Chapter 7 Nonlinear Optimization Models
Sources of nonlinearity
 Non-constant returns
 Demand as a function of price is often nonlinear
 In investment decisions, risk is measured as a nonlinear function such as standard deviation
NLP problem = Nonlinear programming problem
Global optimum:
Local optimum:
A solution that is best in the entire feasible region
A solution that is better than all nearby solutions in the feasible region
Solver always finds global optimum solutions for LP problems. However, Solver often results in
local optimum solutions for NLP problems.
Convex function
Slope is always decreasing
y = cxa where a>0, c>0, x>0
y = cex where c > 0
Concave function
Slope is always increasing
y = cln(x) where c>0, and x>0
y = cxa where 0<a<1, c>0, x>0
Solver guarantees global optimum if:
Objective function
Conditions
Minimization
 Objective function is convex
 Constraints are linear
Maximization
 Objective function is concave
 Constraints are linear
Pricing models
Example 7.1 Pricing decision (Pricing1.xls)
Suppose that the company estimates demand to be 400 when price is $70 and 300 if the price is
$80. Estimate demand curve and determine the best price.
Linear price curve: D = a + b x, where x = price (non-constant demand elasticity)
Power price curve: D = axb, where x = price (constant demand elasticity)
Sensitivity:
 Profit as price changes from $55 to $110 in increments of 5
 Profit as elasticity changes from -2.4 to -1.8 in increments of .1
Example 7.2 Pricing decision with exchange rate (Pricing2.xls)
Suppose that the demand is given by D = 27556760x-2.4, where x = price. The present exchange
rate is $1 = £1.5599999. Determine the best price at which profit in $ is maximized.
Sensitivity:
 What will happen to profit as the exchange rate fluctuates
Example 7.3 Pricing suits at Sullivan’s (Pricing3.xls)
Suppose that the current cost of suits is $320 and the corresponding demand is 300 suits per year.
The elasticity of demand is constant at -2.5. Each purchase of a suit leads to an average purchase
of 2 shirts and 1.5 ties. Profit margins are $25 for shirt and $15 for ties. Determine profit
maximizing price.
Sensitivity:
 What if suits are sold below cost? Also, what if the sales of secondary products shirts
and ties are not the same as assumed?
Example 7.4 Peak load pricing (Pricing4.xls)
Demand is given by,
Dp = 60 – 0.5Pp + 0.1Po, and Do = 40 – Po + 0.1Pp
where, Dp = Peak demand in kwh, Pp = Peak price, per kwh Do = off-peak demand in kwh, and
Po = off-peak price per kwh. Also, cost to main 1 kwh of power per day is $10. Determine
pricing and capacity to maximize profit.
Sensitivity:
 Change unit cost of capacity from $5 to $15 in increments of 1
Advertising response and selection models
Example 7.5 Estimating an advertisement response function (AdvertisingResponse.xls)
Given:
Number of ads Exposures (millions)
1
4.7
8
22.1
20
48.7
50
90.3
100
130.5
Purpose: Estimate a response function that will estimate the number of exposures given number
of ads.
Plot of the given data reveals a nonlinear function which increases at a decreasing rate. There
are many mathematical functions that have this basic shape. One such function is,
f(n) = a(1 – e-bn)
where n = number of ads and a and b are parameters to be found.
Model
Changing cells: a and b
Target cell:
Minimize RSME (Root mean square error)
Constraints:
None
Example 7.6 Advertising selection (AdvertisingSelection.xls)
Given: Advertising exposure = f(n) = a(1 – e-bn)
where n = number of ads and a and b are parameters to be found.
Table for values of parameter “a”
Men 18-35
Men 36-55
Men >55
Women 18-35
Women 36-55
Women >55
Friends
93.061
61.129
33.376
105.803
71.784
56.828
Malcolm
in
Middle
84.772
61.528
9.913
66.998
46.146
8.887
MNF
116.808
76.527
57.84
40.113
26.534
17.209
Sports
Center
43.647
47.749
30.075
22.101
16.151
9.101
TRL
Live
26.711
19.655
10.751
42.451
34.609
8.46
Lifetime
movie
11.99
10.281
11.51
29.403
24.276
31.149
CNN
11.793
9.982
22.218
8.236
10.426
23.105
JAG
11.323
21.759
28.121
8.93
22.849
40.672
CNN
0.029
0.054
0.013
0.039
0.046
0.072
JAG
0.080
0.070
0.036
0.026
0.040
0.030
Table for values of parameter “b”
Men 18-35
Men 36-55
Men >55
Women 18-35
Women 36-55
Women >55
Friends
0.029
0.084
0.071
0.035
0.089
0.010
MNF
0.055
0.050
0.068
0.063
0.057
0.033
Malcolm
in
Middle
0.093
0.085
0.077
0.069
0.061
0.078
Sports
Center
0.071
0.094
0.027
0.074
0.055
0.078
TRL
Live
0.087
0.018
0.039
0.060
0.014
0.035
Lifetime
movie
0.038
0.090
0.051
0.012
0.022
0.050
Cost per ad
Cost
Friends
160
MNF
100
Malcolm
in
Middle
80
Sports
Center
9
TRL
Live
13
Lifetime
movie
15
CNN
8
JAG
85
Required exposures
Men 18-35
Men 36-55
Men >55
Women 18-35
Women 36-55
Women >55
Required
exposures
60
60
28
60
60
28
Model
Changing cells: Number of ads purchased
Target cell:
Minimize total cost
Constraints:
Total exposures for each demographic group is at least equal to the required exposures
Sensitivity analysis
How will the optimal cost vary if all the required exposures change by the same percentage?
Facility Location models
Example 7.7 Warehouse location (WarehouseLocation.xls)
Given: Customer location coordinates and shipments per year
Customer 1
Customer 2
Customer 3
Customer 4
X-coordinate
5
10
0
12
Y-coordinate
10
5
12
0
Annual shipments
200
150
200
300
Model
Changing cells: X and Y coordinate of warehouse location
Target cell:
Minimize total distance-shipment
Constraints:
None
Sensitivity analysis
How will the optimal location change if the annual shipment from a particular customer changes?
Modeling issues
 The distance formula used is for straight line distances. If rectangular distance is more
appropriate, use the formula |a-c| + |b-d| for distance, where (a,b) and (c,d) are coordinates.
 In this example we assumed a warehouses could be built anywhere. But often there may
only be a select set of locations where one could be built. Then, we need to incorporate (0,1)
programming to this model.
Rating teams
Example 7.8 Rating NFL teams (NFL2001Ratings.xls)
Given:
Results of all the games from 2001 season and nominal average rating
Model
Changing cells: Ratings for each team in the league, home team advantage
Target cell:
Minimize sum of squared error between actual and predicted point spread
Constraints:
Average rating is equal to nominal average
Terminologies
Actual point spread = Home team score – Visiting team score
Predicted point spread = Home team rating – visiting team rating + Home team advantage
Where predicted home and visiting team ratings must be determined by using VLOOKUP
functions
Modeling issues
Giving larger weight to more recent games may improve accuracy of prediction. This may be
achieved by multiplying squared error by k, where k = number of weeks the game is old, and 
= a factor between 0 and 1.