Assignment
1. A retail store had sales of $45,000 in April and $56,000 in May. The store employs eight full-time workers who work a 40-hour week. In April the store also had seven part-time workers at 10 hours per week, and in May the store had nine part-timers at 15 hours per week (assume four weeks in each month). Use sales dollars as the measure of output.
a. What is the percentage change in total (multi-factor) productivity from April to May?
b. What is the percentage change in single-factor productivity with respect to the part-time workers from April to May?
2. Dick Holliday can build either a large video rental section or a small one in his store. He can also gather additional information (by conducting market study) or simply do nothing. If he gathers additional information, the results could be either positive or negative, but it would cost him $3,000 to gather the information.
Holliday believes that there is a 50-50 chance that the information will be positive. If the rental market is favorable, Holliday will earn $15,000 with a large section or $5,000 with a small. With an unfavorable video-rental market, however, Holliday could lose $20,000 with a large section or $10,000 with a small section. Without gathering additional information, Holliday estimates that the probability of a favorable rental market is 0.7. A positive report from the study would increase the probability of a favorable rental market to 0.9. Furthermore, a negative report from the additional information would decrease the probability of a favorable rental market to 0.4. Of course, Holliday could ignore these numbers and do nothing at any stage.
By drawing and solving a decision tree, what is the maximum expected payoff that Holliday can achieve? What should he do?
3. Two members of a criminal-gang are arrested and imprisoned. The prosecutors lack sufficient evidence to convict the pair on the principal charge. However, the prosecutors offer each prisoner a bargain simultaneously. Each prisoner is given the opportunity either to: betray the other by testifying that the other committed the crime, or to remain silent. Let us call them Prisoner A and Prisoner B, and let us analyze how Prisoner A should make a decision.
Prisoner A has two alternatives as mentioned above. What he does not know is Prisoner B's action. But he knows that 1) if they both remain silent, then they are both sentenced to a year in prison on a lesser charge, 2) if he betrays and B remains silent, then he goes free while B will be sentenced to three years in prison, 3) if he remains silent and B betrays, then he will be sentenced to three years in prison while B goes free, and 4) if they both betray the other, then they are both sentenced to two years in prison.
Now, remember that they are in solitary confinement with no means of communicating with the other. Furthermore, they are both self-interested and have no loyalty to each other whatsoever. Determine the appropriate decision under uncertainty for Prisoner A using the following decision rules, respectively:
a. Maximax
b. Maximin
c. Equally likely
Hint: The events (states of nature) to A are B's possible actions.
4. Chung Manufacturing is considering the introduction of a family of new products.
Long-term demand for the product group is somewhat predictable, so the manufacturer must be concerned with the risk of choosing a process that is inappropriate. Chen Chung is VP of operations. He can choose among batch manufacturing or custom manufacturing, or he can invest in group technology.
Chen won't be able to forecast demand accurately until after he makes the process choice. Demand will be classified into four compartments: poor, fair, good, and excellent. The table below indicates the payoffs (profits) associated with each process/demand combination, as well as the probabilities of each longterm demand level:
|
Poor |
Fair |
Good |
Excellent |
| Probability |
0.1 |
0.4 |
0.3 |
0.2 |
| Batch |
-$200,000 |
$1,000,000 |
$1,200,000 |
$1,300,000 |
| Custom |
$100,000 |
$300,000 |
$700,000 |
$800,000 |
| Group Tech |
-$1,000,000 |
-$500,000 |
$500,000 |
$2,000,000 |
a. Based on expected value, what choice offers the greatest gain?
b. How much would Chen Chung be willing to pay for a forecast that would accurately determine the level of demand in the future?
5. Toy Story: You are the owner of a toy stand in a mall and you need to decide how many units of a particular toy you will order to sell at your stand. You know that the demand will either be 20, 40, 60, 80 or 100 with equal likelihood. The unit cost of the toy is $4 and every unit you sell earns revenue of $10 per unit. Leftover units cannot be salvaged or re-sold and are worthless. You have only one opportunity to buy in anticipation of demand. Because the toys are shipped in batches, you must order in units of 20.
a. For each combination of your choice (order quantity) and demand realization, find your profit. Construct a decision table with the profits you find.
b. How many toys should you order to achieve highest expected profit?
Hint: you only need to order one of the demand realizations. Hence, both the alternatives and the events are the same, namely the five possible demands. The outcome of any combination of your order quantity and the demand is your profit.