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Mutual effects of defective components in assemblies
Journal of Manufacturing Systems 36 (2015) 1–6
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Journal of Manufacturing Systems journal homepage: www.elsevier.com/locate/jmansys
Technical Paper
Mutual effects of defective components in assemblies Moshe Eben-Chaime ∗ Department of Industrial Engineering & Management, Ben-Gurion University of the Negev, P.O. Box 653, Beer Sheva 84105, Israel
a r t i c l e
i n f o
Article history: Received 13 August 2014 Received in revised form 16 February 2015 Accepted 22 February 2015 Keywords: Assembly Bill of Materials Defect rate Product structure Quality
a b s t r a c t The focus of this study is a known and disturbing actual problem. The industry will soon celebrate a century of quality awareness and efforts. Still, according to field data, many new products exit the manufacturing systems defective. This study proposes mutual effects among assembly’s components as an explanation to this phenomenon – many defective new products. While each item in a serial manufacturing process moves individually, items are joined to others in assemblies. There, a single defective component suffices to disqualify a whole assembled unit! Surprisingly, few studies have focused on the repercussions of defective items on production. Particularly, there appears to be no study that quantifies these mutual effects of components which arrive from different sources with different defect rates. Thus, this study is also a first attempt to analyze and quantify these mutual effects. Apparently, the mutual effects of their components amplify the defect rates of assemblies dramatically, to the extent that defects due to common or random causes become significant. © 2015 The Society of Manufacturing Engineers. Published by Elsevier Ltd. All rights reserved.
1. Introduction This study was motivated by the reality portrayed in Table 1, where the industry averages (IA) of the number of defects per 100 cars detected within 90 days of purchase through the years 2004–2013 (JD Power & Associates, Initial Quality Study [1]), are shown. This work aims to reveal and explain the causes for this phenomenon. May 16, 1924 is the date when Walter A. Shewhart first presented the statistical control chart [2]. It really is a wonder, then, that after 90 years of quality awareness and efforts, there are still so many defective products – each car, on average. It should be noted that both flaws found after 90 days or detected prior to delivery of the cars, are excluded in Table 1. Note also that these findings in the car industry do not imply that other industries are better – “All participating mask shops (in the 2013 photo-mask industry survey) reported shipping masks targeting the 22 nm node or smaller, and on average, yields were greater than 90% at all nodes down to 22 nm.” [3] Feigenbaum [4] minted the term “hidden plant” and defined it as “the proportion of plant capacity that exists to rework unsatisfactory parts, to replace product recalled from the field, or to retest and re-inspect rejected units” – i.e., to deal with the
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consequences incurred after an item is detected as defective. However, extra production capacity is required even before items are actually damaged and for these matters the type of the defect does not matter, whether dimensional, form, surface quality, hardness of contact, or other defects, all consume additional capacity. Feigenbaum estimated that hidden plants amount “to 15 percent to as much as 40 percent of productive capacity”. The present study also aims to develop means to quantify this figure and, more important, to evaluate and compare alternatives. A preliminary remark concerning the focus of this study is in order here. To the best of the author’s knowledge, the type of analysis presented in the sequel has never been done before. One possible explanation may be that the present study integrates different domains and thus represents a novel departure. While the mathematics may seem straightforward, our application is not trivial – as will soon be seen, a single attempt to make a similar application in a simple special case led to erroneous results. Another reason could be confusion with reliability theory – though the mathematics may seem similar, the focus is different. Reliability, e.g. [5] is concerned with products, while the present study considers manufacturing processes. Reliability analyses evaluate the probability that components and systems will remain functional while in use. This study, on the other hand, focuses on the yield of manufacturing processes – the proportion of conforming units that are produced. The practical implications of this distinction are discussed in the sequel, too. Regardless of analytical sophistication, this paper explains a known and disturbing problem, provides means to quantify it, and
http://dx.doi.org/10.1016/j.jmsy.2015.02.008 0278-6125/© 2015 The Society of Manufacturing Engineers. Published by Elsevier Ltd. All rights reserved.
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M. Eben-Chaime / Journal of Manufacturing Systems 36 (2015) 1–6
Table 1 Industry averages of the number of defects per 100 cars.
processes. If Q0 units enter the first activity in a serial process of n operations, the mean number of acceptable units at the end is:
Year
2004
2005
2006
2007
2008
2009
2010
2011
2012
2013
IA
119
118
109
125
118
108
109
107
102
113
Qn = Q0
underlines those attributes that distinguish it from other concerns. Furthermore, the analysis suggests ways of coping with the problem. The paper is organized as follows. Assumptions and notation are provided next. Section 3 presents a brief review of known results on serial processes. This lays the foundations for the analysis of assembly operations in Section 4. The results are discussed in Section 5 and conclusions are drawn in Section 6. 2. Assumptions and notation Let pi denote the average defect rate of operation/activity i. The defect rate depends on both the activity and the station chosen to perform it, but let’s assume the stations for each activity have already been selected. Since performed in different stations, it is reasonable to assume that the operations are (close to be) independent. Note, again, the difference between the defect rate and the failure rate of reliability theory – defect rate characterizes manufacturing/production operations, while failure rate (e.g., [6]) characterizes components, products or systems. Shewhart, in his pioneering book [7], distinguished between assignable and other causes for poor quality. It should also be noted, that here, defects due to common, or random causes, are considered, not quality deterioration due to assignable causes as in [8]. This makes independence between items a reasonable approximation, in addition to activity independence. Finally, in repetitive production, long term averages are proper performance measures to use, rather than first time quality as in [9]. In sum, four assumptions are made: The stations for each activity have already been selected; Activities/operations are independent; The processing of items in a station are independent; Long term averages are proper performance measures to use. Notation: ci – unit processing cost of activity i; Cn – total cost of n activities; mk – the assembly coefficient of the kth component type – the number of units of type k component in an assembly; pi – the average defect rate of operation/activity i; paA – the actual defect rate of an assembly; Qi – the number of units on which activity i is performed; Yi – the yield of – the number of conforming units manufactured by activity i. 3. Defect rates and input/output ratios If activity i is performed on Qi units, the mean number of acceptable units is only (1 − pi )Qi . This is easily extended to serial
2
(1)
This expression was used to develop the following cost model [10]: Cn = c0 x0 + x0
j-1
j
i−1 n
(1 − dj),
ci
i=1
(2)
j=1
where x is the quantity that enters the process (Q in Eq. (1)), dj is station’s j’s defect rate (p in Eq. (1)) and c abbreviates ‘cost’. This cost model leads to the conclusion that “the total costs (Cn ) of production actually decrease. . . if we have quality problems and therefore a defect rate greater than zero” (Ibid)! Certainly, costs do not decrease in the presence of quality problems, on the contrary, costs increase. There are simply, a couple of errors in this cost model. First, it assumes that each defective item is detected and removed as soon as it is damaged. This is not the standard case, as demonstrated in Fig. 1, which portrays a serial manufacturing process where each node represents an operation. Each defective item divides its manufacturing process into three segments. In the first segment the item conforms to specifications; this segment ends when the item is damaged. At this point the second segment starts and it ends when the defective item is detected; while the third segment extends from detection onward. Note that, unless the item is conforming, at least one segment is not empty. Typically, the detection of defective items occurs in later stages, often after delivery as Table 1 indicates. Thus, the second segment is not empty and costs are wasted on the processing of defective items. The second and more severe error in the cost model (2) is in the application of Eq. (1). The cost model applies it forward. Hence, the number of conforming units – the yield, decreases as we move forward in the process. For example, in a moderate size process of 70 operations, with defect rate of 1% each, less than half (49.5%) of the units that enter the process will come out conforming! No one can allow this pattern to occur in practice. There are sales targets and/or orders to deliver; a manufacturer that let this happen loses sales in the better case, and is likely to find himself out of business in most cases due to the failure to timely deliver orders. This discussion demonstrates the issue that rose in the preliminary remark in the introduction – not the mathematics but its application matters. If inspections are performed, defective items that are detected can be removed, leading to savings in terms of costs incurred by and capacity required for future operations. A defective item can either be scrapped, used as is but offered at a lower price, reworked, or repaired. The last two cases involve costs and consume capacity in addition to regular capacity and costs, while in the first two cases capacity is wasted and income is lost, which is equivalent to cost increase. Again, the type of the defect does not matter, whether dimensional, form, surface quality, hardness of contact, or other defects, all require additional capacity and incur additional costs. Production planners know how many end items are needed. From these figures, order quantities are calculated backward, as in
Damaged 1
(1 − pi ).
i=1
Source: JD Power & Associates, Initial Quality Study [1].
1. 2. 3. 4.
n
Detected J+1
k-1
k
k+1
n-1
Fig. 1. A defective item divides the manufacturing process into three segments.
n
M. Eben-Chaime / Journal of Manufacturing Systems 36 (2015) 1–6
4.1. The actual defect rate of an assembly
Assembly
Component 2 – 2 units
Component 1 – 4 units
Following the discussion above, the defect rate of an assembly depends not only on its own defect rate, pA , but also on the defect rates of its components. Consequently, actual defect rates should be calculated for each assembly. The actual defect rate paA of an assembly with K component types is:
Component 3 – 1 unit
Fig. 2. A product structure.
paA = 1 − (1 − pA) material requirements planning (MRP) (see, e.g., [11]). Whenever defective units are not used as intended, more units must be produced to replace them. A reworked unit is just like an additional unit with, perhaps, additional preparation activities, and repair consumes additional capacity. Consequently, Eq. (1) should be inverted as Eq. (3), where pi is the fraction of defective units that are not repaired: Qjin =
Q out
(3)
.
n
(1 − pi )
i=j
This inversion is a key to the distinction between quality and reliability. It turns the defect rates into required quantities, which, in turn, determine capacity requirements for the manufacturing system. In so doing, Eq. (3) quantifies the repercussions of poor quality – Qj in units should enter operation j for an average of Qout units to come out conforming. Eq. (3) accounts for all items that will be defective and not repaired in operation j and subsequent operations up to the last activity, n. In the 70 operations example above, about 2021 units should enter the process to yield 1000 conforming units! Still, however, this is the minimal quantity – larger quantities may pass through activity j if defective items from preceding operations have not been removed earlier. With these minimal quantities the following cost model is obtained, which also assumes that each defective item is detected and removed as soon as it is damaged: C n = Q out
n j=1
n
cj
3
.
(1 − pi )
i=j
(The appearance of the cost model (2) in 2005 is rather surprising in light of the fact that the inversion of Eq. (1) into (3) has been applied correctly for several decades prior to 2005; e.g., [12,13].)
K
(1 − pk )mk
(4)
k=1
Consider for example the assembly depicted in Fig. 2. Suppose the components arrive with defect rates p1 , p2 , and p3 , respectively. The determination of these rates will be considered later, but assuming their values are given, the probability that an assembly will conform is given by: (1 − p1 )4 (1 − p2 )2 (1 − p3 )(1 − pA ). The complement of this probability is the actual defect rate paA of the assembly operation. This example demonstrates the dramatic increase of the actual defect rates of assemblies. If the defect rate of each component is 1% and the defect rate of the assembly operation is 0.1%, then the actual defect rate of the assembly in this example is about 7%! 4.2. Mutual effect of defective components in assemblies The requirement that more than one component should conform to specifications creates mutual effects between different components – it is not necessary that all the components are defective to disqualify an assembly; a single defective component suffices for that to happen! To illustrate, consider Fig. 2, again, with the above defect rates. If, say, 1000 assembly units are required, then by Eq. (3): 1000/(1 − 0.07) = 1074 units should be assembled, containing: 4296 units of component 1; 2148 units of component 2; and 1074 units of component 3. However, on average, only 0.01 × 1074 = 11 of the 74 extra units of component 3 are defective; the rest – 63 units – are assembled with defective units of other components, or a failure occurs during the assembly process. The same holds for the other components – there are, on average, only 44 defective items of component 1 and 22 of component 2. Time and resources are required to diagnose the source of failure of each non-conforming assembly. Additional time and resources are consumed to fix defective assemblies or disassemble and return to use the good components. Of course, there is the “rotten apple” effect – “One rotten apple spoils the bunch” – a defective component causes damage to other components. However, since one has to be defective to start the avalanche, this type of mutual effect is always additional to the simple type described above.
4. Assemblies’ defect rates 4.3. Multiple assemblies Products are usually not structured serially as in Fig. 1, but rather in tree structures – a tree for each product. Fig. 2 portrays a product structure whose Bill of Materials (BOM) tells us that 4 units of component 1, 2 units of component 2, and a unit of component 3 are assembled in each end-item’s unit (e.g., [11]). All these components should of course conform to specifications, but how can defective component be accounted for? The answer lies in the nature of assemblies. An assembly conforms to design only if all its components conform. This is another deviation from reliability theory. In reliability, components of the same type are assembled in parallel to increase reliability – redundant components – it is enough for one of them to function for the whole system to work (e.g., [14]). Redundant components increase the load on the manufacturing system even more than defective items – they are added whether needed or not, and for quality matters, all these components most conform for the system to conform.
Eqs. (3) and (4) are generic and the full effect of defective components in assemblies materializes in large products; e.g., the product in Fig. 3. Defect rates in tree product structures should be calculated via the application of Eq. (4), bottom-up – from the elementary components to the final assembly. The defect rate of component 8 equals p8 = 1 − (1 − p9 )2 × (1 − p10 ) × (1 − pA8 ), where pA8 is the defect rate of the assembly operation of component 8. Then, p2 = 1 − (1 − p7 )2 × (1 − p8 )5 × (1 − pA2 ). Similarly, p1 = 1 − (1 − p4 ) × (1 − p5 )2 × (1 − p6 )3 × (1 − pA1 ), and finally, the overall yield: YFA = (1 − p1 )4 × (1 − p2 )2 × (1 − p3 ) × (1 − pFA ). With the same rates as above: 1% for each component and 0.1% for each assembly operation, YFA = 54%! It may be argued that a defect rate of 1% is too high. It is, indeed high, but lower than the figures reported in; e.g., the Sematech 2013 industry survey. An explanation to these high values is proposed
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M. Eben-Chaime / Journal of Manufacturing Systems 36 (2015) 1–6
Comp. 4 1 unit
Comp. 5 2 units
Comp. 2 2 units
Comp. 3 1 unit
Comp. 6 3 units
Comp. 7 2 units
1880 1870 1860 1850 1840 1830 1820 1810 1800 1790 1780 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.11 0.12 0.13 0.14 0.15 0.16 0.17 0.18 0.19 0.2
Comp. 1 4 units
Required quanty
Final assembly
Comp. 8 5 units Comp. 9 2 units
Comp. 10 1 unit
Fig. 3. A larger product.
in the next section, but higher or lower, defect rates are strictly positive. 5. Discussion – coping with defect components While strictly positive, defect rates can be reduced by improving materials’ quality and processes and using advanced technologies. The outcome of such improvements is quantified in Figs. 4 and 5 in terms of the quantities that need be processed in order to produce 1000 conforming unit of the product of Fig. 3 – 1000/YFA , on average. In Fig. 4, the assemblies’ defect rate is fixed at 0.1% and YFA is changed by changing the components defect rates from 0.1% to 2%, while in Fig. 5, assemblies’ defect rates are changed from 0.01% to 0.2%, for operations’ defect rate of 1%. Evidently, the effect in Fig. 4 is much larger. While the required quantity grows more than 3 times, from about 1080 units to 3350 in Fig. 4, it rises only from 1810 units to 1870 units in Fig. 5. Also evident is the convexity of the curve in Fig. 4, which is maintained for all values of the assemblies’ defect rates. This convexity explains the difference in the effects’ magnitude – the range of assemblies’ defect rates is ten times smaller and corresponds to the lower, flat part of the curve in Fig. 4. In addition, there are more components than subassemblies. This amplifies their joint rate due to the mutual effects among them. In sum, however, the advantage of quality improvements to reduce the defect rates is clearly revealed.
3500
Assemblies' defect rates Fig. 5. The effect of assemblies’ defect rates on the required quantities.
The last example demonstrates another characteristic of assemblies, revealed by the present analysis – two passes through the product structure are required for non-serial products. The reason is that Eqs. (3) and (4) are applied in opposite directions. As just noted, defect rates should be calculated by Eq. (4), bottom-up – from the elementary components to the final assembly. Required quantities, on the other hand, are calculated by Eq. (3) with actual defect rates for assembly operations, in the opposite direction – from the end product to the elementary components. Fig. 6 is used to extend and illustrate. The figure presents a process chart (e.g., [11–13,15]). The operations are numbered – the numbers in the boxes on the top, and the numbers inside are the defect rate, in percent’s – 1 means 1%. An operation with more than one entering arrow is an assembly operation. Thus, this process chart fits the product structure of Fig. 3: ten components, of which 3 are sub-assemblies. The terminal operation of each component is numbered by the number of the corresponding item in Fig. 3, and if each of the tree branches is rolled up to the terminal node, Fig. 3 appears with circles instead of rectangular boxes. Eq. (4) associates assembly operations with their immediate predecessor operations. An extended and generalized perception regards all operations the same, where in Eq. (4), K > 1, for assemblies, while K = 1 for other operations. Then, the actual yield of an operation, say j, with a single predecessor i: Yj = (1 − pj )(1 − pi ). A serial process is then a special case with K = 1 for all operations. The yield calculation can, thus, be continued with the predecessor operation of operation i, its predecessor, etc., until ultimately, Eq. (1) is formed. Fig. 6 is but one of many process charts that fit the product structure of Fig. 3. In particular, operations may follow each assembly operation – between op. #8 and op. #2, between op. #1, #2, or #3
0
3000
Required quanty
0.1
2500
1 0.1
2
3
0.1
0.25
2000 5
8
11
0.33
0.1
0.25
1500 4
12
7
10
13
0.5
0.33
0.5
0.5
0.25
6
1000
9
1
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
1
14
15
16
17
18
0.5
0.33
0. 5
0.5
0.25
Operaons' defect rate % Fig. 4. The effect of components’ defect rates on the required quantities.
Fig. 6. The process chart for the large product.
M. Eben-Chaime / Journal of Manufacturing Systems 36 (2015) 1–6 Table 2 A Comparison of cleaning methods. Op.# The number of units to be processed under
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
perfect process
cleaning after each operation
cleaning just prior to assembly
no cleaning
1000 4000 2000 1000 4000 8000 12,000 4000 10,000 20,000 10,000 1000 8000 1000 4000 8000 4000 10,000 1000
1001 4008 2004 1004 4028 8043 12,145 4028 10,030 20,263 10,080 1006 8070 1009 4048 8097 4048 10,131 1011
1001 4008 2004 1011 4048 8097 12,145 4048 10,030 20,263 10,131 1011 8097 1011 4048 8097 4048 10,131 1011
1840 7362 3681 1840 7362 14,723 22,085 7362 18,404 36,808 18,404 1840 14,723 1840 7362 14,723 7362 18,404 1840
and the final assembly, #0, or following the final assembly. However, in this particular chart the defect rate can be set in agreement with those used above: 1% for each component and 0.1% for each assembly, too. For example, (1 − 0.005)(1 − 0.005) ≈ (1 − 0.01) for components 4, 7 and 10. Similarly, the defect rate of component 5 is (1 − 0.0033)3 ≈ (1 − 0.01) and (1 − 0.0025)4 ≈ (1 − 0.01) for component 3. The actual yield of the product of Fig. 3 is calculated above: YFA = 54%. Thus, in order to produce 1000 conforming units, 1000/0.54 = 1840.4 units should be processed! This requires some 7362 units of component 1, 3681 unit of component 2 and 1840 unit of component 3. These, in turn, amount to 7362 units of component 4, 14,723 units of component 5, 22,085 units of component 6, 7362 units of component 7 and 18,404 units of component 8. The later requires 18,404 units of component 10 and 36,808 units of component 9. All these units flow through the corresponding operations in the process chart of Fig. 6. To demonstrate the potential quality assurance (QA) efforts, perfect cleaning, which allows no defect item to pass from any operation to its successor, is considered. Perfect cleaning is, of course, not realistic but provides lower bounds on the required quantities. It is also useful for demonstration since it makes a special case in which only a single pass through the product structure suffices even for non-serial structures. The reason is that Eq. (3) can be applied backward along each branch of the process chart. For example, since only conforming components are used, 1001 units of the final assembly should be assembled. Of which, 1 unit comes out defective, on average. This requires 1011 units to enter operation 18. On average, 2.5 of these units will be cleaned prior to moving to operation 13, another 2.5 between operations 13 and 11, another 2.5 between 11 and 3 and another 2.5 units before component 3 is completed and moved to the final assembly. Following another branch, 1001 final assemblies require 8097 units to enter operation 15, of which 8070 conforming units move to operation 12. 8043 conforming units move from operation 12 to operation 5, and only 8016 conforming units of component 5 are assembled in 4008 units of subassembly 1. Another 4 defective units of subassembly 1 are cleaned, on average, prior to the final assembly. Perfect cleaning however, requires the inspection of each item following the execution of each operation and the immediate removal of each defective item. The understanding of the mutual effects of defective components in assemblies, suggests a medium alternative: clean defective components just prior to assembly
5
operations. If no defective component enters any assembly operation, the mutual dependencies are detached since the defect rates pk of Eq. (4) are all equal zero and only the self-defect rate of the assembly operations, pA , remain active. Note! This does not imply that other operations produce no defective items, only that all defective items are cleaned. This discussion is summarized in Table 2, where the number of units to be processed in each operation under each cleaning method is presented for comparison. The numbers of units processed in a perfect process, where no defective items are produced, are also listed as a base-line. The differences between the numbers on the right column and the numbers in the other columns clearly demonstrate the value, the contribution and the significance of QA efforts. Further, the small differences between the three middle columns signify that, while QA efforts are unavoidable, there are efficient ways to apply them and the analysis above indicates how.
6. Conclusions This is a first study of the effects of defective components on the quality of the products in which these components are assembled. The study revealed the mutual effects of their components on assemblies’ quality. The foremost conclusion is the dramatic amplification of the defect rates of assemblies due to the mutual effects of their components, to the extent that common or random causes become significant. This observation reveals and strengthens the awareness and the understanding for the need for quality improvements by all means, the significance and contributions such improvements may have. A second conclusion is the difference between the calculations of the defect rates and those of the required quantities and the consequent need to traverse each process chart twice. In a multi-product environment, these calculations can be done for each product independently. A third conclusion is the indication where the focal points of QA should be: just prior to assembly stations. When no defective component is used in an assembly, the mutual dependencies are detached. Unfortunately, there are some undesirable repercussions, too. The disengagement of the mutual relations among assemblies’ components requires to inspect each item – 100% inspection, whenever items are inspected. Moreover, perfect inspection tools are required, since any defective component that slips through restores the mutual effects. Perfect inspection tools do not exist. Even worse, the inspection tools make two types of errors: miss targets – faulty items, or chose wrong targets – a conforming item is designated defective. The latter contributes additional yield reduction!! The integration of inspection errors in the analysis is a subject for future work. Another direction for future research is the effect of the defective components on operations management. Here, average rates were used, which do not fit assemblies in batch production environment. The number of conforming units of each component arriving to the assembly of each batch is a random variable, and the number of units that can be assembled is the minimal allowed by all components. This creates chaos, which sets serious challenges for operations managers. Upon conclusion, a question arises: what has changed? Products have been assembled before. Thus, this question actually closes a cycle returning to the question in the outset: why is it, then, that quality problems still exist? One answer is the increase in products’ complexity. Today’s products consist of many more components and are much more complex than past products. Further, increased product complexity results in larger and more complex manufacturing processes.
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M. Eben-Chaime / Journal of Manufacturing Systems 36 (2015) 1–6
Eq. (4) tells us that even when the defect rates of each step, each operation in the process seems negligible; in large processes these rates accumulate and become significant! This is a price of advanced products – increased complexity which sets hurdles on the preservation of quality; a price that should be considered in the course of product and process design. Another facet of the relations between complexity and quality is considered in [16]. Particularly, the present analysis suggests that it is preferable to concentrate assembly activities to few stations since the required QA efforts are thereby reduced. References [1] JD Power & Associates, Initial quality survey, http://businesscenter. jdpower.com/ [2] “Western electric – a brief history”. The Porticus Centre. Retrieved from: http://www.beatriceco.com/bti/porticus/bell/westernelectric history.html# Western%20Electric%20-%20A%20Brief%20History [2009-04-10]. [3] Sematech. Photo-mask industry survey; 2013 http://www.sematech.org/ corporate/news /releases/20130924.htm [4] Feigenbaum AV. Total quality control. 3rd ed. revised New York, USA: McGrawHill; 1991.
[5] O’Connor PDT. Practical reliability engineering. 4th ed. New York: John Wiley & Sons; 2002. [6] Graves TL, Anderson-Cook CM, Hamada MS. Reliability models for almostseries almost-parallel systems. Technometrics 2010;52(2):160–71. [7] Shewhart WA. Economic control of quality of manufactured products. London, UK: Van Nostrand Company Inc.; 1931. [8] Kim J, Gershwin SB. Analysis of long flow lines with quality and operational failures. IIE Trans 2008;40:284–96. [9] Jingshan L, Blumenfeld DE, Marin S. Production systems design for quality robustness. IIE Trans 2008;40:162–76. [10] Freiesleben J. The economic effects of quality improvements. Total Qual Manag 2005;16(7):915–22. [11] Nahmias S. Production and operations analysis. 6th ed. New York: McGraw Hill; 2009. [12] Apple JM. Plant layout and material handling. New York, USA: The Ronald Press Company; 1951. [13] Moore JM. Plant layout and design. New York: Macmillan; 1962. [14] Garg H, Rani M, Shama SP, Vishwakarma Y. Bi-objective optimization of the reliability-redundancy allocation problem for series–parallel system. J Manuf Syst 2014;33(3):335–47. [15] Francis RL, McGinnis LF, White JA. Facility layout and location: an analytical approach. 2nd ed. Englewood Cliffs, NJ, USA: Prentice-Hall Inc.; 1992. [16] Fast-Berglund T, Fassberg T, Hellman F, Davidsson A, Stahre J. Relations between complexity, quality and cognitive automation in mixed-model assemblies. J Manuf Syst 2013;32(2):449–55.
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Journal of Manufacturing Systems journal homepage: www.elsevier.com/locate/jmansys
Technical Paper
Mutual effects of defective components in assemblies Moshe Eben-Chaime ∗ Department of Industrial Engineering & Management, Ben-Gurion University of the Negev, P.O. Box 653, Beer Sheva 84105, Israel
a r t i c l e
i n f o
Article history: Received 13 August 2014 Received in revised form 16 February 2015 Accepted 22 February 2015 Keywords: Assembly Bill of Materials Defect rate Product structure Quality
a b s t r a c t The focus of this study is a known and disturbing actual problem. The industry will soon celebrate a century of quality awareness and efforts. Still, according to field data, many new products exit the manufacturing systems defective. This study proposes mutual effects among assembly’s components as an explanation to this phenomenon – many defective new products. While each item in a serial manufacturing process moves individually, items are joined to others in assemblies. There, a single defective component suffices to disqualify a whole assembled unit! Surprisingly, few studies have focused on the repercussions of defective items on production. Particularly, there appears to be no study that quantifies these mutual effects of components which arrive from different sources with different defect rates. Thus, this study is also a first attempt to analyze and quantify these mutual effects. Apparently, the mutual effects of their components amplify the defect rates of assemblies dramatically, to the extent that defects due to common or random causes become significant. © 2015 The Society of Manufacturing Engineers. Published by Elsevier Ltd. All rights reserved.
1. Introduction This study was motivated by the reality portrayed in Table 1, where the industry averages (IA) of the number of defects per 100 cars detected within 90 days of purchase through the years 2004–2013 (JD Power & Associates, Initial Quality Study [1]), are shown. This work aims to reveal and explain the causes for this phenomenon. May 16, 1924 is the date when Walter A. Shewhart first presented the statistical control chart [2]. It really is a wonder, then, that after 90 years of quality awareness and efforts, there are still so many defective products – each car, on average. It should be noted that both flaws found after 90 days or detected prior to delivery of the cars, are excluded in Table 1. Note also that these findings in the car industry do not imply that other industries are better – “All participating mask shops (in the 2013 photo-mask industry survey) reported shipping masks targeting the 22 nm node or smaller, and on average, yields were greater than 90% at all nodes down to 22 nm.” [3] Feigenbaum [4] minted the term “hidden plant” and defined it as “the proportion of plant capacity that exists to rework unsatisfactory parts, to replace product recalled from the field, or to retest and re-inspect rejected units” – i.e., to deal with the
∗ Tel.: +972 8 6472206. E-mail address: [email protected]
consequences incurred after an item is detected as defective. However, extra production capacity is required even before items are actually damaged and for these matters the type of the defect does not matter, whether dimensional, form, surface quality, hardness of contact, or other defects, all consume additional capacity. Feigenbaum estimated that hidden plants amount “to 15 percent to as much as 40 percent of productive capacity”. The present study also aims to develop means to quantify this figure and, more important, to evaluate and compare alternatives. A preliminary remark concerning the focus of this study is in order here. To the best of the author’s knowledge, the type of analysis presented in the sequel has never been done before. One possible explanation may be that the present study integrates different domains and thus represents a novel departure. While the mathematics may seem straightforward, our application is not trivial – as will soon be seen, a single attempt to make a similar application in a simple special case led to erroneous results. Another reason could be confusion with reliability theory – though the mathematics may seem similar, the focus is different. Reliability, e.g. [5] is concerned with products, while the present study considers manufacturing processes. Reliability analyses evaluate the probability that components and systems will remain functional while in use. This study, on the other hand, focuses on the yield of manufacturing processes – the proportion of conforming units that are produced. The practical implications of this distinction are discussed in the sequel, too. Regardless of analytical sophistication, this paper explains a known and disturbing problem, provides means to quantify it, and
http://dx.doi.org/10.1016/j.jmsy.2015.02.008 0278-6125/© 2015 The Society of Manufacturing Engineers. Published by Elsevier Ltd. All rights reserved.
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M. Eben-Chaime / Journal of Manufacturing Systems 36 (2015) 1–6
Table 1 Industry averages of the number of defects per 100 cars.
processes. If Q0 units enter the first activity in a serial process of n operations, the mean number of acceptable units at the end is:
Year
2004
2005
2006
2007
2008
2009
2010
2011
2012
2013
IA
119
118
109
125
118
108
109
107
102
113
Qn = Q0
underlines those attributes that distinguish it from other concerns. Furthermore, the analysis suggests ways of coping with the problem. The paper is organized as follows. Assumptions and notation are provided next. Section 3 presents a brief review of known results on serial processes. This lays the foundations for the analysis of assembly operations in Section 4. The results are discussed in Section 5 and conclusions are drawn in Section 6. 2. Assumptions and notation Let pi denote the average defect rate of operation/activity i. The defect rate depends on both the activity and the station chosen to perform it, but let’s assume the stations for each activity have already been selected. Since performed in different stations, it is reasonable to assume that the operations are (close to be) independent. Note, again, the difference between the defect rate and the failure rate of reliability theory – defect rate characterizes manufacturing/production operations, while failure rate (e.g., [6]) characterizes components, products or systems. Shewhart, in his pioneering book [7], distinguished between assignable and other causes for poor quality. It should also be noted, that here, defects due to common, or random causes, are considered, not quality deterioration due to assignable causes as in [8]. This makes independence between items a reasonable approximation, in addition to activity independence. Finally, in repetitive production, long term averages are proper performance measures to use, rather than first time quality as in [9]. In sum, four assumptions are made: The stations for each activity have already been selected; Activities/operations are independent; The processing of items in a station are independent; Long term averages are proper performance measures to use. Notation: ci – unit processing cost of activity i; Cn – total cost of n activities; mk – the assembly coefficient of the kth component type – the number of units of type k component in an assembly; pi – the average defect rate of operation/activity i; paA – the actual defect rate of an assembly; Qi – the number of units on which activity i is performed; Yi – the yield of – the number of conforming units manufactured by activity i. 3. Defect rates and input/output ratios If activity i is performed on Qi units, the mean number of acceptable units is only (1 − pi )Qi . This is easily extended to serial
2
(1)
This expression was used to develop the following cost model [10]: Cn = c0 x0 + x0
j-1
j
i−1 n
(1 − dj),
ci
i=1
(2)
j=1
where x is the quantity that enters the process (Q in Eq. (1)), dj is station’s j’s defect rate (p in Eq. (1)) and c abbreviates ‘cost’. This cost model leads to the conclusion that “the total costs (Cn ) of production actually decrease. . . if we have quality problems and therefore a defect rate greater than zero” (Ibid)! Certainly, costs do not decrease in the presence of quality problems, on the contrary, costs increase. There are simply, a couple of errors in this cost model. First, it assumes that each defective item is detected and removed as soon as it is damaged. This is not the standard case, as demonstrated in Fig. 1, which portrays a serial manufacturing process where each node represents an operation. Each defective item divides its manufacturing process into three segments. In the first segment the item conforms to specifications; this segment ends when the item is damaged. At this point the second segment starts and it ends when the defective item is detected; while the third segment extends from detection onward. Note that, unless the item is conforming, at least one segment is not empty. Typically, the detection of defective items occurs in later stages, often after delivery as Table 1 indicates. Thus, the second segment is not empty and costs are wasted on the processing of defective items. The second and more severe error in the cost model (2) is in the application of Eq. (1). The cost model applies it forward. Hence, the number of conforming units – the yield, decreases as we move forward in the process. For example, in a moderate size process of 70 operations, with defect rate of 1% each, less than half (49.5%) of the units that enter the process will come out conforming! No one can allow this pattern to occur in practice. There are sales targets and/or orders to deliver; a manufacturer that let this happen loses sales in the better case, and is likely to find himself out of business in most cases due to the failure to timely deliver orders. This discussion demonstrates the issue that rose in the preliminary remark in the introduction – not the mathematics but its application matters. If inspections are performed, defective items that are detected can be removed, leading to savings in terms of costs incurred by and capacity required for future operations. A defective item can either be scrapped, used as is but offered at a lower price, reworked, or repaired. The last two cases involve costs and consume capacity in addition to regular capacity and costs, while in the first two cases capacity is wasted and income is lost, which is equivalent to cost increase. Again, the type of the defect does not matter, whether dimensional, form, surface quality, hardness of contact, or other defects, all require additional capacity and incur additional costs. Production planners know how many end items are needed. From these figures, order quantities are calculated backward, as in
Damaged 1
(1 − pi ).
i=1
Source: JD Power & Associates, Initial Quality Study [1].
1. 2. 3. 4.
n
Detected J+1
k-1
k
k+1
n-1
Fig. 1. A defective item divides the manufacturing process into three segments.
n
M. Eben-Chaime / Journal of Manufacturing Systems 36 (2015) 1–6
4.1. The actual defect rate of an assembly
Assembly
Component 2 – 2 units
Component 1 – 4 units
Following the discussion above, the defect rate of an assembly depends not only on its own defect rate, pA , but also on the defect rates of its components. Consequently, actual defect rates should be calculated for each assembly. The actual defect rate paA of an assembly with K component types is:
Component 3 – 1 unit
Fig. 2. A product structure.
paA = 1 − (1 − pA) material requirements planning (MRP) (see, e.g., [11]). Whenever defective units are not used as intended, more units must be produced to replace them. A reworked unit is just like an additional unit with, perhaps, additional preparation activities, and repair consumes additional capacity. Consequently, Eq. (1) should be inverted as Eq. (3), where pi is the fraction of defective units that are not repaired: Qjin =
Q out
(3)
.
n
(1 − pi )
i=j
This inversion is a key to the distinction between quality and reliability. It turns the defect rates into required quantities, which, in turn, determine capacity requirements for the manufacturing system. In so doing, Eq. (3) quantifies the repercussions of poor quality – Qj in units should enter operation j for an average of Qout units to come out conforming. Eq. (3) accounts for all items that will be defective and not repaired in operation j and subsequent operations up to the last activity, n. In the 70 operations example above, about 2021 units should enter the process to yield 1000 conforming units! Still, however, this is the minimal quantity – larger quantities may pass through activity j if defective items from preceding operations have not been removed earlier. With these minimal quantities the following cost model is obtained, which also assumes that each defective item is detected and removed as soon as it is damaged: C n = Q out
n j=1
n
cj
3
.
(1 − pi )
i=j
(The appearance of the cost model (2) in 2005 is rather surprising in light of the fact that the inversion of Eq. (1) into (3) has been applied correctly for several decades prior to 2005; e.g., [12,13].)
K
(1 − pk )mk
(4)
k=1
Consider for example the assembly depicted in Fig. 2. Suppose the components arrive with defect rates p1 , p2 , and p3 , respectively. The determination of these rates will be considered later, but assuming their values are given, the probability that an assembly will conform is given by: (1 − p1 )4 (1 − p2 )2 (1 − p3 )(1 − pA ). The complement of this probability is the actual defect rate paA of the assembly operation. This example demonstrates the dramatic increase of the actual defect rates of assemblies. If the defect rate of each component is 1% and the defect rate of the assembly operation is 0.1%, then the actual defect rate of the assembly in this example is about 7%! 4.2. Mutual effect of defective components in assemblies The requirement that more than one component should conform to specifications creates mutual effects between different components – it is not necessary that all the components are defective to disqualify an assembly; a single defective component suffices for that to happen! To illustrate, consider Fig. 2, again, with the above defect rates. If, say, 1000 assembly units are required, then by Eq. (3): 1000/(1 − 0.07) = 1074 units should be assembled, containing: 4296 units of component 1; 2148 units of component 2; and 1074 units of component 3. However, on average, only 0.01 × 1074 = 11 of the 74 extra units of component 3 are defective; the rest – 63 units – are assembled with defective units of other components, or a failure occurs during the assembly process. The same holds for the other components – there are, on average, only 44 defective items of component 1 and 22 of component 2. Time and resources are required to diagnose the source of failure of each non-conforming assembly. Additional time and resources are consumed to fix defective assemblies or disassemble and return to use the good components. Of course, there is the “rotten apple” effect – “One rotten apple spoils the bunch” – a defective component causes damage to other components. However, since one has to be defective to start the avalanche, this type of mutual effect is always additional to the simple type described above.
4. Assemblies’ defect rates 4.3. Multiple assemblies Products are usually not structured serially as in Fig. 1, but rather in tree structures – a tree for each product. Fig. 2 portrays a product structure whose Bill of Materials (BOM) tells us that 4 units of component 1, 2 units of component 2, and a unit of component 3 are assembled in each end-item’s unit (e.g., [11]). All these components should of course conform to specifications, but how can defective component be accounted for? The answer lies in the nature of assemblies. An assembly conforms to design only if all its components conform. This is another deviation from reliability theory. In reliability, components of the same type are assembled in parallel to increase reliability – redundant components – it is enough for one of them to function for the whole system to work (e.g., [14]). Redundant components increase the load on the manufacturing system even more than defective items – they are added whether needed or not, and for quality matters, all these components most conform for the system to conform.
Eqs. (3) and (4) are generic and the full effect of defective components in assemblies materializes in large products; e.g., the product in Fig. 3. Defect rates in tree product structures should be calculated via the application of Eq. (4), bottom-up – from the elementary components to the final assembly. The defect rate of component 8 equals p8 = 1 − (1 − p9 )2 × (1 − p10 ) × (1 − pA8 ), where pA8 is the defect rate of the assembly operation of component 8. Then, p2 = 1 − (1 − p7 )2 × (1 − p8 )5 × (1 − pA2 ). Similarly, p1 = 1 − (1 − p4 ) × (1 − p5 )2 × (1 − p6 )3 × (1 − pA1 ), and finally, the overall yield: YFA = (1 − p1 )4 × (1 − p2 )2 × (1 − p3 ) × (1 − pFA ). With the same rates as above: 1% for each component and 0.1% for each assembly operation, YFA = 54%! It may be argued that a defect rate of 1% is too high. It is, indeed high, but lower than the figures reported in; e.g., the Sematech 2013 industry survey. An explanation to these high values is proposed
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M. Eben-Chaime / Journal of Manufacturing Systems 36 (2015) 1–6
Comp. 4 1 unit
Comp. 5 2 units
Comp. 2 2 units
Comp. 3 1 unit
Comp. 6 3 units
Comp. 7 2 units
1880 1870 1860 1850 1840 1830 1820 1810 1800 1790 1780 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.11 0.12 0.13 0.14 0.15 0.16 0.17 0.18 0.19 0.2
Comp. 1 4 units
Required quanty
Final assembly
Comp. 8 5 units Comp. 9 2 units
Comp. 10 1 unit
Fig. 3. A larger product.
in the next section, but higher or lower, defect rates are strictly positive. 5. Discussion – coping with defect components While strictly positive, defect rates can be reduced by improving materials’ quality and processes and using advanced technologies. The outcome of such improvements is quantified in Figs. 4 and 5 in terms of the quantities that need be processed in order to produce 1000 conforming unit of the product of Fig. 3 – 1000/YFA , on average. In Fig. 4, the assemblies’ defect rate is fixed at 0.1% and YFA is changed by changing the components defect rates from 0.1% to 2%, while in Fig. 5, assemblies’ defect rates are changed from 0.01% to 0.2%, for operations’ defect rate of 1%. Evidently, the effect in Fig. 4 is much larger. While the required quantity grows more than 3 times, from about 1080 units to 3350 in Fig. 4, it rises only from 1810 units to 1870 units in Fig. 5. Also evident is the convexity of the curve in Fig. 4, which is maintained for all values of the assemblies’ defect rates. This convexity explains the difference in the effects’ magnitude – the range of assemblies’ defect rates is ten times smaller and corresponds to the lower, flat part of the curve in Fig. 4. In addition, there are more components than subassemblies. This amplifies their joint rate due to the mutual effects among them. In sum, however, the advantage of quality improvements to reduce the defect rates is clearly revealed.
3500
Assemblies' defect rates Fig. 5. The effect of assemblies’ defect rates on the required quantities.
The last example demonstrates another characteristic of assemblies, revealed by the present analysis – two passes through the product structure are required for non-serial products. The reason is that Eqs. (3) and (4) are applied in opposite directions. As just noted, defect rates should be calculated by Eq. (4), bottom-up – from the elementary components to the final assembly. Required quantities, on the other hand, are calculated by Eq. (3) with actual defect rates for assembly operations, in the opposite direction – from the end product to the elementary components. Fig. 6 is used to extend and illustrate. The figure presents a process chart (e.g., [11–13,15]). The operations are numbered – the numbers in the boxes on the top, and the numbers inside are the defect rate, in percent’s – 1 means 1%. An operation with more than one entering arrow is an assembly operation. Thus, this process chart fits the product structure of Fig. 3: ten components, of which 3 are sub-assemblies. The terminal operation of each component is numbered by the number of the corresponding item in Fig. 3, and if each of the tree branches is rolled up to the terminal node, Fig. 3 appears with circles instead of rectangular boxes. Eq. (4) associates assembly operations with their immediate predecessor operations. An extended and generalized perception regards all operations the same, where in Eq. (4), K > 1, for assemblies, while K = 1 for other operations. Then, the actual yield of an operation, say j, with a single predecessor i: Yj = (1 − pj )(1 − pi ). A serial process is then a special case with K = 1 for all operations. The yield calculation can, thus, be continued with the predecessor operation of operation i, its predecessor, etc., until ultimately, Eq. (1) is formed. Fig. 6 is but one of many process charts that fit the product structure of Fig. 3. In particular, operations may follow each assembly operation – between op. #8 and op. #2, between op. #1, #2, or #3
0
3000
Required quanty
0.1
2500
1 0.1
2
3
0.1
0.25
2000 5
8
11
0.33
0.1
0.25
1500 4
12
7
10
13
0.5
0.33
0.5
0.5
0.25
6
1000
9
1
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
1
14
15
16
17
18
0.5
0.33
0. 5
0.5
0.25
Operaons' defect rate % Fig. 4. The effect of components’ defect rates on the required quantities.
Fig. 6. The process chart for the large product.
M. Eben-Chaime / Journal of Manufacturing Systems 36 (2015) 1–6 Table 2 A Comparison of cleaning methods. Op.# The number of units to be processed under
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
perfect process
cleaning after each operation
cleaning just prior to assembly
no cleaning
1000 4000 2000 1000 4000 8000 12,000 4000 10,000 20,000 10,000 1000 8000 1000 4000 8000 4000 10,000 1000
1001 4008 2004 1004 4028 8043 12,145 4028 10,030 20,263 10,080 1006 8070 1009 4048 8097 4048 10,131 1011
1001 4008 2004 1011 4048 8097 12,145 4048 10,030 20,263 10,131 1011 8097 1011 4048 8097 4048 10,131 1011
1840 7362 3681 1840 7362 14,723 22,085 7362 18,404 36,808 18,404 1840 14,723 1840 7362 14,723 7362 18,404 1840
and the final assembly, #0, or following the final assembly. However, in this particular chart the defect rate can be set in agreement with those used above: 1% for each component and 0.1% for each assembly, too. For example, (1 − 0.005)(1 − 0.005) ≈ (1 − 0.01) for components 4, 7 and 10. Similarly, the defect rate of component 5 is (1 − 0.0033)3 ≈ (1 − 0.01) and (1 − 0.0025)4 ≈ (1 − 0.01) for component 3. The actual yield of the product of Fig. 3 is calculated above: YFA = 54%. Thus, in order to produce 1000 conforming units, 1000/0.54 = 1840.4 units should be processed! This requires some 7362 units of component 1, 3681 unit of component 2 and 1840 unit of component 3. These, in turn, amount to 7362 units of component 4, 14,723 units of component 5, 22,085 units of component 6, 7362 units of component 7 and 18,404 units of component 8. The later requires 18,404 units of component 10 and 36,808 units of component 9. All these units flow through the corresponding operations in the process chart of Fig. 6. To demonstrate the potential quality assurance (QA) efforts, perfect cleaning, which allows no defect item to pass from any operation to its successor, is considered. Perfect cleaning is, of course, not realistic but provides lower bounds on the required quantities. It is also useful for demonstration since it makes a special case in which only a single pass through the product structure suffices even for non-serial structures. The reason is that Eq. (3) can be applied backward along each branch of the process chart. For example, since only conforming components are used, 1001 units of the final assembly should be assembled. Of which, 1 unit comes out defective, on average. This requires 1011 units to enter operation 18. On average, 2.5 of these units will be cleaned prior to moving to operation 13, another 2.5 between operations 13 and 11, another 2.5 between 11 and 3 and another 2.5 units before component 3 is completed and moved to the final assembly. Following another branch, 1001 final assemblies require 8097 units to enter operation 15, of which 8070 conforming units move to operation 12. 8043 conforming units move from operation 12 to operation 5, and only 8016 conforming units of component 5 are assembled in 4008 units of subassembly 1. Another 4 defective units of subassembly 1 are cleaned, on average, prior to the final assembly. Perfect cleaning however, requires the inspection of each item following the execution of each operation and the immediate removal of each defective item. The understanding of the mutual effects of defective components in assemblies, suggests a medium alternative: clean defective components just prior to assembly
5
operations. If no defective component enters any assembly operation, the mutual dependencies are detached since the defect rates pk of Eq. (4) are all equal zero and only the self-defect rate of the assembly operations, pA , remain active. Note! This does not imply that other operations produce no defective items, only that all defective items are cleaned. This discussion is summarized in Table 2, where the number of units to be processed in each operation under each cleaning method is presented for comparison. The numbers of units processed in a perfect process, where no defective items are produced, are also listed as a base-line. The differences between the numbers on the right column and the numbers in the other columns clearly demonstrate the value, the contribution and the significance of QA efforts. Further, the small differences between the three middle columns signify that, while QA efforts are unavoidable, there are efficient ways to apply them and the analysis above indicates how.
6. Conclusions This is a first study of the effects of defective components on the quality of the products in which these components are assembled. The study revealed the mutual effects of their components on assemblies’ quality. The foremost conclusion is the dramatic amplification of the defect rates of assemblies due to the mutual effects of their components, to the extent that common or random causes become significant. This observation reveals and strengthens the awareness and the understanding for the need for quality improvements by all means, the significance and contributions such improvements may have. A second conclusion is the difference between the calculations of the defect rates and those of the required quantities and the consequent need to traverse each process chart twice. In a multi-product environment, these calculations can be done for each product independently. A third conclusion is the indication where the focal points of QA should be: just prior to assembly stations. When no defective component is used in an assembly, the mutual dependencies are detached. Unfortunately, there are some undesirable repercussions, too. The disengagement of the mutual relations among assemblies’ components requires to inspect each item – 100% inspection, whenever items are inspected. Moreover, perfect inspection tools are required, since any defective component that slips through restores the mutual effects. Perfect inspection tools do not exist. Even worse, the inspection tools make two types of errors: miss targets – faulty items, or chose wrong targets – a conforming item is designated defective. The latter contributes additional yield reduction!! The integration of inspection errors in the analysis is a subject for future work. Another direction for future research is the effect of the defective components on operations management. Here, average rates were used, which do not fit assemblies in batch production environment. The number of conforming units of each component arriving to the assembly of each batch is a random variable, and the number of units that can be assembled is the minimal allowed by all components. This creates chaos, which sets serious challenges for operations managers. Upon conclusion, a question arises: what has changed? Products have been assembled before. Thus, this question actually closes a cycle returning to the question in the outset: why is it, then, that quality problems still exist? One answer is the increase in products’ complexity. Today’s products consist of many more components and are much more complex than past products. Further, increased product complexity results in larger and more complex manufacturing processes.
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Eq. (4) tells us that even when the defect rates of each step, each operation in the process seems negligible; in large processes these rates accumulate and become significant! This is a price of advanced products – increased complexity which sets hurdles on the preservation of quality; a price that should be considered in the course of product and process design. Another facet of the relations between complexity and quality is considered in [16]. Particularly, the present analysis suggests that it is preferable to concentrate assembly activities to few stations since the required QA efforts are thereby reduced. References [1] JD Power & Associates, Initial quality survey, http://businesscenter. jdpower.com/ [2] “Western electric – a brief history”. The Porticus Centre. Retrieved from: http://www.beatriceco.com/bti/porticus/bell/westernelectric history.html# Western%20Electric%20-%20A%20Brief%20History [2009-04-10]. [3] Sematech. Photo-mask industry survey; 2013 http://www.sematech.org/ corporate/news /releases/20130924.htm [4] Feigenbaum AV. Total quality control. 3rd ed. revised New York, USA: McGrawHill; 1991.
[5] O’Connor PDT. Practical reliability engineering. 4th ed. New York: John Wiley & Sons; 2002. [6] Graves TL, Anderson-Cook CM, Hamada MS. Reliability models for almostseries almost-parallel systems. Technometrics 2010;52(2):160–71. [7] Shewhart WA. Economic control of quality of manufactured products. London, UK: Van Nostrand Company Inc.; 1931. [8] Kim J, Gershwin SB. Analysis of long flow lines with quality and operational failures. IIE Trans 2008;40:284–96. [9] Jingshan L, Blumenfeld DE, Marin S. Production systems design for quality robustness. IIE Trans 2008;40:162–76. [10] Freiesleben J. The economic effects of quality improvements. Total Qual Manag 2005;16(7):915–22. [11] Nahmias S. Production and operations analysis. 6th ed. New York: McGraw Hill; 2009. [12] Apple JM. Plant layout and material handling. New York, USA: The Ronald Press Company; 1951. [13] Moore JM. Plant layout and design. New York: Macmillan; 1962. [14] Garg H, Rani M, Shama SP, Vishwakarma Y. Bi-objective optimization of the reliability-redundancy allocation problem for series–parallel system. J Manuf Syst 2014;33(3):335–47. [15] Francis RL, McGinnis LF, White JA. Facility layout and location: an analytical approach. 2nd ed. Englewood Cliffs, NJ, USA: Prentice-Hall Inc.; 1992. [16] Fast-Berglund T, Fassberg T, Hellman F, Davidsson A, Stahre J. Relations between complexity, quality and cognitive automation in mixed-model assemblies. J Manuf Syst 2013;32(2):449–55.