Making Decisions – Vol. X: Fuller Triangle

            Decision-making tools vary in sophistication and in their applicability to some types of decisions.  The Fuller Triangle (FT) method is relatively simple; however, this comes with a tradeoff.  Similar to Analytic Hierarchy Process (AHP), presented in Vol. III (6May 2020) of this series, but less rigorous, FT may be less suitable to highly consequential or complex decisions.
            Multicriteria analysis (MCA) tools facilitate organization of decision information for several alternatives and attributes.  When the rigor of a complex method is not required and a decision is needed quickly, FT may be an excellent compromise.  In this installment of Making Decisions, the Fuller Triangle is presented, including some comparisons to AHP to facilitate selection of a decision-making aid.

            To facilitate comparison of methods and the results obtained, the Fuller Triangle (FT) method is presented with the same hypothetical machinery purchase decision used in Vol. III:  Analytic Hierarchy Process.  Reviewing AHP alongside FT is recommended to clarify when either method may be more appropriate than the other.

The Fuller Triangle Method
            A Fuller Triangle serves as a visualization of a series of pairwise comparisons.  An abstract example is shown in Exhibit 1, where criteria A, B, C, and D are considered.  Each row represents comparisons of the subject criterion to those to which it has not been compared in rows above.  Only half of the rectangle, above the diagonal, is used, leaving blank row/column intersections representing duplicate comparisons (e.g. row B/column A duplicates row A/column B, each comparing A and B).  Thus, a triangle results.

            For the visualization, the criterion of greater importance in each comparison is circled.  If the two are of equal import, a rectangle is drawn around both.  To score the comparisons, one point is assigned to a criterion for each time it is circled in the triangle.  For each rectangle drawn around a criterion in the triangle, one half (1/2) point is assigned.  Note that, in Exhibit 1, a rectangle surrounds the criterion identifiers each time a criterion is compared to itself.  These are greyed out because “self-comparisons” are not normally presented in the Fuller Triangle.  They are included here for explanatory purposes; their relevance becomes clear in the presentation, below, of the entire FT process.

            Returning to the hypothetical machine purchase, the summary of criteria and source alternatives, called the performance matrix in AHP, is reproduced in Exhibit 2.  The information contained in this table is used throughout the FT example presentation.

            The number of comparisons, p, to be performed is found as p = n(n-1)/2, where n is the number of criteria considered.  In our example, n = 3; the number of comparisons needed is 3(3-1)/2 = 3.  The small number of criteria yields a compact Fuller Triangle, shown in Exhibit 3.  The criteria are identified as COST, PROD (productivity), and SVC (service life).  Larger analyses often use single letters or numbers, rather than names or abbreviations, to identify criteria.  When this is done, a legend should accompany the comparison triangle.
            Following the logic of the original example, both productivity and service life are considered more important than cost and productivity is more important than service life, as shown in Exhibit 3.  This results in criteria preference scores (pc) of pCOST = 0, pPROD = 2, and pSVC = 1.
            Though cost is deemed less important than the other criteria, using a preference score of zero could yield misleading results.  To prevent this, the “self-comparison” point is added to each criteria preference score, ensuring none are zero.  This results in the modified preference score, pm.  The total modified preference score is pm = n(n-1)/2 + n, or pm = p + n; in this example, pm = 3 + 3 = 6.  A summary table of criteria comparison scoring is provided in Exhibit 4.

            Criteria weights are now calculated using the preference scores obtained above.  Each criterion weight is determined as wc = pmc/pm.  Therefore, wCOST = 1/6 = 0.167, wPROD = 3/6 = 0.500, and wSVC = 2/6 = 0.333.

            Next, the alternatives are considered, with respect to each criterion, following the same process as that used for evaluation of criteria.  Alternative comparison triangles and scoring summaries are shown in Exhibits 5, 6, and 7.  The self-comparison entries have been eliminated and the comparison triangles and scoring tables are combined, providing a more-conventional presentation of the analysis.  The lengthy subscripts make evident the benefit of shortened identifiers and an accompanying legend to a large analysis.  The small number of comparisons and the explanatory value warrant use of abbreviations here.
            The number of comparisons needed, relative to each criterion, is, as before, p = n(n-1)/2 = 3; pa is the alternative preference score and pma is the modified alternative preference score, pma = p + n = 6.  Our example uses the same number of criteria and alternatives, but this need not be the case and is not typical.

            The score for each alternative, relative to each criterion, is the product of the alternative’s modified preference score on that criterion and the criterion’s weight:  rac = pmac⋅wc, where rac is the ranking score of alternative a with respect to criterion c, pmac is the modified preference score of alternative a with respect to criterion c, and wc is the criterion weight of criterion c.  The total ranking score of each alternative, ra, is the sum of the alternative’s criterion ranking scores:  ra = ∑ rac.
            A summary of the alternative ranking calculations is shown in Exhibit 8.  The rightmost column of the table shows the final ranking of preferences, where the alternative with the highest total ranking score is the preferred option.  The order of preferences obtained – (1) Acme, (2) Wiley, (3) Jones – coincides with the results obtained for this hypothetical example using AHP.

FT vs. AHP
            The consistency of the result obtained using the Fuller Triangle with that of AHP is expected.  The evaluations of criteria were found to be highly consistent in the AHP analysis; therefore, following the same logic with a similar process can be expected to yield similar results.  If FT were to be used in lieu of AHP for a real decision, however, the reverse cannot be assured.  That is, FT lacks the tests of evaluation consistency that ensure valid results in AHP.
            The magnitudes of preferences are not accounted for in FT.  This may also cause results to differ from those obtained by AHP.  While sensitivity analysis could be performed by adjusting criteria weights, interpreting the results is very difficult.  This is because potential causes of criteria weight changes are not clearly defined; each criterion weight is based on a binary choice.  In contrast, AHP allows an adjustment from e.g. “strongly” to “somewhat” preferred.  It is this nuance that is explored in sensitivity analysis and that is lacking in FT.
                The reduced rigor of FT make it both easier to use and less reliable as the size (i.e. number of criteria and alternatives) and criticality of analysis grows.  When conducting a simple analysis, such as that in our example, it is easy to maintain consistency.  Therefore, the additional rigor of AHP may not be warranted in an actual decision of similar complexity.  The threshold at which the additional rigor of AHP is justified is reduced by the use of spreadsheet templates.  Performing calculations in software significantly reduces the workload involved in a large, complex analysis.  The selection of a decision-making aid requires consideration of the implications of the decision, stakeholders’ acceptance of the method, and other variables.


            Use of one MCA tool does not exclude the use of others.  Multiple tools can be employed to increase confidence in a decision or reveal discrepancies in information that must be resolved to maintain decision quality.  Methods may even be intertwined; for example, criteria weights determined using FT could be inserted in AHP or other analysis as a “shortcut,” a form of sensitivity analysis, or simply to provide an alternative perspective on the decision scenario.  Deeper understanding of the tools and methods and rapid computation provided by a computer expands opportunities to explore possible scenarios and potential outcomes.


            For additional guidance or assistance with decision-making or other Operations challenges, feel free to leave a comment, contact JayWink Solutions, or schedule an appointment.

            For a directory of “Making Decisions” volumes on “The Third Degree,” see Vol. I:  Introduction and Terminology (8Apr2020).

References
[Link] “An Approach to Multi-criteria Environmental Evaluation with Multiple Weight Assignment.”  Boris Agarski, Igor Budak, Borut Kosec, and Janko Hodolic.  Environmental Modeling & Assessment; June 2012.
[Link] “The Issue of Multicriteria Decision-Making:  Model Example of Business Partner Risk Assessment.”  Jaromir Vrbka and Vojtech Stehel.  Proceedings of the 13th International Management Conference; January 2020.
[Link] “Application of the Chosen Multi-Criteria Decision-Making Methods to Identify the Autonomous Train System Supplier.”  Ondrej Stopka, Mária Stopkova, Vladimír Ľuptak, and Srećko Krile.  Transport Problems; June 2020.

Jody W. Phelps, MSc, PMP®, MBA
Principal Consultant
JayWink Solutions, LLC
[email protected]