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NETWORK PROBLEMS
THE PROBLEM
Imagine a salesman or a milk vendor or a post man who has to cover certain previously earmarked places to perform his daily routines. It is assumed that all the places to be visited by him are connected well for a suitable mode of transport. He has to cover all the locations. While doing so, if he visits the same place again and again on the same day, it will be a loss of several resources such as time, money, etc. Therefore he shall place a constraint upon himself not to visit the same place again and again on the same day. He shall be in a position to determine a route which would enable him to cover all the locations, fulfilling the constraint. The shortest route method aims to find how a person can travel from one location to another, keeping the total distance traveled to the minimum. In other words, it seeks to identify the shortest route to a series of destinations.
EXAMPLE
Let us consider a real life situation involving a shortest route problem.
A leather manufacturing company has to transport the finished goods from the factory to
the store house. The path from the factory to the store house is through certain
MBA-H2040 Quantitative Techniques for Managers
intermediate stations as indicated in the following diagram. The company executive wants to
identify the path with the shortest distance so as to minimize the transportation cost. The
problem is to achieve this objective.
95 Store house
Factory 40 40 35 65 70 6
Linkages from Factory to Store house
The shortest route technique can be used to minimize the total distance from a node designated as the
starting node or origin to another node designated as the final node.
In the example under consideration, the origin is the factory and the final node is the store house.
STEPS IN THE SHORTEST ROUTE TECHNIQUE
The procedure consists of starting with a set containing a node and enlarging the set by choosing a node in each
subsequent step.
Step 1:
First, locate the origin. Then, find the node nearest to the origin. Mark the distance between the origin and the
nearest node in a box by the side of that node.
In some cases, it may be necessary to check several paths to find the nearest node.
Step 2:
Repeat the above process until the nodes in the entire network have been accounted for. The last distance placed in
a box by the side of the ending node will be the distance of the shortest route. We note that the distances indicated
in the boxes by each node constitute the shortest route to that node. These distances are used as intermediate
results in determining the next nearest node.
SOLUTION FOR THE EXAMPLE PROBLEM
Looking at the diagram, we see that node 1 is the origin and the nodes 2 and 3 are neighbours to the
origin. Among the two nodes, we see that node 2 is at a distance of 40 units from node
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95 Store house
Factory 35 65 70 1 40 100 5 3 20
ITERATION No. 1
Now we search for the next node nearest to the set of nodes {1, 2}. For this purpose, consider those nodes
which are neighbours of either node 1 or node 2. The nodes 3, 4 and 5 fulfill this condition. We calculate the
following distances.
The distance between nodes 1 and 3 = 100.
The distance between nodes 2 and 3 = 35.
The distance between nodes 2 and 4 = 95.
The distance between nodes 2 and 5 = 65.
Minimum of {100, 35, 95, 65} = 35.
Therefore, node 3 is the nearest one to the set {1, 2}. In view of this observation, the set of nodes is enlarged
from {1, 2} to {1, 2, 3}. For the set {1, 2, 3}, there are two possible paths, viz. Path 1 โ 2 โ 3 and Path 1 โ
3 โ 2. The Path 1 โ 2 โ 3 has a distance of 40 + 35 = 75 units while the Path 1 โ 3 โ 2 has a distance of
100 + 35 = 135 units.
Minimum of {75, 135} = 75. Hence we select the path 1 โ 2 โ 3 and display this path by thick edges. The
distance 75 is marked in a box by the side of node 3. We obtain the following diagram at the end of Iteration
No. 2.
95 Store house
Factory 40 40 6 35 65 70 1 40 100 5 3 20
Factory 100
ITERATION No. 3
Now 2 nodes remain, viz., nodes 4 and 6. Among them, node 4 is at a distance of 135 units from the
origin (95 units from node 4 to node 2 + 40 units from node 2 to the origin). Node 6 is at a distance of 135 units
from the origin (40 + 95 units). Therefore, nodes 4 and 6 are at equal distances from the origin. If we choose
node 4, then travelling from node 4 to node 6 will involve an additional distance of 40 units. However, node 6 is
the ending node. Therefore, we select node 6 instead of node 4. Thus the set is enlarged from {1, 2, 3,
5} to {1, 2, 3, 5, 6}. The distance 135 is marked in a box by the side of node 6. Since we have got a path beginning from the start node and terminating with the stop node, we see that the solution to the given problem
has been obtained. We have the following diagram at the end of Iteration No. 4.
95 Store house
Factory 100
MINIMUM DISTANCE
ITERATION No. 4
Referring to the above diagram, we see that the shortest route is provided by the path 1 โ 2
โ 3 โ 5 โ 6 with a minimum distance of 135 units.
MBA-H2040 Quantitative Techniques for Managers
QUESTIONS
- Explain the shortest path problem.
2. Explain the algorithm for a shortest path problem
3. Find the shortest path of the following network:
4. Determine the shortest path of the following network:
