# Use of the PVM Method Computed in Vector Space of Increments in Decision Aiding Related to Urban Development

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Literature

## 3. Methodological Framework

^{th}variant, with reference to the variant ${A}_{k}$. In the columns there are successive k

^{th}decision variants.

^{th}decision variant, whereas ${\overline{x}}_{j}\left(l\right)$ is an element of that vector, while N is the number of decision variants.

## 4. Decision Problem

- C1—spatial order: a criterion on the basis of which the projects were evaluated as to their impact on sorting out the urban space and on how well the constituents of the space are harmonized;
- C2—modernization (revitalization): this criterion accounts for the impact that projects would exert on improving aesthetic assets of the town, estate, quarter, or street to which projects apply, by increasing their value in use and advantageous transformations in the area included in the project;
- C3—environmental and nature protection: a criterion that is used to evaluate projects regarding their impact on nature and the environment, surrounding greenery, inland water, and fauna management in the area concerned;
- C4—sport and tourism: a criterion used in evaluation of the impact of projects on the physical wellbeing of those who live in the area concerned, and to improve the attractiveness of the environs in the eye of tourists in the area;
- C5—culture: a criterion which is used to evaluate the projects in the sense of what bearing they have on spiritual development of people to whom the project may concern.

- A1: construction of a walking path along the river Warta from the East Boulevard to the Lubuski Bridge;
- A2: a swimming pool within the river Warta water current;
- A3: improvement in bicycle urban infrastructure;
- A4: integrative playground for handicapped children;
- A5: pro-eco revitalization of Słowiański Park.

## 5. Results

## 6. Discussion

#### 6.1. Global Sensitivity and Uncertainty Analyses

#### 6.2. Fuzzy AHP Calculations

#### 6.3. Fuzzy TOPSIS Calculations

#### 6.4. NEAT F-PROMETHEE Calculations

#### 6.5. Comparison of Obtained Ranks

_{1}and A

_{4}. The PVM-VSI ranking overlaps with the other variants in terms of the first (A

_{3}), second (A

_{5}), and last (A

_{2}) positions. One of the reasons for this similarity may be the fact that the PVM-VSI method contains elements that are methodically similar to the other discussed methods, namely:

- Initial stages (the use of pairwise comparison matrices inter alia) are close to AHP;
- The motivating and demotivating vector of preference to a certain degree correspond methodically-wise to the values ${\varphi}^{+}$ and ${\varphi}^{-}$, in the method NEAT F-PROMETHEE;
- On certain stages of the authors’ method, an approach based on vector—similar to that in the TOPSIS method—is used.

## 7. Conclusions

_{3}—improvement of the urban bicycle infrastructure. This project was followed in the rank by: A

_{5}—pro-eco revitalization of Słowiański Park; A

_{1}—construction of a walking path along the river Warta from the East Boulevard to the Lubuski Bridge; A

_{4}—integrative playground for handicapped children; and A

_{2}—a swimming pool within the river Warta water current. Accounting for urban sustainable development, citizens can be introduced to the obtained rank so that they are aware which of the projects is recommended by experts as best serving the interests of their town.

_{3}proved to be the best irrespective of the used method. Also, irrespective of the applied method, the final position was taken by project A

_{2}. The proposed PVM-VSI method—as demonstrated—has extra merit. It is possible to analyze the consistency of the solution on the grounds of the standard deviation value. The decision maker has the possibility of pointing out those variants that are well defined in the rank and characterized by having the lowest deviation or the smallest variability or inconsistency of the evaluation scores. This method can play a significant role in the decision-making of problems of urban management, or more broadly, sustainable management. In such problems, we often have to deal with imprecise and subjective assessments, expressed on qualitative scales. Meanwhile, the PVM-VSI method we developed takes into account such imprecise and inconsistent assessments, at the same time examining the degree of imprecision and taking it into account in the final results. This is a completely different approach to, for example, the AHP method, in which if there is an inconsistency ratio greater than 0.1 for the examined pairwise comparison matrix, the decision maker must reassess the variants.

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A. Pairwise Comparison Matrix Split into Criteria

A_{1} | A_{2} | A_{3} | A_{4} | A_{5} | |
---|---|---|---|---|---|

A_{1} | 1 | 4 | 1/3 | 1 | 1/2 |

A_{2} | 1/4 | 1 | 1/3 | 1/3 | 1/2 |

A_{3} | 3 | 3 | 1 | 1 | 2 |

A_{4} | 1 | 3 | 1 | 1 | 2 |

A_{5} | 2 | 2 | 1/2 | 1/2 | 1 |

A_{1} | A_{2} | A_{3} | A_{4} | A_{5} | |
---|---|---|---|---|---|

A_{1} | 1 | 3 | 2 | 1 | 1/3 |

A_{2} | 1/3 | 1 | 1/3 | 1/2 | 1/4 |

A_{3} | 1/2 | 3 | 1 | 1/2 | 1/2 |

A_{4} | 1 | 2 | 2 | 1 | 1 |

A_{5} | 3 | 4 | 2 | 1 | 1 |

**Table A3.**Pairwise comparison matrix for variants according to criterion C

_{3}—environmental and nature protection.

A_{1} | A_{2} | A_{3} | A_{4} | A_{5} | |
---|---|---|---|---|---|

A_{1} | 1 | 2 | 3 | 3 | 1/4 |

A_{2} | 1/2 | 1 | 2 | 2 | 1/4 |

A_{3} | 1/3 | 1/2 | 1 | 4 | 1 |

A_{4} | 1/3 | 1/2 | 1/4 | 1 | 1/3 |

A_{5} | 4 | 4 | 1 | 3 | 1 |

A_{1} | A_{2} | A_{3} | A_{4} | A_{5} | |
---|---|---|---|---|---|

A_{1} | 1 | 1/2 | 1/3 | 1 | 3 |

A_{2} | 2 | 1 | 1/3 | 2 | 3 |

A_{3} | 3 | 3 | 1 | 3 | 4 |

A_{4} | 1 | 1/2 | 1/3 | 1 | 2 |

A_{5} | 1/3 | 1/3 | 1/4 | 1/2 | 1 |

A_{1} | A_{2} | A_{3} | A_{4} | A_{5} | |
---|---|---|---|---|---|

A_{1} | 1 | 1 | 2 | 1/3 | 1/3 |

A_{2} | 1 | 1 | 1 | 1/2 | 1/3 |

A_{3} | 1/2 | 1 | 1 | 2 | 1/2 |

A_{4} | 3 | 2 | 1/2 | 1 | 3 |

A_{5} | 3 | 3 | 2 | 1/3 | 1 |

## Appendix B. Symbols Used Throughout the Paper

^{th}criterion

^{th}decisive variant

^{th}coordinate of the vector $\overrightarrow{{X}_{j}}$

^{th}decision variant

^{th}decision variant

^{th}decision variant

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**Figure 3.**Uncertainty analysis. Output: cumulative shift from the baseline ranking for five variants.

**Figure 4.**A five-stage scale of linguistic evaluations and the respective fuzzy values [15].

**Table 1.**Use of Multi-criteria decision analysis methods in decision-making urban development problems.

No. | Decision-Making Problem | Location | Applied Method(s) | Number of Criteria (Subcriteria) | Number of Variants | Criteria | Reference |
---|---|---|---|---|---|---|---|

1 | Selection of end use of urban lands | Teheran (Iran) | Fuzzy TOPSIS, GIS | 5 | 12 | So, Ec | [20] |

2 | Selection of public parking place | Teheran (Iran) | AHP, GIS | 5 (5) | 13 | So, Ec, Te | [35] |

3 | Selection of a new housing for a firefighting station | Bolzano (Italy) | AHP (cw), ELECTRE (pa) | 5 | 6 | So, Ec, Ei, Sp, Te | [24] |

4 | Site selection for a wind farm | Szczecin (Poland) | AHP (cw), PROMETHEE (pa) | 10 | 4 | So, Ec, Ei, Te | [28] |

5 | Selection of a landfill | Al-Kufa (Iraq) | AHP, GIS | 11 | 6 | So, Ei, Sp, Te | [25] |

6 | Selection of a shopping center site | Istanbul (Turkey) | Fuzzy AHP (cw), Fuzzy TOPSIS (pa) | 8 | 6 | So, Ec, Ei, Sp | [29] |

7 | Selection of the site of an urban logistic center to improve urban sustainable development | Unidentified | AHP (cw), Fuzzy TOPSIS (pa) | 4 (16) | 4 | So, Ec, Ei, Te | [11] |

8 | Selection of a permanent site of a healthcare waste disposal facility | Garhwal (India) | Fuzzy AHP (cw), Fuzzy TOPSIS (pa) | 8 | 7 | Ec, Ei, Sp, Te | [26] |

9 | Selection of an urban distribution center | Unidentified | Fuzzy TOPSIS | 11 | 3 | So, Ec, Ei, Sp, Te | [33] |

10 | Selection of the concept of urban logistic system | Belgrad (Serbia) | Fuzzy ANP + Fuzzy DEMATEL (cw), Fuzzy VIKOR (pa) | 10 | 4 | So, Ec, Ei, Sp, Te | [30] |

11 | Selection of sustainable transport systems | La Rochelle (France) | Fuzzy TOPSIS | 24 | 3 | So, Ec, Ei, Sp, Te | [31] |

12 | Planning the range of spatial zoning in an urban planning scenario | Queensland (Australia) | AHP, Fuzzy AHP | 23 (36) | 4 | So, Sp, Te | [21] |

13 | Selection of a sustainable development of transport systems | Taipei City (Taiwan) | AHP, Fuzzy Cognitive Maps | 10 | 4 | So, Ei, Te | [32] |

14 | Choice of municipal police building construction plans | Taipei City (Taiwan) | Fuzzy AHP | 6 (20) | 5 | Sp, Te | [9] |

15 | Selection of an optimal site to erect a hospital building in an urban space | Teheran (Iran) | Fuzzy AHP, GIS | 5 | 5 | So, Ec, Ei, Sp, Te | [15] |

16 | Selection of a site to construct an urban distribution center | Unidentified | Fuzzy TOPSIS, THOWA | 13 | 4 | So, Ec, Ei, Sp, Te | [34] |

17 | Selection of a site to construct a wastewater treatment plan of the river Anyangcheon | Seul (South Korea) | Fuzzy TOPSIS | 10 | 10 | So, Sp, Te | [27] |

18 | Ranking of projects in a participatory budget | Poznań (Poland) | Fuzzy TOPSIS | 4 | 24 | So, Ec | [10] |

19 | Selection of the site to place new emergency services | Teheran (Iran) | Fuzzy AHP (cw), Fuzzy TOPSIS (pa) | 4 | 22 | So, Sp, Te | [23] |

20 | Selection of the region to locate hospitals or joint-venture healthcare institutions in China | Surrounding Bohai Bay, (China) | Fuzzy AHP | 6 (19) | 4 | Ec, Po | [22] |

A_{1} | A_{2} | A_{3} | A_{4} | A_{5} | |
---|---|---|---|---|---|

A_{1} | 1 | 4 | 1/3 | 1 | 1/2 |

A_{2} | 1/4 | 1 | 1/3 | 1/3 | 1/2 |

A_{3} | 3 | 3 | 1 | 1 | 2 |

A_{4} | 1 | 3 | 1 | 1 | 2 |

A_{5} | 2 | 2 | 1/2 | 1/2 | 1 |

A_{1} | A_{2} | A_{3} | A_{4} | A_{5} | |
---|---|---|---|---|---|

A_{1} | 4 | 0.3333 | 1 | 0.5 | |

A_{1} (A_{2}) | 4 | 1.3333 | 1.3333 | 2 | |

A_{1} (A_{3}) | 1 | 0.3333 | 0.3333 | 0.6667 | |

A_{1} (A_{4}) | 3 | 1 | 1 | 2 | |

A_{1} (A_{5}) | 1 | 0.25 | 0.25 | 0.5 | |

A_{2} (A_{1}) | 0.25 | 0.0833 | 0.25 | 0.125 | |

A_{2} | 0.25 | 0.3333 | 0.3333 | 0.5 | |

A_{2} (A_{3}) | 1 | 0.3333 | 0.3333 | 0.6667 | |

A_{2} (A_{4}) | 0.3333 | 0.3333 | 0.3333 | 0.6667 | |

A_{2} (A_{5}) | 1 | 0.25 | 0.25 | 0.5 | |

A_{3} (A_{1}) | 3 | 12 | 3 | 1.5 | |

A_{3} (A_{2}) | 0.75 | 3 | 1 | 1.5 | |

A_{3} | 3 | 3 | 1 | 2 | |

A_{3} (A_{4}) | 1 | 3 | 1 | 2 | |

A_{3} (A_{5}) | 4 | 4 | 1 | 2 | |

A_{4} (A_{1}) | 1 | 4 | 0.3333 | 0.5 | |

A_{4} (A_{2}) | 0.75 | 3 | 1 | 1.5 | |

A_{4} (A_{3}) | 3 | 3 | 1 | 2 | |

A_{4} | 1 | 3 | 1 | 2 | |

A_{4} (A_{5}) | 4 | 4 | 1 | 2 | |

A_{5} (A_{1}) | 2 | 8 | 0.6667 | 2 | |

A_{5} (A_{2}) | 0.5 | 2 | 0.6667 | 0.6667 | |

A_{5} (A_{3}) | 1.5 | 1.5 | 0.5 | 0.5 | |

A_{5} (A_{4}) | 0.5 | 1.5 | 0.5 | 0.5 | |

A_{5} | 2 | 2 | 0.5 | 0.5 |

A_{1} | A_{2} | A_{3} | A_{4} | A_{5} | |
---|---|---|---|---|---|

C_{1} | 1.5417 | 3.5 | 0.5875 | 0.8292 | 1.2563 |

C_{2} | 1.1042 | 3.3917 | 1.625 | 0.9042 | 0.5472 |

C_{3} | 1.1625 | 1.8396 | 2.1417 | 4 | 0.6479 |

C_{4} | 1.5792 | 0.9417 | 0.3611 | 1.6875 | 3.5 |

C_{5} | 2.1375 | 2.0917 | 1.9458 | 1.0361 | 1.0708 |

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

C_{1} | 1.4288 | 0.4811 | 0.5127 | −0.1366 | 0.6047 |

C_{2} | 0.4811 | 0.7451 | 0.8011 | −0.4659 | 1.0190 |

C_{3} | 0.5127 | 0.8011 | 1.6302 | −0.6122 | 0.7037 |

C_{4} | −0.1366 | −0.4659 | −0.6122 | 1.7413 | −0.5255 |

C_{5} | 0.6047 | 1.0190 | 0.7037 | −0.5255 | 4.2967 |

$\overrightarrow{\mathsf{\Psi}}$ | 2.0313 | 2.0667 | 2.6063 | 2.1406 | 2.1031 |

$\overrightarrow{\mathsf{\Phi}}$ | 0.7688 | 0.8149 | 1.0339 | 0.7965 | 1.0622 |

$\overrightarrow{V}$ | 1.2625 | 1.2517 | 1.5724 | 1.3441 | 1.041 |

$\overrightarrow{{V}^{\prime}}$ | 0.4324 | 0.4288 | 0.5386 | 0.4604 | 0.3566 |

M1 | M2 | M3 | M4 | M5 | |
---|---|---|---|---|---|

M1 | 0.4324 | 0.4288 | 0.5386 | 0.4604 | 0.3566 |

M2 | 0 | 0 | 0 | 0 | 0 |

M3 | 0 | 0 | 0 | 0 | 0 |

M4 | 0 | 0 | 0 | 0 | 0 |

M5 | 0 | 0 | 0 | 0 | 0 |

**Table 11.**Evaluations, values of variances, and standard deviation values obtained by individual decision variants.

${\mathit{u}}_{\mathit{j}}$ | var | σ | Rank | |
---|---|---|---|---|

A_{1} | 3.2554 | 2.6335 | 1.6228 | 3 |

A_{2} | 5.1378 | 4.7668 | 2.1833 | 5 |

A_{3} | 2.9643 | 5.3510 | 2.3132 | 1 |

A_{4} | 4.0469 | 4.4545 | 2.1106 | 4 |

A_{5} | 3.1200 | 1.6044 | 1.2666 | 2 |

S_{i} | S_{Ti} | S_{Ti}-S_{i} | |
---|---|---|---|

C_{1} | 0.1075 | 0.1145 | 0.0069 |

C_{2} | 0.0762 | 0.0822 | 0.0061 |

C_{3} | 0.5522 | 0.5570 | 0.0048 |

C_{4} | 0.1236 | 0.1279 | 0.0043 |

C_{5} | 0.1407 | 0.1455 | 0.0047 |

A_{1} | A_{2} | A_{3} | A_{4} | A_{5} | |
---|---|---|---|---|---|

C_{1} | 1.4288 | 6.0250 | 0.1156 | 0.4427 | 0.4893 |

C_{2} | 0.7451 | 3.8480 | 1.4385 | 0.4481 | 0.0956 |

C_{3} | 1.6302 | 3.4000 | 8.3126 | 10.9188 | 0.5336 |

C_{4} | 1.7413 | 0.6181 | 0.0389 | 1.6582 | 5.1917 |

C_{5} | 4.2967 | 3.5069 | 3.4891 | 1.3708 | 1.9189 |

A_{1} | A_{2} | A_{3} | A_{4} | A_{5} | |
---|---|---|---|---|---|

C_{1} | 0.0770 | 0.4820 | 0.0010 | 0.0010 | 0.0070 |

C_{2} | 0.0210 | 0.1960 | 0.0260 | 0.0010 | 0.0000 |

C_{3} | 0.1000 | 0.1530 | 0.8290 | 0.9590 | 0.0090 |

C_{4} | 0.1140 | 0.0050 | 0.0010 | 0.0210 | 0.8650 |

C_{5} | 0.6900 | 0.1630 | 0.1480 | 0.0140 | 0.1170 |

A_{1} | A_{2} | A_{3} | A_{4} | A_{5} | |
---|---|---|---|---|---|

C_{1} | −0.0070 | 0.0000 | −0.0020 | 0.0010 | 0.0060 |

C_{2} | −0.0070 | 0.0010 | −0.0020 | 0.0010 | 0.0060 |

C_{3} | −0.0070 | 0.0010 | −0.0010 | 0.0010 | 0.0060 |

C_{4} | −0.0040 | 0.0000 | −0.0020 | 0.0010 | 0.0030 |

C_{5} | −0.0030 | 0.0000 | −0.0020 | 0.0000 | 0.0040 |

**Table 16.**Mapping crisp values onto fuzzy ones on the grounds of linguistic evaluations [15].

Crisp Value | Linguistic Value | Fuzzy Value | Inverted Fuzzy Value |
---|---|---|---|

1 | Equally important | (1,1,2) | (1/2,1,1) |

2 | Intermittent value | (1,2,3) | (1/3,1/2,1) |

3 | Weakly important | (2,3,4) | (1/4,1/3,1/2) |

4 | Intermittent value | (3,4,5) | (1/5,1/4,1/3) |

5 | Fairly important | (4,5,6) | (1/6,1/5,1/4) |

6 | Intermittent value | (5,6,7) | (1/7,1/6,1/5) |

7 | Very important | (6,7,8) | (1/8,1/7,1/6) |

8 | Intermittent value | (7,8,9) | (1/9,1/8,1/7) |

9 | Absolutely important | (8,9,9) | (1/9,1/9,1/8) |

A_{1} | A_{2} | A_{3} | A_{4} | A_{5} | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

A_{1} | 1 | 1 | 1 | 3 | 4 | 5 | 1/4 | 1/3 | 1/2 | 1 | 1 | 2 | 1/3 | 1/2 | 1 |

A_{2} | 1/5 | 1/4 | 1/3 | 1 | 1 | 1 | 1/4 | 1/3 | 1/2 | 1/4 | 1/3 | 1/2 | 1/3 | 1/2 | 1 |

A_{3} | 2 | 3 | 4 | 2 | 3 | 4 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 2 | 3 |

A_{4} | 1 | 1 | 2 | 2 | 3 | 4 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 2 | 3 |

A_{5} | 1 | 2 | 3 | 1 | 2 | 3 | 1/3 | 1/2 | 1 | 1/3 | 1/2 | 1 | 1 | 1 | 1 |

Spatial Order | Modernization | Environmental and Nature Protection | Sport and Tourism | Culture | |
---|---|---|---|---|---|

A_{1} | [0.1028 0.1790 0.3510] | [0.1050 0.2066 0.4187] | [0.1150 0.2290 0.4152] | [0.0873 0.1485 0.3043] | [0.0688 0.1390 0.2981] |

A_{2} | [0.0407 0.0774 0.1528] | [0.0402 0.0771 0.1461] | [0.0672 0.1448 0.2839] | [0.1051 0.2212 0.4180] | [0.0716 0.1162 0.2855] |

A_{3} | [0.1552 0.3109 0.5886] | [0.0737 0.1480 0.3060] | [0.1150 0.1830 0.3565] | [0.2337 0.4248 0.7203] | [0.0851 0.1829 0.4083] |

A_{4} | [0.1359 0.2557 0.5264] | [0.1243 0.2345 0.4934] | [0.0399 0.0691 0.1288] | [0.0755 0.1331 0.2827] | [0.1471 0.3001 0.5518] |

A_{5} | [0.0769 0.1769 0.3630] | [0.1714 0.3338 0.6078] | [0.2204 0.3741 0.6247] | [0.0434 0.0725 0.1385] | [0.1236 0.2618 0.4856] |

**Table 19.**Final fuzzy evaluation scores of individual variants, defuzzified final evaluation score, and position in ranking, established via Fuzzy AHP.

W_{l} | W_{m} | W_{u} | Scale Vector | Ranking | |
---|---|---|---|---|---|

A_{1} | 0.0958 | 0.1804 | 0.3575 | 0.1833 | 4 |

A_{2} | 0.0650 | 0.1273 | 0.2572 | 0.1301 | 5 |

A_{3} | 0.1325 | 0.2499 | 0.4759 | 0.2484 | 1 |

A_{4} | 0.1046 | 0.1985 | 0.3966 | 0.2024 | 3 |

A_{5} | 0.1271 | 0.2438 | 0.4439 | 0.2358 | 2 |

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

A_{1} | [0.1747 0.3041 0.3041 0.5964] | [0.1727 0.3399 0.3399 0.6889] | [0.1841 0.3665 0.3665 0.6646] | [0.1211 0.2061 0.2061 0.4224] | [0.1247 0.2518 0.2518 0.5402] |

A_{2} | [0.0692 0.1314 0.1314 0.2595] | [0.0661 0.1268 0.1268 0.2403] | [0.1076 0.2318 0.2318 0.4544] | [0.1458 0.3070 0.3070 0.5803] | [0.1297 0.2106 0.2106 0.5173] |

A_{3} | [0.2637 0.5282 0.5282 1] | [0.1213 0.2435 0.2435 0.5034] | [0.1840 0.2929 0.2929 0.5707] | [0.3244 0.5896 0.5896 1] | [0.1542 0.3314 0.3314 0.7400] |

A_{4} | [0.2308 0.4344 0.4344 0.8944] | [0.2044 0.3857 0.3857 0.8117] | [0.0639 0.1106 0.1106 0.2061] | [0.1048 0.1847 0.1847 0.3924] | [0.2666 0.5439 0.5439 1] |

A_{5} | [0.1307 0.3006 0.3006 0.6167] | [0.2819 0.5492 0.5492 1] | [0.3528 0.5988 0.5988 1] | [0.0602 0.1006 0.1006 0.1922] | [0.2239 0.4745 0.4745 0.8800] |

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

FPIS | [1 1 1 1] | [1 1 1 1] | [1 1 1 1] | [1 1 1 1] | [1 1 1 1] |

FNIS | [0.0692 0.0692 0.0692 0.0692] | [0.0661 0.0661 0.0661 0.0661] | [0.0639 0.0639 0.0639 0.0639] | [0.0602 0.0602 0.0602 0.0602] | [0.1247 0.1247 0.1247 0.1247] |

A_{1} | A_{2} | A_{3} | A_{4} | A_{5} | |
---|---|---|---|---|---|

${\mathit{d}}_{\mathit{i}}^{+}$ | 3.4376 | 3.9019 | 3.0050 | 3.3112 | 3.0245 |

${\mathit{d}}_{\mathit{i}}^{-}$ | 1.4972 | 0.9532 | 2.0920 | 1.7359 | 2.0539 |

CC_{i} | 0.3034 | 0.1963 | 0.4104 | 0.3439 | 0.4044 |

Rank | 4 | 5 | 1 | 3 | 2 |

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

Preference function | V-shape | V-shape | V-shape | V-shape | V-shape |

q | 0 | 0 | 0 | 0 | 0 |

p | 0.3 | 0.3 | 0.3 | 0.3 | 0.3 |

A_{1} | A_{2} | A_{3} | A_{4} | A_{5} | |
---|---|---|---|---|---|

A_{1} | [0 0 0 0] | [0 0.1540 0.2253 0.8838] | [0 0 0.0697 0.7196] | [0 0.1065 0.1168 0.7928] | [0 0.0506 0.0520 0.7677] |

A_{2} | [0 0 0.0484 0.5176] | [0 0 0 0] | [0 0 0 0.4173] | [0 0.0504 0.1091 0.4806] | [0 0.0990 0.0990 0.4008] |

A_{3} | [0 0.1841 0.4292 0.8949] | [0.0016 0.2914 0.5171 0.9700] | [0 0 0 0] | [0 0.2703 0.3127 0.8952] | [0.0634 0.2000 0.4000 0.7703] |

A_{4} | [0 0.1074 0.2697 0.7394] | [0 0.3464 0.6000 0.7594] | [0 0 0.1357 0.6418] | [0 0 0 0] | [0 0 0.1184 0.7595] |

A_{5} | [0 0 0.4819 0.8075] | [0.0168 0.3903 0.5634 0.8222] | [0 0.2512 0.4526 0.7385] | [0.0611 0.2000 0.2662 0.7933] | [0 0 0 0] |

A_{1} | A_{2} | A_{3} | A_{4} | A_{5} | |
---|---|---|---|---|---|

${\mathit{\varphi}}^{+}$ | [0 0.0778 0.1159 0.7910] | [0 0.0373 0.0641 0.4541] | [0.0162 0.2364 0.4147 0.8826] | [0 0.1134 0.2809 0.7250] | [0.0194 0.2103 0.4410 0.7904] |

${\mathit{\varphi}}^{-}$ | [0 0.0729 0.3073 0.7399] | [0.0046 0.2955 0.4765 0.8588] | [0 0.0628 0.1645 0.6293] | [0.0152 0.1568 0.2012 0.7405] | [0.0158 0.0874 0.1673 0.6746] |

${\mathit{\varphi}}_{\mathit{net}}$ | [−0.7399 −0.2295 0.0430 0.7910] | [−0.8588 −0.4391 −0.2313 0.4494] | [−0.6130 0.0719 0.3519 0.8826] | [−0.7405 −0.0877 0.1241 0.7098] | [−0.6551 0.0430 0.3536 0.7745] |

A_{1} | A_{2} | A_{3} | A_{4} | A_{5} | |
---|---|---|---|---|---|

${\mathit{\varphi}}^{+}$ | 0.2914 | 0.1650 | 0.4011 | 0.2971 | 0.3725 |

${\mathit{\varphi}}^{-}$ | 0.2956 | 0.4139 | 0.2383 | 0.3078 | 0.2648 |

${\mathit{\varphi}}_{\mathit{net}}$ | -0.0200 | -0.2542 | 0.1646 | -0.0028 | 0.1142 |

Rank | 4 | 5 | 1 | 3 | 2 |

Method | Rank |
---|---|

PVM-VSI | ${A}_{3}\succ {A}_{5}\succ {A}_{1}\succ {A}_{4}\succ {A}_{2}$ |

Fuzzy AHP | ${A}_{3}\succ {A}_{5}\succ {A}_{4}\succ {A}_{1}\succ {A}_{2}$ |

Fuzzy TOPSIS | ${A}_{3}\succ {A}_{5}\succ {A}_{4}\succ {A}_{1}\succ {A}_{2}$ |

NEAT F-PROMETHEE I | ${A}_{3}\succ {A}_{5}\succ {A}_{1}\approx {A}_{4}\succ {A}_{2}$ |

NEAT F-PROMETHEE II | ${A}_{3}\succ {A}_{5}\succ {A}_{4}\succ {A}_{1}\succ {A}_{2}$ |

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## Share and Cite

**MDPI and ACS Style**

Kannchen, M.; Ziemba, P.; Borawski, M.
Use of the PVM Method Computed in Vector Space of Increments in Decision Aiding Related to Urban Development. *Symmetry* **2019**, *11*, 446.
https://doi.org/10.3390/sym11040446

**AMA Style**

Kannchen M, Ziemba P, Borawski M.
Use of the PVM Method Computed in Vector Space of Increments in Decision Aiding Related to Urban Development. *Symmetry*. 2019; 11(4):446.
https://doi.org/10.3390/sym11040446

**Chicago/Turabian Style**

Kannchen, Marek, Paweł Ziemba, and Mariusz Borawski.
2019. "Use of the PVM Method Computed in Vector Space of Increments in Decision Aiding Related to Urban Development" *Symmetry* 11, no. 4: 446.
https://doi.org/10.3390/sym11040446