MODELING THE CONDITION OF BUILDINGS BY REAL FUZZY SETS

Transcription

1 Int. Journal for Housng Scence, Vol.38, No.1 pp.13-23, 2014 Publshed n the Unted States MODELING THE CONDITION OF BUILDINGS BY REAL FUZZY SETS Ádám BUKOVICS Szécheny István Unversty Department of Structural and Geotechncal Engneerng Győr, Hungary László Tamás KÓCZY Szécheny István Unversty, Department of Automaton Győr, Hungary ABSTRACT It s an mportant task to qualfy and rank resdental buldngs based on varous prorty aspects and to make optmum allocaton of the materal resources avalable for the renewal of the buldngs. To ths end a model based on fuzzy logc was prepared. To construct and to test ths model many detaled techncal-statc expert reports were avalable all related to a stock of resdental buldngs n Budapest. Based on ths report a database was created. Wth the help of ths database a model was prepared, calculatng a so-called status characterstc value between 0 and 1 on the bass of the structures and status of the buldngs. For ths calculaton a fuzzy sngleton sgnature model was prepared. Based on ths a herarchy can be set up related to the stock of buldngs, whch s sutable for supportng the decson-makng on nterventon. The model was examned by usng the created database. Membershp values charactersng the status of the load-bearng structures were defned on the bass of the deteroraton of the structures. In ths paper a new method for the determnaton of the membershp values s descrbed, whch n addton to the deteroratons of the structures takes nto account other parameters of the structures, and the mpact exerted on the qualty of the structure, too. The method s sutable for the determnaton of the membershp values of all prmary and secondary structures. As an example the membershp values of the foundaton structures of the buldngs n the database were defned and the results /01/13-23, 2014 Copyrght 2013 IAHS

2 14 Bukovcs and Kóczy were analysed. The method was elaborated by the use of real fuzzy values (R-fuzzy sets), an extenson of the concept of classc fuzzy sets, the former beng sutable for smultaneously takng nto account varous aspects text. Key words: Buldng Dagnostcs; Structural Deteroratons; Fuzzy Sgnatures; Real Fuzzy Set of Values. Introducton It s mportant from the pont of vew of natonal economc nterest to keep records of and process the characterstcs and status of the exstng stock of buldngs. Durng the past perod n our research we have dealt wth ths problem. In the course of our research work a great many detaled techncal and statc expert reports on resdental buldngs n Budapest bult at the end of the 19th and at the begnnng of the 20th centures had been avalable. A consderable part of the current stock of buldngs n Budapest was bult n a rather short tme perod all buldngs havng smlar structural arrangement and status. Because of ths the examnaton and dagnoss of the structures of these buldngs enable us to draw general conclusons on smlar constructons n other parts of Budapest, too. Havng processed the data ncluded n the expert opnons a database was created, sutable for makng structural and dagnostc analyss of the buldngs [1, 2]. We propose that t s practcal to express the status of the buldngs by a relatvely objectve number, refer to as status characterstcs, whose knowledge enables rankng buldngs based on varous prorty aspects. To acheve the foregong goals the use of a fuzzy sgnature based model would practcal. To set up the model the requred relevance weghts, aggregaton operators and membershp values featurng the status of varous structures were defned n relaton to the examned buldngs. The fuzzy sngleton sgnature was set up n a tree-structure format, and the examned resdental buldngs were qualfed and ranked wth the help of the model [3]. An mproved method of the determnaton of membershp values s descrbed n ths paper. Our am s to work out a method for defnng the membershp values, whch characterse the condton of the examned load-bearng structures, whch, n addton to the detected deteroraton of the structures drectly or ndrectly, takes nto account other parameters of the structure, and the respectve effect, exerted on the qualty of the structure. As a frst step a general form of the method was elaborated, whch s sutable for takng nto account an arbtrary number of postve and negatve effects. The number of factors, nfluencng the condton of a structure, taken nto account n case of a gven load-bearng structure, depends on the nvestgated structure and the avalable database, too. Wth regard to the elaborated method t was nvestgated what actual results can be acheved and how do these relate to the expected results. Ths method s sutable for defnng the membershp values of all prmary and secondary structures. On the bass of the database prepared wth the help of the

3 Modelng the Condton of Buldngs 15 nvestgaton of the gven stock of resdental buldngs of the 13th dstrct of Budapest the membershp values of foundaton structures were defned and the obtaned results analysed. The method was elaborated usng extended, so called real fuzzy values, become correspondng. R-fuzzy operatons are sutable for takng nto account varous aspects as a whole. Introducton of Fuzzy Sets of Real Values and the Appled Operatons Accordng to the orgnal defnton of fuzzy sets the elements of the basc set (unverse of dscourse X) are translated nto the [0, 1] unt nterval by the membershp functon, thereby provdng membershp values for the gven elements of the set [4]. Later on ths defnton was generalsed n many ways, the most approprate one, best sutng our present goals s the membershp functon extended to the whole real lne: A : X R 1 (1) Negatve and greater than one membershp values have no physcal meanng, however the extended set of values enable us to make certan techncal computatons n a way that such magnary membershp values should be generated as ntermedary results (whle the fnal result s a true membershp value between 0 and 1.) Ths extenson s prmarly justfed by the fact that n the so-called I-fuzzy and R-fuzzy axomatc systems t s possble to determne the nverse of the set ntersecton and the unon under certan condtons. In many aspects the features of these nverse operatons are smlar to the algebrac nverse operatons [5]. The smplest and most commonly used generally prevalng fuzzy operatons, fttng R-fuzzy axoms are as follows. Fuzzy dsjuncton (algebrac sum): A B = A + B A B (2) Fuzzy conjuncton (algebrac product): A B = A B (3) On the bass of the axomatcs t becomes possble to ntroduce fuzzy nverse dsjuncton and fuzzy nverse conjuncton, too. The formula of the former one s: A B A B = 1 1 B B If B > A, then A B < 0 ; (4)

4 16 Bukovcs and Kóczy whle the formula of the latter one s: A A B = B B 0 If A > B, then A B > 1 (5) Creatng a Model to Defne Membershp Values In order to defne the membershp values of the nvestgated characterstcs, as a frst step a decson has to be made as to how many factors nfluencng the status of the structure shall be taken nto consderaton. The values of these factors descrbe the ntensty and the extent of the nvestgated effect exerted on the condton of the examned structure of a buldng. Factors affectng the status of the structure n a postve drecton wll be called status mprovng factors and wll be denoted by (=1 n, n s the number of status mprovng factors taken nto account). Those factors affectng the status of the structure n a negatve sense wll be called status deteroraton factors and wll be denoted by (=1 m, m s the number of status deteroraton factors taken nto account). There can be any number of status mprovng and status deteroraton factors, accordng to the nvestgatons the applcaton of 6 (3 status mprovng and 3 status deteroraton) factors s justfed. Snce each factor s supposed to affect the status n each case, the nvestgaton must be restrcted to strctly monotonously behavng fuzzy operators. The above defned fuzzy dsjuncton, whch can be nterpreted n the case of the strctly monotonously behavng fuzzy operators, s sutable for takng nto account all the factors. The factor, whch ncludes the effect of all status mprovng factors, s called the resultant of µ status mprovng factors and s denoted wth P. The factor that ncludes the effects of all status deteroraton factors, s called the resultant of status deteroraton factors and s denoted wth N. The resultant of the status mprovng and status deteroraton factors was defned wth the help of the fuzzy dsjuncton (1). Snce fuzzy dsjuncton satsfes the assocatve property, a one-step calculaton can be made n case of any number of factors affectng the membershp values. In case of arbtrary n postve, and m negatve factors effectng the membershp value, µ usng the fuzzy dsjuncton, the expresson of factors P and N are gven as follows: P N = n A = m A n n = j n n ± ± ( 1) n 1 = 1 (1 ) j m m m m = ± ± ( 1) m 1 = 1 (1 ) j j The above so called seve formulas could be transformed nto closed format, usng De Morgan s form. [6]. µ (6) (7)

5 Modelng the Condton of Buldngs 17 Ths s the general format of the resultant of status mprovng and status deteroraton factors. If three status mprovng and three status deteroraton factors are taken nto consderaton (ths s featurng n most of the load bearng structures), the value of P and N factors can be calculated wth the followng connecton: µ P = = Af + + Af Af Af + Af (8) N = = Af + + Af Af Af + Af (9) In the next we propose two extreme values that descrbe the qualty of the structure of a buldng n two dfferent ways. These two together defne a fuzzy set of nterval value. The two extreme values of the nterval are determned by the membershp values calculated on the bass of what we call optmstc and pessmstc estmaton. FIG 1. Fuzzy set of nterval value. ( λ ) The optmstc estmaton SUP can be obtaned by usng the fuzzy nverse dsjuncton and t means the upper lmt of the membershp value of the examned structure (charactersng ts general status). The resultant of the status mprovng factors s multpled by the complement of the status deteroraton factors and ths value s dvded by the complement of the relatve deteroraton Ths optmstc estmaton (upper value) s gven by: ( µ P N ). λ SUP P (1 N ) = 1 P N (10) ( λ INF ) The pessmstc estmaton s obtaned by usng the fuzzy conjuncton (2) and t means the bottom lmt of the membershp value of the examned structure. The resultant of the status mprovng factors s multpled by the complement of the resultant of status deteroraton factors. The pessmstc estmaton (bottom value) s as follows: λ INF = P (1 N ) (11)

6 18 Bukovcs and Kóczy Membershp values, obtaned as a result of the two estmatons, were compared wth the results of expert evaluaton, and t was found that ths latter one was well (, ) approxmated by the average of the two values λ INF λ SUP. Thus we propose ths ( λ ) nterval for descrbng estmated status of the examned structure of a buldng AVG. The estmated status s thus: λ AVG = λ INF + λ SUP 2 (12) λ AVG Ths value wll be consdered n the future as the overall membershp value assocated wth the condton of the examned structure and we state that these overall values can be appled when usng the fuzzy sngleton sgnature-based status determnng and rankng model. The Applcablty of the R-Fuzzy Algebrac Structure for the Targeted Problem At frst a methodologcal research was conducted, than the applcablty of the method was examned. We found that n each case the upper value, obtaned by the applcaton of fuzzy nverse dsjuncton was greater than the lower one obtaned by usng the fuzzy conjuncton ( λ SUP > λ INF ). We nvestgated whch values may be assumed by the upper and lower membershp functons and the estmated status dependng on the resultant of the status mprovng factors (mp) and the status deteroraton factors (an). In the example n secton 5 the smallest value of the resultant of status mprovng factors may be 0.65, snce the mnmum value of ma1 descrbed n the followng secton, cannot be smaller than ths. The value of mp was nvestgated n the nterval between 0.65 and 1 wth 0.05 parttonng. The resultant of the an status deteroraton factors was examned and plotted n the nterval [0, 1] wth 0.05 parttonng. It can be seen how the value and the character of the examned fuzzy algebrac operators are nfluenced by slght and sgnfcant status deteroraton (negatve) effects. In case of four charactersng values of the status mprovng factors (0.65, 0.75, 0.85, 0.95) the changes of the value of fuzzy algebrac operators was examned n connecton wth the value of the status deteroraton factors. As an example n the case of mp=0.75 the resultng chart can be seen n Fgure 2. The blue lne ndcates the optmstc estmaton, that s the upper value, the yellow lne shows the pessmstc value, that s the lower value, whle the purple lne shows the estmated status of the membershp value of the examned structure. In lne wth the ncrease of the value of the resultant of status mprovng factors the dfference between the upper and lower values of the membershp functon wll ncrease, too. Typcally the dfference between the upper and lower values of the membershp functon (the open scssors ) s great f the value of the resultant of

7 Modelng the Condton of Buldngs 19 status mprovng factors and that of the status deteroraton factors s great alke. In such a case the dfference between the upper and lower value of the membershp functon may exceed If the resultant of the status deteroraton factors s extremely small or great, the value of asup-ainf decreases. If the dfference between the upper and lower values of the membershp functon exceeds 0.45, an ndvdual nvestgaton of the reasons of deteroraton s strongly recommended. FIG 2. Possble values of λ SUP, λ INF and λ AVG subject to N n case of µ P=0.75. Testng the Model on a Load Bearng Structure Examnng a specfc stock of buldngs wth the use of the created model, the membershp values of foundaton structures were defned, and t was nvestgated how the method based on magnary fuzzy values could be appled from the aspect of the qualfcaton and rankng of the load bearng structures of resdental buldngs. The schematc model of the defnton of membershp values, featurng the status of foundaton structures can be seen n Fgure 3. Defnng the Membershp Values Featurng the Foundaton Structures In the case of the examned types of resdental buldngs the foundaton structure s nearly exclusvely strp foundaton. Our examnatons relate to the strp foundaton under the exteror man walls. The membershp value, featurng the foundaton structure, s nfluenced by the factors the followng subsecton.

8 20 Bukovcs and Kóczy FIG 3. Schematc model of the defnton of µ The Factor Dependng on the Materal of Strp Foundaton (maf) Its value was selected n relaton to the materal of strp foundaton of the examned buldngs. For example, n the case of strp foundatons made of stone maf=0.65, n case of strp foundatons made of concrete, whch s more relable and less senstve, maf=0.90. The Factor Dependng on the Conformance of the Wdth of Strp Foundaton (Postve Drecton) (m) In order to determne the conformance of the wdth of the strp foundaton, the rato of the wdth of the actual strp foundaton (wf) and the dealzed, estmated wdth of the strp foundaton (wf) was examned. To the end of defnng the wdth of the dealzed strp foundaton t s necessary to approxmately determne the loads exerted on the strp foundaton, as well as to know the characterstcs of the sol under t. These values are ncluded n the techncal-statc expert reports, whch are necessary for the status-determnng model, and can be determned on that bass. On the bass of the foregong the requred wdth of the strp foundatons under the external man wall of the examned buldngs was determned. Comparng ths to the wdth of the appled strp foundaton, the factor, dependng on the conformance of the wdth of strp foundaton, was obtaned. The Factor Dependng on the Year of Constructon (m) In 1914 Budapest ntroduced the Constructon Rules and Regulatons for all buldngs n ts terrtory. For all buldngs bult after ths regulaton n Budapest had been put

9 Modelng the Condton of Buldngs 21 nto force, 0.10 was appled for the value of factor m, whle n case of buldngs, bult before 1914, the value of the factor m s 0. Ths factor takes two aspects nto consderaton. One s the queston whether n the year of the constructon the cty had ts regulatons n effect or not, the second beng smply the number of years that had passed after constructon because of the process of the natural mparment of the condton of buldngs n terms of tme. Factor Dependng on the Detected Deteroraton of Strp Foundaton ( Af ) Factor aaf s determned by usng the component featurng the status of the formerly determned membershp value of foundaton structure (1111). The bgger the extent of the detected deteroratons s the closer the value of 1111 s to 0. Membershp values are determned on the bass of the fuzzy dsjuncton and the fuzzy conjuncton, therefore n our case the bgger the extent of the deteroraton s, the closer Af s to 1. Thus the complement value Af = was appled. The Factor Dependng on the Conformance of the Wdth of Strp Foundaton (Negatve Drecton) ( ) If the wdth of strp foundaton s smaller than the deal wdth, the value of a may be between 0.05 and 0.40 (n relaton to wf / wf). The Factor Featurng other Effects Deterorated the Qualty of the Structure ( ) Ths factor takes nto account the effect of vbratons mparng the status of the foundaton structures. In case of buldngs stuated by roads wth hgh traffc the value of s 0.15, whle n case of buldngs stuated over the subway tran (Vác út) t s In the close vcnty of buldngs n sde streets there s no consderable traffc, therefore n these cases the value of s 0.00, that s, t has no mpact on the value of 2. Results As descrbed n the above sectons to determne the membershp values, featurng the foundaton structures of the examned stock of resdental buldngs, factors nfluencng the membershp value, were defned frst for each examned buldng. Altogether 6 factors were defned: three status mprovng and three status deteroraton factors. In the next stage the membershp values featurng the status of the foundaton structure of examned buldngs were determned. lsup and linf are the two extreme values of the fuzzy nterval values whle the arthmetc mean of these two gves the estmated overall membershp value (lavg). Fgure 4 ncludes the values of all tree membershp functons (lsup, linf, lavg). The examned buldngs were arranged n monotoncally ncreasng sequence on the

10 22 Bukovcs and Kóczy bass of the estmated value of overall membershp functons, referrng to the status of the foundaton structures. In the case of structures n very good and n very bad condton, the dfference between the upper and the lower values of the membershp value s generally rather small whle the scssors open up more for medan condton buldngs. FIG 4. Values of the membershp functons n case of ranked buldngs. Summary A new method was elaborated n order to defne the membershp values, used for the fuzzy sngleton sgnature-based model, formerly proposed by us, whch s sutable for supportng the decson-makng, relatng to resdental buldngs. In addton to the deteroratons of the structures, detected at the tme of examnaton, ths method s sutable for takng nto account other parameters of the structures, and ther respectve effects thereof, exerted on the qualty of the structure. Snce each factor modfyng the status of the structures s suspected to nfluence the qualfcaton n each case, the nvestgatons were restrcted to strctly monotonc fuzzy operators. The method was developed by usng fuzzy values wth extended values. Such R-fuzzy operatons are sutable for jontly takng nto account varous ponts of vew. From the results of the nvestgaton t was establshed that membershp values featurng the status of varous structures can be determned usng the strctly monotonc fuzzy algebrac operators, as well as the fuzzy nverse dsjuncton that can

11 Modelng the Condton of Buldngs 23 be deduced therefrom. Ths method can be practcally used to evaluate the status of other bult structures and nfrastructural facltes, too. References [1] Bukovcs, Á. Buldng Dagnostc and Pathologcal Analyss of Resdantal Buldngs of Budapest, PATORREB 2009, 3 Encontro Sobre Patologa E Reabltacao de Edfcos, Porto, Portugal, (2009), pp [2] Bukovcs, Á. Pathologcal Analyss of Suspenson Corrdor and Floor Structures of Resdental Buldngs, CIB 2010 World Congress, Unversty of Salford, CIB, Manchester, UK, [3] Bukovcs, Á.; Kóczy L.T. Fuzzy Sgnature-Based Model for Qualfcaton and Rankng of Resdental Buldngs, XXXVIII IAHS World Congress on Housng, Istanbul, Turkey, 2012, pp [4] Kóczy, L.T.; Vámos, T. and Bró, G. Fuzzy Sgnatures, Proceedngs of EUROFUSE-SIC 99, Budapest, 1999, pp [5] Kóczy L. T.; Hajnal M.A. New Attempt to Axomatze Fuzzy Algebra wth an Applcaton Example, Problems of Control and Informaton Theory, Budapest, Vol. 6 (1), (1977), pp [6] Kóczy L. T. Fuzzy graphs n the evaluaton and optmzaton of networks, FUZZY SET SYST 46:(3), 1992, pp [7] Mends, B. S. U., Gedeon, T. D., Botzhem, J. and Kóczy, L. T. Generalsed Weghted Relevance Aggregaton Operators for Herarchcal Fuzzy Sgnatures, Proceedngs of the Internatonal Conference on Computatonal Intellgence for Modellng, Control and Automaton, CIMCA 2006, [8] Molnárka, G.; Kóczy, L.T. Decson Support System for Evaluatng Exstng Apartment Buldngs Based on Fuzzy Sgnatures, Int. J. of Computers, Communcatons & Control, No.3, (2011), pp