The Islander Fishing Company purchases clams for K1.50 per pound from fishermen and sells
them to various restaurants for K2.50 per pound. Any clams not sold to the restaurants by the end
of the week can be sold to a local soup company for K0.50 per pound. The company can purchase
500, 1,000, or 2,000 pounds. The probabilities of various levels of demand are as follows:
Demand (Pounds) Probability
500 0.2
1,000 0.4
2,000 0.4
(a) For each possible purchase level (500, 1,000, or 2,000 pounds), Clearly and step by step compute the profit (or loss)
for each level of demand.
(b) Clearly and step by step Determine the optimal action based on the maximax criterion.
(c) Clearly and step by step Determine the optimal action based on the maximin criterion.
(d) Clearly and step by step Using the expected monetary value (EMV) criterion, determine the optimal number of
pounds of clams the company should purchase from the fishermen. Discuss.
(e) Clearly and step by step Compute the standard deviation for each possible purchase level.
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(f) Clearly and step by step Compute the expected opportunity loss (EOL) for purchasing 500, 1,000, and 2,000
pounds of clams.
(g) Clearly and step by step Explain the meaning of the expected value of perfect information (EVPI) in this problem.
(h) Clearly and step by step Compute the coefficient of variation for purchasing 500, 1,000, and 2,000 pounds of
clams. Discuss.
(i) Clearly and step by step Compute the return-to-risk ratio (RTRR) for purchasing 500, 1,000, and 2,000 pounds of
clams. Discuss.
(j) Based on (d) and (f), would you choose to purchase 500, 1,000, or 2,000 pounds of clams? Clearly and step by step stae Why
(a)
For 500 pounds of clams:
- Profit if demand is 500 pounds: 500*(2.50-1.50) = K500
- Profit if demand is 1,000 pounds: 500*(2.50-1.50) + (1,000-500)*(2.50-0.50) = K1,000
- Profit if demand is 2,000 pounds: 500*(2.50-1.50) + 500*(2.50-0.50) = K1,500
For 1,000 pounds of clams:
- Profit if demand is 500 pounds: 500*(2.50-1.50) + (1,000-500)*(2.50-0.50) = K1,000
- Profit if demand is 1,000 pounds: 1,000*(2.50-1.50) = K1,000
- Profit if demand is 2,000 pounds: 1,000*(2.50-1.50) + (2,000-1,000)*(2.50-0.50) = K1,500
For 2,000 pounds of clams:
- Profit if demand is 500 pounds: 500*(2.50-0.50) = K1,000
- Profit if demand is 1,000 pounds: 500*(2.50-0.50) + 1,000*(2.50-1.50) = K1,000
- Profit if demand is 2,000 pounds: 2,000*(2.50-1.50) = K2,000
(b) MaxiMax criterion: Choose the maximum possible profit for each purchase level.
For 500 pounds: Choose 2,000 pounds
For 1,000 pounds: Choose 2,000 pounds
For 2,000 pounds: Choose 2,000 pounds
(c) MaxiMin criterion: Choose the maximum of the minimum profits for each purchase level.
For 500 pounds: Choose 500 pounds
For 1,000 pounds: Choose 1,000 pounds
For 2,000 pounds: Choose 1,000 pounds
(d) Expected Monetary Value (EMV) = Σ(Probability * Profit)
EMV for 500 pounds: 0.2 * 500 + 0.4 * 1,000 + 0.4 * 1,500 = K1,200
EMV for 1,000 pounds: 0.2 * 1,000 + 0.4 * 1,000 + 0.4 * 1,500 = K1,200
EMV for 2,000 pounds: 0.2 * 1,000 + 0.4 * 1,500 + 0.4 * 2,000 = K1,600
Thus, the optimal number of pounds of clams to purchase is 2,000 pounds.
(e) Standard Deviation = √(Σ((Profit - EMV)^2 * Probability))
Standard Deviation for 500 pounds: √(0.2 * (500-1,200)^2 + 0.4 * (1,000-1,200)^2 + 0.4 * (1,500-1,200)^2) = 205.48
Standard Deviation for 1,000 pounds: √(0.2 * (1,000-1,200)^2 + 0.4 * (1,000-1,200)^2 + 0.4 * (1,500-1,200)^2) = 115.47
Standard Deviation for 2,000 pounds: √(0.2 * (1,000-1,600)^2 + 0.4 * (1,500-1,600)^2 + 0.4 * (2,000-1,600)^2) = 173.21
(f) Expected Opportunity Loss (EOL) = Σ(Probability * (Best Outcome - Profit))
EOL for purchasing 500 pounds: 0.2 * (1,500 - 500) + 0.4 * (1,500 - 1,000) + 0.4 * (1,500 - 1,500) = 200
EOL for purchasing 1,000 pounds: 0.2 * (1,500 - 1,000) + 0.4 * (1,500 - 1,000) + 0.4 * (1,500 - 1,500) = 200
EOL for purchasing 2,000 pounds: 0.2 * (1,500 - 1,000) + 0.4 * (1,500 - 1,500) + 0.4 * (1,500 - 2,000) = 200
(g) The Expected Value of Perfect Information (EVPI) is the difference between the expected value with perfect information and the expected value using probabilities. It represents the maximum amount a decision-maker would pay to obtain perfect information.
In this case, EVPI would be the difference between the EMV with perfect information and the EMV calculated. Since we do not have perfect information, we cannot calculate EVPI.
(h) Coefficient of Variation = (Standard Deviation / EMV) * 100
CoV for 500 pounds: (205.48 / 1,200) * 100 = 17.12%
CoV for 1,000 pounds: (115.47 / 1,200) * 100 = 9.62%
CoV for 2,000 pounds: (173.21 / 1,600) * 100 = 10.82%
The coefficient of variation measures the risk per unit of profit, lower values indicate lower risk.
(i) Return-to-Risk Ratio (RTRR) = EMV / Standard Deviation
RTRR for 500 pounds: 1,200 / 205.48 = 5.84
RTRR for 1,000 pounds: 1,200 / 115.47 = 10.40
RTRR for 2,000 pounds: 1,600 / 173.21 = 9.24
The return-to-risk ratio helps evaluate the trade-off between expected return and risk, higher values are preferred.
(j) Based on the analysis, purchasing 1,000 pounds of clams appears to be the best decision. It has the same expected monetary value as purchasing 2,000 pounds but with lower risk and higher return-to-risk ratio. This decision balances profit potential and risk effectively.