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GENERAL LINEAR PROGRAMMING NOTATION 71

MANAGEMENT SCIENCE IN ACTION

Using Linear Programming for Traffic Control
he Hanshin Expressway was the first urban toll In the first phase of the steady-state case, the
T expressway in Osaka, Japan. Although in 1964 Hanshin system uses a linear programming model
its length was only 2.3 kilometres, today it is a large- to maximize the total number of vehicles entering the
scale urban expressway network of 200 kilometres. system, while preventing traffic congestion and
The Hanshin Expressway provides service for the adverse effects on surrounding road networks. The
Hanshin (Osaka-Kobe) area, the second-most popu- data that drive the linear programming model are
lated area in Japan. An average of 828 000 vehicles collected from detectors installed every 500 metres
use the expressway each day, with daily traffic some- along the expressway and at all entrance and exit
times exceeding one million vehicles. In 1990, the ramps. Every five minutes the real-time data col-
Hanshin Expressway Public Corporation started using lected from the detectors are used to update the
an automated traffic control system in order to max- model coefficients, and a new linear programme
imize the number of vehicles flowing into the express- computes the maximum number of vehicles the
way network. expressway can accommodate.
The automated traffic control system relies on two The automated traffic control system has been
control methods: (1) limiting the number of cars that successful. According to surveys, traffic control
enter the expressway at each entrance ramp; and (2) decreased the length of congested portions of
providing drivers with up-to-date and accurate traffic the expressway by 30 per cent and the duration
information, including expected travel times and infor- by 20 per cent. It proved to be extremely cost
mation about accidents. The approach used to limit effective, and drivers consider it an indispensable
the number of vehicles depends upon whether the service.
expressway is in a normal or steady state of opera-
Based on T. Yoshino, T. Sasaki and T. Hasegawa, ‘The Traffic-Control
tion, or whether some type of unusual event, such as System on the Hanshin Expressway’ Interfaces (January/February
an accident or a breakdown, has occurred. 1995): 94–108.

The Management Science in Action, Using Linear Programming for Traffic Control,
provides just one of many examples of the widespread use of linear programming. In the
next two chapters we will see many more applications of linear programming.

Problems

1 Which of the following mathematical relationships could be found in a linear programming
model, and which could not? For the relationships that are unacceptable for linear
programmes, state why.
a. 1x 1 +2x 2 1x 3 70
b. 2x 1 2x 3 ¼ 50
2
c. 1x 1 2x 2 +4x 3 10
p ffiffiffiffiffi
d. 3 x 1 +2x 2 1x 3 15
e. 1x 1 +1x 2 +1x 3 ¼ 6
f. 2x 1 +5x 2 +1x 1 x 2 25
2 Find the feasible solution points for the following constraints:
a. 4x 1 +2x 2 16
b. 4x 1 +2x 2 16
c. 4x 1 +2x 2 ¼ 16

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