Question

. A bike shop is selling a fashionable newly designed folding bike. The shop is now considering how many of them to order for the coming season. The supplier requires that orders for the bikes must be placed in quantities of 25. The cost per bike is $820, $790, $750 and $700 for an order of 25, 50, 75 and 100 respectively. The bikes will be sold for $1,200 each. Since there will be a new design for folding bikes next season and there is no spare storage space in the store, any bikes left over at the end of this season will have to be sold at a low price of $450 each to another bike shop. However, if the shop runs out of bikes during the season, it will suffer a loss of goodwill among its customers. The shop estimates this goodwill loss to $60 per customer who is not able to buy a bike. From past experience, the shop estimates that the demand for folding bikes this coming season will be 25, 50, 75 and 100 bikes with probabilities of 0.4, 0.3, 0.2 and 0.1 respectively.

(a) Construct the payoff table for the above situation.

(b) Which alternative should be chosen using each of the maximax, maximin, minimax regret, Hurwicz (take α = 0.6), equal likelihood, expected value, and expected opportunity loss criteria?

(c) Find the expected value of perfect information.

Answer 1

a)

We need to consider two angles. One is the loss of good will and the other is the loss from excess order.

If we order 25 bikes and the demand is 50 then we make a total profit of 25*(1200-820). However we have a loss of goodwill that cost (50-25)*60. This means the net profit is 25*(1200-820)-(50-25)*60 = 8000.

If we order 75 bikes and the demand is 50 then we make a total profit of 50*(1200-750). However the excess 25 bikes will provide a profit (loss) of 25*(450-750). The net profit is 50*(1200-750)+25*(450-750) = 15000

This way we create the payoff table. It is shown below

b)

Maximax = Maximum payoff from each decision is shown in the image above. The best decision based on this is to order 100 bikes for payoff of 50000

Maximin = Minimum payoff from each decision is shown in the image above. The best decision based on this is to order 25 bikes for payoff of 5000

Minimax regret = Regret table is shown in the image. Comparing the maximum regrets for each decision, the best decision is 100 bikes. It has the lowest regret at 15750

Hurwicz criteria = Value for each decision using Hurwicz criteria multiplies α with highest payoff and (1- α) with the lowest payoff. Those values are shown in the image. The best decision is to order 100 bikes with an expected Hurwicz value of 27500

Equally likely = When the events of demands are equally likely each possibility of demand has a probability of 0.25. Considering this the value for decisions are shown in the image. The best decision is to order 100 bikes with an expected value of 21875.

Expected value = We do the same approach as equally likely but instead use the given probabilities. The expected values are shown in the image. The best option is to order 75 bikes with an expected payoff of 13350.

Expected opportunity loss = We use the expected value method on the regret table. The opportunity loss for each decision is shown. The lowest is to order 75 bikes with expected opportunity loss of 7400.

c)

We have determined that the expected value method yields a value of 13350. This is expected value without perfect information (EVwoPI). Now, if we have perfect information we will choose the best options. For example if we know that the demand will be 25 bikes we will only buy 25 bikes. This means there is a 0.4 probability to get a payoff of 9500. Same way the 0.3 probability to get 20500, 0.2 probability to get 33750, and 0.1 probability to get 50000. This means the expected value with perfect information (EVwPI) is 0.4*9500 + 0.3*20500 + 0.2*33750 + 0.1*50000 = 21700

This means the value of perfect information is worth EVwPI – EvwoPI = 21700 – 13350 = 8350