A food manufacturer has to decide how many batches of a product to produce next week. If one batch is produced then a profit of $15,000 will be made. If two batches are produced then either a loss of $5,000 will be made if demand only equals one batch or a profit of $20,000 will be made if demand equals two batches. The manufacturer provisionally estimates the probabilities of these two outcomes to be 0.4 and 0.6 respectively. After making these estimates the manufacturer finds that a statistical demand forecasting method suggests that demand will equal two batches. In the past the method has correctly predicted demand in 60% of the weeks, irrespective of what the level of demand turned out to be. To maximize his expected profit, the manufacturer should:
a)produce one batch
b)produce two batches
c) be indifferent between producing 1 or 2 bathces
d) seek further information as it is not possible to computer the expected profits from the information
Next, suppose that the manufacturer had to decide whether it was worth paying for the forecast of the statistical demand forecasting method before making his decision. The expected value of imperfect information obtained from the method would have been:
a) 0
b) $2,700
c) $5,000
d) $7,800
Next, suppose that the statistical demand forecast always gave a correct indication. The expected value of perfect information obtained from the method would have been:
a)0
b) $3,000
c) $6,000
d) $18,000
We do not do your homework for you. Although it might take more effort to do the work on your own, you will profit more from your effort. We will be happy to evaluate your work though.
The problem is I don't even know how to get started. My book is horrible at explaining the concept. My guess is 2 batches for the first question, but then I'm not sure how to figure out the next two.
To solve these questions, we will calculate the expected profits for each decision and then compare them.
1) To maximize expected profit, the manufacturer should compare the expected profits for producing one batch and producing two batches.
For producing one batch:
Expected profit = (probability of demand equaling one batch * profit for one batch) = (0.4 * $15,000) = $6,000
For producing two batches:
Expected profit = (probability of demand equaling one batch * loss for two batches) + (probability of demand equaling two batches * profit for two batches) = (0.4 * (-$5,000)) + (0.6 * $20,000) = $12,000
Since the expected profit for producing two batches ($12,000) is greater than the expected profit for producing one batch ($6,000), the manufacturer should produce two batches. Therefore, the answer is b) produce two batches.
2) To calculate the expected value of imperfect information, we compare the expected profits with and without the forecast.
Without the forecast:
Expected profit = (probability of demand equaling two batches * profit for two batches) = (0.6 * $20,000) = $12,000
With the forecast:
Expected profit = (probability of forecast being correct * expected profit with the forecast) + (probability of forecast being incorrect * expected profit without the forecast)
Expected profit = (0.6 * $12,000) + (0.4 * $12,000) = $7,800
The expected value of imperfect information is the difference between the expected profit with the forecast and without the forecast: $7,800 - $12,000 = -$4,200.
Since the expected value of imperfect information is negative (-$4,200), it is not worth paying for the forecast. Therefore, the answer is a) $0.
3) To calculate the expected value of perfect information, we assume the forecast always gives a correct indication.
Expected profit with perfect information = (probability of demand equaling two batches * profit for two batches) = (0.6 * $20,000) = $12,000
The expected value of perfect information is the difference between the expected profit with perfect information and without the forecast: $12,000 - $12,000 = $0.
Since the expected value of perfect information is $0, the answer is a) $0.
Therefore, the answers are:
a) produce two batches
b) $0
To determine the best decision for the food manufacturer, we need to calculate the expected profit for each option.
For option a) producing one batch, the profit is $15,000 with a probability of 0.4. Therefore, the expected profit for option a) is 0.4 * $15,000 = $6,000.
For option b) producing two batches, if demand equals one batch, the profit is a loss of $5,000 with a probability of 0.6. If demand equals two batches, the profit is $20,000 with a probability of 0.6. Therefore, the expected profit for option b) is 0.6 * (-$5,000) + 0.6 * $20,000 = $9,000.
Comparing the expected profits, we can see that option b) has a higher expected profit ($9,000) compared to option a) ($6,000). Therefore, to maximize expected profit, the manufacturer should produce two batches (option b).
Now let's move on to the second part about the expected value of imperfect information.
To calculate the expected value of imperfect information, we need to compare the expected profit with and without the statistical demand forecasting method.
Without the statistical demand forecasting method, the expected profit of producing two batches is still $9,000.
With the statistical demand forecasting method, if it predicts two batches and demand indeed turns out to be two batches, the profit would be $20,000. However, since the statistical demand forecast only correctly predicts demand 60% of the time, the probability of a correct prediction is 0.6 * 0.6 = 0.36.
Therefore, the expected profit with the statistical demand forecasting method is 0.36 * $20,000 = $7,200.
The expected value of imperfect information is the difference between the expected profit without the statistical demand forecasting method and the expected profit with the statistical demand forecasting method. In this case, it is $9,000 - $7,200 = $1,800.
Thus, the expected value of imperfect information obtained from the statistical demand forecasting method is $1,800 (option a).
Lastly, let's consider the scenario where the statistical demand forecast always gives a correct indication.
With a correct indication, if the demand equals two batches, the profit is $20,000 with a probability of 1. Therefore, the expected profit with perfect information is 1 * $20,000 = $20,000.
The expected value of perfect information is the difference between the expected profit with perfect information and the expected profit without the statistical demand forecasting method. In this case, it is $20,000 - $9,000 = $11,000.
Therefore, the expected value of perfect information obtained from the statistical demand forecasting method is $11,000 (option c).
In summary:
- The manufacturer should produce two batches (option b) to maximize expected profit.
- The expected value of imperfect information is $1,800 (option b).
- The expected value of perfect information is $11,000 (option c).