Computation Section

- Pull/Tree Process

  We recognize two structure alternatives, tree and network, and two drive alternatives, pull and push. To find the flows in the operations of a process, the structure of the process and the driver of the flows must be specified. We first consider the pull/tree process.
Pull/Tree Process

Figure 1

The generic pull/tree process is illustrated in Figure 1. For this structure the flow through each operation goes to a unique following operation, while each operation may have several input flows from other operations. Product is withdrawn or pulled from the operation with the greatest index, operation 5 for the example, in the amount . We usually assume that this amount is 1.

In addition to the final operation of the process, our models also allow flow to be pulled from the other operations. These flows represent intermediate products. In general, we identify the amount pulled from the output of operation i as , the pull 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. The greatest index is m. For the example m is 5.

For the pull tree we identify the proportion, , as the amount of the output of operation i required for each unit of product passing through 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 pull 1 from operation 5 and nothing from 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 1 through 4. This means that operation 3 receives half of its input from each of operations 1 and 2. Further, operation 5 receives half of its input from each of operations 3 and 4. Units are inserted into the operations with no predecessor operations in the amounts required to fullfill the pulled demands.


We represent the data for the example pull 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.

Pull Out
Op. 1
Op. 2
Op. 3
Op. 4
Op. 5

For the pull tree, an operation can send its output to no more than one other operation, so the column labeled Next is sufficient to describe the tree structure. The column labeled Proportion gives the proportion of the input of the next-operation that is obtained from the operation. The number (0.5) in the row for Op. 1, holds the value of .

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

  • = the index of the operation following (or after) operation i. This is the number in the Next column.
  • = 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 that is obtained from operation i.
  • = the time required for one unit to pass through operation i. (not shown in the table)

We use the symbol a as the second subscript on to indicate that it is the proportion of the input of the following operation that must come from operation i. When an operation has no following 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 Next column represents the tree structure of Fig. 1. The Pull Out column shows 1 unit pulled from operation 6. 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 pull/tree process.


Scrap and Flow Removed


Scrap is material entering an operation that is removed from the process. We call the associated proportion, known as the scrap rate, . There are many reasons for scrap. Perhaps the operation involves cutting a pattern from a sheet of material. The material not used and thus discarded is the scrap. If the operation involves inspection, the inspection procedure may discover and discard defective items. Items discarded without further processing are scrap. This example has scrap, but no defects, so the flow removed is entirely due to scrap, and the Flow Removed Column is the same as the scrap column.


Grouping, Flow Removed and Ratio


Consider flow passing between a pair of adjacent operations i and j. Grouping takes place when several items entering operation i are grouped for subsequent processing in operation j. This most frequently occurs in an operation just preceding an assembly. For example, the four legs of a table may be grouped into a single unit before assembly into a complete table. We use the letter to represent the grouping proportion, with 4 the appropriate value for the table example. A grouping factor less than one is appropriate when an item entering an operation is divided for subsequent operations. For example, if a board is cut into six pieces, the item entering the operation is a board and the item leaving is a piece. For each entering item, six will leave and the grouping proportion is 1/6.

We also use the grouping proportion to represent a change in dimension. For example, consider operation i at which some solid material measured in pounds enters the operation and through some treatment is converted to a liquid material measured in gallons. Here would measure the transformation factor pounds per gallon.

We define the flow ratio for an operation as the ratio between the flow leaving an operation and the flow entering. The unit flow entering the operation is and define the flow leaving as . Then the flow ratio is:

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

To illustrate the computation of the unit flows we use an example with three operations as in Fig. 3. The value of is entirely dependent on the pull flow withdrawn at operation i and the amount required by the following operation k. Since the amount required is we have the relation between the unit flows at i and k.

Notice that the unit flow for an operation depends on the unit flow of its unique following operation. For the pull/tree process, the unit flows can be computed recursively, starting with the and continuing for each operation with sequentially decreasing 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 1 is 0.3429. This means that for 1 unit of product produced at operation 5, 0.3429 units must pass through operation 1. If the scrap rates were zero, all ratios would have been 1, and the unit flow for operation 1 would have been entirely determined by the proportions. The unit flow at operation 1 would have been 0.25 in this case.


Unit Time

The time required for operation i per unit of finished product is called the unit time and designated . The unit flow for the operation tells the quantity of flow through the operation for each unit of finished product, so the unit time is simply the product of the operation time and the unit flow.

For the example these quantities are in column M. The sum of the operation unit time values is the total time in the system for each unit of finished good. 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 K2.


Operation Flow


The production volume for the process is designated V is placed in cell B4 at the top of the table. This flow is measured using the Flow Time Interval. In this case the production volume is 100 units per week. The unit flow for the operation tells the quantity of flow through the operation for each unit of finished product. It is measured using the Operation Time Interval, so the total flow through the operation is the product of the flow ratio and the production volume divided by the operation time interval (40 in this case).

For the example these quantities are in column N.


Work-in-Process (WIP)


The work-in-process at an operation is the product of the flow rate through the process and the residence time in the process (Little's Law). For the example, we multiply the quantities in columns M and N to find the WIP for each operation, 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 K3.



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tree roots

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

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