# Pareto Optimal Decisions in Multi-Criteria Decision Making Explained with Construction Cost Cases

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Description of Selected Methods

#### 2.1.1. TOPSIS

- m—the number of available solutions,
- $n$—number of considered criteria,
- ${x}_{ij}$—value of the $i$-th variant according to $j$-th criterion,
- $X$—data matrix [${x}_{ij}]$,
- $Q$—vector of criteria weights $\left[{q}_{1},{q}_{2},\dots ,{q}_{n}\right]$.

- Positive Ideal Solution (PIS), marked as ${A}^{+}$, defined as:$${A}^{+}=\left\{{v}_{1}^{+},{v}_{2}^{+},\dots ,,{v}_{n}^{+}\right\}$$$${v}_{j}^{+}=\{\begin{array}{c}ma{x}_{i}{v}_{ij},j\in I\\ mi{n}_{i}{v}_{ij},j\in J\end{array}$$
- Negative Ideal Solution (NIS), marked as ${A}^{-}$, defined as:$${A}^{-}=\left\{{v}_{1}^{-},{v}_{2}^{-},\dots ,,{v}_{n}^{-}\right\}$$$${v}_{j}^{-}=\{\begin{array}{c}ma{x}_{i}{v}_{ij},j\in J\\ mi{n}_{i}{v}_{ij},j\in I\end{array}$$

#### 2.1.2. AHP

- Step I—hierarchy of the problem
- Step II—definition of preferences by the decision-maker
- Step III—preference matrix consistency testing
- Step IV—creating a summary ranking

_{ij}illustrate the preferences of the decision-maker. The digits 1 are on the diagonal of the matrix A, there is also the reciprocal of the adopted preferences, i.e.,

- ${\lambda}_{max}$—the maximum eigenvalue of the matrix,

- $P$—final score for a given solution variant,
- ${w}_{i}$—criterion weight according to the Formula (12),
- ${k}_{i}$—evaluation of a given criterion.

#### 2.1.3. FUCOM

#### 2.1.4. MARCOS

_{i}.

_{i}).

#### 2.2. Description of Selected Problems to Solve

#### 2.2.1. Choosing the Masonry Wall Material

#### 2.2.2. Choosing the Facade System

^{2}, execution time, aesthetics in visual reception, ease of access to the material, and durability.

## 3. Results

#### 3.1. The Sequences of Solutions for Masonry Wall Materials

#### 3.2. The Sequences of Solutions for Facade System

#### 3.3. Results Summarized

## 4. Discussion

^{2}may significantly influence the financial result of the decision-maker employer.

- It protects the decision-maker from choosing the variant which has a lower value, but its cost is higher
- Every next variant, if considered, brings an increase of the value
- Economic analysis (based on the cost of the analysed variant) can be made on each intermediate stage (not only on the final one)
- As a result of these economic analyses, the decision-maker can stop the procedure at every analysed variant, i.e., he can choose the variant providing fair technical properties paying a fair price for it. The choice of technically the best solution is not compulsory.

- The producers usually price their products based on the properties of the products, but also based on the competitors’ prices, the state of the market, etc. Accepting the price into MCDM analysis means the influence of the market issues on the prepared assessment. Considering the price after MCDM analysis (i.e., applying proposed CCAF), makes the choices more explainable. Features of the product are transformed into the ranking based on the decision maker’s preferences (through the MCDM method), and then, the decision-maker’s price sensitivity is matched to the market price of the products (stages 3 to 6).
- In the traditional MCDM, the variant assessed as the 2nd (or with lower rank) is presented as a worse one (also because of its price). In economic decisions (as the decision based on cost is) the ratio: value (received) to cost (spent on it), is very important. The proposed method is based on this concept
- All advantages of MCDM methods are kept, as MCDM is a part of the 6-stage method (jointly named as MCDM-CCAF).
- Price changes (caused by market processes or achieved in negotiations) can be easily adjusted in the scatterplot (without the necessity of repeating MCDM analysis, as other than price criteria e.g., sound insulation or number of block per m
^{2}are stable).

- Higher analytical effort (if compared to a solely applied, pure MCDM method).
- Only one cost criterion can be considered (if there are more, they should be combined into one).

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 3.**(

**a**) solid bricks, (

**b**) cellular concrete blocks, (

**c**) ceramic blocks, (

**d**) silicate blocks [own photos].

**Figure 4.**(

**a**) ETICS system, (

**b**) clinker cladding, (

**c**) natural stone, (

**d**) cement panels finishing [own photos].

**Figure 5.**Criteria weights obtained in the AHP method in the variant of calculations taking into account the cost criterion.

**Figure 6.**Criteria weights obtained in the AHP method in the variant of calculations without consideration of the cost criterion.

**Figure 8.**Criteria weights obtained in the AHP method in the variant of calculations taking into account the cost criterion.

**Figure 9.**Criteria weights obtained in the AHP method in the variant of calculations without considering the cost criterion.

**Figure 10.**Results from the FUCOM-MARCOS method presented as a discrete function of prices (wall structural materials case).

**Figure 11.**Results from FUCOM-MARCOS method with price criterion considered (wall structural materials case).

**Figure 13.**Results from the FUCOM-MARCOS method presented as a discrete function of prices (facade cladding case).

MCDM Method | Reference |
---|---|

AHP | [12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29] |

ANP | [12] |

ARAS | [10,15,30] |

COPRAS | [30] |

DELPHI | [12] |

DEMATEL | [5,12,31,32,33,34] |

EDAS | [30,33,34,35,36] |

ELECTRE | [12,24,26,37] |

ENTROPHY | [20,38] |

FUCOM | [7,39,40] |

MARCOS | [41] |

MOORA | [9,10,30,42,43] |

PROMETHEE | [14,18,24,30,44,45] |

SAW | [18,22,44,46,47,48,49] |

SWARA | [42] |

TOPSIS | [14,16,24,26,27,30,43,44,46,47,50] |

VIKOR | [51] |

WASPAS | [39,42] |

Comparative, Pairwise Assessment of A against B | Value |
---|---|

Just as good or important | 1 |

A little better or more important | 3 |

Definitely better or more important | 5 |

Much better or more important | 7 |

Extremely better or more important | 9 |

Matrix Dimension n | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |

$RI$ | 0 | 0.52 | 0.89 | 1.11 | 1.25 | 1.35 | 1.40 | 1.45 | 1.49 |

Criteria | Solid Bricks | Cellular Concrete Blocks | Ceramic Blocks | Silicate Blocks |
---|---|---|---|---|

Material cost | 90.21 PLN/m^{2} | 71 PLN/m^{2} | 56 PLN/m^{2} | 64 PLN/m^{2} |

Blocks consumption | 93 items/m^{2} | 8.33 items/m^{2} | 10.7 items/m^{2} | 15 items/m^{2} |

Acoustic insulation | 47 dB | 38 dB | 42 dB | 47 dB |

Thermal insulation | 3.03 W/m^{2} K | 0.68 W/m^{2} K | 1.31 W/m^{2} K | 3.22 W/m^{2} K |

Ease of processing | 2 out of 6 | 5 out of 6 | 4 out of 6 | 3 out of 6 |

**Table 5.**Final grades awarded in terms of the criteria taken into consideration by the decision-maker.

Criteria | ETICS | Clinker Cladding | Natural Stone Panel | Fiber-Cement Panel |
---|---|---|---|---|

Cost in PLN/m^{2} (rank) | 200 (4) | 417 (3) | 656 (2) | 1435 (1) |

Execution time | 4 out of 4 | 2 out of 4 | 2 out of 4 | 3 out of 4 |

Visual reception | 1 out of 4 | 3 out of 4 | 4 out of 4 | 2 out of 4 |

Ease of access | 4 out of 4 | 3 out of 4 | 1 out of 4 | 3 out of 4 |

Durability | 1 out of 4 | 3 out of 4 | 4 out of 4 | 3 out of 4 |

Criteria | Solid Bricks | Cellular Concrete Clocks | Ceramic Blocks | Silicate Blocks |
---|---|---|---|---|

Material cost | 1 | 2 | 4 | 3 |

Blocks consumption | 1 | 4 | 3 | 2 |

Acoustic insulation | 4 | 2 | 3 | 4 |

Thermal insulation | 1 | 4 | 2 | 1 |

Ease of processing | 1 | 4 | 3 | 2 |

MCDM Method | Cost Criterion Considered | Solid Bricks | Cellular Concrete Clocks | Ceramic Blocks | Silicate Blocks | |
---|---|---|---|---|---|---|

AHP | Yes | Result | 0.087 | 0.290 | 0.414 | 0.210 |

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

No | Result | 0.113 | 0.462 | 0.278 | 0.148 | |

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

TOPSIS | Yes | Result | 0.058 | 0.501 | 0.787 | 0.544 |

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

No | Result | 0.010 | 0.921 | 0.544 | 0.260 | |

Rank | 4 | 1 | 2 | 3 |

Ai | Si | With the Cost Criterion Considered | |||||

AAI | 0.266 | Ki− | Ki+ | F(K−) | F(K+) | F(Ki) | Rank |

A1 | 0.297 | 1.117 | 0.297 | 0.210 | 0.790 | 0.281 | 4 |

A2 | 0.723 | 2.723 | 0.723 | 0.210 | 0.790 | 0.685 | 2 |

A3 | 0.836 | 3.147 | 0.836 | 0.210 | 0.790 | 0.791 | 1 |

A4 | 0.617 | 2.322 | 0.617 | 0.210 | 0.790 | 0.584 | 3 |

AI | 1.000 | ||||||

Without Consideration of the Cost Criterion | |||||||

AAI | 0.273 | Ki− | Ki+ | F(K−) | F(K+) | F(Ki) | Rank |

A1 | 0.319 | 1.167 | 0.318 | 0.214 | 0.786 | 0.301 | 4 |

A2 | 0.956 | 3.500 | 0.955 | 0.214 | 0.786 | 0.902 | 1 |

A3 | 0.675 | 2.473 | 0.674 | 0.214 | 0.786 | 0.637 | 2 |

A4 | 0.470 | 1.723 | 0.470 | 0.214 | 0.786 | 0.444 | 3 |

AI | 1.000 |

MCDM Method | Cost Criterion Considered | ETICS | Clinker Cladding | Natural Stone Panel | Fiber-Cement Panel | |
---|---|---|---|---|---|---|

AHP | Yes | Result | 3.125 | 2.750 | 2.538 | 2.008 |

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

No | Result | 2.634 | 2.529 | 2.837 | 2.716 | |

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

TOPSIS | Yes | Result | 0.671 | 0.574 | 0.442 | 0.243 |

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

No | Result | 0.448 | 0.446 | 0.552 | 0.481 | |

Rank | 3 | 4 | 1 | 2 |

Ai | Si | With the Cost Criterion Considered | |||||

AAI | 0.270 | Ki− | Ki+ | F(K−) | F(K+) | F(Ki) | Rank |

A1 | 0.782 | 2.896 | 0.783 | 0.213 | 0.787 | 0.740 | 1 |

A2 | 0.576 | 2.133 | 0.577 | 0.213 | 0.787 | 0.545 | 2 |

A3 | 0.553 | 2.048 | 0.554 | 0.213 | 0.787 | 0.524 | 3 |

A4 | 0.462 | 1.709 | 0.462 | 0.213 | 0.787 | 0.437 | 4 |

AI | 0.999 | ||||||

Ai | Si | Without Consideration of the Cost Criterion | |||||

AAI | 0.367 | Ki− | Ki+ | F(K−) | F(K+) | F(Ki) | Rank |

A1 | 0.656 | 1.786 | 0.656 | 0.269 | 0.731 | 0.597 | 3 |

A2 | 0.633 | 1.723 | 0.633 | 0.269 | 0.731 | 0.576 | 4 |

A3 | 0.712 | 1.937 | 0.712 | 0.269 | 0.731 | 0.648 | 1 |

A4 | 0.679 | 1.848 | 0.679 | 0.269 | 0.731 | 0.618 | 2 |

AI | 1.000 |

MCDM Method | Cost Criterion Considered | Solid Bricks | Cellular Concrete Clocks | Ceramic Blocks | Silicate Blocks | |
---|---|---|---|---|---|---|

AHP | Yes | Result | 0.087 | 0.290 | 0.414 | 0.210 |

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

No | Result | 0.113 | 0.462 | 0.278 | 0.148 | |

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

TOPSIS | Yes | Result | 0.058 | 0.501 | 0.787 | 0.544 |

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

No | Result | 0.010 | 0.921 | 0.544 | 0.260 | |

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

FUCOM-MARCOS | Yes | Result | 0.281 | 0.685 | 0.791 | 0.584 |

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

No | Result | 0.301 | 0.902 | 0.637 | 0.444 | |

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

Cost of variants PLN/m^{2} | 90,21 | 71.00 | 56.00 | 64.00 |

MCDM Method | Cost Criterion Considered | ETICS | Clinker Cladding | Natural Stone Panel | Fiber-Cement Panel | |
---|---|---|---|---|---|---|

AHP | Yes | Result | 3.125 | 2.750 | 2.538 | 2.008 |

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

No | Result | 2.634 | 2.529 | 2.837 | 2.716 | |

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

TOPSIS | Yes | Result | 0.671 | 0.574 | 0.442 | 0.243 |

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

No | Result | 0.448 | 0.446 | 0.552 | 0.481 | |

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

FUCOM-MARCOS | Yes | Result | 0.740 | 0.545 | 0.524 | 0.437 |

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

No | Result | 0.597 | 0.576 | 0.648 | 0.618 | |

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

Cost of variants PLN/m^{2} | 200 | 417 | 656 | 1435 |

Label of Variant | A | B | C | D | E | F |
---|---|---|---|---|---|---|

Cost of variant | 100 | 120 | 125 | 140 | 150 | 155 |

Values from MCDM method (cost excluded) | 0.36 | 0.52 | 0.27 | 0.88 | 0.92 | 0.69 |

Ranking (based on values) | 5 | 4 | 6 | 2 | 1 | 3 |

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## Share and Cite

**MDPI and ACS Style**

Anysz, H.; Nicał, A.; Stević, Ž.; Grzegorzewski, M.; Sikora, K. Pareto Optimal Decisions in Multi-Criteria Decision Making Explained with Construction Cost Cases. *Symmetry* **2021**, *13*, 46.
https://doi.org/10.3390/sym13010046

**AMA Style**

Anysz H, Nicał A, Stević Ž, Grzegorzewski M, Sikora K. Pareto Optimal Decisions in Multi-Criteria Decision Making Explained with Construction Cost Cases. *Symmetry*. 2021; 13(1):46.
https://doi.org/10.3390/sym13010046

**Chicago/Turabian Style**

Anysz, Hubert, Aleksander Nicał, Željko Stević, Michał Grzegorzewski, and Karol Sikora. 2021. "Pareto Optimal Decisions in Multi-Criteria Decision Making Explained with Construction Cost Cases" *Symmetry* 13, no. 1: 46.
https://doi.org/10.3390/sym13010046