运筹第五次作业

10.2-2

The sales manager for a publisher of college textbooks has six travelling salespeople to assign to three different regions of the country.She has decided that each region should be assigned at least one salesperson and that each individual salesperson should be restricted to one lof the regions,but now she wants to determine how mani salespeople should be assigned to the respective regions in order to maximize sales.

The next table gives the estimated increase in sales(in appropriate unites)in each region if ti were allocated various numbers of salespeople. Saleslpersons Region 1 2 3 1 40 24 32 2 54 47 46 3 78 63 70 4

99

78

84

(a)Use dynamic programming to solve this problem.Instead of using the usual tables,show your work graphically by constructing and filling in a network such as the one shows for Prob.10.2-1.Proceed as in Prob.10.2-1by solving for

)(*n n x f for each node (except theterminal

node) and writing its value by the node.Draw an arrowbead to show the optimal link (or links in case of a tie)to take out of each other.Finally,identify the resulting optimal path (or paths)through the network and the corresponding optimal solution.

(b)Use dynamic programming to solve this problem by constructing the usual tables forn=3,n=3 ,and n=1.

○2 27 ○1

99 47 32

78 ○

324 ○2 46 ○

6 54 63 4

7 70 ○0 ○

4 24 ○3 40 78 63 47 84

○

5 24 ○4

Sn=number of traveling salespeople still available for allocation to remaining regions. Xn=number of salespeople to allocation to stage(region)

⎩⎨

⎧-=-=⎪⎩⎪

⎨⎧--=-==2

231

12213

1

21666x S S x S S x x S x S S

{}

)()(max )()

()(),(*1**11n n n n n n n n n n n n n n n n

n n x S f x P S f x S f x P x S f x S S -+=-+=-=+++

n=3

3S

)(3*3S f

*3x

1 3

2 1 2 46 2

3 70 3 4

84

4

n=2

x 2

S 2

)()(),(22*322222x S f x P X S f -+=

)(2*2

S f

*2x

1 2 3 4 2 24 24 1 3 70 79 79 2 4 94 93 95 95 3 5 108

117

109

110

117

2

n=1

)()()1,(11*21111x S f x P x S f -+=

)(1*1x f

*1x

1

2 3 4 6 157

149

157

123

157

3

1

,2,33

,2,1*3

*2

*1

*3*

2*1======∴x x x x x x

The total maximize sales is 157.

10.3-9

Consider the following integer acelinear progressing problem.

3222312153x x x x MaximizeZ -+-=

subject to

4221≤+x x

and

0,021≥≥x x

x 1,x 2 are integers.

Use dynamic programming to solve this problems.

是整数2121213

2

223121,0

,04

253x x x x x x x x x MaximizeZ ⎩⎨

⎧≥≥≤+-+-=

{}32223121253max )4(x x x x f -+-=

4221≤+x x

[]{}312132223max 5max x x x x -+-=

是整数

22024x x ≥-

是整数

2121,24x x x x -≤

{})24(5max 213222x f x x -+-=

2,1,02=x

)}0(12),2(4),4(0max{111f f f +++=

},max{}3max{)(131211x h x x w f =-=

是整数

110x w x ≤≤

是整数

11x w x ≤

1211

63x x dx dh

+-=∴