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180 CHAPTER 4 LINEAR PROGRAMMING APPLICATIONS

Figure 4.10 Logistics of the Leisure Air Problem

Edinburgh
E
Glasgow
G
Flight
Leg 1 Flight
Leg 2
Amsterdam
A
Flight
Leg 4
Flight
Leg 3
Geneva Salzburg
V S

Next we must write the constraints. We need two types of constraints: capacity and
demand. We begin with the capacity constraints.
Consider the Glasgow–Geneva flight leg in Figure 4.10. The Boeing 737–400
aeroplane has a 132-seat capacity. Three possible final destinations for passengers
on this flight (Amsterdam, Salzburg or Geneva) and two fare classes (Q and Y)
provide six ODIF alternatives: (1) Glasgow–Amsterdam Q class; (2) Glasgow–Salz-
burg Q class; (3) Glasgow–Salzburg Q class; (4) Glasgow–Amsterdam Y class; (5)
Glasgow–Geneva Y class; and (6) Glasgow–Salzburg Y class. Thus, the number of
seats allocated to the Glasgow–Amsterdam flight leg is GAQ + GSQ + GVQ +
GAY + GSY + GVY. With the capacity of 132 seats, the capacity constraint is as
follows:
GAQ þ GSQ þ GVQ þ GAY þ GSY þ GVY 132 Glasgow---Amsterdam
The capacity constraints for the Edinburg–Amsterdam, Amsterdam–Geneva and
Amsterdam–Salzburg flight legs are developed in a similar manner. These three
constraints are as follows:

EAQ þ ESQ þ EVQ þ EAY þ ESY þ EVY 132 Edinburgh---Amsterdam
GVQ þ GVY þ EVQ þ EVY þ AVQ þ AVY 132 Amsterdam---Geneva
GSQ þ GSY þ ESQ þ ESY þ ASQ þ ASY 132 Amsterdam---Salzburg
The demand constraints limit the number of seats for each ODIF based on the
forecasted demand. Using the demand forecasts in Table 4.16, 16 demand constraints
must be added to the model. The first four demand constraints are as follows:

GAQ 33 Glasgow-Amsterdam Q class
GSQ 44 Glasgow-Geneva Q class
GVQ 45 Glasgow-Salzburg Q class
GAY 16 Glasgow-Amsterdam Y class
The complete linear programming model with 16 decision variables, four capacity
constraints and 16 demand constraints is as follows:

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