# A New Model for Determining Weight Coefficients of Criteria in MCDM Models: Full Consistency Method (FUCOM)

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Review of Applications of Subjective MCDM Methods in Different Studies

## 3. Full Consistency Method (FUCOM)

_{j}represents the criteria ($j=1,2,\dots ,n$) and A represents the set of the alternatives a

_{i}($i=1,2,\dots ,m$). The values ${f}_{ij}$ of each considered criterion ${f}_{j}$ for each considered alternative ${a}_{i}$ are known, namely ${f}_{ij}={f}_{j}\left({a}_{i}\right),\forall \left(i,j\right);i=1,2,\dots ,m;j=1,2,\dots ,n$. The relation shows that each value of the attribute depends on the jth criterion and the ith alternative.

_{ij}shows the relative preference of criterion i to criterion j) are not based on accurate measurements, but rather on subjective estimates. There is also a deviation of the values ${a}_{ij}$ from the ideal ratios ${w}_{i}/{w}_{j}$ (where ${w}_{i}$ and ${w}_{j}$ represents criteria weights of criterion i and criterion j). If, for example, it is determined that A is of much greater significance than B, B of greater importance than C, and C of greater importance than A, there is inconsistency in problem solving and the reliability of the results decreases. This is especially true when there are a large number of the pairwise comparisons of criteria. FUCOM reduces the possibility of errors in a comparison to the least possible extent due to: (1) a small number of comparisons ($n-1$) and (2) the constraints defined when calculating the optimal values of criteria. FUCOM provides the ability to validate the model by calculating the error value for the obtained weight vectors by determining DFC. On the other hand, in the other models for determining the weights of criteria (the BWM, the AHP models), the redundancy of the pairwise comparison appears, which makes them less vulnerable to errors in judgment, while the FUCOM methodological procedure eliminates this problem.

_{2}> C

_{1}> C

_{3}is being subjected to consideration. Suppose that the scale ${\varpi}_{{C}_{j(k)}}\in [1,9]$ is used to determine the priorities of the criteria and that, based on the decision-maker’s preferences, the following priorities of the criteria ${\varpi}_{{C}_{2}}=1$, ${\varpi}_{{C}_{1}}=3.5$ and ${\varpi}_{{C}_{3}}=6$ are obtained. On the basis of the obtained priorities of the criteria and condition $\frac{{w}_{k}}{{w}_{k+1}}={\phi}_{k/(k+1)}$ we obtain following calculations $\frac{{w}_{2}}{{w}_{1}}=\frac{3.5}{1}$ i.e., ${w}_{2}=3.5\cdot {w}_{1}$, $\frac{{w}_{1}}{{w}_{3}}=\frac{6}{3.5}$ i.e., ${w}_{1}=1.714\cdot {w}_{3}$. In that way, the following comparative priorities are calculated: ${\phi}_{{C}_{2}/{C}_{1}}=3.5/1=3.5$ and ${\phi}_{{C}_{1}/{C}_{3}}=6/3.5=1.714$ (expression (2)).

**Example**

**1.**

_{1}), the manufacturer’s warranty (C

_{2}), the service network (C

_{3}), and the maximum load capacity (C

_{4}). As previously described, FUCOM was implemented through the following steps:

_{1}> C

_{2}> C

_{3}> C

_{4}.

**Example**

**2.**

_{1}), Price (C

_{2}), Comfort (C

_{3}), Safety Level (C

_{4}), and Interior (C

_{5}) were considered.

_{2}> C

_{1}> C

_{4}> C

_{3}> C

_{5}.

_{2}criterion. The comparison was based on the scale $[1,9]$. Thus, the priorities of the criteria (${\varpi}_{{C}_{j(k)}}$) for all of the criteria ranked in Step 1 were obtained (Table 2).

_{i}(i = 1, 2, …, m) of the alternative i, can be obtained using different methods. However, FUCOM can be successfully transformed into a classic multi-criteria model by adding the expression (6) that is presented in the next section. The values of the weight coefficients of the criteria obtained by FUCOM and which meet the condition that ${w}_{j}\ge 0$ and ${\sum}_{j=1}^{n}{w}_{j}}=1$ can also be used to determine the finite values of the criterion functions applying the expression (6)

## 4. Discussion and Comparisons

**Example**

**3.**

_{3}, C

_{5}and C

_{7})) were used. In this example, there are three most influential criteria, so we choose the comparisons made for one of the three criteria. Based on the comparisons for the C

_{7}criterion in Table 4 (the data from the seventh row of Table 4), the criteria are possible to rank in the following manner: C

_{3}= C

_{5}= C

_{7}> C

_{1}> C

_{2}= C

_{4}= C

_{6}= C

_{8}and the priorities of the criteria can be determined (Table 5).

**Example**

**4.**

_{1}), the price (C

_{2}), comfort (C

_{3}), safety (C

_{4}) and the style (C

_{5}). By using the BWM method, the Best–to-Others (BO) and the Others–to-Worst (OW) vectors are obtained [118], ${A}_{B}={(2,1,4,2,8)}^{T}$ and ${A}_{W}={(4,8,2,4,1)}^{T}$. By solving the BWM, the optimal values of the weight coefficients are obtained:

_{AHP}= 0.029, CR

_{BWM}= 0.000).

_{2}> C

_{1}= C

_{4}> C

_{3}> C

_{5}, with only n − 1 comparison, and the priorities of the criteria can be determined (Table 8).

_{ij}shows the relative preference of criterion i to criterion j and a

_{jk}shows the relative preference of criterion j to criterion k). If the pairs are compared, the obtained relation reads ${a}_{13}=3$ and ${a}_{34}=6$; then, in order to meet the condition of transitivity, ${a}_{14}$ should have the value ${a}_{14}=18$. However, since the scale ${a}_{ij}\in [1,9]$ is applied in both models, in the largest number of the cases of the pairwise comparison, ${a}_{14}$ obtains the maximum value from the scale; i.e., ${a}_{14}=9$. From the above example, it is noted that BWM and AHP models in the case of pairwise comparison deliberately allow certain deviations and ignore total transitivity. However, the deviation from transitivity results in a decrease in the consistency of the model, which further affects the reliability of the results. On the other hand, FUCOM always strives for the maximum consistency of results, which is one of the key conditions in a rational judgment. Meeting the conditions of consistency affects the reliability of results; i.e., the optimality of weight coefficients.

_{ij}= 2, it means that the ratio $\frac{{w}_{i}}{{w}_{j}}=2$. This ratio in the BWM is violated only when the degree of consistency is different from zero; i.e., when the condition of the optimality of weight coefficients is violated. In FUCOM, as shown in Example 1, this ratio solely depends on the decision-maker’s preference since the preference is not defined on the basis of a predefined scale, but on the basis of the subjective assessment instead.

**Example**

**5.**

_{4}. For the criterion C

_{4}, the optimal value of the weight coefficient ${w}_{4}=0.1509$ (obtained by FUCOM) is not covered by the interval of the criterion C

_{4}(${w}_{4}^{\ast}=[0.1563,0.1602],{w}_{4}^{\ast}(\mathrm{center})=0.1582$) defined by the BWM. For the remaining criteria, the optimum values of the weight coefficients are within the defined intervals, but they deviate from the central parts of the interval, which are recommended in the BWM as the optimal values of weight coefficients.

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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Criteria | C_{1} | C_{2} | C_{3} | C_{4} |
---|---|---|---|---|

${\phi}_{k/(k+1)}$ | 1.00 | 1.08 | 1.25 | 1.45 |

Criteria | C_{2} | C_{1} | C_{4} | C_{3} | C_{5} |
---|---|---|---|---|---|

${\varpi}_{{C}_{j(k)}}$ | 1 | 2.1 | 3 | 3 | 7 |

**Table 3.**The required number of comparisons in Analytic Hierarchy Process (AHP), Best Worst Method (BWM) and the Full Consistency Method (FUCOM).

MCDM Method | The Number of Criteria (n) and the Required Number of Pairwise Comparisons | |||||||
---|---|---|---|---|---|---|---|---|

n = 2 | n = 3 | n = 4 | n = 5 | n = 6 | n = 7 | n = 8 | n = 9 | |

AHP (n(n − 1)/2) | 1 | 3 | 6 | 10 | 15 | 21 | 28 | 36 |

BWM (2n − 3) | 1 | 3 | 5 | 7 | 9 | 11 | 13 | 15 |

FUCOM (n − 1) | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |

**Table 4.**The pairwise comparison of eight criteria in the AHP model [141].

Criteria | C_{1} | C_{2} | C_{3} | C_{4} | C_{5} | C_{6} | C_{7} | C_{8} | w_{j} |
---|---|---|---|---|---|---|---|---|---|

C_{1} | 1 | 2 | 1/2 | 2 | 1/2 | 2 | 1/2 | 2 | 0.1111 |

C_{2} | 1/2 | 1 | 4 | 1 | 1/4 | 1 | 1/4 | 1 | 0.0556 |

C_{3} | 2 | 1/4 | 1 | 4 | 1 | 4 | 1 | 4 | 0.2222 |

C_{4} | 1/2 | 1 | 1/4 | 1 | 1/4 | 1 | 1/4 | 1 | 0.0556 |

C_{5} | 2 | 4 | 1 | 4 | 1 | 4 | 1 | 4 | 0.2222 |

C_{6} | 1/2 | 1 | 1/4 | 1 | 1/4 | 1 | 1/4 | 1 | 0.0556 |

C_{7} | 2 | 4 | 1 | 4 | 1 | 4 | 1 | 4 | 0.2222 |

C_{8} | 1/2 | 1 | 1/4 | 1 | 1/4 | 1 | 1/4 | 1 | 0.0556 |

CR = 0.000 |

Criteria | C_{3} | C_{5} | C_{7} | C_{1} | C_{2} | C_{4} | C_{6} | C_{8} |
---|---|---|---|---|---|---|---|---|

${\varpi}_{{C}_{j(k)}}$ | 1 | 1 | 1 | 2 | 4 | 4 | 4 | 4 |

Criteria | AHP (w_{j}) | BWM (w_{j}) | FUCOM (w_{j}) |
---|---|---|---|

C_{1} | 0.1111 | 0.1111 | 0.1111 |

C_{2} | 0.0556 | 0.0556 | 0.0556 |

C_{3} | 0.2222 | 0.2222 | 0.2222 |

C_{4} | 0.0556 | 0.0556 | 0.0556 |

C_{5} | 0.2222 | 0.2222 | 0.2222 |

C_{6} | 0.0556 | 0.0556 | 0.0556 |

C_{7} | 0.2222 | 0.2222 | 0.2222 |

C_{8} | 0.0556 | 0.0556 | 0.0556 |

CR | 0.000 | 0.000 | 0.000 |

Criteria | C_{1} | C_{2} | C_{3} | C_{4} | C_{5} | w_{j} |
---|---|---|---|---|---|---|

C_{1} | 1.000 | 0.333 | 3.000 | 1.000 | 5.000 | 0.2017 |

C_{2} | 3.000 | 1.000 | 5.000 | 3.000 | 7.000 | 0.4641 |

C_{3} | 0.333 | 0.200 | 1.000 | 0.333 | 3.000 | 0.0888 |

C_{4} | 1.000 | 0.333 | 3.000 | 1.000 | 5.000 | 0.2017 |

C_{5} | 0.200 | 0.143 | 0.333 | 0.200 | 1.000 | 0.0436 |

Criteria | C_{2} | C_{1} | C_{4} | C_{3} | C_{5} |
---|---|---|---|---|---|

${\varpi}_{{C}_{j(k)}}$ | 1 | 2 | 2 | 4 | 8 |

Criteria | AHP (w_{j}) | BWM (w_{j}) | FUCOM (w_{j}) |
---|---|---|---|

C_{1} | 0.2017 | 0.2105 | 0.2105 |

C_{2} | 0.4641 | 0.4211 | 0.4211 |

C_{3} | 0.0888 | 0.1053 | 0.1053 |

C_{4} | 0.2017 | 0.2105 | 0.2105 |

C_{5} | 0.0436 | 0.0526 | 0.0526 |

CR | 0.029 | 0.000 | 0.000 |

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**MDPI and ACS Style**

Pamučar, D.; Stević, Ž.; Sremac, S.
A New Model for Determining Weight Coefficients of Criteria in MCDM Models: Full Consistency Method (FUCOM). *Symmetry* **2018**, *10*, 393.
https://doi.org/10.3390/sym10090393

**AMA Style**

Pamučar D, Stević Ž, Sremac S.
A New Model for Determining Weight Coefficients of Criteria in MCDM Models: Full Consistency Method (FUCOM). *Symmetry*. 2018; 10(9):393.
https://doi.org/10.3390/sym10090393

**Chicago/Turabian Style**

Pamučar, Dragan, Željko Stević, and Siniša Sremac.
2018. "A New Model for Determining Weight Coefficients of Criteria in MCDM Models: Full Consistency Method (FUCOM)" *Symmetry* 10, no. 9: 393.
https://doi.org/10.3390/sym10090393