Computation Section

- Push/Network Process

  In this section we consider the push/network process. Many of the definitions and computations associated with the various drive/structure alternatives are the same as for the push 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.
Push/Network Process

Figure 1

The push/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 push/tree process. Product is inserted or pushed into any of the operations. We use as the amount pushed into operation i. We assign the value of 1 to . The push flows to 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.

For the push network we identify the proportion, , as the amount of the output of operation i that is passed to operation j for each unit of product passing through operation i. The value of may be any nonnegative amount. Typically for a service system, the sum of the proportions leaving an operation is equal to 1. This means that the flow is split among the several following operations. It may be necessary to use other combinations of proportions to represent different systems.

The example shows an arc passing from operation 5 back to operation 3. In a practical instance, this might represent the reworking of some part. as the proportion of the output of operation 5 returned to operation 3. It is not necessary to define a proportion for the flow leaving the system at operation 5.


Tabular and Matrix Representation


Figure 2

We use Fig. 2 as a numerical example. Here we push 1 unit into operation 1 and nothing into the other operations.

Although we can represent much of the data for a push network with a two-dimensional table as illustrated for the example below, it is necessary to represent the proportions on a square matrix. We call this the proportion matrix P. Notice that we have left out both the Previous and Proportion columns from the table since the following preceding operation is not unique for the network. The matrix P describes both preceding operations and proportions.

For the example, we assume zero scrap rates and grouping factors equal to 1.


Push In
Op. 1
Op. 2
Op. 3
Op. 4
Op. 5

For the push network structure we define the following notation. We use i for the general operation index.

  • = the flow pushed into 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 output of operation i that is sent to operation j.
  • = 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 P. In general

For the case of the example:

The Excel model created by the Process Flow add-in is shown below. The add-in adds dummy operations 0 and 6. Indices are automatically assigned by the add-in, as indicated by the green field. The Previous column is not required. The Push In column shows 1 unit pushed into operation 1. 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 a network.


The structure and proportions are described by the P matrix (Transfer Out 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 P 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 push/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 input flow to operation i depends on the push flow at operation i and the amounts provided by the preceding 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:

The T superscript indicates the matrix transpose operation.

For the example, the matrices are:

Note that the proportions leaving operation i are described by row i of the P 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 K44.


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 B46 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 G45. The computed values are placed in column I of the worksheet.


Work-in-Process (WIP)


This computation is the same for all drive/structure alternatives.

The sum of operation WIP values is computed and stored in cell K45.




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tree roots

Operations Management / Industrial Engineering
by Paul A. Jensen
Copyright 2004 - All rights reserved

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