Time Connection of Subsequent Construction Processes Estimated by Statistical Method

Abstract

An integral part of managing the construction of investment units is the monitoring of labor productivity using statistical methods in combination with construction software. Such a combination includes a number of methods and modeling, among which is a method for determining the probability of the completion of construction processes based on the recorded mean value of labor productivity and its variations. For investment units consisting of two or more objects, it is necessary to assess the probability of the completion of construction processes related to follow-up activities and which are carried out by the same work crews. Based on the selected probability modeling, the aim of this contribution was to show how statistical methods and software support can contribute to savings in resources, both human and time, in construction production. The aspect is documented in a case study of residential buildings. The Lindeberg–Lévy and Moivre–Laplace theory and the Bernoulli principle were used for mathematical modeling. The CONTEC construction software was applied as the software support. The performed modeling showed a decrease in the mean value of performance for all the processes examined compared to the planned values, except for the implementation of reinforced concrete monolithic structures. For these reasons, the working capacities had to be increased in order to meet construction milestones.

1. Introduction

One of the main reasons for not complying with the planned construction cost is the fact that the time norm, i.e., the time needed to install a specific unit of a certain building structure, is not being met. Logically, this fact itself not only increases the wage expenses, which are necessary for installing a given specific unit of a building structure, but it also increases the expenses related to the corresponding prolongation of the overall construction time. That is why it is necessary, for more extensive construction units, to monitor the productivity of individual construction processes, including the time relations of consecutive processes among individual structures, ensuring that the planned construction cost is met. Several international publications have explored the topic of deviances between the planned and actual labor productivity values. This topic is usually explored using deterministic and stochastic network analysis methods which were previously developed during the first half of the 20th century. One of the first publications that explored a network analysis method, which represents the initial construction model based on the deterministic principle, was the introduction of the critical path method (CPM) method developed by Kelley, J. and Walker, M. in 1950 [1]. The method was applied for the first time in 1966 within the frame of the construction of the high-rise buildings of the World Trade Center in New York. Another development stage of the network analysis was represented by the development of the Baukasten netzplanung (BKN) method which introduced four types of relations that represent a condition for minimal time intervals among individual construction processes [2]. A similar topic was explored by research studies that introduced the STSG network analysis method (construction network diagram), also based on the deterministic principle. The STSG method based on the BKN method was designated for modeling construction preparation and implementation processes. Compared with the BKN method, the STSG method is amended by four additional relations, which means that it can more accurately model the course of the construction [3]. One of the creators of the CPM method, Kelley, J. essentially contributed to the development of the PERT method by presenting a definition of the term “critical path”. The PERT method was introduced by Hamilton, B. A., also around 1950 [4]. The CPM method is often combined with the program evaluation and review technique (PERT) method, which is already based on the stochastic principle of the management of construction processes [5].

The described methods and other proposed solutions based on the deterministic as well as stochastic principles were, for the majority of cases, adjusted to the technical possibilities of the already existing construction software systems (for example, MS Project, Primavera or CONTEC). The time analysis prepared using MS Project, including source analysis, budget preparation and risk analysis, is often applied for constructions in the Czech Republic and abroad [6]. The application of the MS Project software also provides other options, such as the preparation of diagrams, which show the budgeted cost of the planned construction works. The Primavera project planning P6 software was able to easily compare the planned course of the construction works with the real progress of the works. This includes the basic progression steps associated with collecting, recording, monitoring and overseeing information related to the actual implementation of the given construction project. Based on these 15 input data, the cause of time delays can be determined. Some of the main advantages of this software include the option to divide large projects into smaller parts which allows for a better control. By using the Primavera project planning P6 software, the risk during the project planning, management and completion stages was reduced. The communication process among the individual participants of the construction process was improved by the means of notes directly recorded in the given time schedule. These notes are available to all partners [7]. The application of the Primavera project planning P3 software affected the three main factors of a successful construction project, which are the subject of the work, represented by the bill of quantities, expenses, and time needed for its implementation [8]. Based on the conducted survey, it was determined that the application of engineering software has a positive effect on the length of the construction time and thus also on the economy of the construction process. The Primavera project manager software belongs to the most frequently used construction software. The second place belongs to the applied Primavera project planning P3 software at a construction in Bahrain, and the third place, from the perspective of usage frequency, belongs to the MS project software. From the quality and performance perspective, the CONTEC computer system [9] was comparable to the other stated computer systems. Its undisputable advantage compared to other computer systems is its ability to calculate construction orders within the frame of a tender, even when a detailed bill of quantities is not available, exclusively based on the number of purposeful specific units of individual structures; for example, 1 m³ of an enclosed area or the characteristic features of a building structure based on the given network type diagrams, which expedites the time model preparation by up to 30 times. Its advantage is also the immediate possibility to create related documents for managing quality, impacts on the environment and occupational health and safety.

A short historical overview of the deterministic and stochastic methods and their implementation into software systems is an example of a fundamental development of monitoring the efficiency of construction processes. Several models that combine an efficient use of statistical methods with classic methods using construction software, a brief list of which will be presented in the next section, have been developed. However, there is only a very small number of studies that explore the development and optimization of the models for the construction management purposes. The proposed solution of the construction process management deals with the planning of construction capacities using the application of statistical methods. Furthermore, it deals with the assessment and evaluation of the collection of data on the performance achieved at the construction site with the aim of shortening the construction time and saving construction costs. Within these statistical analyses, a set of labor productivity data was created and constantly updated, which is retroactively implemented into the database of the CONTEC construction software in order to more accurately predict the time course of the construction and future projects of a similar nature.

2. Literature Review

The professional literature can be divided based on the way of monitoring the efficiency of construction processes into deterministic methods, stochastic methods and simulation techniques. In the case of deterministic methods (see Appendix A for monitoring labor productivity and their deviances, in continuation of literature [10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25]), we can calculate the length of construction processes based on predefined relations which are incorporated into the given construction software. In the case of stochastic methods (see Appendix A, in continuation of literature [26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43]), deviances from the planned performance are assessed using the probability theory and statistical methods in combination with the use of a construction software. Simulation techniques (see Appendix A, in continuation of literature [44,45,46,47]) form a so-called inter-developmental level of the construction management, falling in between the application of a classic method using a construction software and a combination of the statistical methods and a construction software. Integrating fuzzy logic with simulation techniques enhances the capabilities of those simulation techniques, and the resultant fuzzy simulation models are then capable of handling subjective uncertainties in complex construction systems (see Appendix A, [48,49,50,51,52,53,54,55,56]). It is also worth noting the publications relating to the application of various modifications of Petri net models [57,58,59,60,61]. Simulation methods, which are modeled on the actual recorded performance, have been studied by a number of authors, including, for example, Han, S. [62], Lee, T. et al. [63], Shin, Y. et al. [64], Montaser, A. et al. [65], Ko, Y. and Kim, B., [66] and Reinschmidt, K. [67]. Simulation applications formed the subject of the studies published by Han, S. The prediction of construction process duration is based on real data recorded at construction sites that transpire from the assumption of a standardized normal distribution, such as in the case study. The case study also confirmed the validity of the normal distribution by carrying out the Kolmogorov–Smirnov test. A partial simulation model (PSM) published by Shin, Y. et al. overcomes traditional simulation technique limitations by focusing on curtain wall operations. The Bayesian principle of the determination of the mean value performance was presented in the studies published by Ko, Y., Han, S., Kim, B., Reinschmidt, K. and Gardoni, P. Reinschmidt, K. [68]. This result shows that a Bayesian probabilistic approach can be utilized in civil engineering to provide more accurate prediction results based on updated information collecting real data. According to certain studies, it is more important to transpire from recorded real data than from prior information based on a one-time observation. The classical simulation method was improved by Pareek, P. et al. [69] by presenting the model based on Bayes principle. The model published by Smugala, S. and Kubečková, D. [70] was based on an analogous principle, as was the case of Han‘s simulation applications. Compared to Han’s study, the method proposed by authors explored another type of building and evaluated more construction processes, making up the whole structure. Another possibility of estimating the range of the mean value of performance is the application of regression analysis. A simulation model based on the principle of multiple regression analysis was published by Halpin, W. and Han, S. [71].

Examining the mean value of the performance of individual construction processes, such as reinforced concrete monolithic structures, masonry structures and plastering work are the subject of a number of publications. For example, publications dealing with formwork by Moholkar, M. and Patil, V. [72], Hanna, S. and Smith, G. [73] and Karthick, R. [74] can be mentioned. The performance of ironworks is discussed in publications [75,76]. The productivity of ceiling formwork is examined in publications. The productivity of masonry work is documented by Desale, V. and Deodhar, V. [77] Plastering work was the subject of a number of publications [78,79] in which comparable performance was achieved. The results of the average values of tiling work were presented by Idiake, E.J. [80].

3. Material and Methods

The base material is formed by:

• Probability theory;
• Definition and application of suitable methods;
• Real civic amenity structure (case study).

The main objective is:

• Determining the probability of the completion time of the previous process with the objective to determine the probability of the commencement of the subsequent process directly based on the recorded mean performance values of individual construction processes at the construction site at Klementova Street, see Figure 1;
• Implementing personnel measures for ensuring the fluency of the subsequent processes pursuant to the original time schedule based on the probability predictions conducted in compliance with the Lindeberg–Lévy and Moivre–Laplace theorem;
• Creating and updating the set of the performance data of individual construction processes. The set of the recorded mean performance values of the given processes is compared with the performances achieved during previous construction projects;
• The created and constantly updated set of performance data obtained by data collection and examined using the application of statistical methods was implemented in the construction software, which allows a more accurate prediction of the construction time course for similar projects.

3.1. Functionality of the Study

The proposed method of the prediction of work productivity was based on the collection of real performance data at the beginning of individual construction processes, including the application of statistical analyses in sufficient time to enable optimization measures leading to the shortening of the construction time and saving construction costs. Deterministic methods are based on standard performances which do not take into account the specification of the given contract as well as the different quality of the labor force deployed.

3.2. Application Principle, Used Methods

The applied statistical methods will explore the mean value prediction µ and the corresponding standard deviation s based on the random selection principle. By implementing the performance data to the construction software, we obtain the implementation time of the given process. The examination of the sample relative frequency with a 95% confidence interval will provide a percentage probability prediction of the fulfillment of the mean value range of the volume of the works conducted by individual workers during a selected period in relation to the duration of the given process. By testing the left-sided interval of the standard deviation, we achieved the longest process duration with the latest acceptable beginning of the subsequent activity. By implementing these data, we can obtain the pessimistic performance variant. The differences of the percentage prediction of the probabilities of fulfilling the range of the mean value related to the mutually technologically related building structures SO J12 and SO J34 will be compared using the test of the interval difference of the relative frequencies of two populations, using which we assess the necessity to increase the given capacities. A schematic illustration of the application of the statistical method is presented in Figure 2.

A practical example of an application of the statistical methods in combination with a construction software can be an assessment of the implementation of a ceiling formwork by the work squad above a typical SO J34 floor, which, upon the completion of the work, crosses to SO J12. That is why, as part of the assessment, we proceed with the assumption of achieving an identical mean performance value of the work squad at this structure due to the commencement of the iron and steel works conducted by the work squad crossing from SO J34 to SO J12—as can be seen in Figure 3.

3.2.1. Estimation of Mean Value µ

When calculating the mean value estimate, we assume standard distribution, not knowing standard deviation σ, i.e., the performance deviations of individual workers per work shift, in the case of construction processes where the performance of ≥30 workers can be recorded per one day, ensuring that the procedural condition pursuant to the Lin-deberg–Lévy and Moivre–Laplace Theorem was observed. Using the Kolmogorov–Smirnov test, the assumption of whether a population is subject to a standardized normal distribution will be verified. The following formula is used to find the appropriate interval prediction (where σ = s), (1–4):

$$\mathrm{P}(\overline{x}- \mathrm{s}/\surd \mathrm{n}·{\mathrm{z}}_1-{}_{\mathsf{\alpha }/2,\mathrm{n}-1}< \mu <\overline{x}+ \mathrm{s}/\surd \mathrm{n}·{\mathrm{z}}_1-{}_{\mathsf{\alpha }/2,\mathrm{n}-1})=1- \mathsf{\alpha }$$

(1)

$$\mathrm{P}(\overline{x}- \mathrm{s}/\surd \mathrm{n}·{\mathrm{t}}_1-{}_{\mathsf{\alpha }/2,\mathrm{n}-1}< \mu <\overline{x}+ \mathrm{s}/\surd \mathrm{n}·{\mathrm{t}}_1-{}_{\mathsf{\alpha }/2,\mathrm{n}-1})=1- \mathsf{\alpha }$$

(2)

$$\overline{x}={{\displaystyle \sum }}_{í=1}^{\mathrm{n}}{x}_{\mathrm{i}}/\mathrm{n}$$

(3)

$$\mathrm{s}=\surd {\mathrm{s}}^2 {\mathrm{s}}^2={{\displaystyle \sum }}_{í=1}^{\mathrm{n}}(x_{\mathrm{i}}-\overline{x}{)}^2/\mathrm{n}-1$$

(4)

where P—the interval with a 95% probability; s—the performance standard deviation of a worker per a given unit; n—the number of workers whose performance was measured; 1 − α—the confidence of interval prediction = 0.95; α—the significance level; $\overline{x }$—the sample average; s—the standard deviation; s²—the sample dispersion; µ—the range of the performance mean value; and z₁ − _α/2—the selected quantile of the standardized standard distribution [82].

3.2.2. Kolmogorov–Smirnov Test

Based on the information related to the monitored data, a theoretical division (standard distribution) of the given population was expected for individual tests. The correctness of this estimate needs to be verified by the means good conformity tests. The Kolmogorov–Smirnov test will be applied to test conformity between the sample and theoretical distributions. It is used for verifying the hypothesis that the obtained selection comes from a distribution with a continuous distribution function F(x) with the stipulation that the function must be fully specified.

• Selection of the null and alternative hypotheses:

Hypothesis 0 (H0). 

F(x) = F₀ (x).

Hypothesis A (HA). 

F(x) ≠ F₀ (x).

F(x) is the distribution function of the division, from which the random selection comes which represents a theoretical distribution function:

• Selection of the test statistics T(X) including zero distribution.

The statistics D_n is defined as a maximal deviation of the theoretical and empirical distribution function—as can be seen in (5) and (6). The sample empiric distribution function F_n(x) is presented as

F_n(x) = 0

= i/n

= 1

T(X) = D_n = sup |F_n (x) − F0(x)| = max (D₁*, D₂*, … D_n*)

(5)

where

D_i* = max {|F₀(x) – í−1/n|, |i/n − F₀(x_i)|} for i = 1, 2, 3…, n.

(6)

3.2.3. Relative Frequency π

Following Bernoulli’s equation, the relative frequency of a monitored phenomenon statistically converges to its probability. The probability of the given phenomenon—in our case that of a construction process—is basically the limited case of a relative frequency of a certain construction process. The construction work contractor needs to consider the percentage information, which is represented by the probability of the fulfillment of the range of mean value µ, based on which the probability of the completion of the given process and thus also of the entire construction can be determined. Different time intervals are selected based on the duration of individual activities when the performances of the workers were measures and when the percentage fulfillment of the range of mean value µ was assessed. When executing the calculation, it is assumed that the condition of the Moivre–Laplace Theorem will be fulfilled. To find the 95% confidence interval for the relative frequency, we use the following Formula (7):

P (p − √p·(1 − p)/n·z₁ − _α/2,n−1) ˂ π ˂ (p + √p·(1 − p)/n·z₁ − _α/2,n−1) = 1 − α

(7)

where p—the sample relative frequency of workers fulfilling mean value; π—the relative frequency; n—the total number of workers with a performance corresponding to one month; 1 − α—the confidence interval estimation = 0.95; α—the significance level; and z_α—the selected quantile of the standardized normal distribution. The determination of the percentage of fulfillment of the mean value range µ with a 95% confidence interval, respectively, the performance standard per one month, provides important information based on which the given construction site management can adopt decisions related to strengthening the given capacities or increasing work productivity.

3.2.4. Standard Deviation Confidence Interval

The subject under examination in the given section will be a determination of the 95% estimate of the left-sided confidence interval for the dispersion and standard deviation of the achieved performance standard average, which can, in a real situation at a construction site, represent the lowest value of the given performance during a given time period with a certain percentage confidence. We can calculate the left-sided confidence interval, i.e., the lowest deviance of the given performance, using Equation (8). The right-sided interval and highest performance estimate will be calculated following mathematical Formula (9):

P₁ (n − 1)/x_1−α,n−1·s² < σ²

(8)

P₂ (σ ˂ √(n − 1)/x_α,n−1·s) = 1 − α

(9)

where P₁(P₂) is the left-sided (right-sided) interval with 95% probability, s2 is the sample dispersion and σ is dispersion. The reliability of the interval prediction: 1 − α = 0.95, $x_{1-\alpha , n-1}$ is the corresponding distribution quantile s with n − 1 degrees of freedom.

3.2.5. Interval Estimate of the Difference of Relative Frequencies of Two Populations

In our case, the interval estimate of the difference of the mean values will not be explored since construction processes are implemented by identical work capacities and it is thus assumed that identical ranges of mean values µ will be achieved. Nevertheless, different relative frequencies can be obtained from two examined objects due to the different extents of their respective elements. The difference of the relative frequencies of two phenomena is explored from the perspective of technological relations. Similarly to Section 3.2.1., it is assumed that the explored quantities X₁ and X₂ are subject to standard distribution, sample sets with an extent of n₁ and n₂ number of the elements, for which we monitor characteristics x₁ and x₂. The calculation of the sample relative frequency is executed following the given mathematical formulas (10), (11) and (12):

p₁ = x₁/n₁, p₂ = x₂/n₂

(10)

Random quantity P₂ is then of the following mathematical form (11):

P₂ = [(p₁ − p₂) − (π₁ − π₂)]/√p(1 − p)(1/n₁ + 1/n₂), where p = (x₁ + x₂)/(n₁ + n₂)

(11)

The two-sided interval for calculating the difference of relative frequencies is then calculated following mathematical Formula (12):

P((p₁ − p₂) − √p(1 − p)(1/n₁ + 1/n₂)·z₁ − _α/2 ˂ (π₁ − π₂) ˂ (p₁ − p₂) + −√p(1 − p)(1/n₁ + 1/n₂)
·z₁ − _α/2 = 1 − α

(12)

where p₁, p₂ are sample relative frequencies; n₁, n₂ are the range of the sample set; and z₁ − _α/2 quantile of the standardized normal distribution.

3.2.6. Test Hypothesis on Relative Frequency

Another test which will be applied explores the relative frequency values of two populations of the achieved % of the fulfillment of the range of the performance mean values at SO J12 and SO J34. The result is a decision if the value of the probability % of the performance fulfillment at SO J12 represents a statistically significant decrease in relation to the % value of the fulfillment of the range of the performance mean values at SO J34, which will require personnel measures that will ensure the fluency of the subsequent construction processes. The determination of the hypothesis consists of the following steps (13).

• The formulation of null and alternative hypothesis: zero hypothesis H₀ represents a value % of the performance at SO J12, an alternative hypothesis H_A value corresponds to the relative frequency value at SO J34. The state where mean value µ ˂ µ₀ will be assumed;
• Selection of test statistics: we will proceed according to the following relationship (13):

T(X) = P₂ = (p − π)/√π(1 − π)·√n → N (0.1)

(13)

where T(X) = alternative of the statistical characteristic;

• Calculation of the monitored value of the test statistics x_OBS: the p-value will be calculated using Formula (13), and it will then be decided whether H₀ is rejected or adopted.

4. Results and Discussion

The collection of data related to the performance of individual workers during a certain time period was conducted with the objective to determine the ranges of the mean values µ (performances), provided that the confidence interval is 95%. By evaluating these random experiments in the form of the daily performance of workers, a probabilistic estimate of the completion date of the given process was achieved. Optimistic and pessimistic performance variants with the shortest and longest process duration can be obtained using the bottom and top performance values. Since identical work squads will be used at the mutually technologically related SO J12 and SO J34 building structures, it is expected that the range of the performance mean value µ will be identical; however, a test of the difference of the relative frequencies of two populations will be conducted with the objective to verify the time and technological connections of the beginnings of individual parts of the implementation of monolithic structures. Data will be collected and the processes of working with the concrete, iron, masonry, plaster, and tiling a façade, which belong to processes with the highest demand for labor, will be subsequently assessed.

4.1. Reinforced Concrete Structure

The underground and aboveground stories will be built in a horizontal ascending manner, meaning that upon completion of the foundation slab on the second underground floor of t building J34, the work squads immediately transfer to the adjoining building structure J12 where they commence work related to the implementation of the foundation slab of this building, while the armoring and concreting processes of the reinforced concrete monolithic walls commence on the surface of the already completed foundation slab of building J34. The technological connection of the monitored construction processes within the frame of the construction of the J34 and J12 buildings is shown in Figure 4. Because of an immediate connection of these activities, it is necessary to monitor and assess the differences of the relative frequencies of the two populations.

4.1.1. Iron Works—Monolithic Foundation Slab

Based on the database, a standard hour per worker for installing the bars represents 0.0241 Sh/kg → 332 kg/shift. To implement the foundation slab, two squats of iron workers, each of them with six workers, will be assigned to each dilatation.

• Estimated range of mean value µ

Due to the limited number of data (performances of 12 workers, see

Table 1. Iron works of monolithic foundation—performance kg/Sh/man.

Work/Load	1	2	3	4	5	6	7	8	9	10	11	12
kg/Sh/man	300	300	315	315	330	330	290	290	320	320	310	310

, the selected procedure corresponds to the number of the conducted random tests. To calculate the range of the mean value µ, i.e., to determine the given interval estimate, we proceed following mathematical Formula (2) stated in Section 3.2.1., utilizing the symmetry characteristics of the Student’s distribution. According to Section 3.2.2., the zero hypothesis was not rejected, i.e., that the data in the form of worker performances are subject to a standard distribution.

The number of random tests is ˂30.

Conclusion: Based on the executed mean value estimates, it can be stated that the range of the mean value is 306.53 ˂ µ ˂ 325.06 with its center based on the Student’s distribution, assuming an accuracy of the 95% confidence interval amounting to 316 kg/shift/man (see Figure 5), which represents a difference of approximately 6% when compared with the performance standard following the database. The contractor is not forced to take any measures to meet the original time schedule.

• Relative frequency SO J34

The objective of the relative frequency test is to determine the percentage fulfillment success rate of the mean value range bottom limit µ = 306.53 kg/shift during a time interval of approximately 3 weeks. Based on Formula (7), the relative frequency of the fulfillment of the range of the performed construction volume for 5 worked cycles was calculated assuming the validity of the 95% confidence interval. The underground and aboveground stories will be built in a horizontal ascending manner which means that upon the completion of the foundation slab on the second underground floor of building J34, the work squads immediately transfer to the adjoining building structure (J12) where they commence work related to the implementation of the foundation slab of this building, while the armoring and concreting processes of the reinforced concrete monolithic walls commence on the surface of the already completed foundation slab of building J34 (note: the technological connection of the monitored construction processes within the frame of the construction of the J34 and J12 buildings is shown in Figure 3 and Figure 4). Because of an immediate connection between these activities, it is necessary to monitor and to assess the differences of the relative frequencies of two populations.

Conclusion: It can be stated with 95% confidence that the percentage probability of the fulfillment of the performance of the foundation slab reinforcement implementation falls between 86.2% and 101.7% in compliance with the range of the mean value µ following the given calculation (note: see Section 3.2.3 with the stipulation that the center of this percentage range is 93.5%).

• Interval estimate of the difference of relative frequencies between J34 and J12

The implementation of the monolithic structures at both buildings is executed by the same workers. Upon completion of a certain activity at SO J34, the work squads immediately move to SO J12, where they execute the same work. That is why the achieved mean value ranges at the J34 and J12 buildings are assumed to be identical. The test of the difference of the relative frequencies between building J34 and building J12 basically confirms the conformity of the range fulfillment of the mean values of the performances conducted at the aforementioned buildings for a certain time interval. Due to a minimal time reserve between the completion of the foundation slab and the commencement of wall structure, their time development needs to be monitored. The comparison will be conducted in relation to a process optimization, during which the number of workers at the building J12 may be increased in order to start the wall structure according to the original time schedule. For the assessment, mathematical formulas (10)–(12) will be used. A time interval of 1 work cycle (three shifts) was selected for assessing the random tests since it is possible to adopt corrective measures due to the overall implementation time of the foundation slab reinforcement installation process. Based on the range estimate of mean value µ at SO J34, the given worker performance per shift amounts to between 306 kg and 325 kg. The test at SO J12 is considered successful (U) if the bottom limit of the performance range is complied with; incompliance with this limit represents an unsuccessful test (N), which was recorded in four cases.

Conclusion: When the value of the difference is (π₁ − π₂) ˂ 0, it can then be stated that the relative frequency of the percentage fulfillment of the performance mean value range related to the foundation slab reinforcement installation is higher at SO 34. On the other hand, should the value of the difference be (π₁ − π₂) > 0, this would mean a lower percentage fulfillment of the performance range of these works at SO J34. In our case, it cannot be unambiguously demonstrated which relative frequency value of the percentage fulfillment of the mean value range is greater since the interval acquires negative as well as positive values. Nevertheless, 95% of the interval range acquires negative values, which means that it can be expected that the probability of the fulfillment of the mean value range related to the iron work performance at SO J 34 will acquire greater values than at SO J12. Because of the negligible time reserve between the completion of the reinforcement installation with the subsequent concreting of the foundation slab at SO J 12 and the commencement of the next process, which is represented by erecting monolithic walls, it can be claimed that the construction preparedness for commencing this construction activity can be endangered, which was confirmed by the execution of the test hypothesis on relative frequency based on Section 3.2.6. The time progress of the foundation slab installation process at SO J 12 should be assessed on a regular basis, making sure appropriate corrective measures in the form of ordering overtime work can be adopted in the case of time delays. The percentage probability of the fulfillment of the performance of the foundation slab reinforcement implementation falls between 86.2% and 101.7% in compliance with the range of mean value µ with the stipulation that the center of this percentage range is 93.5%.

4.1.2. Monolithic Walls on the Second and First Underground Floor

• Estimated range of mean value µ

Based on the database, the aggregate sum of the performance standard per worker for erecting and removing formwork, including the application of concrete, reinforcement and sealing strips, amounts to 2.27 Sh/m² (0.55—formwork erection; 0.32 formwork removal; 0.4—concreting; 0.9—installation of the reinforcement; 0.1—assembly of the sealing strip) → 3.52 m²/shift/worker. Based on the database, a work hour/worker for installing the bars represents 0.0241 Sh/kg → 332 kg/shift. In our case, to conduct these works, there were two squads of carpenters, each comprising four workers, to whom these works were assigned in order to ensure the timely utilization of the crane technology. The reinforcement of monolithic walls are usually implemented in order to ensure the timely progress of individual processes conducted by identical workers. Contrary to assessing other construction processes, the performance of the entire carpenter squad will be calculated instead of the performance of individual workers, because dividing the entire aggregated sum into individual partial subitems is very difficult to perform from a technical perspective at the actual construction site. The calculated performance of a worker of 3. 52 m²/sh/w, which translates into 13.1 m²/sh/sq, represents a work load of 4.5 PERI TRIO formwork segments/day (9.7 linear meters), which will be implemented every other day together with the given concreting cycle. The perimeter dimension of the second underground floor amounts to 273.12 m [83], which allows the implementation of approximately ≥30 workloads (

Table 2. Monolithic walls—time consumption/wall (unit 13.1 m²)/squad (h).

Work/Load	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15
Time/unit/sq	7.5	8.5	8	7.5	8.5	8	9	7.5	8	7.5	8.5	8	8.5	7.5	8
Work/load	16	17	18	19	20	21	22	23	24	25	26	27	28	29	30
Time/unit/sq	8	8.5	8.5	8	7.5	8.5	8	9	7	8	8.5	8	7.5	8.5	8

) on the standard second underground floor and assessments following the Lindeberg–Lévy and Moivre–Laplace theorems.

The number of random tests is ≥ 30.

Conclusion: Because of the unambiguity of the fulfillment of the performance standard per worker of 3.52 m²/shift, the Kolmogorov–Smirnov test, which verifies validity of the assumption that the given population is subject to a standard distribution, was not conducted. When walls are erected on the second underground floor, monitoring the time progress of individual workloads and random tests demonstrates the fulfillment of the performance standard per worker of 3.52 m²/shift, which translates into 13.1 m²/shift/work squad, provided that the 95% confidence interval is valid. The upper limit of the mean value assessment range ˂ 8.5 h, as can be seen in Figure 6 (7.87 ˂ µ ˂ 8.23), demonstrates the ability to fulfill the performance standard related to the erection of monolithic walls in accordance with the database. The mutual time interconnection of individual construction process cycles at building structures J34 and J12 will be the subject of a separate section, which estimates the difference of the relative frequencies of two populations.

• Relative frequency SO J34

The objective was to research the execution of monolithic wall workloads from the first underground floor to the seventh above-ground floor. The objective of the relative frequency test was to determine the percentage fulfillment success rate of this performance standard for a work shift that lasts 8.5 h. Since this construction process takes 7.5 months (contrary to the foundation slab, which is done in 14 days), the percentage fulfillment of the range of the performance mean value for a longer period (1.5 months in our case) can be verified. The performance of 2 work squads/30 shifts—1 work cycle—will be assessed. The performance standard of 3.52 m²/shift/worker was not fulfilled in two cases out of a total of 30 random tests. The others can be marked as successful tests. Based on mathematical Formula (7), assuming the validity of the 95% confidence interval, the relative frequency of the fulfillment of the performance standard related to the implementation of monolithic walls from the second underground floor to seventh above-ground floor can be calculated.

Conclusion: It has been demonstrated with 95% confidence that the percentage success ratio of the fulfillment of the performance standard for implementing the given wall-related work load (3.52 m²/shift/worker) amounts to between 86.2% and 101.7%, with the stipulation that the center of the percentage range is 93.9%. The corresponding relative frequency test demonstrated a high percentage probability of the fulfillment of the performance standard related to the implementation of the given monolithic wall work load.

• Interval estimate of the difference of relative frequencies between J34 and J12

Capacities at SO J12 are not increased for this work process. The works are conducted by the same workers who conduct the works at SO J34. That is why the performance mean value range will be assumed to be identical. Only the interval estimate of the difference of relative frequencies between J34 and J12 will be verified. The ground plan dimensions of the second underground floor at SO J12 are identical to those of building SO J34, which means that we have 30 workloads, i.e., random tests, available. For the assessment, mathematical formulas (10)–(12) will be used. Based on the implementation time of individual workloads (>30), whether the test in question was successful was determined, i.e., whether the performance standard per shift was fulfilled.

Conclusion: In the case of SO J12, > 30 monolithic wall workloads were performed, and the given standard was not fulfilled only in one case. Therefore, it can be stated that the relative frequency of the fulfillment of the time period range that was necessary for implementing the given work load related to monolithic walls was improving in comparison with SO J34, meaning that (π₁ − π₂) > 0 (SO J12 π₁ = 96.6%, SO J34 π₂ = 93%). It can then be stated with 95% probability that the construction preparedness at SO J12 for commencing the following construction process—which is the assembly of the ceiling structure formwork above the second underground to seventh above-ground floors—is ensured, as confirmed by the execution of the test hypothesis on relative frequency based on Section 3.2.6.

4.1.3. Ceiling Formwork from the Second Underground Floor to the Seventh Above-Ground Floor

• Estimated range of mean value µ

The performance standard for erecting and removing the ceiling formwork, including concreting, amounts to 1.99 Sh/m² per 1 worker (1.31—formwork erection; 0.48—form work removal; 0.20—concreting), representing a performance of 4.02 m² per 1 worker per shift. The subject of the evaluation will only be formed by formwork erection without concreting. The performance standard will thus amount to 6.1 m²/shift per 1 worker, provided we assume 1.31 Sh/m². The assembly and disassembly of the formwork on the second underground floor and on the seventh underground floor will be conducted by four squads of four workers each who will work at individual structures J3–J4 and subsequently J1–J2. The performance of the entire work squad will be monitored since a collection of the performance data of individual workers at the construction site is not feasible. The performance standard that corresponds to the performance of the carpenter squad then amounts to 24.4 m²/shift. The size of the ceiling structure for the second and first underground floors of building J34 is 1803 m² while it is 1344 m² from the first to seventh above-ground floors. The corresponding completion time of this dilatation is 18 working days provided that the performance standard following the database and the assignment of four carpenter squads at the J34 dilatation are observed. Considering the number of work squads working on the dilatation for 18 work shifts, it is possible to record ≥ 30 random tests in the form of carpenter performances, as can be seen in

Table 3. Ceiling formwork—performance m²/Sh/squad.

Work/Load	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15
m2/Sh/squad	23	21	22	22	23	21	22	22	21	23	20	22	23	21	22
Work/load	16	17	18	19	20	21	22	23	24	25	26	27	28	29	30
m2/Sh/squad	23	22	21	22	23	22	21	23	22	21	24	20	21	20	21

. The estimation of the range of the mean value µ will be calculated using mathematical formulas (1)–(4), as can be seen in Section 3.2.1.

The number of random tests is ≥ 30.

Conclusion: The calculation of the center of the range of mean value µ of the work squad’s performance based on a 95% confidence interval amounts to 21.8 m²/shift, as can be seen in Figure 7, which represents a 10.7% difference in comparison with the standardized data following the database. The size of the ceiling structure above the second and first underground floors following the implementation project documentation amounts to 1.803 m². The formwork assembly process requires 18 shifts, provided that four work squads are used and the performance of 97.6 m²/shift is achieved in compliance with the performance standard according to the database. Should four work squads be used with a performance of 87.2 m²/shift, the formwork assembly process would require 20 shifts based on the calculation of the range center of the mean value. To comply with the contracted time schedule, it was determined that the formwork assembly process should be performed in 14 shifts. This is why the capacities must be strengthened by 1 work squad to achieve the target state of five work squads.

The size of the ceiling structure formwork from the first to seventh above-ground floors at J34 following the implementation project documentation amounts to 1344 m². Because of an efficient coverage of this work queue, only four carpenter squads with four workers each can be used. The statistical analyses of the interval estimate of the range of mean value µ of the work squad performance based on the 95% confidence interval, which were conducted for the ceiling structures above the second and first underground floors, are also applicable to the ceiling structures from the first to seventh above-ground floors. The formwork implementation process requires implementation in 14 shifts due to the size of the ceiling structure above a standard floor, provided that four work squads with a performance of 97.6 m²/shift in accordance with the database are used. Based on the calculation of the range of mean value µ, the formwork assembly process requires 15 shifts, provided that four work squads with a performance of 87.2 m²/shift are used. Considering the need to comply with the overall construction completion deadline, the formwork assembly time above a standard floor needs to be shortened to 12 work shifts. This time difference can be only overcome by making the given standards more demanding (overtime work) and by ordering additional work shifts since it is not possible to assign another work squad.

• Relative frequency SO J34

The estimate of the mean value range of the work squad performance is between 21.4 and 22.2 m²/shift according to the calculation performed using mathematical formulas (1–4), 34-assuming validity with a 95% confidence interval. A time interval of approximately 1.5 months was selected. During this time, the fulfillment of the calculated mean value range will be assessed, in which case the compliance with the bottom limit of the mean value range will be considered a successful test. Using the given mathematical Formula (7), we calculate the percentage probability of the fulfillment of the range of the performed construction volume in compliance with the mean value µ.

Conclusion: It was demonstrated with a 95% confidence that the percentage probability of the fulfillment of the mean value estimate of the work squad performance for a period of 1.5 months required to implement the ceiling formwork oscillates between 81.7% and 100.8%. The calculated center of this percentage range amounts to 91.5%, which corresponds to a high percentage probability fulfillment of the mean value range when implementing ceiling formwork.

• Interval estimate of the difference of relative frequencies between J34 and J12

In relation to increasing the carpenter capacities from the original four to five squads on the second underground to first underground floors at SO J 12, the population extent increase at SO J34 from n₁ = 32 to n₂ = 42 at SO J12. Since it is possible to adopt corrective measures in advance by making the standard more demanding, the selected time interval for assessing the performance of the squads is that of eight work shift. When calculating the difference between the relative frequencies, mathematical Formulas (10)–(12) will be used.

Conclusion: Since the two-sided interval with the 95% confidence returns negative as well as positive values, we cannot unambiguously demonstrate which of the relative frequency values of the percentage fulfillment of the mean value range is greater. However, 59% of the interval range returns positive values. Therefore, it can be expected that the relative frequency values of the percentage fulfillment of the mean value range related to the formwork performance at SO J12 will be greater than at SO J34, which means that no personnel measures for ensuring preparedness at SO J12 must be adopted in order to ensure the smooth continuity of the subsequent construction process (the implementation of the iron works). The test hypothesis of relative frequency by the rejection null hypothesis H₀ equal to the percentual fulfillment of the performance at SO J12 confirmed this statement.

4.1.4. Iron Works—Ceiling Structures from the Second Underground to Seventh Above-Ground Floors

• Estimated range of mean value µ

Considering the technical complexity and the reinforcement percentage of individual ceiling structures from the second underground to seventh above-ground floors, only one interval estimate of the performance mean value of the ceiling structure reinforcement above the second underground floor will be used to plan of work capacity. When calculating the worker performance mean value estimate according to mathematical formulas (1)–(4), as in Section 3.2.1, a standard distribution is assumed, as demonstrated by the Kolmogorov–Smirnov test. The Student’s distribution theory with v degrees of freedoms will be applied due to the number of random tests of individual workers’ performances. The reinforcement will be installed by two squads with a total number of 12 iron workers, as can be seen in

Table 4. Iron works of ceiling structure—performance kg/Shift/man.

Work/Load	1	2	3	4	5	6	7	8	9	10	11	12
kg/Sh/man	275	275	280	280	270	270	280	280	260	260	275	275

. To ensure the loadbearing capacity of the ceiling above the second underground floor according to the reinforcement drawing in the implementation documentation, 43.272 kg of meshing and bar reinforcement is proposed.

The number of random tests is ˂ 30.

Based on the calculation of the interval estimate of the performance mean value range of the ceiling panel reinforcement above the second underground floor, a proposal for the iron work capacities will be prepared for the first to seventh above-ground floors when the weight of the loadbearing as well as dividing reinforcement is reduced to 32.256 kg. The performance standard according to the database represents a value of 290 kg/shift/worker, resulting in the expected ceiling structure reinforcement installation time above the second and first underground floors of 13 shifts, provided that 12 iron workers are used.

Conclusion: In the case of ceiling structures above the second and first underground floors, the center of the range of the mean value µ based on 12 recorded performances amounts to 273 kg/shift, as can be seen in Figure 8. The conducted and calculated mean values are approximately 6% lower than the performance standard according to the database. The reinforcement installation process requires the implementation of 12 shifts, provided that 12 workers are used and a performance complying with the database is achieved; however, if the center of the mean value range will amount to 273 kg/shift, the process will last 13 shifts. However, because of the need to comply with the contractual deadline, the time must be shortened to 10 work shifts. In order to comply with this requirement, the work capacity must be increased to 14 iron workers and the standard must be made more demanding by ordering overtime work.

Work capacity planning for the reinforcement installation of ceiling structures from the first to seventh above-ground floors is related to calculations of the ceiling structure above the second underground. Based on the deployment of 12 workers and on achieving the performance standard complying with the database, the construction process requires the implementation of 9 work shifts, provided that 2 squads of iron workers with 12 workers each are deployed, and that the center of the range estimate of the iron work performance related to the installation of the ceiling structure above the second underground floor amounts to 273 kg/shift—the duration of this construction process is 10 work shifts. According to the contracted time schedule, the reinforcement assembly time amounts to eight work shifts. It is thus necessary to increase the number of iron workers by one worker. The final number of the iron workers who will work on the reinforcement installation of the above-ground floors will be 13.

• Relative frequency SO J34

The subject of the relative frequency test is the percentage fulfillment of the range estimates of the ceiling structure reinforcement implementation on the second underground to seventh above-ground floors. To achieve greater accuracy with the relative frequency test, a total of seven work cycles will be monitored, which represents a time period of approximately 1.5 months. Compliance with the bottom range limit of 1900 kg/worker/7 work cycles will be assessed as a successful test (268.9 ˂ µ ˂ 277.1 kg). Incompliance will represent an unsuccessful test. The relative frequency of the fulfillment of the range related to the implementation of the ceiling structure reinforcement will be calculated using mathematical Formula (7).

Conclusion—The percentage fulfillment of the mean value range estimate of the worker performance for an approximate period of 1 month (seven work cycles) related to the ceiling structure reinforcement installation amounts to between 78.8% and 99.1%—where the center amounts to 88.9%, which is a lower value compared with the previous results.

• Interval estimate of the difference of relative frequencies between J34 and J12

The assessment will address the calculation of the difference of relative frequencies related to the reinforcement installation between SO J34 and SO J12 from the first underground to seventh above-ground floors. The intervals with different population extents n₁ = 36 at SO J34 and n₂ = 42 at SO J12 will be assessed in relation to the increase in the number of iron workers from the original 12 to 14 according to mathematical formulas (10)–(12), as in Section 3.2.5. A shorter time interval for assessing the difference of relative frequencies (three shifts–one cycle) was chosen in this case as well due to the necessity to immediately adopt personnel measures by means of increasing capacities or making the standard more demanding by adding overtime work should the performance at SO J12 not correspond to the mean value range according to the calculation.

Conclusion: Based on the estimate of the two-sided interval with 95% confidence, we cannot unambiguously determine which building is associated with a higher percentage fulfillment of the mean value range related to the reinforcement installation performance. Based on the 60% interval range, which returns positive values, we can expect that the percentage fulfillment of the relative frequency of the mean value range related to the reinforcement installation in the ceiling structures from the second underground to seventh above-ground floors at SO J12 is greater than that at SO J34, meaning that the construction preparedness at SO J12 for launching the next process is not endangered.

4.2. Masonry Works

Contrary to the construction process of the reinforced concrete structures at SO J34 and J12, different work squads will be used with a possibility of achieving different labor productivities. For these reasons, the estimation range of mean value µ and relative frequency tests will be conducted for both building structures.

SO J34

• Estimated range of mean value µ

The performance standard according to the CONTEC database is 1.86 m³/Sh. A total of 32 random tests were conducted (

Table 5. Masonry works—performance m³/Sh/man.

Worker/Load	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16
m3/Sh/man	1.6	1.5	1.7	1.5	1.6	1.7	1.6	1.5	1.7	1.5	1.6	1.7	1.6	1.7	1.6	1.6
Worker/load	17	18	19	20	21	22	23	24	25	26	27	28	29	30	31	32
m3/Sh/man	1.6	1.5	1.7	1.5	1.6	1.7	1.6	1.5	1.7	1.5	1.6	1.7	1.6	1.7	1.6	1.6

) to determine the range of the labor productivity interval with 95% reliability which will be calculated based on mathematical formulas (1)–(4), as can be seen in Section 3.2.1.

The number of random tests is ≥ 30.

Conclusion: Based on the estimate of the two-sided interval with 95% confidence, estimate of the mean value can be determined as within the range of 1.582–1.637 m³, while the center of the range applied to the database is lower by 13.5%, as can be seen in Figure 9. To secure the time schedule deadlines, it is necessary to strengthen the capacity by two bricklayers who will be assigned to each floor of a typical story. In SO J12, different working squads were deployed compared to SO J34, simultaneously working to achieve different work productivities. For this reason, the estimation of the mean value range in SO J12 was performed. The tests showed the values of the performance range of SO J12 to be lower by 4.5%, as related to SO J34. The estimation of the mean value µ was between 1.498 and 1.562 m³. In addition to increasing the working capacity by two workers, it is necessary to increase the overtime work standards to fulfill planned time schedule.

• Relative frequency

In this section, the fulfillment of the performance mean value range of the bricklaying works for a period of seven shifts will be examined, which represents approximately 50% of the expected time needed for executing the aforementioned construction process in SO J34 and J12. The mathematical Formula (7) will be used to calculate the relative frequency of masonry works.

Conclusion: In SO J34, the percentual probability of the fulfillment of the estimate of the performance mean value range by a work squad is between 81.7% and 100.8%. Slightly higher values were achieved in SO J12, whereby the percentual probability is between 85.6% and 102.3%, which does not require any further personal measures.

4.3. Plastering Works

The subject of the assessment will be a 15 mm-thick lime-gypsum plaster coating, as made on the surfaces of the masonry of the perimeter cladding and the internal partitions from the first floor to the seventh floor, whose performance standard, i.e., its standardized mean value µ, is based on the database and is equal to 0.73 m²/h, which represents an output of 10.95 m² per shift.

SO J34

• Estimated range of mean value µ

The performance of 16 workers was recorded, who were deployed in the two typical floors forming the working queue in order to achieve the number of ≥30 random trials, as can be seen in

Table 6. Plastering works—performance m²/Sh/man.

Worker/Load	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16
m2/Sh/man	12	11	10	12	11	13	12	11	10	12	11	10	13	10	12	11
Worker/load	17	18	19	20	21	22	23	24	25	26	27	28	29	30	31	32
m2/Sh/man	12	11	11	12	11	13	12	11	12	12	11	12	13	12	10	12

, enabling the procedure according to the Moivre–Laplace theorem, as can be seen in Briš, R. et al. Similarly, as in the evaluation of the performance of masonry structures, the plastering works of the SO J34 and SO J12 buildings were simultaneously carried out; therefore, different working groups are placed in a working queue, which will achieve different performances (ranges of mean values µ). Mathematical formulas (1)–(4) will be used to calculate the mean value range of plastering works’ performance.

The number of random tests is ≥30.

Conclusion: The interval estimate of the range of the mean value with 95% confidence ranges from 11.21 m² to 11.79 m², while the center of this range represents the value of 11.5 m², as can be seen in Figure 10. In relation to the standardized database, this value is 15% lower which requires the reinforcement of two workers, provided that the contractual HMG is complied with. The center of the range of the mean value achieved in SO J12 compared to SO J34 is by 2% lower, which is the minimal difference that does not require any further personal measures.

• Relative frequency

Relative frequency represents the percentual fulfillment of the bottom line estimate of the range of the average value of an individual’s performance. The period of three work cycles over six days was chosen due to the building time period of this construction process. The mathematical Formula (7) will be used to calculate the relative frequency of plastering works.

Conclusion: A percentual probability fulfillment of the bottom line of the mean value range µ of plastering works was between 81.7% and 100.8%, with the middle of this range being 91.2%. In comparison with SO J34, the percentual fulfillment of the mean value range at this building is higher by 2.7% which does not require any personal measures to secure the preparedness of subsequent construction process.

4.4. Tiling Works

In relation to the statistical analyses of the construction processes related to installing interior floor and wall tiles, the corresponding values of the time norms were applied according to the database. The time norm for interior wall tiles amounts to 1.41 Nh/m², while this value for the interior floor tiles amounts to 1.23 Nh/m². The calculated labor production value per worker per shift for interior wall tiles is 5.7 m²/shift, while for interior floor tiles, this number is 6.5 m²/shift. In order to simplify the calculations, we will only use one average value of 6.3 m²/shift. This number takes into account the mutual ratio of the implemented surfaces of the interior wall and floor tiles (1:4). To implement this process, a total of 20 workers were employed.

SO J34

• Estimated range of mean value µ

Considering the number of employed workers, only a small number of random tests are available (≤30). The performance per work cycle, which represents one shift, will be recorded.

Because of the number of tile workers (10) working at each building structure of SO J34 and SO J12, data will be collected from 15 random tests in the form of a workers’ performances per shift—as shown in

Table 7. Tiling works—performance m²/Sh/man.

Worker/Load	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15
m2/Sh/man	5	4.9	5	5	4.9	4.8	5	5.1	5.1	5	5	5.1	5	4.9	5.1

. Mathematical formulas (1)–(4) will be used for the calculation of the mean value range of plastering works’ performance. The estimated range interval of the mean value µ will be calculated for each structure separately, because the tiling works at SO J34 and SO J12 will take place simultaneously. It is thus expected that the achieved performances per shift will be different for each of the structures.

Conclusion: The calculation demonstrated that the worker’s productivity interval falls between 5.14 m² and 5.26 m²/shift. Compared to the planned value stated in database, the center of the range is 17.5% lower—as shown in Figure 11. The achieved labor productivity at SO J12 is 3.5% lower than that at SO J34, where the range of the performance mean value is between 4.95 m² and 5.05 m² of the tiled interior wall and floor tiles.

• Relative frequency

The objective of the relative frequency test is to assess the fulfillment of the estimated range interval of the mean value during one work cycle (two shifts), for which the fulfillment of the bottom performance limit will be marked as a successful test (U), while not fulfilling the limit will be considered an unsuccessful test (N). The relative frequency test will be executed for both structures, SO J34 as well as SO J12. The relative frequency test is supposed to verify the stability of the fulfillment of the range of mean value µ for a certain time. Mathematical Formula (7) will be used for the calculation.

Conclusion: The relative frequency test of the fulfillment of the mean value with a 95% reliability for a period of one work cycle (two shifts) ranges between 82.4% and 105.5% with the stipulation that the center of this range amounts to 93.9%. Compared to SO J34, the relative frequency of the fulfillment of the range of the labor productivity mean value of the tiling works at SO J12 is lower and ranges between 79.3% and 100.7% with a range center at 90%. This value is lower by 3.9%. The execution of the hypothesis test of relative frequency shown in Section 3.2.6. did not require any personal measure to secure the commencement of the subsequent construction process.

4.5. Façade Works

The works will be simultaneously implemented at both structures. We can thus expect that the achieved mean value ranges will be different. To conduct these works, a total of 32 façade workers are available. They will be evenly deployed at SO J34 and SO J12, corresponding to the planned number of workers. The value of the worker’s performance time norm is 1.24 Nh/m² of the implemented façade surface. The corresponding labor productivity per one worker per shift for the façade works amounts to 5.7 m².

SO J34

• Estimated range of mean value µ

To achieve the recording of the number of random tests ≥30, which are necessary for fulfilling the procedure condition according to the Moivre–Laplace theorem, data will be collected on the performances of 16 workers (1 work cycle) over 2 shifts, due to which we will have 32 random tests available in the form of performances of the façade workers at each of the building structures, i.e., SO J34 and SO J12—as shown in

Table 8. Façade works—performance m²/Sh/man.

Worker/Load	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16
m2/Sh/man	3.9	4.3	4.4	4.6	4.4	4.3	4	4.3	4.2	4.1	4.3	4.4	4.3	4.2	3.9	4.3
Worker/load	17	18	19	20	21	22	23	24	25	26	27	28	29	30	31	32
m2/Sh/man	4.2	4.3	4.7	4.2	4.2	4.4	4.2	4.1	4.1	4	4.3	4	4.2	4.1	4.3	4.1

.

The range of the mean value µ will be separately assessed and calculated for each structure, since it is expected that the achieved mean values µ will be different.

Conclusion—For this construction process, the estimated range of the mean value µ with 95% reliability ranges between 4.14 m² and 4.26 m². Compared with the planned value according to the database, the center of this range is lower by 26.4%, which represents a big drop in labor productivity, see Figure 12. The contractor must immediately expand its capacity by five workers to a total of 21 façade workers in order to ensure the fulfillment of partial deadlines according to the original time schedule.

• Relative frequency

The objective of the relative frequency test in the given case is to assess the percentage probability of the fulfillment of the range of the mean value µ during an interval of four work cycles (eight shifts). Fulfillment of the bottom labor productivity limit for this time period will be marked as a successful test (U), while not fulfilling the limit will be considered an unsuccessful test (N). The relative frequency test will be conducted at both structures, SO J34 and SO J12. The percentage probability of the fulfillment of the bottom labor productivity limit of the given process will be determined by applying mathematical Formula (7).

Conclusion: The percentage probability of the fulfillment of the estimate of the labor productivity mean value range of individual workers at SO J34 for the period of two work cycles—eight working days—ranges between 81.7% and 100.8%. The center of this range amounts to 91.2%, while the center of the relative frequency at SO J12 is 93.9%. The fulfillment of the labor productivity range at SO J12 reaches higher values and thus no personnel measures are necessary.

4.6. Comparison

4.6.1. Productivity of Construction Processes

The highest and lowest values of the limits of the performance data were executed for individual construction processes in compliance with the conducted estimation test of the left-sided or right-sided interval with a 95% probability (optimistic and pessimistic variants), see Figure 13, Figure 14, Figure 15 and Figure 16, according to mathematical formulas (8) and (9); they were implemented via the construction software, by means of which the longest and shortest times of the implementation of the given construction process, i.e., the optimistic and pessimistic variants of the course of the selected construction activity, were determined.

4.6.2. Duration of Construction Processes

Figure 13 shows the differences among the durations of individual processes in compliance with labor productivity, which were implemented via the construction software. The above-stated selected processes and the prolongation of their durations obviously have an impact on the overall construction time of structure J34. Should no personnel measures by means of increasing the labor capacities be adopted in compliance with the calculation results of the statistical analyses implemented in the construction software, the overall construction time of the residential project in Prague 13, West Town, Klementova Street, would be prolonged by 19 days (Figure 17) when we compare the time differences of the given processes of the pessimistic and optimistic variants. The comparison of the originally planned work completion time according to the database and the final deadline determined based on the conducted statistical analyses in combination with the use of the construction software in the optimistic variant shows that the time delay would amount to 24 days which represents a significant time difference.

4.6.3. Comparison of Work Productivity Achieved on Various Construction Sites

Some of the most demanding construction activities in terms of importance and from the perspective of the number of deployed workers include masonry and plastering works. This is why their labor productivity at the construction sites is monitored on a long-term basis.

• Plastering works

Figure 18 illustrates a comparison of the labor productivity for the plastering works of constructions undertaken in the Czech and Slovak Republics in recent years. The highest labor productivity achieved at the Jégeho alej construction site in Bratislava (2009) was 12% lower compared to the values according to the database. As we can see in Figure 14, the performance values are even lower during the subsequent years. For the construction of the Prague-Prosek apartment complex, this number amounts to 13%, the project at Klementova Street in Prague (2013) recorded 16%, while the number for the projects in Vsetín was 31% and 29% in FNSP Ostrava (2021).

• Masonry works

Figure 19 illustrates a comparison of the sample averages, mean values of the labor productivity achieved at individual projects. The highest sample average was recorded at the Jégeho alej construction site in Bratislava. This value represents a labor productivity decrease of 7% in comparison with the value complying with the database.

The construction of the Prague-Prosek apartment complex recorded a labor productivity decrease of 13%. This number for the Prague Klementova project was 15% and was 12% for the FNSP Ostrava. The biggest labor productivity decrease was recorded in Vsetín. It amounted to 19.4%.

Conclusion: The assessment of the recorded performances of the construction processes at the monitored construction sites show labor productivity decreases during the last 10 years in comparison with the planned values complying with the database. As we can see in Figure 20, these decreases for plastering works reach higher values, up to 30%, while the decreases for masonry works amount to approximately 20%. The reason for the higher labor productivity decrease in the given process is represented by its higher labor intensity compared to that of bricked structures.

The most common construction production management method which is currently being used is the monitoring of progress using construction software, such as Primavera or MS Project. The utilization of this management method does not allow for signaling abnormal deviances from the partial construction process deadlines in advance, which would allow for adopting personnel measures for ensuring compliance with the planned commencement of the subsequent process. The proposed construction process monitoring method corrects this main deficiency by utilizing a collection of the actually achieved performance data during the initial stage of the given construction process. The proposed method serves construction management who can immediately react to the given situation based on these recorded data by planning the work capacities. Accurate capacity planning aims to shorten the construction time, thus reducing the construction costs. Similar simulation models of the construction time course based on the collection of data on currently achieved performances are dealt with by a number of authors such as Han, S. or Lee, T. et al. The prediction of the construction time course, as in the study, is based on the collection of data on achieved performances on the construction site. A performance database for the Curtain Wall Operation in High-Rise Building Construction was also created, which serves for the prediction of the construction time course of this type. The proposed simulation model (PSM) published by Shin Y et al. overcomes traditional simulation technique limitations by focusing on curtain wall operations. An automated methodology for calculating the productivity of earthmoving operations in near real-time was presented by Montaser, A. et al. The simulation was utilized by optimizing the production of an excavator with that of hauling units. The Bayes principle applied to the determination of the upper and lower performance limits was introduced by Ko, Y., Han, S., Kim, B., Reinschmidt, K. and Gardoni, P. Reinschmidt. Using the Bayes theory, the predictions of the upper and lower performance limits were modified on the basis of the level of performances achieved in the past. The method proposed by Ko, Y., Han, S. provides reliable probabilities by presenting updated results based on continuously collected data rather than focusing on a one-time observation and associated analysis. The solution presented by Kim, B., Reinschmidt, K. integrates prior performance information with observations of new actual performance. The same principle is presented by Gardoni, P. Reinschmidt, i.e., having prior information does not have any practical effect on the forecast when progress on a significant portion of the project has been recorded. Simulation based on the Bayes principle was applied by Pareek, P. et al. using real-time data. This predictability solution improves the classical simulation model. The study presented by Smugala, S. and Kubečková, D. is similar to Han’s, Lee‘s simulation applications, i.e., it is based on continuously collected data. The use of simulation for productivity estimation based on multiple regression analysis was presented by Halpin W., Han S. The productivity estimation of interval estimates of performance ranges using regression analysis forms the so-called confidence band around the regression line. Apart from the verification of the duration of construction processes and their time relations for currently ongoing projects based on the recorded labor productivity values, the creation and updating of the performance dataset of individual construction processes represent a necessary condition for a precise time prediction of future constructions. Initially, an analysis of the monolithic structure was carried out. Based on the recorded performances and based on the subsequently conducted statistical analyses, the labor productivity of wall formworks adhering to the database amounting to 9.19 m²/shift/worker was met. This corresponds to the performances achieved at construction sites abroad as demonstrated by the publications of Moholkar M. and Patil V., where a labor productivity of 9.86 m²/shift/worker was recorded for the MIVAN system wall formwork. When plywood is used, the labor productivity amounts to only 2.62 m²/shift/worker. Labor productivity is also fundamentally affected by the formwork dimensions (2.44 × 0.61). In this case, Hanna S. and Smith G., and Karthick R. recorded a labor productivity of 5.2 m²/shift/worker. In accordance with the publications by Forsythe P.J., the application of the foundation slab reinforcement represents a value of 400 kg/shift, which corresponds to a performance that is 17% higher compared to the database [9]. When it comes to the application of a reinforcement of the ceiling panel, the labor productivity complying with the database amounts to 289 kg/shift, which represents a labor productivity that is 10% lower compared to the values stated in the publications by Forsythe P.J. When compared with the values adhering to the database, the recorded performances for the foundation slab and for the ceiling structure achieved at the construction site of the apartment complex at Klementova Street are lower by 6%. When it comes to iron and steel works, a decisive role is played by the shape of the reinforcement structure, including the size of the reinforcement diameters adhering to the publications by Jarkas M. The achieved labor productivity for implementing the ceiling formwork was 5.45 m²/shift/worker. Due to the complexity of the given structure, this value is 10% lower than the value stated in the database. The labor productivity of individual monolithic structure processes was met with a deviance of up to 10% (note: compare Figure 13 and Figure 14), which meant that no fundamental personnel measures were necessary. The analysis of the process of bricked structures recorded a performance of 1.61 m³/shift/worker, which represents a decrease compared to the values complying with the database. Based on the publications by Desale V. and Deodhar V. [77], the recorded time norm at the apartment project construction site in Dhule was 0.73 Nh/m², which is a performance that is 6% better than the values achieved at the Klementova Street construction site. For bricked structures, an important role is played by the format size of the given masonry material. The labor productivity of the plastering works is explored using various statistical methods. Gypsum wall plasters implemented as a part of the KRG construction projects were assessed using the ANOVA test. In accordance with Abdullah et al., the given statistical analyses demonstrated performances of 9.2 m²/shift, which is a comparable value to the labor productivity of the structure assessed by us, for which the achieved performance value, after considering the conducted patching on a part of the ceiling structure, was also approximately 9.0 m²/shift. A comparable performance of 9.04 m²/shift was achieved for plastering the exterior bricked walls based on the study by Monkaew. Labor productivity of the tiling works was the subject of the publications by Idiake J.E. et al., in which a productivity of 1.156 Nh/m² was determined for the floor tiling work of the shopping center based on the given statistical analyses. It is a comparable performance when compared with the values pursuant to the CONTECT database (1.26 Nh/m²), the productivity of which is the empiric average of the ratio of the implemented floor and wall tiles contrary to the values determined by Idiake J.E., where only the productivity of the implementation of floor tiles with a lower labor intensity was explored. The labor productivity of the façade works at the Klementova Street construction site is 25% lower than the planned values pursuant to the database. This is caused by a lack of qualified labor and a high labor intensity of the given construction process.

5. Conclusions

The subject of this study is statistical analyses of the implementation of main construction processes and their influence on the final construction deadline. Statistical analyses of the labor productivity of the main construction processes demonstrated, with the exception of the monolithic structure processes, greater decreases in the performance values in relation to the planned values pursuant to the CONTEC database. The recorded performance values decreased by 13.5% for masonry works, 15% for plastering works, 17.5% for tiling works, and 26.4% for façade works. The monitoring of the labor productivity development at construction projects during the last 10 years also showed decreases. The decreases amounted to 13% for masonry works and 19% for plastering works. The conducted statistical analyses also demonstrated a direct proportion between the labor intensity values and the decreases in the case of individual construction processes. Using the application of the probability estimate of the completion of the given process and the related commencement of the next process at the construction site of the apartment complex, compliance with the originally planned construction deadlines was achieved. The proposed management manner of the mutually and technologically related construction processes using statistical methods was based on real data collection directly at the construction site. The main practical benefit for construction management is an accurate prediction of the completion of the given processes. The performance dataset of the studied construction processes was created. Their implementation into the construction software allows the accurate planning of the time course of future contracts of a similar nature. By repeated monitoring and assessing the efficiency of the construction processes using the proposed method, the shortening of the construction work implementation time, which led to construction cost savings, has been achieved. The proposed method is applicable in the case of cyclic processes of larger construction objects where more workers can be deployed so that statistical analyses are not limited by the scope of the data performances. A sufficient number of data can then be statistically evaluated within the theory of the probability of the completion of construction processes. The next step is to expand the number of monitored construction processes in order to more accurately predict the time course of the construction objects.