     Pull/Tree Push/Tree Pull/Network Push/Network Download Drive/Structure - Push/Tree Process
In this section we consider the push/tree process. Many of the definitions and computations associated with the various drive/structure alternatives are the same as for the pull/tree. We limit our discussion to those aspects that are different. The principal changes are in the data defining the tree structure, the definition of the proportion parameter and the computation of the unit flow.

 Push/Tree Process Figure 1 The generic push/tree process is illustrated in Figure 1. For this structure the flow through each operation comes from a unique preceding operation, while each operation may have several output flows to other operations. Product is inserted or pushed into the operation with the smallest index, operation 1 for the example, in the amount . We usually assume that this amount is 1. In addition to the initial operation of the process, our models also allow flow to be pushed into the other operations. These products entering the system at intermediate points. In general, we identify the amount pushed into the input of operation i as , the push flow at operation i. For tree structures we require that the operations be indexed so that when flow passes from operation i to operation j, i < j. For the push tree we identify the proportion, , as the proportion of the output of operation i that is sent to operation j. The value of may be any positive amount to represent a variety of manufacturing situations.

Tabular Representation Figure 2 We use Fig. 2 as a numerical example. Here we push 1 into operation 1 and nothing into the other operations. For this illustration, we are assuming that 10% of the units passing through each operation are scrapped. We also assume proportions of 0.5 for operations 2 through 5. This means that operation 1 sends half of its output to each of operations 2 and 3. Further, operation 3 sends half of its output to each of operations 4 and 5. Units are removed from the operations with no successor operations in the amounts required to fulfill the pushed input flows. We represent the data for the example push tree with a two-dimensional table as illustrated below. Here we provide only the columns necessary to show the tree structure and compute flows. Later we add other operational data.

 Name Index Previous Push In Scrap Group Proportion Op. 1 1 0 1 0.1 1 1 Op. 2 2 1 0 0.1 1 0.5 Op. 3 3 1 0 0.1 1 0.5 Op. 4 4 3 0 0.1 1 0.5 Op. 5 5 3 0 0.1 1 0.5

For the push tree, an operation can obtain its input from no more than one other operation, so the column labeled Previous is sufficient to describe the tree structure. The column labeled Proportion gives the proportion of the output of the previous operation that is sent to the operation. The number (0.5) in the row for Op. 2, holds the value of .

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

• = the index of the operation preceding (or before) operation i. This is the number in the Previous column.
• = the flow pushed into the input 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 output of operation that is sent to operation i.
• = the time required for one unit to pass through operation i. (not shown in the table)

We use the symbol b as the first subscript on to indicate that it is the proportion of the output of the preceding operation that is sent to operation i. When an operation has no preceding operation we assign the value 0 to , and has no effect.

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 represents the tree structure of Fig. 1. The Push In column shows 1 unit pushed into operation 1. We have indicated arbitrary times in the Operation Time column. The Scrap Rate, Group Factor and Proportion columns are filled with the data specified for the example.  The columns starting with J are computed using formulas inserted by the add-in. Our purpose in this section is to provide the derivation of these formulas for the push/tree process.

Scrap and Flow Removed This is the same as the pull/tree structure.

Grouping, Flow Removed and Ratio The computations are the same as for the pull/tree structure.

The example shows the flow ratios in the columns labeled Ratio (K). Since all scrap rates are 0.1 and all grouping factors are 1, all ratios are 0.9.

Unit Flow Figure 3 For operation i in Fig. 3, the value of is entirely dependent on the push flow added at operation i and the amount passed from operation j. Since the amount from operation j is we have the relation between the unit flows at i and j. Notice that the unit flow for an operation depends on the unit flow of its unique preceding operation. For the push/tree process, the unit flows can be computed recursively, starting with the and continuing for each operation with sequentially increasing operation index. The results for the example are shown in the Unit Flow column (L) of the Excel display. Notice the unit flow through operation 5 is 0.2025. This means that for 1 unit of product introduced at operation 1, 0.2025 units must pass through operation 5. If the scrap rates were zero, all ratios would have been 1, and the unit at flow operation 5 would have been entirely determined by the proportions. The unit flow at operation 5 would have been 0.25 in this case.

Unit Time The computations are the same as for the pull/tree structure. The results are shown in column M. The sum of the operation unit time values is the total time in the system for each unit entering node 1. For a serial system, this is the cycle time for the product. In the more general context we call it the throughput time. This sum is computed and stored in cell K16.

Operation Flow The computations are the same as for the pull/tree structure. The results are shown in column N.

Work-in-Process (WIP) The computations are the same as for the pull/tree structure. The results are shown in column O.

The sum of the operation WIP values is the average WIP of the entire process. This sum is computed and stored in cell K17.  Operations Management / Industrial Engineering
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by Paul A. Jensen
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