CHAPTER 8
NONLINEAR PROGRAMMING
SOLUTION TO SOLVED PROBLEMS
8.S1 Airline Ticket Pricing Model
Business travelers tend to be less price sensitive than leisure travelers. Knowing this, airlines
have discovered that extra profit can be generated by using separate pricing for these two types
of customers. For example, airlines often charge more for a midweek flight (mostly business
travelers) than for travel that includes a Saturday-night stay (mostly leisure travelers). Suppose
an airline has estimated demand vs. price for midweek travel (mostly business travelers) and for
travel that includes a Saturday-night stay (mostly leisure travelers) as shown in the table below.
This flight is served by a Boeing 777 with capacity for 300 travelers. The fixed cost of operating
the flight is $30,000. The variable cost per passenger (for food and fuel) is $30.
Price
$200
$300
$400
$500
$600
$700
$800
Midweek
150
105
82
63
49
35
27
Demand
Saturday-night Stay
465
210
127
82
60
45
37
1
Total
615
315
209
145
109
80
64
a. One function that can used to estimate demand (D) as a function of price (P) is a linear demand
function, where D = a – bP. For positive values of a and b, this will give lower demand when the
price is higher. However, a nonlinear demand function usually can provide a better fit to the
data. For example, one such function is a constant elasticity demand function, where D = aPb.
For positive values of a and negative values of b, this also will give lower demand when the price
is higher. Graph the above data and use the Add Trendline feature of Excel to find the constant
elasticity demand function that best fits the data in the above table for midweek demand,
Saturday-night stay demand, and total demand
A
B
C
D
E
Business
Travelers
(Mid-Week)
150
105
82
63
49
35
27
Liesure
Travelers
(Sat-Night Stay)
465
210
127
82
60
45
37
Total
615
315
209
145
109
80
64
Airline Demand
1
2
3
4
5
6
7
8
9
10
11
12
Price
$200
$300
$400
$500
$600
$700
$800
Demand Curve
700
600
y = 3287816x
-1.62
500
(Mid-Week)
(Sat-Night Stay)
400
Demand
Total
300
Power ((Mid-Week))
-1.83
y = 7490107x
Power ((Sat-Night Stay))
200
Power (Total)
100
y = 102922x
-1.21
0
$0
$100
$200
$300
$400
$500
Price
2
$600
$700
$800
b. For this part, assume that the airline charges a single price to all customers. Using the demand
function for total demand determined in part a, formulate and solve a nonlinear programming
model in a spreadsheet to determine what the price should be so as to achieve the highest profit
for the airline.
The decision to be made is how much to charge for the tickets. Therefore, define a changing cell
TicketPrice (C12). Based on part a, the demand for tickets is Demand = a * (TicketPrice)b =
3,287,816 * TicketPrice-1.62. The goal is to maximize profit. The profit is based on the ticket
price, variable cost, demand, and fixed cost as follows:
Profit = (TicketPrice – VariableCost) * Demand – Fixed Cost.
This formula is entered into Profit (C16). The Solver information and solved spreadsheet is
shown below.
A
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
B
C
E
Airline Price Model (Uniform Price)
Demand = a * (Price)^b
Total Demand
a
3,287,816
b
-1.62
Variable Cost
$30
Fixed Cost
$30,000
Ticket Price
$311.67
Demand
Profit
300
<=
Capacity
300
$54,501
B
14
15
16
D
C
Demand =a*TicketPrice^b
Profit =(TicketPrice-VariableCost)*Demand-FixedCost
Range Name
a
b
Capacity
Demand
FixedCost
Profit
TicketPrice
VariableCost
3
Cells
C5
C6
E14
C14
C10
C16
C12
C8
c. Now assume that the airline charges separate prices for midweek and Saturday-night stay
tickets. Using the two demand functions for midweek and Saturday-night stay tickets determined
in part a, formulate and solve a nonlinear programming spreadsheet model to determine what
the prices of the two types of tickets should be so as to maximize the profit for the airline.
The solution to part c is similar to part b. There are now two decisions: (1) the price for midweek
and (2) the price for Saturday-night stay tickets. Based on part a, the demand for midweek tickets
is Demand = 102,922* TicketPrice–1.21. The demand for Saturday-night stay tickets is 7,490,107*
TicketPrice–1.83.The solved spreadsheet is shown below.
A
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
B
C
D
E
F
G
Total
300
<=
Capacity
300
Airline Pricing Model
Demand = a * (Price)^b
Mid-Week
a
102,922
b
-1.21
Variable Cost
Sat-Night Stay
7,490,107
-1.83
$30
$30
Fixed Cost
$30,000
Ticket Price
$709.59
$271.49
37
263
Demand
Profit
$58,457
d. How much extra profit can the airline achieve by charging higher prices for midweek tickets
than for Saturday-night stay tickets?
With higher prices for midweek tickets, the profit is $58,457. With uniform pricing, the profit is
$54,501. The extra profit with higher prices for midweek tickets is $3,956.
4