Simulation methods for transport logistics

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Computational Engineering and Technical Physics Technomathematics

Nikita Belyak

SIMULATION METHODS FOR TRANSPORT LOGISTICS

Master’s Thesis

Examiners: Professor Heikki Haario Professor Matti Heili¨o Supervisor: Professor Heikki Haario

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Lappeenranta University of Technology School of Engineering Science

Computational Engineering and Technical Physics Technomathematics

Nikita Belyak

Simulation methods for transport logistics

Master’s Thesis

2017

49 pages, 18 figures and 4 tables.

Examiners: Professor Heikki Haario Professor Matti Heili¨o

Keywords: transport logistics, system dynamics, transport traffic, event-based simulation, differential evolution

The purpose of this study is to describe existing simulation methods for the solutions of transport logistics problems and illustrate their application by the real example. In this paper, the transport logistics and system dynamics approaches were used in order to model the processes of the Finnish company. Based on the simulation results and existing statistical data about the work performance the conclusion about the reliability of these models was done. Also, the differential evolution approach was applied for the optimization of these models.

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ABBREVIATIONS AND SYMBOLS

MAS multi-agent system a(i) i-th element of arraya

A(i, j) element of matrixAlocated at the intersection ofi-th row andj-th column exprnd(λ) random numbers from the exponential distribution with mean parameterλ NC number of the cars which assumed to be served during the day

DE differential evolution

Terminology for the description of the traffic simulation SAT matrix of scheduled arrival time

i the indicator, that corresponds to car represented by the row ofSAT that will be taken to the service in the current moment

ATNi arrival time noise of thei-th type ATN arrival time with noise

STN service time with noise

TStart(i) start of the service time for thei-th car fromSAT TEnd(i) end of the service time for thei-th car fromSAT LQ(i) length of the queue in thei-th step

NA number of the areas

TWT array that contains the total waiting time in queue of all cars for each area after the simulation of one day work

CNS array that contains the total number of cars which were not served for each area after the simulation of one day work

Terminology for the description of loads simulation SAML system with assumed maximal loads

SRL system with real loads

CDF cumulative distribution function

ADGT average daily amount of garbage transported

NCNA number of the cars which are supposed to not arrive this day ALN array of line numbers in arrival timetable

which determine which cars are supposed to not arrive

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1 INTRODUCTION

1.1 What is the transport logistics?

The transportation and logistics are two terms which are closely connected with the development of the mankind. The physical transportation of any kind of goods through the different geographical locations always was one of the important tasks for the human- ity [1]. The classical transport transportation problem was mathematically formulated in 18th century and it was the task of transportation a given good from manufacturers to consumers while minimizing the transportation cost [2]. With the discovering of various modes of transport and extension of target locations, these tasks become more and more complicated. In the 19th century, the railway was invented, and in the 20th century a con- tainer for sea freight and aeroplanes were created and, of course, the technical features of these inventions affected the conditions of transportation tasks. There began to appear questions connected with the possibility of reducing the time, distance and other costs for their implementation. Logistics was the answer to these questions [1].

1.2 Importance of the logistics

Logistics is the key factor in the connection between local economic sectors. The design of different effective logical and transport systems by minimizing the result of logistic operations related to turnover, information between the point of origin and point of con- sumption and other factors carries out the coherence of various inter-dependent produc- tion sectors such as tourism, manufacturing, agriculture and others. Moreover, logistics is a connecting link between the internal and international economy. Special attention can be paid to transport logistics, which effectively solves the problems of manufacturers associated with the safe delivery of their goods to recipients in the most low-cost way and minimization of the time delays [3, 4].

1.3 Importance of the transport logistics

It should be noted that the world economic development is closely linked to the state of transport infrastructure and the availability of various large-scale transport networks.

Consequently, transport logistic in collaboration with transport science which actually in

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part poses new challenges for solutions is one of the key factors for the world economy rise [5]. As an example, we can consider the situation that has developed in Poland, a country which, due to its geographic situation, plays an important role in the logistics system of the European Union. Poland is part of various international freight and transit routes and Silesian province of Poland has a significant attractiveness for foreign investment due to its developed infrastructure, good transport accessibility, proximity to the Czech and Slavic borders and the availability of qualified workforce ready to work for a relatively low work rate in relation to the developed countries of the European Union. In order to support the current state of the transport logistics system of freight transport as well as to improve it and to simplify the investment activities, thereby making this area even more attractive for foreign investment, local authorities pursue a policy of support- ing and expanding the operations of logistics centers [6]. Moreover, we can observe the implementation of transport logistics in natural biological processes. For instance, we can consider logistics of intracellular motion and transport processes as well as their biophys- ical underpinnings inside the pollen tube, which understandings are based on the combination of mechanical and mathematical modelling with cell biological approaches [7].

All the information above indicates the importance of such science as transport logistics.

1.4 Approaches of the transport logistics

Over the past decades, a large number of methods have been developed to solve the urban problem associated with the distribution of goods. These solutions usually supplement or completely change the urban freight system. However, there is a problem associated with the evaluation of the proposed solution, since in the most of cases, the solution found had no analogues in the past and, accordingly, there is no real experience of their use. This fact makes it weaker to evaluate the results of applying the solution in real life, since in a multidimensional environment, together with the multi-parameter character of logistic assessments, its application can lead to negative results, in particular, if not all aspects have been carefully taken into account. To avoid such situations, it is proposed to use modelling as a tool for evaluating the proposed solution before applying it as well as sup- porting the decision-making process. Various models were developed to derive the way of logistic evaluation and they can be grouped in different ways, mostly according to the goals they fulfil. For example, Hicks grouped the models based on their profitability into simulation models, optimization models and simulation-optimization models [8]. And Taniguchi, in turn, categorized them in optimization models and simulation models [9].

Optimization processes are connected with the search for the best solution from a variety of alternative options in accordance with the goals to be achieved. For instance, such

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a model has been used to find solutions for the problems of the minimization of total costs [10]. The aim of the simulation models is the replication of a working system for a total and profound understanding of processes. Then after the correction and validation, a model can be used as a testing tool for various scenarios as well as verification of optimization. Often, simulation models replace optimization since they can be interpreted as so-called ”test machines” that by trial and error try out all possible scenarios and choose the best from them. According to the [9] simulation methods can be grouped in system dynamics, multi-agent systems and traffic simulation [11].

In this research, various simulation models are considered, and their implementation is illustrated by using a real case.

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2 BACKGROUND

2.1 Systems dynamics

System dynamics is a simulation modelling approach which focuses on the internal structure and features of the system [11]. Originally it was developed in the 1950s as a tool for corporate managers which helps them better understand industrial processes. The beginning of the field of the system dynamics is associated with the development of hand simulations of the stock-flow-feedback structure of the General Electric plants. Based on the results obtained by hand calculations J. W. Forrester showed that the instability of the company employment was caused by internal processes of the company and not by exter- nal forces as it was initially supposed. Further, the process of simulation was modernized into the computer modelling which entailed the appearance of computer languages for such kind of modelling. Moreover, the fields of application of the system dynamics approach were increased to the modelling of world dynamics and urban dynamics [12].

The examples of such modelling are the models of the world socio-economic system WORLD1, WORLD2 and WORLD3, each of which is an improved version of the previous [12]. The last one was originally created by D. Meadows, who was one of the Jay W.

Forrester’s former PhD students, in collaboration with his associates and it was published in [13]. The purpose of the WORLD3 model is to identify which kind of the behaviour modes is the most characteristic of the population of the globe and material’s outputs under different conditions and to determine such policies which rather may lead to a stable behaviour mode. Considered in the modelling time parameters start from 1900 and cover two centuries. The period 1900-1970 is used as a test of the reliability of the model by comparing its behaviour with real historical trends. The model consists of five interacting sectors (capital, nonrenewable resources, agriculture, pollution, population) and it is presented as a set of equations which are formulated in a format of the simulation language DYNAMO [14].

The main goal of the system dynamics is to consider complex real-world systems as a set of special ”building blocks”, for better understanding the system behaviour over time and use this knowledge to design and implement more efficient policy [12].

In terms of system dynamics, systems can be divided into the ”open” and ”closed”. The outputs of the ”open” systems respond to the inputs but don’t influence them. In the systems of the second type they do both and such kind of systems is more common in the real world. The main ”building blocks” of system dynamics are stocks and flows, which can be outflows and inflows. One of the key task for modelling is to identify what are stocks and what are flows in the system. Stocks have four main characteristics which determine

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the system behaviour. They have a memory, change the time shape of flows, decouple flows and create delays. The last property, in some sense, the interpretation of the fact that in real life, events do not occur instantly and there is often a gap between cause and effect. Fairly often the stocks and flows are the part of the of another ”building blocks” - feedback loops. The feedback loops, which control closed systems, can be of two types:

positive and negative. In positive loops, some actions create the result which in its turn generates more actions which continue to provide more results. Thus, positive loops make a system unstable and force it to leave the current state. And negative loop has opposite effect, which force system to move toward or to keep at some state, stabilizing the system or, in some cases destabilizing and cause it to oscillate [12]. For the representation of the dynamics casual loop diagrams and stock-and-flow diagrams are often used [15].

2.2 Traffic simulation

Due to the complexity of the transport network simulation in the real-world computer models are typically used for this goal. According to the [9] one of the most common classifications of the traffic simulations can be performed by time approach: continuous or discrete or by the level of detailing: macroscopic, microscopic and mesoscopic. In case of macroscopic simulation model traffic flow is described in a general manner without considering its elements in detail. Microscopic models give attention to individual vehicles and their interactions. The mesoscopic models are the intermediate one [16].

The models described above are widely used for planning, design and analysis of the transport influence on the urban mobility and environment. During the simulations, various scenarios are considered that are based on the interactions between different attributes related to the city, such as transport network infrastructure, the features of the local driving regulations, traffic volumes etc. Different software tools can be used for the modelling, but for each model, which is developed for some specific case, it is necessary to realize the calibration and validation to make sure that the model correctly simulates real traffic situations [11]. One of the traffic modelling examples is presented in the [17]. The authors of the article created the model which is based on the Cell Transmission Model [18, 19], which predicts macroscopic traffic behaviour but includes the features of microscopic traffic model. The result of the paper illustrates the simulation model which can represent traffic of large-scale networks in a real time. Moreover, information about the network can be imported directly from the Geographic Information System data. Another example is described in [20], where the author developed the microscopic traffic simulation model of the toll stations work’s process. The paper analyses various scenarios which consider different types of the drivers, traffic volume, the capacity of toll booths and the configura-

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tion of the toll station. The results of the research show that the volume per track and the way of payment strongly affect the length of the queue as well as delays.

2.3 Multi-agent systems

During a decision-making process the traditional methods of modelling, which are used for urban logistics, as well as statistical and probabilistic methods may be not enough effective due to their inability sufficiently consider heterogeneity, complexity and unpre- dictability of the stakeholders. These methods are not capable to provide knowledge in the whole logistics process and to include dynamics in the system. Consequently, Multi- Agent System (MAS) is used to describe the interconnections between stakeholders and measure the effect of their actions on the urban logistics politics analysis. The technique consists of three stages: specification, validation and analysis. During the first step, the information which relates to the decision structure is collected. Then validation of the created model in respect to base models is performed, after which the analysis, based on the considering different scenarios and results of the evaluation of the model, is realized in order to choose the most suitable solution [11]. As it was mentioned above MAS modelling techniques allows investigating the complex freight transport systems, in which some multiple stakeholders are involved [9]. The MAS consider every stakeholder’s group as an independent organization, that is interested in some specific aspects of the solution by creating special objects - “Agents”. Since usually two or more “Agents” participate in the decision-making process, MAS joins the possibilities, knowledge, goals and points of view of different agents and helps them to achieve their common goal through coopera- tion and coordination. The system doesn’t limit the number of “Agents” thus it is very flexible in terms of the stakeholder’s participation [11].

For the urban areas Taniguchi in [21] classified stakeholders into four groups: shippers, freight carriers, residents and administrators, based on their objectives and different types of behaviour. Shippers focus on the minimization of the costs in the supply chains. Freight carries try to perform the shippers requires which are based on the collection and deliv- ering of goods within strict time frameworks. Residents just wish to live in a noiseless place with clean air and administrators, in their turn, want to keep the sustainability of the transport system to maintain the vitality of the city. The VRPTW-D model, which is presented in the paper, increased profits for freight carriers and decreased costs for shippers due to dynamic correction of the route plan for vehicles to current travel times. Tamagawa in [22] proposed the model, in which five stakeholders were involved: shippers, freight carriers, residents, administrators and motorway operators. They used Q-learning [23] in the decision-making process for the “Agents” considering the outcomes of previous ac-

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tions. The model was applied for testing road network with the implementation of several city logistics measures. Finally, the appropriate network for all stakeholders was created, despite some small disadvantages. Boussier in [24] used MAS to model the distribution of goods in big and small cities with electric vehicles. The paper describes the simulator which considers different scenarios and pays significant attention to the sharing of parking places between freight vehicles and passengers cars.

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3 FORMULATION OF THE PROBLEM

This paper demonstrates the application of simulation methods to transport logistics based on the example of the construction of simulation systems presenting the processes of the work of the company as well as the calibration of system’s parameters in order to produce optimization of the operations that have some disadvantages. The corporation under consideration occupies the major place within the local waste-processing activity. The main components of the work system are the net of seven areas and their interconnections with the major company. Each region is represented by the organization which is responsible for the collection of the waste from the whole region area and its transportation to the major company, where it will be burned, reprocessed or handled in some other way in dependence on the kind of garbage. The research is mostly concentrated on the considering the process of the garbage transit.

The first key term here is common service schedule for the regions. Each area may have several cars which in a specified order come to the corporation, where they are served and finally come back to the point of departure. The first problem may arise in this case due to the assumption that the drivers may not strictly follow the established schedule because of unforeseen circumstances and human factors. The possibility of being late, early or not being able to come at all, may affect the appearance of the queue in which drivers will be forced to spend some time before being served. Moreover, in the case of the big queue length, some vehicles may not even enter the company due to the end of the working time. Thus, as a solution of the problem, it is supposed to study the features of arrival process, define the factors cause the emergence of the queue and influence its characteristics as well as to find the optimal conditions under which the deterioration of the company performance will be minimal. Further, this task will be called as the problem of traffic simulation.

Another important term in the traffic system is the annual plan for the implementation of garbage transportation in the amount determined for each company. Despite the existing formal schedules and plans of the realization of the work performance, the real situation does not always correspond to the assumed. Consequently, it is important to study how big is the difference between required supplies and reality. This problem can be charac- terized as the study of dynamics of the system of loads, where the loads are the amount of garbage.

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4 PROBLEM OF TRAFFIC SIMULATION

4.1 Methodology for solving the problem of traffic simulation

Considering problem-related to the traffic simulation it is logical to assume that the first step of the solution is to determine the empirical model of the traffic system. The model considers the information about the current work system but concentrates on the description of processes affecting the appearance of the queue and its influence on the performance within one working day. The formulated model can be further used for the collection of statistical data and testing, required on the stage of the verifying of the candidates for optimal parameters. The next stage after the model formulation is the quantitative simulations that allow getting a big amount of output data required for analysis of system efficiency. The number of the samples depends on the task conditions. For the current problem, the statistics were collected during one year of work simulation is assumed enough. Then the statistical analysis is used to define the problems and their sources.

If the reasons of the inefficiency and problems in the work of the company can not be uniquely determined, then the hypotheses can be formulated and verified by analyzing the results of the simulation of the model with presumably eliminated sources of problems.

Based on the conclusions about the problems the next step related to the optimization processes is realized.

The implication of the algorithm described above to the problem of traffic simulation is described in further sections.

4.2 Formulation of the empirical model

The characterization and modelling of processes can be realized based on some initial information about the work performed. The daily schedule of service for all regions can be considered as basic data. The titles of the areas don’t provide any significant information for the formulation of the model, consequently, seven areas can be coded as numbers from 1 to 7 for the convenient representation. Using these terms, the data can be written as follows:

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Table 1.Initial day schedule

Reception time Line 1 Line 2 Line 3

7:00 – 8:30 1 6

8:30 – 10:00 2 4

10:00 – 11:30 1 5

11:30 – 13:00 7 1

13:00 – 14:30 3 1

14:30 – 16:00 2

16:00 – 17:30 1 1

17:30 – 19:00 2 5

19:00 – 20:30 3 7

20:30 – 22:00 1 2

Based on this data it is possible to determine the number of cars transporting garbage for each region. Using coded areas this information can be written in the following table:

Table 2.Number of cars in each area

Area 1 2 3 4 5 6 7

Number of cars 7 4 2 1 2 1 2

For the convenience of the model construction, it is better to assume that the major company has just one service line for all regions. Under this condition, the next important step is to define all working aspects that allow organizing the arrival timetable for all areas.

Based on the schedule, it is possible to note that the working hours of the company are from 7:00 to 22:00 and this fact remains unchanged in the model with one service line.

As the addition, it can be assumed that employees need 15 minutes at the beginning of the day to prepare the equipment and same amount of time at the end of the work to switch it off. The time of one car service can be about 30 minutes and it can also be assumed that employees need half an hour break after service of every two vehicles in order to prepare for the next. It is also assumed that the employees need a one-hour break in the middle of the working day, which can be implemented between 15:15 and 16:15. Taking into account all the conditions above the arrival timetable can be written as follows:

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Table 3.Modified day schedule

The area (car) Arrival time

1 7:15

6 7:45

2 8:45

4 9:15

1 10:15

5 10:45

7 11:45

1 12:15

3 13:15

1 13:45

2 14:45

1 16:15

1 16:45

2 17:45

5 18:15

3 19:15

7 19:45

1 20:45

2 21:15

Each row of the first table column is understood as one car from the relevant coded region.

The corresponding rows of the second column, in its turn, determine the arrival time for each vehicle. This timetable is fully agreeing with the original maintenance schedule and thus the constructed data presented above can be used for further modelling and analysis.

4.3 Simulation of the traffic system

As it has been mentioned, the purpose of the model is to describe the processes which relate to the queue creation as well as to evaluate numerically its effect on the other objects of the system and the dependence between queue and efficiency of the work. Thus, the algorithm describing these processes will be formulated as event-based simulation of the traffic system. The components of such systems communicate by generation and receiving

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notifications about the event, which means the occurrence of the happening of interest, such as the change state of one of the component. Depending on the notification each component reacts in some way independent on others [25].The area of application of event-based systems is really wide. As an example, this technique was used in order to develop the model for analyzing the situation, evaluation of alternatives and maximization of utilization of some rail system [26]. Another example is presented in [27], where the authors created the New Hybrid Event-based Multi-Agent simulation model. The system is aimed at the improvement of the quality of transit service by considering the passengers and vehicles as separate classes of agents which interact in a dynamic system.

4.3.1 Parameters of the system

As initial data of event-based queuing system the following items are used:

• Information about the arrival time of each car

• Information about service time for each vehicle

• Information about working hours of a company

For the current system, the variability in the arrival time was assumed in such meaning that the cars may not directly follow the schedule determined. They can arrive earlier agreed time or come late, as it was mentioned before. Thus, apart from the arrival timetable, some additional noise will be generated.

Similar approach can be used in order to add the variability to the service time. Despite it has been already agreed to be about 30 minutes for one car, the unforeseen situations and the human factor may influence the continuation of the working process in both directions, decreasing and increasing. Consequently, the natural way to realize this variability is to consider some additional noise as well.

The common approach for the realization of the purposes mentioned above is to consider the values of the random variable that follows a specified probability distribution as a noise. The random variable is a function f, each outcome ω ∈ Ωof which associates with real number f(ω) ∈ R, where Ω is nonempty point set, representing all possible outcomes of the experiment. Let us denoteΣ as an algebra of subsets ofΩ. And let us P : Σ →R⁺be a mapping, called a probability, defined for all elements ofΣso that the following rules are satisfied

• For eachA ∈Σ,06P(A)andP(Ω) = 1

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• A, B ∈Σ,A∩B =∅, impliesP(A∪B) =P(A) +P(B)

Thus, the distribution function is a mappingF_f(x) :R→R⁺given by

F_f(x) = P(f⁻¹(−∞, x)) = P[ω :f(ω)< x] =P[f < x], x∈R

The probability density function (PDF) isf(x) = F⁰(x)[28]

The PDF of normal distribution is following:

f(x|µ, σ²) = ^√¹

2πσ²

−^(x−µ)2

2σ2

Where:

• µis the mean or expectation of the distribution

• σis the standard deviation

• σ²is the variance

The PDF of exponential distribution is:

f(x, λ) =







λexp^−λx, ifx>0 0, ifx <0

The PDF of continuous uniform distribution is:

f(x) =







1

b−a, fora6x6b 0, forx < aorx > b

Whereaandbare the boundary values [29, 28].

Initially, no sufficient statistical data has been received in order to create a hypothesis about the kind of probability distribution of the noise values. Consequently, for generation of the noise for arrival time it was decided to consider three different distributions and compare their effect on the queue.

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Thus, the first distribution that was chosen is the exponential distribution with mean pa- rameterλ = 30. Physically it means that the majority of cars will arrive just a little bit later than they have to according to the scheduled time, but the situations with significant delays are also possible. However, in this case, ”noisy” random variable may have just positive values that exclude the possibility of earlier arrival and arrival in time as well.

To avoid this the outcomes of another generator that randomly generates integer numbers uniformly distributed at the [-1 1] interval is used. Its results are multiplied by the outcome of exponential noise, so after these manipulations, the noise value is capable to have negative and zero values as well. Further, for the convenience, this kind of noise in arrival time will be marked as ATN1 (arrival time noise of the first type). The PDF and CDF of this exponential distribution are presented in the following figure:

Figure 1.Exponential distribution with parameterλ= 30

Another possible kind of distribution is normal (Gaussian), with parameters µ = 0and σ = 15. In physical terms, it means that for all drivers the maximal deviation from the predefined time can be about 45 minutes. However, such situations are least likely and usually, the arrival time of drivers will just slightly vary from scheduled, but it does not exclude the possibility of appearance too late or too early arrival. This noise is marked as ATN2 (arrival time noise of the second type).

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Figure 2. Normal distribution with parametersµ= 0andσ= 15

The least probable but still possible is noise, the values of which are uniformly distributed at the interval [-30 30]. Such kind of arrival uncertainty means that each car can have fluctuations in arrival time from 0 to 30 minutes and any deviation may occur with equal probability. The ATN3 (arrival time noise of the third type) indication will be used when referring to such kind of uncertainty in arrival time.

Figure 3. Uniform distribution with lower endpoint -30 and upper endpoint 30

Finally, the information about arrival time for each car from each area can be generated a combination of scheduled time and generated noise. For mathematical representation scheduled service time is written as numbers representing the minutes after the beginning

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of the working day. In these terms start time 7:00 corresponds to 0 and the end time which is at 22:00 corresponds to 900. Thus, arrival timetable 3 can be reformulated as a matrix, the first column of which represents the order of cars from different regions and the second one relates to arrival time written in the form mentioned above:

First column - order of cars:

[1 6 2 4 1 5 7 1 3 1 2 1 1 2 5 3 7 1 2]^τ Second column - scheduled arrival time:

[15 45 105 135 195 225 285 315 375 405 465 555 585 645 675 735 765 825 855]^τ

This matrix is denoted as SAT (scheduled arrival time). Thus, in these terms ”noisy” arrival time can be generated as the sum of scheduled and one of the generated noise (ATN1, ATN2 or ATN3, in dependence on the required conditions). The arrival time vector generated in this way is conveniently denoted by ATN (arrival time with noise).

The service time is assumed to be equal to half of an hour for every car since the differences between the vehicles were not considered. However, taking into account hypothet- ical variations which are based on the possible unobserved reasons, the service time may be generated as values of a random variable that follows normal (Gaussian) distribution with parametersµ= 30andσ= 5. The interpretation of this is that the service time will be not exactly 30 minutes but close enough to it with the maximal deviation of about 15 minutes. Like the cases above, the service time generated in this way will be denoted as STN (service time with noise).

As the attributes of the system which dynamically change during the simulation the following items are considering:

• Information about the start of a service time for each vehicle

• Information about the end of a service time for each vehicle

• Information about the queue of the cars, which has been already arrived but were not served at the considering moment

The beginning of a working day, the end of the car service process, and the end of the work are perceived as events in this system. It means that whenever one of these actions occurs the state of dynamic attributes changes.

The output parameters of the system are the total waiting time in queue for all cars and the number of the vehicles which were not served because of the limits of working hours.

Both parameters are computed separately for each area and during one working day.

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4.3.2 Algorithm of simulation of the system work

In the simulation of the work of the company due to the initial data which already contains information about service time for each car, the end time of the service for each vehicle can be calculated at the same time when the car has been taken for the service. Thus, taking the car for the service can be considered an event.

The algorithm of one-day simulation can be described by following steps:

Data: SAT

Result: TWT,CNS

Generation ofATNandSTN;

Adding ATN to the second column of SAT;

InitializationTStartand LQ are zero elements arrays of the sizeNC,TWTand CNSare zero elements arrays of the sizeNA,i= 1,TStart(i) = SAT(1,2);

TEnd(i) =TStart(i) +STN(i);LQ(i) = 0;

fori:= 2to NC do

TStart(i) =max(SAT(i,2),TEnd(i−1));

TEnd(i) =TStart(i) +STN(i);

if TEnd(i)>900then break;

else

LQ(i) =number of the elements inSATwhich corresponded indices in TStartare equal to 0 (that means that they were not served yet) and the value of second column inSATof which is less or equal to the min(900,TStart(i));

end end end

fori:= 1to NC do

TWT(i) =the sum of the differencesTStart(k)−SAT(k,2)where the summation is over allkfor which: TStart(k)is not equal to 0,SAT(k,1) =i andTStart(k)>SAT(k,2);

CNS(i) =the number of the cars indicated by indexk for whichSAT(k,1) =i andTStart(k)is equal to 0;

end

The algorithm above demonstrates one-day working processes related to the queue organization and its characteristics. In order to make the conclusions and define the problems in the organization of the system, it is necessary to repeat this simulation some predefined

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number of times. In this case, the statistics are collected as the sum of the outputs of 365 simulations, that corresponds to one year of the work of the system.

4.3.3 Results of the traffic simulation

The average waiting time in queue for one car for each area is presented in following graphs, which are the output of the one-year simulations with three different kinds of noise in arrival time, based on different kinds of probability distributions.

Figure 4. Daily average amount of waiting time in queue for one car from each area in case of ATN1

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Figure 7. Total number of unserviced cars from each area after the annual simulation in case of ATN1

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The main trend in all three cases is that for two regions the annual number of vehicles that were forced to return because of the lack of opportunity to get into the company is large enough. It is about 20 for one company and about 100 for another. Based on the data obtained, the obvious conclusion is that the system should be optimized in such a way that these numbers will be small enough. As the optimal result, the total amount of not served cars no larger than ten can be considered.

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5 PROBLEM OF DYNAMICS OF THE SYSTEM OF LOADS

5.1 Methodology for solving the problem

The next problem that was raised in this paper relates to the studying of dynamics of the system of loads, where the amount of garbage transported by each region is understood as loads. As in case of a traffic problem, it is logical to assume that the first step in the studying of the features of the system of loads is to formulate the empirical model, that concentrates on the explanation of the processes related to the transportation of garbage.

The model will consider the key factors that affect the total amount of waste that produced by each area and this number will be considered as the output, dynamics of which will be studied under different conditions. The purpose of this study is to analyze the differences between the assumed and really transported amount of garbage as well as to define the dependencies between the components of the system and these differences. Thus, the conditions under which the study will be implemented have to be defined before the further actions. After these settings, the second step in the target goal realization is similar to the second stage of the solution of the first problem. Quantitative simulations can be used in order to collect the statistics for the required analysis. Moreover, in dependence on the kind of conditions, the simulations can be performed separately for each of them, in case if they cannot be joint under one system or if it is necessary to study the unique effect of each of them on the outcome. As it was considered for the first problem, It is assumed that every simulation that characterizes one working day of the company will be repeated 365 times, providing the statistics for one year of company work.

5.2 Formulation of the empirical model

As in the previous case, the formulation of the main components of the model and their interactions is based on the initial data about the work performed. For the current problem the day schedule Table 3 will be used, however additional information about loads is required. This requirement is raised due to the lack of a sufficient amount of data for comparative analysis. In addition to the aforementioned initial data, the information about the amount of garbage transported for each area during some period can be used in order to understand the real distribution of weights in loads for each region. Based on this data, annual work simulations can be constructed. The major company provided the infor-

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mation about the total monthly amount of garbage transported from each region for nine months. Thus, the comparison of the amount of garbage computed as the monthly average of the real data multiplied by twelve months with the output of the simulations based on the real data features can be performed. Moreover, the major company has provided the information about the ”ideal” amount of garbage that is assumed to be transported from each region for one year. In addition, there is knowledge about the maximum loads for each car which is 50 tones. Considering all the information above it is possible to define one more task for data analysis. Comparison of the data related to the expected amount of garbage which is supposed to be transported for one year with the output of simulations realized with using of the information about maximal car loads can be done as well.

The next step is connected with the determination of the important characteristics that will be taken into account during the modelling of the transportation process. Due to the fact that for current problem detailed information related to the interconnections between cars or queue influence on the performance does not have high significance, the system will be concentrated just on the description of the information about the transported amount of garbage. The items that affect this number are the daily schedule of arrival and the loads for each car. Due to the two different sources of initial data for the loads generation, two systems for the simulations will be formulated. The first one will utilize the information about maximum loads for each car during the process of loads generation. The generator for the second system will use initial information about the monthly loads of trucks for nine months to study the distribution of the loads for one car separately for each region.

Resulting loads will be represented by the generated random values that follow this distribution.

In order to approximate the model characteristics to the reality, in parallel with the modelling of the dynamics of systems with the conditions described above, simulations which will take into account the possibility of unforeseen circumstances will be performed. As unforeseen circumstances, the impossibility of arriving for some number of cars due to some unpredictable reasons is perceived.

5.3 Simulations of the systems of loads

As it was mentioned earlier, the simulations of the two systems will be realized in order to analyze the dynamics of the loads. Each of the systems has a generator of the daily load for each car represented by values following some predefined distribution. The simulations activities will be performed for the 365 days. The first system contains a generator that produces values according to the assumption about the maximal load for all cars. It is obvious that in reality, despite the known maximum load, the vehicles can often be filled

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with a smaller amount of garbage. Thus, the generator will produce values sufficiently close to the predetermined maximum but with some variability. This case will be denoted asSAML(system with assumed maximal loads). The simulations of another system will be performed with the generator of values based on the initial information about the real loads for cars. This case can be labelled asSRL(system with real loads).

5.3.1 Parameters of the systems

For both systems, one initial parameter is the arrival timetable 3, from which just the first column that represents the area of the arrived car will be used.

In case of SAML it was decided to use exponential distribution with parameter λ to generate the variability in the values. Consequently, the number representing the amount of garbage for each car can be generated from the formula:

−exprnd(λ) + 50

The parameter λ is unstudied, but it is initially assumed to be equal to 30. The further calibration of the parameters will be performed during the optimization stage. Thus, for theSAMLapart of arrival timetable the initial parameter isλ.

In the case of SRLthe second initial parameter is the data that contains the information about the monthly loads for each area for nine months. This data will be used in order to compute the values of the empirical cumulative distribution function (CDF) that will be used for inverseCDFmethod for simulating from a real distribution.

The algorithm of inverse CDF method is based on the fact that random variable x = F⁽⁻¹⁾(u)has the distribution withCDFF, whereF isCDFof some continuous random variable,uis sampled fromU[0,1],F⁽⁻¹⁾ is the inverse function ofF. The visualization of this method can be performed in such way that uniform random number are ”shootted”

from the y-axis to theCDFcurve and the corresponding points in the x-axis are samples from the correct target. It is illustrated for the exponential distribution in the following figure [30]:

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Figure 10. Producing samples from the exponential distribution using inverse CDF, F(x) = 1−exp(−x)andF⁽⁻¹⁾(u) =ln(1−u)

For both systemsSAMLandSRLthe simulations will be performed twice: the first time based on the conditions mentioned above, the second time with additional settings that approximate the conditions of simulations to real. This means that the system admits the situations when some number of cars have not arrived due to the unforeseen circumstances. To achieve this the random generators are used. One of them each day (simulation) generates the number of the cars which will not arrive. This number was determined to be no more than one-fourth of the total number of cars arriving this day. Another one randomly generates the line numbers in arrival timetable 3 in the total amount equal to the number has been generated. These line numbers determine from which areas the vehicles will not arrive this day.

The initial data about the loads for each area for nine months can be collected in the following table:

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Table 4.Loads for each area for nine months

PP PP

PPP

Month

Area 1 2 3 4 5 6 7

January 3071 3831 1084 1448 901 706 719

February 3489 2350 914 1055 419 739 554

March 5054 2811 1317 1238 425 927 751

April 2476 2040 879 695 1425 607 653

May 3661 1200 781 528 658 530 777

June 2777 1941 875 1560 1132 416 853

July 2754 1598 760 633 849 523 378

August 2401 1335 482 687 408 457 618

September 2498 1368 1130 607 499 521 483

Here all values are presented in tons.

Based on the table above as well as on the table that describes the number of the cars transporting the garbage for each area (Table 3) it is possible to calculate the average daily amount of garbage transported from each region by one car through division of the values in each column by the number of cars working at the corresponding region and by division the values in each row by the number of days corresponding to the month. This structure that contains this information will be denoted as ADGT (average daily amount of garbage transported)

The output of the SAML and SRL are the annual loads computed for each area, that means the total amount of garbage transported from each region for the one year.

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5.3.2 Algorithm of simulation of the system work

The algorithm of one-day simulation can be described by following steps:

Data: arrival timetable, ADGT ifcase ofSAML then

Initialization of the parameterλ;

Generation of the day loads as arrayAof the sizeNCthe values of which follows the distribution:−exprnd(λ) + 50;

end

ifcase ofSRL then

Initialization ofADGT;

Generation of the day loads as arrayAof the sizeNCthe values of which will be obtained from inverseCDF(ADGT) method;

end

According to Table 3 andAcalculation of the amount of garbage that will be transported separately for each area;

GenerationNCNAandALN(NCNA);

Removing of rows in Table 3, line number of which are in theALN;

Repetition of the first two steps but with a modified matrix;

The algorithm above demonstrates one-day working processes related to the dynamics of loads. In order to make the conclusions, it is necessary to repeat this simulation 365 times summing the output.

5.4 Results of simulations of the systems of loads

The results of the work simulations of both systems are combined in the following figure:

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Figure 11.Results of the simulations of loads

Each of the first four columns demonstrates the oscillations in the outputs of the systems simulations, that are the minimum, maximum and average values calculated from one hundred annual simulations.

Firstly, it is possible to analyze the accuracy and validity of simulation techniques, based on the inverseCDFmethod. The results are presented by the third and fourth columns of the bar charts, where the left column represents the system where each day some number of cars has not arrived due to the unforeseen circumstances. The fifth (yellow) columns, that represents the annual real average amount of garbage for each area, are used as the item to compare with in this case. Based on the graphs, it is possible to conclude that despite the insignificant variability in the results due to the stochastic nature of the models, the outputs of the simulation are very close to the averaged values. Moreover, the addition of the realism to the model through the daily dropping of some number of cars does not cause significant changes in the simulation results, which shows the relative stability of the system to the uncertain situations. These facts prove the accuracy and reliability of constructed models as well as the validity of the application of simulation techniques in this case.

The analysis of the simulation results of the system based on the assumption about maximum loads for cars provides less optimistic conclusions. The first and second columns of the bar charts, which represents the outputs of the system with the additional condition about daily change in the number of cars and the classical system, can be compared with the fifth columns representing assumed annual transported amount of garbage for each area. Despite the relatively small influence of additional conditions on the system, which indicates its resistance to uncertain situations, for some areas, the results of the simula-

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tions significantly differ from the expected loads. Moreover, the difference has a different character for each area, that is, for some areas the values of the simulation results are higher than the expected loads, and for other areas, they are lower. The natural conclusion is that this system has to be optimized and, moreover, the optimized model has to consider individual features of the loads generation for each area.

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6 OPTIMIZATION

In order to optimize both problems, it was decided to use the algorithm of differential evolution (DE), which is a method of multi-dimensional mathematical optimization related to the class of stochastic optimization algorithms. Its advantage is that it does not require the computation of derivatives of objective functions, unlike classical optimization methods, such as gradient descent and quasi-newton methods. DEworks just with values of multidimensional real-valued functions, that is very convenient for the current case. DE is a population-based optimizer, where the population is the set of a predefined number N_p of D-dimensional vectors. DE handles three population vectors x_g, v_g, u_g for each generation g, wherex_g is a current population at the beginning of each generation, i.e. a population before any changes;v_g is the intermediate population of mutant vectors;u_g is a population of mutant vectors after the application of crossover step. The size of popu- lationsN_p and vector dimensionDremains unchanged for all generations. Considering example where the current population consists of vectors xg = (xj,i,g)j=1,...,D, i=1,...,Np, wherej denotes the vector component index andithe population member, the evolution steps can be described by following stages:

1. Initialization. For the first generation (g = 0) the initial population is usually generated by the following set: (x_j,i,g) = r_j · (b_j,U − b_j,L) + b_j,L where r_j ∼ U(0,1), b_j,U andb_j,Lare upper and lower boundaries for the corresponding dimen- sion j. For other populations initial generation is the population survived after previous step.

2. Mutation. The most popular way to calculate mutant vector is: v_i,g = x_r₁_,g + F ·(x_r₂_,g−x_r₃_,g), wherex_r₁_,g, x_r₂_,g, x_r₃_,g are mutually different randomly chosen members of current population andF > 0is a scaling factor.

3. Crossover. The process of acceptance of the mutations for the trial populationu_g is based on crossover probabilityC_r ∈[0,1]:

ui,j,g=







v_i,j,g, ifr_j ≤C_rorj =j_r xi,j,g, otherwise

.

4. Selection. The algorithm of selection of a new population is based on the values of the cost functionf

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x_i,g+1 =







u_i,g, ifu_i,g ≤f(x_i,g) x_i,g, otherwise .

Thus, one can see that the selection algorithm is similar to the process of evolution where only the strongest representatives of the old generation and children who surpass their parents survive in new generation.

The sequence of steps described above will be repeated until the population will satisfy the required criteria or a specified number of generations is reached [31].

Thus, the algorithm does not require strict mathematical formulation of the cost function.

It is enough just to define the way of the calculation of its output value and its parameters.

To demonstrate the operation of the algorithm, it can be used to find the minimum of a simple two-parameter function

x²+xy+y²+x−y+ 1 =z

This function has an extremum at the pointx=−1, y = 1, which is the minimum of the function andz(−1,1) = 0. The application ofDEalgorithm is presented in the following table:

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Table 5. DEalgorithm: simple case without noise

x initial y initial z(x,y) → x final y final z(x,y)

0.6521 1.7927 4.6675 → -1.0000 1.0000 1.0e-15·0.1110 0.6897 1.9200 5.2562 → -1.0000 1.0000 1.0e-15·0 1.0559 0.9099 4.0496 → -1.0000 1.0000 1.0e-15·0 0.8733 0.8703 3.2831 → -1.0000 1.0000 1.0e-15·0 1.4336 1.4895 7.3531 → -1.0000 1.0000 1.0e-15·0 1.3928 1.1540 6.1178 → -1.0000 1.0000 1.0e-15·0 0.9142 0.5648 3.0204 → -1.0000 1.0000 1.0e-15·0 1.3835 1.5268 7.2143 → -1.0000 1.0000 1.0e-15·0 1.1588 0.5970 3.9527 → -1.0000 1.0000 1.0e-15·0.1110 1.3209 0.5284 4.5143 → -1.0000 1.0000 1.0e-15·0.1110 0.9858 1.8117 6.2144 → -1.0000 1.0000 1.0e-15·0 0.5245 1.8193 4.2441 → -1.0000 1.0000 1.0e-15·0.2220 0.5664 0.0158 1.8807 → -1.0000 1.0000 1.0e-15·0 1.4127 0.7700 5.3192 → -1.0000 1.0000 1.0e-15·0.1110 1.3336 1.3507 6.3870 → -1.0000 1.0000 1.0e-15·0 1.1322 0.4052 3.6317 → -1.0000 1.0000 1.0e-15·0 0.5917 0.5482 2.0184 → -1.0000 1.0000 1.0e-15·0 1.4162 1.4214 7.0336 → -1.0000 1.0000 1.0e-15·0 0.9926 1.8848 6.5165 → -1.0000 1.0000 1.0e-15·0

This table presents the initial generation (from the left side) where each population was randomly generated and the generation generated on 100^th algorithm iteration (from the right side). As can be seen from the table the optimizer coped well with his task. However, this case is quite artificial and usually cost functions may have a stochastic character. And DEalgorithm is supposed to cope with searching for minimum within this conditions as well. This phenomenon can be illustrated by the application of the algorithm to the previous example with modernized cost function where instead Instead of the last operation of adding +1 some small random noise will be added. The table below represents the results of the algorithm for the case described above, where two objective functions with stronger and weaker noise were used.

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Table 6. DEalgorithm: simple case without noise

x1 final y1 final z1(x1,y1) ↔ x2 final y2 final z2(x,y) -0.9967 1.0268 -0.9975 ↔ 0.0317 0.0081 0.0234 -0.9761 0.9781 -0.9986 ↔ -1.5377 0.9318 -0.6685 -0.9870 1.0153 -0.9980 ↔ -1.0861 1.2945 -0.9324 -0.9889 0.9991 -1.0010 ↔ -1.4768 0.9390 -0.7418 -1.0195 1.0344 -1.0008 ↔ -1.7041 1.1206 -0.5754 -0.9936 0.9846 -1.0011 ↔ -0.7759 0.5003 -0.8123 -1.0002 0.9998 -0.9997 ↔ -2.7427 1.4991 1.4178 -0.9932 0.9941 -0.9998 ↔ -0.7825 1.4400 -0.6638 -1.0054 1.0101 -1.0002 ↔ -0.8345 1.3807 -0.7641 -0.9669 0.9857 -0.9989 ↔ -1.1633 0.6282 -0.7752 -1.0473 1.0261 -0.9975 ↔ -0.5629 -0.4204 0.5884 -0.9993 0.9780 -0.9992 ↔ -1.5567 1.6031 -0.6610 -0.9853 0.9822 -1.0009 ↔ -0.6880 1.1531 -0.8310 -1.0266 1.0326 -0.9989 ↔ -1.1173 0.9875 -0.9836 -0.9997 1.0225 -0.9985 ↔ -2.3992 1.8536 0.4914 -1.0177 1.0022 -0.9993 ↔ -2.0944 1.9560 0.0646 -1.0308 1.0066 -0.9979 ↔ -0.0664 1.3294 0.2879 -1.0085 0.9817 -1.0019 ↔ -1.8162 1.1249 -0.4202 -0.9932 1.0075 -0.9998 ↔ -0.7466 1.5760 -0.4587 -0.9969 1.0104 -1.0012 ↔ -1.4248 1.4393 -0.8136

The left part represents the results of searching for the minimum for cost function with slight noise generated as normally distributed random numbers with zero mean and unit variance scaled by multiplication by 0.001. The right part, respectively, is the result of the optimizer for the cost function with stronger noise generated as normally distributed random numbers with zero mean and unit variance. Based o the results of simulation it is possible to make several conclusions. Firstly, DEalgorithm is able to optimize the objective function under uncertainty. However, the more stochastic the cost function, the less accurate the results of the algorithm work.

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6.1 Optimization of the traffic system

For the calculation of the cost function in case of traffic simulation, it was decided to consider the total number of cars from all areas which were not serviced during annual simulation. The parameters of cost function under optimization are the mean valueµand standard deviationσof the normal distribution which is used for generation of the service time initially where determined as µ = 30andσ = 5 for all variants of noise. The optimal parameters were defined separately for each variant of noise. Taking into account the physical aspects of the real service process, some limitations were introduced into the optimization algorithm. Thus, theµparameter was set to be no less than ten and the de- pendences betweenµandσwere also considered. Consequentially, If the new candidate does not meet the boundary conditions described above, the value of the corresponding parameter increases significantly. This action automatically excludes him from the set of potential candidates for a new generation.

The traffic simulation system is stochastic, with both the service time and the arrival time being indeterminate. For this reason, the result of the algorithm is also not accurate and it strongly depends on the generated noise. Thus, there is no possibility to identify the optimal value of each parameter as a unit number. As an inference, we can only determine the confidence interval for the parameters and demonstrate how the cost function oscillates within the frameworks described above.

In case of exponential noise in the arrival time, the optimal parameters for noisy service time distribution fluctuate within the following intervals: µfrom10,04to21,96 and σfrom0,34to4,31. The change in the cost function, representing the total annual number of cars from all areas that were not serviced, is presented in the following graph:

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Figure 12.Optimized service time in case of ATN1

The blue square in down part represents the confidence interval for theµandσparameters and green square presents changes in the cost function under these constraints.

Considering the average values of the parameters which were calculated after running of the optimization multiple times, it is also possible to estimate how the waiting time in the queue has changed. The average parameters areµ= 13.52andσ = 1.84.

Figure 13. Optimized daily average amount of waiting time in queue for one car from each area in case of ATN1

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As it can be seen from the pictures, in case if the approximate normal service time varies from 10 to 21 minutes and the random approximate deviations are from 1 to 13 minutes, the waiting time does not exceed three minutes for each area in contrast to the original case when it was around seven. And the number of not served cars, which is annually from 54 to 92, decreased approximately twice as compared to the initial simulation. These facts indicate successful optimization.

Considering the case of normal arrival time noise, the optimal parameters for noisy service time distribution fluctuate within approximately the same intervals:µfrom10,02to20,78 and σfrom0,23to3,34. The graphical illustration of the work of optimizer is presented below:

if, similarly with the previous case, the averaged values of the parameters when simulating the waiting time will be considered, which are µ = 13.6 andσ = 1.37, then the graph will be as follows:

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Thus, in case of ATN2 and service time fluctuating from 10 to 21 minutes with inaccura- cies from 1 to 10 minutes, the number of annually unserviced cars variates from 3 to 22 which is significantly less than it was with non-optimized parameters. And the waiting time decreased from 4 to maximum 1 minute. Thus, in this case, the optimization also served well.

The confidential interval in case of uniform distribution of the noise in the service time is following: µfrom11,33to19,95 andσfrom0,23to4,25. The representative graph is located below:

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The averaged waited time with averaged optimal parametersµ= 14.3andσ = 1.5is the following:

Thus, in case of uniformly distributed fluctuations in arrival time from 0 to 30 minutes and service time between 11 and 20 minutes with the delays approximately from 1 to 13

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minutes, just maximum 20 cars will be not serviced during the year instead of 140 before optimization. At the same time, the waiting time will not exceed 1,5 minutes for each car in contrast with averaged 5 minutes of waiting in the non-optimized system.

6.2 Optimization of the system of loads

The value of cost function for theDEoptimization of the system of loads may be calculated as the absolute value of the sum of differences between the outputs of the simulations based on the assumption about the maximal loads and the values of loads which are assumed to be transported by the company. The parameter that effects the loads generation in this case is the value ofλin exponential distribution. Thus, this value has to be considered as the parameter of the cost function. Moreover, as it was concluded in the section 5.4, the parameterλhas to be different for various areas. Thus, the cost function has seven parametersλ₁, λ₂, λ₃, λ₄, λ₅, λ₆, λ₇ each of which corresponds to the relevant field. The system of loads is also stochastic, that makes difficult accurate differentiation of optimal parameters. As in the previous case, it is possible to define confidential intervals for each of the parameters. They are presented in the following table:

Table 7.Optimized parameters for the system of loads λ_i bottom bound upper bound

i= 1 31.55 34.99

i= 2 28.4 31.75

i= 3 29.1 34.15

i= 4 10.93 21.36

i= 5 35.61 42.99

i= 6 19.42 40.26

i= 7 35.42 49.92

The results of the simulations with averaged values ofλ₁, λ₂, λ₃, λ₄, λ₅, λ₆, λ₇from confidential intervals are presented below:

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Figure 18.Results of the simulations of the systems of loads after optimization

The results of the simulation of the system, based on the assumption about maximum loads generated with averaged optimal parameters, which are represented by the first and second columns for each of the regions, are fairly close to those originally suggested by the company (the last red columns). Thus, we can conclude that the optimization was successfully carried out.

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7 CONCLUSIONS

In this paper, the analysis of existing simulation technologies for solving the problems of transport logistics has been carried out. The analysis included the study of existing simulation methods and their application for the simulation of the Finnish company processes.

The classification proposed by Taniguchi in [9] was used in the paper. He divided the methods into the three groups, such as multi-agent systems, system dynamics and traffic simulation. The application of the last two was illustrated by the development of simulation models for the presumably not optimal processes of the Finnish company, related to the traffic simulation and the system dynamics of the loads. The information about the classification is presented in Chapter 2. Based on the simulation results and comparing them with the existing real statistical data we can conclude that, despite the variability in the results, they approximate well the company processes. The models as well as the results of their simulations are presented in the Chapters 4 and 5. The variability in outputs is caused by the stochasticity of the models, which, is in its turn, caused by the lack of information and requirement of the assumptions and consideration of different possible conditions.

However, despite the proved reliability of simulation methods, the results of the simulations confirmed the non-optimality of the processes. For the solution of this problem, it was decided to analyze and define the parameters, initial values of which caused the non-optimality. After their differentiation, the differential evolution optimization was performed. The realization of the optimization is presented in the Chapter 6. Due to the properties of this algorithm, the values of the objective function were determined as the output of the simulations. Thus, the possibility of the effective processes optimization based on simulation methods was illustrated as well.

Based on the above information, we can conclude that the application of simulation technologies for description the processes is a quite new and promising way. The model formulated in this way is flexible enough and allows describing any, even the most insignificant details of the process, which in turn makes it as close to real conditions. The results of the simulations can also be considered as values of the target function, which in turn allows you to optimize the system. In this paper, all steps such as the construction of the simulation model, the evaluation of simulation results and the implementation of optimization were done for the processes of the Finnish company and they prove the correctness of the above statements.

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7.1 Future Work

All the processes in this paper were described by models optimized in the absence of a large amount of initial information. This phenomenon led to the need for constructing hypotheses and considering various scenarios. For this reason, each model has several random parameters. This fact does not allow to make a more precise evaluation of the optimal parameters for the system than the confidence interval. Therefore, one of the options for future work may be to conduct the same research for the company, whose system of work is sufficiently predefined and evaluate the results of optimization within such conditions. Otherwise, another possible future direction is to study and apply methods of robust statistics for the models described in the paper.

Simulation methods for transport logistics