Process Flow Models - Defects
A principal cause of waste in a production process is the introduction of defects. The possibility of defects leads to the necessity of inspection. The results of inspection may be the discard of the product as scrap or routing to a rework facility. The discovery and repair or discard of defective items results in additional variability in the production process. There is also the possibility that a product with an undiscovered defect is shipped to the customer. When the customer ultimately discovers the defect there is the cost of warrantee repair and the loss of customer good will. All these results are waste. This section describes the mathematical models used by the add-in to model the creation, combination and discard of defective items.

 The defect feature of the model is added by checking the Defect checkbox. The figure shows that the Proportion box is also checked as the defect model requires the proportion feature. Figure 1 A number of kinds of defects might be introduced by a production process. A fatal defect when present in a product will cause the product to fail in some way to satisfy its specifications. In this section, we restrict attention to fatal defects. For analysis purposes, we define the probability that operation i introduces a defect into a unit of product as . To compute the defect probability associated with two or more operations, we assume that the events producing defects are statistically independent. Fatal defects are discovered only through inspection. If discovered, a defective item will be discarded as scrap. If not discovered, a product with a defect will pass to other operations to receive subsequent processing. If the finished product leaves the process with a defect, it is ultimately discovered by the final inspector, the customer. Although an item may be made defective by more than one operation, one fatal defect is sufficient to cause the failure of the product. No defects are discovered at non-inspection operations, however, there may be other sources of scrap at these operations. For example, a non-inspection operation may have a scrap factor that represents material removed and discarded. We also may identify a second kind of fatal defect that can be immediately observed by the operator and discarded. For instance the operation may break the item causing immediate discard. Both these mechanisms will be incorporated into the scrap proportion for the non-inspection operation. We assume that discards of this fashion do not affect the defect probabilities or the way in which the probabilities combine to determine the removal rate at inspection operations. To illustrate, consider the process with four serial operations followed by an inspection station as in Fig. 1. Assume that each operation introduces fatal defects in 10% of the items passing through it. The inspection station is perfect in that it finds and removes all defective items. The inspection operation is shown as a rectangle in the figure. The process worksheet with this data is shown below along with the computed results. Note that an inspection station is identified by the word "Inspect" in the Type column. Only the first two letters are used, so alternatives such as "In" or "Inspect 1" also represent inspections. The Defects Out column shows the proportion of the items that are defective at each operation, and the Flow Removed column shows the proportion that are removed from the flow. Note that the defects accumulate in the flow as it passes through the process. When items finally reach the inspection station, the defects are discovered and approximately 35% of the flow is discarded. Since this example assumes perfect inspection, the defect rate at the output of the inspection is zero. The effect of the discard of so many defects is to increase the unit flow to over 1.5 for all the operations. For each unit that passes out of the process, 1.5 units must be processed. The throughput time and WIP is increased proportionally.
The following describes the columns that are important for the discussion of defects.

 Column Title Explanation G Defect Rate This is the percentage of products passing through a processing operation that receive defects. We assume a single defect warrants discard of an item, however, the defects are only discovered by an inspection. Defects do not change the flow except for an inspection operation. For an inspection operation, the defect rate is the proportion of defects not discovered and discarded. H Proportion For a pull system, this is the proportion of the flow entering the next node that comes from the operation specified by this data line. For a push system, this parameter is the proportion of the flow leaving the operation that goes to the next operation. The proportion column is necessary when defects are present. M Defects Out For a processing operation ("Op"), this is the proportion of the items leaving an operation that have defects. We assume that the defect producing mechanisms in different operations are independent. The function in this cell combines the probabilities of defective parts entering the operation with the defect probability for the operation to obtain the probability that an item leaving the operation is defective. For an inspection operation ("In"), the probability of a defective part leaving is zero if the inspection is perfect, that is inspections find and remove all defective items. If the inspection is not perfect, the function computes the probability that a defective item leaves the inspection. N Flow Removed For a processing operation, this cell then contains the adjusted scrap rate for the operation. When the operation is an inspection, defective items are removed as waste. The proportion discarded is shown in this column. O Ratio This is the ratio between output flow and input flow for an operation.

Pull serial process

Fig. 1 is an example of a pull serial process. The flow is not affected by defects until it reaches an inspection. There, some or all the defective items are removed. To compute the amount removed, we must first compute the probability that an item entering an inspection is defective. When inspection is perfect, this probability becomes the removal rate at the inspection operation.

Consider the example in Fig. 1. Assuming the material entering operation 1 has no defects, the probability that a defect is produced is . Since they are not removed, defective units are still processed in operation 2 along with the non-defective ones. Assuming independence, we can compute the following probabilities about the output of operation 2.

These probabilities are computed in the Defects Out column, where we see 0.1 as the probability of a defect at the output of operation 1 and 0.19 at the output of operation 2. There is a 0.01 probability that an item has two defects after passing through the first two operations.

The remaining entries in the Defects Out column are computed similarly using a recursive relation

The Flow Removed column (J) shows the proportion of the flow that is discarded as scrap. For non-inspection operations, no flow is removed. For a perfect inspection, the scrap rate of the inspection operation is set equal to the defect probability at its input and the proportion of defects out becomes 0. When the inspection is not perfect, we use as the probability that the inspection does not detect a defect.

Push serial process

 Figure 2 The effect of defects in a push serial system are the same as for the pull serial system as illustrated in Fig. 2 and in the worksheet for this process. The defect rates are the same as for the pull example, but the flows are different because of the different drive mechanism for the push process. For every unit entering node 1 approximately 0.66 units leave node 5.

Pull non-serial process

 Figure 3 We observe the effects of defects in a non-serial process in Fig. 3. When two or more branches come together, as the outputs of 1 and 2, the probabilities of defects combine. Thus we have at the output of operation 3, the probability of defect: 1 - (1 - 0.1)(1 - 0.1)(1 - 0.1) = 0.271 The results for the example in Fig. 3 are almost the same as the results for the example of Fig. 1. The only difference is for operation 2 that has a defect probability of 0.1 rather than 0.19. This does not affect on the unit flows, because the same number of defects are produced and are not eliminated until the inspection station is reached at the end of the process.

 Figure 4 The process of Fig. 4 is the same Fig. 3 except we add inspection stations before the assembly operation at 5. The operations have been renumbered to meet the restrictions of a tree structure. The Excel worksheet for the example shows that the extra inspection stations reduce the unit flows in all the operations. The throughput time and WIP are increased because of the extra inspection time. For a process with defects, the problem of finding the optimum location and number of inspection stations is relevant. With additional cost information, this add-in can guide the search for an optimum.

Push/tree structures

 Figure 5 The process of Fig. 5 is a push tree. Defects are produced in operations 1, 2 and 4, and removed at the inspections at 3 and 5. The analysis of a push tree is easier than for a pull tree, since each operation in a push tree has at most one predecessor. To illustrate, consider operation 1 in the figure. Assuming the material entering operation 1 has no defects, the probability that a defect is produced is . Although defective units may be produced in operation 1, they are still processed in operations 2 and 4. Assuming independence, we can compute the following probabilities about the output of operation 2. These probabilities are computed in the Defects Out column. We see 0.1 as the probability of a defect at the output of operation 1 and 0.19 at the output of operation 2. There is a 0.01 probability that an item has two defects after passing through the first two operations. The same computations can be made for operation 4 which also receives its input from operation 1. Since all have the same defect probabilities, we find the probability of defect at the output of 4 to be 0.19. The remaining entries in the Defects Out column are computed similarly using the recursive relation The Flow Removed column (J) shows the proportion of the flow that is discarded due to defects. For non-inspection operations, no flow is removed. For a perfect inspection, the removal rate at the inspection operation is set equal to the defect probability at its input and the proportion of defects out becomes 0. When the inspection is not perfect we use as the probability that the inspection does not detect a defect.

Network Structures

 Figure 6 The formulas for defect probabilities are applied recursively, starting at the operations with the lowest indices and passing to operations with higher indices. The indexing for the operations of a tree assures that this recursive approach is successful. For the more general network structures, the recursive process is not valid. The add-in does not disable the defect option for network structures, but it should be used with caution. An example is in Fig. 6. This network structure is the same as Fig. 3, but part of the output of inspection station 5 is routed back to operation 3 for additional processing. We assume that 80% of the input to operation 4 comes from operation 3, while 20% comes from operation 5. Not considering the branch from 5 to the input of 4, the structure is a tree. We model the defects using only that tree. In fact, this is valid for this example, because the items routed from operation 5 do not contain defects. The Excel model is below. The Next and Proportion columns describe the tree structure and the Transfer In matrix defines the network structure. It is important that the two models are consistent.

Proportions

 Figure 7 Most of the examples to this point used branch proportions of 1. Here we consider situations in which proportions may be other than 1. For pull/tree structures it is necessary to adjust the defect formulas. Consider the situation of Fig. 7. The numbers on the branches entering operation 3 imply that 40% of the items passing through operation 3 receive items from operation 1 and 60% receive inputs from operation 2. To compute the defect probability for an operation receiving several inputs we use the following general expression. For the example, we have at the output of operation 3, the probability of defect: 1 - (1 - 0.1)[1 - (0.4)(0.1)][1 - (0.6)(0.1)] = 0.1878 This result is rounded to 18.8% on the worksheet below. For the push tree the defect probabilities are not affected by branch proportions.

Grouping Factors

 Figure 8 In the example of Fig. 8, four items from operation 1 are grouped to form a single item passing out of operation 2. For example, four sheets of plywood might be glued at operation 2 and four individual sheets becomes one assembled item for subsequent processing. At operation 4, the process divides the item in some manner, so that one item becomes four in subsequent processing. Continuing with the plywood example, perhaps operation 4 cuts the assembled plywood sheets into four individual smaller parts. Grouping factors are useful in a variety of contexts and defect probabilities must be adjusted to account for them. The expression assumes that the items are grouped prior to the operation. Any defects produced at the operation are not raised to the power. In the example, we introduce 10% defects in operation 1 with no further defects in the subsequent operations. At operation 2 we group four of the items into one. The defect probabilities for the operations are computed recursively, starting from operation 1.

User Defined Functions

The defect proportions are computed using User Defined functions provided by the add-in. There are two different functions used in pull and push trees respectively. The function used for pull trees is:

= pull_defect(op_type, index, next_range, prop_range, defect_range, d, g)

The arguments labeled range refer to the named ranges on the Excel process definition. Algebraically the function computed is:

For push trees the appropriate function is:

= push_defect(op_type, prev, index_range, prop_range, defect_range, d, g)

The arguments labeled range refer to the named ranges on the Excel process definition. Algebraically the function computed is:

The functions are placed in the Defects Out column. The arguments on the worksheet are references to the cells holding the appropriate parameters.

The process definition also uses functions to compute the amount removed at the operation. These functions combine the effects of scrap removed at an operation and items removed due to defects. Different functions are required for pull and push trees.

= pull_remove(op_type, Index, next_range, prop_range, defect_range, Scrap, d, g)

= push_remove(op_type, prev, index_range, prop_range, defect_range, Scrap, d, g)

The functions are placed in the Flow Removed column of the process definition. The contents of the Flow Removed column is used to compute the Ratio column.

Operations Management / Industrial Engineering
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by Paul A. Jensen