Selection of the Utility Function of the Historic Building, Taking into Account Energy Efficiency

Abstract

The energy efficiency of the building should be understood as the degree of preparation of the building to ensure the comfort of its use in accordance with its intended use with the lowest possible energy consumption of the building. The article presents an in-depth analysis of the possibility of changing the utility function of a historic building in such a way that ensures all aspects of comfort while meeting energy efficiency conditions. Combinatorics methods were used for this purpose. Four possible utility functions were proposed, including the reference state, i.e., the existing state. Five aspects of comfort were considered: thermal comfort, carbon footprint, energy efficiency, noise and vibration. For these five aspects of comfort, boundary conditions were adopted depending on the adopted building class. The selected utility function is therefore the result of comfort, energy efficiency and economic aspects. The purpose of the study was to verify whether the developed methodology for the selection of the utility function for historic buildings, based on combinatorial analyses, would allow the selection of the optimal function from the point of view of energy efficiency, user comfort and environmental impact. The methodology was tested on a historic building located in southern Poland. The new utility function for the analyzed building is to use the historic villa (or some part of it) as an Art Gallery.

1. Introduction

An energy efficiency assessment is an assessment of a set of building properties that affect the energy consumption of the building necessary for its use, including, i.a., an assessment of the thermal insulation of the building partitions and the efficiency of the installations and devices used in it.

As the aspect of energy efficiency is a necessity but also a very big challenge, the approach to building design needs to change [1]. Firstly, in addition to minimizing energy consumption, the criterion of minimizing the building’s impact on the environment must be adopted. Another criterion is to optimize the multi-faceted comfort of the buildings [2,3,4,5,6]. The criterion of minimizing energy consumption is closely related to the environmental impact of the building, i.e., the emission of harmful gases over the entire life cycle of the building [7]. For such purposes, the building should be considered as a whole coherent system that obtains energy from renewable sources [8] and energy storage [9] and uses this form of energy [10]. The first step must, of course, in the case of new buildings, be to ensure that the thermal insulation of the building envelope [11] is excellent and that the building envelope is highly air- and heat-tight. In the case of thermally upgraded buildings, it is also necessary to ensure very good thermal protection and sealing of the envelope. In the case of historic and conservation buildings, heritage protection is at the forefront. This is a specific group of buildings. Historic buildings in Poland are exempted from thermal protection requirements due to the priority of protecting historical heritage. However, many buildings in the heritage group are currently in use [12]. In this paper, the authors showed how the energy efficiency of a historic building can be improved and what measures need to be taken to achieve the standard of a building with almost zero energy demand and to achieve climate neutrality with an overriding respect for cultural heritage. Figure 1 shows the criteria for the thermomodernization of listed buildings and buildings under conservation protection.

Of course, achieving the nearly zero energy (nZEB) standard applicable in European Union countries is defined in each country, depending on the possibilities of the construction market and taking into account the cost-optimal level [13]. In Poland, for new buildings to achieve the nZEB standard, they must meet the minimum thermal insulation requirements and the minimum level specified for Primary Energy. For buildings undergoing thermal upgrading, only the requirement for thermal insulation of the partitions applies. The requirements for the nZEB standard in Poland are not specified [14]. At the end of 2020, approximately 80.000 historic buildings and areas were registered in the Polish Register of Historical Monuments [15]. In addition to the Register of Monuments, communal and provincial registers are kept, which include as many as ca. 700 thousand objects. All historic buildings are subjected to the conservator’s care and it is only in agreement with the conservator and by obtaining the relevant permits that modernization measures and measures improving the energy efficiency of a building can be carried out. At the same time, given the very large number of listed buildings and the fact that they were constructed in different years when most of them were not subjected to thermal protection regulations, listed buildings, especially those that are in use, represent a huge potential for reducing the consumption of heat and electricity and lowering the carbon footprint of the building sector. Historic buildings, constructed over centuries, are usually in a poor state of repair, which can accelerate the process of deterioration and destruction. Every effort should therefore be made to improve the technical condition of the buildings and at the same time, with possible thermos modernization measures, to reduce the energy intensity of this group of buildings. As there are no regulations in Poland concerning the requirements for the nZEB standard for historic buildings, the authors propose that such standards should be created. However, the parameter defining the standard should not be the thermal insulation of partitions. Conservators of historic buildings rarely agree to insulate partitions or replace windows. It can be assumed, however, that the requirement for an nZEB standard for historic buildings would be to carry out any energy efficiency improvement elements permitted by the conservator. In contrast, it makes more sense to set levels for the non-renewable primary energy indicator. This is all the more reasonable given the 2021 amendment to the RES (Renewable Energy Sources). Act in Polish law introduced a number of changes and solutions in the renewable energy sector. Among other things, a new type of renewable energy market participant has emerged—the virtual prosumer. A virtual prosumer is a final customer generating electricity exclusively from renewable energy sources for its own needs in a renewable energy source installation connected to the electricity distribution network in a different place than the place of supply of electricity to that customer. A virtual prosumer, therefore, is a system that allows electricity to be generated outside the place where the energy is consumed. Often, the reason for not investing in photovoltaics is that the roof area is too small or precisely for conservation reasons. The RES investment can be located off-site where the energy is consumed. An additional requirement, according to the authors, should be a leakage test of the building envelope with thermal imaging diagnostics and dampness of the partitions, followed by the sealing of the leaks and elimination of the dampness, if possible, due to the consent of the conservation officer.

In our methodology, we made the assumption that we were calculating the carbon footprint of the materials used in the thermal upgrading of historic buildings and the carbon footprint from emissions over a 30-year period. This was based on the assumptions of the methodology we developed for estimating the carbon footprint of historic buildings. The 30-year operational carbon footprint is based on our assumptions, as we developed the methodology for the Local Government of Małopolska in 2020, adding 30 years, we arrive at 2050, when all buildings should reach a climate-neutral standard. In 2050, there may already be a completely different calculation methodology, other possibilities and most importantly, databases of embedded CO₂ emissions from historical materials may be available. As of today, we can accurately use databases of materials currently available on the market (materials for thermal upgrading). At the Cracow University of Technology, we have just started scientific work on estimating the carbon footprint for historical materials and installations.

The carbon footprint, according to the authors’ proposal, should be calculated as the embedded carbon footprint for the materials used for thermal retrofitting and the operational carbon footprint calculated for 30 years of use of the building. In the methodology, the authors assumed that for the calculation of the embedded carbon footprint, the materials from which the historic building is made would not be taken into account, as this would be a very big complication when it comes to obtaining information.

Table 1. Proposed approach to calculating the carbon footprint of historic buildings.

INFORMATION REGARDING THE LIFE CYCLE OF THE BUILDING

PRODUCT STAGE

CONSTRUCTION PROCESS STAGE

USESGATE

END OF LIFE STAGE

A1

A2

A3

A4

A5

B1

B2

B3

B4

B5

B6

B7

C1

C2

C3

C4

Raw material supply

Transport

Manufacturing

Transport

Construction–installation process

Use, installed products

Maintenance

Repair

Replacement

Refurbishment

Operational energy use

Operational water use

Deconstruction

Transport

Waste processing

Disposall

shows all the phases of a building’s life from the raw material acquisition phase to the recycling phase. In the case of existing buildings and even more so in the case of historic buildings, it is almost impossible to obtain information on the carbon footprint of the materials and technologies from which the building is made. It would be necessary to have knowledge of the production of historic materials, the way they were transported or the technology of constructing historic buildings. Therefore, only phases B5 and B6 are included in the analysis presented here. Phase B5 refers to the CO₂ emissions for the building materials that were used in the thermal refurbishment. Here, it was limited to insulation materials, windows and doors. New installations, automation systems and lighting were not analyzed. There are no CO₂ emission databases in Poland for these technical systems. Table 1 shows the proposed approach to calculating the carbon footprint of historic buildings.

The calculated carbon footprint over the 30-year life cycle of the building should be balanced by purchasing credits or planting trees. In the methodology, the authors adopted a parameter to determine the percentage reduction in the carbon footprint after thermal modernization measures compared to the baseline.

Noise and vibration, as physical phenomena, appear to be similar in nature—both are wave phenomena. Despite their similar nature, they are often considered separately in the standards or regulations, both in national and international standards. Noise and vibration are parts of comfort in rooms. They also increasingly appear as one of the aspects of environmental pollution in the Directive [16]. It could be found that:

-

“1. Description of the project, including in particular:

-

description of the physical characteristics of the project and the land use requirements during the individual construction and operation phases

-

description of the main features of the production processes, for example, type and quantity of materials used,

-

assessment of the type and quantity of expected residues and emissions (water, air and soil pollution, noise, vibration, light, heat, radiation, etc.) resulting from the operation of the proposed project.”

An emission is one of the aspects of pollution.

2. Comfort Aspects

2.1. Thermal Comfort

The assessment of thermal comfort can be carried out by means of a questionnaire method or on the basis of “in situ” surveys. According to the PMV comfort scale developed by Ole Fanger [17], seven evaluation points are defined, where thermal equilibrium conditions are indicated by “0”. Fanger’s thermal comfort scale is based on seven points, which stand for the following: (+) 3—hot, (+) 2—warm, (+) 1—fairly warm, 0—indifferent, (−1)—fairly cool, (−2)—cool and (−3)—cold. For each point on the scale, an expected percentage dissatisfied with the prevailing conditions (PPD) is assigned. The

Table 2. Default design categories for mechanically heated and cooled buildings [18].

Category	Coefficient PPD and PMV
	PPD [%]	PMV [-]
A	<6	−0.2 < PMV < +0.2
B	<10	−0.5 < PMV < +0.5
C	<15	−0.7 < PMV < +0.7

shows the index values depending on the room class.

Another method adopted for the analyses carried out in this paper is to adopt default values for the internal operative temperature in winter and summer of mechanically heated and cooled buildings. The temperatures are summarized in

Table 3. Default values for the internal operative temperature in winter and summer of mechanically heated and cooled buildings.

Type of Building	Category	Operative Temperature [C]
		Minimum for Heating (Winter Season) clo Value 1.0 clo *	Maximum for Cooling (Summer Season) 0.5 clo *
Residential buildings, offices conference rooms, restaurants	A	21.0	25.5
	B	20.0	26.0
	C	18.0	27.0

* clo—insulation factor of clothing.

.

2.2. Acoustic Comfort

Acoustic comfort is a concept that can be characterized as the absence of unwanted sound. In order to achieve acoustic comfort in a building, certain requirements must be met regarding insulation from airborne sounds (R′_w), insulation from impact sounds (L′_n,w) and the level of noise from traffic and building service (L_A,eq). So, it is the result of these three parameters. However, for the article’s purposes, only levels of noise from traffic and building services were taken into account. L_A,eq can be described mathematically by the following equation:

$${\mathrm{L}}_{\mathrm{A},\mathrm{e}\mathrm{q}}=10{\mathrm{l}\mathrm{o}\mathrm{g}}_{10}\frac{1}{{\mathrm{T}}_{\mathrm{M}}}\underset{0}{\overset{{\mathrm{T}}_{\mathrm{M}}}{\int }}{\left(\frac{{\mathrm{P}}_{\mathrm{A}}\left(\mathrm{t}\right)}{{\mathrm{P}}_0}\right)}^2\mathrm{d}\mathrm{t}$$

(1)

where L_eq is the equivalent continuous linear-weighted sound pressure level re 20 µPa, determined over a measured time interval T_M (s); P(t) is the instantaneous sound pressure of the sound signal; P₀ is the reference sound pressure of 20 µPa.

2.3. Vibrational Comfort

The criteria pertaining to human exposure to vibration in buildings are contingent upon the chosen evaluation method. Four primary methods incorporate considerations for the following parameters:

-

cceleration (velocity) of vibration, corrected across the entire frequency range.

-

spectrum (frequency structure) of the root-mean-square (RMS) of acceleration (velocity) of vibration in 1/3 octave bands.

-

vibration dose value (VDV).

-

maximum transient vibration value (MTVV).

The ISO standard [19] designates the RMS method as the fundamental evaluation approach, while acknowledging that supplementary methods may be necessary in certain cases. As additional methods, ISO delineates the MTVV method and the VDV method. Particularly in situations characterized by high crest factors, the VDV method, among these two, is recommended. The RMS method entails the averaging of acceleration values over the duration of time:

$${\mathrm{a}}_{\mathrm{w}}={\left[\frac{1}{\mathrm{T}}{\displaystyle \underset{0}{\overset{\mathrm{T}}{\int }}{\mathrm{a}}_{\mathrm{w}}^2\left(\mathrm{t}\right)\mathrm{dt}}\right]}^{\frac{1}{2}}$$

(2)

where

• a_w(t)—represents the weighted acceleration as a function of time [m/s²],
• T—denotes the duration of measurement [s].

The MTVV method similarly averages acceleration values; however, it exhibits heightened sensitivity, especially towards occasional shocks and transient vibrations, achieved through the utilization of a short integration time constant.

$${\mathrm{a}}_{\mathrm{w}}({\mathrm{t}}_0)={\left[\frac{1}{\tau }{\displaystyle \underset{{\mathrm{t}}_0-\mathsf{\tau }}{\overset{{\mathrm{t}}_0}{\int }}{\mathrm{a}}_{\mathrm{w}}^2\left(\mathrm{t}\right)\mathrm{dt}}\right]}^{\frac{1}{2}}$$

(3)

$$\mathrm{MTVV}={\max [\mathrm{a}}_{\mathrm{w}}{(\mathrm{t}}_0\left)\right]$$

(4)

where

• τ—represents the integration time (for running averaging, it is recommended to use τ = 1 s),
• t₀—denotes the time of observation (instantaneous time).

The VDV method is particularly advantageous for capturing peaks in recorded signals due to its utilization of the fourth power, in contrast to the second power employed in RMS and MTVV methods:

$$\mathrm{VDV}={\left[{\displaystyle \underset{0}{\overset{\mathrm{T}}{\int }}{\mathrm{a}}_{\mathrm{w}}^4\left(\mathrm{t}\right)\mathrm{dt}}\right]}^{\frac{1}{4}}$$

(5)

The corrected value is established through the measurement of vibration at the point of transmission to the individual, employing a correction filter. This straightforward method yields information about comfort exceedances but does not encompass details regarding the frequency of such exceedances. The basic RMS analysis is made in a 1/3 octave band to show in which frequencies the limit is exceeded. This method is mostly shown in a graphic form, in which the threshold of vibration is the lowest line and the comfort limit is the upper line and depends on the room destination, daily time and character of vibrations (how often they appeared). To make comfort limit from the threshold of vibration, this threshold must be multiplied by the so-called coefficient “n” (

Table 4. The coefficient “n” values.

Room Destination	Daily Time	Coefficient, “n” Value
		Continuous or Periodic Vibrations Occurring More than 10 Times a Day	Occasional Vibrations Occurring Less than 10 Times a Day
Hospitals (operating theatres)	Day Night	1	1
Hospitals (patients rooms)	Day	2	8
	Night	1	4
Residential	Day	4	32
	Night	1.4	4
Offices	Day Night	4	64
Workshops	Day Night	8	128

acc. [19]).

The values of the “n” coefficient will be the basis for comfort estimation in the adopted adaptive design variants of the building selected in this article. In this paper, the most popular method RMS is taken into account, but with modification to the so-called HPVR (human perception of vibration factor). This factor was first introduced in 2017 to Polish standards [20]. It is a very useful ratio that shows how many times the threshold of human perception of vibration is exceeded:

$$\mathrm{H}\mathrm{P}\mathrm{V}\mathrm{R}=\left(\frac{{\mathrm{a}}_{\mathrm{R}\mathrm{M}\mathrm{S}}}{{\mathrm{a}}_{\mathrm{w}}}\right)$$

(6)

where a_RMS—acceleration RMS value obtained from the analysis and a_w—acceleration RMS value equivalent to the threshold for the perception of vibration in the same 1/3 octave band as in a_RMS.

The HPVR ratio was used to compare different comfort parameters. An example of such analysis is shown in Figure 2.

Human perception of vibration depends also on the direction of vibration, that is why in Figure 2, red lines are in vertical direction and black lines are in horizontal directions. The lower line corresponds to the threshold of vibration, the middle to the comfort at night and the upper line to comfort during the day. All lines are suitable for residential buildings. The horizontal lines refer to the ordinate axis in the logarithmic system.

3. Methodology

Combinatorics methods are used for the determination of admissible combinations of perspective functions. The idea of using combinatorial methods in architecture, interior and exterior [21,22] appearing in the technology of structural synthesis of electromechanical energy converters with the effect of new properties [22,23], has been successfully applied; therefore, there are reasons to apply these methods in order to optimize and combine different aspects of comfortable living in the restored historical building with cultural heritage and economic aspect.

Combinatorics analysis is a branch of mathematics that deals with the selection and arrangement of elements of some, usually finite, set according to given rules. So, if there is a set consisting of n elements (that is, the power of the set is equal to n), then each such rule determines the way in which a certain combinatorial configuration is constructed from the elements of this original set. Accordingly, the purpose of combinatorial analysis is to determine the algorithms for building combinatorial configurations, their research and the quantitative solution of the list [24].

$$P_n=n!$$

(7)

where n! is a factorial that is equal to the product of all natural numbers from 1 to n inclusive.

Arrangements are called combinatorial configurations consisting of m different elements that were selected from the set with a power of n and for which the order of their arrangement in the combinatorial configuration matters. The number of all possible placements depends on the power n of the set from which these elements were obtained, and on the number m of these elements. The number of placements is calculated according to the formula:

$${\mathrm{A}}_n^m=\mathrm{n}\cdot (\mathrm{n}-1)(\mathrm{n}-2)\dots (\mathrm{n}-\mathrm{m}+1),\mathrm{m}\le \mathrm{n}$$

(8)

The main formulas of combinatorics are used in the theory of probabilities. Note that the elements of the original set are considered different and no repetitions can occur when building combinatorial configurations. Therefore, combinatorial configurations are built without repetitions.

Combinatorial configurations consisting of n different elements are called permutations. These configurations differ from each other only in the order of placement of these elements. The number of all possible permutations for a given set depends on its power n and is calculated by the formula:

$$A_n^m=\frac{n!}{(n-m)!}$$

(9)

Note that when m = n, that is, all elements of the set that has power n are considered, and the order of placement of these elements is important (according to the definition of placement), then we obtain the formula:

$$A_n^n=\frac{n!}{(n-n)!}=\frac{n!}{0!}=n!$$

(10)

While choosing a formula for calculating compounds, two basic rules are used to decide which combinatorial formula should be used (Figure 3).

Whether the formula will be combinations, permutations or placement, depends on the answer to two questions:

(1)

Is the order of placement of elements taken into account? If the answer is “no”, then we apply the combination formula. If “yes”, then we have two options, which depend on the answer to the following question:

(2)

Are all elements included in the sample (combination)? If “yes”, then we calculate through permutations; if “no”, we apply the placement formula.

In order to improve the utility function of the building, namely the level of socio-cultural and economic benefits from the restoration of the historical building, the authors proposed the simultaneous pairwise implementation of promising functions proposed and studied earlier for single implementation on the ‘Stara Polana’ villa [9]:

• Public building—hostel (existing condition (‘Reference variant—RV’);
• Public building—five-star hotel (‘Variant V1’);
• Public building—Zakopane Art Gallery (‘Variant V2’); and
• Public building—conference and training centre with accommodation option (‘Variant V3’), using methods of combinatorics to determine all permissible combinations of the specified functions (

  Table 5. Possible versions of simultaneous supported implementation of the specified functions.

  	‘RV’	‘V1’	‘V2’	‘V3’
  ‘RV’		+	+	+
  ‘V1’	+		+	+
  ‘V2’	+	+		+
  ‘V3’	+	+	+

  ).

The ‘Stara Polana’ building is located in the city centre of Zakopane. The building is owned by the Cracow University of Technology. Currently, the building is used as a shelter with a partial gallery function (painting exhibition). The building is located among low-rise buildings at Nowotarska St. The building is a historic building, Witkiewicz’s villa style, which was built in 1905. The facility is of wooden construction with a stone basement. The basement is made of stone, roofed with a stone vault and topped with brick and steel sectional floorings. The ground floor, first floor and attic, both walls and floors are made of wood.

The “+” sign in Table 5 indicates the possibility of a combination of promising functions, indicated vertically and horizontally. In this way, we receive not only a quantitative characteristic of possible pairwise combinations, which were determined by the Formula (11), but also a qualitative list of options for further analysis.

As can be seen from Table 5, according to Formula (9), calculating the number of compounds takes into account the order of placement of elements for n = 4 promising functions by m = 2 functions in one combination.

$${\mathrm{A}}_{\mathrm{n}}^{\mathrm{m}}=\frac{\mathrm{n}!}{\left(\mathrm{n}-\mathrm{m}\right)!\times \mathrm{m}!}=\frac{4!}{\left(4-2\right)!\times 2!}=6$$

(11)

We can obtain six paired combinations that are on field of practical interest:

• RV + V1—public building—hostel (existing condition (‘Reference variant’) + five-star hotel (‘Variant 1’)
• RV + V2—public building—hostel (existing condition (‘Reference variant’) + Zakopane Art Gallery (‘Variant 2’)
• RV + V3—public building—hostel (existing condition (‘Reference variant’) + conference and training centre with accommodation option (‘Variant 3’).
• V1 + V2 public building—five-star hotel (‘Variant 1’) + Zakopane Art Gallery (‘Variant 2’)
• V2 + V3 public building—Zakopane Art Gallery (‘Variant 2’) + conference and training centre with accommodation option (‘Variant 3’)
• V1 + V3 public building—five-star hotel (‘Variant 1’) + conference and training centre with accommodation option (‘Variant 3’).

Calculation of the values of the criteria for evaluating the pairwise application of prospective functions (F1-thermal comfort expressed in terms of operative temperature, F2-percentage reduction in calculated carbon footprint after thermal upgrading measures compared to baseline, F3-non-renewable primary energy indicator EP, F4-Noise in L_A,eq, F5-Vibrations in HPVR, PA-index-profitability index) can be obtained by calculating the average value of the specified criterion based on the known values of the same criterion for a single implementation of prospective functions. This ratio is expressed by dividing the sum of discounted positive cash flows by the sum of discounted negative cash flows:

$$PA-index=\frac{Present Value \left(PV\right)of future cash flows}{initial investment}$$

(12)

If the value of the utility function is greater than 1 (PA-index > 1), the adaptation of the object is profitable for the considered variant. The higher the value of the indicator, the more profitable the new variant option is.

4. Results of Analysis

A preliminary evaluation of the realization of promising functions of the reconstruction of the “Stara Polyana” villa was carried out by the authors by analyzing the criteria:

F1 Thermal comfort expressed in terms of operative temperature (class A, class B, class C)

F2 Percentage reduction in calculated carbon footprint after thermal upgrading measures compared to baseline [%] (class A, class B, class C)

F3 Non-renewable primary energy indicator EP [kWh/(m²year)] (class A, class B, class C)

F4 Noise in L_A,eq (class A, class B, class C)

F5 Vibrations in HPVR (class A, class B, class C)

PA-index–profitability index, using the method of multi-criteria analysis.

In

Table 6. Upper limits for the three classes of buildings.

	RV	V1	V2	V3
F1	18/27.0—class C	21/25.5—class A	18/27.0—class C	20/26—class B
F2	CF reduction: 20%—class C	CF reduction: 60%—class A	CF reduction: 20%—class C	CF reduction: 40%—class B
F3	100—class C	70—class A	100—class C	70—class A
F4	35—class C	25—class A	30—class B	25—class A
F5	1.4—class C	0.8—class A	1.2—class B	1.2—class B
PAI	1.03	1.06	0.05	0.56

, there are upper limits for the three classes of buildings in which class A is the most demanding building (luxury), class B is the upper-class building and class C is a building that only meets the standard requirements (the typical building that does not require special conditions. Classes A, B and C have been established for the purposes of this article. Class A is the highest class while C is the lowest class of the building. Class C simply means meeting the standard requirements for all aspects of comfort, from thermal through acoustic to vibration. The criteria for classes A and B were established by the authors based on their knowledge and experience.

While analyzing the criteria for the compatible pairwise implementation of the perspective functions of the reconstruction of the “Stara Polyana” villa, the assessments of criteria F1–F5 were made based on the requirement to comply with a stricter condition for choosing a class, which would ensure compliance with the class required for the implementation of a more “demanding” version of the perspective function (

Table 7. Summary table for average values for comfort criteria.

	RV + V1	RV + V2	RV + V3	V1 + V2	V2 + V3	V1 + V3
F1	21/25.5 Class A	18/27.0 Class C	20/26 Class B	21/25.5 Class A	20/26 Class B	21/25.5 Class A
F2	CF Reduction: 60% Class A	CF Reduction: 20% Class C	CF Reduction: 40% Class B	CF Reduction: 60% Class A	CF Reduction: 40% Class B	CF Reduction: >60% Class A
F3	70—class A	100—class C	100—class C	70—class A	85—class B	70—class A
F4	25—class A	30—class B	25—class A	25—class A	25—class A	25—class A
F5	0.8—class A	1.2—class B	1.2—class B	0.8—class A	1.2—class B	1.2—class B
PAI	1.045	0.54	0.795	0.555	0.33	0.81

); the calculation of the PA-index was carried out by calculating its average value for pairwise implementation of perspective functions according to known Profitability Indexes for their single implementation.

Despite the simple method of calculating the values of the criteria for evaluating the pairwise application of prospective functions, the obtained values allow customers (building owners and investors) to estimate the benefits in the PA-index of the simultaneous implementation of two prospective functions compared to their individual implementation (

Table 8. Simultaneous implementation of two prospective functions vs. their individual implementation in the PA-index.

Paired Combination/Single Implementation	PAI Ratio
RV + V1/RV	1.01
RV + V1/V1	0.98
RV + V2/RV	0.77
RV + V2/V2	10.8
RV + V3/RV	0.77
RV + V3/V3	1.42
V1 + V2/V1	0.52
V1 + V2/V2	11.1
V2 + V3/V2	6.1
V2 + V3/V3	0.54
V1 + V3/V1	0.76
V1 + V3/V3	1.43

).

Particular attention should be paid to the increase in the level of the PA-index (the ratio of the total costs for the restoration and renovation of the building to the total prospective profits [24]) in the paired combinations RV + V2, V1 + V2 and V2 + V3 (shown in bold in Table 8) compared with the previously calculated RA-index for a single implementation:

• For the RV + V2 combination, the RA-index is 10.8 times higher than the one for V2
• For the combination of V1 + V2, the RA-index is 11.1 times higher than the similar one for V2
• For the V2 + V3 combination, the RA-index is 6.1 times higher than the one for V2

The indicated ratios show that the option of renovating the original building in Zakopane Art Gallery, which is undoubtedly the best from the point of view of social benefits and preserving the historical authenticity of the building, in the case of implementing only this function, is an extremely unprofitable project (recall that investment profit is possible in the case when PA-index > 1). On the other hand, joint implementation of the public building—five-star hotel (‘Variant 1’) + Zakopane Art Gallery (‘Variant 2’) will increase the profit from investment costs for reconstruction 11.1 times, with public building—hostel (existing condition (‘Reference variant’) + Zakopane Art Gallery (‘Variant 2’) 10.8 times, and public building—Zakopane Art Gallery (‘Variant 2’) + conference and training centre with accommodation option (‘Variant 3’) 6.1 times.

5. Discussion

The utilization of combinatorial methods in architecture, both interior and exterior, has been prevalent since the 1960s and continues to be relevant in contemporary architectural developments. This is evident in the integration of nature into architectural structures, contributing to the emergence of “ecological”, “earthy”, “green” and “nature-integrated” architecture globally. The identified types of combinatorics in nature-integrated buildings at the object level include combinatorics of identical figures, combinatorics of similar figures, ascending combinatorics and the figurative combinatorial method.

Various methods of incorporating the natural component of vegetation into the formation of nature-integrated buildings have been recognized. These methods encompass combinatorics of vegetation on facade planes, layer-by-layer modifications of combined plant and artificial pseudo-grid elements in architectural form, application of plant modules on facades, defragmentation and filling of facade grid cells with green modules or “spots”, green compositions and panels on facade planes, landscaping of external fences forming the “green skin” of a building and functional landscaping of rooftops, balconies, terraces and recessed balconies, facilitating tactile interaction with vegetation.

The combinatorics of nature-integrated architecture demonstrates the diffusion of modern algorithmic methods and form-making techniques, incorporating both contemporary and classical modelling approaches. The continual development and enhancement of architectural form-making methods are evident in the rethinking of these approaches, representing a constructive step in advancing the theory of architectural form making, as articulated in the Combinatorial Technique and Means [21].

The term “combinatorial architecture” has been coined to describe a specific architectural direction that integrates quantitative and qualitative parameters (social, physical, sensorial, cultural and economic) into the architectural design process. This approach is embodied in a three-phased architectural design model, emphasizing the integration of diverse factors in the design process [22].

Combinatorial methods are also successfully applied in the technology of structural synthesis of electromechanical energy converters, resulting in the emergence of new properties. The systemicity of the combinatorial phenomenon, specifically hybridization, summarizes the mechanisms governing structure formation, taxonomy, methodology of synthesis and analysis of hybrid electromechanical objects at intraspecies and intergenic levels. These findings contribute to the development of the theory of genetic evolution of technical systems and interdisciplinary research [23].

The universality of combinatorics methods is demonstrated through their application in optimizing and combining different aspects of comfortable living in the restoration of historical buildings with cultural heritage and economic considerations.

6. Conclusions

The use of combinatorial methods in the analysis of the possibilities and advantages of the common implementation of promising directions for the renovation of the villa “Stara Polana” compared to their individual implementation, carried out in this article, allowed us to determine their full number (having four single perspective functions, using the formulas of combinatorics, we obtained six possible options for their paired implementation) and analyzed all possible variants of such combinations.

The results of the analysis of the economic advantages of the paired implementation of prospective functions (by comparing the PAI value for their implementation with the corresponding PAI for the implementation of single functions) are of interest, first of all, at the stage of substantiating the efficiency of the renovation for potential investors.

As a result of the analyses performed using mathematical combinatorics, very interesting results were obtained, which could be further elaborated. There is great potential for three combinations of functional variants of the historic building:

• Hostel + Art Gallery
• Five-star hotel + Art Gallery
• Conference center + Art Gallery

In all three economically beneficial investment options, there is a direction to use the villa “Stara Polana” (or a certain part of it) as a Zakopane Art Gallery. During renovation, this variant requires special attention due to the acoustic insulation and load-bearing capacity of the floor, which results from the need to maintain silence in art objects of this level (both during the assembly and disassembly of temporary exhibitions and during their visit) and the possibility of placing sculptures permanently or temporarily. To ensure acoustic and vibrational comfort (especially in variants with five-star hotels), it is good to apply the technology of vibroacoustic floor–patent no. P.432694 “method of insulating and soundproofing wooden floors”. It is worth noting that in many historical buildings in Poland, the wooden flooring is the prevalent one.