FACILITY LIFE-CYCLE COST ANALYSIS BASED ON FUZZY SETS THEORY Life-cycle cost analysis J. O. SOBANJO FAMU-FSU College of Engineering, Tallahassee, Florida Durability of Building Materials and Components 8. (1999) Edited by M.A. Lacasse and D.J. Vanier. Institute for Research in Construction, Ottawa ON, K1A 0R6, Canada, pp. 1798-1809. National Research Council Canada 1999 Abstract This paper presents the framework of a fuzzy sets-based methodology for life-cycle cost analysis of facilities, including application to buildings. Life-cycle cost analysis is a major phase of facility management and value engineering. Facility components and materials deteriorate at different rates and require some form of maintenance, rehabilitation, and replacement (improvement efforts) throughout their service lives. It is difficult to estimate the timing of these improvement efforts, i.e., the expected service lives of the materials at the end of which they have to be replaced or a point in the life at which they have to be rehabilitated. The timing of the required improvement efforts can be predicted from deterioration models or estimated subjectively based on expert opinions. The costs of required maintenance, rehabilitation, and replacement may be obtained from historical data but these cost estimates have to be sometimes supplemented with expert opinions in order to perform life-cycle analysis. Representing these variables in the form of crisp numbers and fuzzy numbers, an algorithm is presented to formulate a methodology for lifecycle cost analysis of buildings. Fuzzy sets theory has been proven as a valuable tool for handling uncertainties due to subjective estimates in decision making models. Alternative building designs can be evaluated and compared early in the project development stage. Keywords: facilities, life-cycle costs, fuzzy sets, rehabilitation, facility management.
1 Introduction The importance of facility management, in terms of concern about conditions of buildings has been demonstrated by the recent development of deterioration models [Lee and Aktan 1997] and condition indexing methodology [Uzarski and Burley 1997]. Once the deteriorated state of a building component is known, some form of maintenance or improvement (rehabilitation or replacement) is necessitated. Monitoring the costs and timing of these necessary actions over the life of the building constitute life-cycle cost analysis which in turn is a primary aspect of facility management. Traditionally, estimates of initial costs, maintenance costs and improvement costs, expected service lives, and timing of the actions have been done based on historical analysis of data. It is common however to adjust these estimates, usually the average estimate, with expert opinions, thereby making the estimate subjective. The traditional concepts of probability theory can also be applied to lifecycle costing, treating the costs and timings as a stochastic process. In this case the variables have to be estimated purely from a historical data in order to satisfy the assumptions of the randomness. If any subjectivity is introduced to the estimates, then the uncertainty cannot be handled using the probability theory alone. Fuzzy sets theory is a valuable tool for handling such uncertainties. Representing the estimates of the decision variables required as fuzzy numbers, this paper presents a methodology for handling the subjective uncertainty in life-cycle cost analysis. 2 The concept of fuzzy numbers Consider a set A with elements denoted by x. Under the conventional set theory, a characteristic or membership grade function µ A can be used to define the membership of any element or subset in the set A as follows: 1 if x 0 A µ A (x) = { (1) 0 otherwise Instead of the {0,1} (yes or no) valuation, if the membership grade µ A (x) can have values in the real interval [0,1] according to how much x belongs to this set A, then the set A is a fuzzy set [Zadeh 1965]. If the set A is a set of criteria, µ A (x) is the degree to which x satisfies the conditions of A, or in other words, µ A (x) is the "strength" of the statement : "x belongs to the set A." In fuzzy sets terminology, values that are known precisely, are referred to as crisp ordinary numbers, while imprecise values are represented by fuzzy subsets. In the methodology proposed in this paper, the fuzzy sets concept is applied to deal with the imprecision in quantitative values which are subjectively estimated. Decision variables are represented as fuzzy numbers while algebraic computations will be based on the interval of confidence or interval arithmetic described by Kaufmann and Gupta [1985]. Under this approach, the -cut or intervals of confidence of the fuzzy
numbers are used to perform the necessary algebraic operations. Basic fuzzy sets theory textbooks and papers explain the general form of a fuzzy subset A where is the degree of belief, or in terms of fuzzy sets, the degree of membership. A, the interval of confidence associated with is formally referred to as the -cut. All possible values of A whose degrees of belief are greater than or equal to a specified value constitute the -cut A, such that A = {x µ A (x) } (2) A simple form of fuzzy number is the triangular fuzzy number (TFN) with a triangular distribution as the membership function. Using three parameters l, m, and h, a TFN can be adequately expressed as a triplet [l, m, h] where [l, h] is the largest -cut and m is the modal point. The mode m of a TFN is the most possible value under the distribution while l and h represent the lowest and highest possible values respectively. The support is estimated as (h - l). The membership function of TFN A, where A = [l, m, h], can be defined as: µ A (x) = 0, x < l = x - l, l > x < m m - l = 1, x = m (3) = h - x, m > x < h h - m = 0, x > h In the life-cycle costing methodology, it will be necessary to make decisions based on the ranking or comparison of costs of facility alternatives, estimates which are of the form of TFNs. The comparison of TFNs can be mathematically done using two approaches: first, a linear ordering or ranking of the fuzzy numbers based on an equivalent crisp ordinary number; and a "qualified comparison" approach in which the "strength" or "truth values" of the resulting decisions are indicated by the -cuts of the fuzzy numbers. Kaufmann and Gupta [1988] discussed the linear ordering of fuzzy numbers using an index called the "removal" or "ordinary representative" of each fuzzy number the crisp ordinary number equivalent. The "ordinary representative" of a TFN A = [l, m, h] or A ORD can be computed as A ORD = ( l + 2m +h ) (4) 4 The second approach is the "qualified comparison" approach as mentioned earlier. To compare two TFNs A and B, a "qualified" statement can be made on the relative value of the property represented by these fuzzy numbers. If A and B respectively denotes the TFN distributions of the numerical values of a property being used to measure and compare two objects (figure 1), then by graphically comparing A
and B, a "truth value" can be attached to a statement as to whether one object is better than the other [Watson et al. 1979; Whalen 1987]. Consider the -cut at which the inside reference lines of the fuzzy distribution intersects. This intersection is at a membership µ 1. By studying the possible values whose degrees of membership ( ) are greater than µ 1, i.e., -cut at µ 1, it could be seen that the lowest possible value for B is higher than the highest possible value for A. Using this standard of possibility ( = µ 1 ), it could be said that the property value of B is strictly greater than the property value of A. The "strength" of this statement or its "truth value of strict dominance" is given by the compliment of the lowest degree of membership ( ) above which the statement is true [Whalen 1987]. Thus, the statement "B has a better property than A" has a "truth value" of 1 - µ 1, or 0 as denoted in figure 1. Fig. 1: Comparison of triangular fuzzy numbers Looking again at figure 1, above the -cut of µ 2, an overlap occurs between possible values for these two objects. The highest possible value for B is still always higher than the highest possible value for A, and the lowest possible value for B is still higher than the lowest possible value for A, but the lowest values of B are not higher than the highest values of A. Thus, the statement "B has a property at least as good as A has a "truth value" of 1 - µ 2, or as denoted in figure 1. The "truth value" mentioned above can be computed from the graphical relationship of the TFNs. Consider any two TFNs A and B, where A = [l 1, m 1, h 1 ], and B = [l 2, m 2, h 2 ]. If m 2 > m 1, then the "truth value of strict dominance" as discussed above, can be derived from the possibility level µ 1 at which left reference function of B intersects with the right reference function of A (figure 1). Above this level µ 1, all possible values of B are greater than all possible values of A. The desired "truth value" is simply the complement of µ 1. Looking at figure 1, the x coordinate of the intersection point, say C x, can be computed as follows:
Then, h 1 (m 2 - l 2 ) + l 2 (h 1 - m 1 ) (5) C x = (h 1 - m 1 ) + (m 2 - l 2 ) u 1 = h 1 x h 1 m 1 (6) Therefore, the "truth value of strict dominance" can be computed as: 0 = 1 - µ 1 = 1 { h 1 m 2 - m 1 l 2 - m 1 } (7) h 1 - m 1 (h 1 - m 1 ) + (m 2 - l 2 ) Based on the concept of interval arithmetic [Kaufmann and Gupta 1988], the following fuzzy mathematical operations may be pertinent to the proposed methodology of life cycle costing. Assume a crisp ordinary number, k, a TFN A = [l 1, m 1, h 1 ], and a TFN B = [l 2, m 2, h 2 ]. 1. Multiplication of a TFN by a Crisp Ordinary Number: k θ A = k θ [l 1, m 1, h 1 ] = [kl 1, km 1, kh 1 ] (8) 2. Division of a Crisp Ordinary Number by a TFN k τ A = k θ A -1 = k θ [l 1, m 1, h 1 ] -1 = k θ [1/h 1, 1/m 1, 1/l 1 ] = [k/h 1, k/m 1, k/l 1 ] (9) 3. Division of a TFN by a Crisp Ordinary Number A τ k = A θ k -1 = [l 1, m 1, h 1 ] θ [1/k] = [l 1 /k, m 1 /k, h 1 /k] (10) 4. Division of TFNs A τ B = [l 1 /h 2, m 1 /m 2, h 1 /l 2 ] (11) 5. Addition of TFNs A ρ B = [l 1 + l 2, m 1 + m 2, h 1 + h 2 ] (12) 6. Multiplication of TFNs A θ B = [l 1 * l 2, m 1 * m 2, h 1 * h 2 ] (13)
3 Fuzzy sets algorithm for life-cycle cost analysis Life-cycle cost analysis can be simply defined as an economic evaluation of a facility or alternative facilities over a desired service life, taking into consideration, all the costs incurred and benefits gained by the owner over this period, before computing an equivalent cost estimate. These costs include the following general classes: initial costs, maintenance costs (annual), future costs (singular), and salvage value. The equivalent cost estimate is computed by converting the stream of all the time-related costs to a single equivalent value such as the present worth, annual worth, and future worth. Investment in a facility can then be evaluated or compared with an alternative design, using any of these equivalent costs as found appropriate. This paper will use the present worth as the equivalent cost, to be derived from the following equations which can be found in most economic analysis textbooks: P = F(1 + i) -n (14) (1 + i) n 1 P = A { } i(1 + i) n = [A/i][1 - (1 + i) -n ] (15) P = CI (16) where, P = Present worth equivalent F = Single future cost A = Annual uniform series of costs i = interest rate n = economic horizon being considered CI = Initial cost Based on equations 14, 15, and 16 above, the traditional life cycle cost algorithm for computing the Present Worth Costs of a stream of present, annual, and future costs is given as: P = 3[CI] + 3 [F(1 + i) -n ] + 3 [A/i][1 - (1 + i) -n ] (17) The first component on the right hand side of equation 17 represents the sum of initial costs. The second and third components are anticipated single future costs and annual uniform costs respectively, with each component factorized to convert future costs to a present worth equivalent, i.e., using the interest rate to express the time value of money. Representing all variables as TFNs except the interest rate i and the economic horizon, n, which are treated as crisp ordinary numbers, the following algorithm is
formulated for computing the fuzzy number (TFN) estimate of the Present Worth of a stream of a facility s life cycle costs: [P l,, P m,p h,] = 3 [CI l,, CI m,ci h,] + 3 [F l,, F m,f h ](1 + i) -n ] + 3 [A l, A m,a h ][ i -1 ][ 1 - (1 + i) -n ] (18) The variables are same as defined earlier for equations 14-16 except for the addition of subscripts indicating the triplet l, m, and h for a TFN. 4 Numerical application An example is used in the following section of the paper to demonstrate the applicability and usefulness of the proposed life cycle costing methodology. Dell Isola [1982] provided a simple example for a facility s life cycle costs; initial costs are given for an original design of the facility, along with those of two alternative designs. Improvement efforts such as replacement of facility components or overhauls will be necessary at one or two of three different times for each of the choices. An economic life of 20 years is used, at the end of which the facility will be demolished with an expected salvage (net) value as indicated for each option of facility design. The costs used in the Dell Isola [1982] s original example have been utilized here by adopting the exact values as the modes of TFN but introducing uncertainty through the addition of lower and upper bounds to the modes, thereby generating triangular distributions. Interest rates and timings were retained as crisp numbers. The complete analysis is summarized in table 1. Applying equations 8-13 for pertinent fuzzy sets operations and equations 14-18 as described earlier for the proposed algorithm, calculation of the Present Worth Cost for the Original Design and the two alternatives A and B are presented in Appendix A at the end of the paper. The mode of the Present Worth Cost TFN, defined earlier as the most likely value (m from the triplet [l,m,h]) is actually the value obtained if the life-cycle cost analysis had been done without incorporating any uncertainty, i.e, the traditional algorithm [Dell Isola 1982]). The modes for the original design is computed in this paper as $1,458,558 while those of alternatives A and B are $1,290,230 and 1245.639 respectively. Using equation 4 the ordinary representative (ORD) of PW Costs for the original design is estimated as PW ORD (ORIG) = {1282.240 + 2(1458.558) + 1760.407}/4 = 1489.941
Table 1: Fuzzy sets analysis of facility costs ($1,000's) (Assume Economic Life = 20 yrs, Interest Rate = 10%) Costs Original Design Alternative A Alternative B Initial Costs [900,1000,1200] [600,700,850] [750,900,1100] Rehabilitation I at Year 8 [0,0,0] [150,200,210] [0,0,0] Rehabilitation I at Year 10 [0,0,0] [0,0,0] [18,20,25] Rehabilitation II at Year 16 [8,10,14] [175, 200,240] [0,0,0] Sale/Demolition* at Year 20 [(75),(80),(88)] [(90),(100),(125)] [(71),(75),(89)] Operating (Annual) [26,30,35] [30,35,37] [22,25,29] Maintenance (Annual) [20,25,32] [17,20,26] [12,16,23] Present Worth (PW) Cost [1282.2,1458.6,1760.4] [1094.8,1290.2,1518.0] [1035.9,1245.6,1539.1] Ord. Rep. of PW Cost TFN 1489.9 1298.3 1266.6 Rank by Ord. Rep. 3 2 1 Life Cycle (PW) Savings 191.6 223.3 Mode of PW Cost TFN 1458.6 1290.2 1245.6 Rank by Mode 3 2 1 Life Cycle (PW) Savings 168.4 213.0 Truth value of strict dominance 0.45 0.09 Example adopted and modified from Dell Isola (1982) p126. *Salvage values (positive cash flow) Through similar computations, the ORD of the alternatives A and B were obtained as $1,298,324 and $1,266,571 respectively. Table 1 shows a summary of this analysis along with a ranking of the alternatives. With an objective of minimizing Present Worth Cost and based on the ORD of the computed TFN PW Costs, alternative B is the best, followed by alternative A, and lastly, the Original design. The estimated PW costs as indicated by the ORD implies a higher cost for each facility than the corresponding modes, e.g., for the Original Design, the ORD PW Cost is about $31,000 higher than the mode. Since the ORD is an expected value incorporating uncertainties, this higher cost estimate appears to be more realistic. Considering the modes of the computed TFN PW Costs, with the same objective of minimizing Present Worth Cost, the rank of preference is the same, i.e., alternative B is the best, followed by alternative A, and lastly, the Original design. The issue now is how much better is B over the other two alternatives. This question can be answered by the qualified comparison approach theoretically described earlier in this paper and illustrated in figure 1. For the numerical example just presented, the qualified comparison is shown in figure 2, with TFNs graphically representing the cost distribution of each alternative s cost.
Fig. 2: Comparison of present worth costs (Triangular fuzzy numbers) Considering B versus A as shown in figure 2 and employing equations 5-7, the following parameters can be computed: C x = (1539.1)(1290.2-1094.8) + 1094.8(1539.1-1245.6) (1539.1-1245.6) + (1290.2-1094.8) = 1272.4 µ 1 1539.1-1272.4 (B--A) = 1539.1-1245.6 = 0.91 0 (B--A) = 1-0.91 = 0.09 Considering B versus Original Design, C x = (1539.1)(1458.6-1282.2) + 1282.2(1539.1-1245.6) (1539.1-1245.6) + (1458.6-1282.2) = 1378.6 µ 1 1539.1-1378.6 (B--Orig.) = 1539.1-1245.6 = 0.55
0 (B--Orig.) = 1-0.55 = 0.45 These computed parameters for the truth value of strict dominance indicate that the statement Alternative B is strictly better than A has a truth value of 0.09 despite the amount of life-cycle cost savings of $168,4000. Also the truth value of the Alternative B is strictly better than Original design is 0.45 despite the amount of life-cycle cost savings of $213,000. Obviously, despite the amount of life-cycle cost savings, these truth values are not very comforting. The traditional life-cycle cost analysis, relying on the crisp single value of costs (represented by the mode in the TFN triplet) would have indicated a clear advantage of alternative B over the other choices, with savings in the costs. This may turn out to be a misleading decision. If the subjective uncertainties are incorporated as described in the proposed life cycle cost analysis algorithm, a better and more realistic decision can be made. 5 Conclusion A conceptual methodology has been presented to illustrate how the uncertainties introduced due to subjective estimating of cost variables, typically based on expert opinions, can be handled in life cycle cost analysis of facilities. Utilizing fuzzy numbers to represent decision variables will enhance life cycle cost computations to obtain realistic values for evaluation and comparison of facilities. Simply choosing an alternative as being better than the others, based on the traditional approach of life cycle costing, may not be strictly correct. In other words, the confidence of the decision maker s choice depends on the level of uncertainty in the variables considered. This paper has developed a method to estimate such measure of confidence. By reducing the level of subjective uncertainties in the input variables, a more comfortable truth value of the dominance of one alternative over the other may be obtained. This will be a better and more realistic decision. 6 References Dell Isola Alphonse (1982) Value Engineering in the Construction Industry, Third Edition, Van Nostrand Reinhold Company, New York. Kaufmann, A. and Gupta, M. M. (1985) Introduction to Fuzzy Mathematics - Theory and Applications, Van Nostrand Reinhold Company, New York. Kaufmann, A. and Gupta, M. M. (1988) Fuzzy Mathematical Models in Engineering and Management Science, Elsevier Science Publishers B. V., The Netherlands. Lee, Jin H. and Aktan, Haluk M. (1997) A Study of Building Deterioration, Proceedings of the Infrastructure Condition Assessment: Art, Science, and Practice, American Society of Civil Engineers (ASCE), pp 1-10.
Uzarski, Donald, R. and Burley, Laurence A. (1997) Assessing Building Condition by Use of Condition Indexes, Proceedings of the Infrastructure Condition Assessment: Art, Science, and Practice, American Society of Civil Engineers (ASCE), pp 365-374. Watson, S., Weiss, J., and Donnel, M. (1979) Fuzzy Decision Analysis, IEEE Trans. System, Man., and Cybernetics, SMC-9. Whalen, T. (1987) Introduction to Decision Making Under Various Kinds of Uncertainty, Optimization Models Using Fuzzy Sets and Possibility Theory, Theory and Decision Library, Series B: Mathematical and Statistical Methods, Kacprzyk, J. and Orlovski, S. A. (eds), D. Reidel Publishing Company, Holland. Zadeh, L. A. (1965) Fuzzy Sets, Information and Control, 8, 338-353.
Appendix A Present Worth Cost Calculations with Fuzzy Numbers PW(ORIG.) = [900,1000,1200] + [0,0,0][(1 + 0.10) -8 ] + [0,0,0][(1 + 0.10) -10 ] + [8,10,14][(1 + 0.10) -16 ] - [75,80,88][(1 + 0.10) -20 ] + {[26,30,35]/0.10}{1 - (1 + 0.10) -20 } + {[20,25,32]/0.10}{1 - (1 + 0.10) -20 } = [900,1000,1200] + [0,0,0] + [0,0,0] + [1.741,2.176,3.046] - [11.145,11.888,13.077] + [221.364,255.420,297.990] + [170.280,212.850,272.448] = [1282.240,1458.558,1760.407] PW(A) = [600,700,850] + [150,200,210][(1 + 0.10) -8 ] + [0,0,0][(1 + 0.10) -10 ] + [175, 200,240][(1 + 0.10) -16 ] - [90,100,125][(1 + 0.10) -20 ] + {[30,35,37]/0.10}{1 - (1 + 0.10) -20 } + {[17,20,26]/0.10}{1 - (1 + 0.10) -20 } = [1094.839,1290.230,1517.996] PW(B) = [750,900,1100] + [0,0,0][(1 + 0.10) -8 ] + [18,20,25][(1 + 0.10) -10 ] + [0,0,0][(1 + 0.10) -16 ] - [71,75,89][(1 + 0.10) -20 ] + {[22,25,29]/0.10}{1 - (1 + 0.10) -20 } + {[12,16,23]/0.10}{1 - (1 + 0.10) -20 } = [1035.864,1245.639,1539.141]