# Neutrosophic Completion Technique for Incomplete Higher-Order AHP Comparison Matrices

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Preliminaries on Classical AHP Method

_{ij}represents how much more relevant alternative i is with respect to alternative j for making the final decision according to expert’s knowledge of the problem. The matrix is filled with numerical values from the so-called Saaty’s fundamental scale, which allows for the conversion of semantic values into real numbers. The fundamental scale is a discrete set of integers that ranges from 1 to 9, with 1 meaning that both alternatives are equally relevant, and 9 that the first alternative is extremely more important than the second one.

_{ij}} results in a square and reciprocal matrix, i.e., a

_{ij}= 1/a

_{ji}∀i,j ∈ {1,...,n}, where n is the number of alternatives or criteria to be compared. It should be noted as well that a

_{ii}= 1 ∀i ∈ {1,...,n}. The weights of each alternative are then obtained as the elements of the eigenvector associated to the greatest eigenvalue λ

_{max}of the comparison matrix A.

_{max}− n)/(n − 1)

#### 2.2. Analytic Hierarchy Process in a Neutrosophic Environment

#### 2.2.1. Neutrosophic Logic Fundamentals

_{A}(x), i

_{A}(x), and f

_{A}(x) correspond to the truth, indeterminacy, and falsity functions of the element x ∈ X. Truth, indeterminacy, and falsity membership functions are defined in the unit interval [0, 1] and are independent, i.e., they satisfy:

_{1}, a

_{2}, a

_{3}); t

_{a}, i

_{a}, f

_{a}〉 on the real number set is described with the following membership functions [35]:

_{1}, a

_{2}, a

_{3}); t

_{a}, i

_{a}, f

_{a}〉 and b = 〈(b

_{1}, b

_{2}, b

_{3}); t

_{b}, i

_{b}, f

_{b}〉 are defined as [36,37]:

_{a}≤ t

_{b}, i

_{a}≥ i

_{b}, f

_{a}≥ f

_{b}, t

_{b}≠ 0, and i

_{b}, f

_{b}≠ 1.

#### 2.2.2. On the Construction of the Neutrosophic AHP Comparison Matrix

_{ij}is a single-valued neutrosophic number a

_{ij}= 〈(l

_{ij},m

_{ij},u

_{ij}); t

_{ij}, i

_{ij}, f

_{ij}〉 ∀i,j ∈ {1,...,n}. The diagonal elements of the comparison matrix are defined as a

_{ii}= 〈(1,1,1); 0,0,0〉 ∀i ∈ {1,...,n}. It is important to note that the reciprocal elements of the matrix are defined as a

_{ji}= 1/a

_{ij}= 〈(1/u

_{ij},1/m

_{ij},1/l

_{ij}); t

_{ij}, i

_{ij}, f

_{ij}〉 ∀i,j ∈ {1,...,n} [38].

_{ij}of each matrix element a

_{ij}correspond directly to the judgments emitted by the decision maker expressed according to the fundamental scale introduced by Saaty. The lower and upper bounds {l

_{ij}, u

_{ij}} of each triangular number are expressed in terms of the Saaty’s scale as well. These bounds can be obtained as:

_{ij}± ∆(c

_{ij}) represents the value of the extended Saaty’s fundamental scale after subtracting/adding a certain number of steps ∆(c

_{ij}) to the central value m

_{ij}. The extended fundamental scale is a discrete set that adds to Saaty’s fundamental scale its reciprocal values in an inverse order, thus ranging from 1/9 to 9. ∆(c

_{ij}) is defined as a stepwise function that depends on the certainty c

_{ij}emitted by the decision maker when expressing judgment a

_{ij}(Table 1).

_{ij}, i

_{ij}, f

_{ij}} expressing the reliability of each judgment a

_{ij}emitted by the decision maker is now presented. The truth parameter t

_{ij}can be derived from the expert’s assessment credibility [39]. Here, the expert’s credibility is related to the relevant experience that the decision maker expresses on the fields involved in the decision-making problem [40]:

_{ij}can be obtained from the complementary of the certainty values c

_{ij}expressed by the decision maker for each element of the comparison matrix:

_{ij}of the expert’s statements is derived from the consistency of his/her resulting comparison matrix and is common for every statement:

_{lim}is the limiting consistency ratio for a comparison matrix to be considered acceptably consistent. For more than 5 criteria, CR

_{lim}= 10%.

#### 2.2.3. Derivation of Weights from a Neutrosophic Comparison Matrix

_{l,i}, w

_{u,i}) of a neutrosophic weight w as:

#### 2.2.4. Deneutrosophication Process

_{i}for each of the decision criteria i are derived in the form of triangular neutrosophic numbers, namely w

_{i}= 〈(w

_{l,i}, w

_{m,i}, w

_{u,i}); t

_{i}, i

_{i}, f

_{i}〉. A so-called deneutrosophication technique is required to transform those weights into scalar numbers. Sodenkamp et al. [39] proposed a procedure for single-valued neutrosophic numbers based on two subsequent steps, which was extended for its application on generalized neutrosophic numbers by Navarro et al. [13]. Firstly, the neutrosophic weights are converted into triangular fuzzy numbers. The transfer function converting the neutrosophic membership functions 〈μ

_{i}(x),ν

_{i}(x),λ

_{i}(x)〉 into a single fuzzy membership function η

_{i}(x) is obtained as the distance between each point contained in the triad and the point corresponding to the neutrosophic point of optimum reliability 〈1,0,0〉:

_{x}, CoG

_{y}) of the fuzzy weight. The final scalar weights W

_{i}are obtained as:

#### 2.3. Completion Method for Neutrosophic AHP Comparison Matrices

#### 2.3.1. The Classical DEMATEL Technique

^{k}= {z

_{ij}}, where z

_{ij}is the influence score assigned in accordance with the abovementioned scale. Diagonal elements are set to zero. The final direct influence matrix DIM is obtained as the average of the matrices DIM

^{k}obtained from the experts.

_{ij}by s, defined as:

_{ij}} can be obtained by aggregating direct and indirect effects as:

_{i}and C

_{i}. The influential factors R

_{i}and C

_{i}can be obtained as the sum of each row and column of the TRM, respectively. For a particular factor i, depending on whether R

_{i}− C

_{i}is positive or negative, it can be obtained if factor i falls in the cause or in the effect group, respectively.

#### 2.3.2. Completion Method for Neutrosophic AHP Matrices

_{ij}} defined in a neutrosophic environment, i.e., where elements a

_{ij}are defined by triangular neutrosophic numbers a

_{ij}= 〈(l

_{ij},m

_{ij},u

_{ij}); t

_{ij}, i

_{ij}, f

_{ij}〉 ∀i,j ∈ {1,...,n}. The basic inputs for the problem are an incomplete, conventional comparison matrix AHP M* = {m

_{ij}}, together with an incomplete certainty matrix C* = {c

_{ij}}. It shall be noted that c

_{ij}is known only if m

_{ij}is known. The construction of the complete neutrosophic matrix consists of several steps, as follows:

_{ij}of the incomplete matrix M* are obtained following the technique proposed by Zhou et al. [33]:

_{ij}}, compute DIM = {z

_{ij}}, where z

_{ij}= m

_{ij}if m

_{ij}is known and z

_{ij}= 0 if m

_{ij}is unknown.

_{ij}by s, as in classical DEMATEL (see Equation (23)).

_{ij}} as in classical DEMATEL (see Equation (24)).

_{ij}}, considering the relations revealed between factors by the total relation matrix TRM = {g

_{ij}}:

_{ij}shall be derived.

_{ij}, u

_{ij}} of each element a

_{ij}shall be derived. For the known values of the input AHP matrix, the lower and upper bounds {l

_{ij}, u

_{ij}} shall be obtained according to Equation (12). The estimation of the missing bounds {l*

_{ij}, u*

_{ij}} consists of several steps:

_{ij}} and U* = {u

_{ij}} are constructed, where {l

_{ij}, u

_{ij}} = {0, 0} if m

_{ij}is unknown.

_{ij}, u

_{ij}} as follows:

^{L}= {g

^{L}

_{ij}} and TRM

^{U}= {g

^{U}

_{ij}} as in classical DEMATEL (see Equation (24)).

_{ij}, u*

_{ij}} shall then be derived as:

_{ij}, i*

_{ij}, f*

_{ij}} of the missing entries shall reflect the reliability of the estimates. The expert’s credibility when handling complete neutrosophic AHP matrices is considered equal for every matrix entry according to Equation (13). Here, the expert’s credibility t*

_{ij}associated to the repaired missing elements. a*

_{ij}is proposed to be penalized by the number of missing judgments in the original AHP matrix M* = {m

_{ij}}. The greater the number of missing elements, the less information is available to predict {l*

_{ij}, m*

_{ij}, u*

_{ij}}, and consequently the less reliable these estimates will be. The following formula is proposed for determining the missing truth membership function parameters t*

_{ij}:

_{ij}is zero if no elements are known (i.e., n* = n·(n − 1)/2), and equals to their average credibility if no element is unknown (i.e., n* = 0).

_{ij}of a missing entry is expected to be related to the certainty that the decision maker expresses when making comparisons that involve criteria i and j:

_{ij}for the remaining entries shall be obtained following Equation (14). At last, the falsity membership function parameter f

_{ij}depends on the inconsistencies revealed by the expert’s judgments when filling the comparison matrix. The consistency of the responses shall only be evaluated from a complete matrix. Thus, it is proposed to determine the average expert’s incoherency f

_{ij}from the synthetically completed matrix M’ including the central values m

_{ij}. The inconsistency f*

_{ij}associated to the estimated missing values shall then be calculated as:

_{ij}, Equation (31) has been slightly modified so that when no elements are known (i.e., n* = n·(n − 1)/2), the falsity parameter f*

_{ij}becomes 1, while if every entry is known, f*

_{ij}= f

_{ij}∀i,j ∈ {1,...,n}.

## 3. Problem Definition

## 4. Results

#### 4.1. Scalar Weights Derived from the Baseline Complete Comparison Matrix

#### 4.2. Completion Results

_{i}is the scalar baseline weight of criterion i, and w*

_{i,n}is each of the 1000 weight estimates for criterion i for the incompleteness scenario under evaluation. Here, RMSE obtained for each criterion is normalized by the value of the corresponding baseline criterion. Figure 3 shows the normalized RMSE (N-RMSE) derived for each criterion and for each of the four analyzed scenarios. It can be observed that the mean N-RMSE, falling close to 11% when 22% of the required initial judgments are missing, increases with the number of missing entries. Such an increase is in accordance with the increasing dispersion of the estimates detected for scenarios 3 and 4.

#### 4.3. Comparison of the Results Considering a Different Expert

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A

C1 | C2 | C3 | C4 | C5 | C6 | C7 | C8 | C9 | |
---|---|---|---|---|---|---|---|---|---|

C1 | 1 | 1/3 | 1/6 | 1/6 | 1/6 | 1/3 | 1/4 | 1/4 | 1/4 |

C2 | 3 | 1 | 1/2 | 1/5 | 1/4 | 1/2 | 3 | 1/2 | 1/2 |

C3 | 6 | 2 | 1 | 1 | 1/2 | 3 | 7 | 6 | 6 |

C4 | 6 | 5 | 1 | 1 | 1/2 | 3 | 7 | 6 | 6 |

C5 | 6 | 4 | 2 | 2 | 1 | 2 | 6 | 5 | 5 |

C6 | 3 | 2 | 1/3 | 1/3 | 1/2 | 1 | 2 | 1/2 | 1/2 |

C7 | 4 | 1/3 | 1/7 | 1/7 | 1/6 | 1/2 | 1 | 1/2 | 1/2 |

C8 | 4 | 2 | 1/6 | 1/6 | 1/5 | 2 | 2 | 1 | 1 |

C9 | 4 | 2 | 1/6 | 1/6 | 1/5 | 2 | 2 | 1 | 1 |

C1 | C2 | C3 | C4 | C5 | C6 | C7 | C8 | C9 | |
---|---|---|---|---|---|---|---|---|---|

C1 | 1 | 0.8 | 0.5 | 0.7 | 0.9 | 0.8 | 0.8 | 0.6 | 0.2 |

C2 | 0.8 | 1 | 0.7 | 0.8 | 0.6 | 0.6 | 0.4 | 0.7 | 0.7 |

C3 | 1/2 | 2/3 | 1 | 0.8 | 0.8 | 0.7 | 0.8 | 0.7 | 0.6 |

C4 | 0.7 | 0.8 | 0.8 | 1 | 0.8 | 0.3 | 0.4 | 0.5 | 0.6 |

C5 | 0.9 | 0.6 | 0.8 | 0.8 | 1 | 0.2 | 0.7 | 0.6 | 0.4 |

C6 | 0.8 | 0.6 | 0.7 | 0.3 | 0.2 | 1 | 0.8 | 0.8 | 0.4 |

C7 | 0.8 | 0.4 | 0.8 | 0.4 | 0.7 | 0.8 | 1 | 0.5 | 0.5 |

C8 | 0.6 | 0.7 | 0.7 | 0.5 | 0.6 | 0.8 | 0.5 | 1 | 0.4 |

C9 | 0.2 | 0.7 | 0.6 | 0.6 | 0.4 | 0.4 | 0.5 | 0.4 | 1 |

Expert’s Profile Defining Parameters | Value |
---|---|

Knowledge degree in design of infrastructures | 0.60 |

Expertise in economic assessments | 0.60 |

Expertise in environmental assessments | 1.00 |

Expertise in social life-cycle assessments | 0.80 |

Expert’s credibility. Truth membership parameter t | 0.80 |

Expressed mean self confidence | 0.66 |

Expert’s certainty. Indeterminacy membership parameter i | 0.34 |

Consistency ratio of the comparison matrix | 0.07 |

Expert’s inconsistency. Falsehood membership parameter f | 0.72 |

Scenario | C1 | C2 | C3 | C4 | C5 | C6 | C7 | C8 | C9 |
---|---|---|---|---|---|---|---|---|---|

Expert 1 | 0.049 | 0.019 | 0.179 | 0.262 | 0.168 | 0.024 | 0.046 | 0.152 | 0.101 |

Expert 2 | 0.027 | 0.060 | 0.204 | 0.214 | 0.237 | 0.076 | 0.048 | 0.074 | 0.061 |

## References

- Worrell, E.; Price, L.; Martin, N.; Hendriks, C.; Ozawa, L. Carbon dioxide emissions from the global cement industry. Annu. Rev. Energ. Environ.
**2001**, 26, 303–329. [Google Scholar] [CrossRef] - García, J.; Yepes, V.; Martí, J.V. A hybrid k-means cuckoo search algorithm applied to the counterfort retaining walls problem. Mathematics
**2020**, 8, 555. [Google Scholar] [CrossRef] - Penadés-Plà, V.; García-Segura, T.; Yepes, V. Robust design optimization for low-cost concrete box-girder bridge. Mathematics
**2020**, 8, 398. [Google Scholar] [CrossRef] [Green Version] - Kim, S.; Frangopol, D.M. Multi-objective probabilistic optimum monitoring planning considering fatigue damage detection, maintenance, reliability, service life and cost. Struct. Multidisc. Optim.
**2018**, 57, 39–54. [Google Scholar] [CrossRef] - García-Segura, T.; Yepes, V.; Frangopol, D.M. Multi-objective design of post-tensioned concrete road bridges using artificial neural networks. Struct. Multidiscip. Optim.
**2017**, 56, 139–150. [Google Scholar] [CrossRef] - Van den Heede, P.; De Belie, N. A service life based global warming potential for high-volume fly ash concrete exposed to carbonation. Constr. Build. Mater.
**2014**, 55, 183–193. [Google Scholar] [CrossRef] - García, J.; Martí, J.V.; Yepes, V. The Buttressed walls problem: An application of a hybrid clustering particle swarm optimization algorithm. Mathematics
**2020**, 8, 862. [Google Scholar] [CrossRef] - García-Segura, T.; Penadés-Plà, V.; Yepes, V. Sustainable bridge design by metamodel-assisted multi-objective optimization and decision-making under uncertainty. J. Clean. Prod.
**2018**, 202, 904–915. [Google Scholar] [CrossRef] - Gursel, A.P.; Ostertag, C. Comparative life-cycle impact assessment of concrete manufacturing in Singapore. Int. J. Life Cycle Assess.
**2017**, 22, 237–255. [Google Scholar] [CrossRef] - Penadés-Plà, V.; Martí, J.V.; García-Segura, T.; Yepes, V. Life-cycle assessment: A comparison between two optimal post-tensioned concrete box-girder road bridges. Sustainability
**2017**, 9, 1864. [Google Scholar] [CrossRef] [Green Version] - Navarro, I.J.; Yepes, V.; Martí, J.V. Social life cycle assessment of concrete bridge decks exposed to aggressive environments. Environ. Impact Assess.
**2018**, 72, 50–63. [Google Scholar] [CrossRef] - Sierra, L.A.; Pellicer, E.; Yepes, V. Method for estimating the social sustainability of infrastructure projects. Environ. Impact Assess.
**2017**, 65, 41–53. [Google Scholar] [CrossRef] [Green Version] - Navarro, I.J.; Yepes, V.; Martí, J.V. Sustainability assessment of concrete bridge deck designs in coastal environments using neutrosophic criteria weights. Struct. Infrastruct. Eng.
**2020**, 16, 949–967. [Google Scholar] [CrossRef] - Saaty, T.L. The Analytic Hierarchy Process; McGraw-Hill: New York, NY, USA, 1980. [Google Scholar]
- Tavana, M.; Shaabani, A.; Javier Santos-Arteaga, F.; Raeesi Vanani, I. A review of uncertain decision-making methods in energy management using text mining and data analytics. Energies
**2020**, 13, 3947. [Google Scholar] [CrossRef] - Yannis, G.; Kopsacheili, A.; Dragomanovits, A.; Petraki, V. State-of-the-art review on multi-criteria decision-making in the transport sector. J. Traffic Transp. Eng.
**2020**, 7, 413–431. [Google Scholar] [CrossRef] - Navarro, I.J.; Penadés-Plà, V.; Martínez-Muñoz, D.; Rempling, R.; Yepes, V. Life cycle sustainability assessment for multi-criteria decision making in bridge design: A review. J. Civ. Eng. Manag.
**2020**, 26, 690–704. [Google Scholar] [CrossRef] - Radwan, N.; Senousy, M.; Riad, A. Neutrosophic AHP multi-criteria decision making method applied on the selection of learning management system. Int. J. Adv. Comp. Technol.
**2016**, 8, 95–105. [Google Scholar] - Hedelin, B. Complexity is no excuse. Sustain. Sci.
**2019**, 14, 733–749. [Google Scholar] [CrossRef] [Green Version] - Zadeh, L. Outline of a new approach to the analysis of complex systems and decision processes. IEEE Trans. Sys. Man. Cyb.
**1973**, 3, 28–44. [Google Scholar] [CrossRef] [Green Version] - Zadeh, L. Fuzzy sets. Inform. Control
**1965**, 8, 338–353. [Google Scholar] [CrossRef] [Green Version] - Milošević, D.M.; Milošević, M.R.; Simjanović, D.J. Implementation of adjusted fuzzy AHP method in the assessment for reuse of industrial buildings. Mathematics
**2020**, 8, 1697. [Google Scholar] [CrossRef] - Lin, C.-N. A fuzzy analytic hierarchy process-based analysis of the dynamic sustainable management index in leisure agriculture. Sustainability
**2020**, 12, 5395. [Google Scholar] [CrossRef] - Salehi, S.; Ghazizadeh, M.J.; Tabesh, M.; Valadi, S.; Nia, S.P. A risk component-based model to determine pipes renewal strategies in water distribution networks. Struct. Infrastruct. Eng.
**2019**. [Google Scholar] [CrossRef] - Smarandache, F. A Unifying Field in Logics, Neutrosophy: Neutrosophic Probability, Set and Logic; American Research Press: Rehoboth, DE, USA, 1999. [Google Scholar]
- Liu, P.; Liu, X. The neutrosophic number generalized weighted power averaging operator and its application in multiple attribute group decision making. Int. J. Mach. Learn. Cyb.
**2018**, 9, 347–358. [Google Scholar] [CrossRef] [Green Version] - Peng, J.J.; Wang, J.; Yang, W. A multi-valued neutrosophic qualitative flexible approach based on likelihood for multi-criteria decision-making problems. Int. J. Syst. Sci.
**2017**, 48, 425–435. [Google Scholar] [CrossRef] - Saaty, T.; Ozdemir, M. Why the magic number seven plus or minus two. Math. Comput. Model.
**2003**, 38, 233–244. [Google Scholar] [CrossRef] - Harker, P.T. Incomplete pairwise comparisons in the analytic hierarchy process. Math. Mod.
**1987**, 9, 837–848. [Google Scholar] [CrossRef] [Green Version] - Chen, K.; Kou, G.; Tarn, J.M.; Song, Y. Bridging the gap between missing and inconsistent values in eliciting preference from pairwise comparison matrices. Ann. Oper. Res.
**2015**, 235, 155–175. [Google Scholar] [CrossRef] - Bozóki, S.; Fülöp, J.; Rónyai, L. On optimal completion of incomplete pairwise comparison matrices. Math. Comput. Model.
**2010**, 52, 318–333. [Google Scholar] [CrossRef] [Green Version] - Dong, M.; Li, S.; Zhang, H. Approaches to group decision making with incomplete information based on power geometric operators and triangular fuzzy AHP. Expert Syst. Appl.
**2015**, 42, 7846–7857. [Google Scholar] [CrossRef] - Zhou, X.; Hu, Y.; Deng, Y.; Deng, Y.; Chan, F.T.; Ishizaka, A. A DEMATEL-based completion method for incomplete pairwise comparison matrix in AHP. Ann. Oper. Res.
**2018**, 271, 1045–1066. [Google Scholar] [CrossRef] [Green Version] - Sumathi, I.R.; Antony Crispin Sweety, C. New approach on differential equation via trapezoidal neutrosophic number. Complex Intell. Syst.
**2019**, 5, 417–424. [Google Scholar] [CrossRef] [Green Version] - Deli, I.; Subas, Y. A ranking method of single valued neutrosophic numbers and its applications to multi-attribute decision making problems. Int. J. Mach. Learn. Cyb.
**2017**, 8, 1309–1322. [Google Scholar] [CrossRef] [Green Version] - Ye, J. Subtraction and Division Operations of simplified neutrosophic sets. Information
**2017**, 8, 51. [Google Scholar] [CrossRef] - Liang, R.; Wang, J.; Zhang, H. A multi-criteria decision-making method based on single-valued trapezoidal neutrosophic preference relations with complete weight information. Neural Comput. App.
**2018**, 30, 3383–3398. [Google Scholar] [CrossRef] [Green Version] - Sodenkamp, M.A.; Tavana, M.; Di Caprio, D. An aggregation method for solving group multi-criteria decision-making problems with single-valued neutrosophic sets. Appl. Soft Comput.
**2018**, 71, 715–727. [Google Scholar] [CrossRef] [Green Version] - Sierra, L.A.; Pellicer, E.; Yepes, V. Social sustainability in the life cycle of Chilean public infrastructure. J. Constr. Eng. Manag.
**2016**, 142, 05015020. [Google Scholar] [CrossRef] [Green Version] - Abdel-Basset, M.; Manogaran, G.; Mohamed, M.; Chilamkurti, N. Three-way decisions based on neutrosophic sets and AHP-QFD framework for supplier selection problem. Future Gener. Comput. Syst.
**2018**, 89, 19–30. [Google Scholar] [CrossRef] - Dubois, D. The role of fuzzy sets in decision sciences: Old techniques and new directions. Fuzzy Set. Syst.
**2011**, 184, 3–28. [Google Scholar] [CrossRef] [Green Version] - Buckley, J.J. Fuzzy hierarchical analysis. Fuzzy Set. Syst.
**1985**, 17, 233–247. [Google Scholar] [CrossRef] - Wang, Y.M.; Elhag, T.M. On the normalization of interval and fuzzy weights. Fuzzy Set. Syst.
**2006**, 157, 2456–2471. [Google Scholar] [CrossRef] - Enea, M.; Piazza, T. Project selection by constrained fuzzy AHP. Fuzzy Optim. Decis. Mak.
**2004**, 3, 39–62. [Google Scholar] [CrossRef] - Chu, T.; Tao, C. Ranking fuzzy numbers with an area between the centroid point and original point. Comput. Math. Appl.
**2002**, 43, 111–117. [Google Scholar] [CrossRef] [Green Version] - Gabus, A.; Fontela, E. World Problems, an Invitation to Further Tought within the Framework of Dematel; Battelle Geneva Research Centre: Geneva, Switzerland, 1972. [Google Scholar]
- Goedkoop, M.; Heijungs, R.; Huijbregts, M.; De Schryver, A.; Struijs, J.; Van Zelm, R. ReCiPe 2008: A Life Cycle Impact Assessment Method Which Comprises Harmonised Category Indicators at the Midpoint and the Endpoint Level; Ministerie van Volkshuisvesting: The Hague, The Netherlands, 2009. [Google Scholar]
- UNEP/SETAC. Guidelines for Social Life Cycle Assessment of Products; UNEP/SETAC Life-Cycle Initiative: Paris, France, 2009. [Google Scholar]

**Figure 1.**Neutrosophic membership functions associated to the criteria weights derived from the baseline complete AHP comparison matrix.

**Figure 2.**Dispersion of the results for scenarios 1 (3 random entries of the baseline matrix missing) and 4 (12 random entries of the baseline matrix missing).

**Figure 3.**Normalized RMSE resulting for the weight estimation of each criterion for different incompleteness degrees: scenario 1 (3 random entries of the baseline matrix missing), scenario 2 (5 random entries missing), scenario 3 (8 random entries missing), and scenario 4 (12 random entries missing).

**Table 1.**Steps on Saaty’s extended scale [8].

Certainty c_{ij} | Steps ∆(c_{ij}) on Saaty’s Extended Scale |
---|---|

c_{ij} = 1 | 0 |

0.8 ≤ c_{ij} < 1 | 1 |

0.6 ≤ c_{ij} < 0.8 | 2 |

0.4 ≤ c_{ij} < 0.6 | 3 |

0.2 ≤ c_{ij} < 0.4 | 4 |

0 ≤ c_{ij} < 0.2 | 5 |

c_{ij} = 0 | 6 |

**Table 2.**Steps on Saaty’s extended scale [8].

Number of criteria n | 3 | 4 | 5 | 6 | 7 | 8 | 9 |

Random Index RI | 0.58 | 0.90 | 1.12 | 1.24 | 1.32 | 1.41 | 1.45 |

**Table 3.**Sustainability criteria relevant for bridge design in coastal regions [13].

Sustainability Dimension | Criterion | Comments |
---|---|---|

Economy | C1—Construction costs | Costs associated to materials production and installation, machinery, and workers |

C2—Maintenance and demolition costs | Costs associated to materials production and installation, machinery, and workers involved in maintenance activities | |

Environment | C3—Damage on human health | Emission of pollutants causing respiratory diseases, carcinogenics |

C4—Damage on ecosystems | Land occupation, emission of pollutants | |

C5—Resource depletion | Depletion of natural resources resulting from extraction activities | |

Society | C6—Employment generation | Accounts for gender equity, unemployment, safety, and fair salary |

C7—Development of local economies | Economic investments derived from material production activities and machinery rental | |

C8—Impacts on infrastructure users | Accessibility and drivers’ safety | |

C9—Impacts on local communities | Public opinion considering aesthetics and disturbances |

C1 | C2 | C3 | C4 | C5 | C6 | C7 | C8 | C9 | |
---|---|---|---|---|---|---|---|---|---|

C1 | 1 | 6 | 1/5 | 1/6 | 1/5 | 4 | 1 | 1/4 | 1/3 |

C2 | 1/6 | 1 | 1/7 | 1/7 | 1/6 | 1/3 | 1/3 | 1/6 | 1/7 |

C3 | 5 | 7 | 1 | 1/2 | 1 | 5 | 5 | 2 | 2 |

C4 | 6 | 7 | 2 | 1 | 1 | 7 | 6 | 2 | 5 |

C5 | 5 | 6 | 1 | 1 | 1 | 5 | 4 | 1 | 1 |

C6 | 1/4 | 3 | 1/5 | 1/7 | 1/5 | 1 | 1/4 | 1/6 | 1/6 |

C7 | 1 | 3 | 1/5 | 1/6 | 1/4 | 4 | 1 | 1/5 | 1/3 |

C8 | 4 | 6 | 1/2 | 1/2 | 1 | 6 | 5 | 1 | 2 |

C9 | 3 | 7 | 1/2 | 1/5 | 1 | 6 | 3 | 1/2 | 1 |

C1 | C2 | C3 | C4 | C5 | C6 | C7 | C8 | C9 | |
---|---|---|---|---|---|---|---|---|---|

C1 | 1 | 0.80 | 0.80 | 0.90 | 0.90 | 0.60 | 0.60 | 0.70 | 0.20 |

C2 | 0.8 | 1 | 0.80 | 0.90 | 0.90 | 0.70 | 0.70 | 0.80 | 0.20 |

C3 | 0.8 | 0.8 | 1 | 0.70 | 0.80 | 0.70 | 0.50 | 0.50 | 0.50 |

C4 | 0.9 | 0.9 | 0.7 | 1 | 0.90 | 0.80 | 0.80 | 0.90 | 0.90 |

C5 | 0.9 | 0.9 | 0.8 | 0.9 | 1 | 0.70 | 0.70 | 0.70 | 0.90 |

C6 | 0.6 | 0.7 | 0.7 | 0.8 | 0.7 | 1 | 0.80 | 0.80 | 0.80 |

C7 | 0.6 | 0.7 | 0.5 | 0.8 | 0.7 | 0.8 | 1 | 0.50 | 0.50 |

C8 | 0.7 | 0.8 | 0.5 | 0.9 | 0.7 | 0.8 | 0.5 | 1 | 0.60 |

C9 | 0.2 | 0.2 | 0.5 | 0.9 | 0.9 | 0.8 | 0.5 | 0.6 | 1 |

Expert’s Profile Defining Parameters | Value |
---|---|

Knowledge degree in design of infrastructures | 1.00 |

Expertise in economic assessments | 0.60 |

Expertise in environmental assessments | 0.80 |

Expertise in social life-cycle assessments | 0.40 |

Expert’s credibility. Truth membership parameter t | 0.76 |

Expressed mean self confidence | 0.74 |

Expert’s certainty. Indeterminacy membership parameter i | 0.26 |

Consistency ratio of the comparison matrix | 0.06 |

Expert’s inconsistency. Falsehood membership parameter f | 0.59 |

Scenario | C1 | C2 | C3 | C4 | C5 | C6 | C7 | C8 | C9 |
---|---|---|---|---|---|---|---|---|---|

Baseline | 0.049 | 0.019 | 0.179 | 0.262 | 0.168 | 0.024 | 0.046 | 0.152 | 0.101 |

3 missing | 0.051 | 0.019 | 0.170 | 0.267 | 0.177 | 0.026 | 0.048 | 0.155 | 0.088 |

5 missing | 0.052 | 0.019 | 0.169 | 0.263 | 0.178 | 0.026 | 0.048 | 0.155 | 0.091 |

8 missing | 0.053 | 0.018 | 0.168 | 0.257 | 0.178 | 0.026 | 0.049 | 0.156 | 0.096 |

12 missing | 0.053 | 0.018 | 0.164 | 0.249 | 0.178 | 0.027 | 0.051 | 0.159 | 0.101 |

Max. deviation from baseline | 8.4% | 8.7% | 8.1% | 4.9% | 6.3% | 10.2% | 10.5% | 4.5% | 0.1% |

Scenario | C1 | C2 | C3 | C4 | C5 | C6 | C7 | C8 | C9 |
---|---|---|---|---|---|---|---|---|---|

3 missing | 5.25% | 7.41% | 3.26% | 4.22% | 3.76% | 6.62% | 3.29% | 3.22% | 6.75% |

5 missing | 7.11% | 9.92% | 5.58% | 6.64% | 5.26% | 8.54% | 5.31% | 4.88% | 8.29% |

8 missing | 10.45% | 14.63% | 9.39% | 9.89% | 7.93% | 13.14% | 8.63% | 8.23% | 9.07% |

12 missing | 17.22% | 23.00% | 15.83% | 15.57% | 12.60% | 19.44% | 15.81% | 12.95% | 13.87% |

Scenario | C1 | C2 | C3 | C4 | C5 | C6 | C7 | C8 | C9 |
---|---|---|---|---|---|---|---|---|---|

Baseline | 0.027 | 0.060 | 0.204 | 0.214 | 0.237 | 0.076 | 0.048 | 0.074 | 0.061 |

3 missing | 0.025 | 0.071 | 0.225 | 0.210 | 0.230 | 0.074 | 0.043 | 0.070 | 0.052 |

5 missing | 0.025 | 0.070 | 0.220 | 0.209 | 0.233 | 0.074 | 0.043 | 0.071 | 0.055 |

8 missing | 0.025 | 0.068 | 0.214 | 0.206 | 0.238 | 0.074 | 0.043 | 0.072 | 0.059 |

12 missing | 0.024 | 0.065 | 0.209 | 0.202 | 0.242 | 0.073 | 0.045 | 0.076 | 0.065 |

Max. deviation from baseline | 10.1% | 6.9% | 2.6% | 5.8% | 2.3% | 3.1% | 6.6% | 2.6% | 6.4% |

Max. deviation observed for expert 1 | 8.4% | 8.7% | 8.1% | 4.9% | 6.3% | 10.2% | 10.5% | 4.5% | 0.1% |

**Table 10.**Second expert’s relative standard deviation of the criteria weights obtained for each scenario.

Scenario | C1 | C2 | C3 | C4 | C5 | C6 | C7 | C8 | C9 |
---|---|---|---|---|---|---|---|---|---|

3 missing | 7.34% | 7.08% | 5.86% | 5.43% | 6.61% | 5.34% | 6.35% | 6.90% | 9.06% |

5 missing | 9.55% | 9.49% | 8.14% | 7.53% | 8.26% | 8.07% | 8.73% | 8.35% | 10.21% |

8 missing | 13.56% | 13.85% | 11.85% | 10.86% | 10.31% | 12.67% | 12.88% | 11.07% | 11.48% |

12 missing | 21.73% | 22.79% | 17.63% | 17.54% | 14.69% | 21.64% | 19.19% | 15.62% | 15.21% |

**Table 11.**Second expert’s normalized RMSE resulting in the weight estimation of each criterion for different incompleteness degrees.

Scenario | 3 | 5 | 8 | 12 |
---|---|---|---|---|

Expert 1 | 0.072 | 0.087 | 0.117 | 0.180 |

Expert 2 | 0.105 | 0.112 | 0.133 | 0.191 |

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## Share and Cite

**MDPI and ACS Style**

Navarro, I.J.; Martí, J.V.; Yepes, V.
Neutrosophic Completion Technique for Incomplete Higher-Order AHP Comparison Matrices. *Mathematics* **2021**, *9*, 496.
https://doi.org/10.3390/math9050496

**AMA Style**

Navarro IJ, Martí JV, Yepes V.
Neutrosophic Completion Technique for Incomplete Higher-Order AHP Comparison Matrices. *Mathematics*. 2021; 9(5):496.
https://doi.org/10.3390/math9050496

**Chicago/Turabian Style**

Navarro, Ignacio J., José V. Martí, and Víctor Yepes.
2021. "Neutrosophic Completion Technique for Incomplete Higher-Order AHP Comparison Matrices" *Mathematics* 9, no. 5: 496.
https://doi.org/10.3390/math9050496