1. PROMETHEE for OMCDA

1. PROMETHEE for OMCDA

The PROMETHEE methods belong to the family of outranking methods, which include PROMETHEE I for partial ranking of the alternatives and PROMETHEE II for complete ranking. Other upgrades were also developed in the past decades, such as PROMETHEE III for ranking problems involved interval alternatives, PROMETHEE IV for both partial and complete rankings, and PROMETHEE V for problems with specific constraints.

Consider a problem where a set of possible actions A={a, b, ... } are evaluated on k criteria, i.e. F={f1, f2, ..., fk}. In consideration of the convenience in calculation and without loss of generality, each of the criteria is supposed to be maximized. The problem can be presented in a n×k evaluation matrix, with the actions and criteria being presented in the line and column, respectively. Firstly, a comparison in single criterion is conducted between actions a and b as follows:

Eq. (A.1)

where dk(a,b) is difference between actions a and b under the kth criterion. It is then used in preference function Pj(x). Several types of preference functions were provided in the original PROMETHEE definition, like linear, step or Gaussian preference functions. Among them, the linear criterion preference function is often used in solving MCDA problems, which can be described as follows:

Eq. (A.2)

where qk represents value of an indifference threshold; pk represents value of a strict preference threshold pk. The value of Pk(x) ranges from 0 to 1: when approaching 0, it is considered as negligible by the decision maker and action a is not strictly preferred to action b; when approaching 1, on the contrary, the difference between two actions is significant and action a is highly preferred to action b (as shown in Fig. A.1). When the difference is between two thresholds, an intermediate value is calculated by the linear method as the preference degree.

After estimating the weight for each criterion, the positive and negative flows can be defined, and a multi-criteria preference degree equation is introduced as follows:

Eq. (A.3)

where wk represents weight of criterion fk, assuming the sum value of wk is 1 when

Eq. (A.4)

In order to explore the performance of each alternative ranking against n-1 other ones, the following two outranking flows are defined:

The Positive flow:

Eq. (A.5)

Negative flow:

Eq. (A.6)

The positive flow illustrates how much one decision is outranking over the other alternative. If φ+(a) scores high, then the preference power of action a is distinct. In comparison, the negative flow expresses how much one decision is outranked by the others, and a smaller score of φ-(a) represents strength over other alternatives.

In PROMETHEE I, a partial ranking is defined as follows:

– a is preferred to b iif. φ+(a) φ+(b) and φ-(a) φ-(b);

– a is indifferent to b iif. φ+(a) =φ+(b) and φ-(a) =φ-(b) ;

– a is incomparable to b iif. φ+(a) φ+(b) and φ-(a) φ-(b),or φ+(a) φ+(b) and φ-(a) φ-(b).

PROMETHEE II is evolved on the basis of partial ranking, which combines both positive and negative flows and calculates the net flow by:

Eq. (A.7)

This leads to:

– a is preferred to b iif. φ+(a) φ+(b);

– a is indifferent to b iif. φ+(a) =φ+(b) ;

In addition,

represents net flow for action a under criterion j. All individual net flows of alternatives under each criterion can be summarized in a net flow matrix as follows:

where ai represents action, and each column stands for the net flow of action under a specific criterion. The matrix could provide information related to the performance of alternatives, and is used in the PROMETHEE GAIA process.

2. GAIA for OMCDA

The GAIA method can provide graphical representation for decision makers in a two dimensional view. The conflict among criteria and the impact of the weight on final outranking results can be easily observed in the GAIA plane. Combing with the principal component analysis (PCA) on matrix φ, a plane is define on which only a few information is lost in projection. Before the PCA, the covariance matrix of net flows of all alternatives C was calculated by:

Eq. (A.8)

Where φ’ represents transported matrix of φ. Two k-dimensional eigenvectors are described as u and υ, respectively. In order to minimize the loss by projection, expression u’Cu+v’Cv related to u and υ, has to be maximized by the following description:

Eq. (A.9)

Where cjj is variance ofφj; cjs is covariance matrix between φj and φs; ||γj|| is projection length on the jth criterion; (γj, γs) is scalar product between γj and γs; λ1 and λ2 are the largest and second largest eigenvalues of C, respectively. During the PCA, a multidimensional plane (or say GAIA plane) is built based on λ1 and λ2, in which all the projection of each alternative has the least loss in information.

All projection of criteria γj would be presented as a specific line with individual length and direction. The analysis on projection γj mainly focuses on the following aspects: (1) the length of γj on the GAIA plane represents dominated power of criterion fj; if all alternatives were strongly distinguished under fj, then variance of φj would be very large and the projection of the action would range in an extensive scale, leading to a long projection in the plane, i.e. the longer γj, the more fj distinguishing the alternatives; (2) according to the definition of covariance, criteria expressing the same preference have a large covariance. When two axes of criteria present the same direction in the plane, the criteria are considered as the similar criterion; while when two criteria were independent, covariance cjs between them would close to zero, i.e. axes of two criteria being orthogonal. When two criteria are conflict, cjs would be negative and axes of criteria show an opposite direction in the GAIA plane.

Besides the projection of criterion j on the plane, the PROMETHEE decision axis is also estimated by calculating weight vectors w in the k-dimensional space, which is described as:

Eq. (A.10)

Additionally, the net flows are described in the form of scalar product between vectors ai and w as follows:

Eq. (A.11)

Vector w is decision axis to judge the performance of ai when the weight is taken into account. Since all parameter vectors are discussed in the plane, the projection of weight vectors π is used as the PROMETHEE decision axis. When π is short, the power of the decision axis is weak and the length of π indicates that, some criteria are conflict and the ideal alternative would appear near the origin. When the length of π is long, on the contrary, the alternatives selected in the direction would be desirable. Furthermore, the decision maker could adjust the weight and preference function in the PROMETHEE process, and then the direction and length of decision axes will change correspondingly. Except for the alternatives easy to be distinguished in the GAIA plane, there are some alternatives that are incomparable in PROMETHEE I as their performance is remarkable under different criteria. For these alternatives, PROMETHEE II is introduced for decision makers to make appropriate choice.

Table and Figure Captions List

Fig. A.1 The linear preference function

Table A.1 Preference parameters for scenario 1

Table A.2 Performances of top five pumping-rate groups in scenario 1

Table A.3 Preference parameters for scenario 2

Table A.4 Performances of top five pumping-rate groups in scenario 2

Table A.5 PROMETHEE ranking of five typical pumping-rate groups under 10 years of remediation

Table A.6 Preference parameters for scenario 3

Table A.7 Performances of top five pumping-rate groups in scenario 3

Table A.8 PROMETHEE ranking of five typical pumping-rate groups under 15 years of remediation

Table A.9 Preference parameters for scenario 4

Table A.10 Performances of top five pumping-rate groups in scenario 4

Table A.11 PROMETHEE ranking of five typical pumping-rate groups under 20 years of remediation

Fig. A.1 The linear preference function

Table A.1 Preference parameters for scenario 1

Table A.2 Performances of top five pumping-rate groups in scenario 1

Table A.3 Preference parameters for scenario 2

Table A.4 Performances of top five pumping-rate groups in scenario 2

Table A.5 PROMETHEE ranking of five typical pumping-rate groups under 10 years of remediation

Table A.6 Preference parameters for scenario 3

Table A.7 Performances of top five pumping-rate groups in scenario 3

Table A.8 PROMETHEE ranking of five typical pumping-rate groups under 15 years of remediation

Table A.9 Preference parameters for scenario 4

Table A.10 Performances of top five pumping-rate groups in scenario 4

Table A.11 PROMETHEE ranking of five typical pumping-rate groups under 20 years of remediation