In questions involving networks, a complex route diagram between two particular cities or places is given and the student is required to identify the total number of distinct routes or the total amount of a particular product flow from one city to other through pipelines.

Other variations include complex questions that involve calculation of the shortest distance, the time taken to reach based on speed calculations etc.**Strategies:**

· For any given network, wherever it is logically permitted, divide the network into parts.

· If there are two parts of an operation and these two parts can be performed in m and n ways, then that operation can be performed in m x n ways.

· If using one tool an operation can be performed in x ways, using another tool in y ways, and using yet another tool in z ways, the overall operation can be performed in x + y z ways.

· In case of product flow related questions, a proper understanding of the slack concept can be helpful.

Slack = the total capacity of the pipeline – the amount of product flow within that pipeline**Here's an example**

Given, a network of pipelines carrying oil for distribution among ten towns A, B, C, D, E, F, G, H, I and J.

The maximum capacity of every pipeline shown is 1000 units.

The values given next to a town represents the consumption of oil at that town.

Given

i. The oil flows only in the directions shown by the arrows.

ii. The slack in any pipeline is the extra oil that can flow through it to bring the pipeline to full capacity.

iii. At every town, the requirements are exactly met

The slack details of some of the pipelines are given as follows:

Pipeline |
Slack |

AB |
200 |

AC |
200 |

FH |
900 |

CE |
500 |

DG |
700 |

EF |
800 |

GI |
600 |

1. The difference between the slack in the pipeline connecting A and G and the slack in the pipeline connecting F and I is

(1) 300 units (2) 600 units (3) 400 units (4) 500 units (5) Cannot be determined

2. If the flow in the pipeline connecting I and J is 300 units, then the slack in the pipeline connecting A and E is

(1) 700 units (2) 200 units (3) 500 units (4) 600 units (5) Cannot be determined

3. IF the slack in the pipeline that supplies to town A is 400 units, then the capacity of that pipeline ( assuming that it is the only pipeline supplying to town A) is

(1) 3600 units (2) 2800 units (3) 4200 units (4) 4400 units (5) 4900 units

4. The flow in the pipeline from C to G is

(1) 100 (2) 400 (3) 700 (4) 200 (5) Cannot be determined

**Solution:**

The basic approach here is to balance the inflows and out flows plus consumption at each city.

1. Consider city F. Inflow = outflow + consumption EF + GF = FH + FI + Consumption at F

200 + GF = 100 + FI + 100

GF = FI (a)

Again at city G,

CG + AG + DG = GF + GI + Consumption at G Now CG = AC – 200 – 500 = 100

Hence, 100 + AG + 300 = GF + 400 + 300

AG – GF = 300 (b)

From (a) and (b)

AG – FI = 300 ( Note: Difference in flows = difference in slacks )

2. Using the information given that IJ = 300 units, we can solve for HJ as follows HJ – 800 – 300 = 500 (a) (at city J)

From (a)

EH = (300 + 500) – 100 = 700 (b) (at city H)

Now consider city E

AE = 300 + 500 = 200 + 700 + 200 à AE = 300 ( BE = 800 – 500 = 300)

Hence, slack = 1000 – 300 = 700.

3. The total inflow into the city A has to be equal to the total requirements of all the cities together i.e.

Inflow into A = 300 + 500 + 200 + 600 + 200 + 100 + 300 + 300 + 500 + 800 = 3800

If slack is 400, then the capacity = 3800 + 400 = 4200

4. Given that the flow from AC= 800 and at C we have consumption = 200 Flow in CE = 500

Hence, C to G = 800 – (500 + 200) = 100