Demonstrative Implications of a New Logical Aggregation Paradigm on a Classical Fuzzy Evaluation Model of «Green» Buildings

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1 Demonstrative Implications of a New Logical Aggregation Paradigm on a Classical Fuzzy Evaluation Model of «Green» Buildings Milan Mrkalj Faculty of Organizational Sciences, Belgrade, Serbia mrki2000@mail.bg Abstract. This paper demonstrates the application possibilities of the generalised model for Logical Aggregation (LA) in performance and quality evaluation of Green buildings through establishing a linear order of them by a (scalar) general aggregated quality parameter and sorting them into categories. An existing classical fuzzy model is experimentally modified in order to demonstrate the advanced capabilities of adequate articulation of the partial demands for quality via the new generalised model for logical aggregation. Keywords: Logical Aggregation, Quality management, Green buildings. 1 Introduction The advanced abilities of generalised model for Logical Aggregation presented in [1,4,5,6,7] expand the boundaries of existing fuzzy model presented in [2]. The existing model is in essence an aggregation model since the inputs are multidimensional values and the output is a scalar quantity which provides base for simple linear sorting. In existing fuzzy model only trivial quality attributes are used, without interaction between them through logical functions which should improve the articulation of partial demands for quality. Also, the aggregation operator (explained in [3,6]) is only dot product. The model for Logical Aggregation (LA) introduces the use of logical functions between trivial attributes which emerge from the finite set of possible functions Boolean algebra generated over the set of attributes. The new complex aggregation operators are introduced with the use of different generalised products which belong to a subclass of T-norms with additional axiomatic term of nonnegativity, as shown in [1,4,6]. Implementation of the model for Logical Aggregation [1] is possible in both phases of aggregation in the existing model [2]. The LA paradigm emanates an algebraic structure which provides the basis for arithmetic calculation of the output results. A. Laurent et al. (Eds.): IPMU 2014, Part II, CCIS 443, pp , Springer International Publishing Switzerland 2014

2 Demonstrative Implications of a New Logical Aggregation Paradigm 21 2 Implementaton of Logical Aggregation in the First Phase of Aggregation of the Existing Fuzzy Model Implementation of LA on the first phase of aggregation of existing model is demonstrated, and the values of the output are compared. The aim is to show the ability to create complex attributes which have more power of expressing the partial demands for quality in quality management. The term attributes from [1,3,6] are named as factors in [2] and are organized as two factor sets. Vector of weights of secondary attributes of the primary attribute energy efficient and energy utilization from [2] is: A 2 = [ 0,35 0,35 0,2 0,1 ]. Weights of secondary attributes are respectively w a = 0,35 w b = 0,35 w c = 0,2 and w d = 0,1. In this case, generalised pseudo-boolean polynomial is trivial (contains only single basic attributes). Now we shall unify attributes architecture design (a) and envelope structure (b) into one single new complex attribute on the basis of the fact that these two single attributes are significantly positive correlated in qualitative sense (architecture design and envelope structure which assures the rational heat transfer coefficient of external surfaces) [2]. Single attributes a and b will be dismissed (their respective weights become zero instead of 0,35), and new complex attribute a b (1) will be introduced, with weight w a b = w a + w b = 0.7. ϕ(a,b) = (a b) (1) Generalised Boolean polynomial of this logical expression is given in (2). ϕ (a,b) = ((a b)) = a b (2) New Logical Aggregation operator based on [3] is given in (3). 7 Agg (a,b,c,d) = 10 ϕ (a,b) c d = 7 10 (a b) c d (3) Corresponding aggregation measure [3] is given in (4). μ = 7 10 (σ a σ b ) σ c σ d (4) Aggregation measures (4) are shown in Table 1: Table 1. S a b 0,2 c 0,1 d 0,7 a b μ(s) {0} {a} {b}

3 22 M. Mrkalj Table 1. (continued) {c} ,2 {d} ,1 {a,b} ,7 {a,c} ,2 {a,d} ,1 {b,c} ,2 {b,d} ,1 {c,d} ,3 {a,b,c} ,9 {a,b,d} ,8 {a,c,d} ,3 {b,c,d} ,3 {a,b,c,d} On value level input parameters from R 2 matrix from [2] are processed through operator given in (3). «Energy efficient and energy utilization» - values from R 2 Table 2. one star * CATEGORIES two stars three stars ** *** Architecture design a 0,1 0,2 0,7 Envelope structure b 0,1 0,2 0,7 HVAC c 0,15 0,25 0,6 Lighting system d 0,1 0,25 0,65 In qualitative sense, attributes given in Table 2 are highly positive correlated, so the most suitable generalised product [3,4,6] should be the min function ( := min). Aggregation operator (3) processes one by one column of R 2 matrix and outputs into vector B 2 '. Agg (a,b,c,d) : R 2 B 2 ' B 2 '=[0,11 0,215 0,675] This result of Logical Aggregation coincides with the result of first phase aggregation of the existing fuzzy model developed in [2], so B 2 ' = B 2. Now we shall use dot product as generalised product for LA ( : = *). B 2 '=[0,047 0,115 0,675] Different choice of generalised product obviously induces changes on the level of values, B 2 ' B 2. After first phase of aggregation, the resulting values of quality are different from the values given by existing model.

Demonstrative Implications of a New Logical Aggregation Paradigm 23 3 Simulation of the Use of Aggregation Operator of Existing Fuzzy Model through Logical Aggregation Operator New model of LA

The aim is to show the possibilities of manipulation on the level of internal structure of the attributes which is fragmented via the new LA paradigm and to achieve the same results as with usage of

Weights of secondary attributes a (garbage classification and biologic treatment),b (intellectualized system) and c (all kinds of management system) are respectively w a = 0,3 w b = 0,3 and w c = 0,4.

4 Demonstrative Implications of a New Logical Aggregation Paradigm 23 3 Simulation of the Use of Aggregation Operator of Existing Fuzzy Model through Logical Aggregation Operator New model of LA [1,4,6] will be applied again on the first phase of aggregation of the existing model [2], and the results will be compared. The aim is to show the possibilities of manipulation on the level of internal structure of the attributes which is fragmented via the new LA paradigm and to achieve the same results as with usage of the existing model. Vector of weights of the secondary attributes of the primary attribute operation management [2] is: A 6 = [0,3 0,3 0,4]. Weights of secondary attributes a (garbage classification and biologic treatment),b (intellectualized system) and c (all kinds of management system) are respectively w a = 0,3 w b = 0,3 and w c = 0,4. Generalised pseudo-boolean polynomial [1] of the single attributes is equal to the attributes themselves: ϕ (a) = a v ϕ (b) = b v ϕ (c) = c v From the aspect of immanent structure, single attributes are complex elements (contain more than one atomic attribute [1]). In this particular case, every attribute a, b, or c includes 4 atomic attributes. Some of the atomic attributes (which present internal algebraic structure) are shared among the attributes. Now aggregation measures μ(s) are assigned to the atomic attributes (S) according to weights given in vector A 6. Table 3. w σϕ = 0.3 w σϕ = 0.1 w σϕ = 0.6 Atoms (S) μ(s) {0} {a} {b} {c} {a,b} {a,c} ,7 {b,c} ,7 {a,b,c} In Table 3 one of the possible weights assignment and structures is presented, and it defines the immanent structure σϕ (as explained in [1,4,6,7]) of the new attributes which are used to simulate the aggregation operator of existing fuzzy model [2].

5 24 M. Mrkalj The corresponding generalised pseudo-boolean polynomial is generated by summation of atomic pseudo-boolean polynomials included into structure defined by structure vectors σϕ. This procedure gives the aggregation operator shown in equation (5): Agg (a,b,c) = 3 10 [ a - 2(a c) - 2(a b) + 4(a b c) + b - 2(b c) + c ] c + 3 [ a b - 2(a b c) + a c + b c ] (5) 5 On the level of values (arithmetic level), one by one column of matrix of input parameters R 6 from [2] is processed through the operator given in (5) and put into vector B 6 '. For generalised product, dot product is used. B 6 ' = [0,17 0,285 0,545], B 6 ' = B 6 If the min function is used as the generalised product, the yield is: B 6 ' = [0,17 0,285 0,545], B 6 ' = B 6 As shown above, using the Logical Aggregation operator (5) and generalised product ( := *) and ( := min) respectively, the yield is the same as the one in the first phase of the existing model presented in [2], and the aim of simulating the model from [2] is successfully achieved. 4 Implementation of Logical Aggregation in the Second Phase of Aggregation of the Existing Fuzzy Model The second phase of aggregation in [2] will be modified for the purpose of demonstration of two interesting abilities of the LA model. If there is a need to satisfy only one of the two or more quality demands and not all of them at the same time If there is a quality demand which should swap the relative importance of other quality demands Table 4. PRIMARY ATTRIBUTES (A) WEIGHT VECTOR OF PRIMARY ATTRIBUTES A=[w(a i )] Land-saving and outdoor environment (a) 0,1 Energy efficient and energy utilization (b) 0,5 Water-saving and water utilization (c) 0,2 Material-saving and material utilization (d) 0,1 Indoor environment quality (e) 0,05 Operation management (f) 0,05 wa ( ) i = 1

6 Demonstrative Implications of a New Logical Aggregation Paradigm 25 In Table 4 the primary attributes with their respective weights are shown. The weight vector is A = [0,1 0,5 0,2 0,1 0,05 0,05]. If we wish to have satisfied only one of the two quality demands material-saving and material utilization and indoor environment quality, and not both of them at the same time, this can be achieved by removing single attributes (their respective weights become zero), and introducing new complex attribute d e with weight w d e = w d + w e = 0,15. ϕ(d,e) = (d e) (6) Generalised Boolean polynomial [1] of this logical expression is given in (7). ϕ (d,e) = ((d e)) = d + e - d e (7) If we wish to have a quality demand which can swap the relative importance of other quality demands, it can be done through a logical function of these attributes. For example, if Energy efficient and energy utilization is not highly satisfied, then a complex attribute can be created in a way that it gives less importance to it (overall quality will be less affected by this unsatisfaction), and more importance to Watersaving and water utilization, Material-saving and material utilization and Landsaving and outdoor environment. But if Energy efficient and energy utilization is highly satisfied, then the importance will be brought back to itself (so the overall quality will be more affected by Energy efficient and energy utilization attribute), and importance of other attributes is reduced. Logical expression of such complex attribute is given in (8). ϕ (a,b,c,d) = b (Cb a c d) (8) Generalised pseudo-boolean polynomial [1] of this logical expression is given in (9). ϕ (a,b,c,d) = b + [(1 - b) a c d] ϕ (a,b,c,d) = b + a c d - a b c d (9) Weight of this complex attribute will be taken from the single attribute Energy efficient and energy utilization. w b ( b a c d) = 0,25 w b = 0, = 0.25 After both modifications of the model shown in chapter 4, instead of second phase operator from [2], the new operator of Logical Aggregation [3] is given in (10). Agg (a,b,c,d) = a b c f + 3 (d + e - d e) + (b + a c d - a 20 b c d) (10) On the arithmetic level, the input parameters matrix R from [2] will be processed through operator given in (10).

7 26 M. Mrkalj Agg (a,b,c,d) : R B * or B min R = L NM 0, ,285 0, ,11 0,215 0,675 0,2025 0,2925 0,505 0,22 0,285 0,495 0,265 0,265 0,47 0,17 0,285 0,545 If generalised product is dot product ( := *): O QP B * = [0, , ,641322] If generalised product is min function ( := min): B min = [0, ,269 0,594625] Final performance index T from [2] is calculated as a product of vector [1 2 3] with transposed vector B, and it is also valid for our modified model. It is shown in (11). T = [1 2 3] * B T (transposed) (11) 0, T = 1 2 T 3 * B = * 0, = 2,6796 M P * 0, T T = * B min = * 0,269 = 2,4971 0, L M NM L NM 0, Depending on performance index, the building belongs to one of the three categories (one, two or three stars) [2]. O P QP O QP T [1,0 1,7] T [1,7 2,4] T [2,4 3,0] - one star building - two stars building - three stars building. When modified model presented in chapter 4 of this paper is applied, the building from the example given in [2] stays in the interval T [2,4 3,0], which implies that it belongs to the category of three stars building.

8 Demonstrative Implications of a New Logical Aggregation Paradigm 27 5 Conclusion Models for performance evaluation of Green buildings enable good quality and objective evaluation, minimizing the subjectivity bias. Experts in the subject area contributed as the weight coefficients providers, while the final result (performance index) is achieved through processing on arithmetic level, through the fixed mathematic model. Conventional tools for aggregation are often inadequate due to limitations in the sense of disability of using logical interactions between quality attributes. Improvements of the model s articulation abilities by using advanced techniques of Logical Aggregation significantly expand the possibilities of customization of the model to more specific needs. This is especially noticeable in the ability to include complex logical functions in which nontrivial attributes emerge. New paradigm treats logical functions partial aggregation demands as a generalised Boolean polynomial, which processes values from the unitary real interval [0, 1]. Aggregation in general is a generalised pseudo-logical function. Comparative review of the results of modified improved model presented in this paper, and existing fuzzy model [2] is shown, and the advanced abilities of the new approach to aggregation of quality parameters and performance of Green buildings are demonstrated. References 1. Mirković, M., Hodolič, J., Radojević, D.: Aggregation for Quality Management. Yugoslav Journal of Operations Research 16(2), (2006) 2. Sun, J., Wu, Y., Hao, Y., Dai, Z.: Fuzzy Comprehensive Evaluation Model and Influence Factors Analysis on Comprehensive Performance of Green Buildings. In: ICEBO 2006, Shenzhen, vol. VIII-4-2. China Renewable Energy Resources and a Greener Future (2006) 3. Radojević, D.: Logical Aggregation Based on Interpolative Realization of Boolean Algebra. In: Eusflat Conf., vol. (1), pp (2007) 4. Radojevic, D.: Fuzzy Set Theory in Boolean Frame. Int. J. of Computers, Communications & Control III (2008) 5. Radojevic, D.: New [0,1]-valued logic: A natural generalization of Boolean logic. Yugoslav Journal of Operational Research - YUJOR 10(2) (2000) 6. Radojevic, D.: Logical Aggregation Based on Interpolative Boolean Algebra. Mathware & Soft Computing 15 (2008) 7. Radojević, D.: Interpolative relations and interpolative preference structures. Yugo-slav Journal of Operational Research YUJOR 15(2) (2005)

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