Solving Civil Engineering Problems by Means of Fuzzy and Stochastic MCDM Methods: Current State and Future Research

Review Article Solving Civil Engineering Problems by Means of Fuzzy and Stochastic MCDM Methods: Current State and Future Research Jurgita Antucheviciene,1Zdenjk Kala,2Mohamed Marzouk,3and Egidijus Rytas Vaidogas4 1Department of Construction Technology and Management, Faculty of Civil Engineering, Vilnius Gediminas Technical University, Saul˙ etekio Al˙ eja 11, LT-10223 Vilnius, Lithuania 2Department of Structural Mechanics, Faculty of Civil Engineering, Brno University of Technology, Veveˇ r´ ıStreet95, 602 00 Brno, Czech Republic 3Department of Structural Engineering, Faculty of Engineering, Cairo University, Giza 12613, Egypt 4Department of Labour Safety and Fire Protection, Faculty of Civil Engineering, Vilnius Gediminas Technical University, Saul˙ etekio Al˙ eja 11, LT-10223 Vilnius, Lithuania Correspondence should be addressed to Jurgita Antucheviciene; [email protected] Received 9 May 2015; Accepted 11 June 2015 Academic Editor: Peide Liu Copyright © 2015 Jurgita Antucheviciene et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The present review examines decision-making methods developed for dealing with uncertainties and applied to solve problems of civil engineering. Several methodological difficulties emerging from uncertainty quantification in decision-making are identified. The review is focused on formal methods of multiple criteria decision-making (MCDM). Handling of uncertainty by means of fuzzy logic and probabilistic modelling is analysed in light of MCDM. A sensitivity analysis of MCDM problems with uncertainties is discussed. An application of stochastic MCDM methods to a design of safety critical objects of civil engineering is considered. Prospects of using MCDM under uncertainty in developing areas of civil engineering are discussed in brief. These areas are design of sustainable and energy efficient buildings, building information modelling, and assurance of security and safety of built property. It is stated that before long the decision-making in civil engineering may face several methodological problems: the need to combine fuzzy and probabilistic representations of uncertainties in one decision-making matrix, the necessity to extend a global sensitivity analysis to all input elements of a MCDM problem with uncertainties, and an application of MCDM methods in the areas of civil engineering where decision-making under uncertainty is presently not common. 1. Introduction Decision-making is applied in different areas of human activities. In the case of existence of at least two possible options, a person (i.e., a decision-maker) has to make a decision and to select the one which is best suited for his demands. Complex problems in science, engineering, technology, or management are characterised by multiple criteria. Usually they are hardly measurable, conflicting or interacting with each other. Decision-making (DM) problems based on multiple criteria are objects of MCDM. MCDM is a discipline concerned with the theory and methodology for handling problems common in everyday life. They arise in such areas as business, engineering, social organisation, and so forth [1]. MCDM has grown as a part of operation research pertaining to the design of computational and mathematical tools for supporting the subjective evaluation of performance criteria by decision-makers [2]. As it is mentioned in a review paper [3], the origins of MCDM methods can be dated over 270 years ago. As an individual scientific discipline, MCDM has been widely spreading since the middle of the previous century. Numerous works on MCDM are summarized in a number of review papers [4]. Three main types of review papers related to MCDM can be distinguished: reviews of developments and extensions of a particular method (e.g., [5]) as well as its applications (e.g., [6,7]); reviews of different approaches to modern MCDM methods (e.g., DM under uncertain and Hindawi Publishing Corporation Mathematical Problems in Engineering Volume 2015, Article ID 362579, 16 pages http://dx.doi.org/10.1155/2015/362579 2Mathematical Problems in Engineering Quantitative (deterministic in statistical sense) Qualitative (orderable verbal categories) Matrix with crisp elements Elements of MCDM decision-making matrix Matrix  Cwith elements expressed by probability distributions In Bayesian format In Fisherian format Grey numbers Fuzzy sets Linguistic values are transformed into Matrix ←→ Cwith elements expressed b y intervals C Figure 1: The main types of criteria used to construct the decision-making matrix of MCDM. incomplete information and DM for groups of persons [2,8]); reviews of applications of different MCDM methods for a particular problem [9–11]. Most, if not all, decisions in engineering are made under uncertainty. Accordingly, classical crisp methods, involving deterministic conditions under certain decision environments, are losing their relevance and new extensions for handling various types of uncertainties in MCDM problems appear. In most cases, uncertain elements of MCDM problems express subjective expert opinions and less often they represent stochastic nature of input data. Understanding the significance and impact of different types of uncertainties in input information and its relation to reality can increase the quality of DM process. The present review strives to examine the current state of uncertainty modelling in MCDM and to identify several problems which decision-makers may face in the immediate future.Thereviewisorganizedasfollows.Section 2 presents the scope of the review and identifies several problems relatedtoMCDMunderuncertainty.Section 3 presents the latest references on applications of MCDM approaches in uncertain decision environments of civil engineering. Section 4 summarizes the applications of sensitivity analysis (SA) in fuzzy and stochastic MCDM techniques. Section 5 analysespublishedworkonintroducingmeasuresofrisk and reliability in the MCDM related to civil engineering. Section 6 attempts to identify the need for future research. This section presents a brief discussion on potential applications of MCDM under uncertainty in the areas which currently receive much attention in research and practice. Finally, conclusions and prospective research trends are presented in Section 7. 2. The Scope of the Review Generally, a MCDM problem is defined as follows. Let a= (𝑎1,𝑎2,...,𝑎𝑖,...,𝑎𝑚)Tbeavectorofdecisionalternatives and c=(𝑐 1,𝑐 2,...,𝑐 𝑗,...,𝑐 𝑛)a set of criteria, according to which suitability of alternatives 𝑎𝑖is to be judged. The problemisstatedasa𝑚×𝑛decision-making matrix Cwith the elements 𝑐𝑖𝑗.Thevalue𝑐𝑖𝑗 expresses an impact of the criterion 𝑐𝑗on the alternative 𝑎𝑖.Valuesofcrelatedtothealternatives𝑎𝑖 are the row vectors c𝑖=(𝑐 𝑖1,𝑐 𝑖2,...,𝑐 𝑖𝑗,...,𝑐 𝑖𝑛)and, with these vectors, the matrix Cis formulated as [c1,c2,...,c𝑗,...,c𝑚]T. In many MCDM methods, the importance of the criteria 𝑐𝑗is expressed by the weights 𝑤𝑗which sum up to unity and are usually grouped into the vector w=(𝑤 1,𝑤2,...,𝑤𝑗,...,𝑤𝑛). The purpose of DM is to determine the most preferable alternative among 𝑎𝑖withrespecttoallcriteriaortorankthe alternatives. The result of ranking is expressed by a preference sequence, for example, 𝜋:𝑎𝑥≻𝑎 𝑦≻𝑎 𝑧≻⋅⋅⋅. A large variety of MCDM methods have been developed to date. A universal and comprehensive classification of these methods does not exist. For the purposes of the present review, MCDM methods will be classified according to the nature of the criteria grouped into the decision-making matrix C(Figure 1). In the previous five decades, a development of MCDM was focused mainly on solving MCDM problems with crisp elements of C.Inparallelwiththis process,fuzzyMCDMmethodshavebeendevelopedsince 1970s [12]. These methods allow solving MCDM problems with a decision-making matrix ←→ C, the elements of which are modelled by fuzzy sets. Development and application of fuzzy MCDM methods significantly increased during the last two-three decades. One of the possible representations of 𝑐𝑖𝑗 by fuzzy sets is grey numbers based on interval arithmetic. Several recent publications devoted to a use of grey numbers for decision-making in engineering can be cited here [13– 15]. Finally, in the recent time, several MCDM methods were created and applied to a solution of decision-making problems, in which elements of Care modelled by means of probabilitydistributions.Suchamatrixwillbedenotedbythe symbol  C. The fact that until now uncertainties in MCDM problems were expressed in several different ways generates a problem of choosing among these possibilities. Further problems related to handling uncertainties in decision-making are illustrated in Figure 2. A decision-maker having to rank the alternatives 𝑎𝑖inthepresenceofuncertaintiesmayfacethe following problems: (1) The problem of choice among different representations of uncertainty related to criteria values 𝑐𝑖𝑗 and weights 𝑤𝑗. Mathematical Problems in Engineering 3 (1) Choice among alternative types (2) Specification of uncertain weights or  𝐰 Input information Solution methods MCDM method 1 MCDM method k MCDM method l (3) Choice between alternative (4) Sensitivity of 𝜋kto changes in random decision-making matrix and weights Result (sequences of preferences) ←→ 𝐂with 𝐰or ←→ 𝐰 or  𝐂with 𝐰 or  𝐰 . . . . . . ≻a j≻a k≻ ... ≻a i≻a k≻ ... ≻a i≻a j≻ ... . . . . . . MCDM methods of uncertainty representation ←→ 𝐰 ai aj ak 𝜋1: 𝜋k: 𝜋l: Figure 2: Four problems related to handling uncertainties in MCDM. (2) Specification of the weights 𝑤𝑗in the case where they are uncertain quantities. (3) The problem of choice among available MCDM methods for solving a problem with the decision-making matrices ←→ Cor  C. (4) Analysis of sensitivity of MCDM results to the changes in all elements of input data given by the matrices ←→ Cor  Cand the vectors w,←→ w,orw.Input uncertainties can cause possible permutations in the sequences of preferences, 𝜋𝑘,obtainedbymeansofa specificMCDMmethod𝑘. The first problem has its origin “outside” the domain of MCDM. Discussions of the type “fuzzy versus probabilistic” or “nonprobabilistic versus probabilistic” last many years and donotseemtobefinisheduntilnow(e.g.,[16]). In most applications of MCDM under uncertainty, authors do not trouble themselves to explain why a particular method of uncertainty quantification was preferred over others. The fuzzylogicprevailsoverprobabilisticmodellingbythenumber of MCDM methods developed to date and the number of MCDM applications to practical problems. Currently, one can state the obvious that probability distributions can be specified in Fisherian format for 𝑐𝑖𝑗 inthepresenceofsufficientamountofstatisticaldataon𝑐𝑖𝑗. A representation of 𝑐𝑖𝑗 by fuzzy sets is better suited for 𝑐𝑖𝑗 expressing subjective expert opinions. However, it is important to remember that the Bayesian approach to probabilistic modelling has excellent means of quantifying and updating subjective judgments (e.g., [17]). This approach does not break down when data on 𝑐𝑖𝑗 is sparse or absent. MCDM problems can be solved when uncertainties in 𝑐𝑖𝑗 are represented in any of the aforementioned formats. A true problem will arise when the decision-making matrix will be a “mixture” of the elements 𝑐𝑖𝑗 expressed by different means of uncertainty quantification, probabilistic and nonprobabilistic. The second problem of specifying elements of the vector of weights, w, is obviously an “internal” problem of MCDM. There is a body of literature devoted to specifying values of 𝑤𝑗. Methods used to determine criteria weights are classified into subjective, objective, and hybrid or integrated ones [18– 20]. However, the task of assigning specific values to the components 𝑤𝑗will always be subjective or at least partly subjective, no matter what is the degree of mathematical sophistication behind this assignment. A really intriguing question is how 𝑤𝑗can be interpreted as uncertain quantities. The experts who usually specify the values of 𝑤𝑗can be uncertain (vague) regarding their opinions on 𝑤𝑗.Should weights be modelled by a fuzzy vector ←→ wor a vector wwith components expressed by probability distributions? The third problem reflects the well-known decisionmaking paradox which was first identified by Triantaphyllou and Mann in 1989 [21]. The paradox was exhibited by many MCDM methods developed to deal with the crisp decisionmaking matrix C. Switching to uncertain matrices ←→ Cor  C will not resolve this paradox. It will be likely to persist also in the field of decision-making under uncertainty. The present review will not address this tricky issue. The fourth problem arises naturally, because a MCDM method is in essence a mathematical model relating the input information expressed by Cand wto the output information given by the preference sequence 𝜋. A sensitivity of 𝜋to changes in elements of Cand wcan be estimated (analysed) by standard mathematical means of general use. However, a sensitivity analysis (SA) of a MCDM problem with uncertain matrices ←→ Cor  Cand/or uncertain vectors ←→ wor wis a nontrivial task. The four problems of MCDM under uncertainty listed andbrieflydiscussedaboveareofmethodologicalnature. This kind of DM can face also problems of different nature, namely, an application in such areas of civil engineering as development of sustainable and energy efficient buildings, building information modelling, assurance of security, and safety of built property. Solving MCDM problems in these areas will face the necessity to model uncertainties related to long-term predictions, vague information available in the process of building design, and possibility of rare but extremely damaging events. A particular need for modelling uncertainties in MCDM problems arises at a design of safety critical objects of civil engineering. Failures of such objects or damage to them can cause severe consequences to 4Mathematical Problems in Engineering society and environment. Many of safety critical objects are assessed (designed) by applying methods of reliability theory or probabilistic risk assessment. Measures of reliability and components of risk estimates can be uncertain quantities. An application of MCDM to the design of safety critical objects requires including these quantities into the decision-making matrix C. In most cases, uncertainties related to reliably and risk are modelled by means of probability distributions. Therefore, the decision-making matrix will be formulated as the matrix  C. A combination of SA and MCDM applied to the design of the safety critical objects will allow revealing the most important MCDM criteria related to safety. Thefollowingsectionsofthisreviewwillconsiderthe problems raised above in greater detail. The attention will be focused on both results achieved until now and problematic issues which may require attention in the future. 3. MCDM Techniques under Uncertainty in Civil Engineering Decision-making in the field of civil engineering is increasingly complex and is associated with situations where robust decisions are required to be taken. These decisions are made in different stages of civil engineering projects. For example, decision-making takes place during feasibility study stage prior to design, procurement, and construction stages in order to determine the viability of project undertaken by an investor.Alsoitoftenfacestheneedtodealwithhazardous phenomena, including industrial accidents able to damage built property, structural failures, extreme natural phenomena, and human acts affecting security. Proper decisions made by architects and civil engineers may reduce the risk posed by the aforementioned phenomena. Decision-making in this field can be facilitated by an application of formal methods, such as methods of multiple criteria decision-making. Factors relevant to decisions can be identified using the methods of uncertainty and SA of mathematical model outputs. The result of the evaluation of an alternative according to a given criteria is a single value (e.g., number or verbal expressions) for unambiguous deterministic information. For ambiguous information, the result of the evaluation of variants according to a given criteria is a random variable if the information is stochastic in nature or a fuzzy variable if the information is nonstochastic in nature. The stochastic nature of a phenomenon is associated with the unlimited repeatability of the phenomenon. Unrepeatable phenomena or phenomena with uncertain knowledge significance are nonstochastic in nature. 3.1. Fuzzy MCDM. The fuzzy framework is the most common approach to describe and handle uncertainty in MCDM. Considering the imprecisions and uncertainties, that is, fuzziness of the available data and the decision-making procedures, fuzzy set theory can be applied [22]. In a fuzzy MCDM, the elements 𝑐𝑖𝑗 of Care characterised by fuzzy sets. In what follows, such elements will be denoted by the symbol ←→ 𝑐𝑖𝑗 and a decision-making matrix containing fuzzy elements will be ←→ C. The fuzzy set theory can be used for a MCDM, whenever probability distributions of 𝑐𝑖𝑗 are unknown due to lack of statistical data or there is no wish to express subjective judgments about values of 𝑐𝑖𝑗 in a probabilistic way. The weights 𝑤𝑗can also be modelled as fuzzy variables ←→ 𝑤𝑗and so the vector wcanbeafuzzyone,←→ w. In MCDM problems, three types of fuzziness can be observed: the ratings of each alternative with respect to each criterion are uncertain or imprecise and weights are crisp numbers (←→ 𝑐𝑖𝑗 ,𝑤𝑗); the ratings of alternatives are crisp numbers while fuzzy numbers are used to assess the weights of all criteria (𝑐𝑖𝑗,←→ 𝑤𝑗); both the ratings and the weights are fuzzy (←→ 𝑐𝑖𝑗 ,←→ 𝑤𝑗). Each type of uncertainty has its own characteristics and is appropriate for special cases. A lot of fuzzy extensions of MCDM methods have been proposed and applied in engineering, technology, or management. Fuzzy extensions of MCDM involve application of basic fuzzy logic, using triangular fuzzy numbers and arithmetic operations, also trapezoidal fuzzy numbers, as well as intuitionistic fuzzy relations, interval-valued intuitionistic fuzzy relations, type-2 fuzzy sets, and hesitant fuzzy sets concepts. Nonfuzzy uncertain decision methods include reliability theory and probabilistic and grey-valued formulations for handling incomplete or imprecise information. The most widely applied fuzzy extensions of MCDM methods are fuzzy AHP (analytic hierarchy process), with the origins dated to 1983 [23] and various numerous extensions of fuzzy TOPSIS (technique for order preference by similarity to ideal solution) [2,24]. Several latest extensions of more recently developed methods are worth to be mentioned: extension of weighted aggregated sum product assessment in an intervalvalued intuitionistic fuzzy environment (WASPAS-IVIF) [25], a complex proportional assessment method extended with interval-valued intuitionistic fuzzy numbers (COPRASIVIF), and suitable for group decision-making [26]. The most recent extended versions of MULTIMOORA (multiobjective optimization by ratio analysis plus full multiplicative form) utilizethebasicfuzzylogic[27,28]orarebasedonthe interval 2-tuple linguistic variables [29], intuitionistic fuzzy numbers [30], or hesitant fuzzy numbers [31]. ARAS under fuzzy environment was also presented [32]andfurther applied [33,34]. Numerous researches utilize not a single fuzzy MCDM (FMCDM) method but combine several of the methods. Two groups of researches can be distinguished: either integrating two or more available techniques and proposing so-called hybrid methods or employing several methods for a solution of a problem and comparing ranking results. The study [2] reviews papers on development and applications of MCDM during the last two decades by various aspects. The analysis covers 1081 papers related to MCDM and fuzzy MCDM. Whileadetailedsurveyismadeon403paperspublishedin peer review journals and entirely devoted to fuzzy decisions, involving 217 papers in a field of engineering, it is obvious that engineering applications cover over 50 percent of overall applications. Accordingly, it can be stated that fuzzy MCDM Mathematical Problems in Engineering 5 100 80 60 40 20 0 Number of publications 1994–2005 2006–2008 2009–2011 2012–2014 Years of publication Figure 3: Increase of FMCDM applications in engineering. is useful and applicable technique for complex decisionmaking in civil engineering under uncertainty. The increasing importance of the methods can be based on the findings that the number of developments and applicationsisincreasingeveryyear.Onthebasisofcollected data [2], annual distribution of papers in engineering field is explored. The period is divided into several stages. The first stage covers 1994–2005, when a number of papers are small or even absent in certain years. The later period starting from 2006, when a number of researches began increasing, is grouped in three years. A chart (Figure 3) shows increase of FMCDM applications in engineering, measured by a number of publications included in the review paper [2]onFMCDM techniques and applications during two decades. The last period from the above four covers 46 percent of publications issued during 20 years. The largest number of researches on the whole is observed in 2014. ThelatestFMCDMapproachesareappliedinvarioussubfields of civil engineering field. Subfields which are the most numerously supported by considered approaches are related to sustainability and building lifecycle assessment, supply chain management in construction, technology and inventory selection in construction industries, location selection and infrastructure modelling, and knowledge management. Environmental, social and economic aspects of sustainable building were incorporated when evaluating mining projects and their impact on environment [35], assessing building energy performance [36]orperformanceofpavements with emphasis on sustainability [37]. Supply chain management is handled through proper supplier selection considering multiple criteria simultaneously. Various approaches are applied for supplier evaluation and selection, starting from the most common fuzzy AHP and hybrid methods integrating AHP and TOPSIS [38], AHP and PROMETHEE [39], and so forth. Also novel extensions based on interval type-2 fuzzy sets [40] or intuitionistic fuzzy information [41]areutilized. Selection of technologies, inventory, or materials in construction is also widely supported by FMCDM. Examples ofthemostrecentapplicationsareforselectingmaterial handling equipment [42,43], inventory classification [44], and selection of robots for automated technology operations [45]. Fuzzy decision-making methods are extremely important for handling vagueness in special-purpose building projects and their location. Fuzzy multiple criteria approaches successfully applied for nuclear power plant site selection [46], deep-water see port selection [47], and health monitoring of tunnels [48]. Regarding DM methodology in civil engineering applications, it can be concluded that the most common approach is observed to be fuzzy AHP-TOPSIS hybrid technique [38,49], also combining fuzzy AHP with other classical methods as PROMETHEE [50]andDEMATEL[51,52]. Another group of researches applies several FMCDM methods simultaneously and compares ranking results. An example can be provided of material selection applying a number of hybrid approaches, namely, FAHP-VIKOR, FAHPPROMETHEE, FAHP-TOPSIS, and FAHP-ELECTRE [53], technology selection applying FAHP, TOPSIS-F, VIKOR-F, and COPRAS-G [45], and so forth. 3.2. Stochastic MCDM. The probabilistic framework is the second approach to describe and handle uncertainty in MCDM. In this framework, the elements 𝑐𝑖𝑗 of the decisionmaking matrix Care modelled by random variables  𝑐𝑖𝑗 expressing uncertainty in possible values of 𝑐𝑖𝑗.Theuncertainty is quantified by means of probability distributions which can be specified in the format Fisherian or Bayesian statistics. Presence of  𝑐𝑖𝑗 in Cgenerates a matrix  C,allorsome elements of which are random. Stochastic MCDM methods areusedtosolvethedecisionproblemswiththematrix C. MCDM solved with an emphasis on stochastic uncertainty (SMCDM) is focused on decision problems of the selection of alternatives from several criteria that are mathematically described neither as crisp numbers nor as fuzzy numbers or linguistic variables but as random variables [54]. Stochastic methods for the determination of weights from various types of information on the character of the significance of criteria are effectively implemented especially in the AHP method. The classical AHP method lacks probability values for the distinction of adjacent alternatives in the final ranking [55]. Vargas [56] considered the case where members of the pairwise comparison matrix were random variables. It should be noted that although random variables are considered, purely stochastic uncertainty of input data and decision-making procedures is rare. The occurrence of a random phenomenon is almost always accompanied by a certain degree of personal belief; therefore, the term subjective probability is sometimes used. The reason for the implementation of random variables in MCDM is the attempt tousemethodsofthetheoryofprobabilityandmathematical statistics for the analysis of uncertainty of the results of the decision-making process. Advanced methods of stochastic SA, the equivalent of which is unknown in fuzzy MCDM, are available for stochastic MCDM [57]. There are numerous SMCDM methods and applications in real world situations [58,59]. The classical Monte Carlo (MC) method or its improved variants can be used to tackle most SMCDM problems [60,61]. The theoretical application of the Monte Carlo method is very extensive; however, 6Mathematical Problems in Engineering the results of stochastic analysis are very sensitive to the laws of probability density functions and type of dependencies between input variables resulting in fuzzy (epistemic) uncertainty in decision-making tasks. Analysis of forecast uncertainty based on fuzzy stochastic approaches and its application to a number of problems in civil engineering and related fields is described in detail, for example, in the book [62]. However, the complexity of mathematical approaches and the interpretation of results limit the scientific ingenuity of the application of fuzzy stochastic SA methods. 4. Sensitivity Analysis Applied to MCDM Two types of SA are most often mentioned in the literature: local SA and global SA (e.g., [63,64]). Local SA determines a contribution of a given input parameter of a mathematical model to its output. Local methods do not attempt to fully explore the input parameter space. They examine small perturbations, usually one parameter at a time. Global SA examines a mathematical model in the presence of uncertain input parameters. SA of this type apportions output uncertainty to different sources of uncertainty in the input. Input and output uncertainties are expressed by probability distributions. Global SA applies perturbations of the entire space of input parameters. It is also argued that global SA should be used “in tandem” with uncertainty analysis and the latter should precede the former in practical applications [65, 66]. The aim of uncertainty analysis is to estimate variation in model output and SA apportions this variation to input parameters. Local SA was applied in the recent past for better understanding of MCDM methods. Global SA was used for improving models which are related MCDM but do not fit strictly into its scheme. MCDM constitutes a special class of mathematical models. An application of SA to MCDM rises specific problems.However, any methodological specificity of performing SA for applications of MCDM in civil engineering is not known to us. In addition, combined applications of MCDM and SA to problems of civil engineering are few in number and deal mainly with selecting locations of buildings [67,68]. A certain relation to civil engineering has applications of combined MCDM and SA to geographic information systems (see Malczewski and Rinner [69]and references cited therein). Therefore, the remainder of this section will be a general discussion on SA applications to a better understanding of MCDM methods. 4.1. Applications of SA in Deterministic MCDM and MCDM under Uncertainty. In the light of the aforementioned SA definitions, a deterministic (crisp) MCDM method can be interpreted as follows: elements of the matrix Cand components of the vector wrepresent input parameters, the preference sequence 𝜋is model output, and the procedure relating 𝜋to Cand wis a mathematical model. A sensitivity of 𝜋versus Cand wis called ranking sensitivity or sensitivity to ranking stability (e.g., [70,71]). With the deterministic input Cand w, a sensitivity of 𝜋versus Cand wcan be determined by means of local SA methods. Triantaphyllou and S´ anchez applied local SA to determine the criterion 𝑐𝑗and the decision matrix element 𝑐𝑖𝑗 which is most critical to the ranking expressed by 𝜋[72, 73]. They applied sensitivity measures based on minimum changes in the weights 𝑤𝑗andmatrixelements𝑐𝑖𝑗 which cause changes between ranks of the alternatives 𝑎𝑖.Bevilacquaand Braglia used a simple wide-range variation of the weights 𝑤𝑗 to explore shifts in the ranking 𝜋obtained by means of a deterministic AHP [74]. Any numerical sensitivity measure or graphical representation of SA results was not suggested in this study. Chang et al. and Wu et al. carried out SA of AHP results based on increasing values of 𝑤𝑗up to 35% [67,75]. They expressed SA results graphically. This approach was applied to SA of results obtained with fuzzy decisionmaking matrix ←→ C[71]. The results were produced by means of fuzzy TOPSIS and fuzzy AHP methods. Another kind of perturbations used to reveal sensitivity of 𝜋versus 𝑤𝑗is exchanging positions of the weights within the vector w. Choudhary and Shankar used such perturbations for results obtained with a combined fuzzy AHP and TOPSIS method [76]. A total of 𝑛𝑝perturbations produces a set of preference sequences 𝜋𝑝(𝑝=1,2,...,𝑛𝑝)whichare treated as SA result and expressed graphically. In a series of methodologically similar articles, Awasthi et al. suggested using the sequences 𝜋𝑝for a final ranking of the alternatives 𝑎𝑖[68,77–79]. The sequences 𝜋𝑝were obtained for a problem with fuzzy weights ←→ wand fuzzy matrix ←→ C.Thealternatives 𝑎𝑖were ranked by counting scores for each 𝑎𝑖according to its position in 𝜋𝑝. These authors also tried to use this scoring for a qualitative assessment of ranking sensitivity. However, any quantitative sensitivity measure was not suggested. A quantitative measure of ranking sensitivity used in several studies is known as an average shift in ranks (ASR). Some of these studies fit into the scheme of MCDM and some have common elements with this field. In line with the notations used herein, ASR is expressed as the mean 𝑚−1∑𝑚 𝑖=1|𝑟𝜍 𝑖−𝑟 ref 𝑖|,where𝑟𝜍 𝑖is the rank of the alternative 𝑎𝑖related to a perturbation 𝜍and 𝑟ref 𝑖is the rank of 𝑎𝑖in a reference (base) ranking. Saisana et al. used ASR as a modeloutputforaglobalSA[80]. ASR was computed on the basis of a composite indicator. The reference ranks 𝑟ref 𝑖 were obtained from one specific application of the indicator. Ben-Arieh used ASR in the format of MCDM for sort of alocalSA[81].ASRwasappliedtopairwisecomparisons of alternative rankings obtained with different linguistic quantifiers (probabilities). They were used for calculating the weights 𝑤𝑗.Theranks𝑟𝜍 𝑖and 𝑟ref 𝑖were obtained using pairs of different linguistic quantifiers. Later on, the same approach was used by other authors for assessing ranking sensitivity to fuzzy linguistic quantifiers [82–84]. ASR was applied also to a global SA related to MCDM. Ligman-Zielinska suggested using ASR as a scalar representation of output ranking and computing a sensitivity index based on variance of ASR [65]. Unfortunately, values of ASR depend on the choice of the reference ranks 𝑟ref 𝑖and this introduces certain arbitrariness in the process of SA. Apart from ASR, an alternative and well-elaborated scalar measure expressing a sensitivity of Mathematical Problems in Engineering 7 permutations within 𝜋to input of MCDM models is not knowntous. 4.2. Sensitivity to Model Selection. One of the key problems of MCDM is a selection of method (model) relating input information expressed by Cand wto the output ranking sequence 𝜋. If the MCDM method requires normalisation of the initial matrix C, the analyst will also face the problem of choosing among several normalisation rules (formulas) (e.g., [85]). Using different normalisation rules may also contribute to the variability of ranks within the sequence 𝜋. Anapplicationoffuzzynumbersaselementsofthematrix ←→ Cand components of the vector ←→ wwill introduce an additional problem of model selection. Membership functions of fuzzy numbers, 𝜇←→ 𝑤𝑗(𝑤𝑗)and 𝜇←→ 𝑐𝑖𝑗 (𝑐𝑖𝑗),maynotnecessarilybe triangular as in the most fuzzy MCDM applications (Links 1 and 2, Figure 4). The choice of the type of fuzzy numbers in ←→ Cand components of the vector ←→ wis a subjective exercise and introduces arbitrariness into MCDM process. Similar statements can be made about the use of random variables in  Cand w. A specification of MCDM model input will require selecting specific types of probability densities 𝑓 𝑐𝑖𝑗 (𝑐𝑖𝑗 |𝜇 𝑖𝑗,𝜎𝑖𝑗) and 𝑓 𝑤𝑗(𝑤𝑗|𝛼 𝑗,𝛽𝑗)for decision matrix elements  𝑐𝑖𝑗 and criteria weights  𝑤𝑗(Links 3 and 4, Figure 4). In case of  𝑤𝑗, the selection of distribution type and then specification of its parameters 𝛼𝑗and 𝛽𝑗will be purely subjective task. However, in some cases, statistical data on the random components  𝑐𝑖𝑗 of  Ccanbeavailable.Insuchcases,distributiontypeof 𝑐𝑖𝑗 will be dictated by this data. Consequently, we can speak about three SA problems: (1) Sensitivity of the ranking sequence 𝜋to the selection of MCDM method. (2) Sensitivity of 𝜋tothechoiceofnormalisationrule. (3) Sensitivity of 𝜋to the selection of membership function type in case of fuzzy MCDM and probability density type in case of stochastic MCDM. The first two problems will be present in both deterministic MCDM and MCDM under uncertainty. The third problem will arise with the need to introduce uncertainties in MCDM process. To the best of our knowledge, a systematic, in-depth solution of these three problems is not available in the MCDM literature to date. Apart from model selection problem, the use of the membership functions 𝜇←→ 𝑤𝑗(𝑤𝑗)and 𝜇←→ 𝑐𝑖𝑗 (𝑐𝑖𝑗)and the density functions 𝑓 𝑐𝑖𝑗 (𝑐𝑖𝑗 |𝜇 𝑖𝑗,𝜎𝑖𝑗)and 𝑓 𝑤𝑗(𝑤𝑗|𝛼 𝑗,𝛽𝑗)will give rise to a problem of assessing sensitivity of the ranking sequence 𝜋to parameters of these functions (Links 5 to 8, Figure 4). An association of all or some 𝑤𝑗and 𝑐𝑖𝑗 with two or more parameters of membership function or density function may substantially increase the dimensionality of input space. This can encumber SA process, especially in case of global SA. It will be necessary to assign additional distributions to this parameter in order to carry out Monte Carlo analysis of a MCDM model under study. 5. Measures of Risk and Reliability in the MCDM Related to Civil Engineering Civil engineering systems can be damaged by deliberate assaults, actions induced during industrial accidents and extremes of nature. Damage to components of civil engineering systems induces mechanical and thermal actions called often abnormal or accidental ones. Incidents with abnormal actions are relatively rare, short-lasting, and usually unexpected events. They can cause serious harm and sometimes catastrophic consequences [86]. In terms of civil engineering, such incidents are called “abnormal situations” or “accidental situations.” The latter term is used in the widely known standards ISO 2394 and ENV 1991-1. An adequate design of civil engineering systems for abnormal situations will result in resilient buildings and infrastructure able to avoid or absorb damage without undergoing a complete failure [87– 89]. 5.1. MCDM and Safety in Civil Engineering. The design of components of safety critical civil engineering systems for abnormal situations requires, among other things, comparing alternative design solutions of these components. In terms of MCDM, they can be called alternative designs or simply alternatives 𝑎𝑖(𝑖=1,2,...,𝑚). Interalternative comparisons of 𝑎𝑖must include criteria 𝑐𝑖𝑗 expressing safety of 𝑎𝑖or, alternatively, risk posed by 𝑎𝑖. The row vector of criteria, c𝑖=(𝑐 𝑖1,𝑐 𝑖2,...,𝑐 𝑖𝑗,...,𝑐 𝑖𝑛), will include also elements which are not necessarily related to safety, for instance, economic, functional, and aesthetic criteria. The adequacy of the design of critical objects of civil engineering for abnormal situations is assured by applying methods of reliability theory and probabilistic risk assessment (PRA) [90]. These two fields of engineering are closely related and the criteria 𝑐𝑖𝑗 canbespecifiedbyapplying methods developed in each of them. Values of 𝑐𝑖𝑗 can be probabilities of failure or quantitative estimates of risk related to the designs 𝑎𝑖[91]. Practical applications of MCDM to the design for abnormal situations will face the problem of uncertainty related to failure probabilities and risk estimates. Quantitative measures of this uncertainty can be introduced into MCDM problems. A wellestablished platform of uncertainty modelling in PRA is the Bayesianstatisticaltheoryor,inbrief,theBayesianapproach [92]. In line with this approach, the uncertainty in potential safety-related criteria 𝑐𝑖𝑗 is divided into two kinds: stochastic (aleatory) and state-of-knowledge (epistemic) uncertainty [93–96]. Uncertainties of either kind can be accommodated in MCDM problems. 5.2. An Integration of Reliability Measures and Related Quantities into MCDM. A failure probability 𝑝f𝑖characterising a particular alternative design 𝑎𝑖can be used as a MCDM criterion 𝑐𝑖𝑗 [85,91]. A value of the failure probability 𝑝f𝑖 assigned to the alternative 𝑎𝑖accountsforthepossibilityofits potential failures. The probability 𝑝f𝑖can be “mechanically” included into a MCDM problem as one of components of the vector c𝑖. However, in many cases the probabilities 𝑝f𝑖will be uncertain in the epistemic sense. The uncertainty in 𝑝f𝑖 8Mathematical Problems in Engineering 5 3 1 7 Utilisation of SA results 6 7 8 Fuzzy approach Probabilistic approach 1 5 3 4 Perturbation of input signal (s) MCDM method k + Normalisation rule Output variability (a set of nppermuted sequences of ranks) . . . . . . 𝜋k1 𝜋kp 𝜋kn𝑝 (wj) wj1 wj2 wj3 wj cij,1 cij,2 cij,3 ←→ cij cij wj f  c𝑖𝑗 (cij |𝜇ij ,) f w𝑗(wj|𝛼 j,𝛽 j) (cij ) 𝜇←→ 𝜔𝑗 𝜇←→ c𝑖𝑗 𝜎ij 2 Figure 4: Possible SA links related to the input space of a MCDM problem and a specific MCDM method. may stem primarily from sparse information on accidental actions. Quantitatively this uncertainty can be expressed by modelling the probabilities 𝑝f𝑖as epistemic random variables  𝑝f𝑖[97]. If the uncertain failure probability  𝑝f𝑖of 𝑎𝑖is taken as a MCDM criterion, the vectors c𝑖will contain at least one random component and can be replaced by the stochastic vectors  c𝑖given by ( 𝑝f𝑖1, 𝑝f𝑖2,..., 𝑝f𝑖𝑘𝑖,𝑐 𝑘𝑖+1,𝑐 𝑘𝑖+2,...,𝑐 𝑖𝑛), where 𝑘𝑖isthenumberoffailuremodesof𝑎𝑖.Someorall criteria in the vector just mentioned can be uncertain in the stochastic (aleatory) sense. For instance, such criteria can be time of construction or cost of 𝑎𝑖.Uncertaintyrelated to the stochastic components of  c𝑖canbeexpressedbythe random variables  𝑐𝑖𝑗. With the random vectors  c𝑖,aMCDM problemwillhavetobesolvedbyapplyingadecision-making matrix  Csome or all elements of which are random variables. Stochastic MCDM methods must be applied to deal with the matrix  C. VaidogasandZavadskassuggestedintroducingthefailure probabilities 𝑝f𝑖𝑙 (1 ≤𝑙≤𝑘 𝑖)intoaMCDMproblem indirectly, through comparison of the total (life-cycle) utilities 𝑢tot,𝑖 relatedtothealternatives𝑎𝑖[91]. The utility 𝑢tot,𝑖 is expressed as a difference between expected benefit from 𝑎𝑖and a total cost of 𝑎𝑖. The failure probabilities 𝑝f𝑖𝑙 are incorporated into the total cost of 𝑎𝑖through the cost of failures expressed by the sum ∑𝑘𝑖 𝑙=1𝑝f𝑖𝑙𝑐f𝑙,where𝑐f𝑙is the anticipated cost of failure according to the failure mode 𝑙. Vaidogas et al. applied reliability-oriented MCDM for a selection among alternative construction projects of a building [85]. Reliability of the alternative projects 𝑎𝑖was expressed as a probability that a specified construction time will not be exceeded. A further application of MCDM was a ranking of designs of a reinforced concrete slab with different floorings. Probabilities of two failure modes of the slab were used as the criteria 𝑐𝑖𝑗: probability of collapse and probability of excessive deflection. 5.3. An Integration of Risk Estimates into MCDM. The alternatives 𝑎𝑖can represent hazardous industrial objects which pose risk to people and environment in the form of industrial accidents. The magnitude of consequences of such accidents can range between minor and catastrophic consequences [98].IntheEuropeanUnion,mostofhazardousobjects are regulated by Seveso Directives (currently Seveso III Directive) [99]. Such objects are assessed by means of formal methods developed in the field of PRA [93,100]. A risk related to the design 𝑎𝑖is a very informative characteristic suitable for inclusion into a MCDM problem [98]. In line with PRA, the risk related to 𝑎𝑖is expressed by the set {(𝜆𝑖𝑠,𝑐 𝑖𝑠,m𝑖𝑠),𝑠 = 1,2,...,𝑛𝑎𝑖},inwhich𝜆𝑖𝑠 and 𝑐𝑖𝑠 are likelihood-consequence pairs, m𝑖𝑠 are the vectors of magnitudes (severities) of 𝑐𝑖𝑠,and𝑛𝑎𝑖 is the number of accident scenarios related to 𝑎𝑖.Thevectorm𝑖𝑠 is given by a certain number 𝑛𝑠of magnitudes 𝑚𝑖𝑠𝑗 (𝑗=1,2,...,𝑛𝑠). Zavadskas and Vaidogas suggested expressing the criteria 𝑐𝑖𝑗 in the form of expected magnitudes 𝑚𝑖𝑗 [97]. The component 𝑐𝑖𝑗 of the decision-making matrix Crepresented by 𝑚𝑖𝑗 is computed as the sum ∑𝑛𝑎𝑖 𝑠=1𝜆𝑖𝑠𝑚𝑖𝑠𝑗. Mathematical Problems in Engineering 9 As in the case of uncertain failure probabilities  𝑝f𝑖, the likelihoods 𝜆𝑖𝑠 can be uncertain quantities modelled by epistemic random variables  𝜆𝑖𝑠.Inmostinstances,the variables  𝜆𝑖𝑠 will represent uncertain annual frequencies of the consequences 𝑐𝑖𝑠. The presence of the random likelihood  𝜆𝑖𝑠 will stochastise the expected severities 𝑚𝑖𝑗 and they will turn into random variables, for example,  𝑚𝑖𝑗.Consequently, aMCDMproblemwillturnintoastochasticonewitha random decision-making matrix  C.Therow c𝑖of  Cwill be expressed as ( 𝑚𝑖1, 𝑚𝑖2,..., 𝑚𝑖𝑛𝑠,𝑐 𝑖,𝑛𝑠+1,𝑐 𝑖,𝑛𝑠+2,...,𝑐 𝑖𝑛)with𝑛− 𝑛𝑠≥1, where the components denoted by the letter “𝑐”can be either deterministic or stochastically uncertain quantities. With the vectors  c𝑖including the risk-related components  𝑚𝑖𝑗,thematrix Ccanbeexpressedasatwo-blockmatrix [ C1|C2].The𝑚×𝑛 𝑠matrix  C1reflects risk estimates of the alternatives 𝑎𝑖,whereasthe𝑚×(𝑛−𝑛 𝑠)matrix C2 includes criteria which are not directly related to the risk. Stochastic MCDM methods will be necessary to solve the MCDM problem with the matrix [ C1|C2]. Vaidogas and ˇ Sak˙ enait˙ eappliedtherisk-basedMCDMto a choice among alternative sprinkler systems [101]. Zhou et al. used a safety-oriented MCDM for solving decision-making problems of hydropower construction project management [102]. Catrinu and Nordg˚ ard applied PRA and MCDM to a management of electricity distribution system asset [103]. In recent years, a fairly large number of publications considered an application of MCDM methods for handling managerial risk related to construction projects and running built facilities. Although risk of this type differs by nature from the “pure” risk posed by (to) physical objects, assessments of managerial and “pure” risk are related through a need to deal withuncertaintiesinriskyobjectsorprocesses.Nieto-Morote and Ruz-Vila used fuzzy AHP method for assessing building projects and selection of contractors [104,105]. Xiang et al. applied fuzzy AHP for assessing risk arising at a construction of submerged floating tunnels [106]. Wang et al. used AHP in combination with other decision-making methods to assess risk posed by exploitation of bridges [107]. El-Abbasy et al. applied AHP method together with Monte Carlo simulation forselectingcontractorsofahighwayproject[108]. The studies just listed involve elements of a nonprobabilistic uncertainty quantification based on fuzzy sets. As the “pure” risk is always a part of managerial risk, uncertain criteria  𝑐𝑖𝑗 specified by means of probabilistic methods of PRA canbeincludedintothedecisionmatrixCalongside with “fuzzy” criteria ←→ 𝑐𝑖𝑗 .However,MCDMmethodswhichallow a simultaneous juggling of “probabilistic” and “fuzzy” criteria  𝑐𝑖𝑗 and ←→ 𝑐𝑖𝑗 do not exist at present, to the best of our knowledge. 5.4.MCDMandFireProtectionofCivilEngineeringObjects. Fire is a prevailing hazard in most objects of civil engineering. Fire accidents often occur on construction sites [109,110]. As regards fire protection, MCDM methods were used until now mainly for ranking attributes expressing fire safety of completed buildings. AHP method was applied for developing weights of fire safety attributes in the socalled Edinburgh study [111]. A stochastic AHP was used by Zhao et al. to rank attributes of building fire safety [112]. Wong et al. used attributes of fire detection and alarm systems among a fairly large number of characteristics of an intelligent building. They applied two MCDM methods, AHP and ANP, to rank these characteristics [113,114]. Vaidogas and ˇ Sak˙ enait˙ e formulated a number of MCDM problems, in which building fire safety is considered with respect to economics of fire protection: selection among existing buildings, building projects, and construction materials [115]. Vaidogas and Linkut˙ e considered also problems of decisionmaking in the design of structures used for protection of built property against accidental explosions [116]. 6. MCDM in Innovative Areas of Civil Engineering: A Look at Decision-Making under Uncertainty 6.1. Developing Sustainable and Energy Efficient Building. Sustainability is a natural subject of MCDM, because it automatically includes three subsets of criteria, involving economics, environmental, and social aspects. When solving problems of sustainable building, the fourth subset of criteria, involving engineering-technological dimensions, is also necessary. One of the innovative themes in sustainable construction is related to using materials of low embodied energy and energy efficient applications. However, such things as future and real building cost, environmental impact, and future social status of a constructed facility are very uncertain if considered in a long sight. For instance, large built areas in Hamburg (Germany) lost a lot of image due to a social downgradeofinhabitants.Alsoalotofindustrialandfarming buildings having perfect infrastructure were left abandoned due to political and respective economic changes in postSoviet states in Eastern Europe [117]. These buildings and territories make a great potential for further redevelopment as the recent trends in construction emphasize rehabilitation instead of occupying new territories, wasting building materials, and so forth. Building rehabilitation should be performed in accordance with principles of sustainable development, thus combining a number of usually conflicting and hardly measurable aspects. The usefulness and even necessity of application of decision-making methods under uncertainty for aforementioned problems are summarized below. It is worth mentioning that DM under uncertainty is more characteristic to rehabilitation than to new construction. New construction is more regulated by technical norms, standards, and comprehensive planning. However, aspiration to redevelop a building in the most proper way is certainly a multiple criteria DM problem. Problem related to upgrading abandoned or depreciated buildings as well as physically and morally deteriorated built environment can generate several potential alternatives as demolishing depreciated building and building new structures (technically sound approach but contradicting to principles of sustainability), dismantling, reusing, or recycling of building materials (partly meeting principles of sustainability), renewal according to up-to-date requirements and using a building for previous purposes, 16 Mathematical Problems in Engineering [110] M. A. Alqassim and N. N. Daeid, “Fires and related incidents in Dubai, United Arab Emirates (2006–2013),” Case Studies in Fire Safety,vol.2,pp.28–36,2014. [111] D. J. Rasbash, G. Ramachandran, B. Kandola, J. M. Watts, and M. Law, Evaluation of Fire Safety,JohnWiley&Sons, Chichester, UK, 2004. [112] C. M. Zhao, S. M. Lo, J. A. Lu, and Z. Fang, “A simulation approach for ranking of fire safety attributes of existing buildings,” Fire Safety Journal,vol.39,no.7,pp.557–579,2004. [113] J. Wong, H. Li, and J. Lai, “Evaluating the system intelligence of the intelligent building systems—part 1: development of key intelligent indicators and conceptual analytical framework,” Automation in Construction,vol.17,no.3,pp.284–302,2008. [114]J.Wong,H.Li,andJ.Lai,“Evaluatingthesystemintelligence of the intelligent building systems. Part 2: construction and validation of analytical models,” Automation in Construction, vol. 17, no. 3, pp. 303–321, 2008. [115] E. R. Vaidogas and J. ˇ Sak˙ enait˙ e, “Multi-attribute decisionmaking in economics of fire protection,” Engineering Economics, vol. 22, no. 3, pp. 262–270, 2011. [116] E. R. Vaidogas and L. Linkut˙ e, “Sitting the barrier aimed at protecting roadside property from accidental fires and explosions on road: a pre-optimisation stage,” The Baltic Journal of Road and Bridge Engineering,vol.7,no.4,pp.277–287,2012. [117] E. K. Zavadskas and J. Antucheviciene, “Multiple criteria evaluation of rural building’s regeneration alternatives,” Building and Environment,vol.42,no.1,pp.436–451,2007. [118] E. Krygiel and B. Nies, Green BIM: Successful Sustainable Design with Building Information Modeling,Wiley,Indianapolis,Ind, USA, 2008. [119] AIA, Guide, Instructions and Commentary to the 2013 AIA Digital Practice Documents,TheAmericanInstituteofArchitects, Washington, DC, USA, 2013. [120] B. A. Wayland, Security for Business Professionals. How to Plan, Implement, and Manage your Company’s Security Program, Elsevier,Amsterdam,TheNetherlands,2014. [121] D. Drengenberg and G. Corley, “Evolution of building code requirements in a post 9/11 world,” CTBUH Journal,no.3,pp. 32–35, 2011. 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