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NOTES: REJECT AND REWORK MODELS IN INDUSTRY
Max Newbold: May 2004
INTRODUCTION
The writer has, from observing industry, defined five different reject and rework models. The importance
of these models is how they affect manufacturing lead-time and cost. Any factor that causes random
changes to lead-time in manufacturing will affect the level of uncertainty. This in turn will affect
competitive factors of on time delivery and delivery reliability.
MODEL TYPES
Of the five models listed below the most disruptive to a manufacturing system can be model number five.
This is due to the systems sensitivity to changes in yield.
It is this model that will be discussed more
fully. Diagrams of all five models are shown in the appendix.
1.
Rejected items are scrapped (Model No.1)
2.
The reject item can be allocated to a different part number (Model No.2)
3.
Reject items are reworked as an integral part of the production line (Model No.3)
4.
Rejected items are repaired off line and returned to the line after the point of rejection
(Model No.4)
5.
Rejected items are repaired off line but returned to the process where the item was rejected
or before the point of rejection (Model No 5)
SUMMARY OF IMPACT
The method of handling defective product within the manufacturing system is dependent on the ability of
the product/part to be repaired. Where the part is scrapped or redefined as a different product then the
following needs to be considered.
Rejects from a continuous process are unlikely to greatly disrupt the flow or require large buffers so long
as the yield remains above 90% between processes and the system can support the overall effect of the
rolled yield.
In batch production the effect of rejects can result in over or under supplies as the yield varies, creating
additional uncertainty in the system. If the batch size changes to reflect any yield change, then the time in
the system will also alter, again creating uncertainty.
Where product is reclassified the level of disruption is dependent on whether the batches are treated as
independent identities and move through the system independently. If the batches are not allowed to
reform either by storing until the optimum batch size is achieved or dynamically through a “look ahead”
policy, then lead-time will become unpredictable.
In the case where rework is allowed, then working off line, so long as labour is available to meet changes
in the quality level, little disruption to the flow will occur. A similar result is obtained where all products
are moved to an inspection rework area. The system, where due to limited resources, the product must reenter the line at the point it was rejected is the most sensitive to changes in the first pass yield. Under
these conditions the capacity of the system will quickly deteriorate as the yield reduces. If batching is
required, then the management must consider the approach taken on the smaller rework batches.
MODEL NUMBER FIVE
Continuous Systems
This model is common in the mining industry, where ore is crushed and screened. Over sized particles
are sent back around through the crusher as shown in model five. The equilibrium position is defined
when the input to the system equals the output of the system. In the mineral processing industry the
particle size is considered to be a continuous function, with a normal distribution.
A closer algorithm can be found in a communication’s text (Gunther 1998 The Practical Performance
Analyst: Performance–by–Design Techniques for Distribution Systems McGraw – Hill) where the flow
of messages is considered to be discrete objects entering the system. The feed back models for calls that
can not be serviced in the first attempt are cycled back into the system. This is defined in a text by
Gunther (1998) as 1 =  + pn1 . This approach assumes that the value of “p n” is independent of the new
arrivals, which cannot be assumed to be true for manufacturing. In manufacturing an item is rejected
because of a specific defect and once this has been corrected its chance of failure differs from that of a
new arrival. Thus the probability of failure of an item is dependent on the number of cycles that the item
has been through the system. Modifying Guther’s formula to make the failure rate dependent on the
external arrivals and the times the item has cycled the formula is:
1 =  + pn
This formula would hold true if and only if on the second pass through the system the item passes or is
rejected. From the flow shown in appendix 2, this would make X equal to one. Each pass will have its
own probability of failure; thus the formula can be expressed as follows.
1 = + pn1+ pn1pn2 + pn1pn2pn3 +… + pn1…pnn
1 =(1 + pn1+ pn1pn2 + pn1pn2pn3 +… + pn1…pnn)
When the system is operating at capacity, the above formula can be simplified by dividing both sides by 
1.
1 =U(1 + p1+ p1p2 + p1p2p3 +… + p1…pn)
U = 1/(1 + p1+ p1p2 + p1p2p3 +… + p1…pn) (System Capacity)
When the number of recycles through the system is limited and a further failure results in the item being
scrapped, both passed and scrapped items leave the system allowing equation 20 to be truncated. If only
three repairs are allowed, as shown in figure number five the capacity of the system would be:
U = 1/(1 + p1+ p1p2 + p1p2p3)
While the capacity decreases the overall final yield of the system increases.
Let k1 be the first pass yield, k2 the second pass and kN the yield of the Nth pass.
The final yield (kF) of the system is found by:
k F = k1 + k2 p1+ k3p1p2 + k4p1p2p3 +… + k(N-1)p1…pn
If it assumed that the distribution of faults on the product follows a Poisson distribution and only one
defect is picked up on each examination. This type of situation is found in the testing of PCA boards at
incircuit testing where the test program will stop on finding the first fault and the number of faults per
board can be approximated by to a Poisson distribution. Base on the above assumption the probability
that a board will pass the first time it is presented to the tester is e -u , thus the boards returning for a
second passes is (1- e-z )
The proportion of boards failing on the second pass is the number of boards with 2 or more faults divided
by the number of boards with 1 or more faults.
Boards with one or more faults: (1- e-z )
Boards with 2 or more faults: (1- e-z - [z2 e-z /2!] )
Thus pn2 is
(1- e-z - [z2 e-z /2!] ) / (1- e-z )
The calculation is repeated for each board with 3, 4 to n defects, using the probability of j defects divided
by the probability of the sum of (j-1 to n) defects minus 1.
The behaviour of this system is that that an apparent high yield is achieved when the product is allowed to
recycle through the operation. As the defects increase from  to (z + z) the final yield will be only be
affected marginally affected even with large movements, but the capacity falls away even with small
changes in z and follows a similar path to that of the first pass yield. The affect of this behaviour is
shown in the graph below
Ist Pass Yield vs Final Yield and Capacity
120%
Final Yield
100%
%
80%
Effective Capacity
60%
40%
1st Pass Yield
20%
0%
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2
2.1
Defects/unit
The data used in the construction of the graph was:
S = 0.25 hrs
 = 100 units per hr
D = 40 units per hr.
Q = Initial batch size entering the system
 = Calculated on values of 0.4 and 0.2 defects per unit
Batch System
When the process is batch or lot manufactured the impact will be dependent on:
1.
The ability to allow the batch to be split between passed and recycled product
2.
Can the recycled material wait until the next batch is processed?
However since the batch sizes are altered the manufacturing lead-time will vary causing internal
uncertainty.
Appendix One : Differing Models
MODEL NO.
ONE
Rejects Scrapped
MODEL NO. TWO
Rejects Reclassified
Input Flow
Flow = 
Input Flow
Flow = 
Reject
Found
Reject
Found
Scrap
Flow = 1
Output Flow
Flow = - 1
MODEL NO.
THREE
Repair In-line
Input Flow
Flow = 
Output Flow A
Flow =  - 1
MODEL NO.
FOUR
Repair Off-line
Input Flow
Flow = 
Inspection
and Repair
Output Flow
Flow =
Output Flow B
Flow =1
Repair Flow
Flow = 1
Output Flow
Flow = 
Appendix Two: Rework model five
MODEL NO.5
Returned to the Point of
Rejection
Input

Flow
I
Process
Rework
Output Flow

FLOW DIAGRAM : REWORK TEST CYCLE
Pass
No
p n
YES
C>X
NO
Process
Rework
1
Internal

External
5
Scrap