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Sunny Groves is a citrus grower in a southern state that has several citrus orchards. Weather forecasters are predicting a significant drop in temperatures over the next two days. The forecast calls for a 25% chance of near-freezing temperatures and a 10% chance of a deep freeze. The management of Sunny Groves is concerned that the impending bad weather may damage their crop of citrus fruit. If the weather is fine, the company stands to make $1,000,000 revenue for their harvest. There are three options that the company can take in light of the coming weather: (1) Do nothing and hope for the best. If the orchards experience near-freezing temperatures, the company will lose half of its crop (and half of the revenue). A deep freeze would destroy all of the fruit and lead to no revenue). Obviously, taking no precautions costs nothing (2) Take moderate precautions. This involves having many workers placing heaters and warming tents about the orchards. This would cost $50,000 and reduce the potential losses to $200,000 in a near-freeze and $350,000 in a deep freeze. (3) Take extensive precautions. This involves purchasing and assembling temporary enclosures, as well as heaters and tents. This would cost $150,000 and would reduce potential losses to no loss of revenue in a near-freeze and $150,000 in a deep freeze. Create a three-by-three table calculating the net revenue (revenue minus precaution costs) under all combinations of weather and precautions. Find the expected value of the net revenue under each of the precaution options. Which of the three options would you recommend to the management of Sunny Groves and why?
Decision alternatives |
States of nature |
Expected value of net revenue |
||
Weather remains fine (65%) |
Near - freezing temperatures (25%) |
Deep freeze (10%) |
||
Do nothing and hope for the best |
$1,000,000 |
$500,000 |
$0 |
$775,000 |
Take moderate precautions |
$950,000 |
$750,000 |
$600,000 |
$865,000 |
Take extensive precautions |
$850,000 |
$850,000 |
$700,000 |
$835,000 |
Working notes:
We are given that there are three states of nature/weather:
1. Near-freezing temperatures : 25% probability
2. Deep freeze conditions : 10% probability
3. Weather remains fine : Since total probability for anything is 1 or 100%, probability for weather remains fine’ can be calculated as 100% - (25%+10%) = 65%
Further, there are three alternative actions:
1. Do nothing and hope for the best : We are given that if weather remains fine revenue is $1,000,000, if there is near-freeze temperatures, revenue will be $1,000,000*50% = $500,000, and if there is deep freeze, revenue will be $1,000,000*0% = $0.
2. Take moderate precautions: If the weather remains fine, revenues will be $1,000,000 and cost of implementing this option is $50,000, so net revenue = $1,000,000 - $50,000 = $950,000. If there is near-freeze temperature, loss will be $200,000 and cost of this alternative is $50,000. So, payoff or net revenue will be $1,000,000 - $200,000 - $50,000 = $750,000. If there is deep freeze, loss will be $350,000, so net revenue will be $1,000,000 - $350,000 - $50,000 = $600,000.
3. Take extensive precautions: If the weather remains fine, revenues will be $1,000,000 and cost of implementing this option is $150,000, so net revenue will be $1,000,000 - $150,000 = $850,000. Further, if there is near-freeze temperature, loss will be $0, so net revenue = $850,000 and if there is deep freeze, loss will be $150,000, so net revenue will be $1,000,000 - $150,000 - $150,000 = $700,000.
Now, expected value is the sum of expected payoffs under different states of nature weighted by the probabilities of different states of nature.
So, Expected value of ‘Do nothing and hope for the best’ = ($1,000,000*65%)+($500,000*25%)+($0*10%)
= $775,000
Expected value of ‘Take moderate precautions’ = ($950,000*65%)+($750,000*25%)+($600,000*10%)
= $865,000
Expected value of ‘Take extreme precautions’ = ($850,000*65%)+($850,000*25%)+($700,000*10%)
= $835,000
Since the highest expected value is $865,000 for the option ‘Take moderate precautions’, it is advisable to take moderate precautions.