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RR (miles)

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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RR (miles)

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  1. 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)

  2. 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)

  3. 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)

  4. 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)

  5. 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)

  6. 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)

  7. 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)

  8. Figure 8. Hurricane damage to a pine stand after Hurricane Katrina in 2005.

  9. 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 ($) 400,000 CPB  300,000 200,000 CH + CPB 0 0 200,000 400,000 600,000 800,000 1,000,000 CH ($)

  10. 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)

  11. 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

  12. (Figure for question 7) Roads Streams Streams to be treated with logs or boulders

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