Production flow analysis

Given a binary product-machines n-by-m matrix ${\displaystyle b_{ip}}$, Rank Order Clustering[1] is an algorithm characterized by the following steps:

1. For each row i compute the number ${\displaystyle \sum _{p=1}^{m}b_{ip}*2^{m-p}}$
2. Order rows according to descending numbers previously computed
3. For each column p compute the number ${\displaystyle \sum _{i=1}^{n}b_{ip}*2^{n-i}}$
4. Order columns according to descending numbers previously computed
5. If on steps 2 and 4 no reordering happened go to step 6, otherwise go to step 1
6. Stop

Given a binary product-machines n-by-m matrix, the algorithm proceeds[2] by the following steps:

1. Compute the similarity coefficient ${\displaystyle s_{ij}=n_{ij}/(n_{ij}+u)}$ for all with ${\displaystyle n_{ij}}$ being the number of products that need to be processed on both machine i and machine j, u comprises the number of components which visit machine j but not k and vice versa.
2. Group together in cell k the tuple (i*,j*) with higher similarity coefficient, with k being the algorithm iteration index
3. Remove row i* and column j* from the original binary matrix and substitute for the row and column of the cell k, ${\displaystyle s_{rk}=max(s_{ri*},s_{rj*})}$
4. Go to step 2, iteration index k raised by one

Unless this procedure is stopped the algorithm eventually will put all machines in one single group.