Advances in Civil Engineering

Volume 2018, Article ID 6108680, 9 pages

https://doi.org/10.1155/2018/6108680

## A New Approach to Studying Net Present Value and the Internal Rate of Return of Engineering Projects under Uncertainty with Three-Dimensional Graphs

Centre for Construction Innovation, Department of Construction Engineering and Management, Faculty of Civil Engineering, National Technical University of Athens, Athens 15570, Greece

Correspondence should be addressed to John-Paris Pantouvakis; rg.autn.lartnec@ppj

Received 20 April 2018; Revised 29 June 2018; Accepted 18 July 2018; Published 12 September 2018

Academic Editor: Dujuan Yang

Copyright © 2018 Alexander Maravas and John-Paris Pantouvakis. 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.

#### Abstract

Cost-benefit analysis (CBA) is very useful when appraising engineering projects and examining their long-term financial and social sustainability. However, the inherent uncertainty in the estimation of completion time, final costs, and the realization of benefits often act as an impediment to its application. Since the emergence of fuzzy set theory, there have been significant developments in uncertainty modelling in project evaluation and investment analysis, primarily in the area of formulating a fuzzy version of CBA. In this context, in studying the key indicators of CBA, whereas fuzzy net present value (fNPV) has been investigated quite extensively, there are significant issues in the calculation of fuzzy internal rate of return (fIRR) that have not been addressed. Hence, this paper presents a new conceptual model for studying and calculating fNPV and fIRR. Three-dimensional fNPV and fIRR graphs are introduced as a means of visualizing uncertainty. A new approach is presented for the precise calculation of fIRR. To facilitate practical application, a computerization process is also presented. Additionally, the proposed methodology is exemplified in a sample motorway project whereby its advantages over traditional stochastic uncertainty modelling techniques such as Monte Carlo analysis are discussed. Overall, it is concluded that the new approach is very promising for modelling uncertainty during project evaluation for both project managers and project stakeholders.

#### 1. Introduction

Cost-benefit analysis (CBA) is a valuable decision support tool in project evaluation in both the public and private sectors [1]. It is widely acknowledged that the fundamental principles of CBA are accredited to the work of the French civil engineer and economist Jules Dupuit in the 1840s [2]. After being used systematically in the U.S. in the 1930s, by the end of the 1960s, the use of CBA spread around the world in both developed and developing countries [3]. Its broad purpose is to help decision-making and to make it more rational by the more efficient allocation of available resources [4]. Today, many international financial institutions and international organisations such as the European Investment Bank use CBA to appraise the economic desirability of projects [5].

Reasonably, the fundamental difficulties in the estimation of completion time, final costs, and the realization of benefits often act as an impediment to the application of CBA. Hence, besides its significance and importance, there are limitations to its application because of the underlying approximations, the working hypotheses, and the possible lack of data [6]. Additionally, Belli and Guerrero [7] conclude that when CBA project documents are assessed, risk analysis emerges as one of the weakest areas. Since uncertainty management in CBA is identified as problematic, research is needed to improve existing techniques. It is envisaged that innovations and improvements can increase its importance in engineering decision-making theory and practice.

The primary indicators in CBA are the net present value (NPV) which is expressed in monetary values and the internal rate of return (IRR) [3]. The purpose of this paper is to present an uncertainty management model that applies fuzzy set theory to these indicators. Also, an automation process based on computer processing is presented to facilitate application. Last but not least, a case study is discussed to exemplify both the application of the approach presented in this paper and the introduction of uncertainty due to decisions made during the design and planning phases of an engineering project. Finally, the overall conclusions of this work are presented.

#### 2. Literature Review

##### 2.1. Stochastic and Fuzzy Risk Assessment in CBA

Stochastic risk assessment, in CBA, can be performed primarily with Monte Carlo risk analysis, which is a sophisticated technique that starts by specifying stochastic probability distributions for significant uncertain quantitative assumptions. After that, a trial is taken by taking a random draw from the distribution of each parameter. This step is repeated several times in order to produce a histogram that depicts the realization of net benefits. The underlying assumption is that as the number of trials increases, the frequencies will converge towards the true underlying probabilities [4].

There has been extensive research in applying fuzzy set theory in CBA. Kaufmann and Gupta [8] discussed the discounting problem with fuzzy discounting rates and crisp (nonfuzzy) investment costs. Wang and Liang [9] proposed two algorithms to conduct CBA in a fuzzy environment in which it is difficult to obtain exact assessment data such as investment benefit, expenses, project lifetime, gross income, expenses, and depreciation. Mohamed and McCowan [10] proposed a method for modelling the effects of both monetary (construction cost and annual revenue) and nonmonetary (political, environmental, organizational, competition, and market share) aspects of investment options with possibility theory. Schjaer-Jacobsen [11] set out to examine the possibility of attaining a reasonably useful and realistic picture of the economic consequences of strategic decisions when little is known about the future. He argued that the quality of available information to decision-makers renders traditional decision theory and investment calculations obsolete, while he also demonstrated the representation of economic uncertainties in an investment example with the aid of triangular fuzzy numbers. Dompere [12] studied the discounting process under uncertainty and examined the theory of the fuzzy present value. Chiu and Park [13] developed a fuzzy cash flow analysis for engineering decisions. Sorenson and Lavelle [14] compared fuzzy set and probabilistic paradigms for ranking vague economic investment’s information and concluded that cash flows and interest rates should be modelled by fuzzy sets and ranked with a fuzzy ranking method. Sewastjanow and Dymowa [15] recognized how the obtaining of fuzzy IRR is a rather open problem and to this extent examined a framework for solving fuzzy equations. Tsao [16] presented a series of algorithms to calculate fuzzy net present values of capital investments in an environment with uncertainty. He suggested that the imprecision and uncertainty of the project cash flow are higher than that of the cost of capital.

Beyond the development of a fuzzy version of CBA, there is significant research in applying fuzzy set theory in uncertainty in variables that regard the costs and the cash flow of projects. Kishk [17] applied fuzzy set theory in a whole life costing modelling. Shaheen et al. [18] presented a methodology for extracting fuzzy numbers from experts and processing the information in fuzzy range cost estimation analysis. More so, fuzzy project scheduling (FPS) is based on the application fuzzy set theory in traditional scheduling techniques and is useful in dealing with circumstances involving uncertainty, imprecision, vagueness, and incomplete data [19]. The critical concept is modelling activity duration and cost with fuzzy numbers and thereby calculating project duration and cost, activity start and finish dates, and activity criticality. As such, Maravas and Pantouvakis [20] have shown how fuzzy cost estimates of project activities can be combined with fuzzy project scheduling to yield project cash flow projections.

Regarding project benefits, in the specific case of transportation projects, fuzzy traffic assignment models indicate the region of the expected project benefits. Teodorovic [21] emphasized the importance of fuzzy logic systems as universal approximators in solving traffic and transportation problems. Henn and Ottomanelli [22] applied possibility theory in traffic assignment modelling. Ghatee and Hashemi [23] proposed a traffic assignment model with a fuzzy level of demand. Triangular fuzzy numbers were used to show the imprecise number of travellers who want to travel between origin-destination pairs. Caggiani et al. [24] used fuzzy programming to improve origin-destination matrix estimation based on traffic counts and other uncertain data. De Ona et al. [25] used fuzzy optimization to obtain adjusted values of field traffic volume data to meet consistency constraints.

While the NPV and IRR are the most widespread and accepted indicators when conducting CBA analysis, there are significant developments in the study of IRR. As such, Magni [26] introduced the concept of the average internal rate of return (AIRR) as an alternative to the well-established IRR. While dismissing the IRR equation, he argued about the superiority of the AIRR. Guerra et al. [27] applied fuzzy set theory to the AIRR to study investment appraisal under uncertainty. Jiang [28] presented a particular case of a continuous AIRR, named excess return of time-scaled contributions (ERTC) that can be used in capital budgeting and project finance. Mørch et al. [29] considered the maximization of the AIRR in the renewal of maritime shipping capacity.

Overall, besides the significant research in applying fuzzy set theory to CBA, there are significant issues that need to be researched—primarily, the study of the variation of fuzzy NPV in regard to the discount rate and the calculation of fuzzy IRR. Additionally, the emergence of new fuzzy techniques in cost estimations, cash flow prediction, and benefit analysis provides an opportunity for formulating fuzzy variables that can thereafter be used as base estimates in a holistic risk assessment methodology. Finally, the newly introduced AIRR and its fuzzy equivalent could potentially be adopted in CBA analysis.

##### 2.2. Fundamentals of Fuzzy Set Theory

Fuzzy Set Theory is used to describe and quantify uncertainty and imprecision in data and functional relationships. A fuzzy subset A of a universe of discourse is characterized by a membership function : which associates with each element *x* of a number in the interval which represents the grade of membership of in . In fuzzy set theory, the triangular membership function which is defined by three numbers is encountered very often. Hence, a triangular fuzzy number has the following membership function:

Every fuzzy set A can be associated to a collection of crisp sets known as *α*-cuts (alpha-cuts) or *α*-level sets. An *α*-cut is a crisp set consisting of elements of A which belong to the fuzzy set at least to a degree of *α*. As such, if A is a subset of a universe U, then an *α*-level set of A is a nonfuzzy set denoted by Α_{α} which comprises all elements of U whose grade membership in A is greater than or equal to *α* [30]. In symbols,where is a parameter in the range .

Effectively, an -cut is a means to defuzzify a fuzzy set into a crisp set at desired -levels which reflect the perceived risk. More specifically, every -cut indicates the pessimistic and optimistic values for the same risk level, and in the case of a triangular fuzzy number, it is given by the following formula:

In many cases, it is necessary to compare fuzzy numbers in order to attain a linear ordering. In such cases, the removal number can be defined as the first criterion for the linear ordering. Essentially, it is an ordinary representative of the fuzzy number; in the case of a triangular fuzzy number, it is given by the following formula [8]:

The second criterion is the mode of the fuzzy number, that is, “” for the triangular fuzzy number . Finally, the divergence of the fuzzy number around the mode is the third criterion. It is given by the following formula:

#### 3. Fuzzy NPV and IRR

##### 3.1. Three-Dimensional Graphical Representation of Fuzzy NPV and IRR

The net present value (NPV) is essentially the discounted net cash flow—the sum that results when the discounted expected financial costs of investment are subtracted from the discounted value of the expected benefits:where is the crisp financial discount rate and is the crisp net cash flow at period .

However, in the presence of uncertainty, all values may be modelled with fuzzy numbers. Hence, the fuzzy-net present value (fNPV) is defined as follows [31]:where is the fuzzy financial discount rate and is the fuzzy net cash flow at period .

The crisp internal rate of return (IRR) is defined as the discount rate for which the net present value is equal to zero. In essence, it is the discount rate for which the costs are equal to the benefits. It is a measure of the profitability and the final yield of the investment. Thus, in practice, a project is more desirable if it has a higher value of IRR. However, the IRR cannot be calculated analytically from Equation (6). To this extent, numerical methods are employed to find an acceptable value based on convergence criteria. Since the fuzzy net present value (fNPV) is a fuzzy variable, the fuzzy internal rate of return (fIRR) is expected to be a set of discount rates for which fNPV is equal to zero instead of a single number. Hence, the calculation of this fIRR poses a significant challenge.

To this purpose, it is proposed that Cartesian geometry and a three-dimensional Euclidean space are used to graph fNPV and calculate fIRR. Thus, uncertainty can be represented by a three-dimensional plot in which the *x* axis represents the discount rate, the *y* axis the NPV, and the *z* axis the value of possibility [32]. In effect, these plots give the ability to scan across various discount rates and show the membership functions of fNPV for every such value (Figure 1). The plots can show the change in the uncertainty of fNPV in regard to the value of the discount rate. Thus, examining the slope of this plot and the change of the width of individual fNPV shows the variation of uncertainty in the project.