# A Model Proposal for a Multi-Objective and Multi-Criteria Vehicle Assignment Problem: An Application for a Security Organization

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Literature Survey

#### 2.1. Studies about the Assignment Model

#### 2.1.1. Model with at Most One Task per Agent

#### 2.1.2. Models with Multiple Tasks per Agent

#### 2.1.3. Multi-Dimensional Assignment Problem

#### 2.2. Studies Using Multi-Objective Assignment Models

#### 2.3. Studies Using Mathematical Modeling with the MCDM Techniques

#### 2.4. Studies about Vehicle Selection

#### 2.5. Studies about the Quality of Security Service

## 3. Methods

#### 3.1. Analytic Network Process

#### 3.2. The General Assignment Model in Linear Programming

^{2}− n variables have the value 0. Defining w = {(i,j) | x

_{ij}= 1, x

_{ij}∈ X}, w constitutes the assignment corresponding to X. Often, (1) is practically interpreted as n jobs that must be performed by n workers at minimal total cost, and c

_{ij}is the job cost associated with worker i performing job j. Here, w implies that each job must be assigned to one and only one worker, and vice versa. In the model, all c

_{ij}values are deterministic [58].

## 4. Proposed Multi-Criteria and Multi Objective Assignment Hybrid Methodology

- First step: using the ANP method for the calculation of the main/sub-criteria weight,
- Second step: evaluating the performance values of assigned vehicles for each district,
- Third step: modeling the assignment and realizing the vehicle assignment.

## 5. Illustrative Example

#### 5.1. Forming the Weights of the Criteria with ANP (First Step)

#### 5.1.1. Problem Formulation

- Assigning large-capacity vehicles to populous district
- Assigning large-capacity vehicles to districts with a high density of forensic cases
- Assigning large-capacity vehicles to districts that have many policing personnel
- Assigning 4 × 4 type vehicles to high altitude districts.

- Assigning high-speed vehicles to districts that have plain terrain
- Assigning vehicles that have an adequate level of safety equipment, to the district where the accidents occur more frequently
- Using vehicles with high motor power in high altitudes
- Using vehicle with high motor volume in high altitudes
- Assigning relatively new vehicles to districts with a high density of desired purposes

- Assigning vehicles to district which will have minimum fuel consumption cost
- Minimizing maintenance cost and repair, which can change from district to district

#### 5.1.2. Creating Working Group

#### 5.1.3. Determining Main and Sub-Criteria

#### 5.1.4. Determining the Relationship between Criteria

#### 5.1.5. Making Pair-Wise Comparisons

#### 5.1.6. Creating Unweighted Supermatrix

#### 5.1.7. Creating the Weighted Supermatrix

#### 5.1.8. Computing the Limit Supermatrix

#### 5.1.9. Determining the Weight of Criteria

#### 5.2. Finding the Performance Value of Each Vehicle for Each District (Second Step)

#### 5.2.1. Evaluating the Performances of Alternatives

#### 5.2.2. Creating Assignment Matrix

#### 5.3. Establishing the Assignment Model and Performing the Assignment of Vehicles (Third Step)

#### 5.3.1. Establishing the Mathematical Assignment Model

#### 5.3.2. Solving the Assignment Problem and Evaluating the Results

## 6. Conclusions

## Author Contributions

## Conflicts of Interest

## References

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Criteria | Local Weights | Total | Global Weights | Total |
---|---|---|---|---|

Forensic Case Density of the District (FCDD) | 0.39353 | 1 | 0.23000 | 1 |

Geographical Conditions of the District (GCD) | 0.30940 | 0.18083 | ||

Population Density of the District (PDD) | 0.18427 | 0.10770 | ||

Number of Law Enforcement Staff in the District (LESD) | 0.11280 | 0.06593 | ||

Maintenance Cost (MC) | 0.25218 | 1 | 0.03289 | |

Repair Cost (RC) | 0.15565 | 0.02030 | ||

Fuel Consumption Cost (FCC) | 0.59216 | 0.07723 | ||

Safety Equipment of Vehicle (SEV) | 0.03330 | 1 | 0.00464 | |

Speed of Vehicle (SV) | 0.09306 | 0.01296 | ||

Age of Vehicle (AV) | 0.23671 | 0.03296 | ||

Power of Engine (PE) | 0.25387 | 0.03535 | ||

Volume of Engine (VE) | 0.20404 | 0.02841 | ||

Transmission Type (TT) | 0.17902 | 0.02493 | ||

Time to Return from Malfunction to Use (TRMU) | 0.07367 | 1 | 0.01075 | |

Time to Intervene in Forensic Cases (TIFC) | 0.92633 | 0.13513 |

**Table 2.**Performance values of Vehicle 1 in terms of service efficiency objective for each district. D1, District 1.

Districts | Priorities | FCDD | GCD | PDD | LESD |
---|---|---|---|---|---|

D1 | 0.087359 | 1st Priority | 2nd Priority | 2nd Priority | 1st Priority |

D2 | 0.043439 | 3rd Priority | 1st Priority | 4th Priority | 4th Priority |

D3 | 0.066101 | 3rd Priority | 1st Priority | 1st Priority | 3rd Priority |

D4 | 0.087359 | 1st Priority | 2nd Priority | 2nd Priority | 1st Priority |

D5 | 0.111102 | 1st Priority | 1st Priority | 1st Priority | 2nd Priority |

D6 | 0.051921 | 2nd Priority | 2nd Priority | 2nd Priority | 3rd Priority |

D7 | 0.072010 | 2nd Priority | 1st Priority | 2nd Priority | 3rd Priority |

D8 | 0.023351 | 3rd Priority | 2nd Priority | 4th Priority | 4th Priority |

D9 | 0.046818 | 2nd Priority | 3rd Priority | 1st Priority | 4th Priority |

D10 | 0.075306 | 1st Priority | 3rd Priority | 2nd Priority | 1st Priority |

D11 | 0.054069 | 3rd Priority | 1st Priority | 3rd Priority | 2nd Priority |

D12 | 0.066101 | 3rd Priority | 1st Priority | 1st Priority | 3rd Priority |

D13 | 0.075306 | 1st Priority | 3rd Priority | 2nd Priority | 1st Priority |

D14 | 0.076129 | 2nd Priority | 1st Priority | 2nd Priority | 2nd Priority |

D15 | 0.063629 | 3rd Priority | 1st Priority | 1st Priority | 4th Priority |

**Table 3.**Annual maintenance cost and normalized form of Vehicle 1 (V1) according to district generic data.

Annual Maintenance Costs ($) | ||
---|---|---|

District | Real Costs | Normalized Costs |

D1 | 900 | 0.072 |

D2 | 1,100 | 0.088 |

D3 | 850 | 0.068 |

D4 | 980 | 0.078 |

D5 | 700 | 0.056 |

D6 | 750 | 0.060 |

D7 | 740 | 0.059 |

D8 | 880 | 0.070 |

D9 | 790 | 0.063 |

D10 | 700 | 0.056 |

D11 | 720 | 0.057 |

D12 | 780 | 0.062 |

D13 | 710 | 0.057 |

D14 | 840 | 0.067 |

D15 | 1,120 | 0.089 |

Total | 12,560 | 1 |

Current Assignment | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Vehicle Types | Vehicle | D1 | D2 | D3 | D4 | D5 | D6 | D7 | D8 | D9 | D10 | D11 | D12 | D13 | D14 | D15 | Capacity of Vehicles | |

Administrative Veh. | V1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 10 | 15 | |||||

V2 | 1 | 1 | 1 | 1 | 1 | 5 | ||||||||||||

Traffic Vehicles | V3 | 2 | 2 | 1 | 1 | 2 | 8 | 30 | ||||||||||

V4 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 3 | 13 | |||||||

V5 | 1 | 1 | 1 | 1 | 4 | |||||||||||||

V6 | 1 | 2 | 2 | 5 | ||||||||||||||

Tactical Vehicles | V7 | 2 | 2 | 2 | 6 | 39 | ||||||||||||

V8 | 3 | 4 | 1 | 1 | 2 | 3 | 3 | 4 | 3 | 4 | 2 | 3 | 33 | |||||

Patrol Vehicles (T-I) | V9 | 1 | 3 | 2 | 1 | 2 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 1 | 1 | 4 | 27 | 90 |

V10 | 1 | 1 | 2 | |||||||||||||||

V11 | 1 | 11 | 3 | 1 | 1 | 5 | 5 | 3 | 8 | 7 | 2 | 4 | 10 | 61 | ||||

Patrol Vehicles (T-II) | V12 | 1 | 3 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 12 | 22 | ||||||

V13 | 22 | 1 | 2 | 1 | 1 | 1 | 2 | 10 | ||||||||||

Capacity of Districts | 8 | 30 | 7 | 4 | 8 | 7 | 12 | 20 | 14 | 6 | 16 | 19 | 11 | 8 | 26 | 196 |

Assignment Results | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Vehicle Types | Vehicle | D1 | D2 | D3 | D4 | D5 | D6 | D7 | D8 | D9 | D10 | D11 | D12 | D13 | D14 | D15 | Capacity of Vehicles | |

Administrative Veh. | V1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 10 | 15 | |||||

V2 | 1 | 1 | 1 | 1 | 1 | 5 | ||||||||||||

Traffic Vehicles | V3 | 1 | 1 | 3 | 1 | 1 | 1 | 8 | 30 | |||||||||

V4 | 1 | 5 | 1 | 1 | 5 | 13 | ||||||||||||

V5 | 4 | 4 | ||||||||||||||||

V6 | 2 | 1 | 1 | 1 | 5 | |||||||||||||

Tactical Vehicles | V7 | 4 | 1 | 1 | 6 | 39 | ||||||||||||

V8 | 1 | 1 | 1 | 1 | 1 | 1 | 10 | 10 | 1 | 1 | 3 | 1 | 1 | 33 | ||||

Patrol Vehicles (T-I) | V9 | 3 | 3 | 12 | 9 | 27 | 90 | |||||||||||

V10 | 2 | 2 | ||||||||||||||||

V11 | 26 | 7 | 15 | 13 | 61 | |||||||||||||

Patrol Vehicles (T-II) | V12 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 1 | 12 | 22 | |||||

V13 | 1 | 1 | 1 | 3 | 3 | 1 | 10 | |||||||||||

Capacity of Districts | 8 | 30 | 7 | 4 | 8 | 7 | 12 | 20 | 14 | 6 | 16 | 19 | 11 | 8 | 26 | 196 | ||

wp: 0.584455wc: 0.130425wt: 0.145881wu: 0.13924 | wp: weight of service efficiency objectivewc: weight of cost objectivewt: weight of time objectivewu: weight of the usage of technical capacity objective |

Assignment Results | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Vehicle Types | Vehicle | D1 | D2 | D3 | D4 | D5 | D6 | D7 | D8 | D9 | D10 | D11 | D12 | D13 | D14 | D15 | Capacity of Vehicles | |

Administrative Veh. | V1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 10 | 15 | |||||

V2 | 1 | 1 | 1 | 1 | 1 | 5 | ||||||||||||

Traffic Vehicles | V3 | 1 | 2 | 3 | 1 | 1 | 8 | 30 | ||||||||||

V4 | 4 | 1 | 1 | 4 | 1 | 1 | 1 | 13 | ||||||||||

V5 | 3 | 1 | 4 | |||||||||||||||

V6 | 1 | 4 | 5 | |||||||||||||||

Tactical Vehicles | V7 | 4 | 1 | 1 | 6 | 39 | ||||||||||||

V8 | 4 | 1 | 1 | 1 | 1 | 1 | 1 | 7 | 1 | 1 | 8 | 5 | 1 | 33 | ||||

Patrol Vehicles (T-I) | V9 | 1 | 26 | 27 | 90 | |||||||||||||

V10 | 2 | 2 | ||||||||||||||||

V11 | 3 | 6 | 3 | 12 | 15 | 22 | 61 | |||||||||||

Patrol Vehicles (T-II) | V12 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 12 | 22 | ||||||

V13 | 1 | 1 | 1 | 5 | 1 | 1 | 10 | |||||||||||

Capacity of Districts | 8 | 30 | 7 | 4 | 8 | 7 | 12 | 20 | 14 | 6 | 16 | 19 | 11 | 8 | 26 | 196 | ||

wp: 0.1wc: 0.7wt: 0.1wu: 0.1 | wp: weight of service efficiency objectivewc: weight of cost objectivewt: weight of time objectivewu: weight of the usage of technical capacity objective |

Assignment Results | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Vehicle Types | Vehicle | D1 | D2 | D3 | D4 | D5 | D6 | D7 | D8 | D9 | D10 | D11 | D12 | D13 | D14 | D15 | Capacity of Vehicles | |

Administrative Veh. | V1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 10 | 15 | |||||

V2 | 1 | 1 | 1 | 1 | 1 | 5 | ||||||||||||

Traffic Vehicles | V3 | 4 | 2 | 1 | 1 | 8 | 30 | |||||||||||

V4 | 9 | 1 | 1 | 1 | 1 | 13 | ||||||||||||

V5 | 1 | 2 | 1 | 4 | ||||||||||||||

V6 | 1 | 2 | 1 | 1 | 5 | |||||||||||||

Tactical Vehicles | V7 | 4 | 2 | 6 | 39 | |||||||||||||

V8 | 1 | 1 | 1 | 1 | 1 | 8 | 1 | 9 | 1 | 1 | 1 | 1 | 5 | 1 | 33 | |||

Patrol Vehicles (T-I) | V9 | 4 | 11 | 12 | 27 | 90 | ||||||||||||

V10 | 2 | 2 | ||||||||||||||||

V11 | 18 | 16 | 2 | 15 | 10 | 61 | ||||||||||||

Patrol Vehicles (T-II) | V12 | 1 | 1 | 1 | 1 | 8 | 12 | 22 | ||||||||||

V13 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 10 | |||||||

Capacity of Districts | 8 | 30 | 7 | 4 | 8 | 7 | 12 | 20 | 14 | 6 | 16 | 19 | 11 | 8 | 26 | 196 | ||

wp: 0.1wc: 0.1wt: 0.7wu: 0.1 | wp: weight of service efficiency objectivewc: weight of cost objectivewt: weight of time objectivewu: weight of the usage of technical capacity objective |

Assignment Results | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Vehicle Types | Vehicle | D1 | D2 | D3 | D4 | D5 | D6 | D7 | D8 | D9 | D10 | D11 | D12 | D13 | D14 | D15 | Capacity of Vehicles | |

Administrative Veh. | V1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 10 | 15 | |||||

V2 | 1 | 1 | 1 | 1 | 1 | 5 | ||||||||||||

Traffic Vehicles | V3 | 4 | 1 | 1 | 1 | 1 | 8 | 30 | ||||||||||

V4 | 9 | 1 | 3 | 13 | ||||||||||||||

V5 | 1 | 3 | 4 | |||||||||||||||

V6 | 1 | 1 | 1 | 1 | 1 | 5 | ||||||||||||

Tactical Vehicles | V7 | 2 | 3 | 1 | 6 | 39 | ||||||||||||

V8 | 5 | 1 | 1 | 1 | 1 | 1 | 1 | 11 | 1 | 1 | 7 | 1 | 1 | 33 | ||||

Patrol Vehicles (T-I) | V9 | 11 | 12 | 4 | 27 | 90 | ||||||||||||

V10 | 2 | 2 | ||||||||||||||||

V11 | 26 | 15 | 2 | 18 | 61 | |||||||||||||

Patrol Vehicles (T-II) | V12 | 1 | 1 | 1 | 6 | 1 | 1 | 1 | 12 | 22 | ||||||||

V13 | 1 | 1 | 1 | 3 | 1 | 1 | 1 | 1 | 10 | |||||||||

Capacity of Districts | 8 | 30 | 7 | 4 | 8 | 7 | 12 | 20 | 14 | 6 | 16 | 19 | 11 | 8 | 26 | 196 | ||

wp: 0.1wc: 0.1wt: 0.1wu: 0.7 | wp: weight of service efficiency objectivewc: weight of cost objectivewt: weight of time objectivewu: weight of the usage of technical capacity objective |

© 2016 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC-BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Acar, E.; Aplak, H.S.
A Model Proposal for a Multi-Objective and Multi-Criteria Vehicle Assignment Problem: An Application for a Security Organization. *Math. Comput. Appl.* **2016**, *21*, 39.
https://doi.org/10.3390/mca21040039

**AMA Style**

Acar E, Aplak HS.
A Model Proposal for a Multi-Objective and Multi-Criteria Vehicle Assignment Problem: An Application for a Security Organization. *Mathematical and Computational Applications*. 2016; 21(4):39.
https://doi.org/10.3390/mca21040039

**Chicago/Turabian Style**

Acar, Engin, and Hakan Soner Aplak.
2016. "A Model Proposal for a Multi-Objective and Multi-Criteria Vehicle Assignment Problem: An Application for a Security Organization" *Mathematical and Computational Applications* 21, no. 4: 39.
https://doi.org/10.3390/mca21040039