Four Measures of System Behavior
David Kazmer Associate Professor
David Hatch
Research Assistant
Liang Zhu
Research Assistant
Department of Mechanical & Industrial Engineering
University of Massachusetts Amherst
Abstract
Engineering systems are typically designed to cost and specification. The robustness of the engineered systems is frequently measured with respect to consistency according to a Six Sigma philosophy. However, robustness is just one measure of system behavior, and in insufficient to characterize the behavior of complex systems with respect to extensibility, robustness controllability, and flexibility. The extensibility index measures the extent to which the quality attributes may be adjusted relative to their variation, even if such adjustment causes other attributes to fail specification. The robustness index is representative of consistency of the quality attributes, as measured the relative breadth of the specification to the variance of the quality attributes. The controllability index measures the span of the quality attributes while maintaining the feasibility of the other quality attributes. Finally, the flexibility index measures the range of the quality attributes while holding other quality attributes fixed. All of the measures are valid for linear and non-linear systems, uncoupled and coupled systems, and systems with varying number of decision variables and quality attributes. The four measures are applied to the injection molding of digital video disc substrates. The application example demonstrates some significant short comings in this manufacturing process, and provides insight into general system design requirements and strategies.
Introduction
Engineering design is an unstructured but logic-based process where successive iterations of synthesis and analysis eventually converge to the desired solution [Paz-Soldan, 1989 #701]. This iterative process is necessary since it is normally infeasible to deduce the final design from the initial set of specifications.
There are many difficulties in the design process that prevent direct convergence to a final design. First, there are many potential concept strategies that may result in an adequate solution. The development and validation of each design concept requires the configuration of the components within the engineered system and identification of the design parameters. If the approach appears promising, quantitative relationships may be established between the design parameters and performance attributes. These relationships can be complex and numerous such that the designer may not be able to simultaneously navigate and resolve an adequate solution. At this stage, the designer may become acutely aware of difficulties with the design: inadequate performance, conflicting specifications, and infeasibility. The designer may accept the current design as final, modify the design parameters to negotiate between the different performance attributes, consider the relaxation or constriction of the design specifications, or start again with another concept design. This complex decision-making process prevents direct development of the final design in all but incremental redesigns.
In the context of this article, a system is defined by a set of relations between quality attributes, y i , that are dependent upon the selection of the decision variables, x j . For a system design to be feasible, a set of decision variables must be selected within their lower and upper control limits, i.e. LCL j ≤ x j ≤ UCL j , such that each of the quality attributes is within their lower and upper specification limits, i.e. LSL i ≤ y i ≤ USL i . While the relationship between decision variables and quality attributes is frequently presented via the function matrix, A :
[]X
Y=, (0)
A
there is no requirement in engineering design that a linear mapping exist from the decision variables
to quality attributes, nor that the function matrix, A, is square.
Theory
Recent research in both industry and academe has focused on achieving robustness in system
design (Iyer and Krishnamurty 1998; Koch and Mavris 1998; Suri and Otto 1999; Thornton 1999;
Zhang, Wiecek et al. 1999). While many of the academic approaches do provide significant insight
into the behavior and optimality of systems, they very frequently require information and numerical
methods that are beyond the capability of the research team that published the work. Alternatively,
industry practitioners frequently rely on overly simplified approaches that require little information
and no or minimal optimization. However, such overly simplified approaches often produce
incorrect results, or overly focus on non-critical issues. Robust design is not the only, and perhaps
not even the most important, aspect of system design.
Towards this end, four measures of system behavior are defined: robustness, extensibility,
controllability, and flexibility. Robustness will first be defined since it may be best understood due to
significant emphasis of Six Sigma in the media. Special attention has been paid to ensure the
potential for broad adoption. As such, the only required information is the function matrix, control
limits, specification limits, and standard deviations of the quality attributes. All results of this paper
can be reproduced with simple single objective, constrained optimization techniques.
Robustness
Robustness has been defined as the insensitivity of performance to uncontrolled variation. Lack
of system robustness can lead to unsatisfactory product performance, low production yields, and
increased operating cost in end use. The Six Sigma philosophy (Breyfogle 1999) dictates that the
mean performance should be located six standard deviations from the closest performance
specification. In engineering practice, application of Six Sigma implies a defect rate less than three defects per million. However, it is well known among researchers that normal statistics are rarely valid to this degree of precision (Ref), and that such application is very prone to uncertainty in models and insufficiency of assumptions (Roser and Kazmer 2001).
Rather, the original motivation of the Six Sigma methodology was to ensure a production yield of 99.87% in 99.87% of the applications. As shown in Figure 1, the mean performance should be located at least three standard deviations from the nearest specification limit. To allow for long-term shifts in the mean performance, the targeted mean performance should be another three standard deviations from the allowable mean performance. Together, six standard deviations should be required between the targeted mean performance and the closest specification limit.
Figure 1: Six Sigma Philosophy
Frequently, a process capability index, Cp ,
σ
6LSL USL Cp ?= (1) or the asymmetric process capability index, Cpk ,
????????=σμσ
μ3,3min LSL USL Cpk (2)
is utilized as a measure of robustness (Boyles 1991). The Cpk index has two advantageous properties: 1) that it can be used for one sided metrics, and 2) denotes a loss in quality due to shifts of the mean off the target center. In either case, however, a process capability equal to one generally implies product performance at the target level with three standard deviations to the closest specification limit. The process capability should be at least two to meet Six Sigma guidelines.
Most engineering systems, however, have multiple quality attributes. As such, it is desirable to measure the robustness of the individual quality attributes as the aggregate system. The robustness, Ri, of each quality attribute can be computed directly as the process capability index of eq. (1) and
(2), where the USL , LSL , μ, and σ, are appropriately substituted for each quality attribute. If the mean, μ, and standard deviation, σ, are unknown, then the process capability index of eq. (1) is frequently used with an estimate of σ derived from a moment matching formula (Cacoullos 1982). It has been proposed (Ford 1996; Kazmer 1999) that the robustness of a system with multiple quality attributes may be evaluated via the joint probability of likely acceptance as:
(3) i n i system P P 1=Π=where i P i i ??Φ?=??13
1 (4) which is also valid for the relation between √system and P system . Combining these equations gives: ()()attributes quality of Number function
density cumulative normal Inverse function
density cumulative Normal (1) eq. attribute,quality th -i of Robustness :where ,321212131111≡≡Φ≡Φ≡???
??????Φ??Φ?=
??=?Πn i i n i (5)
Thus, system robustness is an aggregate performance measure that includes the likelihood of each quality attribute being acceptable. It is important to note that internal validation has shown that
the independence assumption is not generally valid, as will be discussed in more detail later on. However, this measure generally provides a reasonable estimate of system robustness, and does possess several beneficial properties useful in system design including (Kazmer 1999): 1) models multiple design objectives; 2) convex behavior allows for global optimization; 3) allows for direct inclusion of different kinds of specifications; 4) consistent with Taguchi’s concept of tolerance design since it promotes central tendencies with small deviations in product properties, rather than a goal post mentality (Devore 1995); and, 5) consistent with many design axioms to minimize information content since the production yield will tend to decline geometrically as the number of requirements rise (Suh 1990).
Extensibility
While robustness is a critical measure of system behavior, other measures are useful in system design. Extensibility is defined as the capability of a quality attribute to be extended to the broadest achievable range relative to uncontrolled variation. The lower extensible limit, LEL , and upper extensible limit, UEL , for each quality attribute may be derived directly from the function matrix and control limits on the decision variables, or from a series of optimizations for more general systems:
UCL
X LCL y UEL y LEL i X i i X
i ≤≤==max min (6)
It should be noted that in achieving the minimum or maximum quality attribute, the decision variables may be selected such that some or all of the other quality attributes are not within specification. The invalidation of other quality specifications is not a limitation of the extensibility measure, since the extensibility is intended to simply measure the total extent to which each quality attribute can be modified.
Once the LEL and UEL are available, the extensibility of each quality attribute, E i , can be evaluated according to the established method for the process capability indices of eq. (1) and (2), e.g.
σ6i i i LEL UEL E ?= (7) ??
??????=σμσμ3,3min i i i i i LEL UEL E (8) The system extensibility, E , may be calculated as an inverse joint probability as previously described in eq. 5, and discussed for system robustness. For the asymmetric index of eq. 8, the extensibility is dependent upon the selection of the decision variables in a system design, and the resulting mean of each quality attribute.
The extensibility measure seems trivial, yet provides valuable information that should be considered during system design prior to robustness issues. Generally, system designs should exhibit an extensibility index significantly greater than the robustness index. If the extensibility is less than one, then the system is largely incapable of being designed, i.e. the entire range of quality attributes is small compared to the variation that the quality attributes exhibit. If the system extensibility is small, then several alternatives are suggested. First, the extensibility measures of each quality attribute should be inspected to identify and eliminate those quality attributes that are not extensible. If inextensible quality attributes cannot be eliminated through negotiation, then development of a different system design concept may provide increased extensibility. One common approach is to augment the decision variables with added degrees of freedom that will significantly affect the behavior of the quality attributes.
Controllability
While the extensibility index provides a measure of the total range of the quality attributes, significant portions of the range will require other quality attributes to fall outside of their
specification. As such, the controllability is defined as the capability of a quality attribute to be extended to the broadest range relative to uncontrolled variation while maintaining all quality attributes within specification. Thus, each quality attribute can only be extended to the point at which other quality attributes become unacceptable. In a broad sense, the controllable space may be considered as the feasible performance space (of an engineering design) or process window (of a manufacturing process). The feasible region is a polytope formed by the joint intersection of the half-spaces corresponding to the constraint surfaces of all the system specifications. While the feasible region can be solved efficiently for moderately sized linear systems via the extensive Simplex Method (Zhu, Zhao et al. 2000), it is generally believed that the solution of the feasible region is infeasible for large linear systems or even small nonlinear systems.
For practical characterization of system behavior, however, the lower global controllable limit, LGL , and upper global controllable limit, UGL , for each quality attribute may generally be resolved from a series of optimizations:
j USL y LSL UCL X LCL y UGL y LGL j
j j i X i i X i ?≤≤≤≤==max min (9)
Once the LGL and UGL are available, the controllability of each quality attribute, C i , can be evaluated according to the established method for the previously defined indices in eq. (1-2) and (7-
8). The system controllability, C , may be calculated as an inverse joint probability as previously described in eq. 5, and discussed for system robustness. Again, the asymmetric controllability is dependent upon the selection of the decision variables in a system design, and the resulting mean of each quality attribute.
It should be noted that in resolving the global controllable limits for each quality attribute, the decision variables will be selected such that all quality attributes are within specification. Since the
LGL and UGL are constrained to the LSL and USL, the controllability index will be less than the robustness index. Thus, it is necessary for a system to be robust to be controllable.
However, the controllability index does provide different information than the robustness or extensibility indices. The LGL and UGL are the broadest limits that can be achieved while ensuring other quality attributes (and the overall system) are within specification. Thus the controllability provides a measure indicative of the extent of the system to be significantly modified within specification. It is possible for a system to be extensible and robust, but not controllable. In such a system, multiple quality attributes will conflict and exhibit negative correlations, i.e. desirable changes in a quality attribute will bring about undesirable changes in other quality attributes. If the controllability index is less than one, it is indicative that the system design may not be able to be successfully modified if significant changes in the mean quality attributes is necessary. As such, a different system design concept with fewer quality attributes, augmented degrees of freedom, or differing system behavior may be necessary.
It should also be noted that the controllability index, unlike the previous two indices, will also reflect the infeasibility of a system design. If the quality attributes conflict significantly and the specification levels are unrealistic, then no selection of the decision variables may result in a set of quality attributes that meet specification. As such, no solution will be obtained for the LGL and UGL, and the controllability index will be undefined.
Flexibility
The controllability index provides a measure of the adaptability of the system while maintaining all quality attributes within specification. However, it is sometimes desired to modify the level of individual quality attributes while maintaining other quality attributes fixed. As such, the flexibility is defined as the capability of a quality attribute to be extended to the broadest feasible range relative to uncontrolled variation while maintaining the level of other quality attributes. In a general sense,
the flexibility of a quality attribute can be derived by successively intersecting the global feasible space with additional equality constraints. This approach is generally infeasible. For practical characterization of system behavior, the lower flexible limit, LFL , and upper flexible limit, UFL , for each quality attribute may generally be resolved from a series of optimizations:
i j y UCL X LCL y UGL y LGL j j i X
i i X
i ≠?=≤≤==τmax min (10) Once the LFL and UFL are available, the flexibility of each quality attribute, F i , can be evaluated according to the established method for the previously defined indices in eq. (1-2) and (7-8). The system flexibility, F , may be calculated as an inverse joint probability as previously described in eq. 5, and discussed for system robustness.
It should be noted that in resolving the flexible limits for each quality attribute, the decision variables will be selected such that all other quality attributes are maintained near their current levels. Since the LFL and UFL are constrained to the LGL and UGL , the flexibility index will be less than the controllability index. Thus, it is necessary for a system to be controllable to be flexible. If a system is infeasible, it will not possess any feasible space and thus both the controllability and flexibility indices will be undefined.
There are some substantive differences in the definition and behavior of the controllability and flexibility indices. The controllability index is derived from the global feasible space that maintains all quality attributes within specification. While the asymmetric controllability index is dependent on the selection of the decision variables, the symmetric and more general controllability index (e.g. eq. 8) is not. As such, the controllability of the system can be estimated without selecting a specific design. However, the flexibility of the system is dependent upon the desired targets for the quality attributes. As previously described, the range of flexibility is derived from the global space with additional
constraints based on maintaining the value of the other quality attributes. Since the polytope is irregularly shaped and certainly not prismatic, the selection of the targets for the quality attributes will result in significant changes in the system flexibility. In other words, the targets for the quality attributes will determine the nearest active constraint as well as the amount of allowable slack for the decision variables. If a target value is selected near the edge of the controllable region, then the system flexibility will quickly degrade.
Application
Concepts of extensibility, robustness, controllability, and flexibility will now be applied to the manufacture of a digital videodisc (DVD). The diversity of quality requirements of the DVD, coupled with numerous formats and manufacturing systems, has made the optical media industry open to process improvement opportunities. In addition, growth in the DVD market has left production facilities searching for cost effective added capacity. All DVD formats require bonding of two 0.60 mm polycarbonate substrates. Injection molding is the primary manufacturing process in optical media creation. Each disc is composed of an optically transparent substrate (typically polycarbonate), with one or more substrates containing a reflective metalized data surface. For prerecorded media, the data is stored on a disc in the form of pits that are molded into the disc during the injection molding process. The data is part of the disc; the data is not written in a secondary operation as in magnetic media. The bonding process combined with the small definition of data pits requires stringent flatness specifications of each DVD substrate. In addition, substrate thickness and birefringence play significant roles in the ability of the DVD laser to properly read the optical media (Oshiro, Goto et al. 1997; Park, Han et al. 1998; Shin, Rhee et al. 1998).
Previous research has focused on methodologies for process characterization through empirical models (Hatch 2000). The results of a main effect DOE identified the critical processing parameters that justified further exploration. The newly recognized parameters were segregated into stages
based on idealized interactions affects. Using a designed experiment that yields nonlinear models, including interaction effects, the system can be then optimized within each stage. DVD experiments were conducted on a Sumitomo SD30 injection-molding machine. All laboratory experimentation was conducted at General Electric Plastics Polymer Processing Development Center, in the Optical Media Development Center (OMDC). In the OMDC, the Sumitomo SD30 is configured as a batch process. The OMDC possesses the analytical testing equipment required to characterize all relevant quality characteristics, including a TopoMetrix Atomic Force Microscope, Dr. Schenk Optical Disk Scanner, and multiple CD Associates Stamper Player Signal Analysis instruments. Multiple replicates were measured for each run of the DOE. Data from designed experiments and/or random data can be used to develop a model of the system.
Solution of the model requires determining the model coefficients, or estimation of parameters, and testing for significance (Myers, Khuri et al. 1989). The parameters βj are the regression coefficients and represent the expected change in response, y , due to the change in x j . To estimate the regression coefficients suppose there are n > p observations on the response variable , y 1, y 2,…, y n . For each response variable y k there is a set of predictors, x ij , where k denotes the k th observation of the x j predictor variable. A general response surface model representing the system can be expressed by a multiple linear regression model:
(11)
,110k p j kj j k x y εββ++=∑?=where the error of the model, ε, is assumed normal and has an expected value, E(ε) = 0, variance, Var(ε) = σ2, and the errors, εi , are uncorrelated random variables (Neter, Kutner et al. 1996). Both the mean and variance are modeled as a function of the processing parameters. Frequently the goodness of fit, or how well the model fits the data is expressed in terms of R 2 or R 2-adjusted:
yy E
yy R S SS S SS R ?==12 (12) ()22111)1()
(1R p n n n Syy
p n SS R E
adj ????????????=???= (13) where S yy = SS R + SS E , and SS R is the sum square due to regression model and SS E is the sum square due to the error.
In this work, the DVD processing space was characterized using two different system models. The first model consists of nine input processing parameters and eight output quality characteristics. The nine input processing parameters are based on the injection molding machine control inputs. The melt temperature and the clamping profile are quantified through these parameters, x i for j = 1, 2, 3..9:
(14)
????????????????????????????=????????????????????????????Time Change Point Transfer V/P 2 Time Clamp 2 Tonnage Clamp 1 Time Clamp 1 Tonnage Clamp Time Cooling Temp Mold Temp Melt 987654321x x x x x x x x x The eight output quality parameters evaluate the flatness and optical quality characteristics for the DVD substrate. The tangential, radial, dishing, and outer diameter (OD) deviation measurements guarantee flatness of the disc. Flatness is critical to ensuring proper bonding for the DVD substrates. Optical specifications are characterized by birefringence measurements. The output parameters and quality specifications are:
Using linear regression techniques, a linear empirical model of the system was generated utilizing quality data from 575 molded DVDs across 115 process settings. This model is the input to the
described process window algorithms. The linear empirical system model is quantified by the matrix:
(15) ?????????????????????????????????????????????????????????
???????????????????????????????????????=??????????????????????????98765432187654321169.1325.0053.0023.0047.0022.0142.0022.00001.067.1775.0138.0004.0036.0130.0000.0095.0006.0003.0773.07.40545.054.6653.0701.0096.051.4300.0035.02.441619.108.12931.01.10815.00.1623.1508.028210568.53.2768.14.1672.23.2634.4439.02303767.986.101.167.2825.456.872.1290.00.76716.0168.0019.0037.0056.0009.0026.0002.0001.0036.0575.0183.0010.0041.0058.0009.0021.0003.0001.0203.0x x x x x x x x x y y y y y y y y For comparison, a full quadratic model was developed and compared to the simplified linear model. The {minimum, median, and maximum} R 2-adjusted value across all eight quality attributes for the full quadratic model was {59.2%, 76.7%, 91.3%}, significantly higher than those of the linear models {26.8%, 57.5%, 82.6%}. It should be noted that the complexity of the process, process variance, and lack of knowledge about the true process behavior prevents the development of perfect process models. However, the linear models are of significant value in understanding the trends and constraints of the DVD manufacturing process.
Robustness
The robustness of the system is evaluated according to eq. (1) and 2, which compares the relative width of the specifications to the observed variance for each quality attribute. The specifications and measured variance for the manufacturing of DVD substrates is shown in Table 1. The robustness, R i , of each quality attribute is very high when measured with the symmetric process capability index (eq. 1). Since there are fifteen standard deviations between the specifications, the aggregate system robustness across all quality attributes is approximately equal to the least robust quality attribute.
Table 1: System Robustness of a DVD Substrate Manufacturing Process
As previously mentioned, however, symmetric measures of robustness disregard the error between the mean and target performance value. As such, the robustness of the system, as measured with an asymmetric definition, is dependent upon the selection of the decision variables. One natural tendency would be to evaluate the system at the central value of the decision variables. Such an approach may be valid since the control limits on the decision variables are usually intentionally set to encompass the useful region of the decisions. Inspection of the asymmetric robustness at the center point of the system design immediately indicates that the system design is infeasible since the maximum birefringence and the tangential deviations are outside the specified limits. Clearly the selected values of the decision variables are poor, and hopefully may be improved.
One approach is to select the decision variables to provide a more robust design. In this case, an optimization approach is frequently utilized to bring all quality attributes far within their specifications. The right most column of Table 1 provides the resulting robustness of this system at the optimal settings of the decision variables. It is evidenced that the several of the quality attributes have a robustness less than two, and the aggregate robustness is 1.36 indicating that approximately four standard deviations (3σ*1.36 ≈ 4σ) lie between the mean performance and the specifications. This level of robustness would likely be acceptable in many manufacturing plants. Further inspection of the optimal decision settings indicates that many decision variables are constrained by their control limits, and that a more optimal solution could be obtained by modifying the control limits, if possible.
Extensibility
The extensibility of the system is described in Table 2. The results are somewhat surprising for several reasons. First, it is observed that the extensibility of the system is not significantly greater than the robustness of the system. This observation indicates that the mean of the individual quality attributes can not be adjusted significantly compared to the specification, even if desired. This observation also indicates which of the quality attributes will never be cause for concern. For instance, the total range of the minimum birefringence is [-19.5,15.7] which is small compared to the range of the specification [-50,50]. As such, the extensibility index for minimum birefringence indicates that the selection of the decision variables will have very little influence. As such, the quality attribute can be dropped from further consideration (if within specification) or an entirely new system design is required (if outside of specification).
Table 2: System Extensibility of a DVD Substrate Manufacturing Process
Another very significant result of the extensibility analysis is obtained by comparing the achievable range of the quality attributes (LEL and UEL) to that of the specification (LSL and USL). The achievable range of most of the quality attributes is not centered within the specification range of the quality attributes. Rather, the achievable range typically extends from just over the mean of the specification limits and to the outside of one of the specification limits. This behavior is indicative that there may be an unknown decision variable that is out of control and causing a shift in the mean of the performance attributes. Alternatively, many of the quality attributes may conflict and the system behavior only provides sufficient overlap to achieve the specifications.
It should also be noted that the extensibility of the system is maximized by selecting the mean decision variables. This phenomenon occurs since this selection of decision variables, by definition, places the quality attributes at the center of their LEL and UEL and thereby reduces the asymmetric extensibility index to the symmetric definition. However, the extensibility is significantly reduced as the decision variables vary from their mean setting. The extensibility of the system is not significant when the decision variables are selected for a robust optimal.
Controllability
The controllability of the system is described in Table 3. Recall that the controllability defines the achievable range relative to variation while maintaining all quality attributes within specification. As such, the global controllable range (LGL and UGL) of all the quality attributes in the system are significantly reduced compared to the specification and extensible ranges. The global controllable ranges have significant value, since they explicitly define the achievable range of the quality attributes while maintaining system feasibility. For instance, it may be desirable to set the minimum birefringence to a given value, say 8 μm. By inspecting Table 3, an engineer can immediately identify that this desired level of minimum birefringence is not feasible without causing other quality attributes to fail their specification.
Table 3: System Controllability of a DVD Substrate Manufacturing Process
The controllability index exhibits the expected properties: 1) that the controllable range of the quality attributes is less than the specification range or the extensible range, 2) the controllability of
the aggregate system is governed by the least controllable quality attribute, and 3) the asymmetric controllability is dependent upon the selection of the decision variables. In this system, the lack of controllability of the minimum birefringence limits the overall controllability of the system. As such, significant changes in minimum birefringence could not be achieved without causing other quality attributes to fall outside of the specification. All other quality attributes possess a controllability index greater than 1.0, which indicates that their mean value should be able to be modified to greater than a 99.87% likelihood without causing other quality attributes to fall outside specifications. Flexibility
Table 4 quantifies the flexibility of the DVD substrate manufacturing process at the robust optimal decision settings. As shown in the table, it is not possible to change the level of dishing or maximum birefringence without causing another quality attribute to deviate from its current quality level. All the other quality attributes may be adjusted slightly, but not significantly compared to the natural variation exhibited by the quality attributes. It should be noted that five of the eight decision variables were at their control limits in achieving the robust optimal design, which significantly reduces the feasible space in the vicinity of the system design. As such, a more flexible yet less robust design may exist away from the control limits of the decision variables.
Table 4: System Flexibility of a DVD Substrate Manufacturing Process
The lack of system flexibility implies that it is unlikely that a significant change to a quality attribute can be obtained without significantly changing other quality attributes or otherwise
modifying the system behavior. Such lack of flexibility is commonly exhibited in engineering design problems and manufacturing processes, and strongly suggests the adoption of a system design philosophy such as axiomatic design to enhance the flexibility of systems (Kazmer, Kapoor et al. Submitted 2000.), or at least to quantify the Pareto optimal boundary for selection of a trade-off between multiple quality attributes (Das and Dennis 1995).
Discussion
Perhaps too much emphasis has been placed on robust design methodologies, since emphasis in one area reduces activity in other related but important areas. In the engineering design community, only the robustness of systems is well understood and practiced in industry. However, other measures of system behavior may be more vital than robustness, since in fact the specification region may be very large compared to the flexible range, controllable range, or even reachable range of the quality attributes. To further emphasize this vital concept, the different ranges for the first quality attribute, OD deviation, are plotted in Figure 2.
Figure 2: Defined Ranges for OD Deviation Extensibility, Robustness, Controllability, and
Flexibility
For this example, it is observed that the specification range is the broadest measure of the behavior. This fact implies that the robustness of this quality attribute is not the underlying issue, and that getting the quality attribute to be within the specification range is a more significant issue (i.e. controllability), as is trading off optimality in this attribute for improvements in other attributes (flexibility). These are significant concepts that have not been well addressed. In fact, these concepts may have been obscured by related arguments regarding uncoupled, partially coupled, and fully coupled design behaviors (Frey, Jahangir et al. 2001), the relative number of decision variables to performance measures (Ref), and the qualitative correlation matrices in Quality Function Deployment (Sullivan 1986; Akao 1990).
It should be noted that the globally feasible region of a system is derived from the function matrix and corresponding set of specifications for the quality attributes and control limits for the decision variables. As such, the resulting achievable regions corresponding to the controllability and flexibility of the system will be governed by the set of active constraints. Thus, the behavior of the system will vary considerable not only with the size and topology of the function matrix, but also with the settings of the decision variables as well.
In theory, a fully uncoupled function matrix with equal number of quality attributes and decision variables will provide controllability and flexibility limited only by the robustness of the system. In reality, such fully uncoupled designs are not usually practical or even possible to achieve. However, the measures of system behavior described herein hold for uncoupled and coupled systems, as well as systems of arbitrary size. Since the measures are normalized according to the breadth of the probability density functions of the quality attributes, they provide transportable insight into extensiveness, consistency, controllability, and flexibility of the investigated systems.
Three significant tendencies of system behavior have been identified in the validation of these measures. First, systems do tend to lose controllability and flexibility as the system’s function matrix