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A new fuzzy multicriteria decision making method and its application in diversion of water
Expert Systems with Applications 37 (2010) 8809–8813
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Expert Systems with Applications journal homepage: www.elsevier.com/locate/eswa
A new fuzzy multicriteria decision making method and its application in diversion of water Mohammad Hossein Alipour a,*, Abolfazl Shamsai a, Nazanin Ahmady b a b
Department of Civil Engineering, Sharif University of Technology, P.O. Box 11365-9313, Tehran, Iran Department of Mathematics, Varamin-Pishva Branch, Islamic Azad University, Pishva, Varamin, Iran
a r t i c l e
i n f o
Keywords: Multicriteria decision making Fuzzy numbers with different shapes Water resource planning and management
a b s t r a c t Taking account of uncertainty in multicriteria decision making problems is crucial due to the fact that depending on how it is done, ranking of alternatives can be completely different. This paper utilizes linguistic values to evaluate the performance of qualitative criteria and proposes using appropriate shapes of fuzzy numbers to evaluate the performance of quantitative criteria for each problem with respect to its particular conditions. In addition, a process to determine the weights of criteria using fuzzy numbers, which considers their competition to gain greater weights and their influence on each other is described. A new fuzzy methodology is proposed to solve such a problem that utilizes parametric form of fuzzy numbers. The case study of diversion of water into Lake Urmia watershed, which is defined using triangular, trapezoidal, and bell-shape fuzzy numbers demonstrates the utility of the proposed method. Ó 2010 Elsevier Ltd. All rights reserved.
1. Introduction A decision in water resource planning and management is typically made in order to satisfy multiple objectives as much as possible. In majority of cases, there are a number of alternatives that each one is able to provide different level of satisfaction for every objective. Multiple criteria decision making includes a body of methods to support such a decision, like PROMETHEE methods by Brans, Vincke, and Mareschal (1986), the Analytical Hierarchy Process by Saaty (1980), the Evamix approach by Voogd (1982), the range of value method by Yakowitz, Lane, and Szidarovszky (1993), and Compromise Programming by Zeleny (1973, 1982). Additionally, there are some methods that are usable specific to particular conditions like two studies by Chuntian and Chau (2001, 2002). There are also many studies using MCDM methods in different case studies (Abrishamchi, Ebrahimian, Tajrishi, & Marino, 2005; Abutaleb & Mareschal, 1995; Eder, Duckstein, & Nachtnebel, 1997; Hyde, Maier, & Colby, 2004). Recent research, including review articles, has recognized that MCDM needs research to find better ways for handling risk and uncertainty (Hajkowicz & Collins, 2007). Fuzzy MCDM methods are much more effective than typical crisp methods to take account of uncertainty and risk. Since Bellman and Zadeh (1970) used fuzzy set theory to reform decision making models, many fuzzy MCDM methods have been developed and used to support * Corresponding author. Tel.: +98 9352 218859; fax: +98 2166 014828. E-mail addresses: [email protected], [email protected] (M.H. Alipour). 0957-4174/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.eswa.2010.06.023
the decision making process (Akter & Simonovic, 2005; Bender & Simonovic, 2000; Fu, 2008; Makropoulos & Butler, 2006; Malczewski, 2006; Simonovic & Akter, 2006; Simonovic & Verma, 2008). In the present study, fuzzy numbers are used to take account of uncertainty arising from weighting objectives and determining performance measures of alternatives. Uncertainty arising from weighting objectives is considered using triangular fuzzy numbers. To assign fuzzy weights, it must be noticed when decision makers are trying to assign a weight to an objective, they must consider the other objectives and then indicate the weight of the objective in comparison with them. Crisp methods usually use the stipulation that the summation of all weights must be equal to one. It is done to make decision maker assign weights considering competition between objectives to gain greater weights. In assigning fuzzy weights, this stipulation may be neglected as a result of present difficulty to satisfy it by decision makers. In such a case, decision maker would probably assign high weights to all objectives because there is no inherent unimportant objective to a decision maker. It is only possible that an objective be unimportant compared to another objective. For instance, in Fu (2008) there were seven linguistic values for evaluation of the weights of criteria, including extremely important, very important, important, fairly important, unimportant, very unimportant, and extremely unimportant. It was not considered any stipulation to weight the objectives. Therefore, decision maker assigned extremely important to one criterion, very important to two criteria, and important to the other two criteria and there was no criteria to which one of the other linguistic values had been assigned.
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In the present study in order to consider competition between objectives, the following process is proposed and is utilized in the case study of diversion of water into Lake Urmia watershed: (1) ask the decision maker to assign crisp weights to objectives required that the summation of weights must be equal to one; (2) use determined values as the center values of fuzzy numbers; (3) ask the decision maker to determine maximum and minimum value of each weight considering its indicated center value; this is done to obtain the decision maker’s uncertainty about each weight; (4) use maximum and minimum values to form triangular fuzzy weights. Uncertainty in determining performance measures of alternatives against different objectives is taken account of by using fuzzy numbers as well. In the case study, bell-shape fuzzy numbers are used to obtain quantitative performance measures because center values are comparatively accurate and close to true values. Triangular fuzzy numbers are used to obtain qualitative performance measures. After completing the decision matrix with performance measures, metric sign distance (Abbasbandy & Asady, 2006) is utilized to calculate the distance between each performance measure and fuzzy zero. Subsequently, calculated distances are converted into commensurate units. Finally, a combination of metric sign distance and a penalty coefficient ranks the alternatives. The penalty coefficient, according to its value, is able to prevent neutralization of poor performance of an alternative against some objectives by its good performance against a few other objectives. Finally, proposed method is used to rank alternatives of case study of diversion of water into Lake Urmia watershed.
xl 6 x 6 xc1 ; xc1 6 x 6 xc2 ; xc2 6 x 6 xr ;
ð2Þ
otherwise;
ðaÞ ¼ and its parametric form is u(a) = xc1 (1 a)(xc1 xl), u xc2 þ ð1 aÞðxr xc2 Þ (see Abbasbandy & Asady, 2006; Dubois & Prade, 2000). In the present study in order to discriminate fuzzy numbers ~. from crisp numbers, a fuzzy number like u is displayed as u 3. Methodology Details of the proposed method to solve a multicriteria decision making problem are presented in this section. 3.1. To obtain objectives, alternatives, and weights of objectives In every MCDM problem, first it is necessary to determine objectives and design alternatives to satisfy determined objectives. In the present study an objective set is shown as follows:
O ¼ fo1 ; o2 ; . . . ; on g;
ð3Þ
where oj is the jth objective, j = 1, 2, . . . , n. And an alternative set is shown as follows:
A ¼ fa1 ; a2 ; . . . ; am g;
ð4Þ
where ai is the ith candidate alternative, i = 1, 2, . . . , m. Weights are assigned to objectives as the process mentioned in Section 2. A weight set is shown as
2. Definitions
Definition 2.1. A fuzzy number is a fuzzy set like u: R ? I = [0, 1] which satisfies: u is upper semi-continuous, u(x) = 0 outside some interval [a, d], There are real numbers a, b such that a 6 b 6 c 6 d and (a) u(x) is monotonic increasing on [a, b], (b) u(x) is monotonic decreasing on [c, d], (c) u(x) = 1, 6x 6 c.
a 6 x 6 b; b 6 x 6 c; c 6 x 6 d;
W ¼ fw1 ; w2 ; . . . ; wn g;
ð5Þ
where wj is the weight of jth objective. 3.2. To obtain performance measures
The membership function u can be expressed as
8 ul ðxÞ > > > <1 uðxÞ ¼ > ur ðxÞ > > : 0
8 xx l > ðxc1 xl Þ > > > <1 uðxÞ ¼ xr x > > ðxr x > c2 Þ > : 0
ð1Þ
otherwise;
where ul: [a, b] ? [0, 1] and ur: [c, d] ? [0, 1] are left and right membership functions of fuzzy number u (Abbasbandy & Asady, 2006). Definition 2.2. An arbitrary fuzzy number u in parametric form is Þ of functions uðaÞ; u ðaÞ; 0 6 a 6 1, which signified by a pair ðu; u satisfy the following conditions: (a) u(a) is a bounded monotonic increasing left continuous function, ðaÞ is a bounded monotonic decreasing left continuous (b) u function, ðaÞ; 0 6 a 6 1. (c) uðaÞ 6 u The trapezoidal fuzzy number u = (xl, xc1, xc2, xr), with two defuzzifier xc1, xc2, and left fuzziness xc1 xl > 0 and right fuzziness xr xc2 > 0 is a fuzzy set where the membership function is as
In a MCDM problem, one of the most challenging steps to define the problem is to determine performance measures of alternatives against qualitative criteria. It is clear that a qualitative performance measure cannot be defined like quantitative performance measures. Linguistic variables are very helpful to define qualitative performance measures. Table 1 displays linguistic variables and their corresponding fuzzy numbers used to evaluate qualitative performance measures in the present study. For quantitative objectives, usually financial objectives, primary estimated values are used as the center value of fuzzy number. Maximum and minimum values are respectively used as right limits and left limits of fuzzy numbers. Regarding how much primary estimated values are reliable, different shapes of fuzzy numbers can be used. In the case study, bell-shape fuzzy numbers are used because primary values are well estimated.
Table 1 Linguistic variables and their corresponding fuzzy numbers. Linguistic variables
Fuzzy numbers
Very low Low Fairly low Medium Fairly high High Very high
(0, 0, 0.1, 0.2) (0.1, 0.2, 0.35) (0.25, 0.4, 0.45) (0.35, 0.5, 0.65) (0.55, 0.6, 0.75) (0.65, 0.8, 0.9) (0.8, 0.9, 1, 1)
M.H. Alipour et al. / Expert Systems with Applications 37 (2010) 8809–8813
3.3. To transform performance measures into commensurate units Performance measures, obtained in the prior step, are measured in different units; consequently, first we need to convert them into a dimensionless unit. In order to obtain commensurate units, first, fuzzy distance between each performance measure and fuzzy zero is calculated using metric sign distance. The distance between a fuzzy number and fuzzy zero using metric sign distance is calculated by the following formula:
~; u ~ 0 Þ ¼ cðu ~ ÞDp ðu ~; u ~0 Þ; dp ðu
ð6Þ
~; u ~ 0 Þ and cðu ~ Þ are defined as follows: where Dp ðu
~; u ~0 Þ ¼ Dp ðu
Z
1
1=p ðaÞjp Þda ðjuðaÞjp þ ju
ðp P 1Þ;
ð7Þ
0
(
cðu~ Þ ¼
1; 1;
R1 0
ðaÞÞda P 0; ðuðaÞ þ u
0
ðaÞÞda 6 0; ðuðaÞ þ u
R1
ð8Þ
~ 0 is a fuzzy origin with zero fuzziness on the left and the right. It is u defined by the parametric form (0, 0), and the following membership function:
1; x ¼ 0; 0;
ð9Þ
x – 0:
In a typical MCDM problem, values of performance measures and weights are defined using positive either crisp or fuzzy numbers; ~ Þ is equal to 1. The value of p in each problem is consequently, cðu selected depending on its particular conditions. In this section, due to the fact that calculated distances will be divided by each other to be converted to the dimensionless unit, the value of p does not signify in the results. As a result, in order to simplify calculations, in this section, the value of p is chosen equal to 1. Therefore, the distances between each fuzzy performance measure ~xij and fuzzy zero using Eqs. (6)–(8) is calculated by the following formula:
~0 Þ ¼ zij ¼ d1 ð~xij ; u
Z
1
ðjxij ðaÞj þ jxij ðaÞjÞda:
ð10Þ
0
After calculating the distances between performance measures and fuzzy zero, the following formula is utilized for benefit criteria, including desirable qualitative criteria, to convert the distances into the dimensionless unit between 0 and 1:
cij ¼
zij maxi zij
ð11Þ
and the following formula is utilized for cost criteria, including undesirable qualitative criteria:
cij ¼
mini zij : zij
ð12Þ
3.4. To multiply weights by their corresponding dimensionless performance measures Fuzzy weights assigned to each criterion j are multiplied by their corresponding cijs, i = 1, 2, . . . , m, that is calculated in the previous section. Therefore, there will be m, equal to the number of alternatives, fuzzy vectors. Each fuzzy vector includes n, equal to the number of criteria, fuzzy numbers. 3.5. To rank alternatives In order to rank alternatives, we need to rank fuzzy vectors obtained in the previous section. Therefore, first, the distance between every element of every vector and fuzzy zero is calculated
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using metric sign distance. The value of p in this step is selected equal to two; This is done in order to soften the effect of very large distances. Eq. (13) calculates the distances:
~ j cij ; u ~0 Þ ¼ Q i ¼ d2 ðw
Z
1
1=2 j ðaÞcij j2 da jwj ðaÞcij j2 þ jw ;
ð13Þ
0
after calculating all distances, in order to prevent the good performance of an alternative against a few criteria from neutralizing its poor performance against the other criteria, a penalty coefficient k is utilized. The value of k can have a great effect on the ranking of alternatives. The calculated distances are first increased to the power of 1/k. Afterwards, the resulted values for each alternative are added together and the summation is increased to the power of k. Resulted numerical values are compared and alternatives with the higher value will have the higher rank. Therefore, the following formula is used to obtain final performance assigned to alternatives to rank them:
Ri ¼
n X
!k Q i1=k
:
ð14Þ
j¼1
4. Case study of diversion of water into Lake Urmia watershed Lake Urmia watershed is located in the northwestern part of Iran. During recent years, Lake Urmia has suffered a considerable water loss due to excessive withdrawal, which has caused part of the lake to dry out. In order to prevent other parts of the lake from drying and recover the dried part of the lake, after studying the present options, the responsible organization has decided to transfer water to the Lake Urmia watershed. Five different locations has been indicated as candidates to construct a dam and withdraw water from to transfer to the lake. In order to distinguish the best location among five candidates, eight criteria are determined to evaluate the performance of each alternative. Total cost, water supply volume, time of construction of water transfer system, environmental loss, social equity, technical feasibility, water supply reliability, and political impact are respectively indicated criteria from one to eight. Weights of criteria are indicated following the proposed method in Section 1. Qualitative performance measures are indicated using linguistic variables that are defined in Table 1, and quantitative performance measures are indicated using bell-shape fuzzy numbers. Table 2 displays the decision matrix. After completing the decision matrix, in order to transform performance measures to commensurate units, first, Eq. (10) and parametric form of trapezoidal fuzzy numbers explained in Definition 2.2 are used to calculate the fuzzy distance between each trapezoidal performance measure, like very low, and fuzzy zero. It is clear ~ ¼ ðxl ; xc ; xr Þ is a particular that a triangular fuzzy number u trapezoidal fuzzy number with the parametric form u (a) = ðaÞ ¼ xc þ ð1 aÞðxr xc Þ; this parametric xc (1 a)(xc xl), u form and Eq. (10) are used for triangular fuzzy numbers. Criteria 1 and 3 are evaluated using bell-shape the pffiffiffiffiffiffiffiffiffiffiffiffi fuzzy numbers with pffiffiffiffiffiffiffiffiffiffiffiffi ðaÞ ¼ xc þ 1 a parametric form uðaÞ ¼ xc 1 aðxc xl Þ; u ðxr xc Þ; this parametric form and Eq. (10) are used for criteria 1 and 3. Table 3 displays the calculated distances. Next, Eq. (11) is used to convert the calculated distances of performance measures of criterion 2, 5, 6, and 7 into the dimensionless unit between 0 and 1. Eq. (12) is used for criteria 1, 3, 4, and 8. Table 4 shows the resulted cijs. Next step includes multiplying fuzzy weights by calculated cijs, which is done using indicated weights in Table 2; 5 shows the results. Afterwards, Eq. (13) is utilized to calculate fuzzy distance
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Table 2 Decision matrix of the Lake Urmia watershed case study. Criteria
Weights
a1
a2
a3
a4
a5
c1 (Million $) c2 (MCM/Year) c3 (Years) c4 c5 c6 c7 c8
(0.19, 0.2, 0.23) (0.3, 0.3, 0.33) (0.14, 0.15, 0.16) (0.095, 0.1, 0.105) (0.057, 0.06, 0.063) (0.048, 0.05, 0.053) (0.085, 0.09, 0.095) (0.047, 0.05, 0.052)
(399, 420, 483) 600 (14.5, 15, 16) H L H M FL
(418, 440, 506) 600 (14.5, 15, 16.5) FH L FH M FL
(536, 564, 649) 608 (17.5, 18, 19.5) M FL M FH L
(635, 668, 768) 613 (17.5, 18, 19.5) M FL M FH L
(638, 719, 826) 634 (17.5, 18, 19.5) M M FL H VL
Table 3 Fuzzy distance between performance measures and fuzzy zero.
Table 6 Fuzzy distance between fuzzy zero and elements of Table 5.
Criteria
a1
a2
a3
a4
a5
Criteria
a1
a2
a3
a4
a5
c1 c2 c3 c4 c5 c6 c7 c8
868 1200 30.33 1.575 0.425 1.575 1 0.75
909.33 1200 30.66 1.25 0.425 1.25 1 0.75
1166 1216 36.66 1 0.75 1 1.25 0.425
1380.66 1226 36.66 1 0.75 1 1.25 0.425
1485.33 1268 36.66 1 1 0.75 1.575 0.15
c1 c2 c3 c4 c5 c6 c7 c8
0.0843 0.1710 0.0451 0.0079 0.0013 0.0051 0.0065 0.0001
0.0763 0.1710 0.0443 0.0128 0.0013 0.0032 0.0065 0.0001
0.0462 0.1745 0.0312 0.0200 0.0041 0.0020 0.0101 0.0006
0.0335 0.1781 0.0312 0.0200 0.0041 0.0020 0.0101 0.0006
0.0283 0.1893 0.0312 0.0200 0.0072 0.0012 0.0162 0.0050
R1 = 1.54106, R2 = 1.6106, R3 = 1.7106, R5 = 1.71106 Ranking of alternatives: a5 > a3 > a4 > a2 > a1
Table 4 Dimensionless performance measures. Criteria
a1
a2
a3
a4
a5
c1 c2 c3 c4 c5 c6 c7 c8
1 0.95 1 0.63 0.425 1 0.63 0.2
0.95 0.95 0.99 0.8 0.425 0.79 0.63 0.2
0.74 0.96 0.83 1 0.75 0.63 0.79 0.35
0.63 0.97 0.83 1 0.75 0.63 0.79 0.35
0.58 1 0.83 1 1 0.48 1 1
R4 = 1.64106,
5. Discussion As it can be seen, in the case study ranking of alternatives is sensitive to the value of k. Among alternatives, a1 and a2 descend to a lower rank as the value of k increases. Table 6 shows that these two alternatives have much better performance than the other alternatives on criteria 1 and 3, and fairly better importance on criterion 6, but their performance on criteria 2, 4, 5, 7, and 8 is not as good as the other alternatives; as a consequence, the higher value of penalty coefficient diminishes their rank. On the other hand, a5 has quite different behavior from a1 and a2 so that it gains a higher rank corresponding to the higher value of penalty coefficient. a5 has the best performance on five criteria and the worst performance on the other three criteria. Overall, it seems that a3 is the best alternative due to its fairly good performance on all criteria as well as its appropriate rank for every value of penalty coefficient. It seems that a4 can be considered as the worst alternative.
between elements of Table 5 and fuzzy zero; Table 6 shows the resulted distances. In order to rank alternatives, Eq. (14) is utilized to calculate final performance of the alternatives. Ris are calculated using three values 1, 2, and 10 as k to consider the uncertainty in the value of penalty coefficient. Resulted values and their corresponding rankings are as follows: k = 1: R1 = 1.041, R2 = 1.049, R3 = 1.015, R4 = 0.987, R5 = 1.006 Ranking of alternatives: a2 > a1 > a3 > a5 > a4 k = 1/2: R1 = 4.54, R2 = 4.638, R3 = 4.663, R4 = 4.523, R5 = 4.654 Ranking of alternatives: a3 > a5 > a2 > a1 > a4 k = 1/10:
6. Conclusion The focus of this study is on taking uncertainty, present in a multicriteria decision making problem, into account as much as possible. In order to do that, a new process to determine weights of criteria is utilized so that weights are determined using fuzzy numbers and at the same time decision makers are compelled to
Table 5 Results of multiplying weights by dimensionless performance measures. Criteria
a1
a2
a3
a4
a5
c1 c2 c3 c4 c5 c6 c7 c8
(0.19, 0.2, 0.23) (0.285, 0.285, 0.314) (0.14, 0.15, 0.16) (0.06, 0.063, 0.066) (0.024, 0.026, 0.027) (0.048, 0.05, 0.053) (0.054, 0.057, 0.06) (0.009, 0.01, 0.01)
(0.181, 0.19, 0.219) (0.285, 0.285, 0.314) (0.139, 0.149, 0.158) (0.076, 0.08, 0.084) (0.024, 0.026, 0.027) (0.038, 0.04, 0.042) (0.054, 0.057, 0.06) (0.009, 0.01, 0.01)
(0.141, 0.148, 0.17) (0.288, 0.288, 0.317) (0.116, 0.125, 0.133) (0.095, 0.1, 0.105) (0.043, 0.045, 0.047) (0.03,0.032,0.033) (0.067, 0.071, 0.075) (0.016, 0.018, 0.018)
(0.12, 0.126, 0.145) (0.291, 0.291, 0.32) (0.116, 0.125, 0.133) (0.095, 0.1, 0.105) (0.043, 0.045, 0.047) (0.03, 0.032, 0.033) (0.067, 0.071, 0.075) (0.016, 0.018, 0.018)
(0.11, 0.116, 0.133) (0.3, 0.3, 0.33) (0.116, 0.125, 0.133) (0.095, 0.1, 0.105) (0.057, 0.06, 0.063) (0.023, 0.024, 0.025) (0.085, 0.09, 0.095) (0.047, 0.05, 0.052)
M.H. Alipour et al. / Expert Systems with Applications 37 (2010) 8809–8813
consider the competition between criteria to obtain greater weights. In this way, a decision maker cannot indicate very high weights for all criteria because a high value as the weight of a criterion diminishes the weight of other criteria. In addition to the process, determining quantitative performance measures is done using seven linguistic variables and their corresponding fuzzy numbers. For quantitative performance measures, this study does not restrict the shapes of fuzzy numbers and proposes using appropriate shapes of fuzzy numbers with regard to particular conditions of each problem. In this paper, also a new fuzzy methodology is proposed to solve a multicriteria decision making problem defined in fuzzy environment with different shapes of fuzzy numbers. The proposed method utilizes parametric form of fuzzy numbers and metric sign distance to deal with such a problem as it is demonstrated in the case study of diversion of water into Lake Urmia watershed. The proposed method can be utilized to solve multicriteria decision making problems defined using diverse shapes of fuzzy numbers in many fields. References Abbasbandy, S., & Asady, B. (2006). Ranking of fuzzy numbers by sign distance. Information Sciences, 176, 2405–2416. Abrishamchi, A., Ebrahimian, A., Tajrishi, M., & Marino, M. A. (2005). Case study: Application of multicriteria decision making to urban water supply. Journal of Water Resource Planning and Management, 131(4), 326–335. Abutaleb, M. F., & Mareschal, B. (1995). Water resource planning in the middle east: Application of the PROMETHEE V multicriteria method. European Journal of Operations Research, 81(3), 500–511. Akter, T., & Simonovic, S. P. (2005). Aggregation of fuzzy views of a large number of stakeholders for multi-objective flood management decision making. Journal of Environmental Management, 77, 133–143. Bellman, R. E., & Zadeh, L. A. (1970). Decision-making in a fuzzy environment. Management Science, 17, 141–164. Bender, M. J., & Simonovic, S. P. (2000). A fuzzy compromise approach to water resource systems planning under uncertainty. Fuzzy Sets and Systems, 115, 35–44.
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Brans, J. P., Vincke, Ph., & Mareschal, B. (1986). How to select and how to rank projects: The PROMETHEE method. European Journal of Operational Research, 24, 228–238. Chuntian, C., & Chau, K. W. (2001). Fuzzy iteration methodology for reservoir flood control operation. Journal of The American Water Resources Association, 37(5), 1381–1388. Chuntian, C., & Chau, K. W. (2002). Three-person multi-objective conflict decision in reservoir flood control. European Journal of Operational Research, 142, 625–631. Dubois, D., & Prade, H. (2000). Fundamentals of fuzzy sets. USA: Kluwer Academic Publishers. Eder, G., Duckstein, L., & Nachtnebel, H. P. (1997). Ranking water resource projects and evaluating criteria by multicriterion Q-analysis: An Austrian case study. Journal of Multi-Criteria Decision Analysis, 6(5), 259–271. Fu, G. (2008). A fuzzy optimization method for multicriteria decision making: An application to reservoir flood control operation. Expert Systems with Applications, 34, 145–149. Hajkowicz, S., & Collins, K. (2007). A review of multiple criteria analysis for water resource planning and management. Water Resource Management, 21, 1553–1566. Hyde, K. M., Maier, H. R., & Colby, C. B. (2004). Reliability-based approach to multicriteria decision analysis for water resources. Journal of Water Resource Planning and Management, 130(6), 429–438. Makropoulos, C. K., & Butler, D. (2006). Spatial ordered weighted averaging: incorporating spatially variable attitude towards risk in spatial multi-criteria decision-making. Environmental Modelling and Software, 21, 69–84. Malczewski, J. (2006). Ordered weighted averaging with fuzzy quantifiers: GISbased multicriteria evaluation for land-use suitability analysis. International Journal of Applied Earth Observation and Geoinformation, 8(4), 270–277. Saaty, T. (1980). The analytic hierarchy process. New York: McGraw-Hill. Simonovic, S. P., & Akter, T. (2006). Participatory floodplain management in the red river basin, Canada. Annual Reviews in Control, 30, 183–192. Simonovic, S. P., & Verma, R. (2008). A new methodology for water resource multicriteria decision making under uncertainty. Physics and Chemistry of the Earth, 33, 322–329. Voogd, H. (1982). Multicriterion evaluation with mixed qualitative and quantitative data. Environment and Planning Bulletin, 9, 221–236. Yakowitz, D. S., Lane, L., & Szidarovszky, F. (1993). Multi-attribute decision making: Dominance with respect to an importance order of the attributes. Applied Mathematics and Computation, 54, 167–181. Zeleny, M. (1973). Compromise programming. In J. L. Cocharane & M. Zeleny (Eds.), Multiple criteria decision making (pp. 262–301). Columbia, S.C.: University of Southern Carolina Press. Zeleny, M. (1982). Multiple criteria decision making. New York: McGraw-Hill.
Contents lists available at ScienceDirect
Expert Systems with Applications journal homepage: www.elsevier.com/locate/eswa
A new fuzzy multicriteria decision making method and its application in diversion of water Mohammad Hossein Alipour a,*, Abolfazl Shamsai a, Nazanin Ahmady b a b
Department of Civil Engineering, Sharif University of Technology, P.O. Box 11365-9313, Tehran, Iran Department of Mathematics, Varamin-Pishva Branch, Islamic Azad University, Pishva, Varamin, Iran
a r t i c l e
i n f o
Keywords: Multicriteria decision making Fuzzy numbers with different shapes Water resource planning and management
a b s t r a c t Taking account of uncertainty in multicriteria decision making problems is crucial due to the fact that depending on how it is done, ranking of alternatives can be completely different. This paper utilizes linguistic values to evaluate the performance of qualitative criteria and proposes using appropriate shapes of fuzzy numbers to evaluate the performance of quantitative criteria for each problem with respect to its particular conditions. In addition, a process to determine the weights of criteria using fuzzy numbers, which considers their competition to gain greater weights and their influence on each other is described. A new fuzzy methodology is proposed to solve such a problem that utilizes parametric form of fuzzy numbers. The case study of diversion of water into Lake Urmia watershed, which is defined using triangular, trapezoidal, and bell-shape fuzzy numbers demonstrates the utility of the proposed method. Ó 2010 Elsevier Ltd. All rights reserved.
1. Introduction A decision in water resource planning and management is typically made in order to satisfy multiple objectives as much as possible. In majority of cases, there are a number of alternatives that each one is able to provide different level of satisfaction for every objective. Multiple criteria decision making includes a body of methods to support such a decision, like PROMETHEE methods by Brans, Vincke, and Mareschal (1986), the Analytical Hierarchy Process by Saaty (1980), the Evamix approach by Voogd (1982), the range of value method by Yakowitz, Lane, and Szidarovszky (1993), and Compromise Programming by Zeleny (1973, 1982). Additionally, there are some methods that are usable specific to particular conditions like two studies by Chuntian and Chau (2001, 2002). There are also many studies using MCDM methods in different case studies (Abrishamchi, Ebrahimian, Tajrishi, & Marino, 2005; Abutaleb & Mareschal, 1995; Eder, Duckstein, & Nachtnebel, 1997; Hyde, Maier, & Colby, 2004). Recent research, including review articles, has recognized that MCDM needs research to find better ways for handling risk and uncertainty (Hajkowicz & Collins, 2007). Fuzzy MCDM methods are much more effective than typical crisp methods to take account of uncertainty and risk. Since Bellman and Zadeh (1970) used fuzzy set theory to reform decision making models, many fuzzy MCDM methods have been developed and used to support * Corresponding author. Tel.: +98 9352 218859; fax: +98 2166 014828. E-mail addresses: [email protected], [email protected] (M.H. Alipour). 0957-4174/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.eswa.2010.06.023
the decision making process (Akter & Simonovic, 2005; Bender & Simonovic, 2000; Fu, 2008; Makropoulos & Butler, 2006; Malczewski, 2006; Simonovic & Akter, 2006; Simonovic & Verma, 2008). In the present study, fuzzy numbers are used to take account of uncertainty arising from weighting objectives and determining performance measures of alternatives. Uncertainty arising from weighting objectives is considered using triangular fuzzy numbers. To assign fuzzy weights, it must be noticed when decision makers are trying to assign a weight to an objective, they must consider the other objectives and then indicate the weight of the objective in comparison with them. Crisp methods usually use the stipulation that the summation of all weights must be equal to one. It is done to make decision maker assign weights considering competition between objectives to gain greater weights. In assigning fuzzy weights, this stipulation may be neglected as a result of present difficulty to satisfy it by decision makers. In such a case, decision maker would probably assign high weights to all objectives because there is no inherent unimportant objective to a decision maker. It is only possible that an objective be unimportant compared to another objective. For instance, in Fu (2008) there were seven linguistic values for evaluation of the weights of criteria, including extremely important, very important, important, fairly important, unimportant, very unimportant, and extremely unimportant. It was not considered any stipulation to weight the objectives. Therefore, decision maker assigned extremely important to one criterion, very important to two criteria, and important to the other two criteria and there was no criteria to which one of the other linguistic values had been assigned.
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In the present study in order to consider competition between objectives, the following process is proposed and is utilized in the case study of diversion of water into Lake Urmia watershed: (1) ask the decision maker to assign crisp weights to objectives required that the summation of weights must be equal to one; (2) use determined values as the center values of fuzzy numbers; (3) ask the decision maker to determine maximum and minimum value of each weight considering its indicated center value; this is done to obtain the decision maker’s uncertainty about each weight; (4) use maximum and minimum values to form triangular fuzzy weights. Uncertainty in determining performance measures of alternatives against different objectives is taken account of by using fuzzy numbers as well. In the case study, bell-shape fuzzy numbers are used to obtain quantitative performance measures because center values are comparatively accurate and close to true values. Triangular fuzzy numbers are used to obtain qualitative performance measures. After completing the decision matrix with performance measures, metric sign distance (Abbasbandy & Asady, 2006) is utilized to calculate the distance between each performance measure and fuzzy zero. Subsequently, calculated distances are converted into commensurate units. Finally, a combination of metric sign distance and a penalty coefficient ranks the alternatives. The penalty coefficient, according to its value, is able to prevent neutralization of poor performance of an alternative against some objectives by its good performance against a few other objectives. Finally, proposed method is used to rank alternatives of case study of diversion of water into Lake Urmia watershed.
xl 6 x 6 xc1 ; xc1 6 x 6 xc2 ; xc2 6 x 6 xr ;
ð2Þ
otherwise;
ðaÞ ¼ and its parametric form is u(a) = xc1 (1 a)(xc1 xl), u xc2 þ ð1 aÞðxr xc2 Þ (see Abbasbandy & Asady, 2006; Dubois & Prade, 2000). In the present study in order to discriminate fuzzy numbers ~. from crisp numbers, a fuzzy number like u is displayed as u 3. Methodology Details of the proposed method to solve a multicriteria decision making problem are presented in this section. 3.1. To obtain objectives, alternatives, and weights of objectives In every MCDM problem, first it is necessary to determine objectives and design alternatives to satisfy determined objectives. In the present study an objective set is shown as follows:
O ¼ fo1 ; o2 ; . . . ; on g;
ð3Þ
where oj is the jth objective, j = 1, 2, . . . , n. And an alternative set is shown as follows:
A ¼ fa1 ; a2 ; . . . ; am g;
ð4Þ
where ai is the ith candidate alternative, i = 1, 2, . . . , m. Weights are assigned to objectives as the process mentioned in Section 2. A weight set is shown as
2. Definitions
Definition 2.1. A fuzzy number is a fuzzy set like u: R ? I = [0, 1] which satisfies: u is upper semi-continuous, u(x) = 0 outside some interval [a, d], There are real numbers a, b such that a 6 b 6 c 6 d and (a) u(x) is monotonic increasing on [a, b], (b) u(x) is monotonic decreasing on [c, d], (c) u(x) = 1, 6x 6 c.
a 6 x 6 b; b 6 x 6 c; c 6 x 6 d;
W ¼ fw1 ; w2 ; . . . ; wn g;
ð5Þ
where wj is the weight of jth objective. 3.2. To obtain performance measures
The membership function u can be expressed as
8 ul ðxÞ > > > <1 uðxÞ ¼ > ur ðxÞ > > : 0
8 xx l > ðxc1 xl Þ > > > <1 uðxÞ ¼ xr x > > ðxr x > c2 Þ > : 0
ð1Þ
otherwise;
where ul: [a, b] ? [0, 1] and ur: [c, d] ? [0, 1] are left and right membership functions of fuzzy number u (Abbasbandy & Asady, 2006). Definition 2.2. An arbitrary fuzzy number u in parametric form is Þ of functions uðaÞ; u ðaÞ; 0 6 a 6 1, which signified by a pair ðu; u satisfy the following conditions: (a) u(a) is a bounded monotonic increasing left continuous function, ðaÞ is a bounded monotonic decreasing left continuous (b) u function, ðaÞ; 0 6 a 6 1. (c) uðaÞ 6 u The trapezoidal fuzzy number u = (xl, xc1, xc2, xr), with two defuzzifier xc1, xc2, and left fuzziness xc1 xl > 0 and right fuzziness xr xc2 > 0 is a fuzzy set where the membership function is as
In a MCDM problem, one of the most challenging steps to define the problem is to determine performance measures of alternatives against qualitative criteria. It is clear that a qualitative performance measure cannot be defined like quantitative performance measures. Linguistic variables are very helpful to define qualitative performance measures. Table 1 displays linguistic variables and their corresponding fuzzy numbers used to evaluate qualitative performance measures in the present study. For quantitative objectives, usually financial objectives, primary estimated values are used as the center value of fuzzy number. Maximum and minimum values are respectively used as right limits and left limits of fuzzy numbers. Regarding how much primary estimated values are reliable, different shapes of fuzzy numbers can be used. In the case study, bell-shape fuzzy numbers are used because primary values are well estimated.
Table 1 Linguistic variables and their corresponding fuzzy numbers. Linguistic variables
Fuzzy numbers
Very low Low Fairly low Medium Fairly high High Very high
(0, 0, 0.1, 0.2) (0.1, 0.2, 0.35) (0.25, 0.4, 0.45) (0.35, 0.5, 0.65) (0.55, 0.6, 0.75) (0.65, 0.8, 0.9) (0.8, 0.9, 1, 1)
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3.3. To transform performance measures into commensurate units Performance measures, obtained in the prior step, are measured in different units; consequently, first we need to convert them into a dimensionless unit. In order to obtain commensurate units, first, fuzzy distance between each performance measure and fuzzy zero is calculated using metric sign distance. The distance between a fuzzy number and fuzzy zero using metric sign distance is calculated by the following formula:
~; u ~ 0 Þ ¼ cðu ~ ÞDp ðu ~; u ~0 Þ; dp ðu
ð6Þ
~; u ~ 0 Þ and cðu ~ Þ are defined as follows: where Dp ðu
~; u ~0 Þ ¼ Dp ðu
Z
1
1=p ðaÞjp Þda ðjuðaÞjp þ ju
ðp P 1Þ;
ð7Þ
0
(
cðu~ Þ ¼
1; 1;
R1 0
ðaÞÞda P 0; ðuðaÞ þ u
0
ðaÞÞda 6 0; ðuðaÞ þ u
R1
ð8Þ
~ 0 is a fuzzy origin with zero fuzziness on the left and the right. It is u defined by the parametric form (0, 0), and the following membership function:
1; x ¼ 0; 0;
ð9Þ
x – 0:
In a typical MCDM problem, values of performance measures and weights are defined using positive either crisp or fuzzy numbers; ~ Þ is equal to 1. The value of p in each problem is consequently, cðu selected depending on its particular conditions. In this section, due to the fact that calculated distances will be divided by each other to be converted to the dimensionless unit, the value of p does not signify in the results. As a result, in order to simplify calculations, in this section, the value of p is chosen equal to 1. Therefore, the distances between each fuzzy performance measure ~xij and fuzzy zero using Eqs. (6)–(8) is calculated by the following formula:
~0 Þ ¼ zij ¼ d1 ð~xij ; u
Z
1
ðjxij ðaÞj þ jxij ðaÞjÞda:
ð10Þ
0
After calculating the distances between performance measures and fuzzy zero, the following formula is utilized for benefit criteria, including desirable qualitative criteria, to convert the distances into the dimensionless unit between 0 and 1:
cij ¼
zij maxi zij
ð11Þ
and the following formula is utilized for cost criteria, including undesirable qualitative criteria:
cij ¼
mini zij : zij
ð12Þ
3.4. To multiply weights by their corresponding dimensionless performance measures Fuzzy weights assigned to each criterion j are multiplied by their corresponding cijs, i = 1, 2, . . . , m, that is calculated in the previous section. Therefore, there will be m, equal to the number of alternatives, fuzzy vectors. Each fuzzy vector includes n, equal to the number of criteria, fuzzy numbers. 3.5. To rank alternatives In order to rank alternatives, we need to rank fuzzy vectors obtained in the previous section. Therefore, first, the distance between every element of every vector and fuzzy zero is calculated
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using metric sign distance. The value of p in this step is selected equal to two; This is done in order to soften the effect of very large distances. Eq. (13) calculates the distances:
~ j cij ; u ~0 Þ ¼ Q i ¼ d2 ðw
Z
1
1=2 j ðaÞcij j2 da jwj ðaÞcij j2 þ jw ;
ð13Þ
0
after calculating all distances, in order to prevent the good performance of an alternative against a few criteria from neutralizing its poor performance against the other criteria, a penalty coefficient k is utilized. The value of k can have a great effect on the ranking of alternatives. The calculated distances are first increased to the power of 1/k. Afterwards, the resulted values for each alternative are added together and the summation is increased to the power of k. Resulted numerical values are compared and alternatives with the higher value will have the higher rank. Therefore, the following formula is used to obtain final performance assigned to alternatives to rank them:
Ri ¼
n X
!k Q i1=k
:
ð14Þ
j¼1
4. Case study of diversion of water into Lake Urmia watershed Lake Urmia watershed is located in the northwestern part of Iran. During recent years, Lake Urmia has suffered a considerable water loss due to excessive withdrawal, which has caused part of the lake to dry out. In order to prevent other parts of the lake from drying and recover the dried part of the lake, after studying the present options, the responsible organization has decided to transfer water to the Lake Urmia watershed. Five different locations has been indicated as candidates to construct a dam and withdraw water from to transfer to the lake. In order to distinguish the best location among five candidates, eight criteria are determined to evaluate the performance of each alternative. Total cost, water supply volume, time of construction of water transfer system, environmental loss, social equity, technical feasibility, water supply reliability, and political impact are respectively indicated criteria from one to eight. Weights of criteria are indicated following the proposed method in Section 1. Qualitative performance measures are indicated using linguistic variables that are defined in Table 1, and quantitative performance measures are indicated using bell-shape fuzzy numbers. Table 2 displays the decision matrix. After completing the decision matrix, in order to transform performance measures to commensurate units, first, Eq. (10) and parametric form of trapezoidal fuzzy numbers explained in Definition 2.2 are used to calculate the fuzzy distance between each trapezoidal performance measure, like very low, and fuzzy zero. It is clear ~ ¼ ðxl ; xc ; xr Þ is a particular that a triangular fuzzy number u trapezoidal fuzzy number with the parametric form u (a) = ðaÞ ¼ xc þ ð1 aÞðxr xc Þ; this parametric xc (1 a)(xc xl), u form and Eq. (10) are used for triangular fuzzy numbers. Criteria 1 and 3 are evaluated using bell-shape the pffiffiffiffiffiffiffiffiffiffiffiffi fuzzy numbers with pffiffiffiffiffiffiffiffiffiffiffiffi ðaÞ ¼ xc þ 1 a parametric form uðaÞ ¼ xc 1 aðxc xl Þ; u ðxr xc Þ; this parametric form and Eq. (10) are used for criteria 1 and 3. Table 3 displays the calculated distances. Next, Eq. (11) is used to convert the calculated distances of performance measures of criterion 2, 5, 6, and 7 into the dimensionless unit between 0 and 1. Eq. (12) is used for criteria 1, 3, 4, and 8. Table 4 shows the resulted cijs. Next step includes multiplying fuzzy weights by calculated cijs, which is done using indicated weights in Table 2; 5 shows the results. Afterwards, Eq. (13) is utilized to calculate fuzzy distance
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Table 2 Decision matrix of the Lake Urmia watershed case study. Criteria
Weights
a1
a2
a3
a4
a5
c1 (Million $) c2 (MCM/Year) c3 (Years) c4 c5 c6 c7 c8
(0.19, 0.2, 0.23) (0.3, 0.3, 0.33) (0.14, 0.15, 0.16) (0.095, 0.1, 0.105) (0.057, 0.06, 0.063) (0.048, 0.05, 0.053) (0.085, 0.09, 0.095) (0.047, 0.05, 0.052)
(399, 420, 483) 600 (14.5, 15, 16) H L H M FL
(418, 440, 506) 600 (14.5, 15, 16.5) FH L FH M FL
(536, 564, 649) 608 (17.5, 18, 19.5) M FL M FH L
(635, 668, 768) 613 (17.5, 18, 19.5) M FL M FH L
(638, 719, 826) 634 (17.5, 18, 19.5) M M FL H VL
Table 3 Fuzzy distance between performance measures and fuzzy zero.
Table 6 Fuzzy distance between fuzzy zero and elements of Table 5.
Criteria
a1
a2
a3
a4
a5
Criteria
a1
a2
a3
a4
a5
c1 c2 c3 c4 c5 c6 c7 c8
868 1200 30.33 1.575 0.425 1.575 1 0.75
909.33 1200 30.66 1.25 0.425 1.25 1 0.75
1166 1216 36.66 1 0.75 1 1.25 0.425
1380.66 1226 36.66 1 0.75 1 1.25 0.425
1485.33 1268 36.66 1 1 0.75 1.575 0.15
c1 c2 c3 c4 c5 c6 c7 c8
0.0843 0.1710 0.0451 0.0079 0.0013 0.0051 0.0065 0.0001
0.0763 0.1710 0.0443 0.0128 0.0013 0.0032 0.0065 0.0001
0.0462 0.1745 0.0312 0.0200 0.0041 0.0020 0.0101 0.0006
0.0335 0.1781 0.0312 0.0200 0.0041 0.0020 0.0101 0.0006
0.0283 0.1893 0.0312 0.0200 0.0072 0.0012 0.0162 0.0050
R1 = 1.54106, R2 = 1.6106, R3 = 1.7106, R5 = 1.71106 Ranking of alternatives: a5 > a3 > a4 > a2 > a1
Table 4 Dimensionless performance measures. Criteria
a1
a2
a3
a4
a5
c1 c2 c3 c4 c5 c6 c7 c8
1 0.95 1 0.63 0.425 1 0.63 0.2
0.95 0.95 0.99 0.8 0.425 0.79 0.63 0.2
0.74 0.96 0.83 1 0.75 0.63 0.79 0.35
0.63 0.97 0.83 1 0.75 0.63 0.79 0.35
0.58 1 0.83 1 1 0.48 1 1
R4 = 1.64106,
5. Discussion As it can be seen, in the case study ranking of alternatives is sensitive to the value of k. Among alternatives, a1 and a2 descend to a lower rank as the value of k increases. Table 6 shows that these two alternatives have much better performance than the other alternatives on criteria 1 and 3, and fairly better importance on criterion 6, but their performance on criteria 2, 4, 5, 7, and 8 is not as good as the other alternatives; as a consequence, the higher value of penalty coefficient diminishes their rank. On the other hand, a5 has quite different behavior from a1 and a2 so that it gains a higher rank corresponding to the higher value of penalty coefficient. a5 has the best performance on five criteria and the worst performance on the other three criteria. Overall, it seems that a3 is the best alternative due to its fairly good performance on all criteria as well as its appropriate rank for every value of penalty coefficient. It seems that a4 can be considered as the worst alternative.
between elements of Table 5 and fuzzy zero; Table 6 shows the resulted distances. In order to rank alternatives, Eq. (14) is utilized to calculate final performance of the alternatives. Ris are calculated using three values 1, 2, and 10 as k to consider the uncertainty in the value of penalty coefficient. Resulted values and their corresponding rankings are as follows: k = 1: R1 = 1.041, R2 = 1.049, R3 = 1.015, R4 = 0.987, R5 = 1.006 Ranking of alternatives: a2 > a1 > a3 > a5 > a4 k = 1/2: R1 = 4.54, R2 = 4.638, R3 = 4.663, R4 = 4.523, R5 = 4.654 Ranking of alternatives: a3 > a5 > a2 > a1 > a4 k = 1/10:
6. Conclusion The focus of this study is on taking uncertainty, present in a multicriteria decision making problem, into account as much as possible. In order to do that, a new process to determine weights of criteria is utilized so that weights are determined using fuzzy numbers and at the same time decision makers are compelled to
Table 5 Results of multiplying weights by dimensionless performance measures. Criteria
a1
a2
a3
a4
a5
c1 c2 c3 c4 c5 c6 c7 c8
(0.19, 0.2, 0.23) (0.285, 0.285, 0.314) (0.14, 0.15, 0.16) (0.06, 0.063, 0.066) (0.024, 0.026, 0.027) (0.048, 0.05, 0.053) (0.054, 0.057, 0.06) (0.009, 0.01, 0.01)
(0.181, 0.19, 0.219) (0.285, 0.285, 0.314) (0.139, 0.149, 0.158) (0.076, 0.08, 0.084) (0.024, 0.026, 0.027) (0.038, 0.04, 0.042) (0.054, 0.057, 0.06) (0.009, 0.01, 0.01)
(0.141, 0.148, 0.17) (0.288, 0.288, 0.317) (0.116, 0.125, 0.133) (0.095, 0.1, 0.105) (0.043, 0.045, 0.047) (0.03,0.032,0.033) (0.067, 0.071, 0.075) (0.016, 0.018, 0.018)
(0.12, 0.126, 0.145) (0.291, 0.291, 0.32) (0.116, 0.125, 0.133) (0.095, 0.1, 0.105) (0.043, 0.045, 0.047) (0.03, 0.032, 0.033) (0.067, 0.071, 0.075) (0.016, 0.018, 0.018)
(0.11, 0.116, 0.133) (0.3, 0.3, 0.33) (0.116, 0.125, 0.133) (0.095, 0.1, 0.105) (0.057, 0.06, 0.063) (0.023, 0.024, 0.025) (0.085, 0.09, 0.095) (0.047, 0.05, 0.052)
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consider the competition between criteria to obtain greater weights. In this way, a decision maker cannot indicate very high weights for all criteria because a high value as the weight of a criterion diminishes the weight of other criteria. In addition to the process, determining quantitative performance measures is done using seven linguistic variables and their corresponding fuzzy numbers. For quantitative performance measures, this study does not restrict the shapes of fuzzy numbers and proposes using appropriate shapes of fuzzy numbers with regard to particular conditions of each problem. In this paper, also a new fuzzy methodology is proposed to solve a multicriteria decision making problem defined in fuzzy environment with different shapes of fuzzy numbers. The proposed method utilizes parametric form of fuzzy numbers and metric sign distance to deal with such a problem as it is demonstrated in the case study of diversion of water into Lake Urmia watershed. The proposed method can be utilized to solve multicriteria decision making problems defined using diverse shapes of fuzzy numbers in many fields. References Abbasbandy, S., & Asady, B. (2006). Ranking of fuzzy numbers by sign distance. Information Sciences, 176, 2405–2416. Abrishamchi, A., Ebrahimian, A., Tajrishi, M., & Marino, M. A. (2005). Case study: Application of multicriteria decision making to urban water supply. Journal of Water Resource Planning and Management, 131(4), 326–335. Abutaleb, M. F., & Mareschal, B. (1995). Water resource planning in the middle east: Application of the PROMETHEE V multicriteria method. European Journal of Operations Research, 81(3), 500–511. Akter, T., & Simonovic, S. P. (2005). Aggregation of fuzzy views of a large number of stakeholders for multi-objective flood management decision making. Journal of Environmental Management, 77, 133–143. Bellman, R. E., & Zadeh, L. A. (1970). Decision-making in a fuzzy environment. Management Science, 17, 141–164. Bender, M. J., & Simonovic, S. P. (2000). A fuzzy compromise approach to water resource systems planning under uncertainty. Fuzzy Sets and Systems, 115, 35–44.
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