Week 12
- The following payoff table shows profit for a decision analysis problem with two decision alternatives and three states of nature:
| State of Nature | |||
| Decision alternatives | S1 | S2 | S3 |
| d 1 | 250 | 100 | 25 |
| d 2 | 100 | 100 | 75 |
- Construct a decision tree for this problem.
- If the decision maker knows nothing about the probabilities of the three states of nature, what is the recommended decision using the optimistic, conservative, and minimax regret approaches?
| Decision | Maximum Profit | Minimum Profit |
| d1 | 250 | 25 |
| d2 | 100 | 75 |
Optimistic approach: select d1
Conservative approach: select d2
Regret or opportunity loss table:
| s1 | s2 | s3 | |
| d1 | 0 | 0 | 50 |
| d2 | 150 | 0 | 0 |
Maximum Regret: 50 for d1 and 150 for d2; select d1
- Video Tech is considering marketing one of two new video games for the coming holiday season: Battle Pacific or Space Pirates. Battle Pacific is a unique game and appears to have no competition. Estimated profits (in thousands of dollars) under high, medium, and low demand are as follows:
| Demand | |||
| Battle Pacific | High | Medium | Low |
| Profit | $1000 | $700 | $300 |
| Probability | 0.2 | 0.5 | 0.3 |
Video Tech is optimistic about its Space Pirates game. However, the concern is that profitability will be affected by a competitor’s introduction of a video game viewed as similar to Space Pirates. Estimated profits (in thousands of dollars) with and without competition are as follows:
| Space Pirates with competition | Demand | ||
| High | Medium | Low | |
| Profit | $800 | $400 | $200 |
| Probability | 0.3 | 0.4 | 0.3 |
| Space Pirates without competition | Demand | ||
| High | Medium | Low | |
| Profit | $1600 | $800 | $400 |
| Probability | 0.5 | 0.3 | 0.2 |
- Develop a decision tree for the Video Tech problem.
- For planning purposes, Video Tech believes there is a 0.6 probability that its competitor will produce a new game similar to Space Pirates. Given this probability of competition, the director of planning recommends marketing the Battle Pacific video game. Using expected value, what is your recommended decision?
EV(node 2) = 0.2(1000) + 0.5(700) + 0.3(300) = 640
EV(node 4) = 0.3(800) + 0.4(400) + 0.3(200) = 460
EV(node 5) = 0.5(1600) + 0.3(800) + 0.2(400) = 1120
EV(node 3) = 0.6EV(node 4) + 0.4EV(node 5) = 0.6(460) + 0.4(1120) = 724
Space Pirates is recommended. Expected value of $724,000 is $84,000 better than Battle Pacific.
27. Fresh Made company must decide how much of its cranberry crop should be harvested wet and how much should be dry harvested. Fresh Made had 5000 barrels of cranberries that can be harvested using either the wet or dry method. Dry cranberries are sold for $32.50 per barrel and wet cranberries are sold for $17.50 per barrel. Both wet and dry cranberries must go through dechaffing and cleaning operations. The dechaffing and the cleaning operations can each be run 24 hours per day for a total of 1008 ours. Each barrel of dry cranberries requires 0.18 hours in the dechaffing operation and 0.32 hours in the cleaning operation. Wet cranberries require 0.04 hours in the dechaffing operation and 0.10 hours in the cleaning operation. Wet cranberries must also go through a drying process. The drying process can also be operated 24 hours per day ad each barrel of wet cranberries must be dried for 0.22 hours.
- develop a linear program.
- suppose that Fresh Made can increase its dechaffing capacity by using an outside firm for this operation. Fresh Made will still use its own dechaffing operation as much as possible, but it can purchase additional capacity from this outside firm for $500 per hours. Should Fresh Made purchase additional decahffing capacity? Why of why not.
- interpret the shadow price for the constraint corresponding to the cleaning operation. How would you explain the meaning of this shadow price to management?
a.
- Below is the sensitivity analysis output for this problem:
From this output, we see that dechaffing is not a binding constraint because only 523.273 of the 1008 available hours are being used. Fresh Made should not purchase additional dechaffing capacity.
- Assuming everything else remains unchanged, each additional hour of cleaning capacity added (subtracted) from the 1,008 available hours increases (decreases) revenues $68.18. This is true for any cleaning capacity value between 1008 – 416 = 592 and 1008 + 592 = 1600.
21. Round Tree Manor is a hotel that provides two types of rooms with three rental classes: Super Saver, Deluxe, and Business. The profit per night for each type of room and rental class is as follows:
| Rental class | ||||
| Super saver | Deluxe | Business | ||
| Room | Type I | $30 | $35 | – |
| Type II | $20 | $30 | $40 |
Type I rooms do not have high-speed Internet access and are not available for the Business rental class.
Round Tree’s management makes a forecast of the demand by rental class for each night in the future. A linear programming model developed to maximize profit is used to determine how many reservations to accept for each rental class. The demand forecast for a particular night is 130 rentals in the Super Saver class, 60 rentals in the Deluxe class, and 50 rentals in the Business class. Round Tree has 100 Type I rooms and 120 Type II rooms.
- Use linear programming to determine how many reservations to accept in each rental class and how the reservations should be allocated to room types.
- How many reservations can be accommodated in each rental class?
Given below output answer the following questions:
The optimal solution is
| Optimal Objective Value | ||
| 7000.00000 |
| Variable cells | ||||||
| Model variable | Name | Final Value | Reduced cost | Objective coefficient | Allowable increase | Allowable decrease |
| S1 | S1 | 100 | 0 | 30 | Infinite | 5 |
| D1 | D1 | 0 | -5 | 35 | 5 | Infinite |
| S2 | S2 | 10 | 0 | 20 | 5 | 20 |
| D2 | D2 | 60 | 0 | 30 | Infinite | 5 |
| B2 | B2 | 50 | 0 | 40 | Infinite | 20 |
| Constraints | ||||||
| Constraint number | Name | Final Value | Shadow Price | Constraint R.H. Side | Allowable increase | Allowable decrease |
| 1 | 1 | 110 | 0 | 130 | Infinite | 20 |
| 2 | 2 | 0 | 10 | 60 | 10 | 20 |
| 3 | 3 | 0 | 20 | 50 | 10 | 20 |
| 4 | 4 | 0 | 30 | 100 | 20 | 100 |
| 5 | 5 | 0 | 20 | 120 | 20 | 10 |
- Management is considering offering a free breakfast to anyone upgrading from a Super Saver reservation to Deluxe class. If the cost of the breakfast to Round Tree is $5, should this incentive be offered?
21. a. Let S1 = SuperSaver rentals allocated to room type I
S2 = SuperSaver rentals allocated to room type II
D1 = Deluxe rentals allocated to room type I
D2 = Deluxe rentals allocated to room type II
B1 = Business rentals allocated to room type II
The linear programming formulation and solution is given.
MAX 30S1+20S2+35D1+30D2+40B2
S.T.
1) 1S1+1S2
2) 1D1+1D2
3) 1B2
4) 1S1+1D1
5) 1S2+1D2+1B2
20 SuperSaver rentals will have to be turned away if demands materialize as forecast.
b. RoundTree should accept 110 SuperSaver reservations, 60 Deluxe reservations and 50 Business reservations.
c. Yes, the effect of a person upgrading is an increase in demand for Deluxe accommodations from 60 to 61. From constraint 2, we see that such an increase in demand will increase profit by $10. The added cost of the breakfast is only $5.
3. Tri-County Utilities, Inc., supplies natural gas to customers in a three-county area. The company purchases natural gas from two companies: Southern Gas and Northwest Gas. Demand forecasts for the coming winter season are as follows: Hamilton County, 400 units; Butler County, 200 units; and Clermont County, 300 units. Contracts to provide the following quantities have been written: Southern Gas, 500 units; and Northwest Gas, 400 units. Distribution costs for the counties vary, depending upon the location of the suppliers. The distribution costs per unit (in thousands of dollars) are as follows:
| To | |||
| From | Hamilton | Butler | Clermont |
| Southern Gas | 10 | 20 | 15 |
| Northwest Gas | 12 | 15 | 18 |
- Develop a network representation of this problem.
- Develop a linear programming model that can be used to determine the plan that will minimize total distribution costs.
