Outranking Methods: Promethee I and Promethee II

AHP is an MCDM technique for establishing and analyzing complex decisions based on psychology and mathematics (Saaty, 1980; 2001; 2008; 2010). It was first developed by professor Thomas L. Saaty in 1980 (Saaty, 1980). The AHP methodology is illustrated briefly in Fig. 1.

Figure 1

The steps involved in AHP for calculating the criteria weightages are given as follows:

Step 1: An n × n pair-wise comparison matrix is created based on Saaty's 9 pair comparison scale (Saaty, 1980), which is presented in Table 3.

This pair-wise comparison matrix compares all the criteria among each other, and the relative importance of one criterion over other is given according to the Saaty pair-wise comparison scale.

Table 5 presents the Saaty's 9 pair comparison scale. Here, n is the number of criteria considered for the analysis. In this case, seven criteria are considered; hence, n = 7.

Step 2: The pair-wise comparison matrix (presented in Table 3) is normalized by dividing all the elements of Table 3 by their respective column sum. Table 4 presents the normalized pair-wise comparison matrix.

The normalized values are calculated and presented in Table 4. All the row averages (priority vector) are found out for every criterion, which are the criteria weightages. The criteria weightages are found out to be as follows: w_processor = 0.37657 or 37.657%, w_{hard disk capacity} = 0.09395 or 9.395%, w_{operating system} = 0.04529 or 4.529%, w_RAM = 0.21594 or 21.594%, w_{screen size} = 0.16932 or 16.932%, w_brand = 0.07647 or 7.647%, and w_color = 0.02247 or 2.247%.

Table 4 presents the normalized decision matrix along with the criteria weightages. Now the next step is to check the consistency whether the decision maker's judgment is true and consistent.

Step 3: Calculate the average consistency (λ_max). The priority vector matrix (priority vector column in Table 4) is multiplied with the pair-wise comparison matrix (Table 3) and λ_max is calculated as shown in details below: [15732791/5131/31/3251/71/311/51/31/331/33512391/2331/21371/71/231/31/3151/91/51/31/91/71/51]×[0.376570.093950.045290.215940.169320.076470.02247]=[2.887260.698810.322921.619991.269950.553850.16167]|2.88726/0.376570.69881/0.093950.32292/0.045291.61999/0.215941.26995/0.169320.55385/0.076470.16167/0.02247|=|7.667337.438217.130757.502037.500187.242847.19579| ‖ProcessorHard disk capacityOperating systemRAMScreen sizeBrandColor\matrix {\left[ {\matrix{ 1 & 5 & 7 & 3 & 2 & 7 & 9 \cr {1/5} & 1 & 3 & {1/3} & {1/3} & 2 & 5 \cr {1/7} & {1/3} & 1 & {1/5} & {1/3} & {1/3} & 3 \cr {1/3} & 3 & 5 & 1 & 2 & 3 & 9 \cr {1/2} & 3 & 3 & {1/2} & 1 & 3 & 7 \cr {1/7} & {1/2} & 3 & {1/3} & {1/3} & 1 & 5 \cr {1/9} & {1/5} & {1/3} & {1/9} & {1/7} & {1/5} & 1 \cr } } \right] \times \left[ {\matrix{ {0.37657} \cr {0.09395} \cr {0.04529} \cr {0.21594} \cr {0.16932} \cr {0.07647} \cr {0.02247} \cr } } \right] = \left[ {\matrix{ {2.88726} \cr {0.69881} \cr {0.32292} \cr {1.61999} \cr {1.26995} \cr {0.55385} \cr {0.16167} \cr } } \right] \cr \left| {\matrix{ {2.88726/0.37657} \cr {0.69881/0.09395} \cr {0.32292/0.04529} \cr {1.61999/0.21594} \cr {1.26995/0.16932} \cr {0.55385/0.07647} \cr {0.16167/0.02247} \cr } } \right| = \left| {\matrix{ {7.66733} \cr {7.43821} \cr {7.13075} \cr {7.50203} \cr {7.50018} \cr {7.24284} \cr {7.19579} \cr } } \right|\,\,\left\| {\matrix{ {{\rm{Processor}}} \cr {{\rm{Hard}}\,{\rm{disk}}\,{\rm{capacity}}} \cr {{\rm{Operating}}\,{\rm{system}}} \cr {{\rm{RAM}}} \cr {{\rm{Screen}}\,{\rm{size}}} \cr {{\rm{Brand}}} \cr {{\rm{Color}}} \cr } } \right.}

The consistencies of all the seven criteria are found out. Now, calculate the averages of these seven values to find out the average consistency (λ_max). Average consistency (λmax)=7.66733+7.43821+7.13075+7.50203+7.50018+7.24284+7.195797=7.38245.{\rm{Average}}\,{\rm{consistency}}\,\left( {{{\rm{\lambda }}_{\max }}} \right) = {{7.66733 + 7.43821 + 7.13075 + 7.50203 + 7.50018 + 7.24284 + 7.19579} \over 7} = 7.38245.

Step 4: Calculate the consistency ratio (CR) value and check for consistency. CR value is calculated using Equation 1 as shown below. (1)Consistency ratio (CR)=CIRI{\rm{Con}}{\mathop{\rm sistency}\nolimits} \,{\rm ratio}\,({\rm{CR}}) = {{{\rm{CI}}} \over {{\rm{RI}}}} where CI is the consistency index and RI is the randomly generated CI.

CI is calculated using Equation 2, and RI value can be obtained from Table 6 according to the number criteria. In this case, n = 7, so the RI value corresponding to n = 7 is 1.32. (2)Consistency Index (CI)=(λmax−n)(n−1){\rm{Consistency}}\,{\rm{Index}}\,({\rm{CI}}) = {{\left( {{{\rm{\lambda }}_{\max }} - {\rm{n}}} \right)} \over {({\rm{n}} - 1)}}

Here, λ_max is the average consistency and n is the number of criteria. In this case, n = 7.

Using Equations 1 and 2, CI and CR values are found to be 0.06374 (CI value) and 0.04829 (CR value) or 4.829%.

Here it can be seen that the CR value is less than 0.1 (0.04829 < 0.1), that is, the inconsistency is within 10%; hence, it can be concluded that the decision maker judgment is true and consistent.

In early 1980s, for the first time, Brans presented the PROMETHEE outranking method in 1982 at a conference organized by R. Nadeau and M. Landry at the University Laval, Québec, Canada (Brans, 1982). After that, developments have been carried out for more than a decade to bring out various versions of PROMETHEE by Brans along with other researchers (Brans, 1982; Brans, et al., 1984, 1986; Brans and Mareschal, 1992, 1994, 1995; Brans and Vincke, 1985). The PROMETHEE methodology are explained briefly in Fig. 2.

Figure 2

The steps involved in PROMETHEE are described as follows.

Step 1: Create an m × n evaluation (decision) matrix according to Equation 3 based on the Hwang and Yoon comparison scale (Hwang and Yoon, 1981), which is presented in Table 8, where m is the number of alternatives and n is the number of criteria. Table 7 presents the decision matrix. (3)X=[Xij]m × n=[X11X12⋯X1nX21X22⋯X2n⋯⋯⋯⋯Xm1Xm2⋯Xmn]{\rm{X}} = {\left[ {{{\rm{X}}_{{\rm{ij}}}}} \right]_{{\rm{m}}\, \times \,{\rm{n}}}} = \left[ {\matrix{ {{{\rm{X}}_{11}}} & {{{\rm{X}}_{12}}} & \cdots & {{{\rm{X}}_{1{\rm{n}}}}} \cr {{{\rm{X}}_{21}}} & {{{\rm{X}}_{22}}} & \cdots & {{{\rm{X}}_{2{\rm{n}}}}} \cr \cdots & \cdots & \cdots & \cdots \cr {{{\rm{X}}_{{\rm{m}}1}}} & {{{\rm{X}}_{{\rm{m}}2}}} & \cdots & {{{\rm{X}}_{{\rm{mn}}}}} \cr } } \right] where i = 1, 2, 3…., m and j = 1, 2, 3…., n.

Step 2: The decision matrix presented in Table 7 is normalized using Equations 4 and 5 according to the nature of the selected criteria.

For beneficial criteria (4)Rij=[xij−min(xij)][max(xij)−min(xij)],{{\rm{R}}_{{\rm{ij}}}} = {{\left[ {{{\rm{x}}_{{\rm{ij}}}} - \min ({{\rm{x}}_{{\rm{ij}}}})} \right]} \over {\left[ {\max ({{\rm{x}}_{{\rm{ij}}}}) - \min ({{\rm{x}}_{{\rm{ij}}}})} \right]}},

For Non-beneficial criteria or cost criteria (5)Rij=[max(xij)−xij][max(xij)−min(xij)],{{\rm{R}}_{{\rm{ij}}}} = {{\left[ {{\rm{max(}}{{\rm{x}}_{{\rm{ij}}}}) - {{\rm{x}}_{{\rm{ij}}}}} \right]} \over {\left[ {\max ({{\rm{x}}_{{\rm{ij}}}}) - \min ({{\rm{x}}_{{\rm{ij}}}})} \right]}}, where i = 1, 2, 3…. m; j = 1, 2, 3…. n.

All the criteria considered for this analysis are beneficial in nature, that is, whose higher values are desired. So Equation 4 is used for the normalization of the decision matrix. Table 9 presents the normalized decision matrix.

Step 3: Calculate the evaluative differences of the i^th alternative with respect to the other alternatives using Equation 6.

The evaluative differences (deviations) are calculated and presented in Table 10. (6)D(Ma−Mb)=(R(ij)a−R(ij)b){\rm{D}}({\rm{Ma}} - {\rm{Mb}}) = \left( {{{\rm{R}}_{{{({\rm{ij}})}_{\rm{a}}}}} - {{\rm{R}}_{{{({\rm{ij}})}_{\rm{b}}}}}} \right)

Step 4: Calculate the preference function P_j (Ma, Mb).

The preference function is calculated using the following two conditions given by Equations 7 and 8. (7)Pj(Ma,Mb)=0 if R(ij)a≤R(ij)b→D(Ma−Mb)≤0\matrix{ {{{\rm{P}}_{\rm{j}}}({\rm{Ma}},{\rm{Mb}}) = 0} \hfill \cr\,\,\,\,\,\,\, {{\rm{if}}\,{{\rm{R}}_{{{({\rm{ij}})}_{\rm{a}}}}} \le {{\rm{R}}_{{{({\rm{ij}})}_{\rm{b}}}}} \to {\rm{D}}({\rm{Ma}} - {\rm{Mb}}) \le 0} \hfill \cr } (8)Pj(Ma,Mb)=(R(ij)a−R(ij)b) if R(ij)a>R(ij)b→D(Ma−Mb)>0\matrix{ {{{\rm{P}}_{\rm{j}}}({\rm{Ma}},{\rm{Mb}}) = \left( {{{\rm{R}}_{{{({\rm{ij}})}_{\rm{a}}}}} - {{\rm{R}}_{{{({\rm{ij}})}_{\rm{b}}}}}} \right)} \hfill \cr\,\,\,\,\,\,\, {{\rm{if}}\,{{\rm{R}}_{{{({\rm{ij}})}_{\rm{a}}}}} > {{\rm{R}}_{{{({\rm{ij}})}_{\rm{b}}}}} \to {\rm{D}}({\rm{Ma}} - {\rm{Mb}}) > 0} \hfill \cr }

From Equations 7 and 8, it is clear that if the difference, that is, D (Ma − Mb) in Table 10 is less than or equal to zero, then substitute the preference function value as zero and if D (Ma - Mb) value is greater than zero, then the differences, that is, (R_(ij)a − R_(ij)b) is used as the preference function value.

In more easy words, if the value in any of the cell in Table 10 is less than or equal to zero (i.e., negative values), then substitute the value as zero and if the value in any of the cell in Table 10 is greater than zero, then leave it untouched. So, by substituting all the negative values of Table 10 by zeroes and keeping all the positive values as it is, thus obtaining the Table 11 as shown above.

Step 5: Calculate the aggregated preference, π (Ma, Mb) using Equation 9 as shown below. (9)π(Ma,Mb)=[∑j=1nwjPj(Ma,Mb)]∑j=1nwj.{\rm{\pi }}({\rm{Ma}},{\rm{Mb}}) = {{\left[ {\sum\nolimits_{{\rm{j}} = 1}^{\rm{n}} {{{\rm{w}}_{\rm{j}}}{{\rm{P}}_{\rm{j}}}({\rm{Ma}},{\rm{Mb}})} } \right]} \over {\sum\nolimits_{{\rm{j}} = 1}^{\rm{n}} {{{\rm{w}}_{\rm{j}}}} }}.

The summation of the criteria weightages is always equal to 1; hence, the denominator of Equation 9 becomes one. Now, multiplying all the criteria weightages with their respective column elements of Table 11 and using Equation 9, find all aggregated preference values π (Ma, Mb).

Table 12 presents the calculated aggregated preference values.

Step 6: Create an aggregate preference function matrix. Depending on the number of alternatives m, m × m matrix is formed. In this present analysis, 6 alternatives are considered so 6 × 6 matrix is formed as shown in Table 13.

For π (M1, M2), the aggregated preference value is 0.06554; this means the aggregated preference value of Model 1 with respect to Model 2 is 0.06554 (from Table 12). So this value (i.e., 0.06554) is allotted in the cell 1, 2 as shown in Table 13.

Similarly, for π (M1, M3), the aggregated preference value of Model 1 with respect to Model 3 is 0.02185 (from Table 12), so this value (i.e., 0.02185) is allotted in the cell 1, 3 as shown in Table 13.

In this way, the aggregated preference function value from Table 12 is allotted in all the cells of the above matrix, thus forming Table 13. When an alternative is compared to itself, no values should be assigned.

Step 7: Determination of the leaving and entering outranking flow of the alternatives.

The leaving and entering outranking flow in case of PROMETHEE I is calculated and presented in Table 14 using Equations 10 and 11 given below. (10)Leaving (positive) flow for ath alternative, φ+,∑b=1mπ(a,b){\rm{Leaving}}\;({\rm{positive}})\;{\rm{flow}}\;{\rm{for}}\;{{\rm{a}}_{{\rm{th}}}}\;{\rm{alternative,}}\;{\varphi ^ + }, \sum\nolimits_{{\rm{b}} = 1}^{\rm{m}} {{\rm{\pi }}({\rm{a}},{\rm{b}})} (11)Entering (negative) flow for ath alternative, φ−,∑b=1mπ(a,b){\rm{Entering}}\;({\rm negative})\;{\rm{flow}}\;{\rm{for}}\;{{\rm{a}}_{{\rm{th}}}}\;{\rm{alternative,}}\;{\varphi ^ - },\sum\nolimits_{{\rm{b}} = 1}^{\rm{m}} {{\rm{\pi }}({\rm{a}},{\rm{b}})} for example, a ≠ b.

The leaving and entering outranking flow in case of PROMETHEE II is calculated and presented in Table 15 using the Equations 12 and 13 given below. (12)Leaving (positive) flow for ath alternative, φ+,1n−1∑b=1mπ(a,b){\rm{Leaving}}\;({\rm{positive}})\;{\rm{flow}}\;{\rm{for}}\;{{\rm{a}}_{{\rm{th}}}}\;{\rm{alternative,}}\;{\varphi ^ + }, {1 \over {{\rm{n}} - 1}}\sum\nolimits_{{\rm{b}} = 1}^{\rm{m}} {{\rm{\pi }}({\rm{a}},{\rm{b}})} (13)Entering (negative) flow for ath alternative, φ−,1n−1∑b=1mπ(a,b){\rm{Entering}}\;({\rm negative})\;{\rm{flow}}\;{\rm{for}}\;{{\rm{a}}_{{\rm{th}}}}\;{\rm{alternative,}}\;{\varphi ^ - },{1 \over {{\rm{n}} - 1}}\sum\nolimits_{{\rm{b}} = 1}^{\rm{m}} {{\rm{\pi }}({\rm{a}},{\rm{b}})} for example, a ≠ b and n is the number of alternatives.

Step 8: Calculate the net outranking flow of each alternative using Equation 14. The net outranking flow of the alternatives are determined only in case of PROMETHEE II for the complete ranking of the alternatives. The net outranking flow of the alternatives is presented Table 16. (14)Net Flow{φ(a)}=Leaving Flow {φ+(a)}−Entering Flow{φ−(a)}{\rm{Net}}\,{\rm{Flow}}\,\{ {\rm{\varphi }}({\rm a})\} = {\rm Leaving}\,{\rm Flow}\,\{ {{\rm{\varphi }}^ + }({\rm a})\} - {\rm Entering}\,{\rm Flow}\,\{ {{\rm{\varphi }}^ - }({\rm a})\}

In case of PROMETHEE I, partial ranking is performed and the decision made by the decision maker is based on the three conditions given below.

In PROMETHEE I, instead of defining the rank, preferences are calculated.

Condition 1: Alternative a is preferred to/outranks alternative b, aPb

aPb if:

ϕ⁺ (a) > ϕ⁺ (b) and ϕ⁻ (a) < ϕ⁻ (b); or

Condition 2: Indifference situation, aIb

aIb if:

ϕ⁺ (a) = ϕ⁺ (b) and ϕ⁻ (a) = ϕ⁻ (b)

Condition 3: Incomparable situation, aRb

aRb if:

ϕ⁺ (a) > ϕ⁺ (b) and ϕ⁻ (a) > ϕ⁻ (b); or

On the basis of the above three conditions, all the alternatives are compared among each other and the decisions are made comparing one laptop model over another, which is presented in Table 17.

As there are 6 alternatives selected for this research purposes, there will be 15 comparisons in total. Table 17 presents the choices and preferences of one model over another.

Ranking is performed according to the decreasing order of net flow values. Table 18 shows the complete ranking of the laptop models. The preference ranking order of the models are given as follows:

Model 4 > Model 3 > Model 5 >