Innovative Computational Techniques for Multi Criteria Decision Making, in the Context of Cultural Heritage Structures’ Fire Protection: Case Studies

Abstract

Fire protection for cultural heritage structures is a challenging engineering task that could benefit from the use of specialized computational tools relying on a performance-based design (PBD) concept rather than on prescriptive-based fire protection codes. In the first part of the present study, the theoretical basis of the proposed computational selection and resource (S and R) allocation model is discussed, related to the assessment of the fire safety index (FSI) and the authenticity preservation index (API). Furthermore, two different multi criteria optimization approaches are proposed to generate optimized fire protection upgrading designs, incorporating the nondominated sorting evolution strategies II (NSES-II) algorithm and the analytic target cascading (ATC) method. In this second part of the present work, the proposed S and R allocation model is implemented in two test cases; Villa Bianca, a famous mansion in Thessaloniki, Greece, and the Monastery of Simonos Petra located in Mount Athos, Greece. Several cases are examined regarding the targeted FSI or API values, taking also into account budget restrictions. In cases where the preservation of the authenticity is considered as an objective within the design process, the need to implement more sophisticated and customized fire protection measures can lead to a significant increase up to almost 200% regarding the total cost, subject to the pursued safety level. Detailed results obtained for each case study are presented and discussed comparatively, demonstrating the efficiency of the proposed S and R allocation model in a wide range of scenarios, as well as its possible utility in multiple applications, facilitating the fire protection design process. Finally, a comparison between the two multi criteria optimization approaches incorporated in the study is also presented and discussed.

1. Introduction

As discussed in the first part (Innovative Computational Techniques for Multi Criteria Decision Making, in the Context of Cultural Heritage Structures’ Fire Protection: Theory) of the current work, a selection and resource (S and R) allocation model aiming to provide the optimized performance-based fire protection upgrading design of cultural heritage buildings is proposed. The model relies on two indexes that denote the building’s performance in fire, namely the fire safety index (FSI), expressing the level of fire safety of the structure, together with the authenticity preservation index (API), expressing the alteration of the structure’s unique characteristics caused through the implementation of the fire safety upgrading measures. Both FSI and API depend on the extent of implementation of the fire protection measures (M), which for the purposes of the study are grouped into 16 categories covering a wide range of fire safety engineering possible options. In this context, the implementation grade (G) of the aforementioned measures can take values in the range [0, 1], where zero corresponds to the total absence of the specific measure and monad denotes full implementation of the measure [1,2]. Given that the values G are provided, the impact of each measure on the FSI depends on the selected analytic hierarchy process (AHP) parameters and weights that link the adjacent levels of the corresponding AHP hierarchy, namely from bottom to top; measures (M), strategies (ST), objectives (OB) and policy (PO), which are expressed by FSI. Accordingly, the impact of the measures on the API depends on weight coefficients of the AHP parameters that connect the levels of the second suggested AHP hierarchy, namely; measures (M), preservation goals (PG) and preservation policy (PPO), which are expressed by API. In order to define the total cost required of fire safety upgrading solutions derived through the optimization procedure, the fire protection measures are divided into four cost categories, in accordance to their approximate implementation cost, which for the need of the study were considered at 40, 30, 20, and 10 M.U./sq.m (monetary units per square meter).

In the second part of this work, two test cases are considered, which demonstrate the application range of the proposed model in real-world structures. Τhe first one refers to Villa Bianca, a famous mansion located in the city of Thessaloniki, Greece, which nowadays houses the Municipal Art Gallery, while the second one refers to Simonopetra, perhaps the most photographed and impressive monastery of Mount Athos, Greece. The two different approaches of the proposed computational model described in the first part of the work that are used to solve the fire protection upgrading optimization problem, namely the multidisciplinary optimization approach, incorporating the analytic target cascading (ATC) method and the multi objective optimization approach, as well as incorporating the nondominated sorting evolution strategies (NSES-II) algorithm, are both implemented in the two test cases.

In the corresponding section of the first part of the study, the general AHP formulation and its parameters are presented both for the case of FSI and API assessment. There is a pool of parameters that covers almost all aspects of the fire safety and authenticity issues involved with cultural heritage buildings; not all elements are required to be considered for every case study. This choice depends on the special characteristics of the building, as well as the priorities of the stakeholders. For example, as will be presented below, for the first case study (Villa Bianca), not all parameters of the AHP models available were employed, while for the second case study (Simonopetra Monastery) all parameters were used during the application of the model. This is due to the simpler structure, as well as to the aims of each application. It is worth mentioning that the test cases were partially examined in the previous works of the authors, to which the readers can refer for more information [2,3,4].

2. Problem Formulations and Solution Approaches

2.1. Multi Disciplinary Problem Approach (ATC)

The main questions that the proposed multi disciplinary computational model aims to answer are the following two, both derived through real world problems: (i) In the first one, when the manager, owner, or anyone involved in the management, coordination, and protection of a historic building, intends to upgrade its fire safety level, they aim to achieve a specific increment at FSI, investing the least amount of budget and reducing as much as possible the modifications on the building. (ii) According to the second one, the purpose is to allocate a specific budget in such a way as to achieve the best fire safety upgrade on one hand and reduce as much as possible the modifications on the building on the other hand. In other words, the model aims to find the best possible combination of fire protection measures, which maximize fire safety and minimize the building alterations of the building. Thus, in the first case, a problem is formulated where both cost and API need to be minimized under the constraint of a desired FSI, while in the second one, a problem is formulated where API is to be minimized while the FSI to be maximized is subjected to budget constraint. Therefore, the problem of the fire protection of a cultural heritage building can be formulated in two distinctive ways:

In Formulation A (F.A) the optimum selection of the fire protection measures is defined through a non-linear programming problem:

(1a)$\{\begin{array}{c}\min C\left(G\right)+\phi \left(G^{Cost}-G^{API}\right)\\ \min API\left(G\right)+\phi \left(G^{Cost}-G^{API}\right)\end{array}$

Subject to the constraint

(1b)$FSI\left(G\right)\ge FS{I}_{\mathrm{Target}}$

where $C$ is the total cost of a specific fire safety upgrading solution, $G\left(k\right)$ is the implementation grade of the fire protection measure $k$, representing the unknowns (design variables) of the problem, $API$ is the authenticity preservation index, $\phi$ is the augmented Lagrangian function described in the first part of the study, while $FSI$ is the fire safety index and $FS{I}_{\mathrm{Target}}$ is the desired (targeted) fire safety index.

In Formulation B (F.B) the optimum budget allocation is also defined through a non-linear programming problem:

(2a)$\{\begin{array}{l}\max FSI\left(G\right)+\phi \left(G^{FSI}-G^{API}\right)\\ \min API\left(G\right)+\phi \left(G^{FSI}-G^{API}\right)\end{array}$

Subject to the constraint

(2b)$C\left(G\right)\le C_{\mathrm{Target}}$

where $C_{\mathrm{Target}}$ is the given budget. It should be noted that the differential evolution (DE) algorithm [5] is incorporated to handle the ATC subproblems, as described in the first part of the study.

2.2. Multi Objective Problem Approach (NSES-II)

For the purposes of the second approach adopted for solving the optimization problem, the NSES-II algorithm is used. In this approach, only the first formulation denoted in the previous section is implemented, and in a similar way, it can also be applied to the second formulation as well. As a result, in this formulation the optimum selection of the fire protection measures is defined as a two-objective non-linear programming problem as follows:

(3a)$\{\begin{array}{l}min C\left(G\right)\\ min API\left(G\right)\end{array}$

Subject to the constraint

(3b)$FSI\left(G\right)\le FS{I}_{\mathrm{Target}}$

In the following section of the study, the two test cases adopted for presenting the capabilities of the proposed computational model are described in detail, i.e., Villa Bianca and the monastery of Simonos Petras, alongside the assessment of FSI and API values based on AHP, while the results obtained by the two optimization approaches are presented in Section 4 and Section 5, respectively.

3. Presentation and Analysis of the Two Test Cases

3.1. Test Case 1—Villa Bianca

3.1.1. The Description of the Structure

Villa Bianca, also known as Casa Bianca or Villa Fernandez (see Figure 1), is a famous mansion located on Vassilisis Olgas street in the city of Thessaloniki, Greece. Constructed from 1911 to 1913, it was initially used as the residence of Dino Fernandez Diaz, a wealthy merchant and industrialist of the city, and his family. The architect who designed Villa Bianca was Pietro Arrigoni [6]. Villa Bianca is a typical example of eclecticistic architecture in the city of Thessaloniki. The house incorporates baroque, Art Nouveau, and Renaissance influences, and stands out for its beautiful balconies, open spaces, and the lack of symmetry in the front side of the building. In 1976, the building was characterized as a piece of art to be protected. During the years 1994–1997, a restoration project took place, and since 2013, it hosts the Municipal Art Gallery of Thessaloniki. The building consists of a slightly elevated ground floor, the first floor, and the attic (mansard), while its total area is about 920 sq.m. The structure consists of 38 and 85 cm thick masonry walls, wooden floors (some floors are mosaic), and a tile-covered wooden roof. It is worth mentioning also that the openings have a significant total area and are covered with wooden windows.

3.1.2. Fire Safety Level (FSI) Assessment

Applying the fire safety level assessment model described in detail in the first part of the current work, for the Villa Bianca case, the following parameters and criteria were used: FSI is engaged in the top level (policy) of the AHP hierarchy. In the second level, the following objectives were selected: OB1—human protection (visitors and occupants), OB2—building fabric protection, OB3—cultural contents’ protection, OB4—built environment protection. In the third level, which represents the strategies that serve the aforementioned four objectives, the following five parameters were designated: ST1—reduce the probability of a fire starting, ST2—limit fire development in the fire compartment, ST3—limit fire propagation out of the fire compartment, ST4—facilitate egress, and ST5—facilitate fire-fighting and rescue operations. In the last level, the following 16 measures were taken into account: M1—compartmentation, M2—fire resistance of structural elements, M3—control of fire load, M4—materials (reaction to fire), M5—control of fire spread outside the building, M6—design of the means of escape, M7—signs and safety lighting, M8—access of the fire brigade, M9—detection and alarm, M10—suppression and extinguishing, M11—smoke control systems, M12—training of the personnel, M13—fire drills and emergency planning, M14—management of fire safety, M15—maintenance of the fire safety system, and M16—salvage operation [1,2]. The scale used in the present work is given in Table 1, while Table 2, Table 3 and Table 4 provide the weight coefficients of the elements described above, required to formulate the AHP tree.

At this point, according to the AHP model, the fire safety index for the case of Villa Bianca is calculated as follows:

(4)$FSI={\displaystyle \sum }_{i=1}^4{\displaystyle \sum }_{j=1}^5{\displaystyle \sum }_{k=1}^{16}OB\left(i\right)\cdot ST\left(i,j\right)\cdot M\left(j,k\right)\cdot G\left(k\right)$

where $G\left(k\right)$ represents the implementation grade of the fire protection measures (see the first part of the study for the symbols). Referring to the present situation (before implementing the fire safety upgrading scheme), most of the sixteen measures are partially implemented. Based on the information presented in Table 5, the fire safety index for the present situation of Villa Bianca is equal to $FS{I}^{initial}=0.48$ [4].

3.1.3. Assessment of the Authenticity Preservation

According to the proposed model for the assessment of the authenticity preservation, API is expressed by means of two preservation objectives, namely PG1—reversibility and PG2—discreteness, while the 16 fire protection measures are those described above. Table 6 and Table 7 provide the weight coefficients of these elements that are required to formulate the AHP tree, while Table 1 gives the scale used in this study. As seen, measures 12 to 16 have no impact on the preservation objectives, and thus no effect on API. It is worth mentioning that the weight coefficients are provided by the stakeholders according to their perceptions and priorities. Thus, the authenticity preservation index (API) is calculated as follows:

(5)$API={\displaystyle \sum }_{\mathcal{\ell }=1}^2{\displaystyle \sum }_{k=1}^{16}PG\left(\mathcal{\ell }\right)\cdot M\left(\mathcal{\ell },k\right)\cdot (G\left(k\right)-G\left(k{)}^{init}\right)$

In Section 4, the application of the ATC and NSES-II approaches is presented, aiming to solve the optimum fire protection upgrading design problem for the Villa Bianca case, followed by the section where the two approaches are compared between each other.

3.2. Test Case 2—Monastery of Simonos Petra

3.2.1. The Building Complex Description

The monastery of Simonos Petra, which is known as Simonopetra, is acknowledged as the most impressive among the 20 monasteries of Mount Athos, situated on a 333 m high steep rock in the southwestern side of Holy Mountain (see Figure 2). Simonopetra was initially founded by hermit St. Simon in the 13th century [7], and after multiple modifications during these years, today constitutes a significant engineering achievement constructed by unreinforced bearing masonry. The main structure extends over seven stories, while the total building complex covers a surface of more than 7000 sq.m. Fire severely hit Simonopetra numerous times during its existence, such as in 1581, 1626, and 1891 when the library and the Catholicon were burnt, and the monastery was reconstructed. The last significant fire incident took place during the summer of 1990, when a massive forest fire approached the building complex, causing several damages. Nowadays, a brotherhood of about 60 monks inhabits the monastery while it can accommodate approximately 30 visitors.

3.2.2. Fire Risk and Authenticity Preservation Assessment (FSI and API Assessment)

The assessment of the fire safety level, alongside an extensive theoretical documentation, can be found in [2]. Based on this study, the initial value of FSI is equal to 0.527, which is used as the base value for fire safety in this test case. According to the proposed model for the assessment of the authenticity preservation, the API (authenticity preservation index) is expressed by means of the four preservation objectives chosen, namely PG1—reversibility, PG2—discreteness, PG3—non-competitiveness and PG4—distinctiveness, while the 16 fire protection measures are the same used for the first test case. The weight coefficients of the above elements, required in order to formulate the AHP tree, are provided in Table 8 and Table 9, while the scale used is provided in Table 1. As it can be noticed, measures 12 to 16 have no impact on the preservation objectives POB1—POB3, and thus their impact on API depends only on the significance of POB4. It is worth mentioning that, similar to the first case study, the weight coefficients are provided by the stakeholders according to their perceptions and priorities. In this context, according to the AHP model, the authenticity preservation index (API) is defined as follows:

(6)$API={\displaystyle \sum }_{\mathcal{\ell }=1}^4{\displaystyle \sum }_{k=1}^{16}PG\left(\mathcal{\ell }\right)\cdot M\left(\mathcal{\ell },k\right)\cdot (G\left(k\right)-G\left(k{)}^{init}\right)$

In the following Section 5.1 and 5.2, we present the application of ATC and NSES-II approaches, respectively, for the solution of the optimum fire protection design problem of Simonopetra Monastery. Finally, in Section 5.3 a comparison between these two approaches is presented.

4. Test Case 1—Villa Bianca: Results

4.1. ATC Results (Formulations A and B)

In this part of the study, the results obtained by means of the ATC approach are presented when used to solve the optimum fire safety upgrading problem for the Villa Bianca test case. It should be noted that the assumption for the present case (i.e., before upgrading) is that API is considered equal to zero, thus the implementation of any measure implies API value will increase based on the AHP model. On the other hand, the initial value of FSI is equal to 0.48, as presented previously for the specific test example [4]. Based on the description provided previously, two formulations are used; the first one refers to the achievement of a desired fire safety level, expressed by FSI, minimizing cost and extent of interventions on the building, while the second one refers to the optimum budget allocation in order to maximize fire safety and preserve building authenticity.

4.1.1. Formulation A

The results achieved for Formulation A (F.A) are provided in Table 10 and are depicted in Figure 3, where the optimized implementation grades of the 16 measures are denoted. In the context of the present study, six different test cases of increasing FSI target values, namely 0.50, 0.60, 0.70, 0.80, 0.90, and 1.00, are examined. The two bottom rows of Table 10 summarize cost (in M.U.) and API values corresponding to the six FSI target values noted on the top row. In the first row, the FSI values that correspond to the solutions achieved by the algorithm are given in brackets.

As indicated in this case study, the implementation of more cost demanding measures, such as compartmentation (M1) and smoke control systems (M11) become part of the solution only for the two higher FSI target values i.e., 0.90 and 1.00. Figure 4 depicts graphically the two bottom rows of Table 10 that correspond to API and cost for the six FSI target values used (i.e., 0.50, 0.60, 0.70, 0.80, 0.90, and 1.00). The left vertical axis represents API, which takes value 0.33 when full implementation of all 16 measures is selected and FSI is equal to the monad. It is worth mentioning that between FSI values 0.60 and 0.80, the gradient of the cost curve remains the same, while the gradient of the API curve is reduced, which means that in terms of authenticity preservation, the budget allocation is much better. In this part, investing M.U. 50,466 results to a 33.3% increase of the FSI value (0.8 from 0.6) accompanied by an 80% worsening of the API value (0.09 from 0.05), while in the FSI increment between 0.8 and 0.9, which correspond to a 12.5% increase, the cost is M.U. 67,943 (34% increase of the budget required) and the API value worsens by 166.7% (0.24 from 0.09). In the ranges between 0.50 to 0.60 and 0.80 to 1.00, both gradients for cost and FSI curves remain the same.

4.1.2. Formulation B

This part of the study includes the results for Formulation B of the problem, which refer to the optimum budget allocation to both maximize safety (FSI) and building authenticity preservation, which is translated into minimized API values. In this context, six different test cases, regarding the available budget (in MU), are examined, namely 30,000, 60,000, 90,000, 120,000, and 150,000 MU. The results achieved are summarized in Table 11 and are depicted in Figure 5 and refer to the optimized implementation grades of the 16 measures of fire protection. The two bottom rows of the table represent FSI and API for the budget case of the first row, while the budget that was finally used, satisfying the restriction of the formula of Equation (2b), is given in the brackets of the first row.

Figure 6 depicts the two bottom rows of Table 11, which represent FSI and API for the six chosen budgets. The left vertical axis represents the FSI scale, while 0.48 corresponds to the fire safety level of the building before the upgrading. The right vertical axis represents the API scale, which starts from zero and goes up to 0.27. As can be noticed, for the two higher budgets available (150,000 and 180,000 MU) the algorithm finally uses a smaller part of the budget in comparison with the other budget target cases, as a result of the demand to keep API in limited values, expressed by formulas of Equation (2a).

4.2. NSES-II Results

In this part of the work, we present the results achieved by ATC for the optimum fire protection for the Villa Bianca test case, with two objective functions. As presented above, the first objective function refers to the cost for upgrading the fire protection measures, while the second one refers to API. Additionally, the implementation grades (G) of the 16 measures of fire protection are the design parameters, while the inequality constraint that was considered set a lower limit for the FSI. For the present situation (before upgrading), the API is considered equal to zero, and the implementation of the measures implies its increase, depending on the AHP model. On the other hand, the initial value of FSI is equal to 0.48, as presented previously. The NSES-II algorithm is incorporated to solve the problem. As in the first case, the formulation refers to the achievement of a desired fire safety level, expressed by FSI, minimizing both cost and alteration of the building. The results of this application are depicted on Figure 7. Six different case studies are also examined with increasing target FSIs, with regard to the fire safety index needed to be achieved: 0.50, 0.60, 0.70, 0.80, 0.90, and 1.00 (0.99).

Contrary to the ATC approach, the multi objective approach results in a Pareto front, which corresponds to a series of optimized designs for each FSI target value. Each front includes 100 designs equal to the population chosen for the NSES-II algorithm. The two end points of each Pareto front refer to the two limit designs for the specific FSI target. For example, referring to FSI = 0.60, design (1) corresponds to design variables, which result in the minimization of the authenticity objective function accompanied with increased cost, while design (2) corresponds to design variables which minimize cost objective function, but at the same time increase the implementation of measures, which decrease the authenticity of the building. The ten designs, which refer to the end points of the five Pareto fronts denoted on Figure 7 with numbers (1) to (10), are summarized on Table 12.

The two bottom rows of Table 12 summarize cost (in M.U.) and API values corresponding to the five FSI target values of the top row. Figure 8, Figure 9, Figure 10, Figure 11 and Figure 12 depict the results summarized in Table 12, per couple referring to the same Pareto front. As noticed both in Figure 7 and Table 12, for the first three FSI targets (0.50, 0.60, and 0.70) the problem can be solved without any building alteration, succeeding an API equal or almost equal to zero. It is worth mentioning that the same FSI targets can be achieved with a lower budget, but in this case, API is increased.

As seen in Table 12, a FSI target value equal to 0.70 was achieved with no authenticity issue implications, when M.U. 56,136 are invested, while the same FSI value can also be reached with 29.6% less cost (M.U. 39,495) if authenticity preservation is not considered as part of the problem.

From the previous figures, useful conclusions can be drawn regarding the fire protection design of the examined building. For example, fire protection measure 11, which refers to the application of smoke control systems, is implemented only at the maximum target value of FSI (FSI = 0.99). This is due to its high cost (40 M.U./sq.m.) and its high effect on the authenticity of the building since the installation of such systems requires extensive interventions to the structure.

4.3. Comparison

In this section, we presented the application of the proposed models in the case of Villa Bianca, a well-known mansion in Thessaloniki, Greece. In order to solve the problem of the optimum fire protection of this structure, two different approaches were followed: the multi disciplinary (ATC) and the multi objective (NSES-II). Both approaches resulted in realistic solutions. These solutions are presented in Figure 13, where one can see the solutions derived from Formulation A of ATC approach, alongside the solution derived from the NSES-II approach. Figure 13 presents API and cost for different FSI targets. The NSES-II case includes two curves, the upper one (minAPI), which corresponds to the solutions where API is minimized, accompanied by increased cost, and the lower one (minCost), which corresponds to the solutions that the minimization of cost is prioritized against API. As it can be noticed, the behavior of the two approaches is similar, and it is worth mentioning that the results of the ATC, as achieved with the DE algorithm, are identical with the lower point of the Pareto fronts, achieved with NSES-II, corresponding to cost minimization.

5. Test Case 2—Monastery of Simonos Petra: Results

5.1. ATC Results (Formulations A and B)

In this part of the study, the results of the ATC approach to the optimum fire protection model for the Simonopetra test case are presented. The two criteria (disciplines) for this case of the ATC approach are FSI and API values. It is noted that for the present situation (before upgrading) the API is considered equal to zero, and the implementation of the measures implies its increase, depending on the AHP model. On the other hand, the initial value of FSI is equal to 0.527, as presented previously. As in the first test case, two formulations are presented; the first one refers to the achievement of a desired fire safety level, expressed by FSI, minimizing cost and deterioration of the alteration of the building complex, while the second one refers to the optimum budget allocation in order to maximize both safety and building complex authenticity preservation.

5.1.1. Formulation A

The results achieved for Formulation A (F.A) are presented on Table 13 and depicted in Figure 14, and include the optimized implementation grades of the 16 measures. In the context of the present study, six different cases are studied using different FSI targets, namely 0.55, 0.60, 0.65, 0.70, 0.75, 0.80, 0.85, 0.90, 0.95, and 1.00 (0.999).

Value 0.999, instead of the monad, was selected as the maximum FSI target for algorithm convergence reasons. For this target value, all fire protection measures are fully implemented, which is translated into implementation grades equal, or almost equal, to the monad. The two bottom rows of Table 13 summarize the cost (in M.U.) and API values corresponding to the five FSI target values noted on the top row. The second row represents the FSI values, which the algorithm managed to reach. As it can be seen in this case study, the implementation of “expensive” measures, such as compartmentation (M1) and smoke control systems (M11) becomes a part of the solution only for the higher than 0.85 FSI targets. Figure 15 depicts the two bottom rows of Table 13, which represents API and cost for the 10 chosen FSI targets. The left vertical axis represents API, which takes the value 0.55 when full implementation of all 16 measures is taking place and FSI comes up to monad. It is worth mentioning that, while for a FSI value lower than 0.8 cost and API follow a normal linear increase, for the FSI increment between 0.80 and 0.85, a “jump” of cost and API is noted, which is expressed by the gradient increase in the two curves that revert to the previous situation for FSI values bigger than 0.85.

5.1.2. Formulation B

This part of the study includes the results for the Formulation B of the problem, which refers to the optimum budget allocation in order to maximize both safety (FSI) and building authenticity preservation, which is translated into minimized API values. In this context, ten different test cases, regarding the available budget (in MU), are examined, namely 150,000, 300,000, 450,000, 600,000, 750,000, 900,000, 1,050,000, 1,200,000, 1,350,000, and 1,500,000 MU. The results achieved are summarized in Table 14 and depicted in Figure 16, and refer to the optimized implementation grades of the 16 measures for the fire protection. The two bottom rows of the table represent FSI and API for the budget case of the first row, while the second row includes the budget that was finally used based on the restriction of Formula (4). Figure 17 depicts the two bottom rows of Table 13, which represent FSI and API for the ten chosen budgets. The left vertical axis represents the FSI scale (0.53 corresponds to the fire safety of the building before the upgrading), while right vertical axis represents the API scale. As it can be noticed, for the two higher budgets available (1,350,000 and 1,500,000 MU) the algorithm finally uses a smaller part of the budget in comparison with the other budget target cases, as a result of the demand to minimize API values, as expressed by the formulas of Equation (2a). Figure 18 depicts the variation in fire safety levels for different budgets available. The curve “considering API” corresponds to the solution derived from the model when the authenticity preservation is considered as a goal, while the curve “not considering API” corresponds to the solution where authenticity preservation was not taken into account. As seen when authenticity preservation is considered as an objective during the fire protection design, the fire safety level achieved for a specific budget is diminished. It is obvious that the necessity to preserve the architectural forms and cultural values involves an additional cost that has to be considered by the designer during the fire protection upgrading of a historic structure.

5.2. NSES-II Results

In this part of the study, we present the results of the optimum fire protection model with two objective functions for the Simonopetra test case. Similarly to the test case of Villa Bianca, the first objective function refers to the cost for upgrading the fire protection measures, while the second one refers to the authenticity preservation index (API). Additionally, the grades of implementation (G) of the 16 measures are the design parameters, while the inequality considered constraint of Equation (6) set a lower limit for the FSI. It is assumed that for the present situation (before upgrading) the API is considered equal to zero, and the implementation of the measures implies its increase, depending on the AHP model. On the other hand, the initial value of FSI is equal to 0.53, as presented previously. The NSES-II algorithm is incorporated to solve the problem. As in the first test case, the formulation refers to the achievement of a desired fire safety level, expressed by FSI, minimizing cost and deterioration of the alteration of the building. The results of this application are depicted on Figure 19. Six different cases are also studied with increasing FSI targets, with regard to the fire safety index required to be achieved: 0.55, 0.60, 0.65, 0.70, 0.75, 0.80, 0.85, 0.90, 0.95, and 1.00 (0.975). It must be mentioned that for the current fire safety level of Simonopetra, based on the existing fire protection measures taken, was assessed it to be equal to 0.53 [2].

Contrary with the ATC approach, the multi objective approach results in a Pareto front, which corresponds to a series of designs for each FSI target value. In particular, each front includes 100 designs equal to the population chosen for the NSES-II algorithm. As noticed in Figure 19, for FSI values bigger than 0.80, all the designs are almost similar, leading to the degeneration of fronts into points. Thus, we chose the first five cases, corresponding to FSI values 0.55, 0.60, 0.65, 0.70, and 0.75 for further study. These cases are depicted in Figure 20, which constitutes a “zoom-in” figure of the noted area of Figure 19. The two end points of each Pareto front refer to the two limit designs for the specific FSI target. For example, referring to FSI = 0.60, design (3) corresponds to design variables, which result in the minimization of the authenticity objective function accompanied with increased cost, while design (4) corresponds to design variables, which minimize the cost objective function, but at the same time, increase the implementation of measures that decrease the authenticity of the building. The ten designs, which refer to the end points of the five Pareto fronts denoted in Figure 20 with numbers (1) to (10) are summarized in Table 15. The two bottom rows of this table summarize cost (in M.U.) and API values corresponding to the five FSI target values of the top row. Figure 21, Figure 22, Figure 23, Figure 24 and Figure 25 depict the results summarized in Table 15, per couple referring to the same Pareto front.

As seen in Table 15, for the FSI target value equal to 0.65, the cost to also achieve the maximized authenticity preservation, thus minimize API, is 58% more (M.U. 167,032 instead of M.U. 105,756) than in the case where authenticity is not an issue for the design. In this respect, M.U. 61,276 are needed to achieve an improvement 30% in authenticity terms (API = 0.07 instead of API = 1.0).

Through the figures presented previously, useful conclusions regarding the fire protection design of the examined building can be drawn. Indicatively, none of the design cases concluded on a solution where the application of smoke control systems (M11) was part of the solution. This observation, noticed for the first test case as well, is attributed to its high cost (40 M.U./sq.m.) and its high effect on the authenticity of the building, since the installation of such systems requires extensive interventions to the structure.

5.3. Comparison

In Section 5 of the present study, we presented the implementation of the models proposed here in for the case of the Simonopetra Monastery of Mount Athos, Greece. In order to solve the problem of the optimum fire protection of this structure, two different approaches were followed: the multi disciplinary (ATC) and the multi objective (NSES-II). Both approaches resulted in realistic solutions. These solutions are presented on Figure 26, where one can see the solutions derived from Formulation A of the ATC approach, accompanied by the solution derived from the NSES-II approach. This figure presents API and cost for different FSI targets. The NSES-II case includes two curves, the upper one (minAPI), which corresponds to the solutions where API is minimized, accompanied by increased cost, and the lower one (minCost), which corresponds to the solutions where the minimization of cost is prioritized against API. As can be noticed, the behavior of the two approaches is similar, and it is worth mentioning that the results of the ATC, as achieved with the DE algorithm, are identical with the lower point of the Pareto fronts, achieved with NSES-II, corresponding to cost minimization.

6. Discussion and Conclusions

Fire protection of historic buildings is usually a complex and challenging engineering task. The difficulty in applying conventional fire protection tactics to such structures was acknowledged by several researchers in the past, which highlights the special requirements and additional risks connected with them [8,9,10,11]. Furthermore, the concern for authenticity preservation is also the subject of a number of studies, but commonly in the context of a generic approach to architectural restoration and typically regarding specific cases (i.e., during the re-use of an existing heritage building) [12,13,14]. Many researchers also proposed numerical methodologies to assess and control fire-related risks, proposing optimized solutions for specific cases and individual issues, not under a holistic approach [15,16,17,18,19]. However, there is not any significant study connecting the two aspects of historic buildings, namely fire safety and authenticity preservation, under a common and universal numerical approach that takes into account the implementation cost, as well as utilizes advanced computational techniques.

In this context, the present study suggests an innovative selection and resource (S and R) allocation model, which incorporates both multicriteria decision-making tools for quantifying fire risk and authenticity preservation, and metaheuristic optimization techniques and algorithms to optimally design the fire safety upgrade of a cultural heritage structure.

Such kinds of existing buildings differ from new structures, thus conventional fire protection codes are not able to efficiently address their special characteristics and succeed in further protecting human life, the structure, and its contents. Additionally, in most relevant cases, another major issue is authenticity, which must be preserved, since it is often linked with invaluable cultural values [20,21]. However, this demand can lead to significant additional costs regarding the applied fire protection measures in order to maintain the cultural or historic character of the structure. Nevertheless, given the fact that the budget available is restricted, it must be distributed in an optimal way, aiming to both maximize fire safety levels and minimize the building alteration, and when authenticity is the case, the designer might face up to decision-making challenges.

In this direction, a performance-based framework is proposed, aiming to deal with the challenging problem of the fire protection of historic buildings. In the first place, the Analytic hierarchy process (AHP) is incorporated to quantify the fire safety level and the authenticity preservation level for a specific structure. Two indices are mobilized to assess the aforementioned parameters, namely the fire safety index (FSI) and the authenticity preservation index (API). Subsequently the S and R allocation model scheme incorporates enhanced computational techniques to optimally address the select and resource (S and R) allocation problem regarding the fire protection measures.

In particular, in the present study, two approaches are tested to deal with the optimization of the multicriteria decision-making problem regarding the budget allocation problem for the fire protection of cultural heritage buildings; the multi objective optimization approach, incorporating the NSES-II (nondominated sorting genetic algorithm) solution algorithm and the multidisciplinary optimization approach, as well as the ATC (analytic target cascading) method, the subproblem of which is solved with the differential evolution algorithm. Furthermore, two different formulations were presented with regard to the objective functions. In the first case, a problem of both cost and API minimization under the constraint of a desired FSI is formulated, while in the second case a problem of API minimization and FSI maximization under the constraint of a given budget is formulated.

The proposed model was successfully applied to two real-world cases, yielding feasible S and R allocation problem solutions for different budget scenarios, different chosen fire safety levels and authenticity preservation levels, and a variety of prioritizations. Specifically, the efficiency of the model is presented for Casa Bianca, a famous mansion in Thessaloniki, Greece, and the monastery of Simonos Petra, which is the most photographed monastery in Mount Athos, Greece. The results obtained through the implementation of the proposed S and R allocation model can successfully lead to optimized selection and resource allocation for any historic building, with respect to fire safety, authenticity preservation, and financial constraints.

It is noted that the main objective of this work was to propose a generally applicable selection and resource (S and R) allocation model for upgrading fire safety levels in the case of cultural heritage buildings. This S and R allocation model is formulated as a decision-making problem with multi criteria that relies on AHP; it should be mentioned that the conclusions achieved correspond to the indicative values considered for the parameter involved and significantly different designs shall be obtained depending on the stakeholders’ objectives and needs.

Author Contributions

Conceptualization, I.A.N. and N.D.L.; methodology, I.A.N., C.C.M. and N.D.L.; software, I.A.N., C.C.M. and N.D.L.; validation, I.A.N. and N.D.L.; formal analysis, I.A.N. and N.D.L.; investigation, I.A.N. and N.D.L.; writing-original draft preparation, I.A.N., C.C.M. and N.D.L.; writing-review and editing, I.A.N., C.C.M. and N.D.L.; visualization, I.A.N. and N.D.L., supervision, C.C.M. and N.D.L.; project administration, C.C.M. and N.D.L. All authors have read and agreed to the published version of the manuscript.

Disclaimer: The content of this paper expresses the views of the author and not the official position of the Ministry of Maritime Affairs & Insular Policy.

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Data Availability Statement

Not applicable.

Conflicts of Interest

The authors declare no conflict of interest.

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Figures and Tables

Figure 1. Front view of Villa Bianca (authors’ archive).

Figure 2. The main building complex of Simonopetra Monastery (west view—authors’ archive).

Figure 3. Optimized implementation grades of the 16 fire protection measures for the six different FSI targets (F.A).

Figure 4. Cost and API representation for different FSI values (Test Case 1).

Figure 5. Optimized implementation grades of the 16 fire protection measures for the six different cost targets (F.B).

Figure 6. FSI and API for different budget available (Test Case 1).

Figure 7. Presentation of Pareto fronts for the six test cases of FSI target value.

Figure 8. Implementation grades of the 16 fire protection measures for designs (1) and (2) of the Pareto front corresponding to FSI = 0.60.

Figure 9. Implementation grades of the 16 fire protection measures for designs (3) and (4) of the Pareto front corresponding to FSI = 0.70.

Figure 10. Implementation grades of the 16 fire protection measures for designs (5) and (6) of the Pareto front corresponding to FSI = 0.80.

Figure 11. Implementation grades of the 16 fire protection measures for designs (7) and (8) of the Pareto front corresponding to FSI = 0.90.

Figure 12. Implementation grades of the 16 fire protection measures for designs (9) and (10) of the Pareto front corresponding to FSI = 0.99.

Figure 13. Results obtained by ATC (Formulation A) and NSES-II approaches for the test case of Villa Bianca.

Figure 14. Optimized implementation grades of the 16 fire protection measures for the 10 different FSI targets (F.A).

Figure 15. Cost and API representation for different FSI values (Test Case 2).

Figure 16. Optimized implementation grades of the 16 fire protection measures for the 10 different cost targets (F.B).

Figure 17. FSI and API for different budget available.

Figure 18. FSI for different budget available, considering and not considering authenticity preservation (Test Case 2).

Figure 19. Presentation of Pareto fronts for the ten test cases of FSI target value.

Figure 20. Presentation of Pareto fronts for the first five test cases of FSI target value.

Figure 21. Implementation grades of the 16 fire protection measures for designs (1) and (2) of the Pareto front corresponding to FSI = 0.55.

Figure 22. Implementation grades of the 16 fire protection measures for designs (3) and (4) of the Pareto front corresponding to FSI = 0.60.

Figure 23. Implementation grades of the 16 fire protection measures for designs (5) and (6) of the Pareto front corresponding to FSI = 0.65.

Figure 24. Implementation grades of the 16 fire protection measures for designs (7) and (8) of the Pareto front corresponding to FSI = 0.70.

Figure 25. Implementation grades of the 16 fire protection measures for designs (9) and (10) of the Pareto front corresponding to FSI = 0.75.

Figure 26. Results obtained by ATC (Formulation A) and NSES-II approaches for the test case of Simonopetra.

Table 1

AHP tree weight coefficients (W) scale according to the importance of the element evaluated [2].

Importance	Weight (W)
None	0
Low	1
Moderate	2
High	3
Very high	4

Table 2

AHP tree weight coefficients (W) and normalized weight coefficients (N.W.) between objectives and policy [4].

		OB1	OB2	OB3	OB4
PO	W.	4	3	3	1
	N.W.	0.36	0.27	0.27	0.09

Table 3

AHP tree weight coefficients (W) and normalized weight coefficients (N.W.) between strategies and objectives [4].

		ST1	ST2	ST3	ST4	ST5
OB1	W .	4	3	2	4	3
	N.W .	0.25	0.19	0.13	0.25	0.19
OB2	W .	4	4	3	1	3
	N.W .	0.27	0.27	0.20	0.07	0.20
OB3	W .	4	4	0	0	3
	N.W .	0.36	0.36	0.00	0.00	0.27
OB4	W .	2	2	4	0	2
	N.W .	0.20	0.20	0.40	0.00	0.20

Table 4

AHP tree weight coefficients (W) and normalized weight coefficients (N.W.) between fire protection measures and strategies [4].

	ST1		ST2		ST3		ST4		ST5
	W.	N.W.	W.	N.W.	W.	N.W.	W.	N.W.	W.	N.W.
M1	0	0.00	4	0.15	1	0.11	3	0.07	3	0.08
M2	0	0.00	4	0.15	1	0.11	3	0.07	4	0.11
M3	3	0.13	3	0.11	2	0.22	2	0.05	3	0.08
M4	4	0.17	3	0.11	0	0.00	4	0.10	2	0.05
M5	0	0.00	0	0.00	4	0.44	0	0.00	1	0.03
M6	0	0.00	2	0.07	0	0.00	4	0.10	1	0.03
M7	0	0.00	0	0.00	0	0.00	4	0.10	1	0.03
M8	0	0.00	2	0.07	1	0.11	1	0.02	4	0.11
M9	3	0.13	2	0.07	0	0.00	3	0.07	2	0.05
M10	3	0.13	2	0.07	0	0.00	3	0.07	4	0.11
M11	0	0.00	1	0.04	0	0.00	4	0.10	4	0.11
M12	4	0.17	1	0.04	0	0.00	3	0.07	3	0.08
M13	0	0.00	0	0.00	0	0.00	4	0.10	3	0.08
M14	4	0.17	1	0.04	0	0.00	2	0.05	2	0.05
M15	3	0.13	2	0.07	0	0.00	2	0.05	1	0.03
M16	0	0.00	0	0.00	0	0.00	0	0.00	0	0.00

Table 5

Implementation grades (G) of the fire protection measures regarding the present situation of Villa Bianca [4].

Measure	M1	M2	M3	M4	M5	M6	M7	M8	M9	M10	M11	M12	M13	M14	M15	M16
G (Present Situation)	0.3	0.4	0.7	0.5	0.8	0.7	1.0	1.0	0.7	0.6	0.0	0.0	0.0	0.2	0.5	0.2

Table 6

AHP tree weight coefficients (W) and normalized weight coefficients (N.W.) between preservation objectives (POB1 and POB2) and preservation policy (PPO).

		PG1	PG2
PPO	W.	2	4
	N.W.	0.33	0.67

Table 7

AHP tree weight coefficients (W) and normalized weight coefficients (N.W.) between 16 measures (M1 to M16) and preservation objectives (POB1 and POB2).

	PG1		PG2
	W.	N.W.	W.	N.W.
M1	4	0.14	3	0.09
M2	4	0.14	1	0.03
M3	2	0.07	4	0.12
M4	3	0.10	2	0.06
M5	3	0.10	3	0.09
M6	4	0.14	4	0.12
M7	1	0.03	4	0.12
M8	3	0.10	3	0.09
M9	1	0.03	3	0.09
M10	2	0.07	3	0.09
M11	2	0.07	3	0.09
M12	0	0.00	0	0.00
M13	0	0.00	0	0.00
M14	0	0.00	0	0.00
M15	0	0.00	0	0.00
M16	0	0.00	0	0.00

Table 8

(W) and normalized weight coefficients (N.W.) between preservation objectives (POB1–POB4) and preservation policy (PPO).

		POB1	POB2	POB3	POB4
PPO	W.	3	2	3	2
	N.W .	0.30	0.20	0.30	0.20

Table 9

AHP tree weight coefficients (W) and normalized weight coefficients (N.W.) between 16 measures (M1 to M16) and preservation objectives (POB1–POB4).

	POB1		POB2		POB3		POB4
	W.	N.W.	W.	N.W.	W.	N.W.	W.	N.W.
M1	4	0.14	3	0.09	3	0.10	1	0.03
M2	4	0.14	1	0.03	2	0.06	3	0.10
M3	2	0.07	4	0.12	3	0.10	0	0.00
M4	3	0.10	2	0.06	2	0.06	2	0.06
M5	3	0.10	3	0.09	2	0.06	1	0.03
M6	4	0.14	4	0.12	4	0.13	0	0.00
M7	1	0.03	4	0.12	4	0.13	0	0.00
M8	3	0.10	3	0.09	2	0.06	1	0.03
M9	1	0.03	3	0.09	3	0.10	1	0.03
M10	2	0.07	3	0.09	3	0.10	1	0.03
M11	2	0.07	3	0.09	3	0.10	1	0.03
M12	0	0.00	0	0.00	0	0.00	4	0.13
M13	0	0.00	0	0.00	0	0.00	4	0.13
M14	0	0.00	0	0.00	0	0.00	4	0.13
M15	0	0.00	0	0.00	0	0.00	4	0.13
M16	0	0.00	0	0.00	0	0.00	4	0.13

Table 10

Optimized implementation grades of the 16 fire protection measures for the six different FSI targets (F.A).

Measure	Cost per sq.m.	0.48 (Initial)	0.50 (0.50)	0.60 (0.60)	0.70 (0.70)	0.80 (0.80)	0.90 (0.90)	1.00 (0.99)
M1	40	0.30	0.30	0.30	0.30	0.30	0.56	0.99
M2	40	0.40	0.40	0.40	0.40	0.40	0.45	1.00
M3	10	0.70	0.85	0.99	0.88	1.00	0.98	0.99
M4	30	0.50	0.50	0.78	1.00	1.00	1.00	1.00
M5	10	0.80	0.80	0.81	0.80	0.80	0.85	1.00
M6	30	0.70	0.70	0.70	0.70	0.70	0.82	0.75
M7	20	1.00	1.00	1.00	1.00	1.00	1.00	1.00
M8	10	1.00	1.00	1.00	1.00	1.00	1.00	1.00
M9	30	0.70	0.70	0.70	0.70	1.00	0.97	0.95
M10	40	0.60	0.60	0.60	0.60	0.60	0.99	1.00
M11	40	0.00	0.00	0.00	0.00	0.00	0.91	0.94
M12	20	0.00	0.01	0.08	1.00	1.00	0.97	1.00
M13	10	0.00	0.00	0.00	0.00	0.91	0.94	1.00
M14	10	0.20	0.20	0.80	0.94	1.00	0.99	1.00
M15	20	0.50	0.51	0.50	0.50	1.00	0.92	1.00
M16	20	0.20	0.20	0.20	0.20	0.20	0.62	0.77
Cost	>	0.00	1587	17,694	40,639	68,160	136,103	178,774
API	>	0.00	0.02	0.05	0.05	0.09	0.24	0.33

Table 11

Optimized implementation grades of the 16 fire protection measures for the six different cost targets (F.B).

Measure	Cost per sq.m.	PresentSituation	30,000 (26,430)	60,000 (56,300)	90,000 (84,512)	120,000 (119,500)	150,000 (127,280)	180,000 (127,280)
M1	40	0.3	0.32	0.33	0.70	0.66	0.70	0.50
M2	40	0.4	0.40	0.65	0.74	0.70	1.00	0.91
M3	10	0.7	0.84	0.86	0.78	0.92	0.81	0.81
M4	30	0.5	0.57	0.88	0.92	0.92	0.88	0.89
M5	10	0.8	0.85	1.00	0.85	0.94	0.80	0.80
M6	30	0.7	0.74	0.82	0.83	0.82	0.70	0.77
M7	20	1.0	1.00	1.00	1.00	1.00	1.00	1.00
M8	10	1.0	1.00	1.00	1.00	1.00	1.00	1.00
M9	30	0.7	0.71	0.99	0.77	1.00	0.85	0.85
M10	40	0.6	0.66	1.00	0.68	0.92	0.82	0.82
M11	40	0.0	0.42	0.01	0.67	0.95	0.21	0.20
M12	20	0.0	0.04	0.00	0.16	0.42	1.00	1.00
M13	10	0.0	0.09	0.28	0.33	0.69	1.00	1.00
M14	10	0.2	0.25	0.26	0.55	0.70	1.00	1.00
M15	20	0.5	0.54	0.66	0.50	0.58	1.00	1.00
M16	20	0.2	0.22	0.21	0.30	0.33	1.00	1.00
FSI	>	0.49	0.55	0.66	0.69	0.80	0.88	0.86
API	>	0	0.07	0.15	0.19	0.27	0.17	0.15

Table 12

Optimized implementation grades of the 16 fire protection measures for the 10 denoted designs of Figure 7.

	(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)	(10)
FSI Target	0.60	0.60	0.70	0.70	0.80	0.80	0.90	0.90	0.99	0.99
M1	0.30	0.30	0.30	0.30	0.30	0.30	0.30	0.56	0.96	0.96
M2	0.40	0.40	0.40	0.40	0.44	0.44	0.99	0.97	0.96	0.96
M3	0.70	0.98	0.70	0.98	0.98	0.99	0.99	1.00	1.00	1.00
M4	0.50	0.50	0.50	0.57	1.00	1.00	1.00	1.00	1.00	1.00
M5	0.80	0.80	0.80	0.80	0.80	0.80	0.86	0.98	0.99	0.99
M6	0.70	0.70	0.70	0.70	0.70	0.70	0.70	0.70	0.60	0.90
M7	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	0.30	0.30
M8	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00
M9	0.70	0.70	0.70	0.70	0.73	0.96	0.99	0.98	0.96	0.96
M10	0.60	0.60	0.60	0.60	0.60	0.60	0.99	0.97	1.00	0.99
M11	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.69	0.44
M12	0.70	0.38	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00
M13	0.00	0.00	1.00	0.03	1.00	0.98	1.00	0.98	0.99	0.99
M14	1.00	0.94	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00
M15	0.50	0.50	0.98	0.98	1.00	1.00	1.00	1.00	1.00	1.00
M16	0.20	0.20	0.87	0.20	0.98	0.20	0.97	0.20	0.76	0.76
API	0.00	0.03	0.00	0.04	0.07	0.09	0.16	0.20	0.32	0.33
Cost	20,228	16,343	56,136	39,495	77,166	68,858	119,329	113,872	180,170	175,950

Table 13

Optimized implementation grades of the 16 fire protection measures for the 10 different FSI targets (F.A).

Measure	Cost per sq.m.	0.53 (Initial)	0.55 (0.55)	0.60 (0.60)	0.65 (0.65)	0.70 (0.70)	0.75 (0.75)	0.80 (0.80)	0.85 (0.88)	0.90 (0.91)	0.95 (0.95)	0.999 (0.999)
M1	40	0.30	0.30	0.30	0.30	0.30	0.30	0.30	0.42	0.86	0.98	1.00
M2	40	0.30	0.30	0.30	0.30	0.30	0.30	0.30	0.67	0.74	1.00	1.00
M3	10	0.60	0.81	1.00	1.00	1.00	1.00	1.00	0.96	1.00	0.94	1.00
M4	30	0.60	0.60	0.60	0.60	0.60	1.00	1.00	0.98	0.92	0.91	1.00
M5	10	0.50	0.50	0.90	1.00	1.00	1.00	1.00	0.91	0.97	0.99	1.00
M6	30	0.60	0.60	0.60	0.60	0.60	0.60	0.60	0.84	0.78	0.64	0.99
M7	20	0.30	0.30	0.30	0.30	0.30	0.30	0.30	0.78	0.71	0.90	1.00
M8	10	0.20	0.20	0.20	0.79	1.00	1.00	1.00	0.90	0.99	1.00	1.00
M9	30	0.60	0.60	0.60	0.60	0.60	0.69	0.99	1.00	0.81	0.99	1.00
M10	40	0.50	0.50	0.50	0.50	0.50	0.50	0.55	0.94	0.97	0.99	1.00
M11	40	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.79	0.96	0.95	1.00
M12	20	0.90	0.90	0.90	0.90	0.90	1.00	1.00	0.96	0.98	0.96	1.00
M13	10	0.50	0.50	0.50	0.50	0.50	1.00	1.00	0.84	0.95	0.92	1.00
M14	10	0.90	0.90	0.90	0.90	0.98	1.00	1.00	0.99	0.97	0.99	1.00
M15	20	0.60	0.60	0.60	0.60	1.00	0.60	1.00	0.84	0.88	0.98	1.00
M16	20	0.40	0.40	0.40	0.40	0.40	0.40	0.40	0.73	0.57	0.74	0.99
Cost	>	0.00	15,049	55,421	104,984	180,315	277,370	409,322	984,505	1,110,830	1,287,804	1,473,340
API	>	0.00	0.02	0.06	0.11	0.14	0.18	0.21	0.38	0.43	0.48	0.55

Table 14

Optimized implementation grades of the 16 fire protection measures for the 10 different cost targets (F.B).

Measure	Cost per sq.m.	0.00 (Initial)	150,000 (145,156)	300,000 (237,524)	450,000 (375,194)	600,000 (565,171)	750,000 (743,845)	900,000 (886,958)	1,050,000 (1,032,732)	1,200,000 (1,173,260)	1,350,000 (622,736)	1,500,000 (654,392)
M1	40	0.30	0.30	0.31	0.39	0.64	0.52	0.55	0.86	0.96	0.42	0.42
M2	40	0.30	0.70	0.30	0.47	0.30	0.64	0.64	0.73	0.99	0.68	0.69
M3	10	0.60	0.70	0.60	0.95	0.85	0.84	0.97	0.96	0.92	0.84	0.84
M4	30	0.60	0.61	0.64	0.78	0.80	0.75	0.95	0.84	0.97	0.84	0.84
M5	10	0.50	0.68	0.69	0.77	0.87	0.83	0.87	0.87	0.56	0.82	0.81
M6	30	0.60	0.60	0.60	0.82	0.60	0.78	0.80	0.81	0.96	0.75	0.68
M7	20	0.30	0.31	0.31	0.36	0.31	0.85	0.76	0.74	0.91	0.69	0.30
M8	10	0.20	0.21	0.26	0.36	0.31	0.28	0.59	0.56	0.60	0.77	0.78
M9	30	0.60	0.61	0.87	0.63	0.77	0.79	0.81	0.92	0.94	0.83	0.83
M10	40	0.50	0.50	0.73	0.76	0.93	0.99	0.99	0.92	0.95	0.75	0.73
M11	40	0.00	0.00	0.04	0.11	0.66	0.49	0.64	0.87	0.70	0.05	0.43
M12	20	0.90	0.90	0.90	0.97	0.97	0.95	0.94	0.98	0.99	1.00	1.00
M13	10	0.50	0.51	0.99	0.75	0.59	0.69	0.67	0.80	0.75	0.73	0.73
M14	10	0.90	0.90	0.92	0.96	0.93	1.00	0.99	0.93	0.98	1.00	1.00
M15	20	0.60	0.64	0.60	0.68	0.74	0.76	0.94	0.83	1.00	1.00	1.00
M16	20	0.40	0.42	0.70	0.45	0.41	0.68	0.60	0.63	0.54	0.68	0.68
FSI	>	0.53	0.58	0.61	0.69	0.73	0.75	0.84	0.85	0.88	0.78	0.79
API	>	0	0.06	0.08	0.16	0.20	0.27	0.34	0.38	0.42	0.25	0.25

Table 15

Optimized implementation grades of the 16 fire protection measures for the 10 denoted designs of Figure 20.

	(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)	(10)
FSI(Target)	0.55	0.55	0.60	0.60	0.65	0.65	0.70	0.70	0.75	0.75
M1	0.30	0.30	0.30	0.30	0.30	0.30	0.30	0.30	0.30	0.30
M2	0.30	0.30	0.30	0.30	0.30	0.30	0.30	0.30	0.30	0.30
M3	0.63	0.80	0.62	1.00	0.99	1.00	1.00	1.00	1.00	1.00
M4	0.60	0.60	0.89	0.60	0.99	0.60	0.90	0.74	0.99	0.98
M5	0.50	0.50	0.50	0.50	0.50	0.70	0.54	0.95	0.63	0.98
M6	0.60	0.60	0.60	0.60	0.60	0.60	0.60	0.60	0.60	0.60
M7	0.30	0.30	0.30	0.30	0.30	0.30	0.30	0.30	0.30	0.30
M8	0.20	0.20	0.20	0.48	0.20	0.89	0.84	0.98	0.99	0.98
M9	0.60	0.60	0.60	0.60	0.60	0.60	0.60	0.60	0.87	0.61
M10	0.50	0.50	0.50	0.50	0.50	0.50	0.50	0.50	0.55	0.50
M11	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00
M12	0.95	0.90	0.96	0.91	0.96	0.91	0.99	0.97	1.00	1.00
M13	0.50	0.50	0.50	0.50	0.50	0.50	0.50	0.50	0.50	0.50
M14	1.00	0.90	1.00	0.99	1.00	0.99	1.00	1.00	1.00	1.00
M15	0.75	0.60	0.98	0.60	0.90	0.65	0.99	0.71	1.00	0.98
M16	0.40	0.40	0.40	0.40	0.40	0.40	0.40	0.40	0.40	0.40
API	0.01	0.02	0.04	0.05	0.07	0.10	0.12	0.14	0.16	0.16
Cost	39,982	14,109	129,704	54,594	167,032	105,756	213,875	175,719	319,978	270,172

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