1. The document describes a nonlinear optimization problem with two decision variables, x and y, and four terms in the objective function: x^2y, xy^3, y^3, and a constraint that x^2 + y^2 <= 150000. 2. The problem is solved from four different initial points, with the optimal solution being x=749.8703 and y=1125.065, achieving an objective value of 3.21E+12. 3. The optimal solution is found to be the same regardless of the initial starting point, demonstrating the problem has a single globally optimal solution.

1. Weighted, all decision variables binary, except tennis courts are integer
1 2 3 4 5 6
Golf 1 0
Swimming 0 0 0 1
Gymnasium 0 0 0 0
Tennis courts 1 1 2 2
45 45 90 119
<= <= <= <=
Land requirement 70 80 95 120
User days 1 2 3 4 5 6
Golf 31 27
Swimming 26 21 32 32
Gymnasium 37 29 28 38
Tennis courts 20 23 22 20
Land requirement 1 2 3 4 5 6
Golf
Swimming 29 29 29 29
Gymnasium 38 38 38 38
Tennis courts 45 45 45 45

2. Construction cost Maintenance cost User days
340 340 80 80 31
300 300 36 36 32
840 0 50 0 0
85 510 17 102 127
1150 218 190
Goal Over Under Goal LHS Goal RHS Weight Weighted sum
1 0 50 1200 1200 1000 0
2 0 10 200 200 2 20
3 18 0 200 200 1 18
38

3. Weighted, all decision variables binary
1 2 3 4 5 6
Golf 1 0
Swimming 0 0 1 1
Gymnasium 0 0 0 0
Tennis courts 0 1 1 1
0 45 74 74
<= <= <= <=
Land requirement 70 80 95 120
User days 1 2 3 4 5 6
Golf 31 27
Swimming 26 21 32 32
Gymnasium 37 29 28 38
Tennis courts 20 23 22 20
Land requirement 1 2 3 4 5 6
Golf
Swimming 29 29 29 29
Gymnasium 38 38 38 38
Tennis courts 45 45 45 45

4. Construction cost Maintenance cost User days
340 340 80 80 31
300 600 36 72 64
840 0 50 0 0
85 255 17 51 65
1195 203 160
Goal Over Under Goal LHS Goal RHS Weight Weighted sum
1 0 5 1200 1200 1000 0
2 0 40 200 200 2 80
3 3 0 200 200 1 3
83

5. Pre-emptive, Round 1, all decision variables binary, except tennis courts are integer
1 2 3 4 5 6
Golf 0 0
Swimming 1 0 0 1
Gymnasium 0 0 0 0
Tennis courts 0 1 1 2
29 45 45 119
<= <= <= <=
Land requirement 70 80 95 120
User days 1 2 3 4 5 6
Golf 31 27
Swimming 26 21 32 32
Gymnasium 37 29 28 38
Tennis courts 20 23 22 20
Land requirement 1 2 3 4 5 6
Golf
Swimming 29 29 29 29
Gymnasium 38 38 38 38
Tennis courts 45 45 45 45
Note: There are multiple optimal solutions for Goal 1. This is what Excel gives if you start from all zeros.

6. Construction cost Maintenance cost User days
340 0 80 0 0
300 600 36 72 58
840 0 50 0 0
85 340 17 68 85
940 140 143
Goal Over Under Goal LHS Goal RHS Weight Weighted sum
1 0 260 1200 1200 1000 0
2 0 57 200 200 2 114
3 0 60 200 200 1 0
114
om all zeros.

7. Pre-emptive, Round 2, all decision variables binary, except tennis courts are integer
1 2 3 4 5 6
Golf 0 0
Swimming 0 0 0 1
Gymnasium 0 0 0 0
Tennis courts 1 1 2 2
45 45 90 119
<= <= <= <=
Land requirement 70 80 95 120
User days 1 2 3 4 5 6
Golf 31 27
Swimming 26 21 32 32
Gymnasium 37 29 28 38
Tennis courts 20 23 22 20
Land requirement 1 2 3 4 5 6
Golf
Swimming 29 29 29 29
Gymnasium 38 38 38 38
Tennis courts 45 45 45 45
Note: There are multiple optimal solutions for Goal 1. This is what Excel gives if you start from all zeros.

8. Construction cost Maintenance cost User days
340 0 80 0 0
300 300 36 36 32
840 0 50 0 0
85 510 17 102 127
810 138 159
<=
Goal 1, 1st round 940
Goal Over Under Goal LHS Goal RHS Weight Weighted sum
1 0 390 1200 1200 1000 0
2 0 41 200 200 2 82
3 0 62 200 200 1 0
82
om all zeros.

9. Pre-emptive, Round 1, all decision variables binary
1 2 3 4 5 6
Golf 0 0
Swimming 0 0 1 1
Gymnasium 0 0 0 0
Tennis courts 1 1 1 1
45 45 74 74
<= <= <= <=
Land requirement 70 80 95 120
User days 1 2 3 4 5 6
Golf 31 27
Swimming 26 21 32 32
Gymnasium 37 29 28 38
Tennis courts 20 23 22 20
Land requirement 1 2 3 4 5 6
Golf
Swimming 29 29 29 29
Gymnasium 38 38 38 38
Tennis courts 45 45 45 45
Note: There are multiple optimal solutions for Goal 1. This is what Excel gives if you start from all zeros.

10. Construction cost Maintenance cost User days
340 0 80 0 0
300 600 36 72 64
840 0 50 0 0
85 340 17 68 85
940 140 149
Goal Over Under Goal LHS Goal RHS Weight Weighted sum
1 0 260 1200 1200 1000 0
2 0 51 200 200 2 102
3 0 60 200 200 1 0
102
om all zeros.

11. Pre-emptive, Round 2, all decision variables binary
1 2 3 4 5 6
Golf 0 0
Swimming 0 0 1 1
Gymnasium 0 0 0 0
Tennis courts 1 1 1 1
45 45 74 74
<= <= <= <=
Land requirement 70 80 95 120
User days 1 2 3 4 5 6
Golf 31 27
Swimming 26 21 32 32
Gymnasium 37 29 28 38
Tennis courts 20 23 22 20
Land requirement 1 2 3 4 5 6
Golf
Swimming 29 29 29 29
Gymnasium 38 38 38 38
Tennis courts 45 45 45 45
Note: There are multiple optimal solutions for Goal 1. This is what Excel gives if you start from all zeros.

12. Construction cost Maintenance cost User days
340 0 80 0 0
300 600 36 72 64
840 0 50 0 0
85 340 17 68 85
940 140 149
<=
Goal 1, 1st round 940
Goal Over Under Goal LHS Goal RHS Weight Weighted sum
1 0 260 1200 1200 1000 0
2 0 51 200 200 2 102
3 0 60 200 200 1 0
102
om all zeros.

13. x y x^2y xy^3 y^3
749.8703 1125.065 6.33E+08 1.07E+12 1.42E+09
2 3 2 3.21E+12
50 100 150000 <= 150000
Initial values Final values
x0 y0 x y obj
0 0 0 0 0
1 1 749.8703 1125.065 3.21E+12
-1000 -1000 -1.8E+10 -5.4E+10 8.3E+42
1000 1000 749.8703 1125.065 3.21E+12

14. x y x^2y xy^3 y^3
749.8703 1125.065 6.33E+08 1.07E+12 1.42E+09
2 3 2 3.21E+12
50 100 150000 <= 150000
Initial values Final values
x0 y0 x y obj
0 0 0 0 0
1 1 749.8703 1125.065 3.21E+12
-1000 -1000 0 0 0
1000 1000 749.8703 1125.065 3.21E+12