AUTHOR COPY
2099
Journal of Intelligent & Fuzzy Systems 29 (2015) 2099–2110 DOI:10.3233/IFS-151686 IOS Press
A new prioritized multi-criteria outranking method: The prioritized PROMETHEE Liuhao Chena and Zeshui Xub,∗ of Communications Engineering, PLA University of Science and Technology, Nanjing, Jiangsu, China School, Sichuan University, Chengdu, Sichuan, China
PY
a College
CO
b Business
OR
Abstract. The Multiple Criteria Decision Aiding (MCDA) has been a fast growing area of operations research and management science during the last decades, while the PROMETHEE (Preference Ranking Organization Method for Enrichment Evaluations) is one of the well-known MCDA methods with ranking the alternatives from the best to the worst. In this paper, we propose a new outranking method called Prioritized PROMETHEE (P-PROMETHEE) method, which is used to deal with the ranking problem of Prioritized Multiple Criteria Decision Making (PMCDM). First, we use the idea of entropy to get the information of each criterion in the same prioritized level, which is used to get the weight of each criterion. Then, we use the weighted average operator to aggregate all the criteria in the same prioritized level. Moreover, we apply Yager’s prioritized measure-guided aggregation operator to the classical PROMETHEE method to get the P-PROMETHEE method. Finally, we give a practical example to illustrate the effectiveness of the P-PROMETHEE method.
1. Introduction
TH
Keywords: Entropy, prioritized measure, PROMETHEE method, weight vector, choquet integral
AU
As we know, the PROMETHEE method developed by Brans et al. [4, 6], is one of the effective outranking methods to solve the MCDM problems. As the PROMETHEE method has the properties of simplicity, clearness and stability, it has been widely used to solve practical multicriteria decision making problems, such as environment management, business and financial management, manufacturing and assembly, etc. [1, 2, 20]. The PROMETHEE method is based on the pairwise comparison of alternatives with respect to each criterion to get the ranking of all the alternatives. The PROMETHEE method includes: PROMETHEE I for partial ranking of the alternatives and PROMETHEE II for complete ranking of the alternatives [4]. Later, many useful extensions have been developed to enrich ∗ Corresponding
author. Zeshui Xu, Business School, Sichuan University, Chengdu, Sichuan, China. Tel.: +86 25 84483382; Fax: +86 25 84483382; E-mails: [email protected]; [email protected]
the PROMETHEE method. For instance, Brans and Mareschal developed PROMETHEE III for ranking based on intervals and PROMETHEE IV for solving a choice problem of an infinite set of alternatives. In 1988, Brans and Mareschal [15] proposed the visual interactive module GAIA (Geometrical Analysis for Interactive Aid) which provides a marvellous graphical representation supporting the PROMETHEE method. PROMETHEE V for MCDA including segmentation constraints and PROMETHEE VI for representation of the human brain was also developed by Brans and Marescha [5, 16] in 1992 and 1995 respectively. However, in order to implement the PROMETHEE method, we require three types of additional information: preference function shape, weights of criteria and thresholds. As Corrente et al. [10] pointed out, the preference information can be obtained through two ways: directly and indirectly. The Decision Maker (DM) can get all the preference information between alternatives when he provides directly all values of parameters in the model. The DM only provides some preference
1064-1246/15/$35.00 © 2015 – IOS Press and the authors. All rights reserved
L. Chen and Z. Xu / A new prioritized multi-criteria outranking method: The prioritized PROMETHEE
PY
of safety arrives some given level. For this situation, some methods have been developed to overcome this drawback [8, 19, 24, 28–30, 32, 33]. For example, Yager [28, 29] used the prioritized weights in which the weights associated with the lower prioritized criteria are related to the satisfaction of the higher prioritized criteria to express the prioritized relation between criteria. Based on Yager’s prioritzed weights, Wei and Tang [24] proposed a generalized prioritized averaging operator and a generalized prioritized OWA operator. Yan [32] proposed a prioritized weighted aggregation operator based on the ordered weighted averaging (OWA) operator and the triangular norms (t-norms) to deal with multi-criteria decision making (MCDM) problems with multiple priorities. Yager [30] developed a prioritized aggregation operator which uses the monotonic measure set to convey the prioritization relationship. Based on Yager [30]’s idea, Chen [8] developed a general prioritized aggregation operator which allows the partial compensation between prioritized criteria. However, the mentioned literature focuses on dealing with the prioritized MCDM problems, few papers pay attention to the outranking method with the prioritized relationship of criteria. As we know, Yu et al. [34] developed a new function which takes into account the influence of expectation levels to the prioritized weights in the PROMETHEE method. However, Yu et al. [34] only investigated the strict ordered prioritized PROMETHEE method. In this paper, we will mainly consider some much more general problems with the weak ordered prioritized criteria. As the above mentioned, the PROMETHEE method is a well-known outranking method. In order to meet the needs of weak ordered prioritized relationship, we will propose a new outranking method called Prioritized PROMETHEE method, which is an integrated approach between the prioritized measure and the PROMETHEE method. The rest of the paper is organized as follows: Section 2 describes the classical PROMETHEE II. In Section 3, we recall Yager’s prioritized measure-guided aggregation operator. In Section 4, we propose a P-PROMETHEE method and employ a practical example to illustrate its effectiveness. Section 5 ends the paper with some conclusions.
AU
TH
OR
information between alternatives from which we get indirectly the compatible preference parameters. Obviously, we can find out that the indirect preference information needs less information from the DM than the direct preference information. Meanwhile, the indirect preference information is also relatively easy given by the DM. For this reason, the study of the PROMETHEE method based on indirect preference information is meaningful. Up to now, many methods have been proposed to elicit the preference parameters. For example, Solymosi and Dombi [22] proposed the centralized weight of compatible preference parameters to represent the relative importance of criteria. Mareschal [14] gave a sensitivity analysis in order to obtain stability interval weights for criteria, in which we can get the same ranking of alternatives. Sun and Han [23] solved a linear programming problem to find the most discriminant set of weights compatible with the preference information provided by the DM. Eppe et al. [12] proposed a bi-objective optimization model which considers the number of inconsistencies and the robustness of parameter values to elicit the PROMETHEE II’s preference parameters. The above mentioned papers mainly consider the MCDM problem with the criteria in the same level. Meanwhile, there are many methods to obtain the total satisfaction degree for each alternative to all criteria, such as the weighted average operator [27], the ordered weighted averaging (OWA) operator [27] and the ordered weighted geometric (OWG) operator [9, 25]. These aggregation operators allow to tradeoff between criteria. For example, if we use the weighted average operator, thetotal satisfaction degree can be obtained by C(x) = ni=1 w i Ci (x), where Ci (x) represents the satisfaction degree of the alternative x to the criterion Ci and w i denotes the weight of the criterion Ci . In this case, we can compensate for a decrease of in satisfaction to the criterion Ci by a gain (w k /w i ) in satisfaction to the criterion Ck . However, there are lots of MCDM problems with the prioritized relation of criteria and we usually permit no compensation between criteria. Yager [28, 29] listed several practical MCDM problems to illustrate this kind of situations, such as selecting a bicycle, an organization decision-making problem, and a document retrieval problem. The typical example is the case of buying a car with respect to the criteria of safety and cost. In this case, the criterion safety has a higher priority than the criterion cost, it indicates that we are not willing to tradeoff the satisfaction of the criterion cost until the satisfaction
CO
2100
2. The classical PROMETHEE methods In what follows, we first introduce the basic principles and some basic concepts related to the PROMETHEE method (see Refs. [4, 6] for more
L. Chen and Z. Xu / A new prioritized multi-criteria outranking method: The prioritized PROMETHEE
(1)
π(ai , aj ) = π(aj , ai ) =
n l=1 n
w l · Pl (ai , aj )
(2)
w l · Pl (aj , ai )
l=1
AU
TH
where π(ai , aj ) represents the total degree of preference of the alternative ai to the alternative aj taking into account all criteria. The weight w l represents the relative importance of the criterion Cl . In order to get the ranking of all the alternatives, the PROMETHEE I method defines the positive outranking flow and negative outranking flow as follows: 1 φ+ (ai ) = π(ai , x) m−1 x∈X\{ai }
φ− (ai ) =
1 m−1
In order to capture the prioritized relationship of criteria, Yager [30] adopted the fuzzy measure and the fuzzy integral to develop a prioritized measure-guided aggregation operator, which has the characteristic that the higher prioritized criteria’s satisfaction can’t be compensated by the lower prioritized criteria’s satisfaction. Now we first give the definition of fuzzy measure: Definition 1. [3, 17] Let N = {1, 2, · · · , n}. A discrete fuzzy measure is a set function µ : 2N → [0, 1] which is monotonic (i.e., µ(A) ≤ µ(B) whenever A ⊂ B ⊆ N) and satisfies µ(Ø) = 0 and µ(N) = 1. Fuzzy measure was first introduced by Sugeno [17], and has usually been used to deal with the MCDM problems associated with fuzzy integral. Meanwhile, fuzzy measures are not only used to represent the independent relationships of criteria, but also the dependent relationships of criteria. Furthermore, Yager and Walker [31] first used a special kind of fuzzy measure to capture the strictly ordered prioritizations. They gave the following definitions: ⎧ ⎨ {Ck | k = 1, 2, . . . , j}, j = 1 , . . . , n Lj = (3) ⎩ Ø, j=0
OR
where dl (ai , aj ) = f (ai ) − f (aj ) and Fl (·) is a monotonically non-decreasing function varying from 0 to 1. In order to facilitate the selection of the preference function, Brans and Vincke [7] proposed six basic types of preference functions: (1) usual criterion, (2) U-shape criterion, (3) V-shape criterion, (4) level criterion, (5) V-shape with indifference criterion, (6) Gaussian criterion. Certainly, the DM has to give the value of an indifference threshold q, and the value of a strict preference threshold p for each criterion. For any two alternatives ai , aj ∈ X, we have
3. Prioritized measure-guided aggregation operator
PY
Pl (ai , aj ) = Fl (dl (ai , aj )), for ai , aj ∈ X
for all alternatives. The PROMETHEE II can get a complete ranking according to the value of net outranking flow for each alternative, which is obtained by φ(ai ) = φ+ (ai ) − φ− (ai ).
CO
details). For simplicity, we suppose that a multi-criteria decision making problem consists of a set of alternatives X = {a1 , a2 , . . . , am } and a set of criteria C = {C1 , C2 , . . . , Cn }. Let fl (ai ) represent the evaluation of the alternative ai on the criterion Cl . For the given criterion Cl , the DM can always give a function Pl (ai , aj ) which represents the degree of preference of the alternative ai to the alternative aj under the criterion Cl . We can write the function as follows:
2101
π(x, ai )
x∈X\{ai }
where the positive outranking flow φ+ (ai ) represents how much the alternative ai prefers to all the other alternatives. The larger φ+ (ai ), the better the alternative ai . Similarly, the negative outranking flow φ− (ai ) represents how much all the other alternatives prefers to the alternative ai . The smaller φ− (ai ), the better the alternative ai . The PROMTHEE I can get a partial ranking according to the value of the positive outranking flow and the value of negative outranking flow
where we assume C1 C2 · · · Cn in the set of criteria. They are associated with each subset Lj the value j/n. Then the prioritized measure m : 2C → [0, 1] can be defined as: j Gj (A) (4) m(A) = Maxj n where A is a subset of C and ⎧ ⎨ 1, Lj ⊆ A Gj (A) = ⎩ 0, Lj ⊂ / A
(5)
Besides, the prioritized measure can also be written as: µ(A) =
n 1 · Gj (A) n j=1
(6)
L. Chen and Z. Xu / A new prioritized multi-criteria outranking method: The prioritized PROMETHEE
Definition 2. [3, 13] The discrete Choquet integral with respect to a fuzzy measure µ is given by Choqµ (a) =
n
a(i) µ({j| aj ≥ a(i) })
i=1
−µ({j|aj ≥ a(i−1) })
(7)
C(x) = Choqµ (C1 (x), C2 (x), · · · , Cn (x)) =
n
(µ(Pj ) − µ(Pj−1 ))Cind(j) (x)
j=1
(8)
AU
TH
where Pj = {Cind(1) , Cind(2) , · · · , Cind(j) }, and ind(j) is the index of the jth largest Cj (x). The measure µ(·) is given by the formula (5). The concept of entropy was first introduced by Shannon [21] to measure the uncertainty of a random variable. The entropy is very important for measuring the uncertain information in information theory. Recently, many scholars take the idea of entropy into the area of decision making, for example, O’Hagan [18] suggested a maximum entropy method to determine the weight vector of the OWA operator, Xu and Hu [26] proposed an entropy-based procedure to derive attribute weights under intuitionistic fuzzy environment. The definition of entropy is given as follows: Definition 3. [3, 21] For a given weighting vector w = (w 1 , w 2 , · · · w n ), where w i ∈ [0, 1], and ni=1 w i = 1, then the entropy of this weighting vector is defined by E(w) = −
n
w i ln w i
i=1
with the convention 0 · ln 0 = 0.
As Section 2 pointed out, the classical PROMETHEE method is a simple and effective outranking method. From the formula (1), we can find that the classical PROMETHEE method allow the tradeoff between criteria in order to get the total preference relationship of alternatives. However, this paper mainly studies a special ranking problem with the prioritized relationship between criteria. Since the higher prioritized criteria’s satisfaction can’t be compensated by the lower prioritized criteria in the prioritized MCDM problems, the classical PROMETHEE method is not suitable to deal with the prioritized MCDM problems. Yager [30, 31] first introduced the prioritized measure and the prioritized measure-guided aggregation operator to deal with the prioritized MCDM problems. Fortunately, Yager’s prioritized measure-guided aggregation operator can capture the prioritized relationship between criteria, which means the lack of satisfaction to the higher prioritized criteria cannot be compensated by the increased satisfaction of those lower prioritized criteria. In order to deal with the prioritized outranking problems, we apply Yager’s prioritized measure-guided aggregation operator into the classical PROMETHEE method, and thus develop a prioritized-PROMETHEE method.
OR
where a = (a(1) , a(2) , . . . , a(n) ) is a non-increasing permutation of the input a, and a(0) = ∞ by convention. As we know, Choquet integral is a typical type of fuzzy integrals, and it can be used to aggregate information from a set of resources with respect to a given fuzzy measure. In order to get the prioritized relationship of criteria, Yager and Walker [31] used Choquet integral to obtain an aggregation based on the prioritized measure. Thus, we can use the prioritized aggregation operator to calculate the overall satisfaction degree C(x) for each alternative x by
4. The prioritized PROMETHEE method
PY
The discrete Choquet integral is an adequate aggregation function that extends the OWA operator by taking the interactions of criteria into account.
CO
2102
(9)
4.1. Problem formulation and the proposed approach Before the introduction of the prioritized outranking problems, we first give the definition of the prioritized MCDM problems as follows: Definition 4. [28, 29] In a MCDM problem, if the set of criteria, C = {C1 , C2 , · · · , Cn }, can be partitioned into q distinct prioritized hierarchies, H1 , H2 , · · · , Hq , such that Hk Hl if k < l, q where Hk = k1 , Ck2 , · · · , Cknk } ⊂ C, C = ∪k=1 Hk {C q (i.e., n = k=1 nk ), and Hk ∩ Hl = Ø for k, l ∈ {1, 2, · · · , n} (“Ø” denotes the null set), then the problem is called a prioritized MCDM problem. As Yager [28, 29] pointed out, the prioritized MCDM problems can be classified into two cases: (1) strictly ordered prioritizations, if each priority level has only one criterion, i.e., nk = 1 for k = 1, 2, · · · , q; and (2) weakly ordered prioritizations, otherwise. For the above prioritized MCDM problems, we mainly investigate the prioritized outranking problems, which aim to obtain the ranking of the alternatives based on the pairwise preference relationships of the
L. Chen and Z. Xu / A new prioritized multi-criteria outranking method: The prioritized PROMETHEE
aj ∈X\{ai }
(ai ) = φkj
φkj (ai ) + 1 , j ∈ {1, 2, · · · , nk } , m i ∈ {1, 2, · · · , m} (15)
Therefore, we have
TH
AU
ai ∈X
For the prioritized level Hk = {Ck1 , Ck2 , · · · , Cknk } ⊂ C, we have ⎞ ⎛ φk1 (a1 ) φk2 (a1 ) · · · φknk (a1 ) ⎟ ⎜ ⎜ φk1 (a2 ) φk2 (a2 ) · · · φknk (a2 ) ⎟ ⎟ ⎜ (13) φk = ⎜ ⎟ .. .. .. .. ⎟ ⎜ . . . . ⎠ ⎝ φk1 (am ) φk2 (am ) · · · φknk (am ) In order to get the total flow in the prioritized level Hk for each alternative ai , we have to aggregate the normed flow φkj (ai ), j ∈ {1, 2, · · · , nk } into a value φk (ai ). Usually, we can use the weight average operator to get the value φk (ai ) defined in the following way: φk (ai ) =
nk j=1
w kj φkj (ai )
(14)
0 ≤ φkj (ai ) < 1,
OR
(11) where the preference value Pl (ai , aj ) can be got by the formula (1). For each criterion Cl , we should select a suitable preference function Pl (·, ·) to depict the preference value between two alternatives. In order to obtain the suitable preference function easily, Brans and Vincke [7] proposed six basic types of preference functions. Therefore, the preference function Pl (·, ·) is the key point to capture the criteria’s characteristics. So the normed flow φl (ai ) represents how much the alternative ai prefers to all the other alternatives under the criterion Cl . The larger φl (ai ), the better the alternative ai under the criterion Cl . Moreover, we have −1 ≤ φl (ai ) ≤ 1, φl (ai ) = 0 (12)
PY
where the preference value Pl (ai , aj ) can be got by the formula (1). In order to visually compare the alternatives for each criterion Cl , Mareschal and Brans [15] introduced the normed flow of each alternative ai for each criterion Cl : 1 φl (ai ) = Pl (ai , aj ) − Pl (aj , ai ) m−1
For the alternative ai , the value φk (ai ) can be seen as the net flow of the pseudo criterion Ck , which is the aggregation of all the kth prioritized criteria. However, the DM can’t give the weight of each criterion in lots of actual problems especially when there are lots of criteria in the ranking problem. The DM can only estimate the criteria whether in the same prioritized level. Below we propose an entropy-based approach to elicit the weights of criteria in the same prioritized level. According to the formula, we can know that the normed flow φkj (ai ) isn’t normal for all alternatives. So we introduce a transformation function to normalize the value φkj (ai ) as follows:
CO
alternatives. For the prioritized outranking problem, we can obtain the preference matrix of alternatives for each criterion Cl : ⎞ ⎛ Pl (a1 , a1 ) Pl (a1 , a2 ) · · · Pl (a1 , am ) ⎟ ⎜ ⎜ Pl (a2 , a1 ) Pl (a2 , a2 ) · · · Pl (a2 , am ) ⎟ ⎟ ⎜ (10) ⎟ ⎜ .. .. .. .. ⎟ ⎜ . . . . ⎠ ⎝ Pl (am , a1 ) Pl (am , a2 ) · · · Pl (am , am )
2103
ai ∈X
φkj (ai ) = 1
(16)
We apply the information entropy formula to = (φ calculate the entropy for the vector φkj kj (a ), · · · , φ (a )): (a1 ), φkj 2 kj m E(φkj )
=−
m
φkj (ai ) ln φkj (ai )
(17)
i=1 ), the less Obviously, the greater the entropy E(φkj deviation between all the alternatives under the criterion ) arrives the biggest value, we Ckj . If the entropy E(φkj (a ) = φ (a ) = · · · = φ (a ) which means have φkj 1 kj 2 kj m that all the alternatives have the same value under the criterion Ckj . It is reasonable to give the criterion Ckj small weight for obtaining a complete rank. Using the formula , the weight of each criterion in the same prioritized level can be obtained by
w kj
1 ) E(φkj = nk 1 ) E(φkj
(18)
j=1
Based on the formulas (13) and (17) , we can have the net flow of each alternative ai in the prioritized level Hk . Meanwhile, all the results φk (ai ) (k = 1, 2, · · · , q; i = 1, 2, · · · , m) are listed in the following matrix:
L. Chen and Z. Xu / A new prioritized multi-criteria outranking method: The prioritized PROMETHEE
⎛
φ1 (a1 ) φ2 (a1 ) · · · φq (a1 )
⎞
⎜ 1 ⎟ ⎜ φ (a2 ) φ2 (a2 ) · · · φq (a2 ) ⎟ ⎜ ⎟ φ=⎜ . .. .. ⎟ .. ⎜ .. ⎟ . . . ⎝ ⎠ 1 2 q φ (am ) φ (am ) · · · φ (am )
(19)
Certainly, we have to aggregate the k-net flows φk (ai ), k = 1, 2, · · · , q into a total net flow for each alternative ai . Due to the net flow in the different prioritized level, Yager’s prioritized measure guided aggregation operator is used to aggregate the total net flow. Utilizing the formula, we have the total net flow φ(ai ) as follows:
the information contained in the ASR information structure can be we use the PAC. For simplicity, converted into a to express all set P = (il , jl ) | l ∈ 1, 2, · · · , KPAC the preference information. The generalized prioritized measure-guided aggregation operator uses the following measure: q µ (A) = αj · Gj (A) (24) j=1
where the quantity αi satisfies αi ∈ [0, 1] and n i=1 αi = 1. Combining the formulas (19) and (23) , we have
PY
2104
φ (ai ) = Choqµ (φ1 (ai ), φ2 (ai ), · · · , φq (ai ))
φ(ai ) = Choqµ (φ1 (ai ), φ2 (ai ), · · · , φq (ai )) (µ(Pj ) − µ(Pj−1 ))φind(j) (ai )
(20)
j=1 {Cind(1) ,
Cind(2) ,···
, Cind(j) }, j largest φ (ai ).
q 1 j=1
Gj (A) =
· Gj (A)
(21)
⎧ ⎨ 1, Lj ⊆ A
(22)
⎩ 0, Lj ⊂ / A
⎧ ⎨ {Ck | k = 1, 2, . . . , q}, j = 1 , . . . , q ⎩ Ø,
j=0
(23)
AU
Lj =
q
TH
µ(A) =
(µ (Pj ) − µ (Pj−1 ))φind(j) (ai )
Therefore, we can get a complete ranking of all the alternatives according to the values of the total net flows φ(ai ), i = 1, 2, · · · , m. Of course, if the DM provides partial information of his preference or experience, we employ Yager [30]’s generalized prioritized measure-guided aggregation operator to construct a model to get the total net flow. As Eppe and Smet [11] pointed out, there are two types of partial preference information: Pairwise Action Comparisons (PAC) and Action Sub-rankings (ASR). The PAC gives the pairwise preference relation of alternatives, denoted by ail ajl , which by the can be expressed a . The · · · , K constraint cl = (il , jl ), with l ∈ 1, 2, PAC
ASR considers a sub-ranking c = i1 , i2 , · · · , iaKASR that expresses that ai1 ai2 · · · aiKa . Obviously, ASR
(25)
j=1
In order to elicit the parameter αi , we establish the following model: max ξ s.t. φ (ai ) − ξ ≥ φ (aj ), (i, j) ∈ P, q (MOD - 1) αl = 1,
OR
where Pj = and ind(j) is the index of the j th The prioritized measure can be obtained by the formulas as follows:
q
CO
=
q
=
(26)
l=1
α1 ≥ α2 ≥ · · · ≥ αq ≥ 0.
Through solving the linear programming model (25), we obtain the optimal value ξ ∗ and the parameter value α∗ = (α∗1 , α∗2 , · · · , α∗q ). If ξ ∗ > 0, then we can get the parameter α∗ that is compatible with the DM’s preference information. If ξ ∗ < 0, then we can’t get the parameter α∗ that is compatible with the DM’s preference information. In this case, we propose the following model to elicit the parameter: ξij max (i,j)∈P
s.t. (MOD - 2)
φ (a
≥ φ (aj ), (i, j) ∈ P, q αl = 1, i ) − ξij
(27)
l=1
α1 ≥ α2 ≥ · · · ≥ αq ≥ 0. With the weight parameter vector α∗ = (α∗1 , , α∗q ) obtained by the above linear programming, we can compute the total net flow of the alternatives and rank the alternative from best to the worst one. α∗2 , · · ·
L. Chen and Z. Xu / A new prioritized multi-criteria outranking method: The prioritized PROMETHEE
2105
4.2. The proposed algorithm Based on the above analysis, we develop a practical algorithm for the prioritized multi-criteria outranking problem in which the weights of criteria are completely unknown and the DM’s preference information is partially known or completely unknown, which involves the following steps:
Step 3. Utilize the formula (10) to get the normed flow of the alternative ai for the criterion Cl . For each prioritized level, we utilize the formulas (17), (16) and (14) to obtain the weight vector w k = (w k1 , w k2 , · · · , w knk ).
CO
Step 2. Apply the preference function (1) to calculate the preference of the alternative ai with regard to the alternative aj with respect to each criterion. The preference function Fl (·) and the corresponding thresholds p and q can be given by the DM.
PY
Step 1. For a prioritized multi-criteria outranking problem, we use the formula dl (ai , aj ) = f (ai ) − f (aj ) to determine the difference between the evaluations of ai and aj with respect to each criterion.
Fig. 1. The prioritized relationship between criteria for the strategic assessment problem of islands and reefs.
Step 5. If the DM’s preference information of alternatives is completely unknown, then we use the formulas (19)-(22) to get the total net flow for each alternative. Based on the total net flow, we determine the rank of all alternatives; if the DM’s preference information of alternatives is partially known, then we use the model (MOD-1 or MOD-2) to obtain the vector of the weight parameter values α∗ = (α∗1 , α∗2 , · · · , α∗q ), turn to next step.
ing three aspects: Sensitivity, Exposure and Defense. Meanwhile, the sensitivity contains criteria: history conflict (S1 ), foreign relation (S2 ), ethnic and religious relation (S3 ) and strategic value (S4 ). The exposure contains criteria: population (E1 ), the value of island’s facility (E2 ), the value of island’s resource (E3 ) and strategic position (E4 ). The defense contains criteria: weapon (D1 ), the number of military bases and troops (D2 ), detection capability (D3 ), military support capability (D4 ). The prioritized relationship of criteria is listed in Fig. 1. While the population (E1 ), the value of island’s facility (E2 ), the value of island’s resource (E3 ) and the number of military bases and troops (D2 ) are all qualitative criteria. The assessments for history conflict (S1 ), foreign relation (S2 ), ethnic and religious relation (S3 ), strategic value (S4 ), strategic position (E4 ), weapon (D1 ), detection capability (D3 ) and military support capability (D4 ) are represented by linguistic terms. The linguistic terms used here and their corresponding INs (integer numbers) are shown in Table 1. The weights of all criteria are completely unknown. For each criterion, we consider the linear preference function and define the thresholds as follows: the indifference threshold q is fixed at 0 and the preference
AU
TH
OR
Step 4. Utilize the formula (13) to determine φk (ai ) which represents the net flow of the alternative ai under the pseudo criterion Ck .
Step 6. Utilize the formulas (19)-(23) to obtain the total net flow for each alternative and rank them from best to worst. 4.3. Illustrative example In order to illustrate our approach, let us consider the strategic assessment of islands and reefs. As we know, China is a coastal country and has many islands and reefs. However, due to the limited comprehensive national strength, most of the islands and reefs are undeveloped. Therefore, it is necessary to rank all the islands and reefs according to the strategic status of islands and reefs. In this paper, the strategic status of islands and reefs are evaluated by the follow-
L. Chen and Z. Xu / A new prioritized multi-criteria outranking method: The prioritized PROMETHEE Table 1 The linguistic scales and their corresponding INs
Linguistic variables and semantics
Integer numbers
s1 : Very Low/none s2 : Low/common s3 : Medium/big s4 : High/serious s5 : Very High/fierce
Alternatives
1 2 3 4 5
The second prioritized level
Parameter p
Parameter q
D1 D2 D3 D4 S1 S2 S3 S4 E1 E1 E1 E1
4 4 2 4 3 3 4 3 22 1246 8781 4
0 0 0 0 0 0 0 0 0 0 0 0
D1
D2
D3
D4
5 3 3 1 5 4
4 0 1 0 1 2
2 3 3 1 2 1
1 2 3 1 5 5
S1
S2
S3
S4
3 5 5 2 4 4
2 3 1 1 3 4
5 1 3 3 1 2
1 4 2 3 1 4
a1 a2 a3 a4 a5 a6
E1
E2
E3
E4
21 5 11 19 24 2
2859 2552 1974 1872 1894 1613
20170 20215 26353 25897 22887 17572
3 5 1 2 1 1
OR
The third prioritized level
Criteria
a1 a2 a3 a4 a5 a6
CO
The first prioritized level
Criteria
a1 a2 a3 a4 a5 a6
Table 2 The indifference and preference thresholds of each preference function Prioritized levels
Table 3 Satisfaction degree of each criterion corresponding to each alternative
PY
2106
AU
TH
threshold p is equal to the highest difference between the islands’ performances on the particular criterion (see Table 2). In this example, there are six islands and reefs represented by a1 , a2 , a3 , a4 , a5 , a6 which are needed to be ranked. The decision data used in the strategic assessment of islands and reefs are shown in Table 3. Fortunately, the DM can give partial information of the alternatives: a5 a1 , a1 a3 , a6 a3 (“” denotes “be superior to”). In the following, we use the P-PROMETHEE method to rank the six islands. The solution process and the computation results are summarized as follows: Firstly, we use the formula (10) to calculate the normed flow for each alternative on each criterion. The results are shown in Table 4. Secondly, For each prioritized level, we utilize the formulas (17), (16) and (14) to obtain the weight vector wk = (w k1 , w k2 , · · · , w knk ) as follows: w 1 = (w 11 , w 12 , w 13 , w 14 ) = (0.2489,0.2455,0.2528,0.2528) w = (w 21 , w 22 , w 23 , w 24 ) 2
= (0.2499,0.2493,0.2466,0.2542) w 3 = (w 31 , w 32 , w 33 , w 34 ) = (0.2522,0.2483,0.2506,0.2490)
Thirdly, we utilize the formula (13) to aggregate all the criteria of same prioritized level for each alternative. The obtained results are listed in Table 5. Fourthly, based on the data presented in table, we use the model (MOD-1) to construct a linear programming model as follows: max ξ
s.t. φ (a5 ) − ξ ≥ φ (a1 ), φ (a1 ) − ξ ≥ φ (a3 ), φ (a6 ) − ξ ≥ φ (a3 ), 3 αl = 1, α1 ≥ α2 ≥ α3 ≥ 0. l=1
By solving the above linear programming model, we can get the priority vector α = (α1 , α2 , α3 ) = (0.7227, 0.2773, 0). At last, we use the formulas (19) (23) to obtain the total net flow of each alternative: φ (a1 ) = 0.2692 · 0 + 0.1694 · 0.7227 −0.0841 · 0.2773 = 0.0991 φ (a2 ) = 0.2247 · 0 + 0.1266 · 0 −0.0471 · 1 = −0.0471
L. Chen and Z. Xu / A new prioritized multi-criteria outranking method: The prioritized PROMETHEE
2107
Table 4 The normed net flow of each alternative with respect to each criterion Normed net flow under each criterion
a1 a2 a3 a4 a5 a6 a1 a2 a3 a4 a5 a6
D2 0.8000 −0.4000 −0.1000 −0.4000 −0.1000 0.2000 S2 −0.1333 0.2667 −0.5333 −0.5333 0.2667 0.6667 E2 0.7047 0.4090 −0.1477 −0.2459 −0.2247 −0.4953
PY
D1 0.4500 −0.1500 −0.1500 −0.7500 0.4500 0.1500 S1 −0.3333 0.4667 0.4667 −0.7333 0.0667 0.0667 E1 0.4000 −0.4727 −0.1455 0.2909 0.5636 −0.6364
a1 a2 a3 a4 a5 a6
A1 A2 A3 A4 A5 A6
0.1694 −0.0471 0.1024 −0.5756 0.2518 0.0991
C3
−0.0841 0.2247 −0.0302 −0.2284 −0.1803 0.2984
0.2692 0.1266 −0.0177 0.1271 0.0233 −0.5285
AU
φ (a3 ) = 0.1024 · 0.7227 − 0.0177 · 0 −0.0302 · 0.2773 = 0.0656
φ (a4 ) = 0.1271 · 0 − 0.2284 · 0 − 0.5756 · 1 = −0.5756
D4 −0.5500 −0.2500 0.0500 −0.5500 0.6500 0.6500 S4 −0.6000 0.6000 −0.2000 0.2000 −0.6000 0.6000 E4 0.2500 0.8500 −0.3500 −0.0500 −0.3500 −0.3500
preference information of alternatives, which is a5 a1 , a1 a3 , a6 a3 .
C2
TH
C1
D3 0 0.6000 0.6000 −0.6000 0 −0.6000 S3 0.7500 −0.4500 0.1500 0.1500 −0.4500 −0.1500 E3 −0.2750 −0.2689 0.5700 0.5076 0.0963 −0.6300
OR
Table 5 The normed net flow of each alternative with respect to each prioritized pseudo criterion
CO
Alternative
φ (a5 ) = 0.2518 · 0.7227 + 0.0233 · 0 −0.1803 · 0.2773 = 0.1320 φ (a6 ) = 0.2984 · 0 + 0.0991 · 1 − 0.5285 · 0 = 0.0991 Therefore, the final ranking order of all the alternatives is a5 a1 ∼ a6 a3 a2 a4 , and the best alternative is a5 . Meanwhile, the ranking order of all the alternatives is compatible with the DM’s partial
4.4. Comparative analysis To illustrate the advantage of the P-PROMETHEE method, we make a comparative analysis with the similar methods. As mentioned previously, Yu et al. [34] proposed an approach based on idea of PROMETHEE to deal with the prioritized MCDM problems. However, Yu et al. [34] only investigated the strict ordered prioritized MCDM problems. Therefore, their method fails to deal with the ranking problem of weak ordered prioritized MCDM. Besides, Yan et al. [32] proposed a prioritized weighted aggregation operator based on the ordered weighted averaging (OWA) operator and the triangular norms (t-norms) to deal with the weak ordered prioritized MCDM problems, which are the same with our considered problems. The main procedures of Yan’s prioritized weighted aggregation operator [32] are summarized as follows: Step 1. Normalize the decision matrix, in which each element indicating degree of satisfaction of a given alternative regarding criterion. Step 2. Based on the DM’s attitudinal character q toward the prioritized level Hq , we obtain the weights of criteria in the prioritized level Hq according to O’Hagan [18]’s OWA weight determination method as follows:
L. Chen and Z. Xu / A new prioritized multi-criteria outranking method: The prioritized PROMETHEE nq
Table 6 The normalized satisfaction degree of each criterion corresponding to each alternative
w qi ln w qi
i=1
(MOD - 3) Subject to
nq nq − i i=1
nq
nq − 1
Alternatives
· w qi = , 0 ≤ ≤ 1
w qi = 1, w i ∈ [0, 1], i = 1, 2, · · · , nq
i=1
Step 3. Based on the obtained weights of criteria in the prioritized level Hq , we get the degree of satisfaction of the prioritized level Hq by using the OWA operator. n
q Satq (·) = OWAq Hq = bqk (·)w qk
(28)
k=1
where bqk (·) is the kth largest satisfaction degree in priority level Hq . Step 4. Induce the priority weight for the prioritized level Hq using the following equation: q−1
(29)
D2
D3
D4
a1 a2 a3 a4 a5 a6
0.2381 0.1429 0.1429 0.0476 0.2381 0.1905
0.5000 0.0000 0.1250 0.0000 0.1250 0.2500
0.1667 0.2500 0.2500 0.0833 0.1667 0.0833
0.0588 0.1176 0.1765 0.0588 0.2941 0.2941
S1
S2
S3
S4
a1 a2 a3 a4 a5 a6
0.1304 0.2174 0.2174 0.0870 0.1739 0.1739
0.1429 0.2143 0.0714 0.0714 0.2143 0.2857
0.3333 0.0667 0.2000 0.2000 0.0667 0.1333
0.0667 0.2667 0.1333 0.2000 0.0667 0.2667
a1 a2 a3 a4 a5 a6
OR
Zq = T (Zq−1 (·), Satq−1 (·)) = Tl=0 Sat(·)
Criteria D1
PY
Maximize −
E1
E2
E3
E4
0.2561 0.0610 0.1341 0.2317 0.2927 0.0244
0.2240 0.1999 0.1547 0.1467 0.1484 0.1264
0.1515 0.1519 0.1980 0.1946 0.1720 0.1320
0.2308 0.3846 0.0769 0.1538 0.0769 0.0769
CO
2108
where T denotes the t-norms and Z0 (·) = Sat0 (·) = 1.
Step 5. Get an aggregated value for each alternative under these prioritized criteria as: Q q=1
Zq (·)Satq (·)
(30)
TH
V (·) =
AU
Step 6. Based on the obtained aggregated values, we get the ranking of all the alternatives. In what follows, we utilize Yan’s method [32] to solve the decision making problem mentioned in Section 4.3. Firstly, we calculate the normalized decision matrix, which is shown in Table 6. For simplicity, we let the DM’s attitudinal character q = 0.5, q = 1, 2, 3. Therefore, we can easily obtain the weights of criteria in the same prioritized level by using O’Hagan’s OWA weight determination method (MOD-3). Then, we use the formula (27) to calculate the degree of satisfaction of each alternative in the prioritized level Hq . The obtained results are displayed in Table 7. Furthermore, the priority weight of the prioritized level Hq can be obtained by using the formula (28), in which the product t-norm is used and the results are also listed in Table 7. Finally, we get the aggregated values of all the alternatives by the formula (29). The results are shown as follows: V (a1 ) = 0.2902, V (a2 ) = 0.1569, V (a3 ) = 0.2044
Table 7 The satisfaction degree and prioritized weight with respect to each prioritized pseudo criterion
A1 A2 A3 A4 A5 A6
Sat1 (·)
Sat2 (·)
Sat3 (·)
Z1 (·)
Z2 (·)
Z3 (·)
0.2409 0.1276 0.1736 0.0474 0.2060 0.2045
0.1683 0.1913 0.1555 0.1396 0.1304 0.2149
0.2156 0.1994 0.1409 0.1817 0.1725 0.0899
1 1 1 1 1 1
0.2409 0.1276 0.1736 0.0474 0.2060 0.2045
0.0405 0.0244 0.0270 0.0066 0.0269 0.0439
V (a4 ) = 0.0552, V (a5 ) = 0.2375, V (a6 ) = 0.2524 Thus, the ranking order of all alternatives is a1 a6 a5 a3 a2 a4 , and the best alternative is a1 . It is not hard to see that compared with Yan’s prioritized weighted aggregation operator, the proposed P-PROMETHEE method can yield the distinct ranking results of the alternatives, i.e., a5 a1 ∼ a6 a3 a2 a4 , which is more reasonable. Because the result obtained by the P-PROMETHEE method is compatible with the DM’s preference information of alternatives, while the result of Yan’s prioritized weighted aggregation operator isn’t compatible with the DM’s preference information of alternatives, which is a5 a1 , a1 a3 , a6 a3 .
L. Chen and Z. Xu / A new prioritized multi-criteria outranking method: The prioritized PROMETHEE
5. Concluding remarks
Acknowledgments
PY
The authors are very grateful to the anonymous reviewers for their insightful and constructive comments and suggestions that have led to an improved version of this paper. The work was supported in part by the National Natural Science Foundation of China (No.61273209). References [1]
OR
(1) The P-PROMETHEE method can obtain the weights of criteria in the same prioritized level by the entropy-based method, which can capture the relation of criteria without additional information of weights. However, Yan’s method needs the DM’s attitudinal character q for obtaining the relative importance of each criterion. (2) The P-PROMETHEE method constructs a linear programming model to determine the prioritized weight of the prioritized level, which is compatible with the DM’s partial preference information of alternatives. However, Yan’s method obtains the prioritized weights based on the satisfaction degrees of the higher prioritized levels, which can’t use the partial preference information of alternatives. Therefore, the result may be not effective.
and reefs has been used to illustrate and verify the PPROMETHEE method. Surely, the P-PROMETHEE can also be used to deal with other prioritized muliticriteria ranking problem, such as document retrieval problems, risk assessment problems and selecting a car, etc. In the future, we will consider the general prioritized multi-criteria outranking method and extend the P-PROMETHEE method to uncertain situations.
CO
Consequently, it is easy to see that the PPROMETHEE method, compared with Yan’s method [32], has the following two desirable advantages:
[2]
AU
TH
In this paper, we have extended the PROMETHEE method to the case of the prioritized MCDA, which had never been considered before. The set of criteria is not considered in the same level, but it has a weakly ordered prioritization relation. Then, we have developed the prioritized PROMETHEE method based on the prioritized measure and entropy to capture the ranking order of all the alternatives. In most of the prioritized MCDM problems, it is difficult to give the weight of each criterion by the DM, especially when the number of criteria is large. For the criteria of same prioritized level, we have proposed an entropy-based method to derive the weight of each criterion. Meanwhile, we have considered two cases where the preference information on alternatives is completely unknown and partial known. For the case where the preference information on alternatives given by the DM is completely unknown, we have employed Yager’s prioritized measure-guided aggregation operator [30] to calculate the total net flow of each alternative. For the case where the preference information on alternatives given by DM is partially known, we have proposed a linear programming method to derive the priority vectors. Based on the priority vectors, we can get the ranking of alternative compatible with the DM’s preference information. At last, the strategic assessment of islands
2109
[3] [4]
[5]
[6]
[7]
[8]
[9]
[10]
[11]
A. Albadvi, S.K. Chaharsooghi and A. Esfahanipour, Decision making in stock trading: An application of PROMETHEE, European Journal of Operational Research 177 (2007), 673–683. O.U. Araz, A simulation based multi-criteria scheduling approach of dual-resource constrained manufacturing systems with neural networks, in: AI 2005: Advances in Artificial Intelligence, Springer, 2005, pp. 1047-1052. G. Beliakov, A. Pradera and T. Calvo, Aggregation functions: A guide for practitioners, Springer-Verlag, Berlin, 2007. J.P. Brans, L’ing´enieurie de la d´ecision-Elaboration d’instruments d’aide a` la d´ecision. La m´ethode Prom´eth´ee–Dans Nadeau R. et Landry M, L’aide a` la d´ecision: Nature, intruments et perspectives d’avenir. Presses de l’universit´e de Laval, Qu´ebec, Canada, 1982, pp. 182-213. J.P. Brans and B. Mareschal, The PROMETHEE VI procedure: How to differentiate hard from soft multicriteria problems, Journal of Decision Systems 4 (1995), 213–223. J.P. Brans and B. Mareschal, PROMETHEE methods, in: Multiple criteria decision analysis: State of the art surveys, Springer, 2005, pp. 163-186. J.P. Brans and P. Vincke, A preference ranking organization method. The PROMETHEE method for MCDM, Management Science 31 (1985), 647–656. L.H. Chen, Z.S. Xu and X.H. Yu, Prioritized measure-guided aggregation operators, IEEE Transactions on Fuzzy Systems 22 (2014), 1127–1138. F. Chiclana, F. Herrera and E. Herrera-Viedma, The ordered weighted geometric operator: Properties and application in MCDM problems, in: Technologies for Constructing Intelligent Systems 2, Physica-Verlag, Heidelberg, 2002, pp. 173-183. S. Corrente, J.R. Figueira and S. Greco, The SMAAPROMETHEE method, European Journal of Operational Research 239 (2014), 514–522. S. Eppe and Y. De Smet, Studying the impact of information structure in the PROMETHEE II preference elicitation process: A simulation based approach, in: Advances in Computational Intelligence, Springer, 2012, pp. 383-392.
[16]
[17]
[18]
[19]
[20]
[21] [22]
[25]
[26]
[27]
[28]
[29]
PY
[15]
[24]
Z.X. Sun and M. Han, Multi-criteria decision making based on PROMETHEE method, in: 2010 International Conference on Computing, Control and Industrial Engineering (CCIE), IEEE Computer Society Press, 2010, pp. 416–418. C.P. Wei and X.J. Tang, Generalized prioritized aggregation operators, International Journal of Intelligent Systems 27 (2012), 578–589. Z.S. Xu and Q.L. Da, The ordered weighted geometric averaging operators, International Journal of Intelligent Systems 17 (2002), 709–716. Z.S. Xu and H. Hu, Entropy-based procedures for intuitionistic fuzzy multiple attribute decision making, Journal of Systems Engineering and Electronics 20 (2009), 1001–1011. R.R. Yager, On ordered weighted averaging aggregation operators in multicriteria decisionmaking, IEEE Transactions on Systems, Man and Cybernetics 18 (1988), 183–190. R.R. Yager, Modeling prioritized multicriteria decision making, IEEE Transactions on Systems, Man, and Cybernetics, Part B: Cybernetics 34 (2004), 2396–2404. R.R. Yager, Prioritized aggregation operators, International Journal of Approximate Reasoning 48 (2008), 263–274. R.R. Yager, On prioritized multiple-criteria aggregation, IEEE Transactions on Systems, Man, and Cybernetics, Part B: Cybernetics 42 (2012), 1297–1305. R.R. Yager, C.L. Walker and E.A. Walker, A prioritized measure for multi-criteria aggregation and its Shapley index, in: Fuzzy Information Processing Society (NAFIPS), 2011 Annual Meeting of the North American, IEEE, 2011, pp. 1–4. H.B. Yan, V.N. Huynh, Y. Nakamori and T. Murai, On prioritized weighted aggregation in multi-criteria decision making, Expert Systems with Applications 38 (2011), 812–823. X.H. Yu and Z.S. Xu, Prioritized intuitionistic fuzzy aggregation operators, Information Fusion 14 (2013), 108–116. X.H. Yu, Z.S. Xu and Y. Ma, Prioritized multi-criteria decision making based on the idea of PROMETHEE, Procedia Computer Science 17 (2013), 449–456.
CO
[14]
[23]
[30]
[31]
[32]
OR
[13]
S. Eppe, Y. De Smet and T. St¨utzle, A bi-objective optimization model to eliciting decision maker’s preferences for the PROMETHEE II method, in: Algorithmic Decision Theory, Springer, 2011, pp. 56-66. M. Grabisch, Fuzzy integral in multicriteria decision making, Fuzzy Sets and Systems 69 (1995), 279–298. B. Mareschal, Weight stability intervals in multicriteria decision aid, European Journal of Operational Research 33 (1988), 54–64. B. Mareschal and J.P. Brans, Geometrical representations for MCDA, European Journal of Operational Research 34 (1988), 69–77. B. Mareschal and J.P. Brans, PROMETHEE V: MCDM problems with segmentation constraints, in, ULB–Universite Libre de Bruxelles, 1992. T. Murofushi and M. Sugeno, Fuzzy measures and fuzzy integrals, in: Fuzzy Measures and Integrals: Theory and Applications, M. Grabisch, T. Murofushi, and M. Sugeno, eds., Physica-Verlag, 2000, pp. 3-41. M. O’Hagan, Aggregating template or rule antecedents in realtime expert systems with fuzzy set logic, in: Signals, Systems and Computers, 1988. Twenty-Second Asilomar Conference on, IEEE, Pacific Grove, CA, 1988, pp. 681-689. D.H. Peng, T.D. Wang, C.Y. Gao and H. Wang, Multigranular uncertain linguistic prioritized aggregation operators and their application to multiple criteria group decision making, Journal of Applied Mathematics 2013 (2013). J.C. Petraˇs, Ranking the sites for low-and intermediate-level radioactive waste disposal facilities in croatia, International Transactions in Operational Research 4 (1997), 237–249. C.E. Shannon, A mathematical theory of communication, The Bell System Technical Journal 27 (1948), 379–423, 623–656. T. Solymosi and J. Dombi, A method for determining the weights of criteria: The centralized weights, European Journal of Operational Research 26 (1986), 35–41.
TH
[12]
L. Chen and Z. Xu / A new prioritized multi-criteria outranking method: The prioritized PROMETHEE
AU
2110
[33]
[34]
Journal of Intelligent & Fuzzy Systems 29 (2015) 2099–2110 DOI:10.3233/IFS-151686 IOS Press
A new prioritized multi-criteria outranking method: The prioritized PROMETHEE Liuhao Chena and Zeshui Xub,∗ of Communications Engineering, PLA University of Science and Technology, Nanjing, Jiangsu, China School, Sichuan University, Chengdu, Sichuan, China
PY
a College
CO
b Business
OR
Abstract. The Multiple Criteria Decision Aiding (MCDA) has been a fast growing area of operations research and management science during the last decades, while the PROMETHEE (Preference Ranking Organization Method for Enrichment Evaluations) is one of the well-known MCDA methods with ranking the alternatives from the best to the worst. In this paper, we propose a new outranking method called Prioritized PROMETHEE (P-PROMETHEE) method, which is used to deal with the ranking problem of Prioritized Multiple Criteria Decision Making (PMCDM). First, we use the idea of entropy to get the information of each criterion in the same prioritized level, which is used to get the weight of each criterion. Then, we use the weighted average operator to aggregate all the criteria in the same prioritized level. Moreover, we apply Yager’s prioritized measure-guided aggregation operator to the classical PROMETHEE method to get the P-PROMETHEE method. Finally, we give a practical example to illustrate the effectiveness of the P-PROMETHEE method.
1. Introduction
TH
Keywords: Entropy, prioritized measure, PROMETHEE method, weight vector, choquet integral
AU
As we know, the PROMETHEE method developed by Brans et al. [4, 6], is one of the effective outranking methods to solve the MCDM problems. As the PROMETHEE method has the properties of simplicity, clearness and stability, it has been widely used to solve practical multicriteria decision making problems, such as environment management, business and financial management, manufacturing and assembly, etc. [1, 2, 20]. The PROMETHEE method is based on the pairwise comparison of alternatives with respect to each criterion to get the ranking of all the alternatives. The PROMETHEE method includes: PROMETHEE I for partial ranking of the alternatives and PROMETHEE II for complete ranking of the alternatives [4]. Later, many useful extensions have been developed to enrich ∗ Corresponding
author. Zeshui Xu, Business School, Sichuan University, Chengdu, Sichuan, China. Tel.: +86 25 84483382; Fax: +86 25 84483382; E-mails: [email protected]; [email protected]
the PROMETHEE method. For instance, Brans and Mareschal developed PROMETHEE III for ranking based on intervals and PROMETHEE IV for solving a choice problem of an infinite set of alternatives. In 1988, Brans and Mareschal [15] proposed the visual interactive module GAIA (Geometrical Analysis for Interactive Aid) which provides a marvellous graphical representation supporting the PROMETHEE method. PROMETHEE V for MCDA including segmentation constraints and PROMETHEE VI for representation of the human brain was also developed by Brans and Marescha [5, 16] in 1992 and 1995 respectively. However, in order to implement the PROMETHEE method, we require three types of additional information: preference function shape, weights of criteria and thresholds. As Corrente et al. [10] pointed out, the preference information can be obtained through two ways: directly and indirectly. The Decision Maker (DM) can get all the preference information between alternatives when he provides directly all values of parameters in the model. The DM only provides some preference
1064-1246/15/$35.00 © 2015 – IOS Press and the authors. All rights reserved
L. Chen and Z. Xu / A new prioritized multi-criteria outranking method: The prioritized PROMETHEE
PY
of safety arrives some given level. For this situation, some methods have been developed to overcome this drawback [8, 19, 24, 28–30, 32, 33]. For example, Yager [28, 29] used the prioritized weights in which the weights associated with the lower prioritized criteria are related to the satisfaction of the higher prioritized criteria to express the prioritized relation between criteria. Based on Yager’s prioritzed weights, Wei and Tang [24] proposed a generalized prioritized averaging operator and a generalized prioritized OWA operator. Yan [32] proposed a prioritized weighted aggregation operator based on the ordered weighted averaging (OWA) operator and the triangular norms (t-norms) to deal with multi-criteria decision making (MCDM) problems with multiple priorities. Yager [30] developed a prioritized aggregation operator which uses the monotonic measure set to convey the prioritization relationship. Based on Yager [30]’s idea, Chen [8] developed a general prioritized aggregation operator which allows the partial compensation between prioritized criteria. However, the mentioned literature focuses on dealing with the prioritized MCDM problems, few papers pay attention to the outranking method with the prioritized relationship of criteria. As we know, Yu et al. [34] developed a new function which takes into account the influence of expectation levels to the prioritized weights in the PROMETHEE method. However, Yu et al. [34] only investigated the strict ordered prioritized PROMETHEE method. In this paper, we will mainly consider some much more general problems with the weak ordered prioritized criteria. As the above mentioned, the PROMETHEE method is a well-known outranking method. In order to meet the needs of weak ordered prioritized relationship, we will propose a new outranking method called Prioritized PROMETHEE method, which is an integrated approach between the prioritized measure and the PROMETHEE method. The rest of the paper is organized as follows: Section 2 describes the classical PROMETHEE II. In Section 3, we recall Yager’s prioritized measure-guided aggregation operator. In Section 4, we propose a P-PROMETHEE method and employ a practical example to illustrate its effectiveness. Section 5 ends the paper with some conclusions.
AU
TH
OR
information between alternatives from which we get indirectly the compatible preference parameters. Obviously, we can find out that the indirect preference information needs less information from the DM than the direct preference information. Meanwhile, the indirect preference information is also relatively easy given by the DM. For this reason, the study of the PROMETHEE method based on indirect preference information is meaningful. Up to now, many methods have been proposed to elicit the preference parameters. For example, Solymosi and Dombi [22] proposed the centralized weight of compatible preference parameters to represent the relative importance of criteria. Mareschal [14] gave a sensitivity analysis in order to obtain stability interval weights for criteria, in which we can get the same ranking of alternatives. Sun and Han [23] solved a linear programming problem to find the most discriminant set of weights compatible with the preference information provided by the DM. Eppe et al. [12] proposed a bi-objective optimization model which considers the number of inconsistencies and the robustness of parameter values to elicit the PROMETHEE II’s preference parameters. The above mentioned papers mainly consider the MCDM problem with the criteria in the same level. Meanwhile, there are many methods to obtain the total satisfaction degree for each alternative to all criteria, such as the weighted average operator [27], the ordered weighted averaging (OWA) operator [27] and the ordered weighted geometric (OWG) operator [9, 25]. These aggregation operators allow to tradeoff between criteria. For example, if we use the weighted average operator, thetotal satisfaction degree can be obtained by C(x) = ni=1 w i Ci (x), where Ci (x) represents the satisfaction degree of the alternative x to the criterion Ci and w i denotes the weight of the criterion Ci . In this case, we can compensate for a decrease of in satisfaction to the criterion Ci by a gain (w k /w i ) in satisfaction to the criterion Ck . However, there are lots of MCDM problems with the prioritized relation of criteria and we usually permit no compensation between criteria. Yager [28, 29] listed several practical MCDM problems to illustrate this kind of situations, such as selecting a bicycle, an organization decision-making problem, and a document retrieval problem. The typical example is the case of buying a car with respect to the criteria of safety and cost. In this case, the criterion safety has a higher priority than the criterion cost, it indicates that we are not willing to tradeoff the satisfaction of the criterion cost until the satisfaction
CO
2100
2. The classical PROMETHEE methods In what follows, we first introduce the basic principles and some basic concepts related to the PROMETHEE method (see Refs. [4, 6] for more
L. Chen and Z. Xu / A new prioritized multi-criteria outranking method: The prioritized PROMETHEE
(1)
π(ai , aj ) = π(aj , ai ) =
n l=1 n
w l · Pl (ai , aj )
(2)
w l · Pl (aj , ai )
l=1
AU
TH
where π(ai , aj ) represents the total degree of preference of the alternative ai to the alternative aj taking into account all criteria. The weight w l represents the relative importance of the criterion Cl . In order to get the ranking of all the alternatives, the PROMETHEE I method defines the positive outranking flow and negative outranking flow as follows: 1 φ+ (ai ) = π(ai , x) m−1 x∈X\{ai }
φ− (ai ) =
1 m−1
In order to capture the prioritized relationship of criteria, Yager [30] adopted the fuzzy measure and the fuzzy integral to develop a prioritized measure-guided aggregation operator, which has the characteristic that the higher prioritized criteria’s satisfaction can’t be compensated by the lower prioritized criteria’s satisfaction. Now we first give the definition of fuzzy measure: Definition 1. [3, 17] Let N = {1, 2, · · · , n}. A discrete fuzzy measure is a set function µ : 2N → [0, 1] which is monotonic (i.e., µ(A) ≤ µ(B) whenever A ⊂ B ⊆ N) and satisfies µ(Ø) = 0 and µ(N) = 1. Fuzzy measure was first introduced by Sugeno [17], and has usually been used to deal with the MCDM problems associated with fuzzy integral. Meanwhile, fuzzy measures are not only used to represent the independent relationships of criteria, but also the dependent relationships of criteria. Furthermore, Yager and Walker [31] first used a special kind of fuzzy measure to capture the strictly ordered prioritizations. They gave the following definitions: ⎧ ⎨ {Ck | k = 1, 2, . . . , j}, j = 1 , . . . , n Lj = (3) ⎩ Ø, j=0
OR
where dl (ai , aj ) = f (ai ) − f (aj ) and Fl (·) is a monotonically non-decreasing function varying from 0 to 1. In order to facilitate the selection of the preference function, Brans and Vincke [7] proposed six basic types of preference functions: (1) usual criterion, (2) U-shape criterion, (3) V-shape criterion, (4) level criterion, (5) V-shape with indifference criterion, (6) Gaussian criterion. Certainly, the DM has to give the value of an indifference threshold q, and the value of a strict preference threshold p for each criterion. For any two alternatives ai , aj ∈ X, we have
3. Prioritized measure-guided aggregation operator
PY
Pl (ai , aj ) = Fl (dl (ai , aj )), for ai , aj ∈ X
for all alternatives. The PROMETHEE II can get a complete ranking according to the value of net outranking flow for each alternative, which is obtained by φ(ai ) = φ+ (ai ) − φ− (ai ).
CO
details). For simplicity, we suppose that a multi-criteria decision making problem consists of a set of alternatives X = {a1 , a2 , . . . , am } and a set of criteria C = {C1 , C2 , . . . , Cn }. Let fl (ai ) represent the evaluation of the alternative ai on the criterion Cl . For the given criterion Cl , the DM can always give a function Pl (ai , aj ) which represents the degree of preference of the alternative ai to the alternative aj under the criterion Cl . We can write the function as follows:
2101
π(x, ai )
x∈X\{ai }
where the positive outranking flow φ+ (ai ) represents how much the alternative ai prefers to all the other alternatives. The larger φ+ (ai ), the better the alternative ai . Similarly, the negative outranking flow φ− (ai ) represents how much all the other alternatives prefers to the alternative ai . The smaller φ− (ai ), the better the alternative ai . The PROMTHEE I can get a partial ranking according to the value of the positive outranking flow and the value of negative outranking flow
where we assume C1 C2 · · · Cn in the set of criteria. They are associated with each subset Lj the value j/n. Then the prioritized measure m : 2C → [0, 1] can be defined as: j Gj (A) (4) m(A) = Maxj n where A is a subset of C and ⎧ ⎨ 1, Lj ⊆ A Gj (A) = ⎩ 0, Lj ⊂ / A
(5)
Besides, the prioritized measure can also be written as: µ(A) =
n 1 · Gj (A) n j=1
(6)
L. Chen and Z. Xu / A new prioritized multi-criteria outranking method: The prioritized PROMETHEE
Definition 2. [3, 13] The discrete Choquet integral with respect to a fuzzy measure µ is given by Choqµ (a) =
n
a(i) µ({j| aj ≥ a(i) })
i=1
−µ({j|aj ≥ a(i−1) })
(7)
C(x) = Choqµ (C1 (x), C2 (x), · · · , Cn (x)) =
n
(µ(Pj ) − µ(Pj−1 ))Cind(j) (x)
j=1
(8)
AU
TH
where Pj = {Cind(1) , Cind(2) , · · · , Cind(j) }, and ind(j) is the index of the jth largest Cj (x). The measure µ(·) is given by the formula (5). The concept of entropy was first introduced by Shannon [21] to measure the uncertainty of a random variable. The entropy is very important for measuring the uncertain information in information theory. Recently, many scholars take the idea of entropy into the area of decision making, for example, O’Hagan [18] suggested a maximum entropy method to determine the weight vector of the OWA operator, Xu and Hu [26] proposed an entropy-based procedure to derive attribute weights under intuitionistic fuzzy environment. The definition of entropy is given as follows: Definition 3. [3, 21] For a given weighting vector w = (w 1 , w 2 , · · · w n ), where w i ∈ [0, 1], and ni=1 w i = 1, then the entropy of this weighting vector is defined by E(w) = −
n
w i ln w i
i=1
with the convention 0 · ln 0 = 0.
As Section 2 pointed out, the classical PROMETHEE method is a simple and effective outranking method. From the formula (1), we can find that the classical PROMETHEE method allow the tradeoff between criteria in order to get the total preference relationship of alternatives. However, this paper mainly studies a special ranking problem with the prioritized relationship between criteria. Since the higher prioritized criteria’s satisfaction can’t be compensated by the lower prioritized criteria in the prioritized MCDM problems, the classical PROMETHEE method is not suitable to deal with the prioritized MCDM problems. Yager [30, 31] first introduced the prioritized measure and the prioritized measure-guided aggregation operator to deal with the prioritized MCDM problems. Fortunately, Yager’s prioritized measure-guided aggregation operator can capture the prioritized relationship between criteria, which means the lack of satisfaction to the higher prioritized criteria cannot be compensated by the increased satisfaction of those lower prioritized criteria. In order to deal with the prioritized outranking problems, we apply Yager’s prioritized measure-guided aggregation operator into the classical PROMETHEE method, and thus develop a prioritized-PROMETHEE method.
OR
where a = (a(1) , a(2) , . . . , a(n) ) is a non-increasing permutation of the input a, and a(0) = ∞ by convention. As we know, Choquet integral is a typical type of fuzzy integrals, and it can be used to aggregate information from a set of resources with respect to a given fuzzy measure. In order to get the prioritized relationship of criteria, Yager and Walker [31] used Choquet integral to obtain an aggregation based on the prioritized measure. Thus, we can use the prioritized aggregation operator to calculate the overall satisfaction degree C(x) for each alternative x by
4. The prioritized PROMETHEE method
PY
The discrete Choquet integral is an adequate aggregation function that extends the OWA operator by taking the interactions of criteria into account.
CO
2102
(9)
4.1. Problem formulation and the proposed approach Before the introduction of the prioritized outranking problems, we first give the definition of the prioritized MCDM problems as follows: Definition 4. [28, 29] In a MCDM problem, if the set of criteria, C = {C1 , C2 , · · · , Cn }, can be partitioned into q distinct prioritized hierarchies, H1 , H2 , · · · , Hq , such that Hk Hl if k < l, q where Hk = k1 , Ck2 , · · · , Cknk } ⊂ C, C = ∪k=1 Hk {C q (i.e., n = k=1 nk ), and Hk ∩ Hl = Ø for k, l ∈ {1, 2, · · · , n} (“Ø” denotes the null set), then the problem is called a prioritized MCDM problem. As Yager [28, 29] pointed out, the prioritized MCDM problems can be classified into two cases: (1) strictly ordered prioritizations, if each priority level has only one criterion, i.e., nk = 1 for k = 1, 2, · · · , q; and (2) weakly ordered prioritizations, otherwise. For the above prioritized MCDM problems, we mainly investigate the prioritized outranking problems, which aim to obtain the ranking of the alternatives based on the pairwise preference relationships of the
L. Chen and Z. Xu / A new prioritized multi-criteria outranking method: The prioritized PROMETHEE
aj ∈X\{ai }
(ai ) = φkj
φkj (ai ) + 1 , j ∈ {1, 2, · · · , nk } , m i ∈ {1, 2, · · · , m} (15)
Therefore, we have
TH
AU
ai ∈X
For the prioritized level Hk = {Ck1 , Ck2 , · · · , Cknk } ⊂ C, we have ⎞ ⎛ φk1 (a1 ) φk2 (a1 ) · · · φknk (a1 ) ⎟ ⎜ ⎜ φk1 (a2 ) φk2 (a2 ) · · · φknk (a2 ) ⎟ ⎟ ⎜ (13) φk = ⎜ ⎟ .. .. .. .. ⎟ ⎜ . . . . ⎠ ⎝ φk1 (am ) φk2 (am ) · · · φknk (am ) In order to get the total flow in the prioritized level Hk for each alternative ai , we have to aggregate the normed flow φkj (ai ), j ∈ {1, 2, · · · , nk } into a value φk (ai ). Usually, we can use the weight average operator to get the value φk (ai ) defined in the following way: φk (ai ) =
nk j=1
w kj φkj (ai )
(14)
0 ≤ φkj (ai ) < 1,
OR
(11) where the preference value Pl (ai , aj ) can be got by the formula (1). For each criterion Cl , we should select a suitable preference function Pl (·, ·) to depict the preference value between two alternatives. In order to obtain the suitable preference function easily, Brans and Vincke [7] proposed six basic types of preference functions. Therefore, the preference function Pl (·, ·) is the key point to capture the criteria’s characteristics. So the normed flow φl (ai ) represents how much the alternative ai prefers to all the other alternatives under the criterion Cl . The larger φl (ai ), the better the alternative ai under the criterion Cl . Moreover, we have −1 ≤ φl (ai ) ≤ 1, φl (ai ) = 0 (12)
PY
where the preference value Pl (ai , aj ) can be got by the formula (1). In order to visually compare the alternatives for each criterion Cl , Mareschal and Brans [15] introduced the normed flow of each alternative ai for each criterion Cl : 1 φl (ai ) = Pl (ai , aj ) − Pl (aj , ai ) m−1
For the alternative ai , the value φk (ai ) can be seen as the net flow of the pseudo criterion Ck , which is the aggregation of all the kth prioritized criteria. However, the DM can’t give the weight of each criterion in lots of actual problems especially when there are lots of criteria in the ranking problem. The DM can only estimate the criteria whether in the same prioritized level. Below we propose an entropy-based approach to elicit the weights of criteria in the same prioritized level. According to the formula, we can know that the normed flow φkj (ai ) isn’t normal for all alternatives. So we introduce a transformation function to normalize the value φkj (ai ) as follows:
CO
alternatives. For the prioritized outranking problem, we can obtain the preference matrix of alternatives for each criterion Cl : ⎞ ⎛ Pl (a1 , a1 ) Pl (a1 , a2 ) · · · Pl (a1 , am ) ⎟ ⎜ ⎜ Pl (a2 , a1 ) Pl (a2 , a2 ) · · · Pl (a2 , am ) ⎟ ⎟ ⎜ (10) ⎟ ⎜ .. .. .. .. ⎟ ⎜ . . . . ⎠ ⎝ Pl (am , a1 ) Pl (am , a2 ) · · · Pl (am , am )
2103
ai ∈X
φkj (ai ) = 1
(16)
We apply the information entropy formula to = (φ calculate the entropy for the vector φkj kj (a ), · · · , φ (a )): (a1 ), φkj 2 kj m E(φkj )
=−
m
φkj (ai ) ln φkj (ai )
(17)
i=1 ), the less Obviously, the greater the entropy E(φkj deviation between all the alternatives under the criterion ) arrives the biggest value, we Ckj . If the entropy E(φkj (a ) = φ (a ) = · · · = φ (a ) which means have φkj 1 kj 2 kj m that all the alternatives have the same value under the criterion Ckj . It is reasonable to give the criterion Ckj small weight for obtaining a complete rank. Using the formula , the weight of each criterion in the same prioritized level can be obtained by
w kj
1 ) E(φkj = nk 1 ) E(φkj
(18)
j=1
Based on the formulas (13) and (17) , we can have the net flow of each alternative ai in the prioritized level Hk . Meanwhile, all the results φk (ai ) (k = 1, 2, · · · , q; i = 1, 2, · · · , m) are listed in the following matrix:
L. Chen and Z. Xu / A new prioritized multi-criteria outranking method: The prioritized PROMETHEE
⎛
φ1 (a1 ) φ2 (a1 ) · · · φq (a1 )
⎞
⎜ 1 ⎟ ⎜ φ (a2 ) φ2 (a2 ) · · · φq (a2 ) ⎟ ⎜ ⎟ φ=⎜ . .. .. ⎟ .. ⎜ .. ⎟ . . . ⎝ ⎠ 1 2 q φ (am ) φ (am ) · · · φ (am )
(19)
Certainly, we have to aggregate the k-net flows φk (ai ), k = 1, 2, · · · , q into a total net flow for each alternative ai . Due to the net flow in the different prioritized level, Yager’s prioritized measure guided aggregation operator is used to aggregate the total net flow. Utilizing the formula, we have the total net flow φ(ai ) as follows:
the information contained in the ASR information structure can be we use the PAC. For simplicity, converted into a to express all set P = (il , jl ) | l ∈ 1, 2, · · · , KPAC the preference information. The generalized prioritized measure-guided aggregation operator uses the following measure: q µ (A) = αj · Gj (A) (24) j=1
where the quantity αi satisfies αi ∈ [0, 1] and n i=1 αi = 1. Combining the formulas (19) and (23) , we have
PY
2104
φ (ai ) = Choqµ (φ1 (ai ), φ2 (ai ), · · · , φq (ai ))
φ(ai ) = Choqµ (φ1 (ai ), φ2 (ai ), · · · , φq (ai )) (µ(Pj ) − µ(Pj−1 ))φind(j) (ai )
(20)
j=1 {Cind(1) ,
Cind(2) ,···
, Cind(j) }, j largest φ (ai ).
q 1 j=1
Gj (A) =
· Gj (A)
(21)
⎧ ⎨ 1, Lj ⊆ A
(22)
⎩ 0, Lj ⊂ / A
⎧ ⎨ {Ck | k = 1, 2, . . . , q}, j = 1 , . . . , q ⎩ Ø,
j=0
(23)
AU
Lj =
q
TH
µ(A) =
(µ (Pj ) − µ (Pj−1 ))φind(j) (ai )
Therefore, we can get a complete ranking of all the alternatives according to the values of the total net flows φ(ai ), i = 1, 2, · · · , m. Of course, if the DM provides partial information of his preference or experience, we employ Yager [30]’s generalized prioritized measure-guided aggregation operator to construct a model to get the total net flow. As Eppe and Smet [11] pointed out, there are two types of partial preference information: Pairwise Action Comparisons (PAC) and Action Sub-rankings (ASR). The PAC gives the pairwise preference relation of alternatives, denoted by ail ajl , which by the can be expressed a . The · · · , K constraint cl = (il , jl ), with l ∈ 1, 2, PAC
ASR considers a sub-ranking c = i1 , i2 , · · · , iaKASR that expresses that ai1 ai2 · · · aiKa . Obviously, ASR
(25)
j=1
In order to elicit the parameter αi , we establish the following model: max ξ s.t. φ (ai ) − ξ ≥ φ (aj ), (i, j) ∈ P, q (MOD - 1) αl = 1,
OR
where Pj = and ind(j) is the index of the j th The prioritized measure can be obtained by the formulas as follows:
q
CO
=
q
=
(26)
l=1
α1 ≥ α2 ≥ · · · ≥ αq ≥ 0.
Through solving the linear programming model (25), we obtain the optimal value ξ ∗ and the parameter value α∗ = (α∗1 , α∗2 , · · · , α∗q ). If ξ ∗ > 0, then we can get the parameter α∗ that is compatible with the DM’s preference information. If ξ ∗ < 0, then we can’t get the parameter α∗ that is compatible with the DM’s preference information. In this case, we propose the following model to elicit the parameter: ξij max (i,j)∈P
s.t. (MOD - 2)
φ (a
≥ φ (aj ), (i, j) ∈ P, q αl = 1, i ) − ξij
(27)
l=1
α1 ≥ α2 ≥ · · · ≥ αq ≥ 0. With the weight parameter vector α∗ = (α∗1 , , α∗q ) obtained by the above linear programming, we can compute the total net flow of the alternatives and rank the alternative from best to the worst one. α∗2 , · · ·
L. Chen and Z. Xu / A new prioritized multi-criteria outranking method: The prioritized PROMETHEE
2105
4.2. The proposed algorithm Based on the above analysis, we develop a practical algorithm for the prioritized multi-criteria outranking problem in which the weights of criteria are completely unknown and the DM’s preference information is partially known or completely unknown, which involves the following steps:
Step 3. Utilize the formula (10) to get the normed flow of the alternative ai for the criterion Cl . For each prioritized level, we utilize the formulas (17), (16) and (14) to obtain the weight vector w k = (w k1 , w k2 , · · · , w knk ).
CO
Step 2. Apply the preference function (1) to calculate the preference of the alternative ai with regard to the alternative aj with respect to each criterion. The preference function Fl (·) and the corresponding thresholds p and q can be given by the DM.
PY
Step 1. For a prioritized multi-criteria outranking problem, we use the formula dl (ai , aj ) = f (ai ) − f (aj ) to determine the difference between the evaluations of ai and aj with respect to each criterion.
Fig. 1. The prioritized relationship between criteria for the strategic assessment problem of islands and reefs.
Step 5. If the DM’s preference information of alternatives is completely unknown, then we use the formulas (19)-(22) to get the total net flow for each alternative. Based on the total net flow, we determine the rank of all alternatives; if the DM’s preference information of alternatives is partially known, then we use the model (MOD-1 or MOD-2) to obtain the vector of the weight parameter values α∗ = (α∗1 , α∗2 , · · · , α∗q ), turn to next step.
ing three aspects: Sensitivity, Exposure and Defense. Meanwhile, the sensitivity contains criteria: history conflict (S1 ), foreign relation (S2 ), ethnic and religious relation (S3 ) and strategic value (S4 ). The exposure contains criteria: population (E1 ), the value of island’s facility (E2 ), the value of island’s resource (E3 ) and strategic position (E4 ). The defense contains criteria: weapon (D1 ), the number of military bases and troops (D2 ), detection capability (D3 ), military support capability (D4 ). The prioritized relationship of criteria is listed in Fig. 1. While the population (E1 ), the value of island’s facility (E2 ), the value of island’s resource (E3 ) and the number of military bases and troops (D2 ) are all qualitative criteria. The assessments for history conflict (S1 ), foreign relation (S2 ), ethnic and religious relation (S3 ), strategic value (S4 ), strategic position (E4 ), weapon (D1 ), detection capability (D3 ) and military support capability (D4 ) are represented by linguistic terms. The linguistic terms used here and their corresponding INs (integer numbers) are shown in Table 1. The weights of all criteria are completely unknown. For each criterion, we consider the linear preference function and define the thresholds as follows: the indifference threshold q is fixed at 0 and the preference
AU
TH
OR
Step 4. Utilize the formula (13) to determine φk (ai ) which represents the net flow of the alternative ai under the pseudo criterion Ck .
Step 6. Utilize the formulas (19)-(23) to obtain the total net flow for each alternative and rank them from best to worst. 4.3. Illustrative example In order to illustrate our approach, let us consider the strategic assessment of islands and reefs. As we know, China is a coastal country and has many islands and reefs. However, due to the limited comprehensive national strength, most of the islands and reefs are undeveloped. Therefore, it is necessary to rank all the islands and reefs according to the strategic status of islands and reefs. In this paper, the strategic status of islands and reefs are evaluated by the follow-
L. Chen and Z. Xu / A new prioritized multi-criteria outranking method: The prioritized PROMETHEE Table 1 The linguistic scales and their corresponding INs
Linguistic variables and semantics
Integer numbers
s1 : Very Low/none s2 : Low/common s3 : Medium/big s4 : High/serious s5 : Very High/fierce
Alternatives
1 2 3 4 5
The second prioritized level
Parameter p
Parameter q
D1 D2 D3 D4 S1 S2 S3 S4 E1 E1 E1 E1
4 4 2 4 3 3 4 3 22 1246 8781 4
0 0 0 0 0 0 0 0 0 0 0 0
D1
D2
D3
D4
5 3 3 1 5 4
4 0 1 0 1 2
2 3 3 1 2 1
1 2 3 1 5 5
S1
S2
S3
S4
3 5 5 2 4 4
2 3 1 1 3 4
5 1 3 3 1 2
1 4 2 3 1 4
a1 a2 a3 a4 a5 a6
E1
E2
E3
E4
21 5 11 19 24 2
2859 2552 1974 1872 1894 1613
20170 20215 26353 25897 22887 17572
3 5 1 2 1 1
OR
The third prioritized level
Criteria
a1 a2 a3 a4 a5 a6
CO
The first prioritized level
Criteria
a1 a2 a3 a4 a5 a6
Table 2 The indifference and preference thresholds of each preference function Prioritized levels
Table 3 Satisfaction degree of each criterion corresponding to each alternative
PY
2106
AU
TH
threshold p is equal to the highest difference between the islands’ performances on the particular criterion (see Table 2). In this example, there are six islands and reefs represented by a1 , a2 , a3 , a4 , a5 , a6 which are needed to be ranked. The decision data used in the strategic assessment of islands and reefs are shown in Table 3. Fortunately, the DM can give partial information of the alternatives: a5 a1 , a1 a3 , a6 a3 (“” denotes “be superior to”). In the following, we use the P-PROMETHEE method to rank the six islands. The solution process and the computation results are summarized as follows: Firstly, we use the formula (10) to calculate the normed flow for each alternative on each criterion. The results are shown in Table 4. Secondly, For each prioritized level, we utilize the formulas (17), (16) and (14) to obtain the weight vector wk = (w k1 , w k2 , · · · , w knk ) as follows: w 1 = (w 11 , w 12 , w 13 , w 14 ) = (0.2489,0.2455,0.2528,0.2528) w = (w 21 , w 22 , w 23 , w 24 ) 2
= (0.2499,0.2493,0.2466,0.2542) w 3 = (w 31 , w 32 , w 33 , w 34 ) = (0.2522,0.2483,0.2506,0.2490)
Thirdly, we utilize the formula (13) to aggregate all the criteria of same prioritized level for each alternative. The obtained results are listed in Table 5. Fourthly, based on the data presented in table, we use the model (MOD-1) to construct a linear programming model as follows: max ξ
s.t. φ (a5 ) − ξ ≥ φ (a1 ), φ (a1 ) − ξ ≥ φ (a3 ), φ (a6 ) − ξ ≥ φ (a3 ), 3 αl = 1, α1 ≥ α2 ≥ α3 ≥ 0. l=1
By solving the above linear programming model, we can get the priority vector α = (α1 , α2 , α3 ) = (0.7227, 0.2773, 0). At last, we use the formulas (19) (23) to obtain the total net flow of each alternative: φ (a1 ) = 0.2692 · 0 + 0.1694 · 0.7227 −0.0841 · 0.2773 = 0.0991 φ (a2 ) = 0.2247 · 0 + 0.1266 · 0 −0.0471 · 1 = −0.0471
L. Chen and Z. Xu / A new prioritized multi-criteria outranking method: The prioritized PROMETHEE
2107
Table 4 The normed net flow of each alternative with respect to each criterion Normed net flow under each criterion
a1 a2 a3 a4 a5 a6 a1 a2 a3 a4 a5 a6
D2 0.8000 −0.4000 −0.1000 −0.4000 −0.1000 0.2000 S2 −0.1333 0.2667 −0.5333 −0.5333 0.2667 0.6667 E2 0.7047 0.4090 −0.1477 −0.2459 −0.2247 −0.4953
PY
D1 0.4500 −0.1500 −0.1500 −0.7500 0.4500 0.1500 S1 −0.3333 0.4667 0.4667 −0.7333 0.0667 0.0667 E1 0.4000 −0.4727 −0.1455 0.2909 0.5636 −0.6364
a1 a2 a3 a4 a5 a6
A1 A2 A3 A4 A5 A6
0.1694 −0.0471 0.1024 −0.5756 0.2518 0.0991
C3
−0.0841 0.2247 −0.0302 −0.2284 −0.1803 0.2984
0.2692 0.1266 −0.0177 0.1271 0.0233 −0.5285
AU
φ (a3 ) = 0.1024 · 0.7227 − 0.0177 · 0 −0.0302 · 0.2773 = 0.0656
φ (a4 ) = 0.1271 · 0 − 0.2284 · 0 − 0.5756 · 1 = −0.5756
D4 −0.5500 −0.2500 0.0500 −0.5500 0.6500 0.6500 S4 −0.6000 0.6000 −0.2000 0.2000 −0.6000 0.6000 E4 0.2500 0.8500 −0.3500 −0.0500 −0.3500 −0.3500
preference information of alternatives, which is a5 a1 , a1 a3 , a6 a3 .
C2
TH
C1
D3 0 0.6000 0.6000 −0.6000 0 −0.6000 S3 0.7500 −0.4500 0.1500 0.1500 −0.4500 −0.1500 E3 −0.2750 −0.2689 0.5700 0.5076 0.0963 −0.6300
OR
Table 5 The normed net flow of each alternative with respect to each prioritized pseudo criterion
CO
Alternative
φ (a5 ) = 0.2518 · 0.7227 + 0.0233 · 0 −0.1803 · 0.2773 = 0.1320 φ (a6 ) = 0.2984 · 0 + 0.0991 · 1 − 0.5285 · 0 = 0.0991 Therefore, the final ranking order of all the alternatives is a5 a1 ∼ a6 a3 a2 a4 , and the best alternative is a5 . Meanwhile, the ranking order of all the alternatives is compatible with the DM’s partial
4.4. Comparative analysis To illustrate the advantage of the P-PROMETHEE method, we make a comparative analysis with the similar methods. As mentioned previously, Yu et al. [34] proposed an approach based on idea of PROMETHEE to deal with the prioritized MCDM problems. However, Yu et al. [34] only investigated the strict ordered prioritized MCDM problems. Therefore, their method fails to deal with the ranking problem of weak ordered prioritized MCDM. Besides, Yan et al. [32] proposed a prioritized weighted aggregation operator based on the ordered weighted averaging (OWA) operator and the triangular norms (t-norms) to deal with the weak ordered prioritized MCDM problems, which are the same with our considered problems. The main procedures of Yan’s prioritized weighted aggregation operator [32] are summarized as follows: Step 1. Normalize the decision matrix, in which each element indicating degree of satisfaction of a given alternative regarding criterion. Step 2. Based on the DM’s attitudinal character q toward the prioritized level Hq , we obtain the weights of criteria in the prioritized level Hq according to O’Hagan [18]’s OWA weight determination method as follows:
L. Chen and Z. Xu / A new prioritized multi-criteria outranking method: The prioritized PROMETHEE nq
Table 6 The normalized satisfaction degree of each criterion corresponding to each alternative
w qi ln w qi
i=1
(MOD - 3) Subject to
nq nq − i i=1
nq
nq − 1
Alternatives
· w qi = , 0 ≤ ≤ 1
w qi = 1, w i ∈ [0, 1], i = 1, 2, · · · , nq
i=1
Step 3. Based on the obtained weights of criteria in the prioritized level Hq , we get the degree of satisfaction of the prioritized level Hq by using the OWA operator. n
q Satq (·) = OWAq Hq = bqk (·)w qk
(28)
k=1
where bqk (·) is the kth largest satisfaction degree in priority level Hq . Step 4. Induce the priority weight for the prioritized level Hq using the following equation: q−1
(29)
D2
D3
D4
a1 a2 a3 a4 a5 a6
0.2381 0.1429 0.1429 0.0476 0.2381 0.1905
0.5000 0.0000 0.1250 0.0000 0.1250 0.2500
0.1667 0.2500 0.2500 0.0833 0.1667 0.0833
0.0588 0.1176 0.1765 0.0588 0.2941 0.2941
S1
S2
S3
S4
a1 a2 a3 a4 a5 a6
0.1304 0.2174 0.2174 0.0870 0.1739 0.1739
0.1429 0.2143 0.0714 0.0714 0.2143 0.2857
0.3333 0.0667 0.2000 0.2000 0.0667 0.1333
0.0667 0.2667 0.1333 0.2000 0.0667 0.2667
a1 a2 a3 a4 a5 a6
OR
Zq = T (Zq−1 (·), Satq−1 (·)) = Tl=0 Sat(·)
Criteria D1
PY
Maximize −
E1
E2
E3
E4
0.2561 0.0610 0.1341 0.2317 0.2927 0.0244
0.2240 0.1999 0.1547 0.1467 0.1484 0.1264
0.1515 0.1519 0.1980 0.1946 0.1720 0.1320
0.2308 0.3846 0.0769 0.1538 0.0769 0.0769
CO
2108
where T denotes the t-norms and Z0 (·) = Sat0 (·) = 1.
Step 5. Get an aggregated value for each alternative under these prioritized criteria as: Q q=1
Zq (·)Satq (·)
(30)
TH
V (·) =
AU
Step 6. Based on the obtained aggregated values, we get the ranking of all the alternatives. In what follows, we utilize Yan’s method [32] to solve the decision making problem mentioned in Section 4.3. Firstly, we calculate the normalized decision matrix, which is shown in Table 6. For simplicity, we let the DM’s attitudinal character q = 0.5, q = 1, 2, 3. Therefore, we can easily obtain the weights of criteria in the same prioritized level by using O’Hagan’s OWA weight determination method (MOD-3). Then, we use the formula (27) to calculate the degree of satisfaction of each alternative in the prioritized level Hq . The obtained results are displayed in Table 7. Furthermore, the priority weight of the prioritized level Hq can be obtained by using the formula (28), in which the product t-norm is used and the results are also listed in Table 7. Finally, we get the aggregated values of all the alternatives by the formula (29). The results are shown as follows: V (a1 ) = 0.2902, V (a2 ) = 0.1569, V (a3 ) = 0.2044
Table 7 The satisfaction degree and prioritized weight with respect to each prioritized pseudo criterion
A1 A2 A3 A4 A5 A6
Sat1 (·)
Sat2 (·)
Sat3 (·)
Z1 (·)
Z2 (·)
Z3 (·)
0.2409 0.1276 0.1736 0.0474 0.2060 0.2045
0.1683 0.1913 0.1555 0.1396 0.1304 0.2149
0.2156 0.1994 0.1409 0.1817 0.1725 0.0899
1 1 1 1 1 1
0.2409 0.1276 0.1736 0.0474 0.2060 0.2045
0.0405 0.0244 0.0270 0.0066 0.0269 0.0439
V (a4 ) = 0.0552, V (a5 ) = 0.2375, V (a6 ) = 0.2524 Thus, the ranking order of all alternatives is a1 a6 a5 a3 a2 a4 , and the best alternative is a1 . It is not hard to see that compared with Yan’s prioritized weighted aggregation operator, the proposed P-PROMETHEE method can yield the distinct ranking results of the alternatives, i.e., a5 a1 ∼ a6 a3 a2 a4 , which is more reasonable. Because the result obtained by the P-PROMETHEE method is compatible with the DM’s preference information of alternatives, while the result of Yan’s prioritized weighted aggregation operator isn’t compatible with the DM’s preference information of alternatives, which is a5 a1 , a1 a3 , a6 a3 .
L. Chen and Z. Xu / A new prioritized multi-criteria outranking method: The prioritized PROMETHEE
5. Concluding remarks
Acknowledgments
PY
The authors are very grateful to the anonymous reviewers for their insightful and constructive comments and suggestions that have led to an improved version of this paper. The work was supported in part by the National Natural Science Foundation of China (No.61273209). References [1]
OR
(1) The P-PROMETHEE method can obtain the weights of criteria in the same prioritized level by the entropy-based method, which can capture the relation of criteria without additional information of weights. However, Yan’s method needs the DM’s attitudinal character q for obtaining the relative importance of each criterion. (2) The P-PROMETHEE method constructs a linear programming model to determine the prioritized weight of the prioritized level, which is compatible with the DM’s partial preference information of alternatives. However, Yan’s method obtains the prioritized weights based on the satisfaction degrees of the higher prioritized levels, which can’t use the partial preference information of alternatives. Therefore, the result may be not effective.
and reefs has been used to illustrate and verify the PPROMETHEE method. Surely, the P-PROMETHEE can also be used to deal with other prioritized muliticriteria ranking problem, such as document retrieval problems, risk assessment problems and selecting a car, etc. In the future, we will consider the general prioritized multi-criteria outranking method and extend the P-PROMETHEE method to uncertain situations.
CO
Consequently, it is easy to see that the PPROMETHEE method, compared with Yan’s method [32], has the following two desirable advantages:
[2]
AU
TH
In this paper, we have extended the PROMETHEE method to the case of the prioritized MCDA, which had never been considered before. The set of criteria is not considered in the same level, but it has a weakly ordered prioritization relation. Then, we have developed the prioritized PROMETHEE method based on the prioritized measure and entropy to capture the ranking order of all the alternatives. In most of the prioritized MCDM problems, it is difficult to give the weight of each criterion by the DM, especially when the number of criteria is large. For the criteria of same prioritized level, we have proposed an entropy-based method to derive the weight of each criterion. Meanwhile, we have considered two cases where the preference information on alternatives is completely unknown and partial known. For the case where the preference information on alternatives given by the DM is completely unknown, we have employed Yager’s prioritized measure-guided aggregation operator [30] to calculate the total net flow of each alternative. For the case where the preference information on alternatives given by DM is partially known, we have proposed a linear programming method to derive the priority vectors. Based on the priority vectors, we can get the ranking of alternative compatible with the DM’s preference information. At last, the strategic assessment of islands
2109
[3] [4]
[5]
[6]
[7]
[8]
[9]
[10]
[11]
A. Albadvi, S.K. Chaharsooghi and A. Esfahanipour, Decision making in stock trading: An application of PROMETHEE, European Journal of Operational Research 177 (2007), 673–683. O.U. Araz, A simulation based multi-criteria scheduling approach of dual-resource constrained manufacturing systems with neural networks, in: AI 2005: Advances in Artificial Intelligence, Springer, 2005, pp. 1047-1052. G. Beliakov, A. Pradera and T. Calvo, Aggregation functions: A guide for practitioners, Springer-Verlag, Berlin, 2007. J.P. Brans, L’ing´enieurie de la d´ecision-Elaboration d’instruments d’aide a` la d´ecision. La m´ethode Prom´eth´ee–Dans Nadeau R. et Landry M, L’aide a` la d´ecision: Nature, intruments et perspectives d’avenir. Presses de l’universit´e de Laval, Qu´ebec, Canada, 1982, pp. 182-213. J.P. Brans and B. Mareschal, The PROMETHEE VI procedure: How to differentiate hard from soft multicriteria problems, Journal of Decision Systems 4 (1995), 213–223. J.P. Brans and B. Mareschal, PROMETHEE methods, in: Multiple criteria decision analysis: State of the art surveys, Springer, 2005, pp. 163-186. J.P. Brans and P. Vincke, A preference ranking organization method. The PROMETHEE method for MCDM, Management Science 31 (1985), 647–656. L.H. Chen, Z.S. Xu and X.H. Yu, Prioritized measure-guided aggregation operators, IEEE Transactions on Fuzzy Systems 22 (2014), 1127–1138. F. Chiclana, F. Herrera and E. Herrera-Viedma, The ordered weighted geometric operator: Properties and application in MCDM problems, in: Technologies for Constructing Intelligent Systems 2, Physica-Verlag, Heidelberg, 2002, pp. 173-183. S. Corrente, J.R. Figueira and S. Greco, The SMAAPROMETHEE method, European Journal of Operational Research 239 (2014), 514–522. S. Eppe and Y. De Smet, Studying the impact of information structure in the PROMETHEE II preference elicitation process: A simulation based approach, in: Advances in Computational Intelligence, Springer, 2012, pp. 383-392.
[16]
[17]
[18]
[19]
[20]
[21] [22]
[25]
[26]
[27]
[28]
[29]
PY
[15]
[24]
Z.X. Sun and M. Han, Multi-criteria decision making based on PROMETHEE method, in: 2010 International Conference on Computing, Control and Industrial Engineering (CCIE), IEEE Computer Society Press, 2010, pp. 416–418. C.P. Wei and X.J. Tang, Generalized prioritized aggregation operators, International Journal of Intelligent Systems 27 (2012), 578–589. Z.S. Xu and Q.L. Da, The ordered weighted geometric averaging operators, International Journal of Intelligent Systems 17 (2002), 709–716. Z.S. Xu and H. Hu, Entropy-based procedures for intuitionistic fuzzy multiple attribute decision making, Journal of Systems Engineering and Electronics 20 (2009), 1001–1011. R.R. Yager, On ordered weighted averaging aggregation operators in multicriteria decisionmaking, IEEE Transactions on Systems, Man and Cybernetics 18 (1988), 183–190. R.R. Yager, Modeling prioritized multicriteria decision making, IEEE Transactions on Systems, Man, and Cybernetics, Part B: Cybernetics 34 (2004), 2396–2404. R.R. Yager, Prioritized aggregation operators, International Journal of Approximate Reasoning 48 (2008), 263–274. R.R. Yager, On prioritized multiple-criteria aggregation, IEEE Transactions on Systems, Man, and Cybernetics, Part B: Cybernetics 42 (2012), 1297–1305. R.R. Yager, C.L. Walker and E.A. Walker, A prioritized measure for multi-criteria aggregation and its Shapley index, in: Fuzzy Information Processing Society (NAFIPS), 2011 Annual Meeting of the North American, IEEE, 2011, pp. 1–4. H.B. Yan, V.N. Huynh, Y. Nakamori and T. Murai, On prioritized weighted aggregation in multi-criteria decision making, Expert Systems with Applications 38 (2011), 812–823. X.H. Yu and Z.S. Xu, Prioritized intuitionistic fuzzy aggregation operators, Information Fusion 14 (2013), 108–116. X.H. Yu, Z.S. Xu and Y. Ma, Prioritized multi-criteria decision making based on the idea of PROMETHEE, Procedia Computer Science 17 (2013), 449–456.
CO
[14]
[23]
[30]
[31]
[32]
OR
[13]
S. Eppe, Y. De Smet and T. St¨utzle, A bi-objective optimization model to eliciting decision maker’s preferences for the PROMETHEE II method, in: Algorithmic Decision Theory, Springer, 2011, pp. 56-66. M. Grabisch, Fuzzy integral in multicriteria decision making, Fuzzy Sets and Systems 69 (1995), 279–298. B. Mareschal, Weight stability intervals in multicriteria decision aid, European Journal of Operational Research 33 (1988), 54–64. B. Mareschal and J.P. Brans, Geometrical representations for MCDA, European Journal of Operational Research 34 (1988), 69–77. B. Mareschal and J.P. Brans, PROMETHEE V: MCDM problems with segmentation constraints, in, ULB–Universite Libre de Bruxelles, 1992. T. Murofushi and M. Sugeno, Fuzzy measures and fuzzy integrals, in: Fuzzy Measures and Integrals: Theory and Applications, M. Grabisch, T. Murofushi, and M. Sugeno, eds., Physica-Verlag, 2000, pp. 3-41. M. O’Hagan, Aggregating template or rule antecedents in realtime expert systems with fuzzy set logic, in: Signals, Systems and Computers, 1988. Twenty-Second Asilomar Conference on, IEEE, Pacific Grove, CA, 1988, pp. 681-689. D.H. Peng, T.D. Wang, C.Y. Gao and H. Wang, Multigranular uncertain linguistic prioritized aggregation operators and their application to multiple criteria group decision making, Journal of Applied Mathematics 2013 (2013). J.C. Petraˇs, Ranking the sites for low-and intermediate-level radioactive waste disposal facilities in croatia, International Transactions in Operational Research 4 (1997), 237–249. C.E. Shannon, A mathematical theory of communication, The Bell System Technical Journal 27 (1948), 379–423, 623–656. T. Solymosi and J. Dombi, A method for determining the weights of criteria: The centralized weights, European Journal of Operational Research 26 (1986), 35–41.
TH
[12]
L. Chen and Z. Xu / A new prioritized multi-criteria outranking method: The prioritized PROMETHEE
AU
2110
[33]
[34]
