# Product Screening to Multicustomer Preferences: Multiresponse Unreplicated Nested Super-ranking

Authors: George J. Besseris
Journal: International Journal of Quality, Statistics, and Reliability
 (2008)

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## Abstract

Modern production methods demand the synchronous multicharacteristic optimization of goods. There is a need to diversify a basic product to the importance placed on its individual quality components by a wide spectrum of concerned customers. This work shows how the super-ranking concept may be utilized taking into account relative weights among the implicated responses. The theoretical development is focused on the difficult situation where the optimization is attempted through unreplicated and saturated fractional factorial designs. A nested super-ranking scheme is devised to accommodate a dual weight assignment, first by setting up a single consolidated response per implicated customer and then, in a subsequent step, by incorporating a customer importance rating thus rendering an overall single master response. A demonstration of the proposed method on a pragmatic problem arising in aluminum milling involves optimization due to seven controlling factors concurrently influencing nine product responses modulated by six preference ratings set by a given customer base, respectively. Key benefits of this method are the offered ease of intermixing numerical and categorical data in mainstream multiresponse optimization problems, and keeping customer preferences in perspective through economical, short-cycle screening while relaxing stringent data normality and possible multidistributional effects among the implicated quality characteristics.


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## Body

## 1. Introduction

It is a rare phenomenon that a single dominant product characteristic is the only reason
that a product is selected out by inquisitive and well-informed consumers.
Consumers are becoming in a blink of an eye aware about new trends and at the
same time they shape themselves to new expectations for goods and services they
desire. Modern quality culture endorses the hunt for discovering customer expectations
that in technical language is converted to merely a host of quality
characteristics (or responses). The number of the published work on
multiresponse optimization is justifiably growing with no end in sight. There
is a multitude of techniques and methods to confront the theoretical and
statistical aspects of such difficult problems. However, among the most
challenging experimental design schemes, especially in industrial process
screenings, are the fractional factorials [1, 2]. When an expeditious experimental effort is amenable to either a lack of resources or other economic
issues, then the unreplication of the scheme is anticipated to be an additional
advantage to that end. Furthermore, squeezing out all information from a
saturated fractional factorial design gives the experimenter the chance to
explore  the maximum likely modes that may affect a test product. This practical
investigation will make use of such unreplicated and saturated fractional
factorial designs to size up statistically product and process parameters that
influence many product characteristics. The concept of super-ranking is utilized
here for providing objective contrasting among the combined effect rendered by
all considered responses concurrently. This last notion is deeply appreciated
when designing new processes for new products or improving existing products
and it is an everyday concern in mass production. However, a greater problem
with immense repercussions in manufacturing is optimizing a product line across
a similar workflow that may institute different customer preferences on a given
list of product traits. This may be interpreted as optimizing several responses
simultaneously while taking in account individual response prioritization for
each implicated customer concurrently. It is stressed that the individual
customer trait hierarchy contributes synchronously to estimating the optimal
setting of statistical significant controlling factors. The proposed technique
relies on the statistical grounds of nonparametric theory enjoying the
well-known benefits rank-ordering statistics offer to cases that demand brief
experimentation. One advantage is the inherited exceptional efficiency at small
sampling rates. Thus, decision making is achieved by a simple statistical
method that competes agreeably with a popular alternative such as the
desirability method. Additional advantages discussed for the proposed technique
is the convenience in mixing up variable and categorical response data alike.
The theory developed here is targeting orthogonal array families members of L4(23),
L8(27), L12(211), and L16(215)
because of their inherent reliance on low sampling count [3]. A case study
drawn from the heavy industry aids in demonstrating the possible usefulness of
the tool developed herein. Aluminum can-making material is modulated by seven
controlling factors reflected by the response of nine quality characteristics
and indicators weighted each time by a preference priority setting of six
customer groups.


## 2. Literature Review

The general theory of design of experiments has been well described in several modern
expositions [1–3]. The method of quality loss functions in achieving robust
consumer products has been outlined by Clausing and
Taguchi [4], Taguchi [5] and Ross [6]. Product multicharacteristic selection
has shown to be solvable by fundamental Taguchi methods [7]. Vining and Myers [8] have attempted to bring together the best
features of two very popular methods, response surface methodology and
Taguchi's robust design in order to produce a superior optimizer hybrid. Along
these lines, Tang and Xu [9] have suggested a unified approach in
handling the dual response surface optimization. However,
the signal-to-noise ratio concept introduced by Taguchi in optimization
problems seems to possess several weaknesses that have received critical
attention [10]. There are two in-depth lengthy reviews about the current status
of robust parameter design that are accompanied with a wealth of references [11, 12].
However, there are earlier contributions that have introduced the general
concept of fitting polynomial regression functions to model parameter
selections in multifactor screening [13].Fractional factorial designs and
design of experiment methods remain popular in manufacturing and in mechanical
engineering applications [14–18]. Clearly, design of experiments is among the
most potent data-mining techniques that may provide quick interpretation to
phenomenological behaviors which are much in need at the shop-floor level [19].The multiresponse optimization problem has been recognized to be a very
difficult one. Therefore, there has been a need for insightful direction if it
is to reach to meaningful solutions [20]. Competitive
business environments do not allow development time for ameliorating product
traits on a one-by-one characteristic basis anymore [21, 22]. It
has become particularly important in complex manufacturing environments to
navigate through the trivial many responses in order to uncover the
indispensable few with respect to the needs of a customer [23]. The
first method commending wide acceptance even at present is the involvement of
the desirability function in a way that each response is weighted individually
while the terminal decision making is derived from a composite desirability
measure [24]. Based on this concept, there were several investigations that through
selected modifications provided better estimations with respect to the original
technique [25]. This method has been extended for predicting operating
windows in product designing [26]. The desirability method has become so
indispensable to product improvement that is adopted directly in modern quality
initiatives such as in Six Sigma for translating and driving towards
operational excellence [27]. Ortiz et al. [28] have employed the deterministic
prowess of a genetic algorithm to decipher difficult multiresponse behavior by
an unconstrained desirability method. It is not a rare event to have several
responses that may be related to some degree of correlation among them. In such
cases, the method of [29] appears to be a practical outlet. Exponential desirability
functions have been developed to counter the cumbersome problem of adjusting
the behavior of several mechanical properties for steel at the same time [30].
Kros and Mastrangelo [31] contrasted mixed-type desirability functions in the
additive and multiplicative approach with respect to input parameters and
response levels. However, no matter the selection of the multiresponse
statistical processor, there is insightful information generated by keeping
track of the sensitivity analysis results for various ways of
solutions [32].The multiple response surfaces have been treated with the quality loss function concept
offering obvious simplicity to the analysis [33]. A compromise approach by
Vining [34] has provided promising results. Multiresponse robust design seems
to be greatly extended by the method of combined array [35]. Recent
research on the area of programming a numerical control machine has turned up
useful accounts on multiple comparisons by introducing a new function that
incorporates location and dispersion effects [36].The Taguchi approach continues to find a niche in manufacturing processes even for the case
where many characteristics are investigated. Rekindled interest for this method
arises because of the convenience this method possesses even for complicated
parameter design problems [37]. However, techniques that maintain convincing factor
resolution for a multiresponse optimization problem essentially make an attempt
to join together the data analysis with a sophisticated decision-making
routine. Successful applications of such expert-system-oriented methods have
been relied on goal programming [38], neural networks [39, 40], and the utility
concept [41]. These methods are solid in their “sieving-through” data
capabilities, nevertheless, they may be considered as arduous to the average
product developer. An alternative approach has been mobilized in reducing the
volume of necessary calculations for the multiresponse optimization. When the
number of quality characteristics is small, that is two, then the Principal Component
Analysis is espoused to reduce the number of responses from two to one so in
this case the problem is cast to an unreplicated fractional factorial design
solved by the Taguchi method and the pooled analysis of variance (ANOVA) [42].
Tong and Wang [43] have revisited the multiresponse optimization problem and
they found that there is significant gain in cohabiting the front-processing
response reduction provided by the Principal Component Analysis with the grey relational
analysis in the back-end for decision making. All these different ways of
attempting to explain the multiresponse optimization issue only prove the
incessant necessity for such methods along with the difficulty in encompassing
realistic matters to a product developer in a robust design study. Such matters
may be an emphasis on economic experimentation, the use of robust statistics at
low sampling ranges, the handling of possible intermixing of variables of
different nature, that is, numerical and categorical data and so forth. It is
expected that there will be enough room for new developments in this critical
field that transfixes both areas regarding quality and new technology [44] while at the same time they will become easier to grasp and apply to everyday
process improvement projects [45].The advantages
of nonparametric statistics over normal estimators and their inherent
versatility and efficiency under any unknown data distribution for short
sampling have been collected in the recent work of Conover [46]. It would be an
interesting outcome if multiresponse optimization may be elucidated by
nonparametric inference methods and this is the subject of the next section.


## 3. The Multiresponse Nested Super-Ranking  Approach in Unreplicated Saturated  Fractional Factorials

The possible usefulness of the super-ranking concept in an engineering application that
involved unreplicated saturated fractional designs has been demonstrated by
Besseris [47, 48]. This concept is further explained in this paper for the case
where multiple responses have to be optimized based on a variety of customer
priorities placed on the considered responses. A simple theory is built here
which borrows elements developed from the super-ranking treatment as previously
stated. We contemplate fractional factorial designs of the family series L4(23), L8(27), L12(211), or L16(215) suitable for low sampling screening experimentation. If there are k factors and
possibly accompanying interactions, denoted as X1,X2,…,Xk, then, there are at
least n experimental trials that should be performed to relay possibly useful
information. For any given arbitrary orthogonal array mentioned above, the
corresponding columns of settings may be represented as X1j,X2j,…,Xkj(j=1,2,…,n). It is sought the concurrent optimization of m distinct,
independent responses, denoted as Y1,Y2,…,Ym, modulated by these factor settings. Each
response then is a vector of dimensionality n, comprising of elements y1j,y2j,…,ymj(j=1,2,…,n) (Figure 1).Figure 1
Generic representation of an orthogonal array depiction with respect to
a group of responses.The super-ranking
concept through its squared Euclidean norm formalism as a multicharacteristic measure
provides a combined rank-ordered transformation of all implicated responses in
a new single manageable master response. Therefore, in a similar fashion to the
super-ranking approach, in this work, each response has its elements rank
ordered (Figure2). The
rank assignment process follows in the ordinary way, that is, (a) for the case
“smaller-the-better,” rank one is attributed to the entry possessing the
smallest magnitude, rank two for the next smaller entry, and so on; (b) for the
“larger-the-better” case, the largest response value gets rank one, the second
larger value gets rank two, and so forth. We remark that for the
“nominal-the-best” case, it is the absolute value of a response entry from the
nominal value that is rank ordered. The rank transformed response elements are
simply assigned as yij→Tij(j=1,2,…,n). The next step is to square
all response ranked elements.Figure 2
Response element rank-order transformation process.However, at
this point, we introduce the customer preference priority concept which is simply
expressed as numerical weight factors,wil(i=1,2,…,mandl=1,2,…,p),
assigned after each concerned customer has declared a quality trait hierarchy for
the product. For every customer, l, out of a total participating cohort of p members,
there are as many as m weight factors, which correspond to each response, i, of the m
considered responses. These weight factors possess the following properties:(1)foreveryl(l=1,2,…,p):0≤wil≤1(i=1,2,…,m),foreveryl(l=1,2,…,p):∑i=1mwil=1. Subject to
these constraints, the squared Euclidean norm (SEN) measure is still formed
just as in the super-ranking formalism but this time each squared-rank element
is firstly multiplied by the respective weight factor for a given customer
preference priority. This is easily understood from the depiction of Figure 3.Figure 3
Rank consolidation for all response elements to a squared Euclidean norm (SEN).The next step
is to form a super ranking (SR) vector by simply rank ordering theSENil elements one more time for each set of customer prioritized weight
factors separately. This transformation is better illustrated in Figure 4.Figure 4
Rank ordering of SEN data to SR vector form per customer,l.TheSRil(l=1,2,…,p) vectors
for every customer preference prioritization may be visualized as a set of p superranked
responses, each containing total weighted response behavior according to the
needs of p customers. It is worthwhile to facilitate, in a simplistic manner, the
optimal process management of a group of product quality traits synchronously under
the various quality response expectations that are reflected by a group of p
customers. Assuming all customers are important to this supplier at the same
degree then the family of superrank vectors depicted in Figure 4 should be
treated concurrently. This problem shapes up to be an application of the super-ranking
concept on a group of formed superranked responses. This technique may be
suitably called the nested super-ranking approach. Therefore, the nested optimization
setup now may be presented in the following form shown in Figure 5.Figure 5
Representation of an orthogonal array against a number ofp
superrank responses.Repeating now
once more the super-ranking technique on thep-customer superrank responses by
adopting again the squared Euclidean norm, this time denoted as Si(i=1,2,…,n) to
distinguish it from the squared Euclidean norm (SEN) which is dedicated to the
product characteristics from the previous step. Rank ordering the squared
Euclidean norm in the usual manner will result in the master superrank
response, MSRi(i=1,2,…,n), that contains the complete information
derived from product quality characteristic performance and customer priority
trait setting (Figure 6).Figure 6
Rank consolidation for all response elements to superranked form.All these
manipulations succeed in casting the original problem to that of a single
response unreplicated and saturated fractional factorial design lending itself
to the treatment outlined in [48]. In Figure 7, it is shown the
terminal form of the design. Now, a robust statistical analysis tool should
assign the significance of each controlling factor by incurring factor-level
contrasts. This is attained by sizing up directly the data grouped for each
factor setting separately. This way, factors are ranked according to the
smallest statistical significance. The ordinary method to sieve through the
effect activeness is to accept a level of significance and reject all those
factors that are not meeting or exceeding the predefined level of significance.
The nonparametric test of Wilcoxon-Mann-Whitney or rank-sum test is selected
for the analysis because of its high efficiency and its nondependency on
statistical distribution functions.Figure 7
Representation of an orthogonal array againstp number of
superrank responses.To enable the
use of Wilcoxon-Mann-Whitney test for data collected by a 2-level orthogonal
array, such as theL8(27) highlighted in the case study,
response entries accrued in the master superrank column of Figure 7 should be
separated in two groups according to the factor level they reflect. The case
study that follows this discussion will assist in comprehending this type of grouping.
Then, what remains to be done is either to sum the elements that belong to
either of the two-factor settings and compare it with a critical value
tabulated for several levels of significance in the appendix of an elementary
statistics textbook, or feed the two created sample groups into a software
package, thus obtaining the statistical significance of their medians. Since
the level of computational complexity for nested super-ranking calculations is
increasingly more demanding than that of a simple super-ranking application, the
mere use of a calculator may prove to be cumbersome in this case. Therefore,
the use of an ordinary spreadsheet will suffice in order to carry out the necessary
comparisons without the need to resorting to professional statistical software
packages. This may assist engineers to assimilate easier the method who,
otherwise, do not feel comfortable with such statistical software products.The hypothesis
testing for this kind of data comparisons reflects the examination of the
predicament that the two formed samples belong in the same common population
for all MSR entries, for each considered factor individually. Since we definedX1,X2,…,Xk the factors (including possible interactions) that need to be examined, then X11 and X12, X21 and X22,…,Xk1
and Xk2 are their
respective medians of their setting outcomes which correspond to levels 1 and 2
of a presumed 2-level orthogonal array. As expected, k is equal to seven for the L8(27) OA
showcased in the next section. Then, for each factor the null hypothesis, Ho and
its alternative H1, will be (2)HO:MedianX11=MedianX12ORH1:MedianX11≠MedianX12,HO:MedianX21=MedianX22ORH1:MedianX21≠MedianX22,⋮HO:MedianXk1=MedianXk2ORH1:MedianXk1≠MedianXk2, where k is equal to seven for the case of an L8 orthogonal array.Customarily, the
statistic that needs to be computed is the smaller sum of ranks when all
observations from each setting separately are pooled into one group while
keeping the tag on for each sum of ranks for each setting. If a critical value
defined for a specific level of significance for the Wilcoxon-Mann-Whitney is
exceeded then the null hypothesis is accepted. Otherwise, the alternative
hypothesis is accepted.


## 4. A Case Study

An important
industry is aluminum products manufacturing and great interest receives the
aluminum-alloy can-body stock which is indispensable material for seamless
one-piece containers in the beverages and soft drinks sectors. The continuous
attention to this product is manifested by the enormous amount of patented work
produced every year necessitated by ever economical and lighter containers
which possess improved strength properties in line with better machinability
during rolling processing and afterwards “drawing and ironing” forming
operations. The case study presented here will focus on the strip production of
such material that involves several phases including aluminum ingot casting,
heat-treatment homogenization, hot rolling, anneal treatment, and cold rolling
to the desired gauge specified by a can maker. For this specific study, nine
responses will be attempted to be optimized synchronously to seven selected
process and product factors. The nine responses regard alloy AA3004 and they
are as follows: (1) tensile yield strength (TYS); (2) tensile ultimate strength
(TUS); (3) tensile total elongation (TTE); (4) earing (Ear); (5) surface
quality (SQ); (6) necking/flange quality (N/F); (7) terminal thickness gauge
(Th); (8) terminal width gauge (W); and (9) buckle strength (BS). The
definitions for the first three responses are those found in standard material
science terminology. TYS and TUS
are measured in kpsi unit, and TTE is measured in terms of percent deformation. TYS and TUS are optimized about their respective
target values with further supplemented lower and upper bounds usually set by
the customer. The TTE is expected to be maximized for improved formability while
there is desirable minimum value which below that the material is regarded as a
failure. Earing occurs after the two-phase process of deep drawn containers stamped
from aluminum textured sheet in the cupping press is further subjected to
ironing in a bodymaker to achieve the specified can height. This property
essentially quantifies the difference between the ears (hills) and the troughs
(valleys) appearing at the edge of a can. Earing is also measured in terms of a
percentage and since it is undesirable because it interferes with flange width consistency
during the lid fitting process, it needs to be minimized. Thickness and width
are tabulated in terms of a difference for each of them reflecting the extent
of the discrepancy; the measured gauge of examined samples possess with respect
to can-maker's specified nominal values, respectively. Coincidentally, this
form of representation serves two equally important purposes. First, measurements
do not need to be manipulated according to a “nominal-is-best” treatment.
Taking simply the difference of each data point to the nominal value, the
condition is transformed, in the “super-ranking” formulation, to the
“smaller-is-better” optimization category. Thus, what are sought are the
optimum conditions that will drive this difference to zero. Second, the identity
of the can maker or a group of them as it may be is not revealed through this
study since the combination of terminal strip thickness and width specifications
usually relates to particular can manufacturing enterprises. Surface quality is
categorical variable in ordinal scale based on alloy surface crystallographic texture
and it is rendered after a metallurgical laboratory examination. The rating
ranges from a scale 1 to 5. “Very good” is reflected by rating 5, “good,”
“average,” “poor,” and “very poor” are set aside for ratings 4, 3, 2, and 1,
respectively. “Necking and flange” characteristic is a quality property that is
evaluated in the customer's can-making systems. It has to do with the
consistency the material has in shaped can form after drawing and ironing. Flange
width is crucial in fitting the lid to the necked can at the specifications
dictated by lid dimensions. Just as with the surface quality characteristic the
measuring scale for N/F is of the same nature, the only difference being that
is an evaluation that it is provided by the customer. Buckle strength (dome
reversal pressure) is measured in psi. It is considered as an added material strength
property that dome-shaped can bottoms should possess to withstand canned
internal pressurized contents. This restriction is communicated by the customer
in terms of a minimum specification pressure value which is the point that the
can bottom inverts its shape from concave downward to convex downward.
Therefore, this characteristic should be maximized.A selected
group of seven factors modulates the responses outlined above. Two of these are
the manganese (Mn) and magnesium (Mg) contents expressed as a percentage by
weight. Homogenization treatment was studied according to two process
parameters which are the furnace temperature (TempH) and duration of slab
heating in the furnace (TimeH). Moreover, the importance of two annealing
treatments was investigated, one after hot rolling and the other after cold
reduction. In both cases, the process parameter was the annealing temperature
(in°C) as the annealing interval was maintained constant all
through the experiments. The temperature for the first annealing treatment is
coded as AnA and for the second treatment as AnB. The last factor is the cold
reduction (CR) referring to the thickness gauge reduction incurred by cold
rolling and it is recorded in terms of a percentage with respect to the
incoming workpiece thickness gauge. The study is oriented towards a screening
phase investigation. Therefore, the management team at this point would prefer
to observe comparative behavior among all engaging factors at some reasonable
process and product parameter extremities that nevertheless are not foreign to
tested recipes in previous patent work. Hence, it was thought that a two-level
L8(27) orthogonal array should be utilized in selecting the appropriate trial run
combinations. Resorting to OAs makes sense from an economical perspective in
this situation because each workpiece resulting to final product may pile up a
cost of the order of $100 000 requiring a combined cycle time of one to two
months to produce it. By combined cycle time, it is meant time-to-product-completion
in both sites, at producer's and customer's, respectively. The comprehensive
input selections are tabulated in Table 1.Table 1
Factor coding and factor settings for the multiresponse optimization study.
Coded symbolsControl factorLevel 1Level 2MnManganese content (%)0.61.5MgMagnesium content (%)0.71.8TemphHomogenization temperature (°C)530580TimehHomogenization
duration (hours)624AnAAnnealing before cold rolling (°C)260370AnBAnnealing after cold rolling
(°C)390560CRCold reduction (%)3565Data gathered in Table2 is given in the form of the L8(27) OA arrangement
to enhance readability. In Table 3, response data received a rank ordering for
each response separately where the event of a tie is honored. Responses of TYS and TUS are ranked according to their differences
from the targeted values of 46 and 49 kpsi, respectively. The smaller the
absolute difference, the lower the rank each response entry receives. Similarly,
the response gauge differences for Th and W are ranked based on their absolute
differences with their nominal values. It is interesting to observe the tie occurrence
for the categorical ordinal data for responses SQ and N/F when translated to
the corresponding rank variable. There is reverse meaning in these values as
the maximum for the ordinal scale of 5 will get the first rank, the rating of 4
will be rank rated as 2 and so forth. However, it is noteworthy to stay at the
convenience this method may offer in mingling categorical data with numerical
data of any dimensional basis in any multiresponse optimization circumstances.
The rank ordering provides uniform scaling to all measures thus simplifying enormously
the data analysis in the subsequent step. The coded names for the transformed
rank-ordered responses mirror those of the original data response variables
with the prefix addition of “R,” for “ranked,” as they appear in Table 3.Table 2
Data collected for the nine responses due to the adjustments of the seven selected factors stated in Table1 in terms of a L8(27)
OA arrangement.
MnMgTempHTimeHAnnAAnnBCRTYSTUSTTEEarSQN/FThWBS111111138412.02.3431192111222251532.32.7510296122112254563.02.9330299122221145483.23.1321195212121244471.71.853−1091212212143452.72.8240192221122150514.03.633−1294221211247502.02.2250193Table 3
Rank ordering of data presented in Table2.
MnMgTempHTimeHAnnAAnnBCRRTYSRTUSRTTEREarRSQRN/FRThRWRBSSSRSR11111117.58.0633.04.56.53.56.5291.25811122226.05.5541.58.02.57.02.0232.75512211227.57.0365.04.52.57.01.0251.75612222111.51.5275.07.06.53.53.0195.00221212123.03.5811.54.56.51.08.0216.00421221214.05.5457.52.02.53.56.5208.25322112215.03.5185.04.56.57.04.0254.75722121121.51.5727.51.02.53.55.0158.251At this point, it would be interesting to discover the factors that may prove to be
statistically important in influencing all responses concurrently taking in
account the various ratings that may be placed on each of the implicated responses
by a host of can makers (customers) simultaneously. This very realistic problem
causes cumbersome alterations in production schedules of this nature.
Extensions of this notion to other massively produced goods are easily
perceived. From the standpoint of the impending statistical analysis, this
calls for what has been described in the methodology section of this article as
nested super-ranking in unreplicated and saturated fractional factorial design.
Indeed, application of the super-ranking method on weighted responses according
to each participating customer preferences will generate a single superranked
response for each of the customers separately. Pooling all individual customers
superranked responses to a new single master superranked response, this time across
all customer base considered and performing the super-ranking methodology one
more time on this single response, it would generate a solution that may be
worth of interpretation.Before proceeding with the computations, it would be beneficial to check the results
shown in Table3 with some other competing scheme. This test would provide
confidence in handling such complex problem as outlined in the previous
paragraph. To this author knowledge, it was not easy to retrieve other
well-tested and easily accessed method on multiresponse optimization worked out
in concert with orthogonal arrays besides the highly regarded technique of desirability
analysis. Since SQ and N/F responses are assigned categorical data and the
“Response Optimizer” tool in MINITAB 15 does not discriminate against
continuous or discrete data, we treated the data shown on Table 3 unaltered. The
input bounds for the MINITAB analysis are given in Table 4 and they are these
that are specified by the average customer. To keep things simple at this stage
there was no attempt to introduce importance ratings or weights, according to
the terminology followed in the desirability analysis. It was thought that this
may provide a better sense of comparison between the two methods. The output
produced by MINITAB 15 is tabulated in Table 5. Clearly, the overall
optimization effort may be characterized as not a successful one as it may be
inferred by the composite desirability of about 0.43. Nevertheless, at this
level of difficulty, the desirability analysis suggests that the responses TYS,
TUS, and W have been predicted in a satisfactory fashion with the Th being
exceptional.Table 4
Required response bound inputs for  MINITAB 15 data analysis by the desirability method.
ResponseLowerTargetUpperOptimization goalTYS404652TargetTUS434955TargetTTE1.53MaximizationEar02MinimizationSQ35MaximizationN/F35MaximizationTh01MinimizationW01MinimizationBS9095MaximizationTable 5
Optimization by the desirability method.
Response optimizationGlobal solutionMn = 2Mg = 1.35TempH = 2TimeH = 1AnA = 1.48AnB = 1CR = 2Predicted responsesTYS = 44.73, desirability = 0.789TUS = 47.77, desirability = 0.796TTE = 1.80, desirability = 0.206Ear = 1.88, desirability = 0.056SQ = 3.91, desirability = 0.457N/F = 3.94, desirability = 0.474Th = −0.74, desirability = 1.000W = 0.17, desirability = 0.823BS = 91.88, desirability = 0.376Composite desirability = 0.429Furthermore, DA forces a solution for Mg and AnA that do not involve one of the two
investigated factor levels. For viewing convenience, the optimized response
graphs produced by MINITAB 15 are shown in Figures8(a) and 8(b). It should be
noted that the solution is portrayed in two parts since the long list of responses
could not be fitted in one view as the generated solution appears in a roll-down
window form hiding some of the responses. In turn, this causes duplication of
some of the response results between the two graphs. However, as it can be
witnessed from Figure 8 by focusing on the subplots for composite desirability,
it may be inferred that factors TempH, TimeH, and AnA definitely may be omitted
from further consideration.Graphical solution for the nine responses
for the no-weights, no-importance-rating
optimization problem based on the data of Table2 (plots (a) and (b) offer the complete
solution according to the desirability method (MINITAB 15)).
(a)(b)The corresponding solution obtained through the original super-ranking method is
presented in Table6. The Mann-Whitney test assigns statistical significance to
median central tendencies between factor levels. It is found that only the homogenization
duration rejects the null hypothesis at a significance level of 0.05. This
seems logical as for level 1 and 2, the median SSRs are 253.3 and 201.6,
respectively. Thus, in contrast to the result provided by the desirability
analysis method, level 2 should be the optimal setting according to super-ranking.
The next three more important factors that happen to possess the same
statistical significance of 0.235 are Mn, Temph, and AnB. This conclusion is in
conflict with that reached by the desirability method in the previous
paragraph. One more attempt in resolving this difficult problem is to accept the
sum of squared individual ranks, that is, the squared Euclidean norm according
to [48], as a single response in Table 3. This response is listed as
SSR in the same table. Therefore, the popular tests for single unreplicated and
saturated response offered by the half-normal method and the Pareto analysis/Lenth test may be enforced here. This action is partly justifiable because as
it can be seen from Table 11,
the normality of all individual factor
levels is found to be reasonable, according to Anderson-Darling test, just as
it is for the total eight points in the SSR column as demonstrated in
Figure 11(a).Table 6
Statistical significance assignment according to the super-ranking concept.
LevelsMedian ranksMann-Whitney test statistic WPMn15.521.235Mn23.5Mg14.520.332Mg24.0Temph16.021.235Temph23.5Timeh16.525.030Timeh22.5AnA14.518.500AnA24.5AnB13.015.235AnB25.5CR15.020.332CR24.5However, the output from MINITAB 15 in Figure9
(plots (a) and (b)) produces admittedly no strong candidate even at the
significance level of 0.10. These results advocate further the difficulty of handling
convincingly these kinds of multiresponse problems thus justifying the
continuous need for advanced probing techniques in the area of
multicharacteristic optimization.Data for SSR from Table3 tested with (a)
the half-normal test and (b) Pareto analysis/Lenth test.
(a)(b)At this point, let us return to the
initial problem of detecting possible factors that influence the total
superranked behavior no matter what the customer prescription for the response
weights it may be. For illustrational purposes, this paper will consider the
optimization problem based on the importance rating given by six can makers.
The relative weights corresponding to each of the customers are summed total to
unity while they are applied on the squared ranks for each response separately.
Therefore, for the six can makers there will be six individual superrank
responses. All these responses will face one more level of squaring and addition
thus creating a new squared Euclidean norm which a final ranking will generate
a master superrank column to be contrasted for factor importance according to
the originalL8(27) OA. The selected input weights for
the six can makers are given at Table 7. Table 10 lists the super-ranking
response for each of the six can makers separately. The responses are coded
according to the customer identification, for example, the super-ranking
response created by the weights set out by can maker A is listed as SRA, for
can maker B as SRB and so forth. On the
same table, it is also listed the combined super-ranking variable, MSR, the acronyms standing for “master
superrank” response, which contains all modulations in the nine responses due
to the six can makers. Therefore, this quantity may guide the discovery of
important controlling factors that would influence simultaneously all
considered responses for all participating customers. The MSR is created by
squaring the individual ranks for every customer super-ranking response on Table 10, that is, by creating a new Euclidean norm that encompasses the customer
dimensionality across all trial run selections of the L8(27)
OA. Thus, this essentially justifies the need for a nested super-ranking
routine. This value is listed as MSSR, for master sum of squared ranks, in Table 9. For convenience, in the same table, it also is listed the sum of squared
ranks for every individual can maker. The coding for these values is attributed
for each can maker individually, that is, for can maker A, the SSRA labeling on
Table 9 stands for sum of squared ranks for can maker A, SSRB for can maker B
and so forth. By ranking the elements of MSSR column (Table 9), the rank-ordering
outcomes for MSR are generated at Table 10. Treating the MSR entries with the
classical method of Wilcoxon-Mann-Whitney, just as it was discussed in Besseris [48], then the statistical significance of the consolidated
behavior of all nine responses due to preferential selection of six can makers owing
to the seven controlling factors is tabulated in Table 8. The first impression from
this analysis is that the homogenization duration is statistical important at a
significance level of 0.10 while is very close to solidify this indication at a
significance level of 0.05. Moreover, it is seen that the factor level Timeh2
amasses the smallest median of the two levels which corresponds to a setting
(from Table 1) of 24 hours. At this stage, there are no other controlling
factors that may provide statistically important influence on the combined
behavior of all implicated responses simultaneously for the traditionally
accepted significance level settings of 0.05 and 0.10. For practical reasons,
while understanding the enhanced complexity of this optimization problem at
hand, one may desire to consider more factors for setting adjustment. If
someone is willing to accept the controlling influence of factors Mn and AnB at
significance level of 0.156, then the desired factor settings should be Mn2 and
AnB1 which physically correspond to a manganese content of 1.5% and an
annealing temperature treatment after the cold reduction of 390°C.Table 7
The nine response weights as listed by the six participating can makers.
CUSTOMERTYSTUSTTEEarSQN/FThWBSTOTALA0.150.150.150.20.10.10.050.050.051B0.120.120.110.20.150.10.050.050.11C0.30.30.30.020.020.010.010.020.021D0.10.10.10.20.10.050.050.150.151E0.20.20.20.30.020.020.020.020.021F0.050.050.10.10.050.050.30.20.11Table 8
Statistical significance assignment according to the super-ranking
concept for simultaneous six-customer optimization.
LevelsMedian ranksMann-Whitney Test Statistic WPMn16.022.156Mn23.5Mg14.521.232Mg23.5Temph15.520.332Temph23.5Timeh16.524.056Timeh23.0AnA15.520.332AnA24.0AnB12.514.156AnB25.5CR15.020.332CR24.0Table 9
Squared Euclidean norm (sum of weighted squared ranks)
for each customer response importance rating.
MnMgTemphTimehAnAAnBCRSSRSSRASSRBSSRCSSRDSSRESSRFMSS1111111291.2533.0030.5248.9529.6236.4731.323411112222232.7526.4823.8029.5024.0125.5622.801271221122251.7531.6729.4636.7629.9535.6823.802611222211195.0021.6523.065.3720.9019.1525.25522121212216.0020.6020.7227.5721.8219.9527.96872122121208.2523.4026.3021.4925.5422.3717.981022211221254.7528.4329.3215.1832.0029.9034.702402212112158.2516.7318.6918.0717.7313.9115.2114Table 10
super-ranking according to each participating customer along with resulting master super-ranking due to
simultaneous customer contributions.
MnMgTemphTimehAnAAnBCRSRSRASRBSRCSRDSRESRFMSR111111188886878111222255464535122112267777747122221123312252212121242253363212212134545424221122176628686221211211131111Table 11
Anderson-Darling Normality Test for the squared Euclidean norm (SSR) shown in
Table3 in the text.
LevelsAnderson-Darling Normality TestA2 (P)Mn10.16 (.842)Mn20.23 (.577)Mg10.40 (.176)Mg20.33 (.294)Temph10.23 (.566)Temph20.29 (.384)Timeh10.25 (.516)Timeh20.19 (.745)AnA10.16 (.847)AnA20.15 (.873)AnB10.23 (.551)AnB20.30 (.360)CR10.24 (.527)CR20.28 (.407)It is noteworthy to mention that the behavior of the super-ranking response for the
case where no specific information for the importance rating provided by the
customers due to the controlling factors (Table6) is essentially the same as
that for super-ranking response modulated by the response priorities of the
group of the can makers in a statistical sense. However, the statistical
significance for the three considered factors above, Timeh, Mn, and AnB are
pointing towards different direction. In other words, the inclusion of customer
multipreferences caused a weakening of the statistical significance of Timeh
from 0.030 to 0.056, while for Mn and AnB contributed an observed increase in
statistical significance from 0.235 for both of them to 0.156 for each of the
two, respectively. Overall, the same conclusion would be reached qualitatively
for establishing the important controlling factors if we had used the super-ranking
information collected under the heading SR at Table 3 (or Table 10). This means
that for the specific group of response priorities assigned to the customer
base of Table 7 the modulation of individual rank responses were not strong
enough to alter the behavior of the composite response ranking. This is quite
surprising if one takes in account how different the relative weights were as
nominated by the six can makers in Table 7. Obviously, it is a coincidental
occurrence attached only the particular combination of weights and can makers.
It is stressed that in order to form the ranked entries for the columns SRA, SRB, and
so forth, at Table 10, first the weighted-squared-rank-sum (squared Euclidean
norm) to customer priorities for each can maker is formed at Table 9, expressed
under the headings SSRA, SSRB, and so forth.Two alternative ways were used to
estimate the effect of the seven controlling factors on the combined
weighted-squared-rank-sum response or the master sum of squared ranks response,
MSS (Table9). First, we observe from the Anderson-Darling normality analysis
of the fourteen-factor settings that only two of them seem to have questionable
indication at a significance level of 0.10, that is, Mg1 and AnB1
(Table 12). Furthermore,
the distribution of the eight elements of response MSS seems to comply with
normality (Figure 11(b)). Therefore, we apply two popular tests for
unreplicated and saturated designs: (a) the half-normal test and (b) the Pareto
analysis/Lenth test by making use the respective module in MINITAB 15. The
results are shown in Figure 10 as plots a and b for the two corresponding
graphical tests. It is clear that both tests are not able to decipher any
dominating contributions from the seven parameters that we examined. Indeed, likely
dependencies on the master sum of weighted squared ranks (MSS) are remote even
at the level of significance of 0.10. In spite of no meaningful active effects,
the Pareto analysis/Lenth test has the parameter Timeh leading the rest of
them. On the other side, the half-normal test cannot distinguish a factor
nomination at all for this particular case study. At this point, no further
comparisons may be made between the nested super-ranking method and the two
classical graphical methods employed here. However, possible reasons for these
severe discrepancies may be partially explained by the discussion in Besseris [47]. Hopefully,
this irreconcilable condition on predicted results between well-known graphical
means and robust permutation-based tests may become ground for additional
investigation in forthcoming publications.Table 12
Anderson-Darling Normality Test for the squared Euclidean norm (MSS) shown in
Table9 in the text.
LevelsAnderson-Darling Normality TestA2 (P)Mn10.20 (.685)Mn20.29 (.404)Mg10.58 (.046)Mg20.39 (.182)Temph10.16 (.847)Temph20.45 (.119)Timeh10.27 (.439)Timeh20.20 (.693)AnA10.21 (.637)AnA20.27 (.437)AnB10.48 (.098)AnB20.36 (.244)CR10.23 (.584)CR20.22 (.616)Data for SSR from Table8 tested with (a)
the half-normal test and (b) Pareto analysis/Lenth test.
(a)(b)Normality test for SSR values from Table3 tested (a) and MSS values from Table 9 (b).
(a)(b)


## 5. Discussion

In this work,
there was an attempt to interpret synchronous multiresponse optimization
information under the added constraint of weighted modulation imposed on the
individual influence for each of the engaging responses. To this end, a simple
statistical model has been developed based on the rank-ordering concept
propounded by Wilcoxon, and Mann and Whitney (WMW). This weighted-response
model accumulates the rank-ordered information for each response in linear
fashion and creates a new superrank response that is processed by the WMW
nonparametric test. Therefore, a terminal single response contains compressed
information from a given number of quality characteristics while representing
all of them with a preassigned level of importance among them. Mainstream
competing techniques that incorporate response weighting in their scheme mainly
spawn around the concept of either the desirability analysis or some indigenous
expert system. Both of these methods quite often require a professional
software package to assist in the decision-making and it is handled
meaningfully by a well-trained specialist. Furthermore, mixing up expert
systems with statistical techniques will always relay useful, well-received
conclusions. However, it is easily anticipated that it is hard to acclimate
production personnel to invest in professing such intriguing methods. Quality
and process control personnel desire simple techniques that they realize the
steps taken to reach to a solution. The technique presented here to resolve a
fairly complicated situation often encountered in manufacturing operations,
that of optimizing process or/and product parameters under multiple
specifications concurrently, has several advantages over the aforementioned
popular alternatives. One striking advantage is that the use of software is not
important with this new method after all. Graphical solutions need not to be
drawn. Simple calculator work will suffice in producing results while the usage
of a spreadsheet may be demanded merely in the most difficult cases. This
convenience emanates from the fact that rank statistics are so simple in
understanding the meaning of them in the first place and making estimations
based on them follows naturally afterwards. This is particularly true for the
low sampling schemes this method is intended to be applied at. Suitable OA
family candidates will be members of the seriesL4(23), L8(27),
L12(211), and L16(215). In
addition, it is easily realizable the extension to nonlinear optimization
problems and this will be the subject of a forthcoming publication.Among the strong points of this
extended technique to the one published recently [48] is the convenience in
incorporating any number of numerical and categorical responses through the
rank-ordering technique. This herein outlined methodology when compared to
previous work on the super-ranking concept becomes now a more versatile tool by
allowing the inclusion, in a trivial manner, of the weighted influence among
the responses according to priorities set by a resourceful customer. Then, in a
subsequent, nested super-ranking step,
the total behavior elicited by the customer priorities, established by for all
considered customers simultaneously, are taken in account in achieving optimal
product and process states. The methodology stretches its utility further by
implicating in the treatment multiplecustomer priorities that via to the
Euclidean norm definition receive a judicious consideration against the
engaging production parameters. This is actually accomplished in the second super-ranking
step that in principle nests-in the consolidated response information created
for each customer separately. Tracking the active effects is of paramount
importance even more so often in lean production. Under such circumstances,
production teams are often left clueless in lack of systematic and yet
convincing solution methods that would otherwise propel them to attain optimal
production levels for specifications set simultaneously from a score of
different customers without altering their production cycle each time. The case
study presented on aluminum milling had that goal in mind to become a useful
example for practical implementation in other applications in manufacturing.The method investigated in this
article could have been the sister method for weighted MANOVA. But this latter
technique is rendered quickly nullified under the inherent constraint of the
unreplication and saturation for fractional factorial schemes imposed
invariably on a experimental plan due to anticipated economical requirements
and trial duration restrictions that result in calling for exploiting in full
OA column occupancy. This nullification stems from the technical complication
arising due to the eliminated unexplainable error and the nonexistent degrees
of freedom for the error. Therefore, being able to judge statistical
significance behind nonrecoverable variability has always been an unwavering
challenge with no consensus in sight for a dominating manner of handling it.The basic assumptions for utilizing the nested super-ranking concept may be
discerned in two parts. The first part has to do with the criteria for
selecting a nonparametric tool for multifactorial processing and this has been
outlined in detail in [47]. In brief, the main motivation for employing
Wilcoxon-Mann-Whitney statistics is the nonreliance on knowing the nature of
the distribution of data points within each factor setting. This simplifies
greatly the task of analyzing the data when the engineer or scientist that is
responsible for carrying out such an endeavour does not possess an in-depth
knowledge on statistics. For a typical working sample range of two to six,
nonparametrics offer a more objective information imprint comparing to normal
statistics as central tendencies based on medians are more robust in contrast
to the means [46]. When factor settings extend to larger sample sizes, benefits
gained by the use of order statistics are dissipating unless the distribution
of the data is evidently deviant from normality, thus obtaining additional
estimations from nonparametrics may be advised. Therefore, the method outlined
herein may also complement other leading techniques such as the half-normal
plot, the Lenth test, the Bayes plot or the pooled ANOVA approach when there is
a discord among their predictions. In as much as there is strong indication that
data normality permeates the factor setting differences then any of the
aforementioned techniques may be suitable for providing reliable inference. The
second part regards the super-ranking concept that it is shown to compete in
effectiveness with more elaborate approaches such as a fuzzy logic type of a
treatment [48]. super-ranking was shown in the last reference to mix up
conveniently and efficiently responses belonging to any of the three categories
without requiring particular modification of the ordering transformation
tactic. The intrinsic mechanism of forming the superrank vector derives
ostensibly but having a theoretical meaning thereof by considering each
participating response entry in the Euclidean norm measure.Another way to treat these problems
could have been through the desirability analysis where the importance of each
of the responses is encompassed in a model. However, when some of the responses
happen to be of categorical nature, like the ones presented in our case study,
it is not an easy feat to handle it by this method. Indeed, the incorporation
of categorical data was tested in the aluminum product improvement study with
limited success. Thus, this is another point where the method of nested super-ranking
presented here may hold an advantage over the desirability method (DM). Rank
ordering assimilates easier the inherent variable mixing while establishing
statistical significance in the classical sense. One obvious advantage of SR
over DM is the way SR establishes a statistical solution irrespectively of some sort of starting values and other
peripheral connotations that may need to fuel the DM model. SR produces
solutions based on available data hinging on both central tendency and variance
effects alike. However, DM operates only on mean estimations that may require
further investigation for their statistical significance in the first place. On
the other hand, it may be mentioned that DM is driven by connotations rendering
each time a new solution.While the methodology outlined in this paper is recommended to assort the employment
of the popularL4–L16 orthogonal array series, there is
no essential restriction for the nonparametric data processing scheme presented
in this work to be espoused for any type of two-level fractional factorial
design. The interested practitioner merely ought to consult Mann-Whitney's
paper for retrieving the appropriate statistical significance limit regarding
each time the proper tested sample size.A great advantage of nested super-ranking is that collapses two kinds of data,
responses and their concomitant weights, to a single master response that is
expressed by entries belonging to the realms of real numbers. Therefore, the
MSS response  is easily treated for multicollinearity effects as if it was
the classical one-response problem. In turn, formal regression methods may be
straightforwardly employed to obtain the estimated regression coefficients (R2)
for every participating factor thus allowing the computation of their
tolerances (=1−R2) leading to the variance inflation factor
(VIF). For VIF values greater than 5 the
multicollinearity effect should then be detected [49]. Hence, overfitting
phenomena may be routinely pinpointed in the proposed method and eliminated.Another obvious advantage that the nested super-ranking technique possesses is that by
encompassing all information in a new single variable by judiciously absorbing
robust representation of data from various sources essentially succeeds in
directing the optimum solution according to all product traits simultaneously.
This is a highly sought feature in a multiresponse technique that may
incorporate multiple correlated responses. When correlated responses are
present employing a technique that analyzes each response separately must be
avoided, otherwise conflicting recommendations about factor activeness are imminent
[50]. Clearly, the suggested method does not ignore a possible correlational
nature among the considered responses thus offers objective view of the
improvement direction.Based on the six-customer group (Table7), the screening optimum factor has been
found to be the homogenization
duration set at level 2 which corresponds to 24 hours. The confirmation
experiment was tested on a cohort of a six-slab batch processing for each
customer order separately. Out of the six-slab batch, a three-slab subgroup was
randomly selected and their nine product properties were measured and recorded.
In a total of eighteen product coils, their properties were analyzed
statistically against the predicted product performance. It was found that
tensile yield strength, tensile ultimate strength, and sheet thickness were
reproducible to their predicted values by an average of 11%, 13%, and 7% (P<.05), respectively. The rest of the
product traits remained unaffected as predicted by the proposed method.Lastly, there is always a need to
reiterate the advantage the rank statistics maintain over normal-oriented
formulations in the low sampling conditions experienced in OA experimentation
engaging the family membersL4(23), L8(27),
L12(211), and L16(215). The
superiority in efficiency and the tight confidence intervals that permutation
techniques such as the WMW test is known to accomplish provide the necessary
and undisputed backbone for robust design through robust statistics. The effect
sparsity assumption is abolished just as with the simple SR scheme [47].
Relying on to this method, the philosophical notion about a sample possessing a
normal or some other more complicated behavior may not need be of concern for
the every day product developer that has to answer pragmatic quality trait
puzzles by day's end. Hopefully, this way of data handling as demonstrated here
may assist those agonizing industrial practitioners that must provide an answer
without feeling apprehended about their lack of an in-depth statistical
background or because there is limited ‘turn-around’ time to place robustness
to their decisions.


## 6. Conclusion

A technique has been developed in this paper to assist in resolving unreplicated and
saturated multiresponse optimization problems when there is available
information about the relative weights among the engaged product
characteristics. The technique embodies possible multiple weight assignments
for the implicated responses reflecting multicustomer requirements. The nested super-ranking
scheme presented here is based on the well-tested nonparametric comparisons
while making use of data obtained by an efficient orthogonal array. The massive
collapse of multiresponses via order transformation to a single master response
grasps multicustomer preferences which in turn are utilized such as production
conditions are optimized concurrently for all customers. The theoretical exposition is illustrated by
a seven-factor, nine-response, six-customer optimization problem encountered in
aluminum milling operations. It is demonstrated that the suggested technique
may outperform standard tests of comparisons including the Lenth test and the
half-normal plot when some factor setting data may exhibit a tendency towards nonnormal
behavior.



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*Source: 156851-2008-12-22.xml*