Chapter 8 - Nonlinear Programming : S-1
————————————————————————————————————————————
Chapter 8
Nonlinear Programming
1.
The GRG algorithm can be used to solve LP problems. If an LP problem has alternate optimal solutions,
the GRG algorithm may terminate at an optimal solution that is not a corner point of the feasible region.
2.
a.
b.
3.
a & b.
c.
Any direction that forms no more than a 180 degree angle with the level curve.
Any direction that forms no more than a 180 degree angle with a line tangent to the level curve.
Chapter 8 - Nonlinear Programming : S-2
————————————————————————————————————————————
4.
a.
25
20
15
Y
10
5
0
1
b.
c.
d.
e.
5.
2
3
4
5
6
7
8
3
4
5
6
7
9 10 11 12 13 14 15 16 17 18 19 20
X
2
3 (including endpoints)
X= 4.5, Y= 13.51
X= 17.13, Y= 24.78
a.
30
25
20
Y
15
10
5
0
1
b.
c.
d.
e.
2
3 (including endpoints)
2
X= 3.86, Y = 13.25
X= 17.55, Y = 18.11
8
9 10 11 12 13 14 15 16 17 18 19 20
X
Chapter 8 - Nonlinear Programming : S-3
————————————————————————————————————————————
6.
a.
b.
c.
Same as in chapter with Xi/(Xi+i)  0.5*Yi
See file: Prb8_6.xlsm
Project
1
2
3
4
5
6
7.
a.
b.
c.
Engineers
Assigned
4
0
6
6
9
0
0.35A + 8.3A + 540  0.60B + 9.45B + 1108  0.47C + 11.0C + 850
3 A + 2 B + 4 C  10,000
2 A + 4 B + 3 C  9,000
3 A + 4 B + 2 C  11,000
A, B, C  0
See file: Prb8_7.xlsm
A=11.857, B=7.875, C=11.702, Profit = $2,648.78
MAX:
ST
2
2
2
8.
See file: Prb8_8.xlsm
Newspaper = $11,323, Radio = $4,568, TV = $3,246, Mail = $863. Profit $129,096.
9.
a.
b.
c.
10. a.
b.
c.
MAX
-0.2(X1)2 - 0.4(X2)2 + 8X1 + 12X2 + 1500
ST
X1 + X2  80
2X1 + 2X2  64
Xi  0 and integer
See file: Prb8_9.xlsm
The optimal solution is: X1= 18, X2= 14. Maximum profit = $1,668.8 (in $000s)
MIN
(SQRT((17 - X)2 + (34 - Y)2) – 29.5)2
+ (SQRT((12 - X)2 + (5 - Y)2) – 4.0)2
+ (SQRT((3 - X)2 + (23 - Y)2) – 17.5)2
See file Prb8_10.xlsm
X=8.0372, Y=6.0545
11. a.
b.
Minimize r . This is a linear objective.
.72*((1+ r/4)4 -1) b1 423
.72*((1+ r/4)4 -1) b2 457
These constraints are nonlinear functions of the decision variable r.
12. a.
b.
See file: Prb8_12.xlsm, Minimum investment = $5,193
This model is linear.
13. See file: Prb8_13.xlsm, Yield = 12.51%
14. See file: Prb8_14.xlsm
a. 1549
b. $7,693
c. Ordering cost = $96.82, Holding cost = $96.82
Chapter 8 - Nonlinear Programming : S-4
————————————————————————————————————————————
15. a. See file: Prb8_15.xlsm
b.
Team
1
2
3
4
5
6
7
8
9
10
A
1
2
3
4
5
6
7
8
9
10
Player From Flight
B
C
5
10
8
4
2
9
1
5
10
7
4
3
9
1
6
8
3
2
7
6
D
7
6
10
8
9
3
1
2
5
4
Optimal objective value (team handicap variance) = 0.277777777777778
16. See file Prb8_16.xlsm
a.
b.
c.
Putting 100% in REREX produces the highest return (31.81%) with a variance of 106.71.
The portfolio with the smallest return is:
RGAEX
3.82%
DODGX
4.38%
SWPPX
5.14%
CVGRX
1.69%
HSVRX
2.29%
HWMAX
2.58%
MSSGX
1.99%
RWIEX
4.58%
REREX
3.72%
PTRAX
22.59%
DODBX
9.21%
Chapter 8 - Nonlinear Programming : S-5
————————————————————————————————————————————
SWBRX
18.34%
SWCRX
8.73%
SWDRX
6.00%
SWERX
4.93%
With an expected retrun and variance of 15.9% and 3.965, respectively.
d.
17. a.
b.
c.
d.
e.
Optimal portfolio with 18% expected return (var = 4.18) is:
RGAEX
4.43%
DODGX
5.34%
SWPPX
5.30%
CVGRX
1.98%
HSVRX
3.21%
HWMAX
3.62%
MSSGX
2.80%
RWIEX
6.14%
REREX
5.26%
PTRAX
15.15%
DODBX
9.47%
SWBRX
16.44%
SWCRX
8.82%
SWDRX
6.48%
SWERX
5.55%
MIN
ST
60*(800/Q1 + 500/Q2 + 1500/Q3) + 0.25*(300Q1 + 1100Q2 + 600Q3)/2
300Q1 + 1100Q2 + 600Q3  45,000
9Q1 + 25Q2 + 16Q3  3,000
Q1 , Q2, Q3  1 and integer
See file: Prb8_17.xlsm
Model 1 = 34, Model 2 = 14, Model 3 = 32
Model 1 = 800/34 = 23.5 , Model 2 = 500/14 = 35.7 , Model 3 = 1500/33= 46.9
Model 1 = about every 15 days , Model 2 = about every 10 days , Model 3 = about every 8 days
18. See file: Prb8_18.xlsm
a. Q=1000, Cost = $48,855
b. Q=107, Cost = $53,061
19. See file: Prb8_19.xlsm
a. Day price = $0.1307, Night price = $0.0805
b. Lines needed = 27,076
20. a.
MAX
ST
(P1-850)X1 + (P2-700)X2
X1 + X2  200
9X1 + 6X2  1566
12X1 + 16X2  2880
1000  P1, P2  1500
where,
X1 = 300 - 0.175P1
X2 = 325 - 0.150P2
Chapter 8 - Nonlinear Programming : S-6
————————————————————————————————————————————
b.
c.
d.
e.
21. a.
b.
c.
See file: Prb8_20.xlsm
Aqua-Spas = $1,282, Hydro-Luxes = $1,433
None! The optimal solution occurs at a point in the interior of the feasible region.
Zero.
MAX
ST
0.6p - 0.002p2 + 0.5f - 0.009f2 - 0.001pf
p + f  600
p + 0.5f  16*60 = 960
p, f  0 & integer
See file: Prb8_21.xlsm
p=93 (or 94), f = 226 (alternate optima exist), Maximum profit = $84.52
22. See file: Prb8_23.xlsm
a. Minimum rate of return  16.6%
b. Monthly contribution = $132
23. a.
b.
c.
See file Prb8_23.xlsm
Assign sales reps to regions 3, 5, 8, 9 and 10. Total variance = 370.
You could minimize the range of the number of doctors assigned. Another objective might consider
geographical size of the regions.
24. See file: Prb8_24.xlsm
a. 399.22 miles of pipe would be needed.
b. X=72.37, Y=93.38, 288.38 miles of pipe is needed
c. Building the substation is best.
d. X=70.371, Y=92.3, 288.44 miles of pipe is needed
25. a.
MIN
130*((X-9)2+(Y-43)2)0.5 +75*((X-2)2+(Y-28)2) 0.5
+90*((X-51)2+(Y-36)2) 0.5 +80*((X-19)2+(Y-4)2) 0.5
ST
((X-9)2+(Y-43)2)0.5  50
((X-2)2+(Y-28)2) 0.5  50
((X-51)2+(Y-36)2) 0.5  50
((X-19)2+(Y-4)2) 0.5  50
b.
c.
d.
e.
26. a.
b.
c.
27. a.
See file: Prb8_25.xlsm
X = 11.97, Y = 35.36, Total shipping miles = 8079.27
X = 12.41, Y = 29.7, Total shipping miles = 8217.63
X = 27.18, Y = 27.82, Total shipping miles = 9249.47
0.5
MIN ( (35-X)2 + (57-Y)2 + (46-X)2 + (48-Y)2 (37-X)2 + (93-Y)2 +(22-X)2 + (67-Y)2 )
The solution is: X=35, Y=57 (which corresponds to the coordinates of Bumpyride)
See file: Prb8_26.xlsm
Vertical distances over the mountains would be a consideration that could be modeled by more
complicated nonlinear functions. Weights could be assigned to the distances to reflect the differences
in the numbers of skiers/accidents at different resorts.
MIN
ST
2(X1)2 - 1X1 + 15 + (X2)2 + 0.3X2 + 10 + CijXij
X11 + X12 + X13 + X14  600
X21 + X22 + X23 + X24  600
X11+ X21  300
Chapter 8 - Nonlinear Programming : S-7
————————————————————————————————————————————
X12+ X22  250
X13+ X23  150
X14+ X24  400
Xij  0
where
X1 = X11 + X12 + X13 + X14
X2 = X21 + X22 + X23 + X24
b.
c.
28. a.
b.
c.
d.
29. a.
b.
See file: Prb8_27.xlsm
Total cost = $887,123
(1-(1-P11)X11) (1-(1-P12)X12) (1-(1-P43)X43) (1-(1-P44)X44)
jXij = 1, for each i
iXij = 1, for each j
Xij binary
See file: Prb8_28.xlsm
Assign Sam to 4, Billie to 3, Sally to 2, Fred to 1
Prob. of receiving all donations = 0.15396
Sam to 4, Billie to 2, Sally to 1, Fred to 3, Expected value = $3.97 million
Assign Sam to 4, Billie to 3, Sally to 2, Fred to 1, Max regret = 0.07
MAX
ST
See file: Prb8_29.xlsm
From
1
1
2
2
3
3
4
5
6
c.
30. a.
b.
To
2
3
4
5
4
5
6
6
1
Flow
80
30
20
60
30
0
50
60
110
Probability of no failure = 0.96
See file: Prb8_30.xlsm
Warehouse1 = (5.8, 14.9), Warehouse 2 = (12.0, 11.0), Warehouse 3 = (17.0, 3.9), Distance = 25.486
Note: It is important to use a good starting solution in this problem; otherwise a local optimum is
likely.
31. See file: Prb8_31.xlsm
a. A = 65.5%, B=6.8%, C=27.6%, Variance = 0.00947
b. A = 41.9%, B=15.3%, C=42.8%, Variance = 0.00462
32. See file Prb8_32.xlsm
a. A=43.8%, B=17.9%, C=4.9%, D=33.4%, variance=0.00088, return=10.68%.
b. C=100%, variance=0.03406, return=15.25%.
c. A=79.6%, B=2.8%, C=17.6%, D=0%, variance=0.00138, return=13.28%.
33. See file: Prb8_33.xlsm
Windsor = 20.7%, Flagship = 33.6%, Templeman = 10.6%, T-Bills = 35.1%
Chapter 8 - Nonlinear Programming : S-8
————————————————————————————————————————————
34. See file: Prb8_34.xlsm
a. This is a fairly straightforward quadratic portfolio selection problem as outlined in the chapter. The
only "twist" is that the decision variables represent the amount of money to invest in each stock. The
percentage invested in each stock can then be calculated easily for use in the objective function. The
optimal solution is: Amalgamated Industries = $11,609, Babbage Computers = $11,660, Consolidated
Foods = $6,732 .
b. This generates $952 in expected earnings, so Barbara should have enough to buy the computer she
wants.
35. a.
b.
c.
d.
See file Prb8_35.xlsm
See file Prb8_35.xlsm
None.
65.78 minutes of travel time is the best solution I’ve found.
36. a. See file: Prb8_36.xlsm
b. 4 mortgage packages of at least $1 million can be created.
37. See file Prb8_37.xlsm
a. Order: 5,4,7,8,3,6,1,2,10,9.
b. Order: 1,3,5,4,2,6,7,8,10,9.
c. Order: 4,5,6,2,10,8,9,3,1,7.
d. Order: 1,3,7,5,4,8,10,2,9,6.
e. Order: 1,2,3,4,5,6,7,8,10,9.
Number late = 5, Days late = 91, Max lateness = 30.
Number late = 6, Days late = 67, Max lateness = 21.
Number late = 4, Days late = 138, Max lateness = 59.
Number late = 5, Days late = 81, Max lateness = 40.
Number late = 8, Days late = 93, Max lateness = 21.
38. a & b. See file Prb8_38.xlsm
c. 0, 3, 1, 10, 8, 2, 7, 5, 6, 4, 9, 0. Tour length = 122.8.
39. a.
b.
c.
d.
See file: Prb8_39.xlsm
Tour length = 7,289.6
Tour: 0, 16, 10, 2, 5, 15, 8, 9, 14, 13, 6, 7, 11, 12, 1, 3, 4, 0. Tour length = 3,415.0
Week 1: 0, 4, 1, 3, 16, 0
Week 2: 0, 13, 14, 9, 8, 0
Week 3: 0, 6, 7, 11, 12, 0
Week 4: 0, 15, 5, 2, 10, 0
Length = 4798.3
40. Teaching note: There are a number of ways to build a model for this problem. Note that the solution given
uses the INDEX( ) function to return a vector of group averages as the second argument within the
SUMXMY2( ) function. (Using a value of zero in the 2 nd (row) or 3rd (column) argument of the INDEX( )
function causes it to return the entire column or row, respectively.) Also note that Solver may run on this
problem for quite some time.
a.
b.
See file: Prb8_40.xlsm, Total distance = 250.60 (better solutions may exist).
Group Averages
Avg
#
Group
Income
Purchase
Purchases
Dependents
1
3.03
1.59
3.29
2.08
2
3.75
1.82
5.62
1.59
3
3.32
1.74
4.88
5.15
Group 1 appears to have the lowest income and lowest number of dependents (recent college graduates
perhaps?).
Chapter 8 - Nonlinear Programming : S-9
————————————————————————————————————————————
Group 2 appears to have the highest average income, fewest dependents, and makes more purchases
than the other groups (retirees perhaps?).
Group 3 appears to represent people with the highest number of dependents and average level of
income (families with kids perhaps?).
41. a.
b.
c.
d.
See file: Prb8_41.xlsm
97 out of 127 or 76%
See file: Prb8_41.xlsm. Note that this requires Solver's evolutionary algorithm.
113 out of 127 or 89%
Case 8-1: Tour De Europe
See file: Case8_1.xlsm
1.
2.
1 -> 6-> 8-> 10-> 1 , Optimal cost = $270
Case 8-2: Electing The Next President
1 & 2. See file: Case8_2.xlsm
3 & 4.
State
Visits
Florida
2
Georgia
1
California
3
Texas
1
Illinois
5
New York
5
Virginia
3
Michigan
1
5. Total votes = 197
Advertising $
1.753
0.868
1.997
2.101
0.500
0.856
1.297
1.627
Case 8-3: Making Windows at Wella
See file: Case8_3.xlsm
1.
825 = 3.77789E+22 = lots
Chapter 8 - Nonlinear Programming : S-10
————————————————————————————————————————————
2.
3.
4.
5.
Used 33 pieces of stock, generated 720.375 inches of scrap.
Used 31 pieces of stock, generated 336.375 inches of scrap.
$4 × 4 pieces saved per hr × 8 hours per day × 5 days per week × 52 weeks per year × 2 plants × 5
production lines per plant = $332,800 annual savings.
i) What does Wella do with the existing scrap? Can it be sold? If so, for how much? Will producing less
of it create a problem?
ii) Scrap could be eliminated entirely if the finger joining process took place in Wella's factory – essentially
creating one infinitely long pice of stock.
Case 8-4: Newspaper Advertising Insert Scheduling
See file: Case8_4.xlsm
This is, in essence, a multiple traveling salesperson problem. The best solution I have found allows all papers to
be completed in 1.948 hours.
Teaching note: I have encountered a bug in Solver's evolutionary algorithm where, at times, it zeros out all the
values cells to which the 'alldifferent' constraint condition is applied. FrontLine is aware of this problem and
suggests using an IF( ) function in the fitness (objective) function to return an arbitrarily bad fitness value if the
fitness function evaluates to an error value. I have implemented this suggestion in the solution to this case.