Applied Mathematical Modelling 36 (2012) 3271–3282

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Applied Mathematical Modelling journal homepage: www.elsevier.com/locate/apm

The practical research on flood risk analysis based on IIOSM and fuzzy a-cut technique Qiang Zou a, Jianzhong Zhou a,⇑, Chao Zhou b, Lixiang Song a, Jun Guo a, Yi Liu a a b

College of Hydropower and Information Engineering, Huazhong University of Science and Technology, Wuhan 430074, PR China Department of Control Science and Engineering, Huazhong University of Science and Technology, Wuhan 430074, PR China

a r t i c l e

i n f o

Article history: Received 18 April 2011 Received in revised form 30 September 2011 Accepted 11 October 2011 Available online 20 October 2011 Keywords: Flood disaster Risk analysis IIOSM Fuzzy risk Fuzzy a-cut Fuzzy expected values

a b s t r a c t Flood disasters are one of the most common and destructive natural hazards all over the world. In this paper, improved interior-outer-set model (IIOSM) based on information diffusion theory is introduced in detail to assess flood risk in an effort to obtain accurate analytical results that represent the actual situation. Then fuzzy a-cut technique is applied to calculate the fuzzy expected values under the possibility–probability distribution (PPD) calculated by IIOSM. Taking the value of a throughout the interval (0, 1], we correspondingly get access to the conservative risk value (RC) and venture risk value (RV). Selection of a, RC and RV is dependent on present technical conditions and risk preference of different people. To illustrate the procedure of IIOSM and fuzzy a-cut technique, we employ them respectively to analyze the flood risk in Sanshui District, located in the center of Guangdong province in China. The results, such as risk value estimations, as well as fuzzy expected values, i.e. RC and RV under the given a-cut level, can reflect the flood risk quite accurately. The outcomes of this research based on IIOSM and fuzzy a-cut technique offer new insights to carry out an efficient way for various flood protection strategies.  2011 Elsevier Inc. All rights reserved.

1. Introduction Flood disasters are among the most frequent and devastating types of disasters around the world [1]. Worldwide statistics indicate that continuously increasing flood damages and losses of human lives remain at high levels. Flooding poses a threat to millions of the citizens in Europe, and it remains the most widely distributed natural hazard leading to significant economic and social impacts [2]. Flooding is also a major problem for the United States. Coastal and riverine inundations affect 6–8% of the nation’s conterminous territories, about 300,000 km2 located in the 100-year floodplain, and more than 6.4 million structures are at risk [3]. The risk of flooding is a main concern in many areas nearly all around the world and especially in China. Up to two-thirds of the land in China are at different types and levels of risk from river and coastal flooding, which results from the combined effects of natural, social and economic factors. Floods occur quite frequently, there being a larger disastrous flood every 2–3 years in China since 1949. In addition, according to statistical data, the direct economic loss caused by floods is 77.94 billion yuan in 1991, 179.65 billion yuan in 1994, up to 228.04 billion in 1996, and as high as 255.09 billion yuan in 1998. Undoubtedly, it is one of the most threatening natural hazards for human society that results in enormous damages in the last 60 years [4,5]. We are supported by sound reasons that reliable flood risk assessment is important, because it is the basis upon which an optimal disaster mitigation and flood defenses strategy can be worked out. Uncertainty always exists and, therefore, risk is inevitable [6]. There is a growing acceptance that the risk associated is generally either given as the probability distribution of an unfavorable value for the measure of effectiveness or measured ⇑ Corresponding author. Tel.: +86 02 78 7543 127. E-mail address: [email protected] (J. Zhou). 0307-904X/$ - see front matter  2011 Elsevier Inc. All rights reserved. doi:10.1016/j.apm.2011.10.008

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by the variance of measure effectiveness [7]. For many years methods have been developed in the flood frequency analysis [8,9], flood routing simulation [10,11], flood control [12], flood disaster evaluation [13–15] and so on for flood risk assessment. A wide variety of methods have been used, and there is much potential for development of advanced measurement which can better define information of given small samples [16]. Traditionally, risk assessment using given samples collected from flood disasters is performed in two ways. One way is the parameter estimation approach, which assumes the probability distribution and calculates distributed parameters. But it is not easy to calculate the probability distribution function with few samples for the complexity of flood disaster system and the incompleteness of data. In this case, an alternative consists of relaxing the assumption that the probability distribution is completely known, and it is difficult to judge whether a hypothesis and shape of probability distribution is suitable, leading to interval estimates instead of point estimates [17,18]. The other way is histogram approach. When nobody knows the shape of probability, as normal or exponential, the histogram can guarantee explicability of a rough estimation [19], but the results are in a large degree of inaccuracy, especially when few samples are available [18]. Fortunately, information diffusion theory can offer more reliable estimation of a probability without any subjective hypothesis, as well as compensate for deficiencies in the conventional approach to perform risk analysis. Information diffusion theory was presented by Huang [20] based on fuzzy set theory [21,22] to process the small-samples (size n < 30), which could change a traditional sample-point into a fuzzy set to partly fill the gap caused by incomplete data, making it extract as much as possible underlying useful information to improve the accuracy of risk recognition. And the principle of information diffusion had been demonstrated by using the fuzziness of incomplete data, which is an application of the fuzzy set theory [20]. And soft histogram is used to improve the accuracy of conventional histogram with the idea of information diffusion [19] to estimate the probability distribution. And derived from the method of information distribution, Huang [23] first proposed the interior-outer-set model (IOSM) to represent the imprecision by possibility–probability distribution (PPD) for fuzzy risk analysis. Afterwards, considering the shortcoming of IOSM, improved interior-outer-set model (IIOSM) is proposed to smoothen the abrupt slopes in the PPD [24]. The methods based on information diffusion have been applied in many fields, such as ordering farming alternatives [24], surveying and mapping [25], grassland fire disaster [26], and fuzzy risk of landfall typhoon [27] and flood disasters [28], to resolve problems in nearly every walk of life. But the methods, such as soft histogram and IIOSM, had not yet been compared with others in one paper, and seldom employed in the flood risk analysis to calculate and compare the risk value estimations together. Furthermore, the exceeding probabilities do not relate to any confidence in the risk estimation, which may ignore the fuzziness and lead to inaccuracy in representing the natural disaster risk [29]. Therefore, fuzzy a-cut technique based on fuzzy set theory and fuzzy logic [21,22,30] is adopted to express the fuzzy imprecision and describe statistically the confidence in the accuracy of risk estimates. With different values of a, we could obtain the corresponding conservative risk value (RC) and venture risk value (RV). This is not only reveals the inaccuracy of the exceeding probability estimation, but also provides a way for the model to accommodate fuzzy information. Therefore, in the proposed way, i.e., with IIOSM and fuzzy a-cut technique, we can represent impression of the estimated probabilities with a small sample size and give better results to support flood risk management [27,28]. In this paper, information diffusion theory and fuzzy a-cut technique are applied together in flood risk analysis, focusing on providing information to help make better decisions in an uncertain world. And the paper is organized as follows. Firstly the introduction of ISOM and IIOSM are given in Section 2, respectively. Section 3 introduces risk value estimation based on IIOSM. And fuzzy expected value calculation based on fuzzy a-cut technique is introduced in Section 4. Then, the procedure of flood risk analysis based on IIOSM and fuzzy a-cut technique is presented in Section 5. And there is a practical application in Section 6. Finally, we conclude the paper in Section 7. 2. Improved interior-outer-set model 2.1. Interior–outer-set model IOSM is a hybrid model that consists of information distribution method and possibility inference. IOSM is capable of calculating a fuzzy probability distribution also called possibility–probability distribution to represent the fuzzy risk, and the outcome of this model has been demonstrated to be more accuracy than the traditional method [31]. In this section, IOSM based on the principle of information diffusion theory is introduced. Let X = {xiji = 1, 2, . . . , n} be a given sample, xi 2 R (set of real numbers), and U ¼ fuj jj ¼ 1; 2; . . . ; mg  R be a discrete universe of X, with a step length D, D  uj  uj1, j = 2, 3, . . . , m. n and m is the number of samples and intervals, respectively. xi and uj are called a sample point and a controlling point, respectively. And let Ij = [uj  D/2, uj + D/2) be the observation interval associated to uj, and uj be the center point of Ij. The intervals are selected so that all of the sample points xi only lie within a certain interval Ij. From the point of view of information distribution, the simplest non-trivial diffusion function among the information diffusion techniques is the linear distribution function, and a sample point xi can allocate its information with a value of qij to point uj, "xi 2 X, "uj 2 U [19]. Thus we distribute the information carried by xi to uj at gain qij by one-dimensional linearinformation-distribution, shown in Eq. (1):

Q. Zou et al. / Applied Mathematical Modelling 36 (2012) 3271–3282

 qij ¼

1  jxi  uj j=D; if jxi  uj j 6 D 0; else

xi 2 X;

uj 2 U;

3273

ð1Þ

where xi is the observation value, uj is a controlling point, and D is the step length of controlling points. qij is called an information gain. When a sample point is subjected to random disturbance, it may depart from the interval Ij, while a point outside may þ also enter the interval. The possibility of sample point xi leaving or joining the interval Ij is expressed as q ij and qij , respectively. For the purpose of computation, the interior set and outer set of Ij are defined as follows: (1) Xinj = X \ Ij is called an interior set of interval Ij. The elements of Xinj are called the interior points of Ij (2) Xoutj = XnXinj is called an outer set of interval Ij. The elements of Xoutj are called the outer points of Ij "xi 2 X, if "xi 2 Xinj, we say that it loses information, by gain at 1  qij, to other interval, we use q ij ¼ 1  qij to represent the loss; if "xi 2 Xoutj, we say that it gives information, by gain at qij to Ij, we use qþ ij ¼ qij to represent the added information. So for Ij, with information gain qij, these possibilities can be formulated as follows Eqs. (2) and (3):

qij ¼

qþij ¼



1  qij ; if xi 2 X inj ; 0;



ð2Þ

if xi 2 X outj ;

qij ; if xi 2 X outj ; 0;

ð3Þ

if xi 2 X inj ;

þ q ij is called the leaving possibility, and qij the joining possibility. The leaving possibility of an outer point is defined as 0 (for it has gone). The joining possibility of an interior point is defined as 0 (for it has been in the interval). n o  Let Q  be the list of membership degrees to the information diffusion distribution from sample points within the ¼ q n o j ij þ observation interval Ij and Q þ be the list of membership degrees to the information diffusion distribution from samj ¼ qij  ple points outside the observation interval Ij. Arrange Q  j in ascending order to obtain the leaving information " Q j ; and arþ þ range Q j in descending order to obtain the joining information # Q j , respectively. For the sample size n, the universe of discourse of probability is

 P ¼ fp0 ; p1 ; . . . ; pn1 ; pn g ¼

 1 n1 n 0; ; . . . ; ; : n n n

ð4Þ

    If Q  j  ¼ nj , we can use Eq. (5) to calculate a possibility–probability distribution (PPD), which is called the interior-outerset model (IOSM).

8 1stðsmallestÞ element of Q j ; > > > > > 2ndðsmallestÞ element of Q j ; > > > > > >  > > > > > LastðlargestÞ element of Q j ; > > < pIj ðpÞ ¼ 1; > > > > 1stðlargestÞ element of Q þj ; > > > > > 2ndðlargestÞ element of Q þ ; > > j > > > >    > > > : LastðsmallestÞ element of Q þj ;

if p ¼ 0; if p ¼ 1n ; if p ¼ if p ¼ if p ¼ if p ¼

nj 1 ; n nj ; n nj þ1 ; n nj þ2 ; n

ð5Þ

if p ¼ 1;

Q j

 where the 1st smallest element of is the first element of " Q  j , the 2nd smallest element of Q j is the second element of  þ þ " Q j , and so on; the 1st largest element of Q j is the first element of # Q j , the 2nd largest element of Q þ j is the second element of # Q þ j , and so on. Therefore, for the sample X, we can obtain a PPD, pIj ðpi Þ; j ¼ 1; 2; . . . ; m; pi 2 P, which is a fuzzy relation on the Cartesian product space {I1, I2, . . . , Im}  {p0, p1, . . . , pn}, written as:

ð6Þ

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2.2. Improved interior-outer-set model The imperfection of the original interior-outer-set model lies in the fact that the membership of this model is confined to be within [0, 0.5] [ {1}, for using a midpoint of interior interval to construct information distribution for fuzzy risk analysis is far from enough. In this case, more points associated to the interval are needed. And it is reasonable for us to construct information distributions centered at these end points so that we can enlarge the range of possibility to [0, 1]. Considering the shortcoming of IOSM mentioned above, IIOSM is proposed to smoothen the abrupt slopes in the PPD [24], by adopting Eqs. (7) and (8) to improve Eqs. (2) and (3) as follows,

qij ¼

qþij ¼



2  ð1  qij Þ; if xi 2 X inj ; 0;



ð7Þ

if xi 2 X outj ;

2  qij ; if xi 2 X outj ; 0;

ð8Þ

if xi 2 X inj ;

and when we adopt the same steps as IOSM, a new PPD will be obtained correspondingly. 3. Risk value estimation In a PPD, the number of intervals is m, while the number of probabilities for each interval is n + 1. We may consider that for a interval Ij, there are n + 1 probabilities pIj ðpi Þ ði ¼ 0; 1; . . . ; nÞ for it. So in order to obtain as much information as possible for risk management, we calculate the weights and expectation of probabilities in the interval Ij (j = 1, 2, . . . , m) by using Eq. (9) as follows:

E½uj  ¼

n X

xi pi ;

ð9Þ

i¼0

where the weight xi (i = 0, 1, . . . , n) is calculated by Eq. (10) as follows [32]:

1 06r6n 0