In essence, this tutorial is the Author's paper entitled Developing a Single
Performance Indicator From Hierarchical Metrics that was peer reviewed and
published in the 1995 National Council on Systems Engineering (NCOSE) Symposium
Proceedings. (In that year, NCOSE became INCOSE, for International.)
Abstract. A contemporary System Engineering task is development of product/process performance metrics. Executive management desires a single performance indicator that combines subsidiary metrics into a meaningful summary representation of organization health with respect to its mission. A purely mathematical approach can produce such an indicator, but when constituent metrics also contain subsidiary metrics, a multiple stage method is required. Because not all metric result customers are fully competent to audit the result, this paper sets forth a generally understandable method that "presells" the output to all involved parties through their approval of each model base before alternatives performance data are entered. The eminent decision support capability inherent in this method suits it for complex tradeoff analyses and Capability Maturity Modeling.
At the end of this tutorial is an opportunity to pose question(s) on this subject.
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any of the listed subsections to quickly reach this option.]
Increasing emphasis on continuous improvement of process, as well as product,
requires ever increasing employment of qualitative measures as the constituent
elements of their definable metrics. But, mixing of subjective with objective
measures (apples and oranges comparison, so to speak) requires their conversion
into a common currency (fruit?). That is why cost/benefit analyses convert the
benefits into cost-like numerics. Non-numeric values must be quantified for
combining with numeric values to establish viable process metrics. Subsidiary
qualitative metrics also must become numeric quantities, for combining into a
higher level metric.
Factors weighting is determined by customer surveys of their relative
importance. Customer rating for each element at its calculated total rating on
a 1 (dissatisfied) to 6 (extremely satisfied) ordinal scale is plotted for a
significant number of projects and a linear "best fit" line overlaid.
Its angle of slope displays aggregate level of customer sensitivity to changes
in the Quality, Schedule, and Cost factors for prioritizing improvement
efforts.
The metric development method must allow customer tailoring of combined metrics
models. If not, regardless of the cost or choice of "operator," the model
won't effectively assist organization performance improvement efforts and
another method should replace it.
A development method should assist cognition (of everyone involved) if it is to
result in the consensus selection of a "best" indicator from potential
solutions. A method for development of process metrics should be easily
computerized, be understandable by all users from naive to technically astute,
be usable anywhere, be supportive of combined judgments, and be logical in
approach from start to finish. Users should know what, why, when, and how to
perform each method step, for adequate effectiveness.
Combined metric quality arises from appropriate and consistent treatment of
every subsidiary element with regard to preferences, data uncertainty, depth of
research, performance data evaluation, and approach to justification. Method
structure and methods should be compatible with normal tasks, organizational
structures, and company policies.
Not all customers of a performance indicator perform each method step, although
someone must. Rather, all parties should have sufficient method understanding
to either control or contribute to the metric development and to evaluate the
results without specialized training and experience. The method should be
neither complex nor hidden from users and should be perceived as both complete
and supportive while used during meetings and by individuals, without requiring
mathematics support from technical people.
The method should allow custom tailoring of models to fulfill metric goals.
With flexibility of solution logic as well as variable input data forms, "What if
... ?" questions are answered quickly by the model.
A few simple ideas underly the steps of a systematic, effective method for
developing single performance indicators. But what is effectiveness, in this
context? Process effectiveness is a measure of how well you (or your
group) identified, specified, accomplished, and verified fulfillment of
organizational purposes after performing the defined set of procedures. It
measures how well you did, compared to what you should have done (assuming
validated requirements).
Method efficiency, on the other hand, is closely related to how much time
you must spend on all of the various procedural inputs to obtain a satisfactory
output of required results.
The combined metric development method described herein is prescriptive, to
help you and those you will counsel to build better performance metrics.
Back in 1981, this author worked with MAUA. FMC proposed a tracked vehicle for
an USMC weapon system design concept study. The government-provided system
evaluation model was hierarchical and modular. It used utility graphs for
qualitative as well as quantitative metrics. It summed coefficient weighted
utility scores to obtain a single figure of merit for evaluating the competing
systems against a hypothetical ideal system. Each designer thereby learned what
effect any design attribute would have on system performance and the resulting
utility score.
MAUA is not fully described here, because adequate theoretical treatment is too
lengthy, but those portions important to the described method have been
retained.
The proprietary monicker for the integrated ensemble of tradeoff study method
steps is Decision Support Incorporating Documented Evaluations (DSIDE^{(tm)}), to be
pronounced as "decide") method. Usefulness, clarity, and extreme
simplicity were the primary objectives for developing the method whose
application to developing a single performance indicator is next set forth.
The few simple steps in the DSIDE^{(tm)} method are:
A. Define the purpose
B. Partition problem into model structure
C. Make utility graphs for model primitives
D. Weight the model elements
E. Sum weighted scores for measurement results
F. Document the method and its conclusions
Although "steps" are listed for a complete trade study method (useful
for the combined metrics development), blind rule-following is neither implied
nor recommended.
You will note general similarity between the listed steps and methods you have
seen before. The major differences are critical enhancements that fulfill the
combined metrics criteria listed on the first section.
Step A initiates "preselling" final results. The second big contributor
(step B) is the understandably structuring the problem. Another "preselling"
contributor is the explicit approval (and retention) of each method product
developed during steps A through D. When required approvals empower those with
relevant knowledge, total method time is minimized. The next subsections of
this paper provide concise overviews of the listed DSIDE^{(tm)} method steps as
used to develop a single performance indicator from subsidiary metrics.
This step resembles System Engineering requirements analysis in a system
development project where you must fully understand a problem before you can
solve it. Identify the purpose to effectively apply your efforts toward solving
the "right" problem. Until you establish the purposes of your study,
you can't know which criteria are important or what priority (how much weight)
to set for each.
Purpose definition is in terms of goals and subsidiary parametric criteria.
Attainment of goals is measured by fulfillment of the defined criteria. This
first step, then, consists of listing the goals to fulfill the purposes and
their constituent criteria, justifying them sufficiently to obtain consensus
among those involved in model design. Approval of the defined purpose list
achieves initial results preselling.
The model framework, the structure of the hierarchy of the listed goals and
criteria, is the output of this step. To form the model structure, use a
top-down method to classify the listed goals and their criteria into categories
for easy comprehension of its parametric elements.
Starting with the problem's overall purpose, you know it has goals which can be
classified into separate categories for grouping parametric criteria. Next,
divide the categorized first level goals into their subsidiary subgoals and/or
criteria and determine interrelationships. Work down from each less certain,
fairly general, parent goals to the next level of subgoals. Continue division
until each branch ends with one definite, primitive (undividable) criterion as
the leaf. As will be shown in the next step, the criterion may be quantitative
or qualitative. The branching hierarchy you create in this way should resemble
an inverted tree structure.
The preferred Model hierarchy is the least complex structure which encompasses
all significant criteria.
The structuring method improves upon simply listing problem criteria because it
divides a complex problem into manageable components which facilitates the next
two method steps of scoring and weighting as well as general comprehension of
the overall model and an earlier attainment of group consensus.
In the TSR metric example, goals and their criteria are parsed into Quality,
Schedule, or Cost categories. Their criteria are collected as child elements.
Each goal or subgoal can have its own subelements, to whatever depth complexity
of the problem requires, ending each branch with a primitive criterion.
Expressing the model hierarchy numerically, the hypothetically combined TSR
metric example elements are:
1.0 Quality Index (Q) [Goal]
1.1 Q Metric 1 [Criterion]
1.2 Q Metric 2 [Subgoal]
1.2.1 Q SubMetric 2.1 [Criterion]
1.2.2 Q SubMetric 2.2 [Criterion]
1.3 Q Metric 3 [Criterion]
2.0 Schedule Index (S) [Goal]
2.1 S Metric 1 [Criterion]
2.2 S Metric 2 [Criterion]
2.3 S Metric 3 [Criterion]
3.0 Cost Index (C) [Goal]
3.1 C Metric 1 [Criterion]
3.2 C Metric 2 [Criterion]
3.3 C Metric 3 [Criterion]
3.4 C Metric 4 [Criterion]
The model building method is iterative, with choice of criteria influencing
likelihood of feasible results. Upon approval of a metric model hierarchy, the
second basis for "preselling" results is established.
A utility graph is a two-axis scale for scoring the utility (worthiness of an
option) for a criterion. The vertical axis is an absolute scale for utility
score output, with a range from zero to maximum value. The horizontal axis is
the criterion performance range. Utility curve span is within performance range
and includes at least one zero and one maximum utility value. Curve shape is
drawn from a rationale for scoring and should accompany the graph for use. For
Figure 1, an example qualitative criterion, "Ugly" has charm and
"Plain" is unforgivable, but "Wow!" attracts both vandalism
and highway patrol attention.) Utility score is the output value that aligns
with the curve intersection directly above the position for attained
performance, as shown by the dashed line.
For each hierarchy criterion, construct an appropriate utility graph. Zero
score is the minimum acceptable performance with respect to the ultimate
purpose. This method of scoring works well with both qualitative and
quantitative performance attributes.
The validity of quantifying qualitative criteria with utility graphs is
established by its equivalence to fuzzy logic definition of linguistic
variables [Schmucker (Returns here)] for ordinal positions
on the utility graph input range [Jones 1994 NCOSE (Returns
here)]. The output utility scores for qualitative performance factors can
be fixed or be given a range of uncertainty to use within best/worst case and
Monte Carlo analyses.
Utility graphs clearly communicate the assumptions and relative values, just
as the model's structure facilitates a common understanding of the single
performance indicator or study purpose.
Mathematical formulas easily accomplish the scoring with quantitative data and
translation tables can convert qualitative values into numbers, but neither
conveys the information as clearly as utility graphs. Remember, the primary
customers for the results must understand the means used to obtain the final
figure of merit. When they have approved a utility graph for each criterion,
the third method output which assists "preselling" the final result is
established.
Another benefit from using utility graphs for option scoring in hierarchical
models is the automatic development of an "ideal" standard for use in
tradeoff analyses. The theoretical option that scores maximum for every utility
graph is, by definition, the ideal solution to the problem. The ideal solution
ignores real-world performance tradeoffs while fulfilling the entire wish list.
Impossible to implement, it still provides a fixed standard for evaluating and
ranking all real-world and proposed options. An "ideal" standard for
comparison of options improves on a pure relative comparison approach to
picking a best solution from many options. Independently comparing each option
to the ideal removes the possibility of ranking changes which sometimes occur
when an option is added to or removed from studies that compare their results
relative to each other.
The hierarchy could be used as is, but each child element to a given parent
currently is equal in importance to its siblings. This is unlikely for
real-world elements, so explicit weighting to set the relative priorities of
each element is the next step in assessment of options. The following manual
methods are proposed for use when computer tools are not available:
Direct assignment of weighting is performed by assigning the total figure of
merit value as 1.0. Assign to each top level goal category the decimal value
representing its weight (relative priority) which will sum to one when combined
with the weights assigned to the other categories. Next, assign relative
weighting to the subelements for each goal category in the same fashion.
Continue descending through the hierarchy levels until every criterion has been
assigned its weight value. Explicit priority is now established for each
element, but a rationale for the weighting must be supplied or auditability is
lacking. Without argument for the priority values or their differences,
justification for the assigned weighting is weak at best.
An alternative to direct weighting that better supports auditing the priority
assignments is algebraic computation of simultaneous equations. Consider
weighting a three element group from an algebraic perspective. Compare elements
in pairs and assign their fractional relationships from 1/1 for equality to any
appropriate ratio. The relationship of three pairs is assigned and the formulas
solved with substitution in the others. The solution set then is normalized
through division of each by its sum. The need to sum to 1 can be one formula.
For instance, if element 2 (E2) = 4/5 E3 and E1 = 2/3 E2, the weighting
relationship simultaneous equations are:
E1 + E2 + E3 = 1.0,
E2 = 4/5 x E3, and
E1 = 2/3 x E2, which leads to:
(0.6667 x E2) + E2 + E3 = 1.0, or
0.6667 x (0.8 x E3) + (0.8 x E3) + E3 = 1.0, or
(0.5334 x E3) + (0.8 x E3) + E3 = 1.0, or
2.3334 x E3 = 1.0, so
E3 = 1.0/2.3334 = 0.4285 and
E2 = 0.8 x 0.4285 = 0.3429 and
E1 = 0.6667 x 0.3429 = 0.2286.
An algebraic approach is recommended for weighting two and three element
groups. As number of elements to be weighted increases, the difficulty (tedium)
multiplies along with the number and length of the simultaneous equation
formulas.
Following weighting, the model is complete for use as shown below. Further,
listing the maximum possible contribution for each element to the TSR as
multiplied weights at the ideal standard maximum utility score of 100 for every
criterion provides automatic sensitivity analysis. For example, as shown for
element 1.1 (the first criterion) in the Maximum Utility column, 12.3 =
100 x .229 x .537. The largest Maximum Utility values provide the greatest
benefit from an improvement effort.
Element Maximum
Weight Utility
1.0 Quality Index (Q) .537
1.1 Q Metric 1 .229 12.3
1.2 Q Metric 2 .327
1.2.1 Q SubMetric 2.1 .631 11.08
1.2.2 Q SubMetric 2.1 .369 6.48
1.3 Q Metric 3 .444 23.84
2.0 Schedule Index (S) .364
2.1 S Metric 1 .179 6.52
2.2 S Metric 2 .439 15.98
2.3 S Metric 3 .384 13.98
3.0 Cost Index (C) .099
3.1 C Metric 1 .313 3.1
3.2 C Metric 2 .222 2.2
3.3 C Metric 3 .306 3.03
3.4 C Metric 4 .159 1.57
When primary customers have approved weighting for all groups of model elements,
the fourth and last decision basis which helps to "presell" the result
is established. You now can gather the performance data, input it to the model,
and determine the results.
Still, even with the pairing relationship assignments recorded for their
justification in the algebraic weighting method, consistency is a remaining
issue for discussion. Consistency of weighting is a measure of the nearness of
implied relationships to explicitly assigned element relationships. Achieving
it becomes increasingly less manageable as the number of group elements
increases. Accurate weighting of five or more model elements with good
consistency requires mental ability few among us can claim.
Paired comparisons input for a group of more than two elements forms an implied
group which can be inconsistent with the input. As an example, if pair (1,2)
element 1 (E1) is more important by 9/1 and for pair (1,3) E1 is more important
by 7/1, then the implied relationship for pair (2,3) is E3 is more important by
9/7. Anything else asserted as the relationship reduces the weighting
consistency in proportion to the difference, such as element 3 is more
important with 5 for priority level. Claiming reverse of the implied
relationship further increases the inconsistency. This example of inconsistency
is used in the next section to illustrate its detection and correction.
A tool which provides auditable and highly consistent element weighting is
supplied by ODDSCO on floppy diskettes for the IBM PC and Apple MacIntosh
computers. A worksheet template for Lotus 1-2-3® and Excel® spreadsheet
programs applies the Analytic Hierarchy Process (AHP) [Saaty
(Returns here)] to convert paired comparisons for up to nine sibling model
elements into an objective weighting set [Note 1 (Returns
here)]. Its unparalleled advantage is simplification of the difficult,
subjective weighting task while indicating level of attained consistency.
To use the weighting template, one merely inputs the relationships of compared
element pairs and it calculates the resulting set of weights. Lists of element
pairings are questionnaires for gathering the input relationships. An Input
Form lists importance level definitions and records the relationship of each
pairing for a set of up to nine elements. The relationships assignments are
mentally "natural" (a scale of 1 to 10) so they are easily comprehended by
the result customers. The relationship record indicates which member of each
pair is most important and by how much, just as fractions are recorded in the
algebraic weighting method.
Input Forms (especially with approval signatures) are useful historical
records. They can be presented during review meetings to continue "preselling"
the results.
Figure 2 is a typical weighting template screen which has relatively consistent
input entries for the maximum set of nine elements. Matrix data input and
weighting output both occur on the same screen, as shown. The input pairings
begin with comparison of elements 1 and 2 (labeled C 1,2: in cell column A row
2 or A2) with element 1 assigned as most important in cell B2 and by how much
as 2/1 in cell C2. Cells A3 through A9 label the remaining element 1
relationships in columns B and C. Input for the other labeled pairings is
similar. For pairing C 8,9:, labeled in cell J15, element 8 is most important
at 3/1 rating.
___A_____B_____C_____D_____E_____F_____G_____H_____I_____J_____K_____L___
0001ºPAIRED-COMPARISONS ENTERED BELOW FROM WORKSHEET PER INSTRUCTIONS, WITH º
0002ºC 1,2: 1 2 MOST IMPORTANT IN FIRST ADJACENT COLUMN AND "BY HOW º
0003ºC 1,3: 1 3 C 2,3: 2 2 MUCH" REPLACING 1 IN NEXT COLUMN. º
0004ºC 1,4: 1 1 C 2,4: 4 2 C 3,4: 4 3 º
0005ºC 1,5: 5 2 C 2,5: 5 3 C 3,5: 5 5 C 4,5: 5 3 º
0006ºC 1,6: 6 4 C 2,6: 6 5 C 3,6: 6 7 C 4,6: 6 5 º
0007ºC 1,7: 7 3 C 2,7: 7 4 C 3,7: 7 6 C 4,7: 7 4 º
0008ºC 1,8: 8 1 C 2,8: 8 2 C 3,8: 8 3 C 4,8: 4 1 º
0009:C 1,9: 1 3 C 2,9: 2 2 C 3,9: 3 1 C 4,9: 4 3 º
0010º 1 1 1 1 1 1 1 1 1 9 1 º
0011ºENTRIES BELOW ARE USED FOR WEIGHTING PROBLEMS UTILIZING MORE THAN FIVE º
0012ºC 5,6: 6 3 PARAMETERS OR HIERARCHY SUBELEMENTS: º
0013ºC 5,7: 7 2 C 6,7: 6 2 º
0014ºC 5,8: 5 3 C 6,8: 6 4 C 7,8: 7 4 ######º
0015ºC 5,9: 5 5 C 6,9: 6 7 C 7,9: 7 6 C 8,9: 8 3 º
0016º º
0017º (PRESS F9 KEY AFTER DATA ENTRY. PARAMETER VALUES WILL APPEAR BELOW.) º
0018º #1 #2 #3 #4 #5 #6 #7 #8 #9 C.I. LIM: º
0019º 0.078 0.048 0.029 0.071 0.15 0.304 0.219 0.073 0.029 0.025 0.125 º
0020º 0.08 0.049 0.03 0.063 0.142 0.309 0.226 0.075 0.025 SIMULT-EQU º
:________________________________________________________________________:
The input importance level is a ratio scale to retain essential proportionality
after normalization. The input rating assignment of 1/1 means equality of
contribution to the parent, that each element is as good as the other.
Assignment of 9.99/1 is an assertion that most important element of the pair is
highly dominant and contributes extremely more value or worthiness to the
parent.
When all the relationships have been entered for the group being weighted,
initiate recalculation and the two bottom rows (19 and 20) fill with result
values for the element numbers in row 18. Row 19 is the weighting set from the
AHP computation, while row 20 is the real "bottom line" or true simultaneous
equation solution for the input element relationships. These are the weights
to use in the model [Note 2 (Returns here)]. Both weighting
sets are shown to provide feedback on effect of revisions to the input
relationships when consistency is deficient as indicated by the AHP algorithm
[Note 3 (Returns here)].
In cell K19 of Figure 2 template screen is the computed Consistency Index
(C.I.) and in cell L19 an associated Limit (LIM:) whose value is related to the
number of elements being weighted [Note 4 (Returns here)].
It is desirable to revise input and recalculate to bring the C.I. below LIM:
because consistency affects reliability of weighting and thereby the output
result.
The consistency testing feature helps guard against input of clerical errors
which might otherwise stay undetected. It also prevents a display of obvious
ignorance regarding one or more of the study elements. It certainly helps bring
about earlier group consensus via discussion of compromises needed to obtain
consistency prior to the synthesized weighting approval.
Weight two or three elements with direct assignment or an algebraic method,
unless you input decimal numbers at places between the defined whole numbers
for importance levels because output set is small with whole numbers importance
level input.
Following groups weighting, transfer the weights into the derived model to
await input of the primitive elements utility scores.
Once every primitive element has a measured (or estimated) attainment for an
option, the utility curve translations provide the earned scores (numbers
between 0 and 100). Multiply each score by its assigned weighting and add that
to the weighted scores from all neighboring children to the same parent element
(its siblings).
Summing the weighted scores for all the children of each parent element
provides its utility value, which is another score for weighting and adding
with weighted scores for the other children at the next parent level in the
hierarchy. Sometimes you must descend another branch to obtain the weighted
score for summation with its siblings.
When all criteria utility scores are weighted and summed to the top of the
model for an option, the final result is a single, easily interpreted scalar
value (from 0 to 100) which indicates the purpose level performance for that
option. Repeat for each option and rank them by their net utility scores to
find the best solution.
When all decision bases produced by the DSIDE^{(tm)} method are in
force and unassailable, agreement with and acceptance of model results is
virtually automatic. No customer approval is required here, because this step
merely applies data to the approved model.
Of course, the model result customer(s) may insist on evidence of option data
validity for each input to the utility graphs or metric formulas. Review of the
option performance data for accuracy and validity is prudent in any case. Then,
the "presold" result should be accepted without argument from the model
customers because all areas of potential contention have been addressed and
approved.
Typically, an analysis Report is required. Assemble records of the actions and
decisions involved in performing method steps A through E and develop an
understandable description of the method and model.
Make the report concise, but with audit trail of method steps to support future
investigation of result and model validity. Clearly show each step, so others
can follow the sequence.
When performing the method for others, as an expert or consultant, you want the
fruits of your labor used. Therefore, an appropriate report has complete
exposition of model design along with rationale and methods used for setting
weights, designing utility graphs, and getting usable performance data. When a
customer or assigned experts or customer representatives helped, say so. The
objective is to show sufficient cause for anyone who may review the report to
generally concur with its conclusions. Describe model foundation development
and any economic justification approach, if employed.
The DSIDE^{(tm)} method collects and organizes relevant data into a logical
structure, communicates the model elements, and gains support for conclusions
from group members. It doesn't manipulate you with the order of presentation
and, instead, helps you to obtain insight into your trade study methods. It
employs a few simple procedural steps that can be addressed separately, so an
uninterrupted and lengthy duration of attention is not required for successful
application. The methodology is effective because it accommodates factors upon
which it often is very difficult to place numerical values.
It doesn't tell you what to think, but helps you to discover what you should
think to achieve your study objectives.
Customer(s) accept results as rationally derived from a method specifically
tailored to solving their problem.
Spreadsheet Decision Modeling. When a computer is available,
installation of the weighted model in a spreadsheet template is quite simple.
A temporal series plot of method net utility scores from each acquisition of
the subsidiary metrics with an overlaid linear best fit line clearly shows
single performance indicator trend, indicating corrective action when slope is
downward.
Summarized merit with level of attainment indicated by assigned net utility
score allows explicitly assigned weighting in accordance with customer
priorities, rather than implicit priority that is adjustable only by revising
evaluation wording. Maturity models should be tailored to anticipated project
tasks. The additive model for a single performance indicator allows an
increased utility score for one metric to compensate for another metric's lower
or decreased score.
Note 1. (For the mathematicians.) Weighting template
input fills a square matrix with unity diagonal whose size is equal to number
of elements. Placement of the priority level value above or below the diagonal
and its inverse in the reflected cell is determined by which element of pair is
most important. This input method preconditions the matrix for orthogonality,
which allows a simple eigenvalues computation algorithm.
After input of final values for the set of pairings and template calculation,
the iterative algorithm calculates the right positive eigenvector with the
largest eigenvalue and normalizes the result. Calculation and normalizing
repeat until convergence to a priority ordered (weighted) solution. In the
positive reciprocal matrix, the largest eigenvalue becomes a measure of the
input relationships consistency whose value to obtaining group consensus has
been described. (Most of us need know only that AHP works for this application
and is particularly valuable for obtaining weighting consistency.)
Return to citing paragraph.
Note 2. The principal critic [Yu] of Saaty's AHP
method correctly points out that its output is not equal to a simultaneous
equations solution. Employing the inconsistent input example, (E1 > E2 by
9/1, E1 > E3 by 7/1, and E3 > E2 by 5/1) the template AHP method outputs
weights of #1 = .7719, #2 = .0546, and #3 = .1735. Simultaneous equation
solution weights are #1 = .7857, #2 = .0357, and #3 = .1786. A significant
difference in the second decimal place, but is it really a problem? If the
implied E3 > E2 by 9/7 is input, the C.I. = 0 with identical weighting sets.
AHP as applied in the template provides clear feedback of example inconsistency,
with a C.I. = .104 for LIM: = .025, so final weights which differ greatly from
the exact solution are unlikely.
Return to citing paragraph.
Note 3. Many sources of potential inconsistency (in
implied versus input relationships) exist when group approaches the nine
elements maximum template design. With time for trial and error work to find a
highly consistent input set expanding to impracticality, AHP weights alone may
be considered deficient. Responding to that criticism and Note 2, the
simultaneous equation solution to the element pairs relationship input also is
provided as the weighting set to use in a model.
Return to citing paragraph.
Note 4. Limit rises exponentially from zero for two
elements (which cannot be inconsistent) through 0.1 for seven elements (which
is the psychological limit for most people during simultaneous consideration of
multiple items) because inconsistency is less important than consistency by one
order of magnitude [Saaty].
Return to citing paragraph.
Delozier, Randall, and Snyder, Neil, Engineering
Performance Metrics, Proceedings of the Third Annual
International Symposium, National Council on Systems
Engineering (NCOSE), Arlington, Virginia, July 1993.
Return to citing paragraph.
Jones, James H., Evaluationg Project Risks With
Linguistic Variables, Proceedings of the Fourth Annual
International Symposium, National Council on Systems
Engineering (NCOSE), San Jose, CA, August, 1994.
Return to citing paragraph.
Saaty, Thomas L., The Analytic Hierarchy Process,
McGraw-Hill, New York, 1980.
Return to citing paragraph.
Schmucker, Kurt J., Fuzzy Sets, Natural Language
Computations, and Risk Analysis, Computer Science
Press, Rockville, Maryland, 1984.
Return to citing paragraph.
Yu, Po-Lung, Multiple-Criteria Decision Making:
Concepts, Techniques, and Extensions, Plenum Press,
New York, 1985.
Return to citing paragraph.
If you have any questions on this tutorial subject, please contact the
author as listed below. A response will occur and a FAQs section may result.
This tutorial is presented by:
E-mail:
Tutorial Author: jonesjh@optants.com
Other subjects: consult@optants.com