برآورد عملکرد پل بر اساس یک مدل رگرسیون خطی فازی ماتریس محور

Determining a reliable bridge maintenance and rehabilitation strategy relies on accurate predictions of bridge conditions. Conventional regression cannot handle visual inspection results that are inherently non-crisp or linguistic. On the other hand, fuzzy regression provides an effective means for coping with such fuzzy data or linguistic variables. However, many of the existing fuzzy regression models require substantial computations due to complicated fuzzy arithmetic. This paper presents a multiple fuzzy linear regression using matrix algebra. The proposed model is capable of dealing with a mixture of fuzzy data and crisp data. Moreover, the approach is intuitive and easy to implement as compared to other related fuzzy regression models. A case study using bridge inspection data is presented to establish estimated fuzzy regression equations produced by the proposed approach and examine the factors contributing to overall bridge performance. The results demonstrate the capability of the approach, which can assist bridge managers to make better maintenance policies based on the future bridge conditions predicted by the model.

Bridges are chief elements in the transportation system. Maintenance of highway bridges plays an important role to assure the desirable service and adequate reliability of highway networks. A primary goal of a Bridge Management System (BMS) is to assist bridge managers in determining the best bridge maintenance, repair and rehabilitation strategy with respect to current or future bridge conditions. In essence, the government funds available for maintaining existing bridges are usually limited. With restricted funds, maximizing the effect of the investment on the improvement of serviceability and safety of the existing highway system is the major challenge for highway agencies. For example, between 2003 and 2007, the percentage of the nation's 599,893 bridges rated functionally obsolete or structurally deficient decreased slightly from 27.1% to 25.59%. To eradicate all bridge deficiencies, it will cost 9.4 billion US dollars a year for 20 years; however, long-term budget is compounded by the shortage of the Highway Trust Fund [1]. Bridges are composed of several components including decks, girders, piers, cap beams, bearings, joints, abutments and foundation, etc. Typically, highway bridges are exposed to increasing traffic volumes and detrimental environmental conditions and have been built some decades ago. As a result, many highway bridges have been deteriorated or damaged. Bridge deteriorations may reduce functional performance such as loss of comfort for the road user, reduce structural reliability and require higher maintenance expenditure. A structurally deficient bridge may be closed or restricted to light vehicles because of its deteriorated structural components. Typical bridge defects include cracking, surface distortion, disintegration, rebar corrosion, expansion joint damage, waterproof deterioration, bearing damage, and foundation erosion, etc [2]. A major cause for bridge deficiencies is inadequate and ineffective maintenance activities. In general, the bridge maintenance decision-making process comprises the steps of (1) assessing bridge condition, (2) forecasting bridge deterioration, (3) determining the most desirable maintenance strategy, (4) prioritizing maintenance actions, and (5) optimizing resource allocations [3] and [4]. Forecasting the future bridge conditions in advance is useful for taking required or urgent repair actions in order to avoid disastrous consequences. Accordingly, accurate predictions of future bridge conditions based on periodic bridge inspection results are essential to develop an optimal maintenance policy. To estimate a bridge's condition in future, approximately 60% of BMS depends on periodic bridge inspection results [5]. However, bridge inspection observations are innately imprecise or fuzzy because they are usually collected from the bridge inspector's visual and subjective assessments using linguistic descriptions such as “The condition of this pier is very good” or “The condition rating of this concrete deck is about 80”. On the other hand, data or variables affecting a bridge's condition including bridge age, traffic load, bridge geometry such as bridge span, and environmental condition (e.g., rainfall and temperature) are numerical or crisp. Consequently, information on current bridge condition is a mixture of crisp data and fuzzy data. Various techniques have been applied to predict bridge condition [6], [7], [8] and [9]. The Markov chain is one of the most widely used probabilistic state-of-the-art techniques for forecasting the performances of bridge infrastructures [10], [11] and [12]. However, the main problem of this approach is the assumption that the future state of infrastructure components depends only on the present condition, which may affect its prediction accuracy. State-of-the-art infrastructure assessment also applies regression analysis [13]. Regression analysis is primarily used to find the best-fitted mathematical model, so that a dependent or response variable can be predicted from independent or predictor variable(s). Descriptions and controls of the cause-and-effect relationship between variables are also major purposes that regression analysis serves. However, conventional regression techniques are suitable for dealing with non-fuzzy data. To analyze data containing ambiguity and imprecision, fuzzy set theory is an effective approach [14]. Recently, numerous studies have been conducted using artificial intelligence techniques such as Artificial Neural Network (ANN) methods [15], [16] and [17], Genetic Algorithms [18] and [19], and fuzzy techniques [20], [21], [22] and [23] to evaluate bridge conditions. Bridge conditions are usually difficult to precisely estimate because of uncertainties and vague information involving the process of inspection. Fuzzy regression analysis is suitable for coping with problems in which human experts rely on subjective judgment or rules-of-thumb. Tanaka et al. [24] proposed the first fuzzy linear regression analysis for crisp input and fuzz output data. Following their work, various developments of fuzzy regression techniques and applications have been accomplished [25], [26], [27], [28], [29], [30] and [31]. Many of the existing fuzzy regression models require a great deal of computations because of difficult fuzzy arithmetic. The regression model proposed by Tanaka et al. is quite popular and useful; however, this model is restricted to symmetric triangular fuzzy numbers. To overcome this limitation, Chang and Lee developed a fuzzy least-squares regression model [28]. However, in their model, the regression coefficients are derived from a nonlinear programming problem that requires considerable computations. This paper presents a matrix-driven multiple fuzzy linear regression model to overcome the difficulties arising from Chang and Lee's approach and other models that require complex fuzzy mathematics. The proposed approach can deal with asymmetric and symmetric triangular membership functions. Furthermore, by the use of matrix algebra, the model is simpler to follow and easier to apply. An illustration using this model for the estimation of overall bridge conditions is presented using actual bridge inspection data from Taiwan.

Bridge assessment results are usually qualitative; thus, ordinary regression cannot deal with such non-crisp data. This paper presents a matrix-driven multivariate fuzzy linear regression model which is capable of fitting a regression equation to fuzzy data represented by asymmetric and symmetric triangular fuzzy numbers, numerical data, and their combination. This approach also enables to adequately reflect the decision-makers' confidence in the collected data and in the established model by employing particular membership values. Moreover, this model is more intuitive and straightforward than other related models. The main purpose of employing the proposed model is to establish fitted regression equations for estimating future conditions and investigating the cause-and-effect relationship between variables. Major advantages to using the method for bridge administration are as follows: (1) Future bridge conditions estimated by the model are particularly useful for planning inspection, establishing rehabilitation, performing necessary repairs, and predicting the impact of undertaking maintenance policies in a bridge system. (2) This model can assist bridge managers in prioritizing maintenance alternatives, determining the most appropriate maintenance strategy, and optimizing resource allocations. (3) The prediction outcomes given by the approach can be used as a signal to warn the bridge maintenance units under the situation when the estimated bridge condition is abnormally bad or close to the limit of allowance. (4) This model can identify noteworthy variables affecting bridge performance and evaluate the magnitude of each variable, which can assist bridge engineers for designing bridges that will reduce the future bridges' maintenance cost. (5) Bridge inspection is an expensive and time-consuming task. Using the predictions from the model, one may save considerably on inspection costs and find potential bridge damage in advance. (6) The approach is practical in dealing with large-sized observations due to its computational efficiency and simplicity of use. Therefore, this model can assist bridge engineers in making better decisions. The consequence of the case study shows that this model has potential for use with similar types of prediction problems or in various areas of decision-making problems. The outcomes given by this model depend on the inspector's assessments; thus, sufficient personnel experience and training is essential. Basically, the use of the model is not restricted to the numbers of independent variables; however, computation time increases as the numbers of predictor variables and sample sizes increase. To facilitate matrix operations in this approach, the use of software such as MicroSoft™-Excel, Maple™ and MathWorks™-MATLAB is helpful. Notably, although the proposed method is suitable for fuzzy regression analysis, it can also be applied for a broad class of nonlinear functions. This can be accomplished by transforming variables to find a linear function involving the transformed variables. This model is restricted to triangular membership functions; thus it can not handle other types of membership functions. In addition, the approach allows fitting a model to crisp-input and fuzzy-output data, but it is incapable of dealing with fuzzy independent variable(s). These are two main limitations of this approach, which are suggestions from the author for future studies.