Making Decisions – Vol. XII: PROMETHEE

            The Preference Ranking Organization METHods for Enrichment Evaluations (PROMETHEE) resemble previously-presented decision-making tools in some fundamental ways.  For example, it is a set of compensatory assessments employing pairwise comparisons.  However, substantial differences in PROMETHEE provide advantages not offered by other methods and, thus, warrant discussion.
            The use of various preference functions in PROMETHEE enable evaluations of criteria that more-accurately reflect the ways decision-makers think about their choices.  The determination of the “net desirability” of each alternative also differs from other compensatory methods discussed.  PROMETHEE methods are presented in this installment of the “Making Decisions” series, highlighting these key differences.

Preference Functions
            The foundation of PROMETHEE is the use of various preference functions to reflect the true nature of criterion assessments.  While any number of preference functions can be defined for a decision environment, the six types summarized in Exhibit 1 are widely used; this set is deemed sufficient to support the vast majority of decisions.

            Each preference function is defined according to the type of criterion.  That is, whether it is a cost to be minimized or a benefit to be maximized.  Qualitative criteria are converted to quantitative measures for use with preference functions.
            Predefined parameters are used to determine function values.  A threshold of indifference (q) is the difference in criterion values between alternatives at which a preference emerges.  The difference in criterion values at which a preference becomes complete is the threshold of preference (p).  For Type II criteria, p = q; therefore, only q is defined.  For Types IV and V, a criterion difference that exceeds the threshold of indifference (i.e. d > q or d < -q) creates a partial preference.
            Type VI criteria require σ, the standard deviation, to be defined.  This function is represented by an inverted “bell curve,” or normal distribution; hence the Gaussian label.
            To demonstrate use of PROMETHEE methods, the hypothetical machinery purchase of previous installments is revisited.  The performance matrix, expanded to include preference function and parameter information for each criterion, is shown in Exhibit 2.  Once again, labels J, W, and A are used to refer to the Jones, Wiley, and Acme alternatives, respectively.  Likewise, labels C, P, and S are used to reference cost, productivity, and service life criteria, respectively.

            To determine preference function values, criterion value differences for each alternative pair must be computed.  Generically, this is written dj (a,b) = gj (a) - gj (b), where dj is the value difference on the jth criterion, a and b are the two alternatives compared, and gj is the performance level of an alternative on the jth criterion.
            For our example, cost differences are calculated as follows:
dC (J,W) = C(J) - C(W) = 1.2 - 1.4 = -0.2
dC (W,J) = C(W) - C(J) = 1.4 - 1.2 = 0.2
dC (J,A) = C(J) - C(A) = 1.2 - 1.8 = -0.6
dC (A,J) = C(A) - C(J) = 1.8 - 1.2 = 0.6
dC (W,A) = C(W) - C(A) = 1.4 - 1.8 = -0.4
dC (A,W) = C(A) - C(W) = 1.8 - 1.4 = 0.4
Though seemingly redundant, the bidirectional difference calculations are needed in subsequent stages of analysis.  Repeating this step for the productivity and service life criteria yields the results summarized in the tables of Exhibit 3.  In each table, the row represents the first alternative in the comparison (i.e. a), while the column represents the second (i.e. b).  Each entry can, therefore, be read as “row minus column.”  Units are omitted for simplicity.

            Preference function values are then determined for each alternative pair by using the criterion differences in the preference function assigned to each criterion, as tabulated in Exhibit 2.
            Cost is a V-shaped criterion with a minimization objective; therefore, the second preference function in the Type III row of Exhibit 1 is used.  Only the (Jones, Wiley) comparison yields an intermediate value, calculated as fC (J,W) = -(-0.2/0.25) = 0.8.  The other two criteria are maximization objectives; therefore, the first preference function in the corresponding row is used for each.
            Productivity is a Level (Type IV) criterion.  The (Wiley, Jones) and (Acme, Wiley) comparisons reside at the intermediate level; only (Acme, Jones) yields a complete preference.  Identical service life projections for Wiley and Acme provide both a partial preference to Jones as a Linear (Type V) criterion.  For both, this value is (d-q)/(p-q) = (2-1)/(4-1) = 0.3.  All preference function values for the example are tabulated in Exhibit 4; values are always in the range 0 to 1, inclusive [0 ≤ f(a,b) ≤ 1].

Preference Indices
            An alternative’s preference index (π) is a cumulative measure that incorporates all criteria.  It takes the form

where π(a,b) is the preference index of alternative a with respect to alternative b, n is the number of criteria considered, fj (a,b) is the preference function value for the (a,b) comparison with respect to the jth criterion, and wj is the relative weight of the jth criterion.  If criterion weights have not been assigned, wj is replaced with (1/n) to reflect equal weighting.  For our example, the normalized criterion weights found in the AHP (Vol. III) and tabulated in Exhibit 2 are used.  Doing so maintains consistency and comparability across installments of the series.  Other methods of assigning criterion weights could be used, however.
            The preference index calculations for our example are as follows:
π(J,W) = (0.064)(0.8) + (0.669)(0) + (0.267)(0) = 0.051
π(W,J) = (0.064)(0) + (0.669)(0.5) + (0.267)(0.3) = 0.424
π(J,A) = (0.064)(1) + (0.669)(0) + (0.267)(0) =0.064
π(A,J) = (0.064)(0) + (0.669)(1) + (0.267)(0.3) = 0.758
π(W,A) = (0.064)(1) + (0.669)(0) + (0.267)(0) = 0.064
π(A,W) = (0.064)(0) + (0.669)(0.5) + (0.267)(0) = 0.335
Note that values are always in the range 0 to 1, inclusive [0 ≤ π(a,b) ≤ 1]; the sum of complementary indices also lie in this range [0 ≤ π(a,b) + π(b,a) ≤ 1].

Outranking Flows
            An alternative’s outranking flows compare it to all other alternatives under consideration.  An alternative’s positive outranking flow, or leaving flow, (φ⁺) reflects its superiority to other alternatives.  This is called its outranking character or power and is expressed as

where φ+(a) is the positive outranking flow of alternative a and m is the number of alternatives under consideration.  The summation is performed for all competing alternatives; that is, all alternatives to a considered in the decision.
            Negative outranking flow, or entering flow, (φ-) represents an alternative’s inferiority to other alternatives, called its outranked character or weakness.  It is expressed as

As for the leaving flow, the summation is performed for all competing alternatives.
            An alternative’s net flow (φ) is simply the difference between its positive and negative outranking flows.  That is, φ(a) = φ⁺(a) – φ⁻(a).
            The outranking flows for the Jones Machinery option of our example are as follows:
φ⁺(J) = [1/(3-1)] (0.051 + 0.064) = 0.058
φ⁻(J) = [1/(3-1)] (0.424 + 0.758) = 0.591
φ(J) = 0.058 – 0.591 = -0.533
A summary of outranking flow calculations for the example is tabulated in Exhibit 5.

PROMETHEE Outranking Methods
            Outranking is a term used to describe the assignment of preference, indifference, or incomparability to alternatives.  Two alternatives found to be incomparable are defined by criteria values that are too contradictory for the method in use to reach a conclusion.  Other methods may be capable of differentiating and ranking them, however.
            Notation used to represent outranking relationships varies among sources.  For purposes of this presentation, a symbol set is adopted to make the information as intuitive as possible.  This symbol set is summarized in Exhibit 6; alternative representations that may be encountered in cited references and elsewhere, including this page due to limitations of the text editor, are also shown.  A Roman numeral in parentheses or superscript identifies the PROMETHEE method by which the outranking relationship was established.

            PROMETHEE I provides a preorder, or partial ranking, of alternatives by comparing positive and negative outranking flows.  The ranking is established according to the following rules:

            Three reminders are useful here: (1) “iff” is read “if and only if;” both conditions in a statement must be true for the corresponding ranking to be assigned.  (2) The third rule is a mathematical expression of the contradictory nature of decision criteria values, i.e. each alternative outranks the other with respect to a subset of criteria. (3) Outranking notation and mathematical relationships are both used to define outranking rules; these sometimes have opposite meaning (see the note in Exhibit 6).
            To rank alternatives, an exhaustive set of comparisons can be created; however, one-half will be inconclusive!  Instead, identify the alternative with the largest positive outranking flow (φ⁺).  Compare this alternative to each of the others, in turn.  This selection reduces the rule set to three statements where φ⁺(a) > φ⁺(b).  Comparing the negative outranking flows (φ⁻) of the alternatives determines which of the remaining statements and, therefore, the outranking relationship that applies.
            For our example, the largest value of φ⁺ is associated with Acme (0.546).  Comparing Acme’s φ⁻ (0.064) to that of Jones (0.591) and Wiley (0.193) reveals that the first rule statement applies in both cases.  Therefore, A >~(I) J and A >~(I) W.  This process is repeated for the second-highest φ⁺, comparing to remaining alternatives.
            The second-highest φ⁺ in our example is associated with Wiley (0.244).  Comparing Wiley’s φ⁻ (0.193) to that of Jones (0.591), the only remaining alternative, the first rule statement once again applies; W >~(I) J.  This gives the final ranking of A >~(I) W >~(I) J with the minimum number of comparisons.
            A larger decision matrix is analyzed in the same manner, by repeating the steps outlined above, progressing through the list of alternatives in order of decreasing φ⁺.  Where equal values of φ⁺ are encountered, the subset of rule statements considered shifts from the three “>” relations to the two “=” relations.  The rule set shifts back to the “>” relations to compare these alternatives to those remaining with lower φ⁺ values.

            PROMETHEE II provides a complete ranking of alternatives by comparing net outranking flows (φ).  Ranking by this method is established according to the following rules:

• a >~(II) b   iff   φ(a) > φ(b).
• a ~~(II) b   iff   φ(a) = φ(b).

Note that, by comparing only net flows, incomparabilities are eliminated.  While this simplifies its application, this loss of information is also commonly cited in caveats related to the use of PROMETHEE II.
            In our example φ(A) > φ(W) > φ(J) (see Exhibit 5), corresponding to the ranking A >~(II) W >~(II) J.  PROMETHEE I and II yield identical results in our relatively-simple example; however, the loss of incomparability information may cause the results of a larger analysis to differ.

            PROMETHEE III provides a complete ranking of alternatives by comparing intervals.  Like the block distances used in ORESTE (Vol. XI), intervals are in “decision space.”  To determine each alternative’s interval, intermediate values must first be computed.
            An alternative’s mean flow is φ̅(a) = φ(a)/n, where φ is the net outranking flow of alternative a and n is the number of decision criteria.  Its standard deviation can then be calculated:

            Each alternative’s interval [X(a), Y(a)] is determined according to X(a) = φ ̅(a) – ασ(a) and Y(a) = φ ̅(a) + ασ(a), where α is a parameter, the derivation of which is omitted for simplicity, with a typical value of 0.15.  Ranking of alternatives is then established according to the following rules:

• a >~(III) b   iff   X(a) > Y(b).
• a ~~(III) b   iff   X(a) ≤ Y(b) and X(b) ≤ Y(a).

            In our example, Jones’ mean flow is φ ̅(J) = -0.533/3 = -0.178.  Its standard deviation is

Using α = 0.15, Jones’ interval is now found:
X(J) = -0.178 – (0.15)(0.319) = -0.225,
Y(J) = -0.178 + (0.15)(0.319) = -0.130;
[X(J), Y(J)] = [-0.225, -0.130].
A summary of interval calculations for our example is tabulated in Exhibit 7; a visual representation of the intervals is provided in Exhibit 8.

            Comparing interval limits to apply the preference rules, we find X(A) > Y(J) and X(A) > Y(W); therefore, A >~(III) J and A >~(III) W.  Also, X(W) > Y(J); therefore, W >~(III) J.  Thus, the final ranking is A >~(III) W >~(III) J, in alignment with the PROMETHEE I and II results.
            Though the final ranking obtained matches those of the previous methods, the addition of a visual element enables decision-makers to rapidly gain insights regarding the alternatives’ relative performance.  Observing Jones’ position on the far left of the interval diagram, its entire range less than zero, and Acme’s interval on the far right makes the outranking character of the Acme alternative much more salient than numbers in a table.  This is a significant contribution of PROMETHEE III to decision-makers’ available information.

            PROMETHEE IV extends PROMETHEE II for applications in which alternatives are defined by continuous variables rather than discrete values.  Calculating flows in PROMETHEE IV requires integration of preference indices, while the ranking rules remain the same as in PROMETHEE II.  Details of this method, and the “Brans simplification,” can be found in cited references or other sources.

            Additional techniques have been developed as part of the family of PROMETHEE methods.  Decision under constraints, advanced visual techniques, and sensitivity analysis are among variants of PROMETHEE methods.  These variants employ sophisticated techniques that are beyond the scope of this series, the charter of which is to present tools that are reasonably straightforward and practical for implementation by a broad spectrum of practitioners in a wide range of applications.


            PROMETHEE methods are valuable additions to one’s decision-making toolbox; they provide perspectives that other tools do not offer.  All of these tools become more valuable when used synergistically, exploring the nuance provided by each.  Similar to other tools discussed, the PROMETHEE methods presented are amenable to development of spreadsheet templates or purpose-built software that minimizes the effort required for each new analysis, further enhancing their value.


            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] Multiple Attribute Decision Making:  Methods and Applications.  Gwo-Hshiung Tzeng and Jih-Jeng Huang.  CRC Press; 2011.
[Link] Multiple Criteria Decision Analysis:  State of the Art Surveys, 2ed.  Salvatore Greco, Matthias Ehrgott, and José Rui Figueira (eds).  Springer; 2016.
[Link] New Methods and Applications in Multiple Attribute Decision Making (MADM).  Alireza Alinezhad and Javad Khalili.  Springer; 2019.
[Link] “Multicriteria Decision-Making Methods.”  M. Pavan and R Todeschini.  In Comprehensive Chemometrics, 2ed ; S. Brown, R. Tauler, and B. Walczak (eds).  Elsevier; 2020.

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