Prateek Singh Bapna
Prof. S. Kannan
Transportation problem is a special kind of LP
problem in which goods are transported from a
set of sources to a set of destinations subject to
the supply and demand of the source and the
destination respectively, such that the total cost
of transportation is minimized.
Formerly, this problem was studied under the
single objective of the minimization of :
total path length
Location of a new rail line between two major
cities of a developing country
In the construction of networks of highways and
In the design of airline routes
A Multiobjective Approach to
Transportation Network Design
Istanbul Technical University, Faculty of
Management, Industrial Engineering Department,
Macka, Istanbul, Turkey
In this article, a two-layer transportation network
design model with three objectives is
The objectives are:
The minimization of the primary path length (or
traveling time) from a predetermined starting node to
a predetermined terminus node.
The minimization of the total distance traversed by the
demand to reach a node on the primary path.
The minimization of the total network construction
Formulation of the Problem
Demand exists at every node.
Demand at every node must be satisfied.
Demand at a node is satisfied if either the node
is on the primary path or is connected to the
primary path via a secondary path.
Flow along all arcs is incapacitated.
All arc costs are non-negative.
There is no budget constraint.
Demands and costs are deterministic.
Primary and secondary arc costs are not
proportional to the arc length.
Transshipment costs are neglected
D = demand at node i
Li,j = length of arc (i,j)
Ti,j = distance (or travel time) via the shortest path from
node i to node j
C = the primary road construction cost of arc (i, j)
C’i,j = the secondary road construction cost of arc (i, j)
Xi,j = 1 if a primary path connects node i to node j
0 if otherwise
Yi,j = 1 if a secondary path connects node i and node
j to reach the primary path
0 if otherwise
Pi = (j/a path from node i to node j is defined)
Ni = (j/arc (i, j) exists)
Mj = (i/arc (i, j) exists)
node s = the starting node
node t = the terminus node
V = the set of nodes
Q = a nonempty subset of V
|Q| = the cardinality of subset Q
Incorporates a K-shortest path algorithm.
By using this algorithm, all primary paths, from a
predetermined starting node to a predetermined
terminus node are determined.
After finding all shortest paths, all secondary path
combinations are determined for each primary path.
The objective values are calculated for all
After eliminating the non-efficient solutions, the
qualitative criteria are included.
5. The decision maker is asked to scale the efficient solutions
alternate transportation facilities (Z )
environmental concerns (Z )
socio-economic structure of the nodes which are covered
by the path ( i e . regional development) (Z6).
For each objective, the best values are chosen to
find the ideal point proposed by Zeleny. The
weighted Euclidian distance between each solution
and ideal point are calculated as follows:
d = [(Z1* - Z1)2 + (Z2* - Z2)2 + … + (Zn* - Zn)2]1/2
The solution having the minimum total weighted
Euclidian distance is selected.
The major drawbacks to solving the model as a
relaxed linear program are the necessity for an
interactive solution technique to eliminate sub
tours and the computational burden imposed by
the branch and bound algorithm.
And also the need to combine all objective
functions in one function in a weighted form.
Because of these drawbacks the authors
proposed an interactive combinatorial solution
method especially preferable for small networks.
In general, the integer programming solution
approach yields only one feasible solution, the
Proposed method, however, yields the whole
set of feasible solutions. Since many network
design problems are multi objective in nature
these additional feasible solutions may be of
interest to the network planner.
But of course the proposed method can be used
efficiently only for small networks.