
In this section we consider
the pull/network process. Many of the definitions and computations
associated with the various drive/structure alternatives are
the same as for the pull tree. The principal changes are in the
data defining the network structure and the proportions. The
computation of the unit flows is quite different for the network
structure as compared to the tree structure. 

Pull/Network
Process
Figure 1 
The generic pull/network process is illustrated
in Fig. 1. For this structure the flow through each operation
may go to more than one operation, and each operation may
have several input flows from other operations. This is
a more general structure than the pull/tree structure.
Product is withdrawn or pulled from any of the operations.
Again we use as
the amount pulled from operation i. With m operations
we assign the value of 1 to .
The pull flows from the other operations are given as relative
to .
Indices are assigned to the operations arbitrarily, however,
it is often convenient to assign the indices to be increasing
in the direction of primary product flow.


Tabular
and Matrix Representation 

Figure 2 
We use Fig. 2 as a numerical example.
Here we pull 1 unit from operation 5 and nothing from
the other operations.
Although we can represent much of the data for a pull
network with a twodimensional table as illustrated for
the example below, it is necessary to represent the proportions
on a square matrix. We call this the proportion matrix Q.
Notice that we have left out both the Next and Proportion columns
from the table since the following operation is not unique
for the network. The matrix Q describes
both following operations and proportions.
For the example, we assume zero scrap rates and grouping
factors equal to 1. 
Name 
Index 
Pull
Out 
Scrap 
Group 
Op.
1 
1 
0 
0 
1 
Op.
2 
2 
0 
0 
1 
Op.
3 
3 
0 
0 
1 
Op.
4 
4 
0 
0 
1 
Op.
5 
5 
1 
0 
1 
For the pull network structure we define the following notation.
We use i for the general operation index.

= the flow pulled from the output of operation i.

= the proportion of flow that is scrapped or removed at operation
i.

= the number of items grouped at operation i.

= the proportion of the input of operation j that
is obtained from operation i.

= the time required for one unit to pass through operation
i. (not shown in the table)
For the network, flow may pass from an operation to any other
operation, so a matrix is required to describe the proportion
information. We call the matrix Q. In general
For the case of the example:
The Excel model created by the Process Flow addin
is shown below. The addin adds dummy operations 0 and 6. Indices
are automatically assigned by the addin, as indicated by the
green field. The Next column is not required. The Pull
Out column shows 1 unit pulled from operation 6. We have
indicated arbitrary times in the Operation Time column.
Since the Scrap Rate and Group Factor are
0 and 1 respectively, we have not included their columns. No
Proportion column is necessary for the network structure. 


The structure
and proportions are described by the Q matrix
(Transfer In matrix). This matrix is on the left of the figure
below and includes the dummy operations 0 and 6. The matrix
on the right is called the Augmented Matrix. Proportion
data is entered in the Q matrix, and the Augmented
Matrix is determined by Excel formulas. The two matrices are
constructed on the same rows of the Excel worksheet as the tabular
data for the process. 

Scrap and Flow Removed 

These features take the
default values, 0 and 1, respectively. They do not affect the
analysis. Nontrivial values would be handled in the same way
as the pull/tree structure. 
Grouping, Flow Removed
and Flow Ratio 

The general expression
for the flow ratio is:
Using the example parameters all ratios are 1,
as shown in column F. The column designations provided in this
discussion (F) refer to the example worksheet above. The column
designations for a different instance will depend on the location
of the process on the worksheet. 
Unit Flow 

Figure 3 
To illustrate the computation of the unit flows we use
an example with three operations as in Fig. 3. The value
of ,
the output flow from operation i is dependent on
the pull flow withdrawn at operation i and the amounts
required by the following operations, j and k. 
We write the equations entirely in terms of the
unit flows by using the flow ratios.
This generalizes to the expression that must
hold for each operation.
We define the augmented proportion matrix as
Also define the column vector u
of unit flows. Then the unit flows are the solution to the linear
set of equations:
For the example, the matrices are:
Note that the proportions entering operation i
are described by column i of the Q
matrix. Solving for the unit flows we find:
The unit flow vector is computed using Excel matrix
operations and is shown in column G of the example worksheet. 
Unit Time 

The time required for operation i per
unit of finished product is called the unit time and designated
.
This computation is the same for all drive/structure
alternatives and is stored for the example in Column H. The
sum of the unit times is the Throughput time. It
is computed and stored in cell K30.

Operation Flow 

Again, this computation is the same for all
drive/structure alternatives. The illustration assumes there
the operating inverval is hours and the demand interval is
weeks with 40 hours used per week.
The value of V is from cell B32 in
the example. The denominator of the expression depends on
the time units selected for the demand and operation intervals.
It is entered in cell G31. The computed values are placed
in column I of the worksheet.

WorkinProcess (WIP) 

This computation is
the same for all drive/structure alternatives.
The sum of operation WIP values is computed and
stored in cell K31. 

