Question 4 (26 marks) Mrs Kwan is the owner of a store selling fruits imported f

Question

Question 4 (26 marks) Mrs Kwan is the owner of a store selling fruits imported from Japan. She has to decide the amount of Shine Muscat grapes to order from Okayama Prefecture at the beginning of each month during the summer. The cost per pack of grapes is $150 and the selling price is $250 per pack. At the end of each month, unsold grapes will be sold at $80 per pack. If there is a shortage of grapes, Mrs Kwan would consider the profit of S100 per pack that she could have earned to be an opportunity cost and, in addition, $30 per pack will be taken as customers' ill will cost. From past experience, the probability distribution of the monthly demand for Shine Muscat grapes is: Demand (number of packs)100 105 110 115120 Probabilities 01 0.2 0.3 0.30.1 a Construct the payoff table for the above situation. (10 marks) Which alternative should be chosen using maximax, maximin and the Bayes'decision rule? b (11 marks) c Find the expected value of perfect information. (5 marks)

Explanation / Answer

(a) The pay-off table is as shown below:

The profit corresponding to an order of 115, but demand of 110 is calculated as 110 x 250 + 5 x 80 - 115 x 150 = $ 10,650

The profit corresponding to an order of 115, but demand of 120 is calculated as 115 x 100 - 5 x 130 = $ 10,850

(b) Maximax Choice: The maximax looks at the best that could happen under each action and then chooses the action with the largest value. They assume that they will get the most possible and then they take the action with the best best case scenario.

We see that the maximum of the columns under the alternatives (how much to order) is 12,000. So we choose to order 120 packs.

Maximin Choice: The maximin person looks at the worst that could happen under each action and then choose the action with the largest payoff. They assume that the worst that can happen will, and then they take the action with the best worst case scenario.

For the 5 alternatives, we see that the minimum values are 7400, 8550, 9300, 8950, 8600. The maximum among these is 9300. So we choose to order 110 packs

Baye's decision rule: We compute the expected value under each action and then pick the action with the largest expected value.

As is seen in the last row of the pay-off table, we have the expected value for each order quantity. We see that the maximum expected profit is 10,585 corresponding to the order of 115 packs. So we choose to 115 packs under Baye's decision rule.

(c) To find EVPI (Expected Value of perfect information), we calculate the profit we can make if we have the perfect information of demand (for each demand scenario, we choose the order value which maximises the profit) and then subtract the expected profit from this to get the EVPI.

The maximum of the row profits are 10000, 10500, 11000, 11500 and 12000 respectively, for demands of 100,105,110,115 and 120 packs respectively.

So, Expected value, given perfect information of demands = 0.1 x 10000 + 0.2 x 10500 + 0.3 x 11000 + 0.3 x 11500 + 0.1 x 12000 = 11,050

We already have calculated the maximum Expected profit as 10,585

So EVPI = 11050 - 10585 = 465

Demand (No. of packs) Probability Alternatives (no. of packs ordered) 100 105 110 115 120 100 0.1 10000 9650 9300 8950 8600 105 0.2 9350 10500 10150 9800 9450 110 0.3 8700 9850 11000 10650 10300 115 0.3 8050 9200 10350 11500 11150 120 0.1 7400 8550 9700 10850 12000 Expected Profit 8635 9635 10335 10585 10385

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