Life Cycle Equivalent Annual Cost (LCEAC) as a comparative indicator in the life cycle cost analysis of buildings with different lifetimes

The paper deals with the issue of the life cycle cost in the construction. It also focuses on the part of the model for the calculation of the life cycle cost of buildings with consideration of the risk that allows the calculation of the life cycle cost of buildings with different lifetimes. For the construction of this part of the model, the authors used the life cycle equivalent annual cost indicator (LCEAC). The operation of this part of the model was presented on the example of three scenarios of the life cycle of pneumatic sports halls for which the estimated service life of the building (ESLB) was ranged from 19 to 30 years.


Introduction
One of the issues directly related to the life cycle in the construction are the life cycle cost of buildings, mentioned by the authors of the publications: [1][2][3][4][5][6][7].
As defined in the work of Boussabaine and Kirkham [8], life cycle cost are expressed in monetary value of all significant costs that are related both to the initial cost of the construction and its subsequent operation, and to the final withdrawal. Estimation of the life cycle cost is a calculation. Therefore, the impact of the risk should be taken into account in the determined value. The ISO 15686-5:2008. "Buildings and constructed assets. Service life planning: Life cycle costing" [9] notes that there is no methodology for calculating the life cycle cost with consideration of the risk, which in the case of construction, would be based merely on the expert knowledge as in many over cases of risk management in construction [8,10].
This study is a continuation of the research work of the authors on the construction of the model for the calculation of the life cycle cost of buildings with consideration of risk factors. The results of the research work on the model were presented by the authors in [6,7] so far. The main aim of this study is to present a part of the model that allows the calculation of the life cycle cost of buildings with different lifetimes using the life cycle equivalent annual cost indicator (LCEAC). The operation of this part of the model is presented on the example of three scenarios of the life cycle of pneumatic sports halls for which the estimated service life of the building (ESLB) is different.

Algorithms and methods used in the construction of the model
The structure of the author's model for calculating the life cycle cost of buildings with consideration of risk factors is based on the possibility theory and fuzzy sets.
The fuzzy sets theory is combined here with the dynamic method used for economic efficiency analysis of construction projects based on discounted cash flows (DCF). This is the net present worth method (NPW) which will be used for the construction of the model in a fuzzy version (as a fuzzy NPW method).
The calculation procedure of the model is based on the following terms, methods and analysis:  discounted cash flows analysis (DCFA) at a given value of discount rate (r),  net present worth method (NPW),  term of a fuzzy number which makes it possible to express a value encumbered with measurable uncertainty (risk),  decomposition theorem of a fuzzy set into α-cuts,  principle of the fuzzy expansion of Zadeh.
The fuzzy NPW method calculation procedure requires the calculation to be carried out in accordance with the decomposition theorem of a fuzzy set and principle of the fuzzy expansion of Zadeh. Without these rules, the performing basic arithmetic operations on fuzzy numbers, i.e. addition, subtraction or multiplication would be impossible [11].
In addition, the calculation procedure of the model is complemented by the following methods and algorithms that the authors described in their publications [6,7]. These are: 1. (DSW) algorithm developed first by the team of Dong, Shah and Wong [12], and later modified by the team of Givens and Tahani [13], involving the implementation of the principle of the fuzzy expansion of Zadeh in the context of performing basic arithmetic operations (addition, subtraction, multiplication and division) on two fuzzy numbers (marked as [a, b] and [c, d] in a given α-cut), it comes down to the use of appropriate mathematical formulas, 2. vertex method (VM) that eliminates errors that could occur during arithmetic operations on fuzzy numbers, in particular on their intervals after the decomposition of fuzzy sets into α-cuts [14], 3. centre of gravity method (CoG) used to sharpen the result.

Division and parametrization of the input data
The input data to the model for calculating the life cycle cost of buildings with consideration of risk factors are divided into the following parameters:  time (global), i.e. the lifetime (T i ), which is equal to the estimated service life of the building (ESLB),  time (local), namely, times (t ik ), (t im ), after which k-th periodic operating cost or m-th periodic benefit (revenue) is calculated, respectively,  financial (global), in the form of a discount rate (r), which is necessary to calculate the present value of a given monetary amount based on its value determined in the future time,  cost (understood as costs that may arise in the life cycle of a building), among which annual costs are distinguished -in particular, annual operating costs (C opA,ij ) and MATEC Web of Conferences 196, 04079 (2018) https://doi.org/10.1051/matecconf/201819604079 XXVII R-S-P Seminar 2018, Theoretical Foundation of Civil Engineering periodic ones -initial costs (C in,i ), periodical operating costs (C opNA,ik ) and withdrawal costs (C wd,i ), respectively,  cost (understood as revenues that may occur in the life cycle of a building), i.e. annual revenues (I opA,il ) and periodic -successively periodical revenues earned in the building operation phase (I opNA,im ) and revenues achieved in the decommissioning phase (I wd,i ). The division of individual parameters according to their global or local character is related to the fact that the building's service life (T i ) or the discount rate (r) are accepted as permanent parameters, conditioning and valid throughout the entire life cycle of the building. The local character for the remaining time and cost parameters is connected with the necessity of taking them into consideration only after those years (during the period of building operation), in which their occurrence is expected..
To parameterize any inputs to the module for estimating the total cost of the building life, convex and normal fuzzy numbers will be used, and the following two cases are distinguished:  if the parameter associated with discount rate, time, cost or income is not burdened with the influence of the impacting risk, the parameter will be modelled as a certain amount in the form of a fuzzy number with a singleton membership function with a degree of membership of 1.0 only for one particular value of the argument (x) (Figure 1a),  if any of the parameters mentioned above take into account the impact of risk, then it will take the form of an uncertain amount, modelled as a fuzzy number, or shaped as a membership function as in Figure 1b   The calculation algorithm for the model of estimating the total cost of living, especially considering the risk allowance in the life cycle of buildings, is presented in the publications of [6,7,15]. In these publications, the authors discussed in detail the parts of the model that concern life cycle cost analysis (LCCA) within the following scope: To be able to estimate the amount of the life-cycle cost of the i-th building (LCC i ) and the life cycle equivalent annual cost indicator (LCEAC i ), it is necessary to find a set of discounting factor values (PWF) which relate the future cost values to the present value, that is until the moment when a the life cycle cost analysis should be done. These coefficients are determined separately for initial, annual operational and periodical costs, as well as for decommissioning costs. Then, the fuzzy value of life cycle costs of the i-th building is calculated, LCC i = LCNPW i C according to the formula: where: cost data (C) is as described in chapter 2.2; PWF AC,i -present worth factor for annual costs of i-th building; n AC,i -number of annual costs of i-th building; n NAC,inumber of non-annual costs of i-th building; PWF NAC,ik -present worth factor for nonannual costs of i-th building in k-th year; PWF WD,i -present worth factor for withdrawal costs of i-th building. When comparing buildings whose liftimes are different, first the fuzzy values of the annual discount coefficient should be calculated (AF i ), and then the fuzzy values for the equivalent annual costs of i-th building (LCEAC i ). The following formulas serve this purpose: where: r -discount rate; T i -lifetime.

Subject of the analysis
The subject of the analysis of the the life cycle equivalent annual cost indicator of buildings (LCEAC) are three alternative life cycle scenarios of pneumatic sports halls used for roofing tennis courts, swimming pools, skate parks or football fields in the winter season (in Poland, usually up to 180 days a year). Sports facilities are often subject to cost analysis. This is discussed by such publications as [16][17][18]. Pneumatic halls are devoid of a heavy and expensive load-bearing structure, which means that the costs of their implementation are much lower than in the case of fixedconstruction halls. The covering of a pneumatic hall maintains a structure in space thanks to the overpressure in the interior with a value of from 250 to 350 (Pa). This overpressure is generated by the warm air coming from a blower device. The value of the heat transfer coefficient for the covering of the pneumatic hall ranged from 2.5 to 2.9 (W/m 2 K).   , and [i = 3]) are secured against breakdown resulting from an act of vandalism with a net of wire ropes, which limits the tear to a maximum of one grid field, and against weather factors, provided that the outer foil layer is replaced every 3-5 years. Table 1 presents results of the analysis of lifetimes (T i ) taking into account the abovedescribed factors affecting (ESLB).  For all the types of costs listed above, the values were determined based on market data from the companies offering pneumatic halls in Poland, i.e. Grimar Katowice, Interhall Katowice, Plastbud Warszawa, and Sporthalls Wrocł aw. Decommissioning costs were assumed to be equal to 0.00 (PLN) due to the assumption of full recycling of all components.
Moreover, it was assumed that future market changes (one type of financial risk factors) were very probable during the lifetime of the building. Therefore, the life cycle cost analysis (LCCA) also involved changes in the discount rate by modelling it in the form of an interval of achievable values. Table 2 presents the all cost, time and financial data assumed for the analysis. , respectively) were calculated with the use of the centre of gravity method (CoG) to sharpen the result. Figure 4 illustrates an output membership function for the life cycle equivalent annual cost (LCEAC) for the three analysed scenarios of pneumatic sports halls.  Table 3 illustrates the output values calculated for all scenarios of pneumatic sports hall in the chosen α-cut equal to 0.5.

Conclusions
The results of the calculations presented in chapter 3.3 indicate that the most favourable solution for a pneumatic sports hall due to the life cycle cost of the building (LCC) is scenario (i = 1). The (LCC) amount is the smallest for a double-layer hall ([LCC 1 ] and is equal to 1,796,800.00 [PLN]), but the scenario predicts that the lifetime of the hall is, however, the shortest of all three scenarios ([T 1 ] ranges from 19 to 30 [years]). On the basis of Figure 4 one can conclude that the most favorable solution for the planned investment is the implementation of scenario (i = 3) which generates the lowest value of the