For example, the ideal scenario is to create *just enough* inventory to satisfy's the period's needs. Creating any more (assuming a perishable good) results in inflated costs related to waste and/or inventory carrying costs. Creating any less results in lost revenue related to stock-outs.

However, in real life, it is unrealistic to assume perfect inventory planning all the time, so chances are there will be some days with over stock and some days with stock-outs.

The basic formula for profit is: Profit = Revenue - Costs

and

Profit Margin = Marginal Revenue - Marginal Cost or Marginal Revenue - Variable Cost

We know that we will incur some sort of inefficiency or uncertainty cost in the form of over / under stocking as mentioned above. However, the idea is to minimize this "capacity cost" we have to understand how operations affect these costs. For example:

A bakery sells donuts for $1.00. Donuts cost 10c to make. Therefore, over-stocking results in a cost of 10c per donut due to wastage. However, stock-outs cost $1.00 per donut due to lost sales. Therefore there is a 10 to 1 cost per unit on either side of the ideal capacity target. Let's say on any given day, the average sales is approximately 1000 donuts.

To minimize the cost side of the profit equation, we have to look at the probability of capacity distributions above and below the target.

Let's make a HUGE assumption (for simplicity) and say that there is a uniform distribution about the target (not normal, but uniform). Let's say there is a 10% chance of the actual daily sales being:

- 950
- 960
- 970
- 980
- 990
- 1000
- 1010
- 1020
- 1030
- 1040

How many donuts should the bakery produce?

I would propose that you should overlay the capacity with the associated cost of capacity management. What do I mean? If you produced 1000 donuts and you only sold 950, you're capacity related costs would be amount of capacity variance x cost per unit of variance. Generally this would be expressed as:

Capacity Related Cost = Capacity Variance x Cost per unit of Variance

So in this case:

Capacity Related Cost = 1000-950 x (10c)

=$5.00

What about baking 1000 donuts and then selling all 1000, but having an additional 20 donut customers who go unserved? Then:

Capacity Related Cost = 1000-1020 x ($0.90)

= $20.00

Notice that for a smaller number of donuts not sold, there is a much larger effect on Capacity Related Costs. This is a reflection on the profit margin (profit from one lost sale = marginal revenue - variable cost). That is to say, it is much worse to not sell 1 donut rather than have 10 donuts go stale.

To optimize planning (create a capacity level which will optimize profits) it would make sense to minimize the Capacity Related Costs. Since we have the probabilities of the capacity distribution and the associated costs with under production, what target capacity creates the minimal expected capacity related cost? Below is a chart outlining the capacity related costs for given target and actual production levels.

You'll notice that the optimal production level is actually 1040 or 1030. This sort of makes sense as the costs for missed sales are so high. Generally, because of the structure of the model, there are two major factors affecting the end result for capacity planning:

- Volatility and variablity of demand
- Profit margin (difference between cost of wastage and cost of lost sales)

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