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Case Study: Logistical Behavior in the Use of Urban Transport Using the Monte Carlo Simulation Method

  • Lorenzo Cevallos-TorresEmail author
  • Miguel Botto-Tobar
Chapter
Part of the Studies in Computational Intelligence book series (SCI, volume 824)

Abstract

This study presents a proposal to determine solutions to the models of queue theory through the use of simulation. The main objective is to evaluate the number of people who arrive at a public transport service station in order to be able to minimize monetary losses, the product of the defection of the people of the waiting line of this station. To evaluate the model, we proceeded to use tools that allow simulating random values based on probability distributions; such as the Log-Normal probability distribution, and the Binomial distribution.

This study presents a proposal to determine solutions to the models of queue theory through the use of simulation. The main objective is to evaluate the number of people who arrive at a public transport service station in order to be able to minimize monetary losses, the product of the defection of the people of the waiting line of this station. To evaluate the model, we proceeded to use tools that allow simulating random values based on probability distributions; such as the Log-Normal probability distribution, and the Binomial distribution. Our study case was a public taxi transport stop located in Victor Manuel Rendon and Pedro Moncayo streets in Guayaquil city, where it was likely to observe all people waiting for the taxi service. We used simulation methods to obtain estimations from real cases.

6.1 Introduction

The search for explanations for phenomena that happen randomly has led the human being to the need to use mechanisms that allow him to quantify in an imperfect way the possibility of an event occurring. A tool that measures results under some uncertainty event is the probability. To model real situations under probabilistic reasoning, it is essential to use of procedures that allow reasonably to emulate these situations; A resource that helps to obtain estimates or approximations of cases of a real situation is the simulation [1, 2, 3, 4, 5].

The construction of a simulation tool has allowed giving feasible and optimal solutions to a problem that is presented daily in a public transport service station, and the citizens very request that at different times of the day. This tool constitutes an alternative to a certain extent economic; for the evaluation of the service quality, and it is useful for supporting decision making, and moreover, allows to define and evaluate common performance measures of waiting for lines, such as the arrival and waiting time of people in the transport station [6, 7, 8].

6.1.1 Related Work

Martinez et al. [9] proposed obtaining a quantitative predictive model as an effective way to address the problem of daily demand of passengers in a bus line. In order to solve the problem, the author has used SQL Server Management to tackle and manage simulated data through a database. However, this environment requires users with enough knowledge in this tool.

In the study carried out by Lojano et al. [10] used a hybrid model combining, as they are, the multi-indicator and multiple-cause (MIMIC) model, and the theory of random utility. In order to simulate the demand for passengers, they use the Quito-Cable, the data were obtained by predictions of quantifiable variables such as time, service operational costs, and service prices. However, this research proposed the inverse transformed and Monte Carlo method for obtaining the data; and a computational tool was used for processing them.

6.2 Case Study

A circumstance is given on a daily basis in a taxi’s company that transport passenger to Duran city. By using the observation [11, 12] in order to reach this goal, we started by taking data such as the arrival time (time in which people arrive at the transport station), during an hour per day, and also identify the number of people who left the queue for different reasons; this results are marked by “*” and are presented in Table 6.1.
Table 6.1

Arrivals data to the station

Time

Conversion

Time

Conversion

1

0:00:31

0.00861

26

0:00:40

0.01111

2

0:00:45

0.01250

27

0:01:13

0.02028

3

0:00:27

0.00750

28

0:02:06

0.03500 *

4

0:01:02

0.01722

29

0:01:02

0.01722

5

0:00:50

0.01389

30

0:01:08

0.01889

6

0:01:10

0.01944 *

31

0:00:06

0.00167

7

0:00:59

0.01639

32

0:00:15

0.00417

8

0:00:40

0.01111

33

0:00:34

0.00944 *

9

0:00:52

0.01444 *

34

0:01:02

0.01722

10

0:01:08

0.01889

35

0:01:04

0.01778

11

0:02:03

0.03417

36

0:00:43

0.01194

12

0:00:35

0.00972

37

0:00:19

0.00528 *

13

0:00:27

0.00750

38

0:00:53

0.01472

14

0:00:51

0.01417

39

0:01:32

0.02556

15

0:01:25

0.02361 *

40

0:02:08

0.03556

16

0:01:04

0.01778

41

0:01:21

0.02250

17

0:01:30

0.02500

42

0:01:03

0.01750 *

18

0:01:16

0.02111

43

0:02:02

0.03389

19

0:00:58

0.01611 *

44

0:01:02

0.01722 *

20

0:01:08

0.01889

45

0:01:15

0.02083

21

0:00:44

0.01222

46

0:01:07

0.01861

22

0:00:11

0.00306 *

47

0:00:55

0.01528

23

0:00:17

0.00472

48

0:02:37

0.04361 *

24

0:00:54

0.01500

49

0:00:42

0.01167

25

0:00:37

0.01028

50

0:01:05

0.01806

Table 6.1 also presents the time conversion in hours by applying the following formula:
$$\begin{aligned} conversion = \frac{hour(cell)+ minutes(cell)}{\frac{60+seconds(cell)}{3600}} \end{aligned}$$
(6.1)
Reducing the formula is:
$$\begin{aligned} conversion = cell * 24 \end{aligned}$$
(6.2)

6.2.1 Queue Theory Model

Queue management or line-of-waiting models consist of some elements that generate waits on entities or agents waiting to be served. In general, queue models have two components. The elements that are processed in the system are entities, and these entities are handled according to a defined criterion [13].

Two phases were followed for the construction of the simulation model. The first was the identification of the cases to be modeled, and then data time was taken corresponding to the times to arrivals to the system. To adjust the data in a statistical distribution, we proceeded to use stat-fit. that was very helpful to be able to identify which is the probability allocation that best fits our problem [14, 15].

6.2.2 Using Computer Tools: Stat-Fit

It is used to analyze and determine the type of probability distribution of a dataset, in such a way that allows to compare the results between several distributions analyzed by a qualification. Once the data have been compiled, an analysis is performed, and then the generated data are shown.
Fig. 6.1

Verification of rejected and non-rejected distributions. Log-normal Distribution Graph

Figure 6.1 shows Stat-Fit probability distributions results, in both distributions that are accepted and those rejected. For this case study, we will use Log-Normal probability distribution [16, 17, 18].

6.2.3 Log-Normal Distribution

The log-normal distributions was used in [19] to determine arrivals times of buses. Therefore, in this we applied similarly, to simulate the arrival time people to the waiting queue [17].

The log-normal distribution is obtained when the normal distribution describes the logarithms of a variable [20, 21].

Where:

\(\mu \) \(=\) It is the mean of ln(x).

\(\sigma \) \(=\) It is the standard deviation of ln(x).

ln(x) \(=\) It is a random variable that has a normal distribution.

We built a simulator using a log-normal distribution, to this, the verification process was done that consisted of making the pilot runs and observe the behavior of these.

Then, we determine a procedure that allows creating pseudo values by applying the probability log distribution formula-Normal and the Monte Carlo method [22]. Later, the final results are made the conversion in time format (hh:mm:ss), and it was obtained the probability of each outcome of the arrival people (see Algorithm 1).

Once the verification is carried out, this behavior is contrasted with the information provided in Table 6.2, where the process of the log-normal distribution is presented using random numbers to generate arrival times for people with their respective probability percentage (Fig. 6.2).
Fig. 6.2

Data generated from the Log-normal distribution

Table 6.2

Data generated from the log-normal distribution

No

Random

Log-normal formula

Arrive PERs

PROB (%)

ACUM (%)

1

0.683

0.00852698

0:00:31

2.38

2

2

0.165

0.00333393

0:00:12

0.93

3

3

0.753

0.00913470

0:00:33

2.55

6

4

0.091

0.00204823

0:00:07

0.57

6

5

0.388

0.00526711

0:00:19

1.47

8

6

0.592

0.00660240

0:00:24

1.84

10

7

0.422

0.00536830

0:00:19

1.50

11

8

0.316

0.00277153

0:00:10

0.77

12

9

0.426

0.00647585

0:00:23

1.81

14

10

0.076

0.00069473

0:00:03

0.19

14

11

0.861

0.01592467

0:00:57

4.44

18

12

0.450

0.00602272

0:00:22

1.68

20

13

0.618

0.00738569

0:00:27

2.06

22

14

0.471

0.00462775

0:00:17

1.29

23

15

0.139

0.00170623

0:00:06

0.48

24

16

0.092

0.00162126

0:00:06

0.45

24

17

1.000

0.01798504

0:01:05

5.02

29

18

0.641

0.00968650

0:00:35

2.70

32

19

0.587

0.00983501

0:00:35

2.74

35

20

0.213

0.00584133

0:00:21

1.63

37

21

0.724

0.01254751

0:00:45

3.50

40

22

0.713

0.00514247

0:00:19

1.43

41

23

0.443

0.00896722

0:00:32

2.50

44

24

0.636

0.00512337

0:00:18

1.43

45

25

0.040

0.00057840

0:00:02

0.16

46

26

0.859

0.01065522

0:00:38

2.97

48

27

0.711

0.00723388

0:00:26

2.02

51

28

0.815

0.00567573

0:00:20

1.58

52

29

0.353

0.00320361

0:00:12

0.89

53

30

0.084

0.00090140

0:00:03

0.25

53

31

0.668

0.00712541

0:00:26

1.99

55

32

0.449

0.00976155

0:00:35

2.72

58

33

0.580

0.01360042

0:00:49

3.79

62

34

0.807

0.02654637

0:01:36

7.40

69

35

0.446

0.00417016

0:00:15

1.16

70

36

0.296

0.00289687

0:00:10

0.81

71

37

0.001

0.00002162

0:00:00

0.01

71

38

0.384

0.00342935

0:00:12

0.96

72

39

0.330

0.00765315

0:00:28

2.13

74

40

0.279

0.00652828

0:00:24

1.82

76

41

0.279

0.00358722

0:00:13

1.00

77

42

0.691

0.02696885

0:01:37

7.52

85

43

0.379

0.00661347

0:00:24

1.84

86

44

0.688

0.00495803

0:00:18

1.38

88

45

0.589

0.01350218

0:00:49

3.77

92

46

0.613

0.01027003

0:00:37

2.86

94

47

0.305

0.00825954

0:00:30

2.30

97

48

0.033

0.00029565

0:00:01

0.08

97

49

0.492

0.00642460

0:00:23

1.79

99

50

0.273

0.00500267

0:00:18

1.40

100

0:21:31

100.00

After taking the data, the service times were analyzed with Algorithm 2. We determined the procedure allow to create pseudo values by applying the binomial probability distribution formula and the Montecarlo method. Later, the final results are compared with the probability obtained, if it is greater than 0.5, it is added a counter, which is counting all interactions, and finally, it shows the number of people who leave the queue for some reason.

6.2.4 Binomial Probability Distribution

It is a discrete probability distribution that counts the number of successes in a sequence of n trials of Bernoulli independent among themselves, with a fixed probability of occurrence of success between trials [23, 24].

It is applied the distribution of binomial probability to know the probability of success of finding algae within 1 quadrant. In our study, we implemented the same distribution to determine the probability of a person leaving the waiting line at the taxi station [25, 26].
$$\begin{aligned} p(x) = \bigg (\frac{n}{x}\bigg )p^x(1-p)^{n-x} \forall x \in \{0,1,\ldots ,n\} \end{aligned}$$
(6.3)
where:
  • n. It’s the number of tests.

  • k. It’s the number of hits.

  • p. It’s the probability of success.

  • q. It’s the probability of failure.

According to the result generated by Algorithm 2. Table 6.3 shows the process of binomial distribution using random numbers to generate the probability each person has when leaving the queue and the total number of people of leaving it (Fig. 6.3).
Table 6.3

Data from people leaving the queue

No

Random

Binomial formula

Prob (%)

RESULT

Contd

1

0.843

0.01381154

0.38

0

21

2

0.019

0.01783901

0.54

1

3

0.774

0.01424622

0.39

0

4

0.275

0.01739878

0.52

1

5

0.281

0.01739878

0.52

1

6

0.704

0.01466523

0.41

0

7

0.955

0.01243539

0.32

0

8

0.380

0.01698638

0.51

1

9

0.142

0.01776783

0.54

1

10

0.453

0.01645167

0.49

0

11

0.005

0.01783901

0.54

1

12

0.563

0.01580689

0.46

0

13

0.142

0.01776783

0.54

1

14

0.320

0.01720853

0.52

1

15

0.818

0.01381154

0.38

0

16

0.297

0.01739878

0.52

1

17

0.580

0.01580689

0.46

0

18

0.518

0.01614218

0.47

0

19

0.456

0.01645167

0.49

0

20

0.041

0.01783901

0.54

1

21

0.954

0.01243539

0.32

0

22

0.907

0.01290393

0.34

0

23

0.314

0.01720853

0.52

1

24

0.610

0.01544764

0.44

0

25

0.111

0.01776783

0.54

1

26

0.011

0.01783901

0.54

1

27

0.500

0.01614218

0.47

0

28

0.894

0.01336338

0.36

0

29

0.370

0.01698638

0.51

1

30

0.016

0.01783901

0.54

1

31

0.591

0.01580689

0.46

0

32

0.579

0.01580689

0.46

0

33

0.583

0.01580689

0.46

0

34

0.497

0.01645167

0.49

0

35

0.888

0.01336338

0.36

0

36

0.148

0.01776783

0.54

1

37

0.641

0.01544764

0.44

0

38

0.882

0.01336338

0.36

0

39

0.710

0.01466523

0.41

0

40

0.243

0.01755600

0.53

1

41

0.376

0.01698638

0.51

1

42

0.637

0.01544764

0.44

0

43

0.325

0.01720853

0.52

1

44

0.593

0.01580689

0.46

0

45

0.416

0.01673361

0.50

0

46

0.083

0.01782119

0.54

1

47

0.756

0.01424622

0.39

0

48

0.555

0.01580689

0.46

0

49

0.683

0.01506640

0.43

0

50

0.181

0.01767926

0.54

1

0.80184529

23

Fig. 6.3

Data from people leaving the queue

6.3 Results

By using Algorithm 3, the results of the simulation processes are presented with the purpose of showing the effect of people arriving at the station and those who leave the queue, for different reasons, using Binomial probability distribution and log-normal probability distribution.

Tables 6.4, 6.5, and 6.6 present the total number of people entering the queue and leaving the line for different reasons marked with *.
Table 6.4

Simulation results of month 1

Monday

Tuesday

Wednesday

Thursday

Friday

Sum

Prob

Week 1

129

55

129

60

148

521

0,33

Leave

21*

24*

17*

22*

15*

99*

Week 2

112

48

44

46

51

301

0,19

Leave

18*

20*

18*

15*

22*

93*

Week 3

82

56

39

63

59

299

0,19

Leave

14*

21*

20*

17*

18*

90*

Week 4

117

92

70

115

50

444

0,28

Leave

14*

18*

12*

24*

26*

94*

Subtotal

1565

1

Lost

376*

0,24

Total

1189

0,76

In the simulation of the first month, the total number of people entering the queue is 1,565 of which 376 leave the queue for different reasons; being 1,189 the number of people using the service, generating $1,189 gain and $376 loss.

In the simulation of the second month, the total number of people entering the queue is 1,844 of which 381 leave the queue for different reasons; being 1,463 the number of people using the service, generating $1,463 gain and $381 loss.
Table 6.5

Simulation results of month 2

Monday

Tuesday

Wednesday

Thursday

Friday

Sum

Prob

Week 1

59

113

150

79

76

477

26

Leave

19

23

20

21

24

107 *

Week 2

58

143

35

65

128

429

23

Leave

17

25

14

23

19

98 *

Week 3

60

39

97

38

139

373

20

Leave

18

12

17

20

15

82 *

Week 4

115

150

54

129

117

565

31

Leave

20

27

13

19

15

94 *

Subtotal

1844

1

Lost

381 *

21

Total

1463

0,79

Table 6.6

Simulation results of month 3

Monday

Tuesday

Wednesday

Thursday

Friday

Sum

Prob

Week 1

32

44

101

52

94

323

19

Leave

14

19

22

24

20

99 *

Week 2

138

90

145

123

59

555

0,32

Leave

16

22

20

15

17

90 *

Week 3

64

114

72

69

140

459

27

Leave

21

14

17

20

18

90 *

Week 4

100

50

65

33

129

377

22

Leave

22

24

19

17

21

103 *

Subtotal

1714

1

Lost

382 *

22

Total

1332

0,78

In the simulation of the third month, the total number of people entering the queue is 1,714 of which 382 leave the queue for different reasons; being 1,332 the number of people using the service, generating $1,332 gain and $382 loss.

Monthly the taxi station loses about 23% of the profits being equivalent to $380.

6.4 Conclusion

Analyzing the results obtained in 3 months, we can conclude that the taxi cooperative can lose about 23% of the profits in each month If you keep the same way of managing. It can be solved in different ways, for example, the acquisition of new vehicles in order to meet daily demand, preventing people from leaving the queue.

Finally, it can be concluded that using the log-normal probability distribution tools and the Binomial probability distribution, Montecarlo and Visual Basic for Excel, the simulated data of the consecutive three months can be obtained and their possible losses. Thus, it is recommended for future work a meticulous analysis of one year to make the results more accurate.

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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Faculty of Mathematical and Physical SciencesUniversity of GuayaquilGuayaquilEcuador
  2. 2.Eindhoven University of TechnologyEindhovenThe Netherlands
  3. 3.University of GuayaquilGuayaquilEcuador

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