# Methodology for Optimizing Factors Affecting Road Accidents in Poland

^{*}

## Abstract

**:**

## 1. Characteristics of the Issue

- Good atmospheric conditions are as follows:
- ○
- air temperature > 3 deg;
- ○
- no precipitation;
- ○
- wind < 5.5 m/s;
- ○
- visibility > 10 km;
- ○
- daily differential pressure < 8 hPa.

- Bad weather conditions (if one of the following factors is met) are as follows:
- ○
- slippery pavement (temperature < 3 °C and occurrence of precipitation);
- ○
- driving rain (temperature >0 °C, precipitation > 3 mm);
- ○
- snowstorm (temperature <0 °C, precipitation > 3 mm);
- ○
- strong wind (wind > 10 ms/s);
- ○
- dense fog (visibility < 300 m).

## 2. Multi-Criteria Optimization Model

^{N}

_{1}(x),…, F

_{n}(x),…, F

_{N}(x)) ∈ R

^{N}

_{n}(x)—the value of the n-th quality indicator (n-th criterion function for the solution x ∈ X).

- A—the space of solutions;
- B—the space of solution evaluations;
- F: A ⇒B—a criterion function, assigning to each solution X⊂A its grade Z∈B and assuming that the set of possible solutions A is not empty, a certain subset X (the set of acceptable solutions) can be selected, whereby

_{1},…,x

_{n}}—a set of possible solutions;

^{N}

_{1}(X), f

_{2}(X),),…, f

_{n}(x),…, f

_{N}(x))

^{2},

_{1}(X), f

_{2}(X))

_{1}(X), f

_{2}(X) can have a dominance relation preference structure, i.e., Φ MAX or MIN, respectively, where the dominant relationship Φ has a preference MAX:

_{1}, c

_{2},…, c

_{n}, …, c

_{N}) ∈C × C}

_{1}(x), f

_{2}(x)) ∈ R

^{2}: x∈ X}

_{1}, c

_{2}—points of space C;

_{1}, d

_{2}, …, d

_{n}, …, d

_{N}) ∈D × D}

_{1}(x), f

_{2}(x)) ∈ R

^{2}: x∈ X}

_{1}, d

_{2}—points of space D.

_{1}, F

_{1}, Φ

_{1})

_{1}—the set of admissible solutions defined as

_{1}= {x

_{1,1}, x

_{1,2}, x

_{1,3}, x

_{1,4}}

_{1}—quality indicator defined as F

_{1}: X

_{1}⇒ R

^{2}

_{1}(X

_{1}) = (f

_{1,1}(x), f

_{1,2}(x))

_{1}—dominance relationship of preference, e.g., MAX, MAX.

_{D}

^{Φ1}of the optimization task, find the product of the following sets X

_{1}

^{1}and X

_{1}

^{2}:

_{1}

^{1}= {x

^{*}∈ X

_{1}: f

_{1,1}(x

^{*}) = max f

_{1,1}(x)}; x∈ X

_{1}

_{1}

^{2}= {x

^{*}∈ X

_{1}: f

_{1,2}(x

^{*}) = max f

_{1,2}(x)}; x ∈X

_{1}

_{1,1}(x), f

_{1,2}(x) are defined by appropriate relations, e.g.,

_{1},

_{1}(x) = ej (x) and f

_{1},

_{2}(x) = kj (x)

- maximize the function,

_{1,1}(x) = e

_{j}(x), x ∈X

_{1}; j = 1,…,n

- maximize the function,

_{1,2}(x) = k

_{j}(x), x∈X

_{1}; j = 1,…,n

_{1}

^{1}and X

_{1}

^{2},

_{1}

^{1}= {x∈

^{*}X

_{1}: e

_{j}(x

^{*}) = max e

_{j}(x)} for x ∈X

_{1}

_{1}

^{2}= {x

^{*}∈X

_{1}: k

_{j}(x

^{*}) = max k

_{j}(x)} for x ∈X

_{1}

_{1}

^{1}and X

_{1}

^{2},

_{D}

^{Φ1}= X

_{1}

^{1}∩ X

_{1}

^{2}

_{D}

^{Φ1}is empty, the set of non-dominated solutions X

_{N}

^{Φ1}and the set of compromise solutions X

_{K}

^{Φ1}are determined.

^{*}= (c

_{1}

^{*}, c

_{2}

^{*}):

_{1}

^{*}= max e

_{j}(x); c

_{2}

^{*}= max k

_{j}(x)

x ∈X

_{1;}x ∈X

_{1}

_{1}= {f

_{1,1}, f

_{1,2}}, it follows that for c* the maximum value of e

_{j}is demanded and the maximum value of k

_{j}is demanded.

_{1}

^{*}(x) = {f

_{1,1}

^{*}(x), f

_{1,2}

^{*}(x)}

_{1}

^{max}= max f

_{1,1}(x), c

_{2}

^{max}= max f

_{1,2}(x)

x ∈X

_{1}x∈X

_{1}

^{**}= (c

_{1}

^{**}, c

_{2}

^{**})

_{1}(discreteness) to determine the set of its non-dominated solutions X

_{N}

^{Φ1}and compromise solutions X

_{K}

^{Φ1}, a method is proposed to determine the approximate result (and therefore the solution) of the compromise for the norm |●|, which is a measure of the distance of the results c

^{*}∈C

^{*}from the ideal point c

^{**}[54,55].

^{**}denote the ideal point determined by relation (29) and C

^{*}the known set of normalized results:

^{*}= {c

^{*i}}, i = 1,…,n

^{*i}= (c

_{1}

^{*i}, c

_{2}

^{*i}), whereby

^{o}, which would minimize the calculated values of r

_{i}norms, e.g., x

_{1}

^{o}= x

_{1,3}

_{1}

^{o}= c

^{o}= min r

_{i}

## 3. Optimization of Factors Affecting the Number of Traffic Accidents

_{1}}

_{1}= {x

_{1,1},…,x

_{1,n}}—is a set of atmospheric factors affecting the number of traffic accidents,

_{1}= (x

_{1,1}, x

_{1,2},

_{…}, x

_{1,35})

- x
_{1,1}—good weather; - x
_{1,2}—fog, smoke; - x
_{1,3}—rainfall; - x
_{1,4}—snowfall, hail; - x
_{1,5}—blinding sun; - x
_{1,6}—cloudy; - x
_{1,7}—strong wind;

- x
_{1,8}—Monday; - x
_{1,9}—Tuesday; - x
_{1,10}—Wednesday; - x
_{1,11}—Thursday; - x
_{1,12}—Friday; - x
_{1,13}—Saturday; - x
_{1,14}—Sunday;

- x
_{1,15}—Lower Silesia; - x
_{1,16}—Kujawsko-pomorskie; - x
_{1,17}—Lubelskie; - x
_{1,18}—Lubuskie; - x
_{1,19}—Lodzkie; - x
_{1,20}—Lesser Poland; - x
_{1,21}—Mazovian; - x
_{1,22}—Opolskie; - x
_{1,23}—Subcarpathian; - x
_{1,24}—Podlaskie; - x
_{1,25}—Pomeranian; - x
_{1,26}—Silesian; - x
_{1,27}—Swietokrzyskie; - x
_{1,28}—Warmian-Masurian; - x
_{1,29}—Greater Poland; - x
_{1,30}—Zachodniopomorskie.

- x
_{1,31—}highway; - x
_{1,32}—expressway; - x
_{1,33}–with two one-way roadways; - x
_{1,34}—road—one-way; - x
_{1,35}—two-way, single carriageway.

_{1}, we can define the vector solution quality index F as

_{1}= F

_{1}(X

_{1}) = (f

_{1,1}(X

_{1}), f

_{1,2}(X

_{1}), f

_{1,3}(X

_{1}))

_{1}criterion functions, for optimizing weather conditions affecting the number of traffic accidents in Poland, for example, as [54,55]:

_{1}= (f

_{1,1}, f

_{1,2,}f

_{1,3})

_{1,1}—method of maximum relative change in factor affecting lwd;

_{1,2}—method of maximum change in gradient of factor affecting lwd;

_{1,3}—method of maximum change in factor affecting lwd;

_{max}—maximum lwd value of analyzed measurements;

_{i})—number of traffic accidents over time t

_{i};

_{i+1})—number of traffic accidents over time t

_{i}

_{+1}.

_{1}—criteria dominance relationship vector indicator of solution quality F ([51,57]:

_{1}= {Φ

_{1,1}, Φ

_{1,2}, Φ

_{1,3}}

_{1,1}—dominance relationship in terms of f

_{1,1}(MAX);

_{1,2}—dominance relationship in terms of f

_{1,2}(MAX);

_{1,3}—dominance relationship in terms of f

_{1,3}(MAX).

_{1}, F

_{1}, Φ

_{1}>

- Normalization of criterion space—space D
^{*}

^{*}.

^{*}= {d

^{*i}}, i = 1,…,n; d

^{*i}= (d

_{1}

^{*i}, d

_{2}

^{*i}, d

_{3}

^{*i})

- 2.
- Determination of the coordinates of the ideal point—d
^{**}.

^{**}= (d

_{1}

^{**}, d

_{2}

^{**}, d

_{3}

^{**})

d

_{1}

^{**}= max f

^{*}

_{1,1}(x), d

_{2}

^{**}= max f

^{*}

_{1,2}(x)

d

_{3}

^{**}= max f

^{*}

_{1,3}(x), x ∈ X

_{1}

- 3.
- Calculation of the value of the standard with parameter p = 2—r
_{j}(D^{*}).

^{*}∈ D

^{*}results from the ideal point d

^{**}.

_{1}

^{o}in an optimization task—for example, if x

_{1}

^{o}= x

_{1,4},

_{1}

^{o}= d

^{o}= min r

_{i}; because d

^{o}= min r

_{3}

_{i}∈ X

_{1}, for example, is the factor x

_{1,4}(the optimal set of one-element solutions is obtained—one factor).

_{i}} [57] and, based on them, determine a multi-element set of solutions (the optimal set of factors significantly affecting the number of traffic accidents in Poland).

## 4. Example of Optimization of Factors Affecting the Number of Road Accidents in Poland

- the presentation of a set X
_{j}and selection of elements x_{i}X_{j}; - the presentation of the set F
_{j}and selection, by the computer program operator, of the elements f_{i}F_{j}and the dominance relation_{i j}; - data entry according to two options (option 1—manual data entry (f
_{i}F_{j}values), option 2—calculation of f_{i}F_{j}values) based on data obtained during experimental or simulation studies;

_{av}determined from the set of values {r

_{j}(x

_{i}), j = 1,…, 35} was determined, which is the criterion for classifying the admissible solutions x

_{i}∈X

_{j}into the set of optimal solutions x

_{i}

^{o}∈X

_{j}

^{o}according to the principle expressed by the relation

_{i}

^{o}∈X

_{j}

^{o}= r

_{j}<= r

_{av}

_{av}= 15.33, then the elements of the set of solutions of optimal factors significantly affecting the number of traffic accidents X

_{j}

^{o}, according to the above classification criterion, respectively, are as follows:

- fog, smoke;
- rainfall;
- snowfall or hail;
- cloud cover;

- Monday;
- Tuesday;
- Wednesday;
- Thursday;
- Friday;
- Saturday;
- Sunday;

- Lower Silesian;
- Lubelskie;
- Lodzkie;
- Małopolskie;
- Mazovian;
- Opolskie;
- Podkarpackie;
- Pomeranian;
- Silesian;
- Warmian-Masurian;
- Greater Poland;

- with two one-way carriageways;
- a two-way, single carriageway road.

## 5. Summary

_{1}) and elements of the dominance relation

_{1}) allows us to conclude that it can be used to optimize factors affecting the number of traffic accidents in Poland. The main advantage of the presented algorithm is its versatility; it follows that it will probably be possible to apply the procedures of the presented methodology in situations where the elements of the criterion function will be quantitative and qualitative, and when there is a need to obtain a multi-element or single-element optimal set of solutions.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Number of road accidents in Poland between 1990 and 2021 [3].

**Figure 3.**Graphical interpretation of the solution to the optimization task [56].

**Figure 4.**Multi-criteria optimization scheme for determining the optimal set of factors influencing road accidents in Poland.

x_{1,1} | x_{1,2} | x_{1,3} | x_{1,4} | x_{1,5} | x_{1,6} | x_{1,7} | x_{1,8} | x_{1,9} | x_{1,10} | x_{1,11} | x_{1,12} | |
---|---|---|---|---|---|---|---|---|---|---|---|---|

f_{1,1} | 0.73 | 0.53 | 0.70 | 0.50 | 0.78 | 0.68 | 0.53 | 0.73 | 0.74 | 0.74 | 0.74 | 0.75 |

f_{1,2} | 1622.45 | 132.20 | 700.40 | 539.85 | 69.15 | 1541.80 | 62.50 | 305.60 | 297.50 | 323.70 | 261.55 | 338.45 |

f_{1,3} | 9721.62 | 0.53 | 2041.29 | 1188.33 | 178.81 | 3600.67 | 269.00 | 2162.38 | 1911.57 | 1993.81 | 1981.33 | 2168.48 |

x_{1,13} | x_{1,14} | x_{1,15} | x_{1,16} | x_{1,17} | x_{1,18} | x_{1,19} | x_{1,20} | x_{1,21} | x_{1,22} | x_{1,23} | x_{1,24} | |

f_{1,1} | 0.70 | 0.72 | 0.74 | 0.59 | 0.60 | 0.78 | 0.85 | 0.75 | 0.61 | 0.23 | 0.77 | 0.43 |

f_{1,2} | 397.00 | 277.75 | 229.80 | 96.50 | 146.25 | 40.65 | 224.60 | 406.60 | 431.40 | 247.20 | 148.80 | 96.75 |

f_{1,3} | 2492.81 | 1935.95 | 914.71 | 1100.14 | 1205.62 | 223.14 | 726.57 | 1267.81 | 3399.33 | 3942.71 | 543.86 | 1301.67 |

x_{1,25} | x_{1,26} | x_{1,27} | x_{1,28} | x_{1,29} | x_{1,30} | x_{1,31} | x_{1,32} | x_{1,33} | x_{1,34} | x_{1,35} | Φ_{1} | |

f_{1,1} | 0.76 | 0.69 | 0.71 | 0.73 | 0.68 | 0.73 | 0.64 | 0.49 | 0.76 | 0.83 | 0.73 | MAX |

f_{1,2} | 184.05 | 297.05 | 81.20 | 140.05 | 392.75 | 103.20 | 40.70 | 23.60 | 294.25 | 81.40 | 1732.15 | MAX |

f_{1,3} | 883.10 | 2189.29 | 631.62 | 599.95 | 1709.76 | 598.05 | 171.05 | 229.05 | 1620.90 | 253.00 | 12,116.14 | MAX |

F/X | f_{1,1} | MAX (f_{1,1}) | f_{1,1}* | f_{1,1}** | f_{1,2} | MAX (f_{1,2}) | f_{1,2}* | f_{1,2}** | f_{1,3} | MAX (f_{1,3}) | f_{1,3}* | f_{1,3}** |
---|---|---|---|---|---|---|---|---|---|---|---|---|

x_{1,1} | 0.73 | 0.85 | 1.17 | 3.66 | 1622.4 | 1732.15 | 1.07 | 73.4 | 9721.62 | 12,116.14 | 0.8 | 1 |

x_{1,2} | 0.53 | 1.61 | 132.2 | 13.1 | 0.53 | 0 | ||||||

x_{1,3} | 0.7 | 1.21 | 700.4 | 2.47 | 2041.29 | 0.17 | ||||||

x_{1,4} | 0.5 | 1.7 | 539.85 | 3.21 | 1188.33 | 0.1 | ||||||

x_{1,5} | 0.78 | 1.09 | 69.15 | 25.05 | 178.81 | 0.01 | ||||||

x_{1,6} | 0.68 | 1.25 | 1541.8 | 1.12 | 3600.67 | 0.3 | ||||||

x_{1,7} | 0.53 | 1.59 | 62.5 | 27.71 | 269 | 0.02 | ||||||

x_{1,8} | 0.73 | 1.16 | 305.6 | 5.67 | 2162.38 | 0.18 | ||||||

x_{1,9} | 0.74 | 1.14 | 297.5 | 5.82 | 1911.57 | 0.16 | ||||||

x_{1,10} | 0.74 | 1.15 | 323.7 | 5.35 | 1993.81 | 0.16 | ||||||

x_{1,11} | 0.74 | 1.15 | 261.55 | 6.62 | 1981.33 | 0.16 | ||||||

x_{1,12} | 0.75 | 1.13 | 338.45 | 5.12 | 2168.48 | 0.18 | ||||||

x_{1,13} | 0.7 | 1.22 | 397 | 4.36 | 2492.81 | 0.21 | ||||||

x_{1,14} | 0.72 | 1.19 | 277.75 | 6.24 | 1935.95 | 0.16 | ||||||

x_{1,15} | 0.74 | 1.14 | 229.8 | 7.54 | 914.71 | 0.08 | ||||||

x_{1,16} | 0.59 | 1.45 | 96.5 | 17.95 | 1100.14 | 0.09 | ||||||

x_{1,17} | 0.6 | 1.41 | 146.25 | 11.84 | 1205.62 | 0.1 | ||||||

x_{1,18} | 0.78 | 1.1 | 40.65 | 42.61 | 223.14 | 0.02 | ||||||

x_{1,19} | 0.85 | 1 | 224.6 | 7.71 | 726.57 | 0.06 | ||||||

x_{1,20} | 0.75 | 1.14 | 406.6 | 4.26 | 1267.81 | 0.1 | ||||||

x_{1,21} | 0.61 | 1.39 | 431.4 | 4.02 | 3399.33 | 0.28 | ||||||

x_{1,22} | 0.23 | 3.66 | 247.2 | 7.01 | 3942.71 | 0.33 | ||||||

x_{1,23} | 0.77 | 1.1 | 148.8 | 11.64 | 543.86 | 0.04 | ||||||

x_{1,24} | 0.43 | 1.97 | 96.75 | 17.9 | 1301.67 | 0.11 | ||||||

x_{1,25} | 0.76 | 1.12 | 184.05 | 9.41 | 883.1 | 0.07 | ||||||

x_{1,26} | 0.69 | 1.23 | 297.05 | 5.83 | 2189.29 | 0.18 | ||||||

x_{1,27} | 0.71 | 1.19 | 81.2 | 21.33 | 631.62 | 0.05 | ||||||

x_{1,28} | 0.73 | 1.16 | 140.05 | 12.37 | 599.95 | 0.05 | ||||||

x_{1,29} | 0.68 | 1.25 | 392.75 | 4.41 | 1709.76 | 0.14 | ||||||

x_{1,30} | 0.73 | 1.17 | 103.2 | 16.78 | 598.05 | 0.05 | ||||||

x_{1,31} | 0.64 | 1.32 | 40.7 | 42.56 | 171.05 | 0.01 | ||||||

x_{1,32} | 0.49 | 1.75 | 23.6 | 73.4 | 229.05 | 0.02 | ||||||

x_{1,33} | 0.76 | 1.11 | 294.25 | 5.89 | 1620.9 | 0.13 | ||||||

x_{1,34} | 0.83 | 1.03 | 81.4 | 21.28 | 253 | 0.02 | ||||||

x_{1,35} | 0.73 | 1.16 | 1732.15 | 1 | 12116,14 | 1 |

x_{1,1} | x_{1,2} | x_{1,3} | x_{1,4} | x_{1,5} | x_{1,6} | x_{1,7} | x_{1,8} | x_{1,9} | x_{1,10} | x_{1,11} | x_{1,12} | |
---|---|---|---|---|---|---|---|---|---|---|---|---|

r_{j} | 72.37 | 13.20 | 2.76 | 3.63 | 25.07 | 1.70 | 27.76 | 5.79 | 5.94 | 5.48 | 6.72 | 5.24 |

x_{1,13} | x_{1,14} | x_{1,15} | x_{1,16} | x_{1,17} | x_{1,18} | x_{1,19} | x_{1,20} | x_{1,21} | x_{1,22} | x_{1,23} | x_{1,24} | |

r_{j} | 4.53 | 6.35 | 7.62 | 18.01 | 11.93 | 42.63 | 7.78 | 4.41 | 4.26 | 7.91 | 11.69 | 18.01 |

x_{1,25} | x_{1,26} | x_{1,27} | x_{1,28} | x_{1,29} | x_{1,30} | x_{1,31} | x_{1,32} | x_{1,33} | x_{1,34} | x_{1,35} | ||

r_{j} | 9.48 | 5.96 | 21.37 | 12.42 | 4.59 | 16.83 | 42.58 | 73.42 | 5.99 | 21.30 | 1.83 |

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## Share and Cite

**MDPI and ACS Style**

Gorzelanczyk, P.; Tylicki, H.
Methodology for Optimizing Factors Affecting Road Accidents in Poland. *Forecasting* **2023**, *5*, 336-350.
https://doi.org/10.3390/forecast5010018

**AMA Style**

Gorzelanczyk P, Tylicki H.
Methodology for Optimizing Factors Affecting Road Accidents in Poland. *Forecasting*. 2023; 5(1):336-350.
https://doi.org/10.3390/forecast5010018

**Chicago/Turabian Style**

Gorzelanczyk, Piotr, and Henryk Tylicki.
2023. "Methodology for Optimizing Factors Affecting Road Accidents in Poland" *Forecasting* 5, no. 1: 336-350.
https://doi.org/10.3390/forecast5010018