# The Task Assignment of Vehicles for a Production Company

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Literature Analysis

## 3. The Mathematical Model of the Assignment Problem

- -
- The production company starts the production process in the case when all the cargo has been gathered in the warehouses. It means that the cargo is transported from the warehouses to the production company. The relation: suppliers and a production company does not exist. All the cargo which is necessary for the production must be stored in the warehouses. This situation is to ensure the continuity of the production process.
- -
- The amount of the cargo which is offered by all the suppliers is bigger than the demand of the production company.
- -
- The organization of transport is on the side of the productize the total costs which result from transporting the on company. The aim of the production company is to minimalicargo from the suppliers to the production company.
- -
- The model assumes that the distances between the suppliers and the warehouses and the warehouses and the production company are the same as between the warehouses and the suppliers, and the production company and the warehouses.
- -
- The warehouses belong only to the production company. This situation lets the production company know the capacity of the warehouses.

**V**= {v: v = 1, 2, …, v’, …, V} of the transport network with has interpretation of (are subset of

**V**): suppliers

**DS**= {v: α(v) = 0 for v ∈

**V**}, warehouses,

**MS**= {v: α(v) = 1 for v ∈

**V**}, for which defined capacity

**CAP**= [cap(v): cap(v) ∈

**R**

^{+}, v ∈

**MS**], and also production companies (recipients),

**P**= {v: α(v) = 2 for v ∈

**V**}. The volume of deliveries is also associated with suppliers

**Q1**= [q1(v): q1(v) ∈

**R**

^{+}, v ∈

**DS**], while with production companies, demand

**Q2**= [q2(v): q2(v) ∈

**R**

^{+}, v ∈

**P**].

**VOL**= {1, …, vol, …, VOL}, and each of them is characterized by a capacity in pallets

**PA**= [pa(vol): pa(vol) ∈

**R**

^{+}, vol ∈

**VOL**], fuel consumption

**C**= [c(vol): c(vol) ∈

**R**

^{+}, vol ∈

**VOL**] and permissible speed s(vol).

**D1**= [d1(v,v’): d1(v,v’) ∈

**R**

^{+}, v ∈

**DS**, v’ ∈

**MS**] and warehouses-enterprises,

**D2**= [d2(v,v’): d2(v,v’) ∈

**R**

^{+}, v ∈

**MS**, v’ ∈

**P**]. Similarly, time dependencies were defined

**T1**= [t1(v,v’): t1(v,v’) ∈

**R**

^{+}, v ∈

**DS**, v’∈

**MS**],

**T2**= [t2(v,v’): t2(v,v’) ∈

**R**

^{+}, v ∈

**MS**, v’ ∈

**P**]. Time matrices includes for example traffic jams in cities.

**UMZ**= [δ(v): δ(v) ∈

**R**

^{+}, v ∈

**MS**], number of vehicles of each type

**NV**= [nv(vol): nv(vol) ∈

**R**

^{+}, vol ∈

**VOL**], transition costs

**KZ**= [kz(v): kz(v) ∈

**R**

^{+}, v ∈

**MS**] and the purchase cost of the cargo at the suppliers

**KB**= [kb(v): kb(v) ∈

**R**

^{+}, v ∈

**DS**].

**X**= [

**X1**,

**X2**], determine the quantity of cargo flow between suppliers—warehouses

**X1**= [x1(v,v’): x1(v,v’) ∈

**R**

^{+}, v ∈

**DS**, v’ ∈

**MS**] an warehouses—production companies

**X2**= [x2(v,v’): x2(v,v’) ∈

**R**

^{+}, v ∈

**MS**, v’ ∈

**P**]. The second type of the decision variable

**Y**= [

**Y1**,

**Y2**], determines the use of the given type of the vehicle in the task and takes the following form (1—a given type is used in the relation):

**Y1**= [y1(v,v’,vol): y1(v,v’,vol) ∈ {0,1}, v ∈

**DS**, v’ ∈

**MS,**vol ∈

**VOL**],

**Y2**= [y2(v,v’,vol): y2(v,v’,vol) ∈ {0,1}, v ∈

**MS**, v’ ∈

**P**, vol ∈

**VOL**]. The third type of the decision variable

**N**= [

**N1**,

**N2**], determines the number of the vehicles which perform the task

**N1**= [n1(v,v’,vol): n1(v,v’,vol) ∈

**R**

^{+}, v ∈

**DS**, v’ ∈

**MS,**vol∈

**VOL], N2**= [n2(v,v’,vol): n2(v,v’,vol) ∈

**R**

^{+}, v ∈

**MS**, v’ ∈

**P**, vol ∈

**VOL**].

**F1**(1) containing costs and the second task execution time

**F2**(6).

**F1**(1) consists of two parts: the fuel consumption costs among the facilities of the logistic network

**F1a**(2) and the transition costs of the cargo via the warehouses and the purchase costs of the cargo

**F1b**(3). Relation (4) has an interpretation of the number of tasks in a given relation, in turn (5) has an interpretation of the number of the courses in which all the tasks on a given relation are performed. The criterion function

**F2**(6) determines the time needed to perform the tasks takes. The solutions to be included in the permissible set must meet a number of defined constraints: (7) the production capacity of the suppliers cannot be exceeded—the suppliers can provide the cargo to the warehouses, (8) the recipients’ demands must be met—the cargo can flow to the recipients from the warehouses, (9) the warehouse capacities cannot be exceeded—the cargo can flow to the warehouses from the suppliers, (10) the cargo flowing out from the warehouse is equal to the cargo flowing into the warehouse, (11) the minimal stream of the cargo flowing into the warehouses decides about the choice of the warehouses to the supply network for the production company, (12) and (13) many types of vehicles can exist on a given relation, (14) the number of the vehicles must be met. For the above model, a unique algorithm was developed which allows effective solution determination.

## 4. The Genetic Algorithm for the Assignment Problem

#### 4.1. Main Assumptions

#### 4.2. The Structure of the Genetic Algorithm

**M**(t,k), which shows the flow of the cargo among the particular elements of the transport network (Part I), the type of the vehicles (Part II and Part III or more parts depend on the number of the types of the vehicles), the number of the vehicles (the Part IV for the first type of the vehicles and Part V for the second type of the vehicles) in t-th iteration, k-th structure in the population, Figure 4. The lines and the columns of this matrix in each part define the facilities of the transport network structure. In order to determine the flow of the cargo, the lines were defined as the starting points from which the cargo flows out to the other facilities. The matrix cells are located in the following sequence: the suppliers (D1–D2), the warehouses (MS1–MS4) and the production company (P1). The tasks are designated in Part I, II and III, while the task assignment of the vehicles is carried out in Part IV and V.

- -
- Step 1: Setting the values of all the cells of the matrix to 0. This value determines, e.g., the connections for which it is not possible to transport the cargo, e.g., among the suppliers.
- -
- Step 2: Setting the cells of the structure (Part I): D1, MS1—D2, MS4 in a random way (the relation: the suppliers—the warehouses). The values in these cells must meet the limits: the production capacity of the suppliers (7), the warehouse capacity (9), the minimal stream of the cargo flowing into the warehouses (11). It should be underlined that the sum of the cargo flowing out from the suppliers must be equal the demand of the production company. The cells which do not meet these limits take the value 0. In the case when the demand of the production company is met, other cells which were not designated take the value 0 as well. In the case when the demand is not fulfilled and all the cells from D1, MS1—D2, MS4 are designated, the cells are selected in a random way and their values are increased until the moment when the demand is met.
- -
- Step 3: Setting the cells of the matrix: MS1, P1—MS4, P1. It should be remembered that the cargo flowing out from the warehouse is equal to the cargo flowing into the warehouse (10) and the recipients’ demands must be met (8).
- -
- Step 4: Setting the cells of the matrix in the Part II and III. In the presented matrix two types of the vehicles were considered (the first type of the vehicles—Part II and the second type of the vehicles—Part III). These cells are designated in a random way and take the value 1 or 0. 1—a given type of vehicles is used on the connection, 0—otherwise. In this step the limits: many types of the vehicles can exist on a given relation (12), (13) must be met.
- -
- Step 5: Setting the cells of the matrix in the Part IV and V in a random way. It should be remembered that the number of the vehicles must be met (14).

#### 4.3. The Adaptation Function

**M**(t,k) must take the following form (

**K**= {1, …, k, …, K}-the set of the structures

**M**(t,k) in the population, t—iteration):

**F1**min(t) determines the minimum value of the structure calculated according to the first criterion function (1) from the whole population in a given iteration of algorithm,

**F1**(k,t) determines the value of criterion function (1) for k—the structure of the matrix

**M**(t,k),

**F2**min(t) determines the minimum value of the structure calculated according to the second criterion function (6) from the whole population in a given iteration of the algorithm. The function

**F**(k,t) for k—the structure of the matrix

**M**(t,k) will reach the maximum value in the case when each function, e.g.,

**F1**(k,t) reaches

**F1**min,

**F2**(k,t) reaches

**F2**min and so on. The structure of maximal value of the function

**F**(k,t) after all iterations is the solution of the algorithm.

#### 4.4. The Crossover Process

**DIV**which comprise rounded up average values from both parents, and the matrix

**REM**containing the information whether the rounding up was indeed necessary. Assuming that the value of the matrices

**M1**and

**M2**(parents) in all the cells assume determination ${m}^{1}{}_{v,v\u2019}$, ${m}^{2}{}_{v,v\u2019}$, the values of the elements of the matrices

**DIV**and

**REM**are calculated from the following dependencies:

**REM**are added to the matrix

**DIV**. As a result of this operation two new structures are developed. In the case when the constraints for each part of the matrix are not met, the new structures take the parents’ forms for each part.

#### 4.5. The Mutation Process

## 5. The Results and Discussion

_{cross}—crossover parameter, p

_{mut}—mutation parameter (Table 2). The number of the iterations was set to 100. The linear scaling factor for the genetic algorithm accordance with the recommendations of the literature [49] assumes the value 2.0. The results of all the tests for the parameters for two variants arre presented in Table 3. The graphical presentation of work of the genetic algorithm is shown in Figure 12 (p

_{cross}= 0.8, p

_{mut}= 0.03).

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 5.**The substructures (Part I of the structure) in the crossover process (

**M1**,

**M2**—the substructures of two chromosomes).

**Figure 8.**The substructures (Part II of the structure) in the crossover process (

**M1**,

**M2**—the substructures of two different chromosomes.

**Figure 9.**The new substructures (Part II of the structure) after the crossover process (

**M1**,

**M2**—the substructures of two different chromosomes).

**Figure 15.**An example graph of a searching the solutions set by methods (GA, SA, PSO) due to the adaptation function for one of the tests for variant 2.

**Figure 16.**An example graph of algorithms convergence (GA, SA, PSO) due to the adaptation function for one of the tests for variant 2.

Data | Warehouses [km/km/min/min] | - | |||
---|---|---|---|---|---|

1 | 2 | 3 | Q1 Pallet/KB | ||

Suppliers [km/km/min/min] | 1 | 50/70/130/90 | 40/25/70/40 | 20/30/30/40 | 300/25 |

2 | 60/40/100/60 | 15/20/40/45 | 10/15/20/40 | 250/30 | |

3 | 40/55/70/70 | 20/23/35/30 | 10/30/20/40 | 350/15 | |

4 | 20/40/40/70 | 80/40/100/60 | 22/15/35/25/ | 150/35 | |

5 | 10/30/20/40 | 40/60/50/70 | 10/20/30/35 | 200/40 | |

6 | 30/40/50/60 | 10/40/20/60 | 30/20/40/30 | 150/10 | |

7 | 10/20 /40/30 | 70/50/90/70 | 20/10/30/20 | 100/15 | |

Company [km/km/min/min] | - | 25/35/40/50 | 20/30/30/40 | 40/50/60/70 | - |

Q2 pallet | - | - | - | - | 360 |

CAP pallet | - | 200 | 150 | 200 | - |

UMZ pallet | - | 30 | 30 | 30 | - |

KZ PLN/ pallet | - | 12 | 10 | 11 | - |

**Q1**: volume of deliveries;

**Q2**: demand of production companies;

**CAP**: capacity of warehouses;

**UMZ**: the minimum volume of the cargo;

**KZ**: transition cost of the units via warehouses.

Test | p_{cross} | p_{mut} | Test | p_{cross} | p_{mut} | Test | p_{cross} | p_{mut} |
---|---|---|---|---|---|---|---|---|

1 | 0.2 | 0.01 | 6 | 0.2 | 0.03 | 11 | 0.2 | 0.05 |

2 | 0.4 | 0.01 | 7 | 0.4 | 0.03 | 12 | 0.4 | 0.05 |

3 | 0.6 | 0.01 | 8 | 0.6 | 0.03 | 13 | 0.6 | 0.05 |

4 | 0.8 | 0.01 | 9 | 0.8 | 0.03 | 14 | 0.8 | 0.05 |

5 | 1 | 0.01 | 10 | 1 | 0.03 | 15 | 1 | 0.05 |

Test | The Best Value of Population | Test | The Best Value of Population | Test | The Best Value of the Structure of Population |
---|---|---|---|---|---|

1 | 0.53/0.49 | 6 | 0.57/0.41 | 11 | 0.42/0.3 |

2 | 1.3/1.2 | 7 | 1.4/1.0 | 12 | 1.3/1.22 |

3 | 1.43/1.39 | 8 | 1.52/1.2 | 13 | 1.6/1.5 |

4 | 1.6/1.55 | 9 | 1.73/1.81 | 14 | 1.5/1.77 |

5 | 1.7/1.75 | 10 | 1.63/1.7 | 15 | 1.55/1.63 |

**Table 4.**Comparison of results obtained with different methods (variant 2) repeated 50 times using the genetic algorithm (GA), simulated annealing (SA), and particle swarm optimization (PSO) methods.

Number of Iterations | GA | SA | PSO | ||||||
---|---|---|---|---|---|---|---|---|---|

Adaptation Function | Average Computation Time (s) | Adaptation Function | Average Computation Time (s) | Adaptation Function | Average Computation Time (s) | ||||

Max | Average | Max | Average | Max | Average | ||||

20 | 0.740 | 0.690 | 131.867 | 1.204 | 1.100 | 0.501 | 0.890 | 0.810 | 34.209 |

50 | 1.814 | 1.801 | 371.762 | 1.330 | 1.272 | 1.020 | 1.020 | 1.012 | 62.127 |

500 | 1.831 | 1.829 | 6259.782 | 1.673 | 1.632 | 14.560 | 1.807 | 1.805 | 651.020 |

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**MDPI and ACS Style**

Jacyna, M.; Izdebski, M.; Szczepański, E.; Gołda, P.
The Task Assignment of Vehicles for a Production Company. *Symmetry* **2018**, *10*, 551.
https://doi.org/10.3390/sym10110551

**AMA Style**

Jacyna M, Izdebski M, Szczepański E, Gołda P.
The Task Assignment of Vehicles for a Production Company. *Symmetry*. 2018; 10(11):551.
https://doi.org/10.3390/sym10110551

**Chicago/Turabian Style**

Jacyna, Marianna, Mariusz Izdebski, Emilian Szczepański, and Paweł Gołda.
2018. "The Task Assignment of Vehicles for a Production Company" *Symmetry* 10, no. 11: 551.
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