Int J Civ Eng (2017) 15:641–652

DOI 10.1007/s40999-017-0177-8

RESEARCH PAPER

Evaluation of Planned Construction Projects Using Fuzzy Logic

Marcin Gajzler1 · Krzysztof Zima2 

Received: 2 May 2016 / Revised: 25 September 2016 / Accepted: 11 December 2016 / Published online: 4 April 2017
© The Author(s) 2017. This article is an open access publication

Abstract  The study presents a model for the evaluation analysed when the exact parameters of the project in
of construction projects from the point of view of the cli- the planning and preparation stage of the project are not
ent (e.g. developer company). The problem lies in choosing known.
the best solution from the point of view of many criteria.
The proposed model is based on a multi-criteria compara- Keywords  Fuzzy logic · Construction investment ·
tive analysis using fuzzy logic. The first part of the paper Investment evaluation · Multi-criteria analysis
presents a selection of criteria describing the construction
project along with their description. The set of attributes
describing the analysed object was determined on the basis 1 Introduction
of the synthesis of specific proposals for the parameters of
construction projects. The set of criteria has been divided Evaluation of construction projects is particularly difficult
into two groups: technical, technological and organisa- due to their complexity. The difficulty of this issue consists
tional criteria and separately the economic criteria. Then, of determining the criteria for the evaluation of the planned
the number of variables describing the observations was construction project, which consists of a number of eco-
checked using principal component analysis (PCA). Course nomic, technical and commercial parameters. A client con-
of action was presented in the event of multiple criteria structing a building for sale or rent is primarily interested
analysis using the fuzzy set theory. Both the weights and in the cost of construction, the quality and the duration of
the evaluations of individual criteria were modelled using construction. Despite this, he must also take into account
membership functions due to the fact that when describ- the preferences of future users of the building. Reconciling
ing a construction project, or the validity of the criteria of the client’s own interests with that of the user is a difficult
describing variables, they are approximate. An analysis of task for the client. In the evaluation of a project, we can
the correlation of selected project criteria was presented. meet many possible variants of the final implementation of
The proposed decision support model of assessing a con- the project. For example, analysing such evaluation criteria
struction project makes possible to compare various vari- as cost, quality and delivery time, we can make a differ-
ants based on 11 factors identified. The use of fuzzy logic ent choice of solutions for the planned project in accord-
has enabled more accurate description of the phenomenon ance with each of these criteria. The purpose of the article
is to present a comprehensive methodology for evaluating
construction projects. The methodology and the evaluation
* Krzysztof Zima criteria are shown using the example of a project for the
kzima@izwbit.pk.edu.pl construction of residential buildings.
1 Trying to describe the construction project, it often hap-
Faculty of Civil and Environmental Engineering, Poznan
University of Technology, Piotrowo 5, 60‑965 Poznań, pens that the values for some variables are approximate
Poland and subjective. Construction projects are often described
2
Faculty of Civil Engineering, Cracow University in very vague terms. This is expressed in such statements
of Technology, Warszawska 24 st., 31‑155 Kraków, Poland as “substantially”, “good”, “almost”, etc. For example, in

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642 Int J Civ Eng (2017) 15:641–652

describing the location of the project, we can say that: The criteria for evaluating the effectiveness of a construction
location is good, or comparing two projects: The location project.
of the building project A is much better than that of con- Zavadskas et al. [14] have used one of the methods of
struction project B. Also, assessing the impact of a factor multi-criteria analysis ELECTRE III for the evaluation of
on the proposed project, we can say that: The impact of a construction project from the point of view of many cri-
this factor is almost none. This is partly due to the fact that teria. Šarka et  al. [15] used a multi-criteria comparative
the experts, in order not to make a mistake, prefer to give analysis in the evaluation of public investment, and Usti-
answers only in reference to quality, with a certain degree novichius and Šarkienė [16] applied mathematical meth-
of generalisation. Of course we can interpret such state- ods in the assessment of construction projects involving
ments and use knowledge formulated in this way to solve the construction of apartment buildings.
problems involving the evaluation of a construction project. The concept and the mathematical tool of the fuzzy set
However, it is difficult to accurately interpret them. Their theory proposed by Zadeh [17] have become very pop-
vagueness is the cause hindering the sufficiently accurate ular. The task of multi-criteria evaluation in the condi-
determination of the value of individual evaluation criteria. tions of non-statistical uncertainty, often using linguis-
The problem lies in the specific determination of what is tic expressions, is very effectively formulated using the
actually meant when the given expert says the “location is theory invented by Zadeh. Thus, methods have been cre-
good” or the “location is much better”. ated, which employ optimisation methods using the fuzzy
The multi-criteria optimisation problem consists of find- sets theory. An example may be the method used by
ing the optimal solution, which is kind of a compromise Baas–Kwakernaak [18], showing both the evaluation cri-
between all the adopted criteria of evaluation of a project. teria and their validity in fuzzy form. Based on the theory
In order to describe approximate (vague) values, methods of fuzzy sets, Guneri et al. [19] described the use of the
based on the theory of fuzzy sets are used. Fuzzy logic fuzzy analytic network process (fuzzy ANP) to choose
is very well suited to seek solutions to problems, which the best location of a shipyard. Zima [20] presented an
include the human element of subjectivity, such as making example of estimating costs in the early phase of the pro-
decisions about choosing the best variant of the planned ject using fuzzy case-based reasoning and Kaveh et  al.
construction project. [21] used two metahuristic algorithms for solving fuzzy
The client should get a clear answer that solution vari- resource allocation project scheduling problem.
ants are efficient from the point of view of many criteria. Risk factors, similarly to individual types of cost,
The aim of the study should, therefore, be to prioritise the may be ascribed to subsequent stages of a building’s life
planned variants of solutions from best to worst. cycle [22]. Minasowicz [23] presented the NPV analysis
and the investment risk analysis at the stage of strategy
assessment and the feasibility study. Analysis of the value
2 Literature Review of a project allowing for the specification of the probabil-
ity of a given value of cash flows and NPV was also car-
There have been many attempts to identify the success fac- ried out with the use of the fuzzy set theory. Chen [24]
tors of a construction project. Yong Mustaffa [1] identified proposes a hybrid knowledge-sharing model that inte-
37 factors determining the success of a construction pro- grates the concepts of self-organisation of the optimisa-
ject, of which in their study they distinguished 15 critical tion function, fuzzy logic control and hyper-rectangular
factors for the success of a construction project in Malay- composite neural networks, in order to answer the ques-
sia, similarly to Takim and Adnan [2] who have identified tion of whether to perform or refrain from construction
30 such factors. The effectiveness and strategies of a con- projects carried out by foreign companies. The study was
struction project were also investigated, among others in based on 520 quarterly financial reports of all construc-
[3–8]. Frequently, factors are highlighted in the literature tion companies operating in Taiwan. Nasirzadeh et  al.
affecting the cost of construction of a building, for exam- [25] proposed model accounts for both the client and
ple, in [9–11]. the contractor costs using cooperative-bargaining game
Thomas Ng et  al. [12] on the basis of experience in model for quantitative risk allocation to perform the
Hong Kong have developed two modelling approaches: a quantitative risk allocation process.
vector error correction (VEC) and the multiple regression There have also been attempts of mathematical model-
model, and have compared their accuracy with respect to ling of the quality management process in the construc-
public projects, gross domestic product (GDP) and unem- tion project [26], integrated design management system
ployment rate. (IDMS) [27] or even assessment of a conceptual cost esti-
Models supporting decision-making, for example [13], mation [10].
are also created, taking into account the most important

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3 Evaluation Model for a Construction Project 3.2 Determination of an Initial Set of Criteria

Using Fuzzy Logic
A decisive action in regard to the correctness of the analy-
Decision problem faced by a client to choose the optimal sis is determining the elements of the initial set of criteria
variant of the solution of the planned development project explaining the analysed object. This set should contain all
in the housing sector. The proposed model for the assess- the variables fully representing the characteristics of the
ment of the building project uses elements of fuzzy logic. designed object. The set of attributes that describe the ana-
The adopted course of action is illustrated in Fig.  1 and lysed object was determined on the basis of the synthesis of
briefly characterised in the following sections. specific proposals for the parameters of construction pro-
jects. The initial set of criteria has been divided into those
3.1 Determining the Object and Purpose that describe technical, technological and organisational
of the Analysis criteria, as well as customer preferences and those describ-
ing the economic criteria. A detailed analysis of the eco-
The main task is to describe the analysed object in detail. nomic criteria has been carried out separately.
Determining the analysed object is to indicate the studied A descriptive model showing the dependence of the ana-
object and provide its characteristics (a set of attributes lysed object on the explanatory variables (characteristics of
describing the analysed object). The analysed object is the analysed object) in general form can be written as:
therefore a construction project, involving the implemen-
Y = f (x1 , x2 , … , xk , 𝜖), k = 1, 2, … , n.
tation of a multi-family residential building intended for
sale. The aim of the analysis is a certain final state of the where Y analysed object; x1, x2, …, xk characteristics of the
designed object assumed in the course of the study, in this analysed object (explanatory criteria); ε random deviation
case distinguishing from the analysed set the best solution (sum of partial random deviations ε1, ε2, …, εk); n num-
or a subset of a number of sufficiently good solutions, or ber of finally adopted criteria for evaluating the variants of
prioritising these solutions from best to worst. a construction project; f analytical form of the function of

Fig. 1  Evaluation model for

variants of a construction pro-
ject using fuzzy logic

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explanatory criteria, which is determined during the con- directed to each criterion. The introduction of a global
struction of the model. criterion causes a consistency of criteria graphs, condi-
The inclusion of all explanatory variables in the model is tioning the use of the following calculation algorithm.
impossible. Therefore, random deviations arise from a lim- Explanatory variables (criteria) are analysed in terms
ited number of variables used in the model. The better the of information capacity. If in the analysed pair of crite-
model reflecting the reality, the less random are the devia- ria, information on the object of study overlaps at least
tions. Introduction of hierarchical coefficients (weights) in part, the relationship between the criteria is studied,
corresponding to each of the evaluation criteria will, how- defining the direction of this relationship. In this case for
ever, cause the assignment of partial random deviations to the pair of criteria Ki, Kj, if Kj includes a part or all of
individual criteria. the information of Ki, the element of the matrix of arcs
of the graph of criteria G(j, i) is assigned the value of 1,
3.3 Analysis of Selected Criteria and the element G(i, j) the value of 0. If the information
contained in the criteria Ki, Kj is independent, both ele-
The initially accepted set of criteria is a set with several ments G(i, j) and G(j, i) are assigned the value of 0. After
elements, which significantly hinders the process of com- the construction of the matrix G(n + 1, n + 1), we should
parative analysis. With such a large collection of input be able to arrange the resulting graph of criteria in layers.
data, it could be that a part of the data would be difficult For the resulting matrix G={gij}, we calculate:
to determine in the planning and preparation of a construc-
n+1
tion project, and a part of the evaluation criteria would have ∑
b0j = gij
a too small significance for the overall assessment of the
i=1
construction project. This implies the need to reduce the
b0j number of arcs entering the jth vertex.
initial set of explanatory variables. The initially adopted set
of explanatory variables will be analysed in terms of [28]: If for the vertex Kj, b0j = 0, then vertex Kj assumes the
order of 0.
• informational capacity, Then we calculate in sequence:
• mutual relationship between the explanatory variables,
• level of detail of the description, bkj for j = 1, … , n + 1; k = 1, … , r
• level of variation (in order to eliminate variables charac- �
0 when bk−1 =0
terised by too low level of variation). and k−1
bj = j
k−1
bj − i=Zk−1 g(i, j) when bk−1
∑
j
≠0
In order to reduce the initial set of criteria, a matrix Zk−1 set of vertices, for which bkj = 0, assign the verti-
of arcs of the criteria graph G(n + 1, n + 1) was built,
ces of the kth order.
where n is the number of explanatory variables (criteria)
The analysis is completed at the moment of the assign-
and the additional element “n + 1” is the so-called global
ment of all the criteria to the individual layers.
criterion, connected to each of the “n” criteria by arcs

Table 1  Final set of criteria describing the construction project in the housing sector
Criterion Criterion description Valuation type

Distance from the city centre Distance from a fixed point in the city centre Destimulant
Immediate environment External factors influencing the increase/decrease of property value Stimulant
Housing structure Determines the client’s offer compatibility with the current demand in the market, the determi- Nominant
nation of the criteria value is dependent on the statistics of the current structure of the sale
Floorage use ratio Ratio of living area to floorage of a civil structure Stimulant
Design solutions Evaluation of design solutions, the applied technologies and materials Stimulant
Land-use ratio Ratio of a footprint area to a plot area Stimulant
Modishness (trend) Evaluation of the attractiveness of the location from the point of view of the social environment Stimulant
of a project
Scheduled execution time ratio Ratio of the scheduled execution time to the building volume Destimulant
Building layout Shape of the building defined as the ratio of the circumference of the outer walls to the area Nominant
obtained from the building view
Number of storeys Defines the height of the building, depends on the legal constraints (Zoning decision) Nominant
Additional offer Factors affecting attractiveness of the offer Stimulant

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The coefficients of variation for individual criteria have according to eigenvalues. This provided the eigenvalues
also been determined, so that the criteria that will be con- (sorted in descending order), as well as their associated
sidered as quasi-fixed (have too little variability) will be eigenvectors forming the matrix W = [w1, w2, …, w11] of
eliminated from the set of explanatory variables, due to the PCA transformation and the diagonal matrix L com-
their unsuitability in terms of the impact on the global posed of the eigenvalues λ1, λ2, …λ11 of the correlation
result of the evaluation of a construction project. matrix for the set of input data Rxx.
Main components take the following form:
3.4 Creating the Final Set of Criteria y1 = − 0.2838 × X1 + 0.0097 × X2 + 0.4006 × X3 + 0.1202
× X4 0.6072 × X5 + 0.3521 × X6 − 0.2055
The set of criteria resulting from prior analysis of criteria × X7 + 0.0467 × X8 + 0.0154 × X9 + 0.1731 × X10 − 0.4209 × X11
describing the object of multi-criteria evaluation should be y2 = − 0.6210 × X1 + 0.1142 × X2 + 0.0712 × X3 − 0.0094
sufficiently complete and representative. The final set of
× X4 + 0.2365 × X5 − 0.1990 × X6 + 0.3705
criteria specified in this way completes the initial part of
the analysis (Table 1). × X7 − 0.2160 × X8 + 0.0101 × X9 − 0.4040 × X10 − 0.3917 × X11
Due to the type of valuation, the nature of the individual y3 = − 0.0599 × X1 − 0.3318 × X2 − 0.5789 × X3 + 0.1235
criteria is specified in Table 1. The type of stimulant evalu- × X4 − 0,1502 × X5 + 0.0045 × X6 − 0.0276
ation is specified by a criterion, whose higher value also × X7 + 0.4003 × X8 − 0.4901 × X9 − 0.1662 × X10 − 0.2909 × X11
causes a higher global assessment of the analysed variant
y4 = + 0.2897 × X1 + 0.5577 × X2 + 0.0626 × X3 + 0.2240
of the solution. Destimulant is a criterion whose increase
results in the deterioration of the global assessment of the × X4 + 0.1528 × X5 − 0.2358 × X6 + 0.1410

analysed variant of the solution. Nominant is the criterion × X7 + 0.4897 × X8 + 0.0911 × X9 + 0.1905 × X10 − 0.4095 × X11
for which the values falling within a prescribed range, or y5 = − 0.2278 × X1 + 0.0473 × X2 + 0.1221 × X3 + 0.3133
equal to a certain value, indicate the maximum assessment × X4 + 0.0090 × X5 − 0.5619 × X6 − 0.6844
of the given solution, and any deviations—both upward and
× X7 + 0.0405 × X8 − 0.0501 × X9 − 0.1418 × X10 + 0.1558 × X11
downward—lower the global assessment of the solution
variant analysed. y6 = − 0.2170 × X1 + 0.4007 × X2 − 0.3833 × X3 + 0.2943

Using the analysis of main components, the number of × X4 − 0.2043 × X5 + 0.3436 × X6 + 0.0100
variables describing the observations was checked. The × X7 + 0.1612 × X8 + 0.3807 × X9 − 0.3240 × X10 + 0.3507 × X11
correlation between the variables occurring in the data set y7 = + 0.1338 × X1 + 0.1309 × X2 − 0.4014 × X3 + 0.3101
was determined. Variables should be the least correlated
× X4 − 0.3339 × X5 − 0.2817 × X6 + 0.1295
with each other. The procedure for the determination of the
main components is as follows: × X7 − 0.6266 × X8 + 0.0156 × X9 + 0.2928 × X10 − 0.1464 × X11
y8 = − 0.3456 × X1 − 0.1169 × X2 − 0.3613 × X3 − 0.4344
1. The correlation matrix of the input data set Rxx is deter- × X4 + 0.0575 × X5 − 0.1484 × X6 − 0.1829
mined. × X7 + 0.1668 × X8 + 0.4984 × X9 + 0.4402 × X10 − 0.1375 × X11
2. The eigenvalues λii and eigenvectors wi of the correla-
y9 = − 0.3157 × X1 + 0.5288 × X2 − 0.0649 × X3 − 0.3031
tion matrix are determined.
3. The eigenvalues are ranked from largest to smallest. × X4 − 0.0110 × X5 + 0.0625 × X6 − 0.0236

4. The matrix W = [w1, w2, …, wi] T for the PCA transfor- × X7 − 0.0214 × X8 − 0.5934 × X9 + 0.3146 × X10 + 0.2616 × X11
mation and the diagonal matrix L = diag [λ1, λ2, .... λi] y10 = − 0.0220 × X1 + 0.0407 × X2 − 0.1393 × X3 + 0.2815
are determined. × X4 + 0.5939 × X5 + 0.4910 × X6 − 0.3839
5. The main components described in the equation
× X7 − 0.2521 × X8 − 0.0487 × X9 + 0.1546 × X10 + 0.3047 × X11
yi = wiTxi are determined.
6. The relative contribution of each of the main compo- y11 = − 0.3313 × X1 − 0.3014 × X2 + 0.1356 × X3 + 0.5285

nents into the total variance of the data is calculated, × X4 + 0.1416 × X5 − 0.0254 × X6 + 0.3649
according to the formula: × X7 + 0.1974 × X8 − 0.0236 × X9 + 0.4642 × X10 + 0.3047 × X11

After calculation, the relative contribution of each of

𝜆 the main components into the total variance of the tested
mi = ∑k i ,gdzie j = 1,2, … ,k sample is therefore: m1 = 0,2024; m2 = 0,1523; m3 = 0,1312;
j=1
𝜆j m4 = 0,1162; m5 = 0,1122; m6 = 0,0758; m7 = 0,0753;
Analysing selected 11 criteria for assessing the develop- m8 = 0,0452; m9 = 0,0394; m10 = 0,0337; m11 = 0,0164.
ment project, matrix of correlation ­Rxx was decomposed

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The largest principal component obtained for the sam- importance of each of them. When filling out the matrix, lin-
ple shows only 20.24% of participation in the total variance guistic assessment tables are used. The process of filling out
of the data. The percentage cumulative participation of the the table consists of asking the expert to compare in order the
first five values is only 71.42%, and the first eight values validity of all pairs of criteria, e.g. “What is the relation of the
91.05% of participation in the total variance of the data. validity of criterion x to criterion y?”. To which the expert
Adoption of only the first few criteria for the evaluation of responds in accordance with the five-point linguistic assess-
the project may result in a large error in the approximation ments table:
of results.
All the analysed criteria satisfy the basic requirements • Equally important (numerical value −1).
of a linear econometric model. The analysed criteria have a • Moderately more important (numerical value −2).
sufficiently high variability and are poorly correlated with • More important (numerical value −3).
each other, and the individual criteria are strongly corre- • Much more important (numerical value −4).
lated with other criteria which are not criteria of the evalua- • Definitely more important (numerical value −5).
tion they represent, and are also sufficiently correlated with
the global evaluation of the project. After filling out the matrix of comparisons, the weights for
each criterion are calculated. The size of individual weights
can be calculated using, for example, the arithmetic mean
3.5 Determination of Tolerance Limits for the Criteria method, and then by standardising the coefficients. The over-
all validities of the criteria given by P experts are modelled
Analysing the various available solution variants for the using triangular membership functions (Fig. 2).
construction project or variants for carrying out different The values of characteristic points for determining the
projects, we must pre-eliminate those for which even a sin- validity of the criteria are set, for example, by a group of
gle explanatory variable does not meet the requirements. experts. Characteristic points of the triangular membership
Therefore, for the individual variables (if possible) we function are determined as follows:
must determine the minimum and maximum values that a
variable can take. Exceeding these values will eliminate the vi ( min ) = min vij
variant or will lead to its correction already in the initial P
stage of the analysis. 1∑
vi (average) = v
P j=1 ij
3.6 Determination of the Validity Criteria vi ( max ) = max vij
where vij standardised assessment of the validity of the ith
The proposed calculation algorithm enables the decision-
criterion carried out by the jth expert; P number of experts
maker (the client) to determine their own, subjective weights,
making the evaluation.
but in the created model we will use the calculation of hier-
Table  2 summarises the assessment criteria weights,
archical coefficients by expert evaluation using the pair-
as well as the adopted shapes of the membership function
wise comparison matrix. A pairwise comparison matrix
and characteristic points. The shapes of the membership
is a square matrix allowing, on the basis of the comparison
function and characteristic points were determined on the
criteria in pairs, to determine the coefficients of the relative

Fig. 2  Characteristic points
of the triangular membership
function [28]

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Table 2  Shapes and characteristic points of the membership function for individual criteria
Criterion Weight of crite- Shape of membership Function formula Characteristic points
rion (%) function

Distance from the city centre 16.02 Shape class γ a = 1

⎧ 0 for x ⩽ a
⎪ x−a b = 5
𝜇(x) = ⎨ b−a
for a < x < b
⎪ 1 for x ⩾ b
⎩
Immediate environment 14.51 Trapeze shape a = 1 storey
⎪ 0x−afor x ⩽ aix ⩾ d
⎧
b = 4 storey
⎪ for a < x < b c = 5 storey
𝜇(x) = ⎨ b−a
x−d
⎪ c−d for c < x < d d = 17 storey
⎪ 1 for b ⩽ x ⩽ c
⎩
Housing structure 11.55 Shape class γ a = 1
⎧ 0 for x ⩽ a
⎪ x−a b = 5
𝜇(x) = ⎨ b−a
for a < x < b
⎪ 1 for x ⩾ b
⎩
Floorage use ratio 9.70 Shape class γ a = 1
⎧ 0 for x ⩽ a
⎪ x−a b = 3
𝜇(x) = ⎨ b−a
for a < x < b
⎪ 1 for x ⩾ b
⎩
Design solutions 9.57 Shape class L a = 2 km
⎧ 1 for x ⩽ a
⎪ b−x b = 10 km
𝜇(x) = ⎨ b−a
for a < x < b
⎪ 0 for x ⩾ b
⎩
Land-use ratio 7.62 Shape class L a = 4.24
⎧ 1 for x ⩽ a
⎪ b−x b = 7
𝜇(x) = ⎨ b−a
for a < x < b
⎪ 0 for x ⩾ b
⎩
Modishness (trend) 7.10 Shape class γ a = 1
⎧ 0 for x ⩽ a
⎪ x−a b = 4.5
𝜇(x) = ⎨ b−a
for a < x < b
⎪ 1 for x ⩾ b
⎩
Scheduled execution time ratio 6.80 Trapeze shape a = 18 m2 usable area
⎪ 0x−afor x ⩽ aix ⩾ d b = 50 m2 usable area
⎧
⎪
𝜇(x) = ⎨ b−a
for a < x < b c = 55 m2 usable area
d = 150 m2 usable area
x−d
⎪ c−d for c < x < d
⎪ 1 for b ⩽ x ⩽ c
⎩
Building layout 5.94 Shape class L a = 0.04 day/m3
1 for x ⩽ a
b = 0.20 day/m3
⎧
⎪ b−x
𝜇(x) = ⎨ b−a
for a < x < b
⎪ 0 for x ⩾ b
⎩
Number of storeys 5.73 Shape class γ a = 0.63
⎧ 0 for x ⩽ a
⎪ x−a b = 0.85
𝜇(x) = ⎨ b−a
for a < x < b
⎪ 1 for x ⩾ b
⎩
Additional offer 5.46 Shape class γ a = 0.2
⎧ 0 for x ⩽ a
⎪ x−a b = 0.3
𝜇(x) = ⎨ b−a
for a < x < b
⎪ 1 for x ⩾ b
⎩

basis of own research in the Polish market for multi-fam- In the case of economic criteria considered separately,
ily housing, research of customer preferences and experts’ the analysis concerns the two most commonly used indica-
evaluations. tors NPV and IRR. The choice of characteristic points of

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the NPV criterion consists of determining two character- m

∑
(vi ⋅ Ki )
istic points: limit value N­ PVmin = 0, dividing the planned i=1
projects into profitable and unprofitable ones, and N
­ PVmax, Zk = m
which corresponds fully to the expectations of the client
∑
vi
(the value corresponding to 30% of the value of the project
i=1

was determined). Assuming this value, we used our own where μ(vi) membership function for the validity of crite-
surveys of the actual achieved by companies operating in ria; μ(Ki) membership function for the assessment of the
the housing sector in Poland. solution variant according to the ith criterion.
The surveys conducted by the author show that 46% of To benefit directly from the above relationship is not
construction companies in the housing sector in Poland simple; hence, simplified methods of carrying out opera-
determined the actual value of the IRR achieved within the tions on fuzzy numbers are applied, using α-sections of
range of 15–20%. On the other hand, only 4% of the sur- fuzzy sets (Fig. 3). The value of the evaluation of solution
veyed companies determined the value of the IRR achieved variants can be determined by converting fuzzy evaluations
as less than 5% and more than 30%. The first characteristic of solution variants for individual criteria in relation to the
point adopted is the value of ­IRRmin = 5%, and the second corresponding value functions.
characteristic point is ­IRRmax = 20%. Each of the variants of the construction project is evalu-
Assessment of the different variants has been carried ated according to the previously selected criteria. With
out, adopting the assessment criteria for the NPV and IRR these assumptions, we define the value of the replacement
criteria in the class γ. criterion for each of the variants of solutions from set A, as
the fuzzy number with the membership function:
3.7 Comparative Analysis of Variants ∼
Z (Ak ) = 𝜇Zk (zk ) = 𝜇Wki (wki ) × 𝜇vi (vi );zk ∈ [0,1],
k
Both the weight and the assessment of individual criteria where 𝜇Wki (wki ) membership function for the criterion;
were modelled using the membership function. Assuming 𝜇vi (vi ) membership function for the weights of the criterion.
that the values of the evaluations of solutions and the valid- As a result of such procedure, we get the fuzzy evalua-
ities of criteria are standardised and defined in the range tion of the replacement criterion assigned to each project
[0, 1], the replacement criterion value for each variant of variant.
the solution is a fuzzy number. It can be described using a
membership function according to the formula: 3.8 Defuzzification of Variants
∑m
(𝜇Vi (vi ) ⋅ 𝜇Ki (Ki ))
𝜇Zk (Zk) =
i=1
∑m The aim of the analysis is to prioritise solutions from best
i=1 𝜇Vi (vi ) to worst, assuming that the client expects precise informa-
which according to the general principles of operations on tion on the evaluation of the analysed solution variants of
fuzzy numbers leads to the relationship: the proposed construction project. For this purpose, it is
necessary to “sharpen” (defuzzify) the analysed variants
𝜇Zk (Zk) = sup min [ min 𝜇Vi (vi ), min 𝜇Ki (Ki )] according to the relationship:
i=1,…,m i=1,…,m

for

Fig. 3  Triangular membership
function and its decomposition
into α-sections [28]

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1 3.9 Comparison of Solutions
∫ z × 𝜇Zk (zk ) × dz
0
Zk (Ak ) =
1
Sharpened values for both the technical and operational cri-
∫ 𝜇Zk (zk ) × dz teria and economic criteria for the individual variants are
0 compared in a chart. For example, variant A has a higher
For this purpose, the method of centre of gravity can be rating because of the expected profits of 0.8 (variant B
used, assigning a real number to the fuzzy number. This −0.7), but the evaluation of the project in terms of the tech-
value determines the centre of gravity of the area below the nical and operational criteria is lower for variant A (0.4)
graph of the membership function for the given replace- than for variant B (0.5). Despite the expected higher prof-
ment criterion Zk. its for variant A, there is a greater risk of mismatch of the
The centre of gravity method consists in determining the offer to the market conditions (and thus the risk of not sell-
centre of gravity of the area below the graph of the mem- ing all apartments), but there are also more opportunities to
bership function. improve individual technical indicators.
A graphical example of the result of determining the
centre of gravity Fc as the sum of the individual centres of 3.10 Prioritising Variants
gravity Fxi designated for subsequent trapezoidal areas des-
ignated by the α-sections of the sample membership func- Prioritising projects from best to worst in the model, the
tion for replacement criterion Zk is shown in Fig. 4. following scale was proposed:
Analysis of economic criteria is similar to the analysis
of technical criteria. It will be restricted to the analysis of a. For economic criteria—[0; 0.2] project is unprofitable;
the basic methods of assessing the effectiveness of the pro- (0.2; 0.4] project is little profitable, (0.4; 0.6] project
ject—NPV and IRR. is moderately profitable (0.6;0.8] project is very profit-
Having defined two specific criteria, we can define a able, (0.8, 1], project is highly profitable.
replacement criterion: b. For technical and operational criteria —[0, 0.2] pro-
ject is very badly planned; (0.2; 0.4] project is badly
D = min [𝜇(NPV)𝛼1 ,𝜇(IRR)𝛼2 ] planned; (0,4; 0,6] project is moderately planned; (0.6;
where α1, α2 weights of individual criteria; µ(NPV), µ 0.8] project is well planned; (0.8, 1] project is very
(IRR) membership functions for the assessment of criteria well planned.
NPV and IRR.
Sharpening the resulting fuzzy set is also achieved using Project variants can be ranked from best to worst vari-
the centre of gravity method. ant, or the best variant can be highlighted and adopted for
completion.

Fig. 4  Fuzzy interval, its

decomposition into α-sections
and the location of the centres
of gravity for the sample solu-
tion variant of the planned
construction project

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4 Correlation Analysis of Selected Investment variables tested from the normal distribution. Therefore,
Criteria this method is used, although it can be demonstrated that
a number of variables do not have normal distribution. A
The main problem is determining whether there is any point of view was adopted that although the variables do
relationship between the variables and whether it is more not have a strictly normal distribution, their distribution is
or less accurate. Analysing the correlation relationship sufficiently close to it.
between the studied attributes, it is necessary to prepare The resulting correlation matrix is presented in Table 4.
scatter plots that show the relationship between the vari- Based on the literature, the author described the level of
ables graphically. significance α = 0.05. The value of the confidence level is
In the example, the correlations between the criteria therefore equal to p = 95%. As is clear from the data pre-
were examined. The size of the analysed group is N = 41 sented in Table  4, the majority of pairs of the calculated
of construction projects carried out in Kraków in the multi- correlations determine the strength of the interdependence
family housing sector. Before the correlation analysis, it as faint or weak. Only the pairs of criteria 2–9; 2–11; 4–8
was examined whether the criteria selected for analysis determine the correlation between the two variables as
have sufficient variability. The critical value of the variation average.
coefficient v* = 0.1 was adopted. The calculated values of It was attempted to analyse the reasons that cause the
the variation coefficient are shown in Table 3. average correlation in the aforementioned pairs of evalua-
All variation coefficients calculated for individual crite- tion criteria. Scatter diagrams were used for this purpose.
ria are greater than the critical value v* and therefore are These diagrams were prepared for all pairs of criteria and
used in further analysis. did not show significant correlations between the criteria.
The strength of the interdependence of two variables Here the author presents an example of a scatter diagram
was calculated using Pearson’s correlation coefficient, for a pair of criteria, for which the strength of correlation is
marked rXY, which adopts the values in the range [–1]. greater than 0.3.
Analysis of correlation coefficients by this method requires The strength of correlation between the criteria 2 and
that the tested variables have normal distributions. It is 9 is −0.41 and has been described as average. Analysing
known, however, that this particular method to a large Fig.  5a, it can be seen that there are two extreme values
extent is “resistant” to the deviations of distributions of the

Table 3  Variation coefficients Criterion

of evaluation criteria for the
development project 1 2 3 4 5 6 7 8 9 10 11

Arithmetic average 1.75 5.27 3.20 2.27 4.71 5.29 4.01 60.36 0.09 0.73 0.25
Deviation 1.12 2.70 1.10 0.74 1.18 0.72 0.45 32.00 0.10 0.13 0.10
Coefficient of variation vi 0.64 0.51 0.34 0.33 0.25 0.14 0.11 0.53 1.19 0.18 0.40

Table 4  The correlation matrix Rxx of the development project evaluation criteria

Criterion 1 2 3 4 5 6 7 8 9 10 11

1. Additional offer 1 0.01 0.23 0.30 −0.13 −0.24 0.18 0.18 −0.15 0.21 −0.24
2. Number of storeys 0.01 1 0.18 0.22 −0.24 −0.19 0.12 −0.24 −0.41 0.05 −0.41
3. Modishness (trend) 0.23 0.18 1 0.30 −0.00 −0.23 −0.18 −0.22 0.09 0.02 −0.20
4. Immediate environment 0.30 0.22 0.30 1 −0.14 −0.27 −0.07 0.34 −0.19 0.07 0.06
5. Distance from the city centre −0.13 −0.24 −0.00 −0.14 1 0.05 −0.11 −0.03 0.02 0.00 −0.19
6. Building layout −0.24 −0.19 −0.23 −0.27 0.05 1 −0.29 −0.10 −0.22 −0.26 0.01
7. Design solutions 0.18 0.12 −0.18 −0.07 −0.11 −0.29 1 0.11 0.04 0.09 0.03
8. Housing structure 0.18 −0.24 −0.22 0.34 −0.03 −0.10 0.11 1 −0.21 0.12 0.10
9. Scheduled execution time ratio −0.15 −0.41 0.09 −0.19 0.02 −0.22 0.04 −0.21 1 −0.02 0.30
10. Floorage use ratio 0.21 0.05 0.02 0.07 0.00 −0.26 0.09 0.12 −0.02 1 −0.17
11. Land-use ratio −0.24 −0.41 −0.20 0.06 −0.19 0.01 0.03 0.10 0.30 −0.17 1

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Fig. 5  The scatter diagram for the criteria 2—the number of storeys and 9—factor a of the planned pace of construction, b of the planned pace
of construction after the removal of the suspicious value

whose rejection can change the size of the correlation. In The use of fuzzy logic in the model has enabled more
the figure, they are marked with red circles. accurate description of the phenomenon analysed when
After removing the two extreme values marked in the exact parameters of the project in the planning and
Fig.  5a, we obtain a new scatter diagram (Fig.  5b) and a preparation stage of the project are not known. Knowl-
change of the correlation coefficient. The result of re-exam- edge of the factors and their characteristics affecting
ination after the rejection of extreme values increases the the profitability of the project undertaken is essential
strength of correlation between criteria 2 and 9 by only for effective planning and preparing of the development
0.01. The effect of the rejected values is small and does not project. There is a need to create professional tools for
cause an error in the analysis carried out. clients to support investment decision-making. This arti-
cle presents a proposal for an algorithm supporting the
evaluation of development projects.
5 Conclusions
Open Access  This article is distributed under the terms of the
Creative Commons Attribution 4.0 International License (http://
Decisions taken during the planning and preparation of creativecommons.org/licenses/by/4.0/), which permits unrestricted
a development project have a crucial impact on its prof- use, distribution, and reproduction in any medium, provided you give
itability. Lack of proper coordination by the client of the appropriate credit to the original author(s) and the source, provide a
construction process (arrangement and synchronisation of link to the Creative Commons license, and indicate if changes were
made.
mutual action between the parties, harmonisation of the
prevailing relations, coordination of activities related to
risk reduction) can lead to a significant prolongation of the
duration of the investment, to the build up of problems,
conflicts, and even the failure to achieve the intended pur-
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