Consistency& Redundancy in Project Networks

Dr. Jitesh J. Thakkar

INDIAN INSTITUTE OF TECHNOLOGY (IIT) KHARAGPUR

KHARAGPUR – 721 302

CONSISTENCY CHECK

• A project network should not contain any cycles

1 4

2

3

A

BC

D

E

INCONSISTENT NETWORK

1 C 4

2

3

A

B

D

E

The presence of the loop A-C-B shows an inconsistency or a logical error in the project network.

METHODS OF CONSISTENCY CHECKING

• Topological Ordering of activities

• Fulkerson’s Numbering Rule

• Squaring Adjacency Matrix

• Marimont’s Procedure

TOPOLOGICAL ORDERING

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.

Topological Ordering (Continued)

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..

EXAMPLE 1

Job Predecessor

a1 --

a2 --

a3 a1

a4 a1

a5 a2, a3

Topologically ordered list

a1

a2

a3

a4

a5

EXAMPLE 2

Job Predecessor

a1 --

a2 --

a3 a2, a5

a4 a1,a3

a5 a4

Inconsistent project network

a1

a2

a4

a5

a3

FULKERSON’S NUMBERING RULE

Each arc (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.

Fulkerson’s Rule (Continued)

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

1

MATRIX REPRESENTATION OF PROJECT NETWORKS

ADJACENCY MATRIX

1 2 3 4

1 0 1 1 0 2

2 0 0 1 1 2

3 0 0 0 1 1

4 0 0 0 0 0

0 1 2 2

2

3

4

PROPERTIES OF THE ADJACENCY MATRIX

• It is a square matrix (n x n) of 0s and 1s.

• There is no entry on the diagonal.

• The matrix is upper triangular if nodes are numbered according to Fulkerson’s rule.

• Each entry of 1 indicates an arc in network.

• Row (column) sum indicates the no. of arcs emanating from (converging into) the node.

Adjacency Matrix(Continued)

• 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 any tree on the graph.

2

NODE ARC INCIDENCE MATRIX

Matrix of size (n x m) with entries -1, 0 & 1

a1 a2 a3 a4 a5

1 -1 -1 0 0 0

2 1 0 -1 -1 0

3 0 1 1 0 -1

4 0 0 0 1 1

1 4

2

3

a1

a2a3

a4

a5

CIRCUIT MATRIX

Matrix of circuits vs arcs in the network

a1 a2 a3 a4 a5L1 1 -1 1 0 0 L2 0 0 -1 1 -1L3 1 -1 0 1 -1

1 4

2

3

a1

a2

a3

a4

a5

L1 L2

L3

MATRIX SQUARING PROCEDURE

• Multiply the adjacency matrix with itself at most (n-1) times.

• 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, but M(k-1) is not = 0).

MATRIX SQUARING (Inconsistent Network)

M x M = M2 x M = M3

0 1 0 1 0 1 0 1 0 1 10 0 1 0 1 0 0 1 10 0 1 0 0 0 1 0 0 0 0 1 0 0 1 0 0 1 0 00 0 0 1 0 0 0 1 0 1 0 0 0 0 0 1 0 0 1 00 1 0 0 0 1 0 0 0 0 1 0 0 1 0 0 0 0 0 1

1

2

4

3

MARIMONT’S PROCEDURE(Network)

• Identify source and sink

• Cut the arc and remove that portion from source and sink

• If nothing is left, there is no inconsistency, otherwise focus on small network left and remove inconsistency

Network

Source

Sink

Source

Sink

Iteration - 1

Iteration - 2

Iteration - 3

Redundancy Removal

REDUNDANCY CHECK(Example)

Job Immediate Predecessors

A --B AC AD B, CE B, DF C, DG B, D, E, F

REDUNDANCY CHECK(Tabular Method)

REDUNDANCY CHECK(Tabular Method)

SUMMARY - I

• Consistency in a network implies absence of loops. This is tested by various methods:– Topological ordering of jobs– Fulkerson’s numbering rule– Squaring of adjacency matrix– Marimont’s procedure

SUMMARY - II

• Redundancy in project networks is the presence of unnecessary additional precedence information that tends to clutter the network. Can be removed by– inspection, for small networks– the tabular method to generate the predecessor

-successor matrix.

SUMMARY III

• Consistency checks and redundancy removal are part of Preliminary Network Manipulations, which yield a valid, well pruned network for further analysis.

• These are generally a part of a good computer package.

Discussion 6