named as FDSS (Fuzzy Decision Support System). FDSS is applied to ... Both the methods suggested the same irrigation sub system as the best. It is concluded.
6th International Conference on Hydroinformatics - Liong, Phoon & Babovic (eds) © 2004 World Scientific Publishing Company, ISBN 981-238-787-0
FUZZY DECISION SUPPORT SYSTEM FOR MULTICRITERION ANALYSIS K. SRINIVASA RAJU Civil Engineering Department, Birla Institute of Technology and Science Pilani 333 031, Rajasthan, India D. NAGESH KUMAR Civil Engineering Department, Indian Institute of Science Bangalore 560 012, Karnataka, India Multicriterion Decision Making (MCDM) has emerged as an effective methodology as it can integrate quantitative and qualitative criteria for selection of the best alternative. On the other hand, fuzzy logic is gaining importance due to its flexibility in handling imprecise subjective data obtained due to uncertain environment. In the present study two fuzzy logic based MCDM methods are adopted. The two selected methods are implemented in Visual Basic environment to develop a decision support system and named as FDSS (Fuzzy Decision Support System). FDSS is applied to a case study of Sri Ram Sagar Project (SRSP), India for selecting best performing irrigation sub system. Both the methods suggested the same irrigation sub system as the best. It is concluded that integration of MCDM with fuzzy logic methodology for real world decision making is found to be effective. INTRODUCTION Multicriterion Decision Making (MCDM) has emerged as an effective methodology due to its ability to integrate quantitative and qualitative criteria for selection of the best of the available alternatives. On the other hand fuzzy logic is gaining importance due to its flexibility in handling imprecise subjective data obtained due to uncertain environment. In the present study fuzzy logic and MCDM are integrated and applied to a case study for selecting the best performing irrigation subsystem. Numerous studies on fuzzy logic are reported by various researchers for decision making analysis. Anand Raj and Nagesh Kumar [1,2] used fuzzy multi criterion decision making to select the best reservoir configuration for the Krishna river basin in India. They used maximizing and minimizing set concept. Yin et al. [3] employed fuzzy relation analysis for multi criteria water resources management for a case study of Great Lakes- St. Lawrence river basin, U.S.A. Raju and Nagesh Kumar [4] applied MCDM approach for selection of suitable irrigation planning strategy for a case study in Andhra Pradesh and employed PROMETHEE and EXPROM for ranking. Bender and Simonovic [5] applied fuzzy compromise programming to water resources systems planning under uncertainty and compared the same with ELECTRE. In the present study fuzzy MCDM methodology is applied to a case study of Sri Ram Sagar Project for selecting the best performing irrigation sub
2 system amongst four. Sri Ram Sagar Project (SRSP) is an irrigation project on Godavari River in Andhra Pradesh, India. This project has three canal systems, namely, Kakatiya, Saraswati and Lakshmi serving number of irrigation sub systems. Crops grown in the command area are Paddy, Jowar, Maize, Groundnut, Sugarcane and Pulses in both summer and winter seasons. There is extreme variation in temperature and relative humidity from one time period to another. FUZZY MULTICRITERION DECISION MAKING APPROACH Two fuzzy multicriterion decision making approaches viz., Similarity Measure (M1) and Vague Set Theory (M2) are adopted for this study. Similarity measure (M1) This methodology uses the concept of degree of similarity measure and the alternative with a higher degree of similarity with respect to a reference alternative is considered to be the best [6]. In this methodology, criteria are represented by interval valued fuzzy sets (real interval) as compared to crisp real values between zero and one. Characteristics of the alternative a [a=1,2,…A] for various criteria C1, C2, ……., Cj (with weightage of the criteria W=w1, w2, …..wj) are represented as interval-valued fuzzy sets as below. a = {(C1[ya1, y’a1]), (C2[ya2, y’a2], …….., Cj[yaj, y’aj])}
(1)
where [yaj, y’aj] represents fuzzy interval for ath alternative for jth criteria within the ranges of [0 ≤ yaj ≤ y’aj ≤ 1] with 1 ≤ a ≤ A. Here A and j represent the number of alternatives and criteria. Eq. (1) can also be represented in matrix notation as below: A= [ya1, y’a1], [ya2, y’a2], …….., [yaj, y’aj]
(2)
The objective is to choose such an alternative as the best, whose characteristics are most similar to the interval-valued fuzzy reference alternative set, R, which is expressed in the matrix notation as below. R= [x1, x’1 ], [x2, x’2], …….., [xj, x’j]
(3)
where [xj, x’j] represents fuzzy interval for reference alternative for Jth criteria. Similarity measure S (A,R,W) of alternative A with reference to R is given by
3 J
S ( A, R, W ) =
∑ [(1 − (| y j =1
aj
− x j | + | y ' aj − x j ' |) / 2) * w j ] (4)
J
∑w j =1
j
Higher degree of similarity measure is aimed at for selection of the best alternative. Vague set theory (M2) This methodology uses the true and false membership functions to indicate the degrees of satisfaction and dissatisfaction of each alternative with respect to a set of criteria. Degree of suitability of a given alternative is analysed using weighting function [7]. In this methodology, characteristics of the alternative a [a=1,2,…A] for various criteria C1, C2, ……., Cj are represented by vague set as shown below. a = {(C1[ta1, t*a1], (C2[ta2, t*a2], …….., Cj[taj, t*aj])}
(5)
where taj indicates the degree that a satisfies criteria Cj (true membership value) and faj indicates the degree that a does not satisfy the criteria Cj (false membership value) and t*aj = (1- faj ) such that t*aj + faj ≤ 1. If decision maker wants to choose an alternative which satisfies the criteria C1, C2, …, and Ci, or Cj ., this can be represented as C1, C2, …, and Ci, or Cj .
(6)
Weights of the criteria C1, C2, …, and Ci given by the decision maker are w1, w2, …, and wi and w1+ w2+…+ wi=1. Degree of suitability by which the alternative a, satisfies the decision maker’s requirement can be measured by the weighting function W(a), where W(a)= Max [S(ta1, t*a1) w1+ S(ta2, t*a2) w2+...+ S(tai, t*ai) wi+...+ S(taj, t*aj)] or W(a)=Max[(ta1+ t*a1-1)w1+ (ta2+ t*a2-1) w2+…+(tai+ t*ai-1) wi +...+(tai+ t*ai-1)
(7) (8)
where score function S (ta1, t*a1) is computed as (ta1+ t*a1-1). The alternative with a higher value of weighting function is considered to be the best. FORMULATION OF PAYOFF MATRIX In the present study four irrigation sub systems denoted as I1, I2, I3, I4 are considered. Each sub system is the irrigated area under a different distributory of the irrigation canal.
4 Farmers' response survey is conducted to understand the irrigation management characteristics and to identify performance indicators. Six performance criteria, namely, environmental impact (C1), conjunctive use of surface and ground water resources (C2), participation of farmers (C3), social impact (C4), productivity (C5) and economic impact (C6) are formulated and evaluated for selecting the best irrigation subsystem. Brief descriptions of the criteria are presented below. C1: Environmental impact issues analysed after introduction of irrigation facilities are rise in ground water table and salinity level. C2: Conjunctive use of surface and ground water is essential to provide more reliable supply of water to crops when needed as well as to reduce water logging. C3: Participation of farmers: Farmers knowledge of technology and new developments and participation are essential for optimum utilization of the available resources. It is the way in which farmers use irrigation water, that determines the success of an irrigation project. C4: Social impact includes labour employment, which is measured in terms of man days employed per hectare for each crop grown. C5: Productivity of various crops for various seasons for various land holdings are to be determined. C6: Economic impact includes farmer's income and revenue collected for supply of irrigation water. Information on these criteria is obtained from primary sources such as marketing societies, irrigation, ground water and agricultural departments. Additional information on some of the criteria is also obtained from secondary sources such as by interviews with farmers, discussions with officials of the project and from economics & statistics reports. These criteria are evaluated for each irrigation subsystem (termed as payoff matrix) on a fuzzy rating basis. Three experts who are monitoring the project are requested to fill up the payoff matrix with the evaluations ranging from 1 for excellent to 0 for unsatisfactory. Experts are given the flexibility to choose any intermediate evaluation between excellent and unsatisfactory. Table 1 presents payoff matrix corresponding to the four irrigation subsystems and the six performance criteria on a fuzzy rating basis for the three experts. RESULTS AND DISCUSSION Two fuzzy MCDM methods, viz., similarity measure (M1) and vague set theory (M2) are programmed in Visual Basic environment [8] in the form of Decision support system and named as FDSS (Fuzzy Decision Support System). Figures 1 and 2 present sample screens of similarity measure and vague set theory modules of FDSS respectively. The program is capable of handling any number of alternatives and criteria. Option is also given whether to consider the weights of the criteria or not. Output/Result option
5 executes the program and ranks will be displayed. View option displays ranks of alternatives in the form of bar chart. Flexibility is given to the user to change the value of any input at any time. Reset option is also provided to start a new problem. In both the modules common inputs are number of alternatives, criteria and payoff matrix. Provision for changing the values in the payoff matrix and weights is also incorporated in both the modules. Table 1. Payoff matrix on fuzzy rating basis given by individual experts Irrigation sub system
Expert
C1
C2
C3
C4
C5
C6
I1
1 2 3
0.2 0.2 0.2
0.4 0.2 0.2
1.0 1.0 0.8
1.0 0.8 1.0
1.0 0.8 0.8
1.0 0.8 1.0
I2
1 2 3
0.4 0.6 0.4
0.2 0.0 0.2
0.8 0.6 0.6
0.8 0.8 0.6
0.8 0.8 1.0
0.6 0.4 0.6
I3
1 2 3
0.4 0.4 0.4
0.2 0.0 0.0
0.4 0.0 0.6
0.8 0.6 0.6
0.6 1.0 0.6
0.8 1.0 0.8
I4
1 2 3
0.4 0.6 0.4
0.6 0.4 0.2
0.6 0.4 0.4
0.8 0.8 0.6
0.6 0.6 0.4
0.8 0.8 0.6
Similarity measure module (M1) Based on the evaluations given by the three experts for each criterion for each alternative (i.e., 3 values), the lowest and highest values are considered for the interval for that scenario. For example, for alternative 1 and criterion 2, three experts have given their fuzzy rating as 0.4, 0.2 and 0.2. Accordingly interval was given as [0.2, 0.4]. If all the experts give same rating such as 0.2, 0.2 and 0.2 then the interval will be [0.2, 0.2]. Table2 presents payoff matrix in the interval form. Weights of the criteria are taken as same. Reference alternative for each criterion is taken as (1,1). Module computes the degree of similarity between the given alternative and the reference alternative (using Eq.4). Higher degree of similarity of an alternative with respect to the reference alternative is considered to indicate the better alternative. Similarity measure for irrigation sub systems I1 to I4 are computed and found to be 0.6832, 0.5666, 0.5333 and
6 0.5499 respectively indicating that I1 is the best. Table 2 presents the degrees of similarity measure and corresponding ranking pattern of the four irrigation sub systems. Table 2. Payoff matrix in the fuzzy interval form Irrigation sub system I1 I2 I3 I4
C1
C2
C3
C4
C5
C6
[0.2, 0.2] [0.4, 0.6] [0.4, 0.4] [0.4, 0.6]
[0.2, 0.4] [0.0, 0.2] [0.0, 0.2] [0.2, 0.6]
[0.8, 1.0] [0.6, 0.8] [0.0, 0.6] [0.4, 0.6]
[0.8, 1.0] [0.6, 0.8] [0.6, 0.8] [0.6, 0.8]
[0.8, 1.0] [0.8, 1.0] [0.6, 1.0] [0.4, 0.6]
[0.8, 1.0] [0.4, 0.6] [0.8, 1.0] [0.6, 0.8]
Figure 1. Sample screen of similarity measure module
Degree of similarity and rank 0.6832 (1)
Weight function and rank 0.8000 (1)
0.5666 (2)
0.1600 (3)
0.5333 (4)
0.8000 (1)
0.5499 (3)
0.4000 (2)
7 Vague set theory module (M2) This methodology requires two values as inputs i.e., taj, t*aj i.e., (1- faj,) for each criteria and for each alternative which relate to true and false membership functions respectively. In the present study these are taken to be the same as the fuzzy interval payoff matrix as shown in Table 2 based on discussions with experts. Weights of the criteria are assumed to be same. The alternative having the highest value of weight function is taken to be the best (Eq. 8). Remaining alternatives are ranked accordingly. It is observed that irrigation systems I1 and I3 are found to be the best with weight function values of 0.8000 followed by I4 in second position and I2 in third position. This study may be pursued further with additional criteria and such study will be taken in the future.
Figure 2. Sample screen of vague set theory module
8 CONCLUSIONS A decision support system, FDSS, is developed involving two fuzzy multi criteria decision making methods and applied to an existing irrigation system in India. From the case study and sensitivity analysis (effect of changing the weights of the criteria on the ranking pattern) irrigation sub system, I1, is found to be the best by both the fuzzy MCDM methods. It is also observed from sensitivity analysis that any change of parameters has visible impact on the outcome. However, the first position remained unchanged for the irrigation sub system, I1. It is observed that integration of fuzzy logic with real world irrigation planning problem is very much effective particularly with multiple experts and subjective data environment. FDSS is found to be useful due to its interactive nature, flexibility of approach, possibility of evolving graphical features and adoptability for any situation to rank conflicting alternatives. REFERENCES [1] Raj P.A. and Nagesh Kumar D., “Ranking multi-criterion river basin planning alternatives using fuzzy numbers”. Fuzzy Sets and Systems, Vol.100 (1998), pp 89-99. [2] Raj P.A. and Nagesh Kumar D., “Ranking alternatives with fuzzy weights using maximizing set and minimizing”. Fuzzy Sets and Systems, Vol. 105 (1999), pp 365-375. [3] Yin Y.Y., Huang G.H. and Hipel, K.W., “Fuzzy relation analysis for multicriteria water resources management”. Journal of Water Resource Planning and Management, ASCE. , Vol. 125 (1999), pp 41-47. [4] Raju K.S. and Nagesh Kumar., “Multicriterion decision making in irrigation development strategies”, Journal of Agricultural Systems, Vol. 62 (1999), pp 117-129. [5] Bender M.J. and Simonovic, S.P., “A Fuzzy Compromise Approach to Water Resource Systems Planning Under Uncertainty”, Fuzzy Sets and Systems, Vol. 115 (2000), pp 35-44. [6] Chen S.M., “A New Method for Handling Multicriteria Fuzzy Decision Making Problems”, Cybernetics and Systems, Vol. 25 (1994), pp 409-420. [7] Chen S.M. and Tan J.M., “Handling Multicriteria Fuzzy Decision-Making Problems Based on Vague Set Theory”, Fuzzy Sets and Systems, Vol. 67 (1994), pp 163-172. [8] Cornell G., “Visual Basic 6 from the ground up”, Tata McGraw-Hill, (2001).
