Let's solve each part step-by-step: --- **Given:** | | S1 (p=0.25) | S2 (p=___) | S3 (p=0.2) | S4 (p=0.25) | |------|:-----------:|:----------:|:----------:|:-----------:| | A1 | 30 | 10 | 30 | 30 | | A2 | 0 | 65 | 35 | 15 | | A3 | 15 | 45 | 0 | 10 | | A4 | 35 | 23 | 15 | 10 | We need to find the missing probability for S2. The total probability must be 1: \[ p_1 + p_2 + p_3 + p_4 = 1 \implies 0.25 + p_2 + 0.2 + 0.25 = 1 \] \[ p_2 = 1 - (0.25 + 0.2 + 0.25) = 1 - 0.7 = 0.3 \] So, S2 has a probability of 0.3. --- ### a. **Expected Value (Best Alternative using Expected Value Criterion)** For each alternative, calculate the expected cost (lower is better): \[ \text{EV} = p_1C_1 + p_2C_2 + p_3C_3 + p_4C_4 \] #### A1: \[ = 0.25(30) + 0.3(10) + 0.2(30) + 0.25(30) = 7.5 + 3 + 6 + 7.5 = 24 \] #### A2: \[ = 0.25(0) + 0.3(65) + 0.2(35) + 0.25(15) = 0 + 19.5 + 7 + 3.75 = 30.25 \] #### A3: \[ = 0.25(15) + 0.3(45) + 0.2(0) + 0.25(10) = 3.75 + 13.5 + 0 + 2.5 = 19.75 \] #### A4: \[ = 0.25(35) + 0.3(23) + 0.2(15) + 0.25(10) = 8.75 + 6.9 + 3 + 2.5 = 21.15 \] **Best Alternative (minimum expected cost):** **A3** with **19.75** ### **Answer for (a):** > The best alternative is **A3**, with an expected cost of **19.75**. --- ### b. **Regret Table** First, for each state, find the minimum cost. Then, for each cell, compute Regret = Cost - Min Cost (for that state). - **S1:** min(30, 0, 15, 35) = 0 - **S2:** min(10, 65, 45, 23) = 10 - **S3:** min(30, 35, 0, 15) = 0 - **S4:** min(30, 15, 10, 10) = 10 Now, fill the regret table: | | S1 (p=0.25) | S2 (p=0.3) | S3 (p=0.2) | S4 (p=0.25) | |------|:-----------:|:----------:|:----------:|:-----------:| | **A1** | 30-0=30 | 10-10=0 | 30-0=30 | 30-10=20 | | **A2** | 0-0=0 | 65-10=55 | 35-0=35 | 15-10=5 | | **A3** | 15-0=15 | 45-10=35 | 0-0=0 | 10-10=0 | | **A4** | 35-0=35 | 23-10=13 | 15-0=15 | 10-10=0 | ### **Regret Table** | | S1 (0.25) | S2 (0.3) | S3 (0.2) | S4 (0.25) | |------|:---------:|:--------:|:--------:|:---------:| | A1 | 30 | 0 | 30 | 20 | | A2 | 0 | 55 | 35 | 5 | | A3 | 15 | 35 | 0 | 0 | | A4 | 35 | 13 | 15 | 0 | --- ### c. **EOL (Expected Opportunity Loss)** Calculate EOL for each alternative (weighted sum of regrets): #### A1: \[ = 0.25(30) + 0.3(0) + 0.2(30) + 0.25(20) = 7.5 + 0 + 6 + 5 = 18.5 \] #### A2: \[ = 0.25(0) + 0.3(55) + 0.2(35) + 0.25(5) = 0 + 16.5 + 7 + 1.25 = 24.75 \] #### A3: \[ = 0.25(15) + 0.3(35) + 0.2(0) + 0.25(0) = 3.75 + 10.5 + 0 + 0 = 14.25 \] #### A4: \[ = 0.25(35) + 0.3(13) + 0.2(15) + 0.25(0) = 8.75 + 3.9 + 3 + 0 = 15.65 \] **Best (lowest) EOL:** **A3** with **14.25** --- ### **Answer for (c):** > The best alternative using EOL is **A3**, with an expected opportunity loss of **14.25**. --- ### d. **Expected Value of Perfect Information (EVPI)** \[ \text{EVPI} = \text{Expected Value with Perfect Information} - \text{Minimum Expected Cost} \] - **Expected Value with Perfect Information (EVwPI):** - For each state, choose the minimum cost for that state, then multiply by the probability and sum. \[ = 0.25 \times 0 + 0.3 \times 10 + 0.2 \times 0 + 0.25 \times 10 = 0 + 3 + 0 + 2.5 = 5.5 \] - **Minimum Expected Cost (from part a):** 19.75 So, \[ \text{EVPI} = 19.75 - 5.5 = 14.25 \] --- ## **Summary of Answers** **a.** The best alternative is **A3**, with an expected cost of **19.75**. **b.** Regret Table: | | S1 (0.25) | S2 (0.3) | S3 (0.2) | S4 (0.25) | |------|:---------:|:--------:|:--------:|:---------:| | A1 | 30 | 0 | 30 | 20 | | A2 | 0 | 55 | 35 | 5 | | A3 | 15 | 35 | 0 | 0 | | A4 | 35 | 13 | 15 | 0 | **c.** The best alternative using EOL is **A3** with an EOL of **14.25**. **d.** The expected value of perfect information (EVPI) is **14.25**.