Samacheer Kalvi Class 12 Business Maths Solutions Chapter 10 Operations Research Exercise 10.3

 

Question 1. Given the following pay-off matrix (in rupees) for three strategies and two states for nature. Select a strategy using each of the following rule (i) Maximin (ii) Minimax

Answer:(i) Maximin:
First, we find the minimum pay-off for each strategy: for \( S_1 \) it is 40, for \( S_2 \) it is -20, and for \( S_3 \) it is -40. Then, we choose the maximum of these minimum values, which is Max(40, -20, -40) = 40. So, strategy \( S_1 \) is selected. The maximin approach aims to maximize the minimum possible gain, focusing on the safest outcome.
(ii) Minimax:
To use the minimax rule, we first identify the maximum possible pay-off for each strategy. For \( S_1 \), the maximum is 60. For \( S_2 \), the maximum is 10. For \( S_3 \), the maximum is 150. Then, we select the minimum among these maximum values: min(60, 10, 150) = 10. This means strategy \( S_2 \) is chosen. The minimax approach focuses on minimizing the maximum possible loss or risk, aiming for the least bad outcome.
In simple words: For maximin, find the lowest profit for each plan and pick the highest of those lowest profits. For minimax, find the highest profit for each plan and pick the lowest of those highest profits.

🎯 Exam Tip: Remember to clearly define strategies (rows) and states of nature (columns) to correctly apply decision criteria like Maximin and Minimax.

 

Question 2. A farmer wants to decide which of three crops he should plant on his 100 acre farm. The profit from each is dependent on the rainfall during the growing season. The farmer has categorized the amount of rainfall as high medium and low. His estimated profit for each is shown in the table. In the farmer wishes to plant only crop, decide which should be his best crop using (i) Maximin (ii) Minimax

Answer:(i) Maximin:
For the maximin rule, we first look at the minimum profit for each rainfall category across all crops. For High rainfall, the minimum profit is Rs. 3500. For Medium rainfall, the minimum is Rs. 4500. For Low rainfall, the minimum is Rs. 2000. We then find the maximum of these minimums: Max(3500, 4500, 2000) = Rs. 4500. Therefore, the decision would be based on the 'Medium' rainfall scenario, which gives the highest minimum profit. This method focuses on maximizing the minimum guaranteed profit, even if it means not choosing the highest possible overall profit.
(ii) Minimax:
For the minimax rule, we first find the maximum profit for each rainfall category across all crops. For High rainfall, the maximum profit is Rs. 8000. For Medium rainfall, the maximum is Rs. 5000. For Low rainfall, the maximum is Rs. 5000. Then, we select the minimum of these maximums: min(8000, 5000, 5000) = Rs. 5000. This means the decision points towards both the 'Medium' and 'Low' rainfall scenarios as optimal, suggesting crops performing well in those conditions. The minimax approach is often chosen by decision-makers who are risk-averse and want to avoid the worst possible outcome.
In simple words: For Maximin, you find the smallest profit in each row (rainfall) and then pick the largest of these smallest profits. For Minimax, you find the largest profit in each row and then pick the smallest of those largest profits.

🎯 Exam Tip: When applying Maximin or Minimax, make sure you consistently apply the rule to either rows (strategies) or columns (states of nature) based on what the question asks you to decide.

 

Question 3. The research department of Hindustan Ltd. has recommended to pay marketing department to launch a shampoo of three different types. The marketing types of shampoo to be launched under the following T estimated pay-offs for various level of sales. What will be the marketing manager's decision if (i) Maximin and (ii) Minimax principle applied?

Answer:(i) Maximin:
To apply the maximin principle, we first find the minimum sales for each type of shampoo. For Egg shampoo, the minimum sales are 10 units. For Clinic shampoo, the minimum is 5 units. For Deluxe shampoo, the minimum is 3 units. We then choose the maximum of these minimums: Max(10, 5, 3) = 10. Therefore, Egg Shampoo is the chosen product. The maximin strategy helps ensure the best possible outcome even in the worst-case sales scenario.
(ii) Minimax:
For the minimax rule, we identify the maximum sales for each type of shampoo. For Egg shampoo, the maximum sales are 30 units. For Clinic shampoo, the maximum is 40 units. For Deluxe shampoo, the maximum is 55 units. Then, we find the minimum of these maximum values: min(30, 40, 55) = 30. This leads to the selection of Egg Shampoo, as it minimizes the maximum potential sales value. This method helps in choosing an option that performs relatively well, even if the absolute best outcome is not achieved.
In simple words: For maximin, find the lowest sales for each shampoo type and pick the highest of these. For minimax, find the highest sales for each shampoo type and pick the lowest of these.

🎯 Exam Tip: When using payoff matrices, clearly list the minimums (for maximin) or maximums (for minimax) for each strategy before making the final selection.

 

Question 4. Following pay-off matrix, which is the optimal decision under of the following rule (i) maximin (ii) minimax

Answer:(i) Maximin:
For the maximin rule, we first find the minimum pay-off for each action. For \( A_1 \), the minimum is 5. For \( A_2 \), it is 7. For \( A_3 \), it is 9. For \( A_4 \), it is 8. Then, we find the maximum of these minimum values. The calculation provided is Max(5, 7, 9, 8) = 5. Based on this, alternative \( A_1 \) is chosen. This method focuses on achieving the highest guaranteed return, even if it means foregoing potentially higher, but riskier, gains.
(ii) Minimax:
For the minimax rule, we first find the maximum pay-off for each action. For \( A_1 \), the maximum is 14. For \( A_2 \), it is 11. For \( A_3 \), it is 11. For \( A_4 \), it is 13. Then, we select the minimum of these maximum values: min(14, 11, 11, 13) = 11. Since both \( A_2 \) and \( A_3 \) have this minimum maximum pay-off, both are considered optimal alternatives. The minimax strategy helps to minimize the potential for the worst possible outcome, making it suitable for risk-averse decision-makers.
In simple words: For maximin, find the smallest outcome for each action and pick the action with the largest of these smallest outcomes. For minimax, find the largest outcome for each action and pick the action with the smallest of these largest outcomes.

🎯 Exam Tip: Always check if there are multiple optimal alternatives for minimax or maximin criteria, as shown here with \( A_2 \) and \( A_3 \).