[MUSIC] Now that we can evaluate the expected NPV of a capacity decision, we can identify the optimal capacity size by maximizing expected net present value. Instead of resorting to a spreadsheet, we review basic optimization logic to understand the factors driving optimal capacity sizing. Marginal analysis investigates whether a small change to a decision variable is desirable. Continuing the example of the previous section, would it be desirable to increase the capacity from 75 to 76? >> This is easiest calculated by considering the increased operating profit in each scenario. In the low demand scenario, the original capacity of 75 was sufficient to fill all demand. So that an additional capacity unit is worthless. In the two other scenarios however, an additional capacity unit would reduce the shortage and yield an additional operating profit of $10. The shortage probability equals 75%. So that the expected increase in operating profit is 10 x 75% = 7.5. The increase in capital expenditure for the additional capacity unit is $5, so that the expected change in NPV equals to $0.50 Given that the expected NPV increases, it is desirable to increase capacity to $76. >> Repeating this argument shows that it is optimal to increase capacity from 75 to 100. Because it increases expected NPV by 2.5 times 25 is $62.5 dollars to a total of $437.5 dollars. Further increasing capacity beyond 100 would reduce value. Indeed, the shortage probability would fall from 75% to 25%. And the expected value of one additional capacity unit would fall to 10 times 25% is two and a half dollars, which is less than the marginal capacity cost. >> The logic above reveals that the marginal value of capacity, stems from the value of reducing shortages relative to the cause of having excess capacity or overage. In a single product, single asset setting, there is a simple formula for the optimal capacity size, if demand is a continuous random variable level three uncertainty. Then the optimal capacity size equates marginal value to marginal cost and solves the following equation. The optimal shortage probability equals the marginal capacity cost divided by the operating profit of marginal capacity unit. This expression is called the newsvendor solution. With discreet scenarios, it is sub-optimal to reduce the shortage probability below the optimal level. For example above, the optimal shortage probability is $5 divided by $10 equals 50%. >> Specifying an optimal shortage probability is equivalent to specifying an optimal service level. The optimal service probability is the probability that d is less then the capacity. And that equals one minus the probability that d exceeds the capacity. The optimal service probability is often expressed in terms of the marginal cost of underage, CU, and the marginal cost of overage, CO. For a continuous demand, the formula is easily remembered using marginal analysis. Which specifies the optimal capacity case star as Co times the probability that demand meets capacity must equal Cu times the probability that demand exceeds capacity. A quick substitution yields that the probability that demand is less than capacity is the fraction of Cu divided by the sum of Cu plus Co. To summarize, a capacity strategy is the long-term plan for developing resources and involves decisions on sizing, timing, types, and location of capacity under uncertainty. The key trade-off in capacity sizing is between the cost of excess capacity and opportunity cost of capacity shortfalls due to lost sales, customer waits, or substitution to other products or competitors. In this module, we demonstrated how a firm that tries to scale its operations needs to use its vision regarding the future to make such decisions. [MUSIC]