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Figure 1. Initial development of a two-variable graph for the road construction problem, with the miles of rocked roads to be built on the Y-axis, and the amount of woods roads to be built on the X-axis. 10. 8. 6. RR (miles). 4. 2. 0. 0. 2. 4. 6. 8. 10. WR (miles).

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Presentation Transcript
slide1

Figure 1. Initial development of a two-variable graph for the road

construction problem, with the miles of rocked roads to be built on

the Y-axis, and the amount of woods roads to be built on the X-axis.

10

8

6

RR

(miles)

4

2

0

0

2

4

6

8

10

WR

(miles)

slide2

Figure 2. The budget constraint for the road construction problem.

10

8

6

RR

(miles)

30,000 WR + 50,000 RR  300,000

4

Feasible region

2

0

0

2

4

6

8

10

WR

(miles)

slide3

Figure 3. A graph of the entire set of constraints to the road construction

problem, and the areas related to the constraints where solutions are feasible.

10

8

WR  2.5

WR  6

6

RR

(miles)

RR  4

4

30,000 WR + 50,000 RR

 300,000

2

RR  1.5

0

0

2

4

6

8

10

WR

(miles)

slide4

Figure 4. Identification of the optimal solution to the road construction

problem using a family of objective functions.

10

8

6

RR +

WR = 8

RR

(miles)

4

RR + WR = 8.4

2

RR +

WR = 4

0

0

2

4

6

8

10

WR

(miles)

slide5

Figure 5. The graphed constraints to the snag development problem, and the

identification of the feasible region (gray area).

2,000

CS  250

1,600

1,200

DS

(trees)

800

DS  600

100 DS + 50 CS

 80,000

400

DS  100

0

0

400

800

1,200

1,600

2,000

CS

(trees)

slide6

Figure 6. The optimal solution to the snag development problem.

2,000

1,600

1,200

DS

(trees)

CS + DS = 1,500

800

400

0

0

400

800

1,200

1,600

2,000

CS

(trees)

slide7

Figure 7. The constraints and feasible region (gray area) associated with the

fish habitat problem.

Boulders  2.5

25

20

Boulders  7.5

15

Logs

(miles)

10,000 Logs + 21,000 Boulders

 250,000

10

Logs  5

5

0

0

5

10

15

20

25

Boulders

(miles)

slide9

Figure 9. Identification of the feasible region and optimal solution to the

hurricane clean-up problem of cost minimization using a family of objective

functions.

CH  400,000

1,000,000

CH + CPB  1,000,000

800,000

600,000

CPB  500,000

CPB ($)

400,000

CPB  300,000

200,000

CH + CPB

0

0

200,000

400,000

600,000

800,000

1,000,000

CH ($)

slide10

Figure 10. A modified fish habitat problem, with multiple optimal solutions.

Boulders  2.5

25

Boulders + Logs  15

20

Boulders  7.5

15

Logs

(miles)

A

10,000 Logs + 21,000 Boulders

 250,000

10

B

Logs  5

5

0

0

5

10

15

20

25

Boulders

(miles)

slide11

Figure 11. An example of efficient, feasible, inefficient, and infeasible

solutions to a broad timber harvest and wildlife habitat management problem.

B

D

Timber

volume

A

C

(Feasible region solutions)

Wildlife

habitat

slide12

(Figure for question 7)

Roads

Streams

Streams to be treated with logs or boulders