Abstract
Fuzzy Inf. Eng. (2011) 2: 169-182 DOI 10.1007/s12543-011-0075-8 ORIGINAL ARTICLE Scheme Choice for Optimal Allocation of Water Resources Based on Fuzzy Language Evaluation and the Generalized Induced Ordered Weighted Averaging Operator Fei Ding· Takao Yamashita· Han Soo Lee · Jian-hua Ping Received: 31 October 2010/ Revised: 20 April 2011/ Accepted: 15 May 2011/ © Springer-Verlag Berlin Heidelberg and Fuzzy Information and Engineering Branch of the Operations Research Society of China Abstract The choice of scheme for the optimal allocation of water resource (OAW R) is a fuzzy multiple-attribution decision that is determined using information from many ﬁgures and fuzzy language regarding several evaluated factors, such as invest- ment, daily water supplying, fee of contaminated water disposal, water conservation, and the development of economy. In this paper, the evaluation system employed to choose an OAWR scheme is es- tablished based on the evaluation of fuzzy language and the generalized induced or- dered weighted averaging (GIOWA) operator. Considering economic aspects and a sustainable water supply, the ﬁve following constituents are chosen: 1) Invest- ment (Yuan), 2) Daily water supply (ton/day), 3) Fee of contaminated water disposal (Yuan), 4) Water conservation (fuzzy language), and 5) Development of economy (fuzzy language). The analytic hierarchy process (AHP) method is used to determine the weighting vector. A case study on the choice of OAWR in the northern area of Shenyang city, China was conducted by a multiple-attribution decision based on the GIOWA operator. The results shows that the system employed was able to choose the best scheme of OAWR in which fuzzy and multiple-attribution decision-making should be performed. Keywords Optimal allocation of water resource· Fuzzy language evaluation· GIOWA operator· Fuzzy multiple-attribution decision making 1. Introduction Fei Ding () · Takao Yamashita · Han Soo Lee Graduate School for International Development and Cooperation, Hiroshima University, Hiroshima, Japan email: dingfei98@sina.com Jian-hua Ping College of Water Conservancy and Environmental Engineering, Zhengzhou University, Zhengzhou, Henan, P.R.China 170 Fei Ding · Takao Yamashita· Han Soo Lee · Jian-hua Ping (2011) As the population increases and the economy develops rapidly in China, the predatory exploitation and over-use of surface and sub-surface water resources have led to a drastic imbalance between the supply and demand of water resources. There are many problems regarding water resources and the environment in the northern area of Shenyang city, China (Figure 1). For example, cross-border and cross-year water issues regarding the uneven quantity of river water have been the subject of large disputes between the upstream and downstream communities in the watershed of the Liao River, the biggest river in Shenyang city, where the dry season lasts nearly ﬁve months. Another example is the problem of excess concentrations of ferrous and man- ganese ion in the groundwater supply system. If the concentrations of ferrous and manganese ion had not been taken into account, the groundwater quality in Shenyang city from May to September, 2008 would have been considered good in 7.2% of total area, better in 76.6%, and bad in 16.2% of the total area using the comprehensive methods of water quality assessment in which an index ≤ 2.50 indicates good, and index between 2.50 and 4.25 is better, and an index between 4.25 and 7.20 is bad (Water Resource Research Report of Shenyang, China, 2008). A third example is the over-exploitation of groundwater by industry and agricul- ture, which are in a heated competition for water access. Shenyang city must pay more than 300,000Yuan/year for the compensation of the damage caused by agri- culture. Moreover, the over-exploitation of groundwater may also negatively impact water quality due to immersion and salty soil, which may be derived from the rise in the groundwater level related to the water storage in the Shi Fosi Reservoir. Considering the situation of overall water resources in this area, there is an urgent need for a method to determine the optimal allocation of water resources (OAWR). As the choice of scheme for the OAWR is a fuzzy multi-attribution decision, the weighted impact of each attribute should be considered. In the conventional method for decision-making, the maximum and the minimum arithmetic operators are typi- cally used; however, these often neglect important fuzzy language information. There- fore, further research on scientiﬁc evaluation systems must be conducted. Most multi-criteria decision analysis problems require neither strict ‘anding’ (min- imum) nor strict ‘oring’ of the s-norm (maximum). For mutually exclusive and inde- pendent probabilities in fault tree analysis, there are two extremes corresponding to multiplication (and-gate) and summation (or-gate). To generalizing the idea, Yager [1] introduced a new family of aggregation tech- niques collectively called the ordered weighted average (OWA) operator, which con- sist of general mean-type aggregators. The OWA operator could provide the ﬂexi- bility in utilizing the range of “anding” or “oring” and capture the attitude of a deci- sion maker in the aggregation process. In ordered weighted averaging, input criteria (sub-indices) are typically assumed to be equally important, and the OWA weight is assigned based on the ordinal position. To deal with criteria of varying importance in the aggregation process, Tora [2] introduced the concept of weighted OWA (WOWA) operators, which initially assign signiﬁcance weights to the input values; then, OWA aggregation is performed in a regular manner. Fuzzy Inf. Eng. (2011) 2: 169-182 171 Filev and Yager [3] introduced the induced OWA (IOWA) operator, which, unlike the OWA operator, allows ordering through an inducing parameter that is associated with the input values. Fig. 1 Map of Shenyang city and the north area of Shenyang city, China The utility of the inducing parameter is related only to ordering and not the aggre- gation process. Thus, Xu and Da [4] proposed a generalized IOWA operator. Now, the GIOWA operator has been extensively used in many ﬁelds, such as selecting salt cavity gas storage sites [5], evaluating environment quality [6], measuring the effec- tiveness of evaluating vehicle maintenance equipment [7] and location decision of logistics center [8]. Since undergoing the aforementioned developmental processes, the GIOWA oper- ator has been extensively used in many ﬁelds. However, it has not yet been applied to research on the choice of scheme for the OAWR. In this paper, the evaluation of scheme selection for the optimal allocation of water resources is established based on fuzzy language evaluation and the GIOWA operator. Then, a case study of north- ern Shenyang city is presented, the results of which demonstrate that the evaluation system is accurate. This paper consists of four sections. In Section 2, the evaluation of scheme se- lection for the OAWR and the steps of the evaluation system are described brieﬂy. In Section 3, a case study is presented in which the choice scheme for the OAWR is applied in Shenyang, China. Section 4 includes discussions and conclusions of the evaluation system . 2. Evaluation of Scheme Choice for OAWR and Steps of the Evaluation System The details of OWA operator [1,3,9], GIOWA operator [10] and Group multiple- attribute decision-making based on GIOWA [4,10] are described in the appendix. 2.1. Evaluation System of Scheme Choice for OAWR 172 Fei Ding · Takao Yamashita· Han Soo Lee · Jian-hua Ping (2011) The evaluation system of scheme choice for the OAWR was established based on the fuzzy linguistic evaluation and GIOWA operator. The following two domains are assumed to exist: G = [G ,··· , G ], (1) 1 m S = [S ,··· , S ] (2) 1 m and G delegates “the multiple factors gathering” for syntheses evaluation, and S del- egates “the comment gathering”. The syntheses evaluation mathematic model is as follows: B× A = C, (3) where A is an n × n modules fuzzy matrix, B-the fuzzy vector for scheme G, the gathering of weights of every evaluation factor, C-evaluation result, which is one of the fuzzy subclasses (fuzzy vector) in the domain S. In fact, it is rather easy to evaluate a single-factor entity. Moreover, if there are multiple factors in an entity, we can obtain an evaluation result from each factor, and thus, many real evaluation results can be obtained after synthesizing all factors. 2.1.1. The Decision of the Evaluating Factors and Weights In this paper, the selection principles of the choice of scheme for the OAWR concern economics and a sustainable water supply. For the economic aspect, these principles include investment, contaminative wa- ter disposal fees, and the development of economy. For the sustainable water supply aspect, these principles include daily water supply and water conservation. The syn- theses evaluating factors can be gathered as follows: G=[investment, daily water supplying, fee of contaminative water disposal, water conservation, development of economy]. Assuming that the weights are determined by some methods, (such as the analytic hierarchy process method or the arithmetic average method), which are used in this paper and the weights are unitary. 2.1.2. The Decision of the Comments Gathering In selecting of scheme choice for the OAWR, the comment domain which has nine comments in one factor G will be obtained, where S = [best, better, good, less good, generic, less bad, bad, worse, worst] is called the fuzzy linguistic scale [4,10]. The triangular fuzzy numbers corresponding to the above scale are as follows: best = [0.8,0.9,1.0], better = [0.7,0.8,0.9], good = [0.6,0.7,0.8], less good = [0.5,0.6,0.7], generic = [0.4,0.5,0.6], less bad = [0.3,0.4,0.5], bad = [0.2,0.3,0.4], worse = [0.1,0.2, 0.3], worst = [0.0,0.1,0.2]; where best better good less good generic less bad bad worse worst. 2.1.3. One-factor Evaluation System Based on the domain S, a one-factor evaluation matrix can be created; then the can- didate scheme based on fuzzy calculation can be evaluated to obtain the best scheme. Fuzzy Inf. Eng. (2011) 2: 169-182 173 ⎡ ⎤ ⎢ x x ··· x ⎥ 11 12 15 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ x x ··· x ⎢ 21 22 25⎥ ⎢ ⎥ C = [G ,··· , G ]× ⎢ ⎥ = [S ,··· , S ]. (4) 1 5 ⎢ ⎥ 1 5 ⎢ ⎥ ⎢ ⎥ ··· ··· ··· ··· ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ x x ··· x 51 52 55 2.2. The Steps of the Evaluation System The speciﬁcs steps of the evaluation system are as follows: Let X be the layout gathering, G the attribute gathering, and D the decision maker gathering. Step 1: The decision maker. d ∈ D gives the fuzzy linguistic evaluating information (k) r of the i-th evaluated object x in the attribute G . The evaluating matrix R is j j k ij obtained. Step 2: Use GIOWA operator g to assemble the i-rank fuzzy linguistic evaluating (k) information in the evaluating matrix R . The synthesizing evaluating value r (i ∈ N, k = 1, 2,··· , t), (k) (k) (k) (k) (k) (k) r = g[ r , u1,α ˆ ,··· , r , um,α ˆ ] = ω b , (5) i i1 i1 im im il (k) (k) (k) r ∈ S, G ∈ G,ˆα is the triangular fuzzy numbers corresponding to r , ω = ij ij ij (k) [ω ,··· ,ω ] is the associated weighting vector of g, ω ∈ [0, 1], b is the third 1 m l ij component of a three dimensional pair corresponding to the l-th largest element of (k) r ( j ∈ {1, 2,··· , m}). ij Determine the weighting vector ω. To obtain the weighting vector objectively, we may use the AHP method to determine the weighting vectorω = [ω ,··· ,ω ] . 1 m (k) Step 3: Use r (i ∈ N) to arrange an order and excellent choice regarding the scheme choice for the OAWR. If the triangular fuzzy numbers corresponding to the water resources’ optimal allocation of conceptual A and B are [a , a , a ] and [b , b , b ], if 1 2 3 1 2 3 [a , a , a ] [b , b , b ], then scheme A is better than scheme B. 1 2 3 1 2 3 3. A Case Study: Water Resources Allocation Problem in Shenyang For the scheme choice for the OAWR in the north of Shenyang city, according to the level of socio-economic development and water conservation ability to determine the six daily water-supplying schemes, see Table 1. The evaluation scheme in the northern area of Shenyang, see Figure 2. There are six evaluation schemes x (i = 1, 2,··· , 6) in the evaluation system and ﬁve attributes: investment (G ), daily water supplying (G ), fee of contaminative 1 2 water disposal (G ), water conservation (G ) and development of economy (G ). 3 4 5 174 Fei Ding · Takao Yamashita· Han Soo Lee · Jian-hua Ping (2011) Table1: The six daily water supplying schemes. Development of economy Water conservation Moderation Strengthen High Scheme V Scheme VI Moderation Scheme I Scheme II Low Scheme III Scheme IV Shi Foshi reservior Yiniu Liao river Six Shi Foshi Shi fo Seven Huang jia A B Scheme 1 the north of Xin Daily water Shen Yang chengzi supplying city 4 91.2x10 Five Ying jia ton/day Legend Pu river Nong water plant pipe line Shen Yang gao sewage river treatment city sewage treatment Hun river well drainage line Fig. 2 Evaluation scheme in the north area of Shenyang The round dotted line area is the north of Shenyang, the long dash dotted dot line area is Shenyang city. The solid line means that is consist now, the dash line means that is planning construct. From these schemes, the important attributes can be obtained, which are listed in Table 2 (based on the research project ‘Controlling Mode of Water Supply System of Shi Foshi Reservoir and its Risk Analysis’). 1) In this paper,we used the analytic hierarchy process (AHP) method to determine the weighting vectorω = [0.04, 0.14, 0.08, 0.37, 0.37] . 2) Now, there are three experts d (k = 1, 2, 3) to evaluate the six candidate schemes x , x , x , x , x , x , respectively through the factors gathering u one by one. The 1 2 3 4 5 6 l fuzzy evaluating matrix R can then be obtained; see Tables 3-5. k Fuzzy Inf. Eng. (2011) 2: 169-182 175 Table 2: Attributes of optimal allocation of water resources. G Investment Daily water Fee of contaminated Water Development (10 yuan/yr) supplying water disposal conservation of economy 4 8 x (10 ton/day) (10 yuan/yr) 1 9.6 91.02 0.83 moderation moderation 2 8.97 77.33 0.7 strengthen moderation 3 6.8 63.8 0.58 moderation low 4 6.01 54.15 0.49 strengthen low 5 13.22 134.2 1.24 moderation high 6 12.9 114.73 1.06 strengthen high Table 3: Evaluating matrices by d . G Investment Daily water Fee of contaminated Water Development of x supplying water disposal conservation economy 1 less good less good generic less good less good 2 good good less good bad less good 3 best better better less good bad 4 best better best bad bad 5 worse worse worst less good bad 6 bad bad worse bad bad Table 4: Evaluating matrices by d . G Investment Daily water Fee of contaminated Water Development of x supplying water disposal conservation economy 1 generic less good less bad good good 2 less good better generic bad good 3 better better better good bad 4 better better better bad bad 5 worst worse worst good bad 6 worse bad worse bad bad 176 Fei Ding · Takao Yamashita· Han Soo Lee · Jian-hua Ping (2011) Table 5: Evaluating matrices by d . G Investment Daily water Fee of contaminated Water Development of x supplying water disposal conservation economy 1 less good less good less good better better 2 good less good less good bad better 3 better good better better bad 4 better good better bad bad 5 worse bad worse better bad 6 worse bad worse bad bad By using the GIOWA operator, we can concentrate the fuzzy language evaluation information in the row of the evaluation matrix R and obtain the syntheses attribute (k) evaluation r (i = 1, 2, 3, 4, 5, 6, k = 1, 2, 3) of the decision-making layout x by expert d . At ﬁrst, we may calculate the syntheses attribute evaluation information of every (1) (1) (1) (1) layout provided by expert d ,as r = less good, r = less good, r = generic, r = 11 12 13 14 (1) (1) (1) (1) (1) (1) less good, r = less good. r ∼ r ∼ r ∼ r r . From the fuzzy language 15 11 12 14 15 13 criterion mentioned above, we know that the triangle fuzzy functions corresponding (1) to r ( j = 1, 2,··· , 5) are: 1 j (k) (k) (k) (k) (k) (k) r = g[ r , u ,α ˆ ,··· , r , u ,α ˆ ] = ω b , 1 m l i i1 i1 im im il (1) (1) (1) (1) (1) α ˆ = α ˆ = α ˆ = α ˆ = [0.5, 0.6, 0.7], α ˆ = [0.4, 0.5, 0.6]. 11 12 14 15 13 So, (1) (1) (1) (1) (1) (1) (1) (1) ˆ ˆ ˆ ˆ b = b = b = b = α ˆ = α ˆ = α ˆ = α ˆ = [0.5, 0.6, 0.7], 11 12 14 15 11 12 14 15 (1) (1) b = α ˆ = [0.4, 0.5, 0.6]. 13 13 We can also obtain the following expression through the GIOWA arithmetic operator g and the calculation principle of the triangle fuzzy number: (1) (1) (1) (1) (1) (1) (1) r = g[ r , u ,α ˆ , r , u ,α ˆ , r , u ,α ˆ , 1 2 3 1 11 11 12 12 13 13 (1) (1) (1) (1) r , u ,α ˆ , r , u ,α ˆ ] 4 5 14 14 15 15 (1) = ω b 1l ⎡ ⎤ ⎢ 0.50.60.7⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ 0.50.60.7 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ = 0.04 0.14 0.08 0.37 0.37 × 0.50.60.7 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ 0.50.60.7 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ 0.40.50.6 = [0.5, 0.6, 0.7]. Analogously, we can obtain: Fuzzy Inf. Eng. (2011) 2: 169-182 177 (1) (1) (1) (1) (1) (1) (1) r = g[ r , u ,α ˆ , r , u ,α ˆ , r , u ,α ˆ , 1 2 3 2 21 21 22 22 23 23 (1) (1) (1) (1) r , u ,α ˆ , r , u ,α ˆ ] 4 5 24 24 25 25 (1) = ω b 2l ⎡ ⎤ ⎢ 0.60.70.8⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ 0.60.70.8⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ = 0.04 0.14 0.08 0.37 0.37 × 0.50.60.7 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ 0.50.60.7 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ 0.20.30.4 = [0.4, 0.5, 0.6], (1) (1) (1) (1) (1) (1) (1) r = g[ r , u ,α ˆ , r , u ,α ˆ , r , u ,α ˆ , 1 2 3 3 31 31 32 32 33 33 (1) (1) (1) (1) r , u ,α ˆ , r , u ,α ˆ ] 4 5 34 34 35 35 (1) = ω b 3l ⎡ ⎤ ⎢ 0.80.91.0⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ 0.70.80.9 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ = 0.04 0.14 0.08 0.37 0.37 × 0.70.80.9 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ 0.50.60.7 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ 0.20.30.4 = [0.4, 0.5, 0.6], (1) (1) (1) (1) (1) (1) (1) r = g[ r , u ,α ˆ , r , u ,α ˆ , r , u ,α ˆ , 1 2 3 4 41 41 42 42 43 43 (1) (1) (1) (1) r , u ,α ˆ , r , u ,α ˆ ] 4 5 44 44 45 45 (1) = ω b 4l ⎡ ⎤ ⎢ 0.80.91.0⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ 0.80.91.0 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ = 0.04 0.14 0.08 0.37 0.37 × 0.70.80.9 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ 0.20.30.4 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ 0.20.30.4 = [0.4, 0.5, 0.6], (1) (1) (1) (1) (1) (1) (1) r = g[ r , u ,α ˆ , r , u ,α ˆ , r , u ,α ˆ , 1 2 3 5 51 51 52 52 53 53 (1) (1) (1) (1) r , u ,α ˆ , r , u ,α ˆ ] 4 5 54 54 55 55 (1) = ω b 5l ⎡ ⎤ ⎢ 0.50.60.7⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ 0.20.30.4 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ = 0.04 0.14 0.08 0.37 0.37 × 0.10.20.3 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ 0.10.20.3 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ 00.10.2 = [0.1, 0.2, 0.3], 178 Fei Ding · Takao Yamashita· Han Soo Lee · Jian-hua Ping (2011) (1) (1) (1) (1) (1) (1) (1) r = g[ r , u ,α ˆ , r , u ,α ˆ , r , u ,α ˆ , 1 2 3 6 61 61 62 62 63 63 (1) (1) (1) (1) r , u ,α ˆ , r , u ,α ˆ ] 4 5 64 64 65 65 (1) = ω b 6l ⎡ ⎤ ⎢ 0.20.30.4⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ 0.20.30.4 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ = 0.04 0.14 0.08 0.37 0.37 × 0.20.30.4 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ 0.20.30.4 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ 0.10.20.3 = [0.2, 0.3, 0.4]. In the same way we can obtain the syntheses evaluation information of d and d . 2 3 (2) (2) (2) r = [0.4, 0.5, 0.6], r = [0.4, 0.5, 0.6], r = [0.4, 0.5, 0.6], 1 2 3 (2) (2) (2) r = [0.3, 0.4, 0.5], r = [0.1, 0.2, 0.3], r = [0.2, 0.3, 0.4], 4 5 6 (3) (3) (3) r = [0.5, 0.6, 0.7], r = [0.4, 0.5, 0.6], r = [0.5, 0.6, 0.7], 1 2 3 (3) (3) (3) r = [0.3, 0.4, 0.5], r = [0.1, 0.2, 0.3], r = [0.1, 0.2, 0.3]. 4 5 6 (k) 3) Finally, we can concentrate on the syntheses attribute evaluation r (k = 1, 2, 3) of the layout x provided by the three experts through the GIOWA arithmetic oper- ator g.If ω = [0.3, 0.4, 0.3] , then we can obtain the colony syntheses attribute evaluation r (i = 1, 2, 3, 4, 5, 6) of decision-making layout x : i i (1) (1) (2) (2) (3) (3) r = g [ r , d ,α ˆ , r , d ,α ˆ , r , d ,α ˆ ] 1 1 2 3 1 1 1 1 1 1 ⎡ ⎤ ⎢ 0.60.70.8⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ (1) ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ = ω b = 0.30.40.3 × 0.50.60.7 = [0.5, 0.6, 0.7], 2 ⎢ ⎥ 1l ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ 0.40.50.6 (1) (1) (2) (2) (3) (3) = g [ r , d ,α ˆ , r , d ,α ˆ , r , d ,α ˆ ] 2 1 2 3 2 2 2 2 2 2 ⎡ ⎤ ⎢ ⎥ 0.40.50.6 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ (1) ⎢ ⎥ ⎢ ⎥ = ω b = 0.30.40.3 × 0.40.50.6 = [0.4, 0.5, 0.6], 2 ⎢ ⎥ 2l ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ 0.40.50.6 (1) (1) (2) (2) (3) (3) r = g [ r , d ,α ˆ , r , d ,α ˆ , r , d ,α ˆ ] 3 1 2 3 3 3 3 3 3 3 ⎡ ⎤ ⎢ 0.50.60.7⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ (1) ˆ ⎢ ⎥ ⎢ ⎥ = ω b = 0.30.40.3 × 0.40.50.6 = [0.4, 0.5, 0.6], 2 ⎢ ⎥ 3l ⎢ ⎥ ⎣ ⎦ 0.40.50.6 (1) (1) (2) (2) (3) (3) r = g [ r , d ,α ˆ , r , d ,α ˆ , r , d ,α ˆ ] 4 1 2 3 4 4 4 4 4 4 ⎡ ⎤ ⎢ ⎥ 0.40.50.6 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ (1) ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ = ω b = 0.30.40.3 × 0.30.40.5 = [0.3, 0.4, 0.5], 2 ⎥ 4l ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ 0.30.40.5 (1) (1) (2) (2) (3) (3) r = g [ r , d ,α ˆ , r , d ,α ˆ , r , d ,α ˆ ] 5 1 2 3 5 5 5 5 5 5 ⎡ ⎤ ⎢ ⎥ 0.10.20.3 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ (1) ⎢ ⎥ = ω b = × ⎢ ⎥ 0.30.40.3 0.10.20.3 = [0.1, 0.2, 0.3], 2 ⎢ ⎥ 5l ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ 0.10.20.3 (1) (1) (2) (2) (3) (3) r = g [ r , d ,α ˆ , r , d ,α ˆ , r , d ,α ˆ ] 6 1 2 3 6 6 6 6 6 6 Fuzzy Inf. Eng. (2011) 2: 169-182 179 ⎡ ⎤ ⎢ 0.20.30.4⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ (1) ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ = ω b = 0.30.40.3 × 0.20.30.4 = [0.2, 0.3, 0.4]. 2 ⎢ ⎥ 6l ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ 0.10.20.3 4) Ordering the candidate sites through r (i = 1, 2, 3, 4, 5, 6), r r ≈ r r r r . 1 2 3 4 6 5 Thus, we obtain x x ≈ x x x x . 1 2 3 4 6 5 Therefore, the best scheme is x . From Table 2, it is obvious that the investments associated with Schemes 3 and 4 are much greater than that of Scheme 1. However, the development of economy under Schemes 3 and 4 are so low as to be unacceptable for the economic situation in Shenyang. On the other hand, the economic development under Schemes 3 and 4 in- dicate quite quick progress. This may cause a crisis of the scheme from an economic point of view. As the investment of Scheme 1 is not so high, the development of economy under this scheme will be moderate, which is considered to be agreeable as the sustainable choice. From the comprehensive judgment of the results estimated by the fuzzy language evaluation and GIOWA operator, it can be concluded that Scheme 1 is acceptable as the best choice for the OAWR in Shenyang. For Scheme 1, the total investment is 9.6 hundred million yuan and the total fee of contaminated water disposal is 0.83 hundred million yuan. These are not particularly high. Moreover, the daily water supply is 91.02 ten thousand ton/day, which is enough to meet the daily water demand in Shenyang for the next 20 years. Water conservation and the development of economy are moderate under Scheme 1. Thus, we can come to the conclusion that Scheme 1 is acceptable as the most sustainable choice. 4. Discussions and Conclusions Group multi-attribution decision-making problems have been widespread in practical use. Choosing a scheme for the OAWR is one of them. In this paper, using fuzzy language evaluation and the GIOWA operator, an evaluation system for choosing a scheme for the OAWR was established. It consists of the ﬁve constituents: invest- ment (Yuan), daily water supplying (ton/day), contaminated water disposal (Yuan) fee, water conservation (fuzzy language), and the development of economy (fuzzy language). Moreover, the analytic hierarchy process (AHP) method was used to de- termine the weighting vector. A case study was conducted to verify the applicability of the evaluation system. The result showed that the system used to evaluate the choice of scheme for the OAWR was feasible and useful. In this research, alternate water supply sources and the reliability of the water sup- ply were not considered. It is better to add a factor of reliability concerning water supply in the syntheses evaluation factor gathering. It is also better to consider the total investment in alternate water supply sources. Finally, it should be noted that 180 Fei Ding · Takao Yamashita· Han Soo Lee · Jian-hua Ping (2011) the method used in this study still requires improvement in obtaining a better weight- ing vector and a reasonable weighting vector. Improving these parameters are very important and will be the focus of study in the future. Acknowledgments I am very grateful to Prof. Jun Pan, the supervisor of my master degree study in Shenyang Jianzhu University, for his valuable suggestions and comments on this re- search that were helpful to improve the quality of research. References 1. Yager R R (1988) On ordered weighted averaging aggregation operators in multi-criteria decision making. IEEE Transactions on Systems, Man, and Cybernetics 18: 183-190 2. Torra V (1997) The weighted OWA operator. International Journal of Intelligent Systems 12: 153-166 3. Filev D, Yager R R (1998) On the issue of obtaining OWA operator weights. Fuzzy Sets and Systems 94: 157-169 4. 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Bordogna G, Fedrizzi M, Pasi G (1997) A linguistic modeling of consensus in group decision making based on OWA operators. IEEE Transactions on Systems, Man, and Cybernetics-Part A, 27: 126-132 10. Xu Z S (2002) A method based on fuzzy linguistic assessments and linguistic ordered weighted averaging (OWA) operator for multi-attribute group decision-making problems. Systems Engineering 20(5): 79-82 11. Van Laarhoven P J M, Pedrycz W (1983) A fuzzy extension of Saaty’s priority theory. Fuzzy Sets and Systems 11: 229-241 Fuzzy Inf. Eng. (2011) 2: 169-182 181 Appendix 1. GIOWA Operator L M U L M U Deﬁnition 1 Let a ˆ = [a , a , a ],where 0 < a < a < a . Then, a ˆ is called a triangular number and is identiﬁed by its characteristic function [11]: ⎪ (x− a ) L M , a ≤ x ≤ a , ⎪ M L ⎪ (a − a ) ⎨ U (x− a ) U = (6) a ˆ M U , a ≤ x ≤ a , M U (a − a ) 0 , others. For convenience, we give the following operational laws related to triangular fuzzy numbers. L M U L M U L L M M U U 1) a ˆ + b = [a , a , a ]+ [b , b , b ] = [a + b , a + b , a + b ], L M U 2)λa ˆ = [λa ,λa ,λa ], whereλ> 0. L M U L M U L L M M U U Deﬁnition 2 Let a ˆ = [a , a , a ], b = [b , b , b ]. If a < b , a < b , a < b . Then it is said to be a ˆ < b. Deﬁnition 3 F is called an induced ordered weighted averaging operator (IOWA), if F[u ,α ,··· ,u ,α ] = ω b , (7) 1 1 n n j j j=1 where ω = [ω ,··· ,ω ] is the associated weighting vector of F, ω ∈ [0, 1]( j ∈ 1 n j N), ω = 1,u,α is called an OWA pair, and b is the second component of the j i i j j=1 OWA pair having the j-th largest u (i ∈ 1, 2,··· , 6) value. u is called the inducing i i component andα the numerical component [1,3,9]. Deﬁnition 4 g is called an n-dimension generalized IOWA operator (GIOWA opera- tor) [10], if g[ξ ,π ,α ,··· ,ξ ,π ,α ] = ω b , (8) 1 1 1 n n n j j j=1 where vector ω = [ω ,··· ,ω ] is the associated weighting vector of g,ω ∈ 1 n j [0, 1]( j ∈ N), ω = 1, v, u,α ∈ Φ × Ψ × Θ is a three-dimensional pair; the j i i i j=1 ﬁrst component v shows an important level or characteristic of the second compo- nent u , that is, the main body of the third component α ; b is the third component i i i that corresponds to the j-th largest of v (i ∈ 1, 2,··· , n) value in the three-dimensional pair;Φ ,Ψ andΘ are a set of the ﬁrst, the second and the third component respectively in all three dimension pair. b can be obtained by the following method. All three-dimensional pairs are ordered according to the big or small of the ﬁrst component v (i ∈ 1, 2,··· , n). b is the third component of the j-th ordered three- i i 182 Fei Ding · Takao Yamashita· Han Soo Lee · Jian-hua Ping (2011) dimensional pair. The radical trait of GIOWA operator is that there is no relation between v, u,α i i i and ω , ω is only related to the i-th place in the assembled process. The element i i α (i ∈ 1, 2,··· , n) weighted assembled is not in accord with the big or small of oneself value and is based on v (i ∈ 1, 2,··· , n), a value that corresponds with in v, u,α . i i i i u is the generalized problem attribute. v is the important level and property of u , i i i such as weight, number, or achievement. u and v are expressed in words or as a i i number. α is a generalized attribute value or other quantity that indicates α .It is i i expressed with a number, for example, a real number, interval number, or triangular fuzzy number. In particular, if v = v , then α and α are averaged in the assembled process for i j i j three-dimensional pairs v, u,α and v , u ,α . Three-dimensional pairs v, u, i i i j j j i i α +α α +α i j i j and v , u , are obtained. The same method is used to solve the situation j j 2 2 in which the ﬁrst component in a three-dimensional pair of three or more than three is equal. 2. Group Multiple Attribute Decision-making Based on GIOWA The scheme choice of the optimal allocation of water resources is a group multiple at- tribute decision-making problem. Every schemes of allocation is regarded as an eval- uated object. Let X , U and D be an evaluated object set, an attribute set and an evalu- ating unit set, respectively. The evaluating unit d ∈ D gives the evaluated object x ∈ k i (k) X a fuzzy linguistic evaluating value r in the attribute u ∈ U. The evaluating ma- ij (k) (k) trix R = r (r ∈ S) is obtained, where S = [best, better, good, less good, generic, ij ij less bad, bad, worse, worst] is called the fuzzy linguistic scale [4,10]. The triangular fuzzy numbers corresponding to the above scale is as follows: best =[0.8,0.9,1], better =[0.7,0.8,0.9], good =[0.6,0.7,0.8], less good =[0.5,0.6,0.7], generic =[0.4,0.5,0.6], less bad =[0.3,0.4,0.5], bad =[0.2,0.3,0.4], worse =[0.1,0.2,0.3] worst =[0,0.1,0.2]; where bestbettergoodless goodgenericless badbadworseworst.
Journal
Fuzzy Information and Engineering
– Taylor & Francis
Published: Jun 1, 2011
Keywords: Optimal allocation of water resource; Fuzzy language evaluation; GIOWA operator; Fuzzy multiple-attribution decision making