Question 1. The World Health Council is devoted to improving health care in under-developed countries of the world. It now has five medical teams available to allocate among three such countries to improve medical care. The council needs to determine how many teams (if any) to allocate to each of these countries to maximize the total effectiveness of the five teams. The measure of performance being used is the additional person-years of life (increased life expectancy times the country's population). The table below gives the estimated additional person-years of life (in multiples of 1,000) for each country for each possible allocation of medical teams. Set this problem up as a dynamic programming problem and solve to determine the allocation which maximizes overall additional person-years of life.
| Thousands of Additional Person- Years of Life |
|
Country |
| Medical Teams |
1 |
2 |
3 |
| 0 |
0 |
0 |
0 |
| 1 |
45 |
20 |
50 |
| 2 |
70 |
45 |
70 |
| 3 |
90 |
75 |
80 |
| 4 |
105 |
110 |
100 |
| 5 |
120 |
150 |
130 |
Assume you have a marketing firm and can invest $7 million in marketing in four different market segments. The optimal solution for each of the four markets showing increased profitability (in millions) is shown in the tables below.
Let xn = $ amount (in millions) to invest in market n
pn(xn) = profit earned from allocating xn to market n
sn = amount of money left for consideration for market n
fn(sn) = max { pn(xn) + fn+1*(sn-xn) }
| S4 |
f.4* (S 4) |
X4* |
| 1 |
6 |
1 |
| 2 |
7 |
2 |
| 3 |
9 |
3 |
| 4 |
9 |
4 |
| S3 |
1 |
2 |
3 |
4 |
f3* (S3) |
X3* |
| 2 |
8 |
|
|
|
8 |
1 |
| 3 |
9 |
10 |
|
|
10 |
2 |
| 4 |
11 |
11 |
13 |
|
13 |
3 |
| 5 |
11 |
13 |
14 |
14 |
14 |
3,4 |
| S2 |
1 |
2 |
3 |
4 |
f2* (S2) |
x2* |
| 3 |
13 |
|
|
|
13 |
1 |
| 4 |
15 |
13 |
|
|
15 |
1 |
| 5 |
18 |
15 |
14 |
|
18 |
1 |
| 6 |
19 |
18 |
16 |
17 |
19 |
1 |
| S1 |
1 |
2 |
3 |
4 |
f1* (S1) |
x1* |
| 7 |
22 |
23 |
21 |
20 |
23 |
2 |
Question 2. Determine the amount of money to invest in each of the 4 markets to maximize overall profitability.
Question 3. Suppose the marketing firm mistakenly invests $2 million each in markets 1 and 2. What is the optimal amount to invest in markets 3 and 4?
Question 4. You are given a multiplicative formulation for a reliability problem. Specifically, you wish to maximize the reliability of an 8 stage manufacturing process.
Max n=1Π8Pn(Xn)
Where Pn(Xn) is the probability that stage n in the process produces a good product, and Xnis 1, 2, or 3 for each stage. Thus the probability of success in stage n is dependent on decision Xn. You are given the follow recursive relationships.
Sn+1 = Sn - Xn
Fn(Sn) = Pn(Xn)*f*n+1(Sn -Xn)
You are given the following tableau for stage 7.
S7 1 2 3 F*7 X*7
2 .81 .81 1
3 .78 .75 .78 1
4 .75 .81 .78 .81 2
5 .65 .78 .75 .78 2
6 .65 .78 .81 .81 3
You are also given the following information for stage 6
X6 P6(X6)
1 0.6
2 0.8
3 0.9
Compute the tableau for stage 6 (-- = not possible).
S6 1 2 3 F*6 X*6
3 -- --
4 --
5
6
7
Question 5. Scout Keegan wishes to pack his knapsack for an upcoming backpack trip. After packing food and essentials, he has set aside 3 additional items which he could take but wishes to keep the weight down to no more than 10 lbs. For each item, he has estimated a numerical benefit value and a weight for each item. He may take up to two units of each item.
Item 1 2 3
Value 6 8 11
Weight 3 7 5
Use dynamic programming to determine how many units of each he should take in order to maximize the benefit while keeping the total weight at 10 or fewer pounds.
Question 6. Jane plays a game for which her probability of winning is 0.6 and her probability of losing is 0.4. Jane has $50 and will play the game twice. At each play of the game, she will bet either $10 or the amount that she currently holds. If she wins, the amount will be added to her total. If she loses the amount will be deleted from her total. Thus on play 1 she may bet $10. If she wins her total will be $60. If she loses her total will be $40. If, on play 1, she bets $50, she will either have $100 or $0. Use dynamic programming to determine her optimal strategy to maximize her holdings after two plays.