = \$0 (critical point). So, as long as the probability of selling that extra unit is 40% or above, you should keep stocking inventory. Most auto-buying systems are designed in this way to stock inventory until the probability of selling that unit of inventory goes below the critical point.

Also, the probability of selling that unit is within what time period? It is within the planning cycle (order cycle time plus lead time from PO to delivery). Your days on hand should always be as lean as possible and close to the planning cycle. Credit period is the maximum limit of the planning cycle, but ideally shorter the planning cycle the better the benefit is for the business

The one mistake that we did in the above calculation is we wrongly assumed that profit that we will make out of the sale is only from that particular unit. We should also include the profit that we could’ve made by that particular customer in all future purchases at our store – customer lifetime value (CLV). So, the tradeoff has to be between the customer lifetime value of the customer vs. the loss you’d incur if the customer doesn’t turn up at all. Also, the above calculation should take ‘time’ into account in the form of cost of capital (inventory holding costs) because many times the loss is not the entire cost of the product.

In conclusion, the value of having that inventory in stock changes basis a lot of parameters such as CLV, importance of that category or brand to the image of the retail store, brand equity of the store, type of store and many other business parameters.

The one question that we are yet to answer is: how do we determine the probability of sale?

Intuitively you know that the probability of selling that first unit of a TV is 100%, the 100th unit is say 80%, 500th unit is 40% and 1000th unit is 10%. Probability changes by quantity.

To get the probability you should simulate a demand distribution basis past historical sales data. Demand in retail scenarios are usually normal distributions. So, we will need the average and standard deviation. It is what we learnt in our MBAs – you will calculate the probability for the random variable (sellout of a TV model) to take the value 500 units and that is P (X=500). One of the ways is for you to calculate the Z value =  (500 – average)/SD and then lookup for that Z value in the Z table to get the probability. This is if the distribution is gaussian or normal distribution. There are other ways too to get this estimate basis various distribution fits such as chi-square, t distribution (for small sample size of 30-100), gamma distribution, beta distribution, pareto distribution and others. Each of these distributions has a method to calculate the probability like the way we have for normal distribution as mentioned above. Typically, in the pre-computer era, because it was so difficult to calculate the area under the curve manually statisticians used to categorize the data they have into one of the 20-30 templatized distributions and use one of the methods to calculate the probability. In the computing era of today, we can actually plot the data and calculate the actual area under the curve with the help of a computer quickly. Therefore, if you don’t know which distribution the plot is looking like it is better to just calculate the area under the curve manually and get the probability. This method is called the ’empirical method’ or ’empirical distribution’. So, if you don’t know which distribution is your data looking like, you can opt for empirical probabilities.

Thank you.