# Decision Making for Project Appraisal in Uncertain Environments: A Fuzzy-Possibilistic Approach of the Expanded NPV Method

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methods

#### 2.1. Fuzzy Sets Principles

**Definition**

**1.**

**Definition**

**2.**

**Definition**

**3.**

- The fuzzy set A is normalized;
- The fuzzy set A is convex;
- The fuzzy set A is upper semi-continuous;
- The support of A is compact;
- The fuzzy set A is normalized if there exists x ∈ X, such that$A(x)=1$;
- The fuzzy set A is a convex fuzzy set if$\forall t\in [0,1]$and${x}_{1},{x}_{2}\in X$,$A(t{x}_{1}+(1-t){x}_{2})\ge \mathrm{min}\left\{A({x}_{1}),A({x}_{2})\right\}$;
- The$supp(A)$=$\cup {}_{a\in \left(0,1\right]}A[a]=\left\{x:A(x)0\right\}$.

**Definition**

**4.**

**Definition**

**5.**

#### 2.2. Fuzzy Arithmetic

**Property**

**1.**

**Property**

**2.**

#### 2.3. Non-Asymptotic Fuzzy Estimators

**Proposition**

**1.**

_{1}, X

_{2}, …, X

_{n}be a random sample, and let x

_{1}, x

_{2}, …, x

_{n}be the sample values assumed by the sample. Additionally, let us say that γ ∈ (0.1) if the size is small. In that case, then

^{2}, and the a-cuts of this fuzzy number are the closed intervals:

**Proposition**

**2.**

_{1}, X

_{2}, …, X

_{n}be a random sample, and let x

_{1}, x

_{2}, …, x

_{n}be sample values assumed by the sample. Additionally, let us say that γ ∈ (0.1) if the size is large. In that case, then

^{2}, and the a-cuts of this fuzzy number are the closed intervals:

## 3. Assumprions for the Present Research Work

#### 3.1. NPV Formulas

- The real cash flows have to be converted to nominal cash flows (the use of a nominal discount rate is also necessary);
- The cash flows are estimated in real values (the use of a real discount rate is also necessary).

^{t}× Real Cash flows

- The time value of money is represented by the opportunity cost of capital, calculated through the weighted average cost of capital (WACC);
- The equity cost is determined through the possibilistic set-up of CAPM;
- The inflation factor is also included in the estimation of the NPV;
- The value from the expansion of the project is calculated through the fuzzy binomial model.

#### 3.2. Possibilistic Discount Rate via Possibilistic CAPM

_{D}is considered to be the gross redemption yield (GRY%) [37]. See Abbreviations for the basic notation.

## 4. Fuzzy Possibilistic Net Present Value

**Definition**

**6.**

_{t}and COF

_{t}and the possibilistic rate resulting from fuzzy data for the discount rate r

_{poss}, is defined as

**Definition**

**7.**

_{t}, COF

_{t}and f

_{t}and the possibilistic rate resulting from the fuzzy data for the discount rate r

_{poss}, is defined as

**Definition**

**8.**

_{t}, COF

_{t}and f

_{t}and the possibilistic rate resulting from the fuzzy data for the discount rate r

_{poss}, is defined as

**Definition**

**9.**

_{t}, cof

_{t}, x

_{t}, COF

_{t}, f

_{t}and L and the possibilistic rate resulting from the fuzzy data for the discount rate r

_{poss}, is defined as

**Definition**

**10.**

_{t}, cof

_{t}, x

_{t}, COF

_{t}, f

_{t}and L and the possibilistic rate resulting from the fuzzy data for the discount rate r

_{poss}, is defined as

**Definition**

**11.**

_{t}, cof

_{t}, x

_{t}, COF

_{t}, f

_{t}and L and the possibilistic rate resulting from the fuzzy data for the discount rate r

_{poss}, is defined as

## 5. Fuzzy Binomial Model

#### 5.1. The Classic Binomial Model

_{u}(greater than 1) with a probability p or fall by a multiple of e

_{d}(less than 1) with a probability q = 1 − p.

_{t}is call option price at time t, P

_{t}is the put option price at time t, j is the number of upward jumps, k − j is the number of downward jumps, S

_{t}is the cash flow at time t and X is the strike price.

#### 5.2. Fuzzy Up and Down Probabilities

^{rf(t)}where $f:(0,+\infty )\to (0,+\infty )$:

_{u}is obtained for u

_{max}= u

^{l}(α) (resp. d

_{max}= d

^{l}(α)) and the minimum for u

_{min}= u

^{r}(α) (resp. d

_{max}= d

^{r}(α)).

_{d}is obtained for u

_{max}= u

^{r}(α) (resp. d

_{max}= d

^{r}(α)) and the minimum for u

_{min}= u

^{l}(α) (resp. d

_{max}= d

^{l}(α)).

_{d}, the solution is

_{ΥΖ}

_{,}is the fuzzy estimator for the standard deviation of the asset.

#### 5.3. Fuzzy Volatility

^{2}), CP

_{0}is the closing price of the previous period (at time 0), OP

_{1}is the opening price of the current period (at time f), HP

_{1}is the the current period’s high during the trading interval (between [ f, 1]), LP

_{1}is the current period’s low during the trading interval (between [ f, 1]), CP

_{1}is the closing price of the current period (at time 1), o = ln OP

_{1}− ln CP

_{0}, the normalized open, u = ln HP

_{1}− ln OP

_{1}, the normalized high, d = ln LP

_{1}− ln OP

_{1}, the normalized low, and, c = ln CP

_{1}− ln OP

_{1}, the normalized close.

_{o}and V

_{c}and finally derived the fuzzy estimator for the Yang and Zhang variance.

_{1}, O

_{2}, …, O

_{n}is exactly the (1 − γ)100% confidence interval for σ

^{2}, and the a-cuts of this fuzzy number are the closed intervals

_{h(a)}, h

_{(a)}and Φ have already been defined.

_{1}, C

_{2}, …, C

_{n}is exactly the (1 − γ)100% confidence interval for σ

^{2}, and the a-cuts of this fuzzy number are the closed intervals

_{h(a)}, h

_{(a)}and Φ have already been defined.

_{1}, O

_{2}, …, O

_{n}is exactly the (1 − γ)100% confidence interval for σ

^{2}, and the a-cuts of this fuzzy number are the closed intervals

_{h(a)}, h

_{(a)}and Φ have already been defined.

_{1}, C

_{2}, …, C

_{n}is exactly the (1 − γ)100% confidence interval for σ

^{2}, and the a-cuts of this fuzzy number are the closed intervals

_{h(a)}, h

_{(a)}and Φ have already been defined.

## 6. Example (Revisited and Substantially Extended for the Illustration of the FPeNPV Method)

**Step 1.**

Project P1 | Project P2 | Project P3 |

equity beta | equity beta | equity beta |

1.5600 | 1.7423 | 0.9870 |

**Step 2.**

_{asset}= E

_{Β}× β

_{equity}/[E

_{Β}+ D

_{Β}× (1 − Τ

_{C})]).

Project P1 | Project P2 | Project P3 |

asset beta | asset beta | asset beta |

1.2000 | 1.4126 | 0.8003 |

**Step 3.**

_{equity}= [E

_{A}+ D

_{A}× (1 − Τ

_{C})] × β

_{asset}/Ε

_{A}).

Project P1 | Project P2 | Project P3 |

equity beta | equity beta | equity beta |

1.7040 | 2.0060 | 1.1364 |

**Step 4.**

Project P1 | Project P2 | Project P3 |

CAPM equity cost | CAPM equity cost | CAPM equity cost |

0.0283 | 0.0333 | 0.0189 |

**Step 5.**

Project P1 | Project P2 | Project P3 |

WACC | WACC | WACC |

4.39% | 4.71% | 3.80% |

**Step 6.**

**Step 7.**

**Step 8.**

**Step 9.**

## 7. Results

## 8. Discussion

_{3}= 0.987, and after the process P

_{3}= 1.1364); (2) the use of the possibilistic beta for each project does not require statistical information (historical data), which could be biased, and it is based on expert opinions; and (3) the possibilistic beta is a safe translation of fuzzy information. Expert fuzzy estimates conclude to a possibilistic crisp value for the equity cost (from CAPM), which is incorporated in the calculation of the WACC.

## 9. Conclusions and Further Work

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

$NPV$ | net present value |

$TFN$ | triangular fuzzy number |

$FPNPV$ | fuzzy possibilistic net present value |

$t$ | time index |

R_{t} | stock return in t |

P_{t} | stock price at the end of t |

P_{t − 1} | stock price at the end of t − 1 |

D_{t} | stock dividend in t |

R_{Mt} | market portfolio return on t |

I_{t} | level of the index at the end of t |

I_{t − 1} | level of the index at the end of t − 1 |

d_{t} | dividend paid on the index in t |

${R}_{f}$ | risk-free rate |

$GRY$ | gross redemption yield |

${T}_{C}$ | corporate tax |

${E}_{company}$ | equity part of a company’s equity/debt ratio |

${D}_{company}$ | debt part of a company’s equity/debt ratio |

${V}_{company}$ | total market value of firm (the sum of E_{Company} + D_{Company}) |

$WACC$ | weighted average cost of capital |

${I}_{0}$ | initial investment outlay |

${p}_{t}$ | sales price in period t |

$TCO{F}_{t}$ | total cash outflows in t |

$co{f}_{t}$ | production- and sales-dependent (variable) cash outflows per unit in period t |

${x}_{t}$ | production and sales volume in period t |

$CI{F}_{t}$ | cash inflows in t |

$CO{F}_{t}$ | production- and sales-independent (fixed) cash outflows in period t |

${r}_{poss}$ | discount rate resulting from possibilistic CAPM |

${L}_{t}$ | liquidation index |

$T$ | the last year when cash flows take place |

$FNPV$ | fuzzy net present value |

## Appendix A

- ➢
- The extension principle:$$\begin{array}{l}FPNPV=-{I}_{0}\\ +{\displaystyle \sum _{i=1}^{T}([CI{F}_{t}^{l}(a),CI{F}_{t}^{r}(a)]-[TCO{F}_{{}_{t}}^{l}(a),TCO{F}_{t}^{r}(a)]){(1+{r}_{poss})}^{-t}}\to \end{array}\phantom{\rule{0ex}{0ex}}\begin{array}{l}FPNPV=-{I}_{0}\\ +{\displaystyle \sum _{i=1}^{T}[(CI{F}_{t}^{l}(a)-TCO{F}_{t}^{r}(a),CI{F}_{t}^{r}-TCO{F}_{{}_{t}}^{l}(a)]{(1+{r}_{poss})}^{-t}}\end{array}\phantom{\rule{0ex}{0ex}}\begin{array}{l}FPNPV=-{I}_{0}\\ +{\displaystyle \sum _{i=1}^{T}[(CI{F}_{t}^{l}(a)-TCO{F}_{t}^{r}(a)){(1+{r}_{poss})}^{-t},(CI{F}_{t}^{r}(a)-TCO{F}_{{}_{t}}^{l}(a)){(1+{r}_{poss})}^{-t}]}\end{array}$$

- The basic formula:$$FPNPV=-{I}_{0}+{\displaystyle \sum _{i=1}^{T}(CI{F}_{t}^{}-TCO{F}_{t}){(1+r)}^{-t}\to}$$
- Convert the real cash flows and real discount rate to nominal values:$$FPNPV=-{I}_{0}+{\displaystyle \sum _{i=1}^{T}(CI{F}_{t}^{}-TCO{F}_{t}){(1+f)}^{t}{((1+r)(1+f))}^{-t}\to}\phantom{\rule{0ex}{0ex}}FPNPV=-{I}_{0}+{\displaystyle \sum _{i=1}^{T}(CI{F}_{t}^{}-TCO{F}_{t}){(1+f)}^{t}{(1+r+f+rf)}^{-t}\to}$$
- The extension principle:$$\begin{array}{l}FPNPV=-{I}_{0}\\ +{\displaystyle \sum _{i=1}^{T}\begin{array}{l}([CI{F}_{t}^{l}(a)-CI{F}_{t}^{r}(a)]-[TCO{F}_{{}_{t}}^{l}(a),TCO{F}_{t}^{r}(a)])\\ {[1+{f}_{t}^{l}(a),1+{f}_{t}^{r}(a)]}^{t}{[1+{r}_{poss}+{f}_{t}^{l}(a)+{r}_{poss}{f}_{t}^{l}(a),1+{r}_{poss}+{f}_{t}^{r}(a)+{r}_{poss}{f}_{t}^{r}(a)]}^{-t}\end{array}}\to \end{array}\phantom{\rule{0ex}{0ex}}\begin{array}{l}FPNPV=-{I}_{0}\\ +{\displaystyle \sum _{i=1}^{T}\begin{array}{l}[CI{F}_{t}^{l}(a)-TCO{F}_{t}^{r}(a),CI{F}_{t}^{l}(a)-TCO{F}_{{}_{t}}^{l}(a)]\\ [{(1+{f}_{t}^{l}(a))}^{t},{(1+{f}_{t}^{r}(a))}^{t}]\\ [{(1+{r}_{poss}+{f}_{t}^{r}(a)+{r}_{poss}{f}_{t}^{r}(a))}^{-t},1+{({r}_{poss}+{f}_{t}^{l}(a)+{r}_{poss}{f}_{t}^{l}(a))}^{-t}]\end{array}}\end{array}$$
- Next, the operation of multiplication on intervals is applied, depending on the sign of the operation $CI{F}_{t}[a]-TCO{F}_{t}^{}[a]$.

- The basic formula:$$FPNPV=-{I}_{0}+{\displaystyle \sum _{i=1}^{T}(CI{F}_{t}-TCO{F}_{t}){(1+r)}^{-t}\to}$$
- Convert the nominal cash flows and nominal discount rate to real values:$$FPNPV=-{I}_{0}+{\displaystyle \sum _{i=1}^{T}(CI{F}_{t}-TCO{F}_{t}){(1+f)}^{-t}{((1+r){(1+f)}^{-1})}^{-t}\to}\phantom{\rule{0ex}{0ex}}FPNPV=-{I}_{0}+{\displaystyle \sum _{i=1}^{T}(CI{F}_{t}{[a]}_{t}-TCO{F}_{t}){(1+f)}^{-t}{(1+\frac{r-f}{1+f})}^{-t}\to}$$
- The extension principle:$$\begin{array}{l}FPNPV=-{I}_{0}\\ +{\displaystyle \sum _{i=1}^{T}\begin{array}{l}([CI{F}_{t}^{l}(a)-CI{F}_{t}^{r}(a)]-[TCO{F}_{{}_{t}}^{l}(a),TCO{F}_{t}^{r}(a)])\\ {[1+{f}_{t}^{l}(a),1+{f}_{t}^{r}(a)]}^{t}{(1+\frac{[{r}_{poss}-{f}_{t}^{r}(a),{r}_{poss}-{f}_{t}^{l}(a)]}{[1+{f}_{t}^{l}(a),1+{f}_{t}^{r}(a)]})}^{-t}\end{array}}\to \end{array}\phantom{\rule{0ex}{0ex}}\begin{array}{l}FPNPV=-{I}_{0}\\ +{\displaystyle \sum _{i=1}^{T}\begin{array}{l}([CI{F}_{t}^{l}(a)-TCO{F}_{t}^{r}(a),CI{F}_{t}^{l}(a)-TCO{F}_{{}_{t}}^{l}(a)])\\ {[1+{f}_{t}^{l}(a),1+{f}_{t}^{r}(a)]}^{t}{[1+\frac{{r}_{poss}-{f}_{t}^{r}(a)}{1+{f}_{t}^{r}(a)},1+\frac{{r}_{poss}-{f}_{t}^{l}(a)}{1+{f}_{t}^{l}(a)}]}^{-t}\end{array}}\end{array}$$
- Next, the operation of multiplication on intervals is applied, depending on the sign of the operation $CIF[a]-TCO{F}_{t}^{}[a]$.

- The extension principle:$$\begin{array}{l}FPNPV=-{I}_{0}\\ +{\displaystyle \sum _{i=1}^{T}\begin{array}{l}(([{p}_{t}^{l}(a),{p}_{t}^{r}(a)]-[co{f}_{t}^{l}(a),co{f}_{t}^{r}(a)])[{x}_{t}^{l}(a),{x}_{r}^{r}(a)]-[CO{F}_{{}_{t}}^{l}(a),CO{F}_{t}^{r}(a)])\\ {(1+{r}_{poss})}^{-t}+[{L}_{t}^{l}(a),{L}_{t}^{r}(a)]{(1+{r}_{poss})}^{-T}\end{array}}\to \end{array}\phantom{\rule{0ex}{0ex}}\begin{array}{l}FPNPV=-{I}_{0}\\ +{\displaystyle \sum _{i=1}^{T}\begin{array}{l}[({p}_{t}^{l}(a)-co{f}_{t}^{r}(a)){x}_{t}^{l}(a)-CO{F}_{t}^{r}(a),({p}_{t}^{r}(a)-co{f}_{t}^{l}(a)){x}_{t}^{r}(a)-CO{F}_{{}_{t}}^{l}(a)]\\ {(1+{r}_{poss})}^{-t}+[{(1+{r}_{poss})}^{-T}{L}_{t}^{l}(a),{(1+{r}_{poss})}^{-T}{L}_{t}^{r}(a)]\end{array}}\end{array}$$
- Next, the operation of multiplication on intervals is applied, depending on the sign of the operation $({p}_{t}^{}[a]-co{f}_{t}^{}[a]){x}_{t}^{}[a]-CO{F}_{t}^{}[a]$.

- The basic formula:$$FPNPV=-{I}_{0}+{\displaystyle \sum _{i=1}^{T}(({p}_{t}-co{f}_{t}){x}_{t}-CO{F}_{t}){(1+r)}^{-t}+L{(1+r)}^{-T}\to}$$
- Convert the real cash flows and real discount rate to nominal values:$$FPNPV=-{I}_{0}+{\displaystyle \sum _{i=1}^{T}\begin{array}{l}(({p}_{t}-co{f}_{t}){x}_{t}-CO{F}_{t}){(1+f)}^{t}{((1+r)(1+f))}^{-t}+\\ L{(1+f)}^{t}{((1+r)(1+f))}^{-T}\end{array}}\to \phantom{\rule{0ex}{0ex}}FPNPV=-{I}_{0}+{\displaystyle \sum _{i=1}^{T}\begin{array}{l}(({p}_{t}-co{f}_{t}){x}_{t}-CO{F}_{t}){(1+f)}^{t}{(1+r+f+rf)}^{-t}\\ +L{(1+f)}^{t}{(1+r+f+rf)}^{-T}\to \end{array}}$$
- The extension principle:$$\begin{array}{l}FPNPV=-{I}_{0}\\ +{\displaystyle \sum _{i=1}^{T}\begin{array}{l}(([{p}_{t}^{l}(a),{p}_{t}^{r}(a)]-[co{f}_{t}^{l}(a),co{f}_{t}^{r}(a)])[{x}_{t}^{l}(a),{x}_{r}^{r}(a)]-[CO{F}_{{}_{t}}^{l}(a),CO{F}_{t}^{r}(a)])\\ {[1+{f}_{t}^{l}(a),1+{f}_{t}^{r}(a)]}^{t}{[1+{r}_{poss}+{f}_{t}^{l}(a)+{r}_{poss}{f}_{t}^{l}(a),1+{r}_{poss}+{f}_{t}^{r}(a)+{r}_{poss}{f}_{t}^{r}(a)]}^{-t}\\ +[{L}_{t}^{l}(a),{L}_{t}^{r}(a)][1+{f}_{t}^{l}(a),1+{f}_{t}^{r}(a)]\\ {[1+{r}_{poss}+{f}_{t}^{l}(a)+{r}_{poss}{f}_{t}^{l}(a),1+{r}_{poss}+{f}_{t}^{r}(a)+{r}_{poss}{f}_{t}^{r}(a)]}^{-T}\end{array}}\to \end{array}\phantom{\rule{0ex}{0ex}}\begin{array}{l}FPNPV=-{I}_{0}\\ +{\displaystyle \sum _{i=1}^{T}\begin{array}{l}[({p}_{t}^{l}(a)-co{f}_{t}^{r}(a)){x}_{t}^{l}(a)-CO{F}_{t}^{r}(a),({p}_{t}^{r}(a)-co{f}_{t}^{l}(a)){x}_{t}^{r}(a)-CO{F}_{{}_{t}}^{l}(a)]\\ {[1+{f}_{t}^{l}(a),1+{f}_{t}^{r}(a)]}^{t}{[1+{r}_{poss}+{f}_{t}^{l}(a)+{r}_{poss}{f}_{t}^{l}(a),1+{r}_{poss}+{f}_{t}^{r}(a)+{r}_{poss}{f}_{t}^{r}(a)]}^{-t}\\ +[[{L}_{t}^{l}(a),{L}_{t}^{r}(a)][1+{f}_{t}^{l}(a),1+{f}_{t}^{r}(a)]\\ {[1+{r}_{poss}+{f}_{t}^{l}(a)+{r}_{poss}{f}_{t}^{l}(a),1+{r}_{poss}+{f}_{t}^{r}(a)+{r}_{poss}{f}_{t}^{r}(a)]}^{-T}\end{array}}\end{array}$$
- Next, the operation of multiplication on intervals is applied, depending on the sign of the operation $({p}_{t}^{}[a]-co{f}_{t}^{}[a]){x}_{t}^{}[a]-CO{F}_{t}^{}[a]$.

- The basic formula:$$FPNPV=-{I}_{0}+{\displaystyle \sum _{i=1}^{T}(({p}_{t}-co{f}_{t}){x}_{t}-CO{F}_{t}){(1+r)}^{-t}+L{(1+r)}^{-T}\to}$$
- Converting nominal cash flows and nominal discount rate to real values:$$FPNPV=-{I}_{0}+{\displaystyle \sum _{i=1}^{T}\begin{array}{l}(({p}_{t}-co{f}_{t}){x}_{t}-CO{F}_{t}){(1+f)}^{-t}{((1+r)(1+f))}^{-t}\\ +L{(1+f)}^{t}{((1+r)(1+f))}^{-T}\to \end{array}}\phantom{\rule{0ex}{0ex}}FPNPV=-{I}_{0}+{\displaystyle \sum _{i=1}^{T}(({p}_{t}-co{f}_{t}){x}_{t}-CO{F}_{t}){(1+f)}^{-t}{(1+\frac{{r}_{pss}-f}{1+f})}^{-t}+L{(1+f)}^{t}{(\frac{{r}_{poss}-f}{1+f})}^{-T}\to}$$
- The extension principle:$$\begin{array}{l}FPNPV=-{I}_{0}\\ +{\displaystyle \sum _{i=1}^{T}\begin{array}{l}(([{p}_{t}^{l}(a),{p}_{t}^{r}(a)]-[co{f}_{t}^{l}(a),co{f}_{t}^{r}(a)])[{x}_{t}^{l}(a),{x}_{r}^{r}(a)]-[CO{F}_{{}_{t}}^{l}(a),CO{F}_{t}^{r}(a)])\\ {[1+{f}_{t}^{l}(a),1+{f}_{t}^{r}(a)]}^{t}{(1+\frac{[{r}_{poss}-{f}_{t}^{r}(a),{r}_{poss}-{f}_{t}^{l}(a)]}{[1+{f}_{t}^{l}(a),1+{f}_{t}^{r}(a)]})}^{-t}\\ +[{L}_{t}^{l}(a),{L}_{t}^{r}(a)][1+{f}_{t}^{l}(a),1+{f}_{t}^{r}(a)]{(1+\frac{[{r}_{poss}-{f}_{t}^{r}(a),{r}_{poss}-{f}_{t}^{l}(a)]}{[1+{f}_{t}^{l}(a),1+{f}_{t}^{r}(a)]})}^{-T}\end{array}}\to \end{array}$$
- Next, the operation of multiplication on intervals is applied, depending on the sign of the operation $({p}_{t}^{}[a]-co{f}_{t}^{}[a]){x}_{t}^{}[a]-CO{F}_{t}^{}[a]$:$$\begin{array}{l}FPNPV=-{I}_{0}\\ +{\displaystyle \sum _{i=1}^{T}\begin{array}{l}[({p}_{t}^{l}(a)-co{f}_{t}^{r}(a)){x}_{t}^{l}(a)-CO{F}_{t}^{r}(a),({p}_{t}^{r}(a)-co{f}_{t}^{l}(a)){x}_{t}^{r}(a)-CO{F}_{{}_{t}}^{l}(a)]\\ {[1+{f}_{t}^{l}(a),1+{f}_{t}^{r}(a)]}^{t}{[1+\frac{{r}_{poss}-{f}_{t}^{r}(a)}{1+{f}_{t}^{r}(a)},1+\frac{{r}_{poss}-{f}_{t}^{l}(a)}{1+{f}_{t}^{l}(a)}]}^{-t}\\ +[{L}_{t}^{l}(a),{L}_{t}^{r}(a)][1+{f}_{t}^{l}(a),1+{f}_{t}^{r}(a)]{[1+\frac{{r}_{poss}-{f}_{t}^{r}(a)}{1+{f}_{t}^{r}(a)},1+\frac{{r}_{poss}-{f}_{t}^{l}(a)}{1+{f}_{t}^{l}(a)}]}^{-T}\end{array}}\end{array}$$

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Company B | Company C | ||
---|---|---|---|

Variable | Project P1 | Project P2 | Project P3 |

Dt [α] | [2.00,2.50] | [2.20,2.60] | [3.00,3.50] |

Pt [α] | [12.00,14.50] | [11.00,12.00] | [14.00,15.00] |

Pt−1[α] | [11.50,11.50] | [10.50,12.00] | [11.70,12.00] |

dt [α] | [3.50,4.00] | [3.50,4.00] | [3.50,4.00] |

It [α] | [12.00,14.50] | [12.50,14.50] | [12.00,14.50] |

It−1[α] | [15.00,15.00] | [15.00,15.00] | [15.00,15.00] |

Type of Variable | Variable | Left Tail of A-Cut | Right Tail of A-Cut |
---|---|---|---|

Crisp | I | 4000.00 € | |

Fuzzy | pt | 15.00 € | 15.50 € |

Fuzzy | coft | 6.00 € | 6.30 € |

Fuzzy | xt | 1500.00 units | 1800.00 units |

Fuzzy | COFt | 5000.00 € | 5400.00 € |

Fuzzy | ft | 4.00% | 4.20% |

Crisp | economic life T | 3 years | |

Fuzzy | Lt | 1000.00 € | 1,250,00 € |

Type of Variable | Variable | Left Tail of A-Cut | Right Tail of A-Cut |
---|---|---|---|

Crisp | I | 4000.00 € | |

Fuzzy | pt | 16.00 € | 16.70 € |

Fuzzy | coft | 5.00 € | 6.50 € |

Fuzzy | xt | 2000.00 units | 2100.00 units |

Fuzzy | COFt | 4000.00 € | 4300.00 € |

Fuzzy | ft | 4.00% | 4.20% |

Crisp | economic life T | 3 years | |

Fuzzy | Lt | 1100.00 € | 1400.00 € |

Type of Variable | Variable | Left Tail of A-Cut | Right Tail of A-Cut |
---|---|---|---|

Crisp | I | 6000.00 € | |

Fuzzy | pt | 12.00 € | 12.50 € |

Fuzzy | coft | 4.00 € | 5.00 € |

Fuzzy | xt | 3000.00units | 3200.00 units |

Fuzzy | COFt | 4000.00 € | 4300.00 € |

Fuzzy | ft | 4.00% | 4.20% |

Crisp | economic life T | 3 years | |

Fuzzy | Lt | 1200.00 € | 1300.00 € |

Period | Net Cash Flow | ||
---|---|---|---|

Right Tail of A-Cut | Left Tail of A-Cut | ||

Project P1 | 1 | 3911.45 € | 6159.80 € |

2 | 1998.97 € | 3134.81 € | |

3 | 1045.33 € | 1625.05 € | |

FPNPV | 2955.75 € | 6919.67 € | |

Project P2 | 1 | 7423.65 € | 10,383.77 € |

2 | 3748.72 € | 5241.33 € | |

3 | 1919.12 € | 2678.76 € | |

FPNPV | 9091.49 € | 14,303.87 € | |

Project P3 | 1 | 8471.05 € | 11,742.01 € |

2 | 4296.05 € | 5942.02 € | |

3 | 2208.11 € | 3038.73 € | |

FPNPV | 8975.22 € | 14,722.76 € |

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## Share and Cite

**MDPI and ACS Style**

Chrysafis, K.A.; Papadopoulos, B.K.
Decision Making for Project Appraisal in Uncertain Environments: A Fuzzy-Possibilistic Approach of the Expanded NPV Method. *Symmetry* **2021**, *13*, 27.
https://doi.org/10.3390/sym13010027

**AMA Style**

Chrysafis KA, Papadopoulos BK.
Decision Making for Project Appraisal in Uncertain Environments: A Fuzzy-Possibilistic Approach of the Expanded NPV Method. *Symmetry*. 2021; 13(1):27.
https://doi.org/10.3390/sym13010027

**Chicago/Turabian Style**

Chrysafis, Konstantinos A., and Basil K. Papadopoulos.
2021. "Decision Making for Project Appraisal in Uncertain Environments: A Fuzzy-Possibilistic Approach of the Expanded NPV Method" *Symmetry* 13, no. 1: 27.
https://doi.org/10.3390/sym13010027