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Information-based inspection allocation for real-time inspection systems
Journal of Manufacturing Systems Vol. 2a/No. 1 2001
0 SMe
a
Information-Based Inspection Allocation for Real-Time Inspection Systems Adan Verduzco, Tyco Electronics, Mesquite, Texas, USA J. Rene Villalobos, Dept. of Industrial Engineering, Arizona State University, Tempe, Arizona, USA Benjamin Vega, Capital One, Richmond, Virginia, USA
Abstract
bly line, a situation that is difficult to achieve when human inspectors are utilized. Another example of added flexibility is the possibility of using different inspection routines for the same board as it advances along the assembly line. For instance, the inspection strategy, or the way the unit is inspected, for each of the units in production would not be fixed but would be determined as the unit advanced through the production line. That is, the inspection strategy would be determined using the information about the manufacturing process, the inspection process, and the time available to perform the inspection. This concept of not having a fixed inspection routine but to generate the routine as the unit advances through the assembly, accumulating information, is termed Flexible Inspection and has been explored by Villalobos,’ Villalobos, Foster, and Disney,2 and Villalobos, Ramachandran, and Baeza.3 While the inspection routine generation problem can be considered as a particular case of inspection effort allocation, it should be treated differently because inspection effort allocation is dynamic and on-line. This paper departs from previous approaches by proposing the generation of the inspection routine based on the maximization of the information gained by performing an extra inspection on each one of the elements on a printed circuit board (PCB). Specifically, this approach considers inprocess fallible AVI systems used in surface-mounted technology (SMT) PCB manufacturing processes, where cycle time restrictions call for the partial inspection of the product. The strategy proposed in this paper enables AVI systems to perform efficient in-process partial inspections. In addition, the inspection routine generated takes into account the quality system information needs that monitor, in real time, the different sources of variability present in the SMD assembly process. The system that generates the inspection routine and the quality moni-
The use of real-time Automated Visual Inspection (AU) in the electronics assembly industry is subject to cycle time constraints determined by the manufacturing system. These constraints must be met to maintain the nominal production rate. A problem thus created is determining which components to inspect at each one of the AVI stations such that the inspection time available is used in an optimal way. This paper presents a real-time inspection allocation that is based on the information gained by inspecting one additional component. The selection of which components to inspect is modeled as an information maximization problem. A modified Knapsack greedy heuristic is used to find near-optimal solutions to this optimization problem within the required time constraints.
Keywords: Inspection,
Inspection Allocation, Automated Visual Information Gain, Flexible Inspection Systems
1. Introduction The miniaturization of electronics devices has made the use of Automated Visual Inspection (AVI) systems a necessity within the electronics assembly industry. This is particularly evident in the assembly of surface-mounted devices (SMD), where human inspectors are being replaced rapidly by AVI systems. The use of AVI presents some obvious advantages over the human inspectors, such as less inspection subjectivity. However, it also presents some other less evident advantageous characteristics. One of these characteristics is the potential use, by these systems, of the inspection information for the continuous improvement of the assembly process. This could be achieved by using the inspection information as input for the statistical monitoring of the different sources of variability in the assembly process. Another important advantage provided by the use of AVI systems in the SMD assembly process is the added flexibility that they lend to the overall production environment. For instance, several products could be assembled sequentially in the same assem-
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Journal of Manufacturing Vol. 2orNo. 1 200 1
Systems
Many authors have addressed the problem of finding an optimal solution for POMDPs. The problem has been found to be computationally hard in the strong sense. l3 For a detailed discussion of this subject, the reader is referred to Monahan14 and Lovej oy. l5 The problem of finding an optimal solution for the inspection routine generation is not only very complex, given that the problem is a POMDP (Villalobos, Foster, and Disney’), but its complexity is compounded by inspection time constraints. Thus, generating real-time, optimal solutions even for a small-sized problem is not feasible. Thus, heuristic approaches must be used to generate inspection routines of meaningful size. An example of heuristic approaches to solve the inspection routine generation problem is given by Villalobos, Ramachandran, and Baeza.’ This paper proposes a heuristic approach to inspection allocation that departs from previous approaches by using a figure of merit, to be introduced as “information gain,” to perform the inspection allocation. This figure of merit does not necessarily rely on the traditional cost parameters but instead relies on factors such as the minimization of inspection errors.
toring systems are components of an Integrated Quality Environment (IQE) for SMD assembly. For a more detailed description of this integrated environment, the reader is referred to Villalobos and Verduzco.4 The main objective of this paper is to introduce the methodology behind the informationbased inspection allocation method utilized to generate the inspection routine as the PCB advances along the assembly line. In particular, near-optimal solutions to the allocation problem are achieved within the given time constraint by applying a modified Knapsack greedy heuristic.
2. Related Works One of the earliest works to address the problem of the optimal allocation of inspection was written by Lindsay and Bishop.5 They proved that to determine the minimum cost per unit either none or an entire batch should be inspected. Pruzan and Jackson6 used the zero- 100% result of Lindsay and Bishop’s work to develop two models for optimal For both models, they inspection allocation. assumed perfect inspection and the dependence of the inspection cost on the number of stations between two inspection points. Eppen and Hurst’ were the first to model the allocation problems assuming imperfect inspection. Their model aims to minimize the total expected cost. They conclude that the total cost is a concave, piecewise-linear function of the state variable (probability of a nondefective unit at a stage). Other papers related to optimal inspection allocation include Britney,* Ballou and Pazer,’ and Tang. lo In addition, Raz” and Tang and Tang” have published excellent surveys in the area of inspection allocation. Villalobos, Foster, and Disney’ used the concept of a Flexible Inspection System (FIS) for the generation of an inspection strategy applied to a serial manufacturing line. They modeled a unit passing through different stages and developed dynamic inspection strategies for its inspection. The model included the incorporation of time constraints and the use of information provided by defective inspection operations (type I and II) to determine a dynamic inspection strategy. They also showed that the problem of optimal inspection allocation is a particular instance of a partially observable Markov decision process (POMDP) whose cost function is piecewise linear.
3. Inspection Allocation Model To understand the inspection allocation model, consider a serial production system for the assembly of SMT components to PCBs. Each one of the workstations in the system is in charge of placing a given number of SMT components on the PCB within the cycle time, which will be assumed to be T seconds. In this system, an automated visual inspection (AVI) system is available after each assembly workstation. Thus, to prevent the overall system from slowing down, the inspection system should not take more than T seconds to inspect some or all of the components already assembled. Given the speed of currently available inspection systems and the dense population of modern PCBs, full-board inspection is not possible in most cases. Thus, a partial inspection list must be generated for each AVI station. This list must specify which components of a board will be inspected at each inspection station. Thus, the problem becomes one of selecting those components that should be inspected without exceeding the cycle time and while optimizing a global objective. The mix of components to inspect at each workstation
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Journal of Manufacturhg Systems Vol. 2o/ivo. 1 2001
for each PCB being processed is known as inspection routine generation. A typical objective to achieve in inspection routine generation is the minimization of total production costs. However, information on costs is often difficult to obtain. Thus, alternative objectives should be sought. In this paper, the maximization of an information gain factor (G) is proposed as the objective to optimize. The use of this factor will allow the implementation of efficient algorithms for inspection routine generation, a characteristic that is indispensable for the implementation of real-time inspection control.
at the beginning of the process may become defective at later stages of the manufacturing process. Let the inspection outcome be represented by the random variable Z8:j,which takes a value of 0 if the element (i,j) fails the inspection performed on it at stage k, and 1 otherwise. Let $j be defined as the probability that the element (ij) is nondefective at the k manufacturing stage or: ?fj =
Pr[M#;j= I]
(1)
Because the inspection operations will be assumed fallible, the following conditional probabilities describe the possible outcomes of the inspection process:
3.1 Nomenclature and Definitions To model the inspection allocation problem, it is necessary to introduce some terminology. This terminology parallels that introduced in Villalobos.’ A PCB is said to be composed of components (i.e., capacitors, resistors, etc.), and the components are said to be composed of elements (i.e., capacitor no. 1, resistor no. 3, etc.). It is assumed that components share common probabilistic characteristics. For instance, the elements of a component share the same inspection characteristics, such as probabilities of type I and II errors, and manufacturing characteristics, such as the probability of being placed or assembled in a defective manner. For each element and component, two different probabilities are defined: a long-term, or process-dependent, probability and a short-term, or unit, probability. The short-term probability is associated with the individual board and is attached to each element and updated in real time with the information provided by the inspection stations. The long-term probability is associated with the process and gives the probability of an element being placed in a defective manner or not placed at all. At the beginning of the assembly process (stage 0), short-term probabilities are the same as long-term probabilities. Let N be the number of components (resistors, capacitors, etc.) composed of L elements (1,2,3, etc.) and K be the number of stages in the assembly process, while k is a given manufacturing stage in the process. Thus, K represents an upper boundary for the value of k. Furthermore, define ~~~ as a binary random variable that represents the actual quality of elementj of component type i, or element (ij), at manufacturing stage k. This random variable takes the value of 0 if the element is defective and 1 if it is not defective. Notice that an element that is not defective
Pr[qj
=s
ai
=
lM,Tj
if
1
=m]
s=O,m=l
l-ai
if s=l,m=l
pi
ifs=l,m=O
l-pi
(2)
if s=O,m=O
If the inspection outcomes are known after each inspection operation, I;!~can be updated according to three different cases.‘** Case 1 The element (i,j) is inspected inspection operation:
and passes the
Case 2 The element (i,j) is inspected and fails the inspection operation: k-l
(4)
The element (i,j) is not inspected: (5) where ai and pi are the respective probabilities of an
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Journal of Manufacturing Systems Vol. 2omo. 1 2001
formed. This expected cost is given by the line that intercepts the y-axis at PC, + (1 - g)C,,, assuming that the inspection cost is negligible. Thus, for this scenario the optimal decision is to immediately accept, perform an inspection on, or immediately reject the element (ij) if I;::I $,rl, 5 I;: I riPj,r:I r,:,, respectively. I;:/I I$ I ryj represents the area in which the inspection of the component would provide additional information that would result in a lower expected cost. Therefore, performing the inspection of a component with a nondefective probability rrj I 75 . 5 raj . would result in an information gain. The shaded area of Figure 2 represents the information gain area. Given the linearity of the functions, it is clear from this figure that the maximal reduction in expected cost to be obtained by performing the inspection occurs at r:, (Villalobos’). Thus, performing an extra inspection on the component becomes more attractive the closer 5,: is to CT,.Therefore, the information gained is directly related to the reduction in expected cost if an extra inspection is performed and can be used as a surrogate measure of the cost reduction achieved by performing an extra inspection on any component. This information gain can be obtained for each component on the board before the inspection is performed. Once the individual inspection time requirements and information gain values are available for all elements, the determination of which components to inspect becomes a problem of inspection allocation. In particular, it becomes a problem of optimal control of partially observable Markov decision processes,’ which are inherently very difficult to solve. A simplification of the problem is to model it as a Zero-One Knapsack problem for each stage in which an inspection routine has to be generated. The formulation of the inspection allocation problem as a Knapsack problem is presented next.
Figure I Expected Cost as a Function of r:
inspection error type I and II for a type i component. Because ?1;,gives a probabilistic measure of each component’s quality, it can be used as the basis to calculate the information gain factor. 3.2 Information Gain To illustrate the concept of information gain, consider a board at the last stage of the assembly process (stage K) after the last assembly operation has been performed. At this point of the process, a decision must be made about whether the element (i,J should be inspected or not. If the element is not inspected, a decision about whether it is defective should be made with the information that is available. Villalobos’ proposed the following decision policy: declare the element (itj) as nondefective if $ > ryj , where c:, is the nondefective probability that would render a breakeven cost between declaring a component as defective (rejection) and as nondefective (acceptance). The cost function is depicted in Figure 1 where the yaxis gives the cost of making a decision (accept, reject, or inspect) and the x-axis gives the probability that the element (ij) is nondefective. C,, Ci,, C,, and C,+ are the costs associated with declaring a nondefective component as defective, a nondefective component as nondefective, a defective component as nondefective, and a defective component as defective, respectively. Of course, in a practical situation the costs C,, and C,. a would be negative (a profit) or zero. However, for the sake of completeness, they are included in the model. The expected acceptance cost is represented by the straight line that intercepts the y-axis at C,, while the expected rejection cost intercepts the y-axis at C,_a. Now consider the scenario where an additional inspection is performed on the element (i,j). Figure I depicts the expected cost if this inspection is per-
4. Knapsack Formulation Once the information gain per component has been determined, the inspection allocation problem can be expressed as: Given: A knapsack with capacity T (Total time available) and N components with L elements, where each element (i,j) is defined by the quantities Gij (the information gain for element i,j) and tij (the time to inspect element i,j).
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Journal of Manufacturing Systems Vol. 20040. 1 2001
Define: The quantity & to be a binary value such that YiJ= 0 if element i,j is not placed in the knapsack (not inspected), and YiJ= 1 if it is placed in the knapsack (inspected).
surrogate measure that would allow the computation of the information gain factor used in the inspection allocation model. First, the cost model parameters will be set based on the nondefective probabilities (rij> of the components so that the classification errors are minimized. Second an evaluation of the inspection allocation system will be conducted using the cost model parameters found. Finally, a quantitative comparison between the inspection allocation system presented in this paper and a random inspection allocation approach will be presented. Two simple cases will be used to represent two different types of printed circuit boards:
Then, the problem becomes: Maximize the Total Information Gain (Z), where: Z = i $ Gi,jy,,j i=l,=I
(6)
Subject to: g&jq,j
i=lj=1 ’
ST
CASE A: This unit consists of three different component types: A, B, and C. The nondefective probabilities (rij) of these components are 0.5, 0.8, and 0.9, respectively. Components A, B, and C have 100 elements each.
(7)
yl,j = 0 or Yi j = 1 Although optimal solutions can be obtained by using existing algorithms, the required computational time restricts their use in the context of the inspection allocation strategy described in this paper (online inspection). On the other hand, near-optimal solutions that require feasible computational times can be obtained using heuristics. For example, the greedy heuristic proposed by Askin and Standridgei6 sorts all elements in the knapsack in the order of their gain-to-cost ratio and incrementally assigns these elements to the knapsack while sufficient capacity (T) exists. A slight modification of this heuristic was used to determine the components to be inspected. For a detailed explanation of this heuristic and its results, the reader is referred to the Appendix of this paper. To model the problem using the Zero-One Knapsack approach, the cost parameters for each stage need to be determined. Because these costs are difficult to obtain, the approach proposed in this paper is to set these costs to values that would optimize a global measure, such as the total number of inspection errors, by maximizing the information gain. Two different approaches to setting these costs are presented in Section 5.
CASE B: This unit consists of three different component types: X, Y, and Z. The nondefective probabilities (~ij> of these components are 0.5, 0.8, and 0.9, respectively. Components X andY have 50 elements each, and component Z has 200 elements. These nondefective probabilities were selected to test the different approaches under pessimistic scenarios. 5.1 Cost Model Parameter Optimization To simplify the cost model introduced in Section 3.2, let R = C,, A = CD,and C,_, = Cr_, = 0. The first step for parameter optimization is to set the values of the cost coefficients A and R to accomplish a given objective. For example, if it is desired to maximize the probability of inspection for a component whose nondefective probability (rid) is P’ (see Figure 2), then the values of A and R should be set so that point P’ lies exactly below the point of maximum information gain. Then the best location for the accept and reject lines on a Cartesian coordinate system is such that their intersection lies on the X= P’ line. However, there are an infinite number of combinations for A and R that intersect on this line. Thus, a base cost, such as M in Figure 2, must be specified to standardize the cost coefficients across all the different components of a unit. For instance, suppose that points P’ and P in Figure 2 correspond to nondefective probabilities for two different components. Thus, the cost coefficients A and R should
5. Computer Simulation Results In the previous section, it was assumed that the cost parameters (C,, Cla, C,, Ci+) were known. In this section, that assumption is relaxed by finding a
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Journal of Manufacturing Vol. 2omo. 1
Systems
2001
Cost Model Parameters
Table I for Separate and Aggregate Approaches (M=lO)
CASE A CASE A CASE A CASE B CASE B CASE B (Q) = 0.5 (rii) = 0.8 (Q) = 0.9 (Q) = 0.5 (rii) = 0.8 (F-J = 0.9
be set such that they represent the same information gain if either of these two components is inspected. This can be accomplished by setting the intersection of the accept and reject lines for P’ and P to yield, respectively, a specified common maximum information gain (M). If the value of A4 is available, then the cost coefficients (A and R) can be obtained by using Eqs. (8) and (9).
(lqj)
M
=(l-
20
50
100
20
50
100
Separate Approach R
20
12.5
11.12
20
12.5
11.12
Aggregate Approach A 37.03
37.03
37.03
55.5
55.5
55.5
Aggregate Approach R 13.7
13.7
13.7
12.2
12.2
12.2
gle value of A and R for all the components of a unit. The former approach will be termed the separate approach, while the latter will be referred to as the aggregate approach for the remainder of this paper. These two approaches were evaluated used the information previously provided as Case A and Case B. Table I presents the cost model parameters for both the separate and the aggregate approach for cases A and B. Note that in the case of the separate approach, the A and R coefficients are different for each component and are determined by the nondefective probability (~ij> of each component. These parameters, on the other hand, are equal for every component in the aggregate approach and were determined using the average of the group’s nondefective probability (rij). To determine which approach (separate or aggregate) minimizes the number of classification errors, an evaluation experiment was performed. Case A and Case B were used in this experiment for a simulated, fallible inspection system with OL= 0.05 and B = 0.02 that inspects 40% to 80% of all elements. A weighted average was used to model the aggregate approach. The results of this experiment were analyzed by using the Analysis of Variance (ANOVA) technique and the Least Significant Difference (LSD) multiple range test. Figures 3 and 4 show the results of this experiment for Cases A and B, respectively. The results from the Analysis of Variance indicate (with 95% confidence) that there is a statistically significant difference between the aggregate approach and separate approach for both cases (A and B). In both cases, the aggregate approach resulted in less classification errors than the separate approach. Thus, to reduce the number of classification errors, an aggregate cost model should
Figure 2 Modified Cost Model
A = Max.Gain
Separate Approach A
(8)
(9)
The value of A4 in Eqs. (8) and (9) can be set at any value greater than zero as long as all the components of a unit have the same value. The advantage of using this approach to determine the cost model parameters is that the expected information gain calculated for different components (or elements) is standardized to a value that depends only on M and not on the individual nondefective probabilities (Q). One important issue to consider is whether to determine the values of A and R for each component of a unit separately or as a group; that is, whether to use the nondefective probability of the components to compute a value for A and R for each component, or to aggregate these probabilities to determine a sin-
18
Journal of Manufacturing Systems Vol. 2a/No. 1 2001
44
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Figure 3 Aggregate vs. Separate Cost Model Mean Response (Case A)
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Figure 4 Aggregate vs. Separate Cost Model Mean Response(Case B)
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Figure 5 Errors for Different Summary
I
AVG
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Statistics (Case A)
Total Classification
be used for units with elements having different nondefective probabilities. However, this experiment evaluated the aggregate approach using only one summary statistic, the Weighted Average. Thus, further analyses of other summary statistics were performed to evaluate their effectiveness. The summary statistics that were evaluated included Maximum (p,,&, Minimum timi,,), Simple Average (p,&, and a Weighted Average (P,_J. The measure of performance selected for this comparison was the total number of classification errors. Three scenarios were evaluated: classification without inspection, classification after one fallible inspec-
Figure 6 Errors for Different Summary
Statistics (Case B)
tion, and classification after two equally fallible inspections. The inspection system simulated was set to inspect 60% of all elements with the probability for a type I and type II error of OL= 0.05 and p = 0.02, respectively. Figures 5 and 6 present the results of the second experiment. The experimental results presented in Figures 5 and 6 indicate that the Minimum @min) and the Weighted Average (s,,,,,J summary statistics yielded a lower classification error rate than the Simple Average. Further analysis is necessary to assess if this difference in classification errors is statistically significant. A LSD multiple-range analysis of the
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Journal of Manufacturing Systems Vol. 2a/No. 1 2001
Total dassification error
Total classification error
60
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Figure 7 Errors for Zero, One, and Two Inspections
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(Case A)
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Figure 8 Errors for Zero, One, and ‘Ibvo Inspections
(Case B)
5.2 Quantitative Comparison with Random Allocation This section presents a quantitative comparison between the inspection allocation system proposed in this paper and a random inspection allocation approach. The latter is defined as a random selection of elements to inspect without concern for the expected information gain. However, this randomly allocated inspection will use the same cost model, discussed in previous sections, as the information-based allocation. Thus, the evaluation discussed in this section serves as a comparison between the two different allocation approaches and not as an evaluation of the effectiveness of the cost model because both approaches are based on the same cost model. Figures 9 and IO present the results of the evaluation. This experiment considered single and double fallible inspections (a = 0.05 and l3 = 0.02) and explored the effects that the constraint level (i.e., time) has on the effectiveness of the random and information-based approaches. Note that for the case of double inspection, the same elements may or may not be inspected on the second inspection. This is because, like the first inspection, the second allocation of elements utilizes the expected information gain, which is a function of the updated (or short-term) nondefective probabilities. The experimental results presented in Figures 9 and 10 indicate that there is a significant difference between random and information-based allocation for time-constrained inspections in both Case A and Case B. A LSD multiple-range test established, with
experimental data revealed (with 95% confidence) that there is a statistically significant difference between the Simple Average andp,, orp,% for Case B. However, because this difference was not detected for Case A, the superiority of pmi,, or pnvryg over pmg cannot be confirmed. Nevertheless, because the Minimum (Pmi,,)summary statistic consistently produced the lowest number of classification errors for both cases and it is the simplest statistic to calculate, this statistic will be used throughout the remainder of this paper. In addition to evaluating the different summary statistics, this second experiment was used to estimate the added value of information-based allocated inspection by comparing three different classification scenarios. These scenarios are: classification without inspection, classification after one fallible inspection, and classification after two equally fallible inspections. Figure 7 presents the experimental results for Case A. This figure confirms the anticipated results regarding the number of classification errors for the three scenarios. Furthermore, a 95% LSD multiple-range analysis indicates that three significantly distinct groups, which correspond to the three scenarios, can be identified in the experimental data. Similar results were obtained for Case B. Figure 8 shows these results. The experimental results obtained so far are promising and represent a good indication of the system’s capability. However, the system must be compared to alternative approaches. This comparison is presented next.
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Journal of Manufacturing Systems Vol. 2ONo. 1 2001
30
”
20
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10
-’
Total
classification error
Total
i
error
r 0
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Figure 9 Classification Errors for Random (R) and Information-Based (I) Allocation (Case A)
Figure IO Classification Errors for Random (R) and Information-Based (I) Allocation (Case B)
95% confidence, that the difference is statistically significant for both cases. Furthermore, this difference seems to decrease in a linear fashion as the allocation problem becomes less constrained. The significant difference between both approaches justifies the use of information-based inspection allocation.
tion and to present its real-time implementation within the environment of electronics assembly. However, the paper also opened a series of questions that will have to be addressed as part of future research. For instance, the benefits provided by the method are influenced by the total inspection time available and the probabilities of defective production and/or inspection. To determine the exact benefit of the method to a particular situation is necessary to either obtain structural analytical results or to conduct careful simulation analysis of these factors.
6. Conclusions
and Future Research
The information gain method presented in this paper allows the generation of real-time inspection routines. In evaluating the performance of the information-based inspection allocation strategies described in this paper, it was found that the approach presented yields feasible solutions to the fallible-inspection allocation problem. This information-based approach is composed of two fundarnental parts. First, an expected information-gain value is estimated by using a cost model. Then, a selection of elements that maximizes the overall expected information gain is performed. With respect to the first part, the amount of classification errors attained by the information gain approach was found to be significantly lower than a random allocation approach. Thus, the information-based approach presented in this paper yields solutions to the fallible-inspection allocation problem that are not only feasible but also advantageous because the number of classification errors is significantly reduced. The main objective of this paper was to introduce the concept of information-based inspection alloca-
Acknowledgments We gratefully acknowledge support from the National Science Foundation (NSF Grant DMI9502897) and Thomson Consumer Electronics for the realization of this research. The authors also would like to thank Vernon Dickson and the anonymous referees for their input in the preparation of the final version of this paper. References 1. J.R. Villalobos, “Inspection Models for Flexible Inspection Systems,’ PhD dissertation (unpublished) (College Station, TX: Dept. of Industrial Engg., Texas A&M Univ., 199 1). 2. J.R. Villalobos, J.W. Foster, and R.L. Disney, “Flexible Inspection Systems for Serial Multistage Production Systems,” IIE Tmns. (~25, 1993), ~~16-26. 3. J.R. Villalobos, V. Ramachandran, and V Baeza, “Inspection Routine Generation for a Flexible Inspection System,” Working Paper (Tempe, AZ: Dept of Industrial Engg., Arizona State Univ.). 4. J.R. Villalobos and A. Verduzco, “Integration of Quality and Process Planning Activities in SMD Assembly,” Proc. of 6th Industrial Engg. Research Conf., Miami Beach, FL, May 17-l 8, 1997. 5. G.F. Lindsay and A.B. Bishop, “Allocation of Screening Inspection
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Journal of Manufacturing Vol. 2a/No. 1 2001
Systems
Effort: A Dynamic Programming Approach,” Mgmt. Science (~10, 1964), ~~342-352. 6. PM. Pruzan and J.T. Jackson, “A Dynamic Programming Application in Production Line Inspection,” Technometrics (~9, 1967), ~~73-8 1. 7. G. Eppen and E. Hurst, “Optimal Location of Stations in a Serial Multistage Production Process,” Mgmt. Science (~20, 1974), ppl194-1200. 8. R. Britney, “Optimal Screening Plans for Non-serial Production Systems,” Mgmt. Science (~18, 1972), ~~550-559. 9. DP Ballou and H.L. Pazer, “The Impact of Inspector Fallibility on the Inspection on the Inspection Policy in Serial Production Systems,” Mgmt. Science (~28, 1982), ~~387-399. 10. K. Tang, “Design of Multi-Stage Screening Procedures for a Serial Production System,” European Journal of Operational Research (~52, 1990), ~~280-290. 11. T. Raz, “A Survey of Models for Allocating Inspection Effort in Multi-Stage Production Systems,” Journal qf Quality Technology (~18, 1986), ~~239-247. 12. K. Tang and J. Tang, “Design of Screening Procedures: A Review,’ Journal of Quality Technology (~26, 1994), ~~209-226. 13. J. Larsen and S. Dyer, “Using Extensive Form Analysis to Solve Partially Observable Markov Decision Problems,” Working Paper 90/91-33 (Dept. of Mgmt. Sciences and Information Systems, The University of Texas at Austin, 1991). 14. G.E. Monahan, “A Survey of Partially Observed Markov Decision Processes: Theory, Models, and Algorithms,” Mgmt. Science (~28, 1982), ~~1-16. 15. W.S. Lovejoy, “Computationally Feasible Bounds for Partially Observed Markov Decision Processes,” Opemtions Research (~39, 1991), ~~162-175. 16. R.G. Askin and C.R. Standridge, Modeling and Analysis of Manufacturing Systems (New York: John Wiley & Sons, 1993).
pass reduces the probability of assigning unusuallylarge-gain elements instead of a combination of elements that yield a higher combined gain. The steps of the two-pass greedy heuristic are:
Authors’ Biographies
A quantitative comparison of the Knapsack greedy heuristic and the modified two-pass greedy heuristic with respect to optimal allocation was conducted to evaluate the efficiency of these two allocation approaches. One thousand boards, with 10 elements each, were generated for this evaluation. Table Al presents the results of this evaluation. These results were obtained by comparing the total gain for optimal allocation (achieved by exhaustive combinatorial searches) with the total gains of both the greedy and the two-pass greedy heuristics. The results are presented in terms of the difference in total expected information gain, maximum percentage error (worst case), average total expected information gain, and average percentage error for the maximum. The results in Table AI indicate that the two-pass greedy heuristic yielded a total gain closer to the optimal total gain than the greedy heuristic yielded.
1. Determine a gain-to-cost ratio for all elements. 2. Sort all elements in decreasing order of their individual gain-to-cost ratio. 3. Assign the first element from the sorted list and subtract its time requirement from the total available time. 4. Repeat step 3 until no assignments can be made. 5. For every element assigned to the knapsack, repeat: 5.1 Extract the element from the knapsack. 5.2Increase the available time by the amount of time required for this element. 5.3 Repeat steps 2 through 4. 5.4 If the combined gain at this point is greater than the combined gain at step 5, keep this knapsack assignment; otherwise, discard this knapsack solution.
Adan Verduzco is a test development engineer with Tyco Electronics in Mesquite, TX. He holds a BS in electrical engineering and an MS in manufacturing engineering from the University of Texas at El Paso. J. Rene Villalobos is an associate professor with Arizona State University. His areas of research include manufacturing systems, on-line quality control, and applied operations research. He holds a BS in mechanical engineering from the Tecnologico de Chihuahua in Mexico, an MS in industrial engineering from the University of Texas at El Paso, and a PhD in industrial engineering from Texas A&M University. Benjamin Vega is an operations analyst with Capital One in Richmond, VA. He holds a BS in industrial engineering from the University of Texas at El Paso and an MS in industrial engineering from Arizona State University.
Appendix: Two-Pass Greedy Heuristic Once all elements have been sorted in order of their gain-to-cost ratio and assigned to the knapsack given the capacity constraints, a second pass is performed to improve the current solution. This is achieved by extracting one element from the knapsack at a time and executing the greedy heuristic again. This second Evaluation
Table AI of Knapsack Heuristic
Maximum Difference in Total Gain (gain units)
Maximum Percentage Difference
Average Difference in Total Gain (gain units)
Average Percentage Difference
Greedy
4
0.6%
1
0.2%
0.228 0.012
0.015%
Two-pass greedy
Knapsack Heuristic
22
0.001%
0 SMe
a
Information-Based Inspection Allocation for Real-Time Inspection Systems Adan Verduzco, Tyco Electronics, Mesquite, Texas, USA J. Rene Villalobos, Dept. of Industrial Engineering, Arizona State University, Tempe, Arizona, USA Benjamin Vega, Capital One, Richmond, Virginia, USA
Abstract
bly line, a situation that is difficult to achieve when human inspectors are utilized. Another example of added flexibility is the possibility of using different inspection routines for the same board as it advances along the assembly line. For instance, the inspection strategy, or the way the unit is inspected, for each of the units in production would not be fixed but would be determined as the unit advanced through the production line. That is, the inspection strategy would be determined using the information about the manufacturing process, the inspection process, and the time available to perform the inspection. This concept of not having a fixed inspection routine but to generate the routine as the unit advances through the assembly, accumulating information, is termed Flexible Inspection and has been explored by Villalobos,’ Villalobos, Foster, and Disney,2 and Villalobos, Ramachandran, and Baeza.3 While the inspection routine generation problem can be considered as a particular case of inspection effort allocation, it should be treated differently because inspection effort allocation is dynamic and on-line. This paper departs from previous approaches by proposing the generation of the inspection routine based on the maximization of the information gained by performing an extra inspection on each one of the elements on a printed circuit board (PCB). Specifically, this approach considers inprocess fallible AVI systems used in surface-mounted technology (SMT) PCB manufacturing processes, where cycle time restrictions call for the partial inspection of the product. The strategy proposed in this paper enables AVI systems to perform efficient in-process partial inspections. In addition, the inspection routine generated takes into account the quality system information needs that monitor, in real time, the different sources of variability present in the SMD assembly process. The system that generates the inspection routine and the quality moni-
The use of real-time Automated Visual Inspection (AU) in the electronics assembly industry is subject to cycle time constraints determined by the manufacturing system. These constraints must be met to maintain the nominal production rate. A problem thus created is determining which components to inspect at each one of the AVI stations such that the inspection time available is used in an optimal way. This paper presents a real-time inspection allocation that is based on the information gained by inspecting one additional component. The selection of which components to inspect is modeled as an information maximization problem. A modified Knapsack greedy heuristic is used to find near-optimal solutions to this optimization problem within the required time constraints.
Keywords: Inspection,
Inspection Allocation, Automated Visual Information Gain, Flexible Inspection Systems
1. Introduction The miniaturization of electronics devices has made the use of Automated Visual Inspection (AVI) systems a necessity within the electronics assembly industry. This is particularly evident in the assembly of surface-mounted devices (SMD), where human inspectors are being replaced rapidly by AVI systems. The use of AVI presents some obvious advantages over the human inspectors, such as less inspection subjectivity. However, it also presents some other less evident advantageous characteristics. One of these characteristics is the potential use, by these systems, of the inspection information for the continuous improvement of the assembly process. This could be achieved by using the inspection information as input for the statistical monitoring of the different sources of variability in the assembly process. Another important advantage provided by the use of AVI systems in the SMD assembly process is the added flexibility that they lend to the overall production environment. For instance, several products could be assembled sequentially in the same assem-
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Journal of Manufacturing Vol. 2orNo. 1 200 1
Systems
Many authors have addressed the problem of finding an optimal solution for POMDPs. The problem has been found to be computationally hard in the strong sense. l3 For a detailed discussion of this subject, the reader is referred to Monahan14 and Lovej oy. l5 The problem of finding an optimal solution for the inspection routine generation is not only very complex, given that the problem is a POMDP (Villalobos, Foster, and Disney’), but its complexity is compounded by inspection time constraints. Thus, generating real-time, optimal solutions even for a small-sized problem is not feasible. Thus, heuristic approaches must be used to generate inspection routines of meaningful size. An example of heuristic approaches to solve the inspection routine generation problem is given by Villalobos, Ramachandran, and Baeza.’ This paper proposes a heuristic approach to inspection allocation that departs from previous approaches by using a figure of merit, to be introduced as “information gain,” to perform the inspection allocation. This figure of merit does not necessarily rely on the traditional cost parameters but instead relies on factors such as the minimization of inspection errors.
toring systems are components of an Integrated Quality Environment (IQE) for SMD assembly. For a more detailed description of this integrated environment, the reader is referred to Villalobos and Verduzco.4 The main objective of this paper is to introduce the methodology behind the informationbased inspection allocation method utilized to generate the inspection routine as the PCB advances along the assembly line. In particular, near-optimal solutions to the allocation problem are achieved within the given time constraint by applying a modified Knapsack greedy heuristic.
2. Related Works One of the earliest works to address the problem of the optimal allocation of inspection was written by Lindsay and Bishop.5 They proved that to determine the minimum cost per unit either none or an entire batch should be inspected. Pruzan and Jackson6 used the zero- 100% result of Lindsay and Bishop’s work to develop two models for optimal For both models, they inspection allocation. assumed perfect inspection and the dependence of the inspection cost on the number of stations between two inspection points. Eppen and Hurst’ were the first to model the allocation problems assuming imperfect inspection. Their model aims to minimize the total expected cost. They conclude that the total cost is a concave, piecewise-linear function of the state variable (probability of a nondefective unit at a stage). Other papers related to optimal inspection allocation include Britney,* Ballou and Pazer,’ and Tang. lo In addition, Raz” and Tang and Tang” have published excellent surveys in the area of inspection allocation. Villalobos, Foster, and Disney’ used the concept of a Flexible Inspection System (FIS) for the generation of an inspection strategy applied to a serial manufacturing line. They modeled a unit passing through different stages and developed dynamic inspection strategies for its inspection. The model included the incorporation of time constraints and the use of information provided by defective inspection operations (type I and II) to determine a dynamic inspection strategy. They also showed that the problem of optimal inspection allocation is a particular instance of a partially observable Markov decision process (POMDP) whose cost function is piecewise linear.
3. Inspection Allocation Model To understand the inspection allocation model, consider a serial production system for the assembly of SMT components to PCBs. Each one of the workstations in the system is in charge of placing a given number of SMT components on the PCB within the cycle time, which will be assumed to be T seconds. In this system, an automated visual inspection (AVI) system is available after each assembly workstation. Thus, to prevent the overall system from slowing down, the inspection system should not take more than T seconds to inspect some or all of the components already assembled. Given the speed of currently available inspection systems and the dense population of modern PCBs, full-board inspection is not possible in most cases. Thus, a partial inspection list must be generated for each AVI station. This list must specify which components of a board will be inspected at each inspection station. Thus, the problem becomes one of selecting those components that should be inspected without exceeding the cycle time and while optimizing a global objective. The mix of components to inspect at each workstation
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Journal of Manufacturhg Systems Vol. 2o/ivo. 1 2001
for each PCB being processed is known as inspection routine generation. A typical objective to achieve in inspection routine generation is the minimization of total production costs. However, information on costs is often difficult to obtain. Thus, alternative objectives should be sought. In this paper, the maximization of an information gain factor (G) is proposed as the objective to optimize. The use of this factor will allow the implementation of efficient algorithms for inspection routine generation, a characteristic that is indispensable for the implementation of real-time inspection control.
at the beginning of the process may become defective at later stages of the manufacturing process. Let the inspection outcome be represented by the random variable Z8:j,which takes a value of 0 if the element (i,j) fails the inspection performed on it at stage k, and 1 otherwise. Let $j be defined as the probability that the element (ij) is nondefective at the k manufacturing stage or: ?fj =
Pr[M#;j= I]
(1)
Because the inspection operations will be assumed fallible, the following conditional probabilities describe the possible outcomes of the inspection process:
3.1 Nomenclature and Definitions To model the inspection allocation problem, it is necessary to introduce some terminology. This terminology parallels that introduced in Villalobos.’ A PCB is said to be composed of components (i.e., capacitors, resistors, etc.), and the components are said to be composed of elements (i.e., capacitor no. 1, resistor no. 3, etc.). It is assumed that components share common probabilistic characteristics. For instance, the elements of a component share the same inspection characteristics, such as probabilities of type I and II errors, and manufacturing characteristics, such as the probability of being placed or assembled in a defective manner. For each element and component, two different probabilities are defined: a long-term, or process-dependent, probability and a short-term, or unit, probability. The short-term probability is associated with the individual board and is attached to each element and updated in real time with the information provided by the inspection stations. The long-term probability is associated with the process and gives the probability of an element being placed in a defective manner or not placed at all. At the beginning of the assembly process (stage 0), short-term probabilities are the same as long-term probabilities. Let N be the number of components (resistors, capacitors, etc.) composed of L elements (1,2,3, etc.) and K be the number of stages in the assembly process, while k is a given manufacturing stage in the process. Thus, K represents an upper boundary for the value of k. Furthermore, define ~~~ as a binary random variable that represents the actual quality of elementj of component type i, or element (ij), at manufacturing stage k. This random variable takes the value of 0 if the element is defective and 1 if it is not defective. Notice that an element that is not defective
Pr[qj
=s
ai
=
lM,Tj
if
1
=m]
s=O,m=l
l-ai
if s=l,m=l
pi
ifs=l,m=O
l-pi
(2)
if s=O,m=O
If the inspection outcomes are known after each inspection operation, I;!~can be updated according to three different cases.‘** Case 1 The element (i,j) is inspected inspection operation:
and passes the
Case 2 The element (i,j) is inspected and fails the inspection operation: k-l
(4)
The element (i,j) is not inspected: (5) where ai and pi are the respective probabilities of an
15
Journal of Manufacturing Systems Vol. 2omo. 1 2001
formed. This expected cost is given by the line that intercepts the y-axis at PC, + (1 - g)C,,, assuming that the inspection cost is negligible. Thus, for this scenario the optimal decision is to immediately accept, perform an inspection on, or immediately reject the element (ij) if I;::I $,rl, 5 I;: I riPj,r:I r,:,, respectively. I;:/I I$ I ryj represents the area in which the inspection of the component would provide additional information that would result in a lower expected cost. Therefore, performing the inspection of a component with a nondefective probability rrj I 75 . 5 raj . would result in an information gain. The shaded area of Figure 2 represents the information gain area. Given the linearity of the functions, it is clear from this figure that the maximal reduction in expected cost to be obtained by performing the inspection occurs at r:, (Villalobos’). Thus, performing an extra inspection on the component becomes more attractive the closer 5,: is to CT,.Therefore, the information gained is directly related to the reduction in expected cost if an extra inspection is performed and can be used as a surrogate measure of the cost reduction achieved by performing an extra inspection on any component. This information gain can be obtained for each component on the board before the inspection is performed. Once the individual inspection time requirements and information gain values are available for all elements, the determination of which components to inspect becomes a problem of inspection allocation. In particular, it becomes a problem of optimal control of partially observable Markov decision processes,’ which are inherently very difficult to solve. A simplification of the problem is to model it as a Zero-One Knapsack problem for each stage in which an inspection routine has to be generated. The formulation of the inspection allocation problem as a Knapsack problem is presented next.
Figure I Expected Cost as a Function of r:
inspection error type I and II for a type i component. Because ?1;,gives a probabilistic measure of each component’s quality, it can be used as the basis to calculate the information gain factor. 3.2 Information Gain To illustrate the concept of information gain, consider a board at the last stage of the assembly process (stage K) after the last assembly operation has been performed. At this point of the process, a decision must be made about whether the element (i,J should be inspected or not. If the element is not inspected, a decision about whether it is defective should be made with the information that is available. Villalobos’ proposed the following decision policy: declare the element (itj) as nondefective if $ > ryj , where c:, is the nondefective probability that would render a breakeven cost between declaring a component as defective (rejection) and as nondefective (acceptance). The cost function is depicted in Figure 1 where the yaxis gives the cost of making a decision (accept, reject, or inspect) and the x-axis gives the probability that the element (ij) is nondefective. C,, Ci,, C,, and C,+ are the costs associated with declaring a nondefective component as defective, a nondefective component as nondefective, a defective component as nondefective, and a defective component as defective, respectively. Of course, in a practical situation the costs C,, and C,. a would be negative (a profit) or zero. However, for the sake of completeness, they are included in the model. The expected acceptance cost is represented by the straight line that intercepts the y-axis at C,, while the expected rejection cost intercepts the y-axis at C,_a. Now consider the scenario where an additional inspection is performed on the element (i,j). Figure I depicts the expected cost if this inspection is per-
4. Knapsack Formulation Once the information gain per component has been determined, the inspection allocation problem can be expressed as: Given: A knapsack with capacity T (Total time available) and N components with L elements, where each element (i,j) is defined by the quantities Gij (the information gain for element i,j) and tij (the time to inspect element i,j).
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Journal of Manufacturing Systems Vol. 20040. 1 2001
Define: The quantity & to be a binary value such that YiJ= 0 if element i,j is not placed in the knapsack (not inspected), and YiJ= 1 if it is placed in the knapsack (inspected).
surrogate measure that would allow the computation of the information gain factor used in the inspection allocation model. First, the cost model parameters will be set based on the nondefective probabilities (rij> of the components so that the classification errors are minimized. Second an evaluation of the inspection allocation system will be conducted using the cost model parameters found. Finally, a quantitative comparison between the inspection allocation system presented in this paper and a random inspection allocation approach will be presented. Two simple cases will be used to represent two different types of printed circuit boards:
Then, the problem becomes: Maximize the Total Information Gain (Z), where: Z = i $ Gi,jy,,j i=l,=I
(6)
Subject to: g&jq,j
i=lj=1 ’
ST
CASE A: This unit consists of three different component types: A, B, and C. The nondefective probabilities (rij) of these components are 0.5, 0.8, and 0.9, respectively. Components A, B, and C have 100 elements each.
(7)
yl,j = 0 or Yi j = 1 Although optimal solutions can be obtained by using existing algorithms, the required computational time restricts their use in the context of the inspection allocation strategy described in this paper (online inspection). On the other hand, near-optimal solutions that require feasible computational times can be obtained using heuristics. For example, the greedy heuristic proposed by Askin and Standridgei6 sorts all elements in the knapsack in the order of their gain-to-cost ratio and incrementally assigns these elements to the knapsack while sufficient capacity (T) exists. A slight modification of this heuristic was used to determine the components to be inspected. For a detailed explanation of this heuristic and its results, the reader is referred to the Appendix of this paper. To model the problem using the Zero-One Knapsack approach, the cost parameters for each stage need to be determined. Because these costs are difficult to obtain, the approach proposed in this paper is to set these costs to values that would optimize a global measure, such as the total number of inspection errors, by maximizing the information gain. Two different approaches to setting these costs are presented in Section 5.
CASE B: This unit consists of three different component types: X, Y, and Z. The nondefective probabilities (~ij> of these components are 0.5, 0.8, and 0.9, respectively. Components X andY have 50 elements each, and component Z has 200 elements. These nondefective probabilities were selected to test the different approaches under pessimistic scenarios. 5.1 Cost Model Parameter Optimization To simplify the cost model introduced in Section 3.2, let R = C,, A = CD,and C,_, = Cr_, = 0. The first step for parameter optimization is to set the values of the cost coefficients A and R to accomplish a given objective. For example, if it is desired to maximize the probability of inspection for a component whose nondefective probability (rid) is P’ (see Figure 2), then the values of A and R should be set so that point P’ lies exactly below the point of maximum information gain. Then the best location for the accept and reject lines on a Cartesian coordinate system is such that their intersection lies on the X= P’ line. However, there are an infinite number of combinations for A and R that intersect on this line. Thus, a base cost, such as M in Figure 2, must be specified to standardize the cost coefficients across all the different components of a unit. For instance, suppose that points P’ and P in Figure 2 correspond to nondefective probabilities for two different components. Thus, the cost coefficients A and R should
5. Computer Simulation Results In the previous section, it was assumed that the cost parameters (C,, Cla, C,, Ci+) were known. In this section, that assumption is relaxed by finding a
17
Journal of Manufacturing Vol. 2omo. 1
Systems
2001
Cost Model Parameters
Table I for Separate and Aggregate Approaches (M=lO)
CASE A CASE A CASE A CASE B CASE B CASE B (Q) = 0.5 (rii) = 0.8 (Q) = 0.9 (Q) = 0.5 (rii) = 0.8 (F-J = 0.9
be set such that they represent the same information gain if either of these two components is inspected. This can be accomplished by setting the intersection of the accept and reject lines for P’ and P to yield, respectively, a specified common maximum information gain (M). If the value of A4 is available, then the cost coefficients (A and R) can be obtained by using Eqs. (8) and (9).
(lqj)
M
=(l-
20
50
100
20
50
100
Separate Approach R
20
12.5
11.12
20
12.5
11.12
Aggregate Approach A 37.03
37.03
37.03
55.5
55.5
55.5
Aggregate Approach R 13.7
13.7
13.7
12.2
12.2
12.2
gle value of A and R for all the components of a unit. The former approach will be termed the separate approach, while the latter will be referred to as the aggregate approach for the remainder of this paper. These two approaches were evaluated used the information previously provided as Case A and Case B. Table I presents the cost model parameters for both the separate and the aggregate approach for cases A and B. Note that in the case of the separate approach, the A and R coefficients are different for each component and are determined by the nondefective probability (~ij> of each component. These parameters, on the other hand, are equal for every component in the aggregate approach and were determined using the average of the group’s nondefective probability (rij). To determine which approach (separate or aggregate) minimizes the number of classification errors, an evaluation experiment was performed. Case A and Case B were used in this experiment for a simulated, fallible inspection system with OL= 0.05 and B = 0.02 that inspects 40% to 80% of all elements. A weighted average was used to model the aggregate approach. The results of this experiment were analyzed by using the Analysis of Variance (ANOVA) technique and the Least Significant Difference (LSD) multiple range test. Figures 3 and 4 show the results of this experiment for Cases A and B, respectively. The results from the Analysis of Variance indicate (with 95% confidence) that there is a statistically significant difference between the aggregate approach and separate approach for both cases (A and B). In both cases, the aggregate approach resulted in less classification errors than the separate approach. Thus, to reduce the number of classification errors, an aggregate cost model should
Figure 2 Modified Cost Model
A = Max.Gain
Separate Approach A
(8)
(9)
The value of A4 in Eqs. (8) and (9) can be set at any value greater than zero as long as all the components of a unit have the same value. The advantage of using this approach to determine the cost model parameters is that the expected information gain calculated for different components (or elements) is standardized to a value that depends only on M and not on the individual nondefective probabilities (Q). One important issue to consider is whether to determine the values of A and R for each component of a unit separately or as a group; that is, whether to use the nondefective probability of the components to compute a value for A and R for each component, or to aggregate these probabilities to determine a sin-
18
Journal of Manufacturing Systems Vol. 2a/No. 1 2001
44
4,
j
_(_ ^.
:
38_I
Total classification error
].
_I
.
3S_;
_^_.
.
:
..“..
.
.
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.
:_
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:_
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:_
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.
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.; ‘
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. .
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.; _ . .
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_:
:
:
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Figure 3 Aggregate vs. Separate Cost Model Mean Response (Case A)
108
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..;
1
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. . :.
25;
Figure 4 Aggregate vs. Separate Cost Model Mean Response(Case B)
1
-
I
:
aa Total classification error
68
48
Total classification error
-
I
. . . . .
_.;.
I.
,
60
,: 30
28
-
1
1
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I MN
0
I WAVG
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Total Classification
Figure 5 Errors for Different Summary
I
AVG
I
MAX
I
I
MIN
WAVG
Summary statistic
Statistics (Case A)
Total Classification
be used for units with elements having different nondefective probabilities. However, this experiment evaluated the aggregate approach using only one summary statistic, the Weighted Average. Thus, further analyses of other summary statistics were performed to evaluate their effectiveness. The summary statistics that were evaluated included Maximum (p,,&, Minimum timi,,), Simple Average (p,&, and a Weighted Average (P,_J. The measure of performance selected for this comparison was the total number of classification errors. Three scenarios were evaluated: classification without inspection, classification after one fallible inspec-
Figure 6 Errors for Different Summary
Statistics (Case B)
tion, and classification after two equally fallible inspections. The inspection system simulated was set to inspect 60% of all elements with the probability for a type I and type II error of OL= 0.05 and p = 0.02, respectively. Figures 5 and 6 present the results of the second experiment. The experimental results presented in Figures 5 and 6 indicate that the Minimum @min) and the Weighted Average (s,,,,,J summary statistics yielded a lower classification error rate than the Simple Average. Further analysis is necessary to assess if this difference in classification errors is statistically significant. A LSD multiple-range analysis of the
19
Journal of Manufacturing Systems Vol. 2a/No. 1 2001
Total dassification error
Total classification error
60
1: 1.
6.
4o
30
i
.
” .;
.’
I
.
‘1 2.
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.
.
.
.
.
.
.
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.;
.
.
.
.
.
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. ,
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Figure 7 Errors for Zero, One, and Two Inspections
Classification
(Case A)
.
-
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_
._...__._..: i
1 Number of inspections
-
.
.
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t:
o
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-
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.
;
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2
Figure 8 Errors for Zero, One, and ‘Ibvo Inspections
(Case B)
5.2 Quantitative Comparison with Random Allocation This section presents a quantitative comparison between the inspection allocation system proposed in this paper and a random inspection allocation approach. The latter is defined as a random selection of elements to inspect without concern for the expected information gain. However, this randomly allocated inspection will use the same cost model, discussed in previous sections, as the information-based allocation. Thus, the evaluation discussed in this section serves as a comparison between the two different allocation approaches and not as an evaluation of the effectiveness of the cost model because both approaches are based on the same cost model. Figures 9 and IO present the results of the evaluation. This experiment considered single and double fallible inspections (a = 0.05 and l3 = 0.02) and explored the effects that the constraint level (i.e., time) has on the effectiveness of the random and information-based approaches. Note that for the case of double inspection, the same elements may or may not be inspected on the second inspection. This is because, like the first inspection, the second allocation of elements utilizes the expected information gain, which is a function of the updated (or short-term) nondefective probabilities. The experimental results presented in Figures 9 and 10 indicate that there is a significant difference between random and information-based allocation for time-constrained inspections in both Case A and Case B. A LSD multiple-range test established, with
experimental data revealed (with 95% confidence) that there is a statistically significant difference between the Simple Average andp,, orp,% for Case B. However, because this difference was not detected for Case A, the superiority of pmi,, or pnvryg over pmg cannot be confirmed. Nevertheless, because the Minimum (Pmi,,)summary statistic consistently produced the lowest number of classification errors for both cases and it is the simplest statistic to calculate, this statistic will be used throughout the remainder of this paper. In addition to evaluating the different summary statistics, this second experiment was used to estimate the added value of information-based allocated inspection by comparing three different classification scenarios. These scenarios are: classification without inspection, classification after one fallible inspection, and classification after two equally fallible inspections. Figure 7 presents the experimental results for Case A. This figure confirms the anticipated results regarding the number of classification errors for the three scenarios. Furthermore, a 95% LSD multiple-range analysis indicates that three significantly distinct groups, which correspond to the three scenarios, can be identified in the experimental data. Similar results were obtained for Case B. Figure 8 shows these results. The experimental results obtained so far are promising and represent a good indication of the system’s capability. However, the system must be compared to alternative approaches. This comparison is presented next.
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Journal of Manufacturing Systems Vol. 2ONo. 1 2001
30
”
20
j. . .
10
-’
Total
classification error
Total
i
error
r 0
O”.__
I
,
40
50
I
60
>
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I
I
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90
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50
:
:
“‘. t
I
I
1
>
60
70
60
90
100
(
Percentage inspected
Figure 9 Classification Errors for Random (R) and Information-Based (I) Allocation (Case A)
Figure IO Classification Errors for Random (R) and Information-Based (I) Allocation (Case B)
95% confidence, that the difference is statistically significant for both cases. Furthermore, this difference seems to decrease in a linear fashion as the allocation problem becomes less constrained. The significant difference between both approaches justifies the use of information-based inspection allocation.
tion and to present its real-time implementation within the environment of electronics assembly. However, the paper also opened a series of questions that will have to be addressed as part of future research. For instance, the benefits provided by the method are influenced by the total inspection time available and the probabilities of defective production and/or inspection. To determine the exact benefit of the method to a particular situation is necessary to either obtain structural analytical results or to conduct careful simulation analysis of these factors.
6. Conclusions
and Future Research
The information gain method presented in this paper allows the generation of real-time inspection routines. In evaluating the performance of the information-based inspection allocation strategies described in this paper, it was found that the approach presented yields feasible solutions to the fallible-inspection allocation problem. This information-based approach is composed of two fundarnental parts. First, an expected information-gain value is estimated by using a cost model. Then, a selection of elements that maximizes the overall expected information gain is performed. With respect to the first part, the amount of classification errors attained by the information gain approach was found to be significantly lower than a random allocation approach. Thus, the information-based approach presented in this paper yields solutions to the fallible-inspection allocation problem that are not only feasible but also advantageous because the number of classification errors is significantly reduced. The main objective of this paper was to introduce the concept of information-based inspection alloca-
Acknowledgments We gratefully acknowledge support from the National Science Foundation (NSF Grant DMI9502897) and Thomson Consumer Electronics for the realization of this research. The authors also would like to thank Vernon Dickson and the anonymous referees for their input in the preparation of the final version of this paper. References 1. J.R. Villalobos, “Inspection Models for Flexible Inspection Systems,’ PhD dissertation (unpublished) (College Station, TX: Dept. of Industrial Engg., Texas A&M Univ., 199 1). 2. J.R. Villalobos, J.W. Foster, and R.L. Disney, “Flexible Inspection Systems for Serial Multistage Production Systems,” IIE Tmns. (~25, 1993), ~~16-26. 3. J.R. Villalobos, V. Ramachandran, and V Baeza, “Inspection Routine Generation for a Flexible Inspection System,” Working Paper (Tempe, AZ: Dept of Industrial Engg., Arizona State Univ.). 4. J.R. Villalobos and A. Verduzco, “Integration of Quality and Process Planning Activities in SMD Assembly,” Proc. of 6th Industrial Engg. Research Conf., Miami Beach, FL, May 17-l 8, 1997. 5. G.F. Lindsay and A.B. Bishop, “Allocation of Screening Inspection
21
Journal of Manufacturing Vol. 2a/No. 1 2001
Systems
Effort: A Dynamic Programming Approach,” Mgmt. Science (~10, 1964), ~~342-352. 6. PM. Pruzan and J.T. Jackson, “A Dynamic Programming Application in Production Line Inspection,” Technometrics (~9, 1967), ~~73-8 1. 7. G. Eppen and E. Hurst, “Optimal Location of Stations in a Serial Multistage Production Process,” Mgmt. Science (~20, 1974), ppl194-1200. 8. R. Britney, “Optimal Screening Plans for Non-serial Production Systems,” Mgmt. Science (~18, 1972), ~~550-559. 9. DP Ballou and H.L. Pazer, “The Impact of Inspector Fallibility on the Inspection on the Inspection Policy in Serial Production Systems,” Mgmt. Science (~28, 1982), ~~387-399. 10. K. Tang, “Design of Multi-Stage Screening Procedures for a Serial Production System,” European Journal of Operational Research (~52, 1990), ~~280-290. 11. T. Raz, “A Survey of Models for Allocating Inspection Effort in Multi-Stage Production Systems,” Journal qf Quality Technology (~18, 1986), ~~239-247. 12. K. Tang and J. Tang, “Design of Screening Procedures: A Review,’ Journal of Quality Technology (~26, 1994), ~~209-226. 13. J. Larsen and S. Dyer, “Using Extensive Form Analysis to Solve Partially Observable Markov Decision Problems,” Working Paper 90/91-33 (Dept. of Mgmt. Sciences and Information Systems, The University of Texas at Austin, 1991). 14. G.E. Monahan, “A Survey of Partially Observed Markov Decision Processes: Theory, Models, and Algorithms,” Mgmt. Science (~28, 1982), ~~1-16. 15. W.S. Lovejoy, “Computationally Feasible Bounds for Partially Observed Markov Decision Processes,” Opemtions Research (~39, 1991), ~~162-175. 16. R.G. Askin and C.R. Standridge, Modeling and Analysis of Manufacturing Systems (New York: John Wiley & Sons, 1993).
pass reduces the probability of assigning unusuallylarge-gain elements instead of a combination of elements that yield a higher combined gain. The steps of the two-pass greedy heuristic are:
Authors’ Biographies
A quantitative comparison of the Knapsack greedy heuristic and the modified two-pass greedy heuristic with respect to optimal allocation was conducted to evaluate the efficiency of these two allocation approaches. One thousand boards, with 10 elements each, were generated for this evaluation. Table Al presents the results of this evaluation. These results were obtained by comparing the total gain for optimal allocation (achieved by exhaustive combinatorial searches) with the total gains of both the greedy and the two-pass greedy heuristics. The results are presented in terms of the difference in total expected information gain, maximum percentage error (worst case), average total expected information gain, and average percentage error for the maximum. The results in Table AI indicate that the two-pass greedy heuristic yielded a total gain closer to the optimal total gain than the greedy heuristic yielded.
1. Determine a gain-to-cost ratio for all elements. 2. Sort all elements in decreasing order of their individual gain-to-cost ratio. 3. Assign the first element from the sorted list and subtract its time requirement from the total available time. 4. Repeat step 3 until no assignments can be made. 5. For every element assigned to the knapsack, repeat: 5.1 Extract the element from the knapsack. 5.2Increase the available time by the amount of time required for this element. 5.3 Repeat steps 2 through 4. 5.4 If the combined gain at this point is greater than the combined gain at step 5, keep this knapsack assignment; otherwise, discard this knapsack solution.
Adan Verduzco is a test development engineer with Tyco Electronics in Mesquite, TX. He holds a BS in electrical engineering and an MS in manufacturing engineering from the University of Texas at El Paso. J. Rene Villalobos is an associate professor with Arizona State University. His areas of research include manufacturing systems, on-line quality control, and applied operations research. He holds a BS in mechanical engineering from the Tecnologico de Chihuahua in Mexico, an MS in industrial engineering from the University of Texas at El Paso, and a PhD in industrial engineering from Texas A&M University. Benjamin Vega is an operations analyst with Capital One in Richmond, VA. He holds a BS in industrial engineering from the University of Texas at El Paso and an MS in industrial engineering from Arizona State University.
Appendix: Two-Pass Greedy Heuristic Once all elements have been sorted in order of their gain-to-cost ratio and assigned to the knapsack given the capacity constraints, a second pass is performed to improve the current solution. This is achieved by extracting one element from the knapsack at a time and executing the greedy heuristic again. This second Evaluation
Table AI of Knapsack Heuristic
Maximum Difference in Total Gain (gain units)
Maximum Percentage Difference
Average Difference in Total Gain (gain units)
Average Percentage Difference
Greedy
4
0.6%
1
0.2%
0.228 0.012
0.015%
Two-pass greedy
Knapsack Heuristic
22
0.001%