A business is considering two sizes for its new warehouse: small and large. An analysis of the profits realized given two states of demand, low and high, are summarized in the table below. Figures are in tens of thousands of dollars per month
Demand
Low High
Size
Large 70 105
Small 85 90
Management is considering conducting a study to determine which warehouse size is the most profitable. The outcomes of such a study are favorable (F) and unfavorable (U). The following probabilities have been established:
Use the decision tree to answer the following questions.
i) What is the optimal decision if the study is not used? Use the expected value method. (4 points)
ii) Find the expected value of perfect information. (6 points)
iii) What is the optimal decision strategy if the study is used? (8 points)
iv) Suppose the cost of the study is estimated to be $1,500. Should it be conducted? Why or why not? (5 points)
v) Find the efficiency of sample information. (5 points)
To answer the questions and determine the optimal decision strategy, we will use the concept of decision trees and expected values.
i) What is the optimal decision if the study is not used? Use the expected value method.
To find the optimal decision without conducting the study, we can use the expected value method.
For the "Large" size warehouse:
Expected value for low demand = (70 * 0.5) + (105 * 0.5) = $87.5 thousand
Expected value for high demand = (70 * 0.5) + (105 * 0.5) = $87.5 thousand
For the "Small" size warehouse:
Expected value for low demand = (85 * 0.5) + (90 * 0.5) = $87.5 thousand
Expected value for high demand = (85 * 0.5) + (90 * 0.5) = $87.5 thousand
Since the expected values for both sizes are the same, either option is equally optimal if the study is not conducted. The optimal decision would be to choose the smaller warehouse size (small) as it offers the same profit with less investment.
ii) Find the expected value of perfect information.
The expected value of perfect information represents the maximum possible expected value achievable with perfect information about the future states of demand.
To calculate it, we need to consider each state of demand and their associated probabilities.
Expected value for low demand:
Maximum profit = max(70, 85) = $85 thousand
Expected value = Maximum profit * Probability = $85 * 0.5 = $42.5 thousand
Expected value for high demand:
Maximum profit = max(105, 90) = $105 thousand
Expected value = Maximum profit * Probability = $105 * 0.5 = $52.5 thousand
Expected value of perfect information = Expected value for low demand + Expected value for high demand = $42.5 + $52.5 = $95 thousand
iii) What is the optimal decision strategy if the study is used?
If the study is used, we can make a decision based on the information obtained.
First, compare the expected values of conducting the study for each warehouse size.
For the "Large" size warehouse:
Expected value of conducting the study = ($87.5 - $95) = -$7.5 thousand (Loss)
For the "Small" size warehouse:
Expected value of conducting the study = ($87.5 - $95) = -$7.5 thousand (Loss)
Since conducting the study results in a loss for both sizes, it is not advisable to conduct the study.
Therefore, the optimal decision strategy if the study is used is to choose the smaller warehouse size (small) without conducting the study since the expected values remain the same.
iv) Should the study be conducted if it costs $1,500? Why or why not?
Considering that the cost of the study is estimated to be $1,500, and it results in a loss of $7.5 thousand if conducted, it is not advisable to conduct the study. The cost of the study outweighs the potential benefit and would result in a net loss.
v) Find the efficiency of sample information.
The efficiency of sample information measures the reduction in risk achieved by obtaining sample information (such as study results) compared to the initial decision strategy.
Efficiency of sample information = (Expected value without sample information - Expected value with sample information) / (Expected value without sample information) * 100
Without sample information (study):
Expected value without sample information = $87.5 thousand
With sample information (study):
Expected value with sample information = $87.5 thousand
Efficiency of sample information = (87.5 - 87.5) / 87.5 * 100 = 0%
The efficiency of sample information in this case is 0%, indicating that the sample information (study results) does not reduce the risk or improve the decision-making process.