In: Operations Management
DYNAMIC PROGRAMMING
A county chairman of a certain political party is making plans for an upcoming presidential election. He has received the services of six volunteer workers for precinct work, and he wishes to assign them to four precincts in such a way as to maximize their effectiveness. He feels that it would be inefficient to assign a worker to more than one precinct, but he is willing to assign no workers to any one of the precincts if they can accomplish more in other precincts.
The following table gives the estimated increase in the plurality (positive or negative) of the party’s candidate in each precinct if it were allocated various numbers of workers.
Number of Workers |
Precinct 1 |
Precinct 2 |
Precinct 3 |
Precinct 4 |
0 |
0 |
0 |
0 |
0 |
1 |
18 |
20 |
23 |
13 |
2 |
38 |
32 |
33 |
24 |
3 |
55 |
54 |
47 |
32 |
4 |
63 |
65 |
50 |
39 |
5 |
69 |
73 |
52 |
45 |
6 |
74 |
80 |
53 |
50 |
Define
Stage 1 = Precinct 1 only Stage 3 = Precincts 1, 2, and 3
Stage 2 = Precincts 1 and 2 Stage 4 = Precincts 1, 2, 3, and 4
State = the number of workers used in total
gn(x) = plurality from x workers in precinct n
e.g. g2(3) = plurality from 3 workers in precinct 2 = 54
fn(x) = plurality from x workers in precincts 1, 2, … , n
e.g. f3(4) = plurality from 4 workers in precincts 1, 2, and 3
Calculations
f1(x) = g1(x) for x >= 0 fn(x) = -1027 for x < 0
fn(x) = max [gn(y) + fn-1(x-y)] = max [gn(y) + fn-1(x-y)]
0 <= y <= x 0 <= y <= 6
Please solve in excel & dynamic programming
Question: A firm has just hired eight new employees and would like to determine how to allocate their time ...
A firm has just hired eight new employees and would like to determine how to allocate their time to four activities. The firm has prepared the following table, which gives the estimated profit for each activity as a function of the number of new employees allocated to it.
Activities/Days | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
Activity 1 | 22 | 30 | 37 | 44 | 49 | 54 | 58 | 60 | 61 |
Activity 2 | 30 | 40 | 48 | 55 | 59 | 62 | 64 | 66 | 67 |
Activity 3 | 46 | 52 | 56 | 59 | 62 | 65 | 67 | 68 | 69 |
Activity 4 | 5 | 22 | 36 | 48 | 52 | 55 | 58 | 60 | 61 |
a. Use
dynamic programming to determine the optimal allocation of
new employees to the activities. (Zero credit will be given
for eyeball or linear programming solutions.)
b. Suppose only six new employees were hired. What activities would you assign to these employees?