CONSISTENCY AND REDUNDANCY IN PROJECT NETWORK

Step 1 Select a job which has no predecessors and place it on the top of the list. Step 2 Delete the job just placed from the predecessor list of all remaining jobs. Step 3 Identify any New sources (jobs without predecessors) so generated for placement on the list and go to step 2. Step 4 Continue until one of the following happens: (a) There are no sources on the list of remaining jobs. This shows the presence of an inconsistency. (b) All jobs are placed on the list, which is now topologically ordered, showing there are no inconsistencies.

AND REDUNCANCY

IN PROJECT NETWORK

Define Inconsistency

Inconsistency is define as that a project network which contain a loop or cycles

METHODS OF CONSISTENCY CHECKING

 Topological ordering of activities

 Fulkerson’s numbering rule

 Squaring adjacency matrix

 Marimont’s procedure

Step 3 Identify any New sources (jobs without predecessors) so generated for

placement on the list and go to step 2.

Step 4 Continue until one of the following happens:

(a) There are no sources on the list of remaining jobs This shows the presence of an inconsistency.

(b) All jobs are placed on the list, which is now topologically ordered, showing there are no inconsistencies

Topologically ordered list

Inconsistent project network

FULKERSON’S NUMBERING RULE

Each are (i,j) of the project network is numbered

Such that i<j, by using the following procedure:

Step 1 Identify the source node(s)in the project network and number

them sequentially from 1 onwards.

Step 2 For each numbered node delete the outgoing arcs and identify

new sources.

Step 3 Number the newly discovered sources sequentially.

Step 4 Continue till one of the following happens:

(a) All the nodes are numbered indicating a consistent network.

(b) The absence of sources in the unnumbered nodes indicating an

inconsistency.

An application of Fulkerson’s numbering rule

MATRIX REPRESENTATION OF PROJECT

NETWORKS

ADJACENT MATRIX

MATRIX REPRESENTATION OF PROJECT

Properties of adjacency matrix

It is a square matrix (nxn) of 0s and 1s

There is no entry on the diagonals

The matrix is upper triangular if nodes are numbered according to the Fulkerson's rule

Each entry of 1 indicates an arc in network

Properties of adjacency matrix

Row sum indicates the no of arc outgoing from the node

Column sum indicates the no of arc incoming from the node

A vacant column indicates a source node

A vacant row indicates a sink node

The rank of the adjacency matrix is (n-1) as defined by an tree

of the graph

NODE ARC INCIDENCE MATRIX

MATRIX OF SIZE (nxm) WITH ENRIES -1, 0& 1

1

2

43

a1

a4

a3

a4

a3L1 L2L3

a1 a2 a3 a4 a5

Matrix Squaring Procedure

 Multiply the adjacency matrix with itself at most 1) times

(n- If the matrix is consistent, it must be nilpotent of

index n or less, since in the absence of loops the

maximum spacing between any two nodes is (n-1)

 A matrix M is nilpotent of index k if Mk=0, butMk-1 is not = 0

Matrix Squaring Procedure

M x M = M2 x M = M3

1

2

34

Matrix Squaring (M2)

1

2

43

M x M = M2

Matrix Squaring (M3)

1

2

43

M2 x M = M3

Matrix Squaring (M4)

1

2

43

M3 x M = M4

Marimont’s Procedure (Network)

a7

a8

a9

Marimont’s Procedure (Network)

a7

a8

a9

Redundancy Check (Example)

Redundancy Check (Tabular Method)

Redundancy Check (Tabular Method)

Redundancy Check (Network)

Thank You