Abstract
Project investment decision making with fuzzy information (PIDMFI) has been investigated in a lot of literature. The aim of this paper is to discover the features, state of the art, interrelations, and research directions of existing methodologies for PIDMFI. To do this, a literature review of the methodologies including theories, methods and models as well as applications is conducted from a taxonomic perspective of methodology. The hierarchical cluster analysis in SPSS V19 is employed to construct a methodological taxonomy framework which consists of fuzzy discounted cash flow (FDCF), fuzzy real option (FRO) and fuzzy multi-criteria decision making (FMCDM). Some hybrids of these methodologies are also shown. Additionally, a discussion is presented. Finally, a conclusion with new research directions is delineated.
Keywords
Introduction
Selecting the best from a set of available investment alternatives is a key issue in project investment decision making (PIDM) which has caught much attention of researchers in project management. Traditional methodologies are based on discounted cash flow (DCF) [2–35]. They, however, have been subject to lots of critics because of requiring a crisp “expected scenario” of cash flows [3] and ignoring the value of managerial flexibility [36, 54] and non-monetary factors [63–65, 81]. The DCF-based methodologies have to be extended to uncertain environment.
Probability distribution of a random event may be estimated upon the related historical data or experience of decision makers and be used to depict cash flow [7]. So, it is natural to use probability theory to model uncertain DCF-based PIDM. In fact, subjective probabilities represent vague information rather than probability of cash flow. Hence, the fuzzy DCF (FDCF) methodologies have been generalized by using fuzzy sets theory [2–35].
Additionally, another kind of uncertainty is the flexibility of PIDM. Real option models, analogous to financial option, have been developed [36, 54] to valuate the flexibility. Besides, fuzzy sets are also integrated with real option to form fuzzy real option (FRO) methodologies [36–62].
With the increasing complexity of investment project and the changing of external context, some non-money factors also highly affect PIDM. These factors are often strategic, qualitative and vague so that fuzzy multi-criteria decision-making (FMCDM) methodologies are developed for PIDM [63–91].
Methodologies for PIDM with fuzzy information (PIDMFI) have been studied in a lot of literature so far. The aim of this paper is to disclose the features, state of the art, interrelations, and research directions of existing methodologies for PIDMFI by a literature review from a taxonomic perspective of methodology.
The rest of this paper is organized as follows. Section 2 introduces literatures retrieval polices in Web of Science and taxonomy by SPSS V19. A finer review of three categories of methodologies for PIDMFI is conducted in Section 3. Some interest results are discussed in Section 4. Finally, a conclusion with some new directions is presented inSection 5.
Literature retrieval and taxonomy
Literature retrieval
Recently, Merigó, Gil-Lafuente and Yager [103] contributed an interesting general review of fuzzy search from such perspectives as author, paper, journal, institute and country, by using bibliometric tool and Web of Science (WOS) Database. Our focus is on the taxonomy of methodologies applying fuzzy set theory to PIDM. For implementing this review effectively, we conduct literature retrieval in WOS Core Database to select 91 journal papers closely related to the methodologies for PIDMFI. These papers are indexed by the Science Citation Index (SCI) and/or Social Science Citation Index (SSCI) in a span of time from 1978 to 2013. To do this, we develop the following search policies: To login of WOS core database. On the Basic Search webpage with a search filed corresponding to a default topic option, we can add a new search filed. The term “fuzzy” is input in the first field and “investment” in the second. As a result, 1119 papers are retrieved. Next, we take the place of “investment” by “project investment” (or “project selection”) in the second search field and 275 (or 581) papers are captured. Furthermore, we select the option “Title” related to the second search field and use “investment”, “project investment” and “project selection” in the second search field, respectively. The corresponding search results are 297, 56 and 104 papers, respectively. Finally, we use the terms “fuzzy” in the firstsearch field and “project investment” (or “project selection”) in the second search field. The options as for both fields are “Title” and then 32 (or 75) related papers are found.
In retrieving process, we record and examine the important information involving the authors, publish years, titles, abstracts, keywords, citations, and applications of these papers obtained by different search policies. These terms used are keywords closely related to PIDMFI. Following these policies, we select 91 papers to be reviewed in detail. In addition, we supplement 11 papers as evidences of fundamental concepts, theories, methods or models. Some of them are also indexed by SCI and/or SSCI.
Literature taxonomy
We observe the theories, methods and models in the 91 papers. As a result, three basic categories of methodologies, such as FDCF, FRO, and FMCDM are easily revealed and assumed. With the information mention above, the Hierarchical cluster in SPSS V19 is utilized to classify the 91 papers into three methodological categories. The steps of the cluster method are delineated as follows: Step 1. Start SPSS V19 installed in PC and define a data table for implementing the cluster analysis; Step 2. Let FDCF, FRO, and FMCDM be three variables and the name of the first author of paper be label case. The variables’ value type, size, etc. are also defined. Then the data table is completed; Step 3. Input values for the three variables and label case. The names of the first authors of the 91 papers are input in the column with respect to the label case in the data table, respectively. As for the values of FDCF variable, we set up a rule that if the methodologies used in the paper whose label case value is in ith row are associated with FDCF, then the value of the ith row and FDCF column is 1, otherwise 0. We can do the same to the FRO and FMCDM variables. Finally, we create a complete binary data table for clustering 91 papers; Step 4. Utilize the Hierarchical cluster in SPSS V19 to cluster the 91 papers. We choose the Hierarchical cluster in the submenu Classify under the main-menu Analyze. In the dialog box of the Hierarchical cluster analysis, we choose the three variables into variable box and the label case into label case box. We select the “Cases” in Cluster option box, and “Statistics” and “Plots” in Display option box. Then, we click the button “Method”. Next, in the dialog of clustering analysis method, we choose “Between-groups linkage” as cluster method and select “Simple match”, “Jaccard” and “Squared Euclidean distance” as measure, respectively, within Binary option.
After the parameters required are configured well, we start the cluster analysis. As a result, three plots of hierarchical cluster of the 91 papers are yielded. The three plots are closely similar. According to the plots and the key information of the 91 papers, we construct three tables (see Tables 1–3, respectively) to illustrate the clustering results. Based on the clustering results, we establish a methodological taxonomy framework in which our literature review can be conducted in detail.
Extending statistic analysis
We also illustrated some trends of research by implementing extending statistic analysis of the 91 papers. First at all, the changes of the numbers of papers published (NPP) in different intervals of years with different methodologies are drawn in Fig. 1. Then, the changes of the citations of papers published (CPP) in different intervals of years with different methodologies are illustrated in Fig. 2.
Figure 1 tells that the NPP from 1987 to 2013 in each methodology grows faster in FRO than that in FMCDM. It is worth noting that a great number of papers appeared after the year 2000.
However, the changes of citations are a little complex in terms of Fig. 2. As to FDCF, the peak of the CPP appears in the interval from 1992 to 1996, and then jumps down to the bottom in the interval from 1997 to 2001. Afterward, it keeps up slowly in the interval from 2002 to 2006. Then, it keeps down slowly in the interval from 2007 to 2013. As for FRO, there is an obviously increase from 1987 to 2013. With respect to FMADM, it is growing up greatly from year 2002 to 2006, and then jumps down from year 2007 to 2013. It is worth noting that the CPP in FMCDM are twice as much as that of FMCDM or FRO in the interval from 2002 to 2006. The FMCDM and FRO are equal with respect to the citations and are nearly twice as much as that of FDCF in the interval from year 2007 to 2013.
In terms of Fig. 1, we can roughly tell that there is an increasing trend of research on FDCF, FRO and FMADM methodologies from 1987 to 2013. The three methodologies become more attractive than ever and are worth investigating in future.
In addition, some combinations of the three methodologies are also illustrated by Fig. 3 in which the number and total citations of papers clustered in specific methodology are denoted by (number/citations). For example, “FDCF + FRO + FMCDM” represents a hybrid methodology of FDCF, FRO and FMCDM, and the (3/8) means the number and total citations of papers belonging to the hybrid methodology are 3 and 8, respectively. Based on the information in Tables 1–3, the relationship between the number of papers and the family of fuzzy sets used in the three taxonomic methodologies is also shown in Fig. 4 where some shorthand notations are employed for convenience of discussion, i.e. FNs refer to fuzzy numbers, IFSs denote intuitionistic fuzzy sets, IVIFSs mean interval-valued intuitionistic fuzzy sets, Other FSs represent other fuzzy sets, and FP is fuzzy programming.
These results of statistics analysis on the literature retrieved not only construct a new taxonomic framework of review on methodologies for PIDMFI, but disclose the features, state of the art, interrelations, and research directions of these methodologies.
Literature review from taxonomic perspective of methodologies
Upon the methodological taxonomy framework, a general literature review is implemented elaborately in this section.
Review of FDCF methodologies available for PIDMFI
FDCF is a natural extension of traditional discount cash flow theory by using fuzzy sets theory with an assumption that a decision maker’s knowledge about future cash flow and discount interest rate is vague. Obviously, both the discounted cash flow theory and fuzzy sets theory form the foundation of FDCF methodology. In 1965, Prof. L. Zadeh first presented the notion of fuzzy sets as an extension of classic set theory to depict the vagueness in human being’s perception on the world. Based on the fuzzy sets, Zadeh (1978) first introduced possibility theory which was treated as a correspondence to probability theory used to deal with random problem [1]. Fuzzy numbers are convex and normative fuzzy subsets, and trapezoidal fuzzy numbers are the typical form of the family of fuzzy numbers and triangular fuzzy numbers are the special case of trapezoidal fuzzy numbers [93]. Dubios and Prade [93] established the operations of fuzzy numbers. Dubois et al. [94] pointed out that the most usual possibility distribution is the triangular fuzzy number or its interval forms derived by its α-cut sets. On the other hand, axiomatic economic evaluation theory of project investment which belongs to engineering economics is on the basis of discounted cash flow theory. From the theory, several canonical economic evaluation methods are derived as: net present value/worth (NPV), internal rate of return (IRR), equivalent uniform annual value (EUAV), benefit-cost ratio (BCR), payback period (PP), etc.
Buckley [2] first introduced the fuzzy sets theory to finance by the possibility distribution constructed by intervals and fuzzy numbers. And then, Calzi [3] established a general framework for the fuzzy mathematics of finance. Buckley [4] presented fuzzy IRR. By combining fuzzy set theory and discounted cash flow analysis, the FDCF methodology is formed and described as following general steps: Step 1. To use the family of fuzzy sets to represent or estimate the cash flow information and discount rate for evaluating projects; Step 2. Based on the fuzzy financial information attained in Step 1, one or a combination of the several canonical economic evaluation methods are developed under fuzzy setting as fuzzy evaluation criteria; Step 3. By the operations of fuzzy parameters, the solutions to the models are attained, which belong to the family of fuzzy sets. Step 4. Using ranking methods of fuzzy sets, the wanted project(s) will be selected finally.
Through our statistics analysis, there are 34 related journal papers retrieved from ISI Web of science by year 2013, which contribute to the FDCF methodology for PIDMFI and can be classified as: fuzzy present value/worth method [2–35], fuzzy future value/worth method [10, 29], fuzzy equivalent uniform annual value method [11, 35], fuzzy payback period method [9, 35], fuzzy rate of return method [10, 32], fuzzy benefit-cost ratio method [11, 13]. In particular, a time-variable discount rate was first introduced to model fuzzy log present value method [17] and fuzzy inflation rate was taken into account in discounted cash flow methodology [15].
All of these fuzzy economic evaluation methods are based on fuzzy discounted cash flow profiles, and most of them are modeled by triangular fuzzy numbers (TFNs) [2–9, 31–34], trapezoidal fuzzy numbers (TrFNs) [12, 35], possibility distribution [5–7, 32], and/or interval numbers (INs) [5, 35]. Besides, some FNs in other forms are also introduced to model the methods, which can be listed as: sextuplet FNs [10], random TFNs [21, 27], extending adaptive FNs [31], interactive FNs [33], ordered FNs [34]. A random fuzzy variable is a function mapping a set of random variables to [1] [104]. E.g., let x i ∈ R (i = 1, 2, 3) be a TFNs variable, if then ξ is a random fuzzy variable taking normally distributed variable values [21].
It is worth nothing that fuzzy logic operator [23] and possibility/probability distribution transformation [7, 19] are utilized to extend and enrich the FDCF methodology. Additionally, a variety of fuzzy mathematic programming models are employed to facilitate the FDCF, such as multi-objective programming model [6, 29], possibilistic linear programming model [6, 33], fuzzy linear programming model with duality and complementary slackness analysis and sensitivity analysis [23], fuzzy chance-constraint programming model [20, 24] and random fuzzy chance-constraint programming model [21, 27], fuzzy mean-variance programming model [22], fuzzy optimal programming with objective of minimum of the composite risk-return index[28].
As for application, 34 papers clustered in the FDCF methodology roughly focus on the topic of project investment. By the classification of terminology, these applications can be listed as: general problems of finance and investment [2–6, 30], project investment assessment/evaluation [11, 35], project investment/selection [7–10, 26–34].
In terms of the type of project, these applications can be classified as: project portfolio/combination [22–24, 33–35], research and development (R&D) project [29], construction project [12, 13], advanced manufacture system project [11, 16], computer aid manufacture system project [15], capital project [2–6, 30], technological project [27].
Review of FRO methodologies availablefor PIDMFI
Besides ignoring the vagueness, the canonical discounted cash flow methodology also pays little attention to the value of flexibility of investment decision [36]. The flexibility is called real option or strategic option [46, 54]. Two major families of methods have been introduced for evaluating the option value of a project, which are Black-Scholes models [95] and binomial lattice models [96]. The analytical Black-Scholes models generally deal with simple and single option, but the binomial models can tackle multi-option situation [46, 54] and only requires algebra solution. In addition, in considering real option value, the value of an investment project is the sum of the NPV of the project and value of the real option [46, 54]. That is so-called extended NPV [46, 54]. Apparently, the FRO methodology is formed by combining real option methods and fuzzy NPV methods. The general valuating process of fuzzy Black-Scholes models can refer to Section 3 in [36] specifically, and that of fuzzy binomial lattice model can refer to Subsection 3.2 in [46] or [54] in detail.
Through our statistics analysis, 27 related journal papers can be found under our retrieving policies from ISI Web of science by year 2013, which contribute to the FRO methodology for PIDMFI and can be classified in detail as: fuzzy Black-Scholes method [36–44, 59], fuzzy binomial lattice method [45, 54], fuzzy trinomial lattice method [48], fuzzy pay-off method [58, 62], Extended least square method [56]. We can employ a pie chart to illustrate the shares of the five classes of methods (see Fig. 5).
All of fuzzy tools used in the 27 papers are, respectively, TFNs [37, 62], TrFNs [36, 61], INs [37, 45], random fuzzy variables (RFV) [37, 51]. Some useful methods are also introduced for ranking fuzzy numbers and they are possibility mean and variance [36, 58], credibilistic mean [62], and centroid method [44].
In addition, fuzzy zero-one integer programming model [39, 60] and fuzzy mix integer programming model [41] are constructed and resolved to capture the best project portfolio. It is worth noting that [37] used INs derived from the α-cut sets of TFNs to model an optimal programming formula with the objective of maximum value of α.
From the viewpoint of application, the 27 papers can be classified as: option valuation [37, 61], project investment selection [36, 60], project evaluation [46, 62].
From the perspective of project type, the 27 papers can be also categorized in detail as: research and development project [39, 60], project portfolio [39, 60], energy project [44, 57], product development project [54], mineral project [55], information system project [58, 59], aerospace project [62], general project [36, 56], others [37, 61].
Review of FMCDM methodologies available for PIDMFI
Both of the traditional discounted cash flow and real option analysis methodologies take only the financial or monetary aspects into account in evaluating projects. With the increase of the size and complexity, investment projects are exercising tremendous influence on society, environment, nature, resources, and the like. The influence should be paid much attention.
That is, the non-monetary aspects or strategic aspects of project investment should be also treated as good as the monetary ones. MCDM becomes a sufficient tool for dealing with project investment decision making with consideration of both of monetary aspects and strategic aspects [63, 97]. Classical MCDM process basically consists of two phases [99]: Aggregation phase of the performance values with respect to all the criteria for obtaining a collective performance value for the alternatives; Exploitation phase of the collective performance value for obtaining a rank ordering, sorting or choice among the alternatives.
Because decision makers always evaluate investment projects based on their own subjective preferences depicted by fuzzy sets from different criteria, it is better to regard this situation as a FMCDM problem [100].
Chen and Hwang [98] developed a typical methodological framework for FMCDM. Chou, Chou and Tzeng [73] outlined their FMCDM process by 10 steps. Mardani et al. [100] freshly provided an elaborate literature reviews on the methods and applications of FMCDM. With difference from their contributions, we confine our review on FMCDM methodology to project investment decision making and selection. And our focuses lie not only in quantitative financial criteria, but in qualitative non-monetary criteria as well.
Obviously, 30 articles retrieved in discussion basically articulate the scenario of the FMCDM methodology and its applications to PIDMFI. In our review, we categorize the basic priority or ranking methods in FMCDM, which are illustrated as: fuzzy analytic hierarchy process (FAHP) [67, 89], fuzzy analytic network process (FANP) [68, 88], fuzzy technique for order of preference by similarity to ideal solution (FTOPSIS) [66, 91], fuzzy elimination and choice expressing the Reality (FELECTRE) [84], fuzzy VIKOR [83], fuzzy integrated index method (FIIM) [63, 65], extend analysis (EA) [66, 77], least square method (LSM) [72], geometric means (GM) [73, 87], centroid index method [78, 83], fuzzy data envelop analysis (FDEA) [79, 87], fuzzy comprehensive evaluation (FCE) [89], 2-tuple linguistic approach (2-TLA) [90], others [70].
We also employ a pie chart to illustrate the shares of these methods (see Fig. 6).
The members in family of fuzzy sets used in the 30 articles are, respectively, TFNs [63–69, 88–90], TrFNs [70, 87], INs (alpha-cut sets of fuzzy number) [66, 82], Type 2 fuzzy sets (T2FSs) [72], IFSs [85], IVIFSs [92], others [91]. Additionally, a variety of fuzzy mathematic programming models are employed to facilitate the FMCDM, such as stochastic dynamic programming [64], fuzzy objective mathematical programming [64, 91], fuzzy linear programming models [79, 87].
From the viewpoint of application, the 30 papers can be classified as: project evaluation [63, 90], project investment/selection [64, 92]. From the perspective of project type, the 30 papers can be also categorized as: general project [68, 84], R&D project [67, 81], portfolio or combination [71, 91], AMS [63, 65], facility location [66, 92], construction project [78, 85], transit project [89], water resources [64], IS/IT project [73], supplier selection [75, 80], new product development [79, 90].
Hybrid methodologies availablefor PIDMFI
By simple bibliometrics analysis in Section 2, there are three papers [42, 59] involving the integration of the three methodologies altogether and the interesting thing is that the first author of the three papers is the same person, A.Çagri Tolga. Among the three papers, Tolga [42], Tolga and Kahraman [43], and Tolga [59] integrated fuzzy discounted cash flows, real option valuation based on Black-Scholes model and fuzzy multiple criteria decision making altogether to form a hybrid methodology. There, however, exist some apparent distinctions among the three papers from the perspective of the priority or ranking method of fuzzy MCDM. Tolga [42] adopted fuzzy TOPSIS, Tolga and Kahraman [43] used fuzzy AHP, and Tolga [59] utilized fuzzy ELECTRE. These contributions enlarge the board of research and diversify the approaches to PIDMFI.
Discussion
In terms of the review, FNs, particularly TFNs or TrFNs, are widely used in the three categories of methodologies for PIDMFI. To some extent, they are superior to other members of the family of fuzzy sets to depict financial criteria and non-monetary or strategic criteria because TFNs and TrFNs are normative and convex fuzzy subsets whose membership functions are with linear mathematic structure, so they are easy and straightforward to model PIDMFI; By the concept of α-cut sets of TFNs and TrFNs, a group of INs are formed with different confidence, α, which make the usage of fuzzy sets in PIDMFI more flexible; TFNs have been treated as the best forms of possibility distribution, a counterpart of probability distribution. By some principles, the transformation between possibility distribution and probability distribution is realized [7, 94]. Accordingly, some random problems can be treated as vague ones reasonably; A great number of methods to operating TFNs (TrFNs or INs) have been established and their good properties have been investigated, which make the usage of TFNs more convenient; TFNs can efficiently quantify the linguistic preference information provided by decision makers or experts.
Additionally, the attention to the FDCF methodology is still stable up to now and we believe that it will keep the pace in near future. As for the FRO methodology, it starts late but increases fast, and we think that it will catch more attention of researchers in this field. The FMCDM methodology performs very well according to our review and it should be a mainstream methodology in PIDMFI with increasing attentions paid on the influence of investment project on economy, society, environment, ecology and technology, etc.
Specifically, fuzzy NPV and IRR methods play the core role in the FDCF methodology; Black-Scholes option valuating model is the fundamental method in the FRO methodology; the FMCDM methodology seriously relies on fuzzy AHP and fuzzy TOPSIS.
Besides the three major methodologies, some others methodologies highlight their role in combining with the three major methodologies, one of them is the fuzzy mathematic programming. Utilizing a variety of fuzzy mathematic programming models, the optimal or best investment projects or portfolios are chosen.
With new extensions of the family of fuzzy sets and other operational research tools, such as interval-valued fuzzy sets [92], intuitionistic fuzzy sets [101], hesitant fuzzy sets [101, 102], etc. the three methodologies will become more diverse not only in theory but in application as well.
Conclusion
Classification is a basic and effective process of recognizing the complex world for human beings. The taxonomic framework developed in this paper provides the literature review with a distinctive, fine and effective perspective. Project investment decision making is one of key issues in the initial phase of project management and decides whether or not the project is successful. Due to the pervasive uncertainty in future and limited knowledge of decision makers, PIDMFI has attracted much more attention of researchers, and a lot of related papers published have contributed to this issue. To effectively and efficiently identify the theories, methods, models and applications investigated and developed for PIDMFI issues in academic papers available, we conduct the literature review from a new taxonomic perspective of methodology.
In the present review, a total of 91 articles, published from 1987–2013, are retrieved from ISI WOS Core Database according to the specific polices. The 91 articles are closely related to PIDMFI issues and indexed by SCI and SSCI of WOS Core Database which has exerted a crucial influence on academic society around the world so far. By observing the basic information of the 91 papers, such as abstract, keywords, citations, etc., we assume that the theories, methods and models used in the 91 papers can be roughly classified into fuzzy discounted cash flow, fuzzy real option and fuzzy multi-criteria decision making methodologies. Then we take use of the hierarchical cluster in SPSS V19 and an extending statistic analysis with spreadsheet, i.e. MS Excel. The clustering result verifies the assumption and categorizes the 91 papers into the three methodologies. From the perspective of the three taxonomic categories, we review the theories, methods, models and applications in the papers of each category in detail. Additionally, the relationships among these methodologies are also investigated. The main contributions of this paper are that we form a new and simple methodology for bibliometrics with statistic analysis tools, construct a new and relatively complete perspective of methodology based on taxonomy for literature review, specify the areas of applications and the types of projects, and point out that the direction of these methodologies for PIDMFI is closely related to the hybrid of them and also closely related to the new extensions of fuzzy sets.
It is important to emphasize that an interesting book “Fuzzy Engineering Economics with Applications” edited by Prof. Kahraman and published by Springer in 2008 provides detailed discussions on extensions of fuzzy sets to classic engineering economic analysis, such as fuzzy NPV, fuzzy EUAV, fuzzy IRR, fuzzy B/C and fuzzy pay-off, etc. It displays a different perspective to understand the theories and methodologies of PIDMFI.
We have to acknowledge that there two main limitations in the study. First at all, the 91 papers are published in academic journals which are confined in ISI WOS Core Database. Those papers or bibliographies published in other journals which are excluded from WOS Core Database, or in the conferences, or in other forms, are not included in the present review. The other limitation is that papers published after 2013 are not considered in question. These limitations may result in some loss of information, but do not influence the structure, content and significance of the review.
Footnotes
Acknowledgments
The authors cordially appreciate the editor and the anonymous referees for their constructive comments and suggestions that led up to an improved version of this paper. This work was supported by grants from the National Natural Science Foundation of China (Nos. 61005042, 60703117 and 11071281), the Natural Science Foundation of Shaanxi Province (Nos. 2014JQ8348) and the Fundamental Research Funds for the Central Universities.