https://ijmri.de/index.php/jmsi
volume 4, issue 3, 2025
740
PROBABILITY AND OPTIMAL CONTROL APPROACH IN ANALYSIS OF TAXI
DRIVER'S MOTION STRATEGY
Mamatova Zilolakhon Khabibullokhonovna
Associate Professor, Doctor of Philosophy (PhD) in Pedagogical
Sciences, Fergana State University
Orcid: 0009-0009-9247-3510
E-mail: mamatova.zilolakhon@gmail.com
Gulchiroy Shuhatjonova Shuhratjon kizi
Fergana State University
E-mail:shuhratjonovagulchiroy@gmail.com
Abstract:
The Taxi Driver Issue is a practical model of process research and optimal
management science based on Markov decision processes. This issue aims to find the optimal
strategy that will give the driver the maximum benefit based on the different options of action in
each city situation. Using an iterative algorithm, the optimal stationary strategy is determined
based on probabilities and rewards. This article analyzes the movements between towns A, B and
C and identifies the strategy of waiting for the most useful decision – queue waiting. The issue is
used in the service management, transportation and logistics management sectors.
Keywords:
Process research, optimal management, Markov decision process, iterative algorithm,
probability, benefit function, strategic planning, taxi system, status and action, stationary strategy,
service system, probability matrix, maximum profit, cost-effectiveness.
Annotatsiya:
Taksi haydovchisi masalasi jarayonlar tadqiqoti va optimal boshqaruv fanining
Markov qaror jarayonlariga asoslangan amaliy modelidir. Ushbu masala haydovchining har bir
shahar holatida turli harakat variantlari asosida maksimal foyda keltiradigan optimal
strategiyasini topishga qaratilgan. Iteratsion algoritm yordamida ehtimolliklar va mukofotlar
asosida optimal statsionar strategiya aniqlanadi. Mazkur maqolada A, B va C shaharchalari
o‘rtasidagi harakatlar tahlil qilinib, eng foydali qaror – navbat kutish strategiyasi aniqlangan.
Masala xizmat ko‘rsatish, transport va logistikani boshqarish sohalarida qo‘llaniladi.
Kalit so‘zlar:
Jarayonlar tadqiqoti, optimal boshqaruv, Markov qaror jarayoni, iteratsion
algoritm, ehtimollik, foyda funksiyasi, strategik rejalashtirish, taksi tizimi, holat va harakatlar,
statsionar strategiya, xizmat ko‘rsatish tizimi, ehtimollik matritsasi, maksimal foyda, iqtisodiy
samaradorlik.
Аннотация:
Задача таксиста — это практическая модель исследования процессов и науки
оптимального управления, основанная на марковских процессах принятия решений. Эта
задача направлена на поиск оптимальной стратегии водителя, которая максимизирует
прибыль на основе различных вариантов действий в каждом городе. С помощью
итерационного алгоритма определяется оптимальная стационарная стратегия на основе
вероятностей и вознаграждений. В этой статье анализируются перемещения между
городами A, B и C и определяется наиболее выгодное решение — стратегия ожидания в
очереди. Задача используется в сферах обслуживания, транспорта и управления
логистикой.
Ключевые слова:
Исследование процессов, оптимальное управление, марковский
процесс принятия решений, итерационный алгоритм, вероятность, функция прибыли,
https://ijmri.de/index.php/jmsi
volume 4, issue 3, 2025
741
стратегическое планирование, система такси, состояние и действия, стационарная
стратегия, система обслуживания, матрица вероятностей, максимальная прибыль,
экономическая эффективность.
Introduction: Process research and optimal control are the sciences that serve to make decisions
in complex systems and ensure the most efficient use of available resources. Markov decision
processes play an important role in this science, as they allow to ensure the maximum expected
benefit through a sequence of actions. In this study, the goal is to find the optimal strategy by
modeling a real-life situation in which a taxi driver may be in different urban situations (A, B, C).
In each situation, there are several solutions (actions), each of which is represented by different
probabilities and rewards. The goal is to find the strategy that will bring the driver the greatest
benefit.
Literature review
A review of scientific research on process research and optimal control shows that decision-
making systems based on probabilistic models are effectively used in the service and transport
sectors. Dynamic programming and Markov processes developed by R. Bellman create the main
theoretical basis in this regard. G. Dantzig's work is focused on determining optimal strategies
using linear and nonlinear programming. L. Kantorovich, on the other hand, connected the
theory of optimal resource allocation with economic models. At the moment, models using
Markov processes provide effective solutions in service systems, especially in transport and taxi
systems.
Research methodology
The study is aimed at determining the most profitable strategy for a taxi driver based on the
probabilities that can be in three cities - A, B and C. The methodological approach includes the
following stages:
1. Mathematical modeling - a model is built based on the probabilities and rewards
corresponding to each action;
2. Iterative calculation algorithm (value iteration) - the maximum expected profit is calculated
for each case;
3. Optimal strategy selection - the profit for each solution is evaluated and the best action is
determined;
4. Stationary strategy determination - when the selected solutions do not change in all cases, this
strategy is considered optimal.
The study found that for all cases, the most profitable strategy is to go to the nearest taxi stand
and wait in line. These results reveal opportunities to improve the efficiency of service systems
and save time and resources. This methodology can be applied to transport systems, logistics and
resource allocation.
Analysis and results
A taxi driver operates in an area consisting of three towns A, B and C. If the driver is in town A,
he has 3 options:
1) drive from one place to another in the town in order to meet a random passenger;
2 ) go to the nearest taxi stand and wait in line;
3) wait for a call while wearing a radio headset.
If the driver is in town C, there are three possibilities as above. However, since radio
communication is not established in town B, the driver is deprived of the last possible option. For
each city (state) and option (solution), the probability of the next trip being in cities A, B, and C
https://ijmri.de/index.php/jmsi
volume 4, issue 3, 2025
742
and the corresponding net benefit (gain) are given. For example, when calculating the net benefit
corresponding to solutions 1 and 2, the cost of traveling from one place to another, to the nearest
taxi stand, must be taken into account. In this case, the transition probabilities and gains depend
on which solution is adopted, because for each solution. The driver faces different density
distributions of passengers. We express this information in the following table:
Assume q = 0.9. Let the initial plan be taken from the point of maximizing the points based on
the statement P, that is, P =.
Table
Holat Yechim
Extimollik
Foyda, yutuq
�
�1
�
�
�2
�
�
�3
�
�
�1
�
�
�2
�
�
�3
�
�
�
�
=
�=1
3
�
�,�
�
�
��
�
A
1
2
3
1/2
1/16
1/4
22221/4b
gfy1/4
3/4
1/8
1/4
3/16
5/8
10
8
4
4
2
6
8
4
4
8
2,75
4,45
B
1
2
1/4
1/16
0
7/8
1/2
1/16
14
8
0
16
18
8
16
15
C
1
2
3
1/4
1/8
3/8
1/4
3/4
1/16
1/2
1/8
3/8
10
6
4
2
4
0
8
2
8
7
4
4,5
According to step 1 of the iterative algorithm, we construct the following system of equations
with respect to the unknowns , i = 1, 2, 3:
8 + 0,9
1
2
�
1
+
1
4
1
4
�
2
+
1
4
1
4
�
3
= �
1
16 + 0,9
1
2
�
1
+
1
2
1
2
�
2
= �
2
7 + 0,9
1
4
�
1
+
1
4
1
4
�
2
+
1
2
1
2
�
3
= �
3
From this it is clear that = 91.26; = 97.55; = 89.97.
Now, moving on to the second step, we define the sets f(i, k).
For case A: when k = 2, 2.75 + 0.9 ( + + )= 88.90 < = 91.26;
When k = 3, 4.25 + 0.9(+ + ) =86.37 < <91.26;. Hence, f ( l , k) =.
For case B: When k = 2, 15 + 0 ,9(+ = 102.01<=97.55. Therefore, f(2, k) = 2.
For case C: When k = 2, 4 + 0 , 9( )=90.23>=89.97; When k = 3, 4.5+0.9(++)= 8 6 ,77 < W3 =
89.97. Therefore, f(3, k) =2.
Since f(1,k) =, f(2,k) = 2 and f(3,k) = 2,
the new stationary plan is as follows: = . Based on this plan, we proceed to step 1 and construct
the following system of equations:
8 + 0,9
1
2
�
1
+
1
4
1
4
�
2
+
1
4
1
4
�
3
= �
1
15 + 0,9
1
16
�
1
+
7
8
7
8
�
2
+
1
16
1
16
�
3
= �
2
4 + 0,9
1
4
�
1
+
1
4
1
4
�
1
+
1
2
1
2
�
3
= �
3
.
https://ijmri.de/index.php/jmsi
volume 4, issue 3, 2025
743
The system is: = 119.44; = 134.48; = 121.93 . Moving to the second step, we check the stationary
strategy for optimality and determine the sets f(i, k). For case A: when k = 2, 2.75 + 0.9 (+ )=
127.68 > = 119.44;
when k = 3, 4.25+0. 9 () = 114.84 < =119.44;. Hence, f ( l , k) = 2. For case B: when k = 1, 16+
0.9() = 124.62 < = 134.48. Hence, f(2,k) = .
For case C: when k=1, 7+0 ,9() =119.00 = 121.93; when k = 3, 4.5 + 0 ,9 () =113.26<= 121.93.
Therefore, f(3, k) = 0.
Since f( 1 , k) = 2, f(2 , k) = and f(3, k) = ,
the new stationary plan looks like this: .
Based on this plan, we proceed to step 1 and construct the following system of equations:
2,75 + 0,9
1
16
�
1
+
3
4
3
4
�
2
+
3
16
3
16
�
3
= �
1
15 + 0,9
1
16
�
1
+
7
8
7
8
�
2
+
1
16
1
16
�
3
=
4 + 0,9
1
4
�
1
+
1
4
1
4
�
2
+
1
2
1
2
�
3
= �
3
.
�
2
The solution to this system of equations is: = 121.66; = 135.31; = 122.84.
Moving to the second step, we check the stationary strategy for optimality and determine the sets
f(i, k). It is easy to show that for all i= 1, 2, 4, f(i, k) = . Therefore, the stationary q is the optimal
strategy. It follows that, regardless of the state in which the taxi driver is, it is better to choose
solution 2, that is, to go to the nearest taxi stand and wait in line.
Then his average payoff in the states will be the largest and will be
= 121.66; = 135.31; = 122.84.
The optimal strategy is determined based on the probabilities and rewards of the taxi driver's
actions. In the end, the second solution — waiting in line — is always the most profitable. This
strategy gives the maximum average profit in each case (A, B, C):
Case A — 121.66,
Case B — 135.31,
Case C — 122.84.
This issue is important in real life, especially in the efficient allocation of resources in service
systems.
Conclusion
During the study, the strategic decision problem faced by a taxi driver in the process of operating
in an area consisting of three towns was analyzed based on process research and optimal control
theory. The problem was mathematically modeled based on Markov decision processes, the
probability and expected profit (profit) values for each case and strategy were
determined, and a way to find the optimal strategy was developed using an iterative algorithm.
The analysis showed that among the available strategies for a taxi driver, the option that gives
the highest average profit is the strategy of going to the nearest taxi stand and waiting in line.
This strategy gives the most optimal result in all cases (A, B, C), that is, it maximizes the driver's
expected profit.
Also, this study showed the practical importance of Markov processes and optimal control
approaches in optimizing decision-making, efficient use of resources, and increasing operational
https://ijmri.de/index.php/jmsi
volume 4, issue 3, 2025
744
efficiency in the service sector, in particular in transport systems.
The results of the study can be applied in the future to other service sectors, including logistics
systems, queuing theory, and automated control systems.
References
1) Karimov M.Q., Yuldashev M.Yu. (2019). Process research and optimal management.
Tashkent: TDYU Publishing House.
2) Bashmakov I.G. (2005). Optimal production systems. Tashkent: Publishing House of the
Ministry of Economy of the Republic of Uzbekistan.
3) Bellman R. (1957). Dynamic Programming. Princeton University Press.
4) Dantzig G.B. (1963). Linear Programming and Extensions. Princeton University Press.
5)Kantorovich L.V. (1959). Mathematical methods of organization and production planning.
Moscow: Fizmatgiz.
6)Dorfman R., Samuelson P., Solow R. (1958). Linear Programming and Economic Analysis.
McGraw-Hill.
7)Nemchinov V.S. (1972). Economic and mathematical methods and models. Moscow: Nauka.
8)Gofurov M.A. (2014). Fundamentals of mathematical programming. Tashkent: Fan Publishing
House.
9)Mirzayev B. (2017). Optimization of transport systems: theory and practice. Tashkent: TJU
University Publishing House.
10)Abdullayev A.A., Yusupov A.A. (2008). Fundamentals of operations research. Tashkent:
Economics Publishing House.
11)Puterman M.L. (1994). Markov Decision Processes: Discrete Stochastic Dynamic
Programming. Wiley-Interscience.
