运筹学Problem 10.3-4 _p459_

10.3-4. A political campaign is entering its final stage, and polls indicate a
very close election. One of the candidates has enough funds left to
purchase TV time for a total of five prime-time commercials on TV
stations located in four different areas. Based on polling information, an estimate has been made of the number of additional votes that can be won
in the different broadcasting areas depending upon the number of commercials run. These estimates are given in the following table in thousands of votes:
Area
4
1 2 3
Commercials
0 0 0 0
3
1 4 6 5
7
2 7 8 9
12
3 9 10 11
4 12 11 10
14
16
5 15 12 9
Use dynamic programming to determine how the five commercials
should be distributed among the four areas in order to maximize the
estimated number of votes won.
Solution.
1. Formulation.
A dynamic programming formulation for this problem is
Stage n = area n (n = 1, 2, 3, 4)
x n = commercials allocated to area n
State s n = commercials left to begin stage n .
c xn = the number of additional votes that can be won when x n
commercials are allocated to area n .
f n (s n , x n ) = the total number of additional votes won to begin stage n when
the state is s n and x n days are allocated to course n .
f n *(s n ) = n
x max f n (s n , x n ), in which f n (s n , x n ) = c xn + f n +1*(s n – x n ) and f 5*(s ) = 0 for any s .
2. Solution procedure.
The resulting dynamic programming calculations are as follows:
n = 4: s 4 f 4*(s 4) x 4*
0 0 0
1 3 1
2 7 2
3 12 3
4 14 4
5 1
6 5
n = 3:
s3 f3*(s3)=max x3{c x3+ f4*(s3–x3)} x3*
0 0 0
1 max{0+3,
5+0}=5 1
9+0}=9 2
2 max{0+7,
5+3,
2
3 max{0+12,5+7,9+3,11+0}=12 0,1,or
4 max{0+14,5+12,9+7,11+3,10+0} = 17 1
5 max{0+16,5+14,9+12,11+7,10+3,9+0} = 21 2
n = 2:
s2 f2*(s2)=max x2{c x2+ f3*(s2–x2)} x2*
0 0 0
1 max{0+5,6+0}=6 1
2 max{0+9,6+5,8+0}=11 1
3 max{0+12,6+9,8+5,10+0}=15 1
4 max{0+17,6+12,8+9,10+5,11+0}=18 1
5 max{0+21,6+17,8+12,10+9,11+5,12+0}=23 1
n = 1:
s1 f1*(s1)=max x1{c x1+ f2*(s1–x1)} x1*
5 max{0+23,4+18,7+15,9+11,12+6,15+0}=230
An optimal solution for the entire problem can now be identified from the four tables. Results for the n = 1 problem indicate that the there should be no commercial for area 1 (x1* = 0), no immediate additional vote should be obtained (c x1 = 0), and thus 5 commercials are left for the remaining stages. For the n = 2, s2 = 5, one commercial should be allocated to area 2 (x2* = 1), 6 immediate additional votes can be won (c x2 = 6), and 4 commercials are left for stages 3 and 4. For the n = 3, s3 = 4, one commercial should be allocated to area 3 (x3* = 1), 5 immediate additional votes can be won (c x3 = 5), and 3 commercials are left for the last stage. For the n = 4, s4 = 3, 3 commercial should be allocated to area 4 (x4* = 3), 12 immediate additional votes can be won (c x4 = 12). Based on this optimal decision, the total additional votes of 23 can be won.。