Barkley and Motard Algorithm and Basic Tearing Algorithm

1. Chapter 4: Partitioning and Tearing
Topic: Barkley and Motard Algorithm , Basic Tearing Algorithm
Prof. Bansi Kansagra
Adhoc Assistant Professor
Sarvajanik College of Engineering and Technology

2.  Barkley and Motard (B&M) (1972) suggest an alternate representation
of a digraph by interchanging nodes and edges known as a ‘signal flow
graph (SFG)’.
The following steps are involved in decomposition of a network using the
B&M algorithm:
 Step 1 Convert the digraph into an equivalent SFG.
 Step 2 Eliminate any node with a single precursor, since it belongs to that
precursor. Replace such a node whenever it appears as a precursor by its
header node.
 Step 3 If any self-loops appear, cut those nodes and assign them to the cut-
set. Eliminate such nodes, which may render the graph reducible again,
i.e., nodes may now appear with a single precursor.
2Bansi_Kansagra

3.  Step 4 Two-way edges can also
prevent complete reduction of
the SFG. A two-way edge is
shown in Figs 12.3 and 12.4.
 In the figure, the situation
presented is for the jth node.
Before the SFG is totally
reduced, two-way edges as
shown below may appear; in
such a situation, assign either the
ith or the yth node to the cut-set.
Fig. 12.3 Simple two-way edge
Fig. 12.4 Compound two-way edges
3Bansi_Kansagra

4.  Step 5 If all other conditions fail to appear, select the node with the
maximum number of output edges for cutting.
 Step 6 When SFG is totally reduced, the cut-set is found.
4Bansi_Kansagra

5. Using the Barkley and Motard (1972) method (B&M algorithm) for the
decomposition of a maximal cyclic net (MCN), find out the streams that
are to be teared (i.e., cut-set) for the digraph (Fig. 12.1) of a process:
(decomposition is by a minimum number of tear streams, all of them
assigned unit weight).
5Bansi_Kansagra

6.  First reduction Eliminate any
node with a single precursor,
since it belongs to that precursor.
 Then, replace such a node
whenever it appears as a
precursor by its header node.
 Note Replace the nodes which
are eliminated with their
precursors.
 Replace node 5 with its precursor
7, node 1 with its precursor 2,
and node 8 with its precursor 7
6Bansi_Kansagra

7.  Remaining nodes are 2, 4, 6, and
7
 The remaining nodes are shown
above in which two ‘self- loops’
are formed at nodes 2 and 7.
 The SFG is totally reduced and
the cut-set found consists of
nodes 2 and 7 (where self-loops
are formed) or equivalent
streams 2 and 7
7Bansi_Kansagra

8.  This is also based on the concept of SFG.
 Pho and Lapidus (1973a, 1973b) presented an alternative algorithm which
offers the advantage of assigning any arbitrary to the edges or streams of a
digraph.
 It locates the cut-set of tear streams by minimizing the sum of their
weights.
 This is equivalent to finding minimum weights of tear streams when all of
them are assigned unit weights.
 The features of the basic tearing algorithm (BTA) are as given below:
1. The reduced digraph (one that excludes feed and product streams) is
first converted into an equivalent SFG.
2. Information on the SFG is provided to the algorithm in the form of an
adjacent matrix.
8Bansi_Kansagra

9. 3. Ineligible nodes are found according to the following criterion: If the
weight of node i is greater than or equal to the sum of weights of non-
zero elements in the first rows of the matrix, then that node is
ineligible.
4. The reduction of ineligible node I is performed as follows: Shift the
elements of column I to those columns in row I containing non-zero
elements forming a Boolean sum. Make row I and column I null.
5. Primarily the cut node is found by locating a self-loop.
6. If no ineligible nodes are found, then two-way edges are searched.
7. When all attempts fail, the branch and bound method is ultimately
applied to totally reduce the SFG.
9Bansi_Kansagra

10. 10

11. Like Share and Subscribe
Transcripts available Google classroom and slideshare