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Citation: Stoilova, S. An Integrated
SIMUS–Game Theory Approach for
Sustainable Decision Making—An
Application for Route and Transport
Operator Selection. Sustainability 2024,
16, 9199. https://doi.org/10.3390
/su16219199
Academic Editor: Georges Zaccour
Received: 26 August 2024
Revised: 13 October 2024
Accepted: 18 October 2024
Published: 23 October 2024
Copyright: © 2024 by the author.
Licensee MDPI, Basel, Switzerland.
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4.0/).
sustainability
Article
An Integrated SIMUS–Game Theory Approach for Sustainable
Decision Making—An Application for Route and Transport
Operator Selection
Svetla Stoilova
Faculty of Transport, Technical University of Sofia, 8 Kl. Ohridski Blvd., 1000 Sofia, Bulgaria; stoilova@tu-sofia.bg
Abstract: The choice of management strategy for companies operating in different sectors of the
economy is of great importance for their sustainable development. In many cases, companies are
in competition within the scope of the same activities, meaning that the profit of one company is at
the expense of the other. The choice of strategies for each of the firms in this case can be optimized
using game theory for a non-cooperative game case where the two players have antagonistic interests.
The aim of this research is to develop a methodology which, in non-cooperative games, accounts for
the benefits of different criteria for each of the strategies of the two participants. In this research a
new integrated sequential interactive model for urban systems (SIMUS)–game theory technique for
decision making in the case of non-cooperative games is proposed. The methodology includes three
steps. The first step consists of a determination of the strategies of both players and the selection
of criteria for their assessment. In the second step the SIMUS method for multi-criteria analysis
is applied to identify the benefits of the strategies for both players according to the criteria. The
model formation in game theory is drawn up in the third step. The payoff matrix of the game is
formed based on the benefits obtained from the SIMUS method. The strategies of both players are
solved by dual linear programming. Finally, to verify the results of the new approach we apply
four criteria to make a decision—Laplace’s criterion, the minimax and maximin criteria, Savage’s
criterion and Hurwitz’s criterion. The new integrated SIMUS–game theory approach is applied to
a real example in the transport sector. The Bulgarian transport network is investigated regarding
route and transport type selection for a carriage of containers between a starting point, Sofia, and a
destination, Varna, in the case of competition between railway and road operators. Two strategies for
a railway operator and three strategies for a road operator are examined. The benefits of the strategies
for both operators are determined using the SIMUS method, based on seven criteria representing
environmental, technological, infrastructural, economic, security and safety factors. The optimal
strategies for both operators are determined using the game model and dual linear programming. It is
discovered that the railway operator will apply their first strategy and that the road operator will also
apply their first strategy. Both players will obtain a profit if they implement their optimal strategies.
The new integrated SIMUS–game theory approach can be used in different areas of research, when
the strategies for both players in non-cooperatives games need to be established.
Keywords: game theory; SIMUS method; non-cooperative game; multi-criteria analysis; benefits;
sustainable decision making; route selection; container carriage
1. Introduction
Sustainable decision making is of great importance when selecting strategies for the
operation and development of systems in various fields of research. The decision-making
process involves the consideration of various factors that are related to solving a given
problem. Game theory is applied in various fields when there are conflict situations and
strategic interactions. Game theory involves two or more players, each of which has
strategies. A payoff matrix is constructed considering the strategies of both players. The
Sustainability 2024,16, 9199. https://doi.org/10.3390/su16219199 https://www.mdpi.com/journal/sustainability
Sustainability 2024,16, 9199 2 of 31
beginning of game theory was established by Neumann and Morgenstern in [
1
]. Later,
John Hash influenced the development of game theory as a tool for strategic decisions [
2
,
3
].
There are two main groups of games—cooperative and non-cooperative games. The former
deal with how coalitions interact; the payoffs are known. The latter include strategic games
with competition between the players. These can be zero-sum-games and non-zero-sum
games. This research studies non-cooperative games. In zero-sum games, the gain of one
player is equal to the loss of the other player. These games are solved mathematically using
the dual problem of linear programming by determining the strategies of both players. Such
a game is Pareto optimal. The payoff matrix is the basis on which the dual mathematical
model is built. The payoff matrix can only consider one factor. Usually, the payoff matrix
takes into account the profit of one player. As the players have conflicting interests, the
matrix shows the other player’s losses accordingly. The non-zero-sum games permit all
participants to win or lose at the same time. There is no competition between the players.
The hypothesis of this research is that the payoff matrix of non-cooperative games
could account for the influence of many criteria expressed as benefits for each player. The
benefits can be determined using the methods of multi-criteria analysis. A suitable method
for determining the benefits is the sequential interactive model for urban systems (SIMUS)
method [
4
]. The main advantages of the SIMUS method, which make it preferable to other
multi-criteria methods are the following:
•It uses linear optimization.
•It is not a subjective method; it is objective because it is based on optimization.
•It does not use experts’ assessment.
•It allows both quantitative and qualitative decision criteria to be explored.
•It does not use weights of criteria.
•
After performing the optimizations for each of the criteria, the weights of the objectives
can be defined, which serve only for analysis and not for follow-up actions.
•
It allows one to analyze the sensitivity of the input data, i.e., to determine the upper
and lower values for each of the criteria under which the obtained ranking of the
alternatives is preserved. The values of the upper and lower limits of the criteria are
determined as results of the linear optimization models.
•
It permits one to undergo a preliminary analysis of results. The preliminary analysis
permits one to compare the values for each objective and the optimum values obtained
by using the SIMUS method. This makes it possible to analyze the real state of the
studied system and opportunities for its improvement.
The determination of the payoff matrix based on the scores of the SIMUS method
permit us to assess the influence of predetermined quantitative and qualitative crite-
ria on the choice of strategy and of obtaining common benefits. The integration of the
SIMUS and game theory methods enable us to use their advantages and opportunities to
make decisions.
The goal of this study is to expand scientific knowledge regarding sustainable decision
making based on the advantages of the SIMUS method and game theory for choosing
strategies in non-cooperative games.
The research questions for this paper consist of the following problems:
•Assessment of the benefits of the strategies of both competing players.
•Establishment the impact of different criteria on the behavior of the players.
•
Formation of the payoff matrix so that it considers the benefits of all the quantitative
and qualitative criteria that influence the players’ choice of strategies.
•
Making decisions about the most suitable strategies of both competing players consid-
ering the benefits of different criteria.
The advantages of the integrated SIMUS–game theory approach are as follows:
(1)
It can assess the common benefits of different quantitative and qualitative criteria for
each strategy and interactions between the players.
Sustainability 2024,16, 9199 3 of 31
(2)
The determination of the benefits of the criteria is carried out by linear optimization,
which is an objective method of decision making.
(3)
The strategies for both players are formed while taking into account a complex of
quantitative and qualitative criteria.
This paper presents a unique study which integrated the power of game theory with
the advantages of the SIMUS method. Here, this approach is presented for the first time.
Game theory models can be used to solve problems in various fields such as eco-
nomics, business, finance, politics, military affairs, logistics, transport, systems science,
information technology.
In this study the transport sector is chosen to present the new integrated approach. In
transport, game theory can be used to find a solution in the cases of competition between
modes of transport that utilize carriages on parallel lines for highway or city transport,
competition between carriers of the same mode of transport, selection of transport equip-
ment for servicing logistics centers, management of congestion in cities, route selection,
pricing, and determining transport company management strategy, when allocating trans-
port services among carriers. In all cases, the main objective is to obtain maximum utility
or maximization of possible profit and to make a decision about a management strategy.
Both cooperative and non-cooperative games can be applied in the transportation
sector. In the first case, for example, transport carriers from one mode of transport can
coalesce to achieve a common goal. In the second case, in non-cooperative games, transport
firms have antagonistic interests, with one’s gain being the other’s loss. The goal here is for
both players to gain utility.
In this paper, a new approach, which makes use of SIMUS game theory, is presented
through a real example about route and transport type selection for the carriage of contain-
ers between a starting point and a final point in the case of competing transport operators.
This is an important problem because it is part of sustainable transport development. The
choice of transport strategy is important for the sustainability of transport companies
in the transport services market and depends on a complex of indicators that need to
be considered.
In this research a case of the Bulgarian transport network is studied.
At present, the interests of the railway and road carriers on the Bulgarian transport
network are in conflict in the field of freight transportation.
(1)
The motorways, first- and second-class roads in Bulgaria are overloaded by trucks
of different types—from 3.5 to over 40 tons. The emitted harmful emissions and
fine dust particles from trucks pollute the air. At the same time, a small number
of freight trains run on the railway highways, although this type of transport is
environmentally friendly.
(2) Transportation by trucks is tolerated by the state. For example, according to data from
the National Statistical Institute of Bulgaria, in 2023 for inland carriages, 11.26 million
tons of cargo were transported by freight rail transport, while 42.70 million tons were
transported by road with trucks over 25 t. The total length of running track is
4029 km
,
and the total length of motorway and category I roads is 3719 km. Furthermore, the
state invests in more road repair projects.
(3) There is the possibility of transportation on parallel lines, i.e., by rail and road. In these
cases, both modes of transport are in a state of competition and have antagonistic
interests, i.e., the gain of one is at the expense of the other. The purpose of both types
of transport is to make a profit.
This paper investigates the selection of route and transport type for the carriage
of containers between a starting point, Sofia, and destination, Varna, on the Bulgarian
transport network in the case of competing railway and road operators. Two route selection
strategies for a railway operator and three route strategies for a road operator were studied.
This paper expands the literature as follows:
(1)
The game relations between rail and road transport are considered.
Sustainability 2024,16, 9199 4 of 31
(2)
An approach is proposed to evaluate the total utility of quantitative and qualitative
criteria for evaluating strategies for both players.
(3)
The strategies for route selection for both players are determined based on the total
utility, which is applied in the game non-cooperative model.
This paper is organized as follows: Section 2presents the literature review. Section 3
introduces materials and methods where the novel integrated SIMUS–game theory ap-
proach is explained. Section 4presents the results obtained by applying the new approach
for route and transport operator selection. Section 5discusses the obtained results and the
advantages of the model. Finally, the conclusions are given.
2. Literature Review
Game theory has been successfully applied in different areas of research. A detailed
analysis of the game models in business areas is given in [
5
]. In transport, game theory has
been applied to make decisions about transport problems [
6
,
7
], route choices in transport
network [
8
,
9
], transport policy [
10
–
12
], travel demand management [
13
,
14
], high-speed rail
and air transport [
15
–
17
], railway companies [
18
–
20
], transport companies [
21
–
27
], urban
public transport [
28
–
32
], logistics and supply chains [
33
,
34
], and intermodal transport [
35
].
In [
6
], the authors describe the applications of non-cooperative games theory models
for solving transport problems. Four groups of games were proposed, according to the
relevant players. The first group includes games against a demon. In these games there
is one player that wishes to maximize the objective function and the other player wants
to minimize it. The second group consists of games between travelers. The third group
covers games between authorities. In this group at least one player is an authority. The
fourth group includes travelers and authorities as players.
An analysis of game theory applications in transport was conducted in [
7
]. The authors
classified these applications into two groups: macro-policy analysis and micro-behavior
simulation. The authors discussed the game theory applications in both cases.
Route choices on the international transport network were studied by applying game
theory and a congestion game [
8
]. The model was experimented with regard to three ports
and an intermodal terminal in Budapest. The players were drivers and logistics operators
who are not in full conflict. As a criterion for choosing a route in the game, the authors used
minimum transportation costs. Game theory non-cooperative approach in the assessment
of network reliability is presented in [
9
]. The players are the network user and the state of
the network. The criterion of optimization is the link cost. An example of a network with
six paths is presented. The problem was defined as a linear programming model.
In [
10
], a concept, based on cooperative game theory, for EU policy to improve the
international cooperation of countries along rail freight corridors is elaborated. The case of
a transferable utility game with characteristic function was studied. The author defined
as players the governments along the corridors, each of which have a common goal and
have to act together. The research proposes a new concept of allocation rules for transport
networks based on the Shapley value and cooperative game theory.
In [
11
], the authors present and discuss the game models and their application to
transport markets. Four groups of models were analyzed, including common oligopoly
models, traffic assignment models, spatial models, and auctioning related to transport. The
authors analyzed the road, rail, aviation, port and multi-modal literature to describe the
real-word application with a game theory component.
A game theory model was elaborated in [
12
] to optimize strategy for the transport
sector in Iraq. The players are public transport and private transport. Both forms of
transport include different types of carriage, such as sea, air, land, rail and port transport.
The minimax–maximin principle and linear programming model were applied to determine
the strategies of both operators.
In [
13
], game theory was applied for the travel demand management of urban trans-
port systems. The model investigated three districts, three passenger flows, and public
transport, meaning that the game model includes seven players. Passengers’ travel time,
Sustainability 2024,16, 9199 5 of 31
environmental conditions, residents’ travel time, and an optimal proportion between using
a private car and public transport were studied in the game model.
In [
14
], a gaming model about the mode choice behavior of travelers is presented. The
model includes two players, the parking operator and transit agency. The application of
the model is presented for Isfahan, Iran.
A game model to analyze the competition between high-speed rail and air transport
was elaborated on in [
15
]. The model includes low cost and network carriers. A generalized
profit function for the different operators was determined and the utility function in terms
of total trip time, fares and frequencies was applied in the model. The methodology was
applied to the analysis of four trans-European networks.
In [
16
], a game model was applied to study the competition between high-speed rail
and airlines in India. The fare and frequency were chosen as the parameters of the game.
The speed was investigated at three levels as an additional parameter.
The competition between a high-speed rail operator and aviation operator was studied
using game theory [
17
]. The game was presented in two stages. In the first stage, based on
a linear programming model, the game gave results regarding the costs. The second stage
included a game between each operator and the goal was to maximize the profit.
In [
18
], the authors presented a concept for applying game theory in some sectors of the
railway systems, such as railway privatization and maintenance of railway infrastructure
and rolling stock. The game model includes three players (track operator, rolling stock
operator and track maintenance contractor). The railway privatization processes in Sweden
and the UK were studied.
In [
19
], the authors present a two-stage model for evaluating the strategic decisions
made by rail operators. The first stage includes a game model with which to solve whether
the railway operators decide to participate in the market. In the second stage, the game
model includes the railway operators and the new operators. The model was studied for
the Madrid–Barcelona European high-speed rail corridor.
In [
20
], a game model for a railroad and a shipper was studied. The game model in-
vestigated two strategies for the railroad and three strategies for the shipper. The strategies
for the railroad refer to the pricing to maximize the profits and to avoid litigation. The
strategies of the shipper are related to the prices offered by the railway line.
Game theory was used to determine an alternative strategy for two players—online
transportation companies and drivers [
21
]. Six strategies were studied. The values of
strategies were determined through a survey. Both players have antagonistic interests,
i.e., the online transportation companies aim to minimize the maximum loss, while the
drivers want to maximize the minimum profit. Linear optimization was applied to solve
the problem under investigation.
A game theory model that takes into account the competition between two oil-product
transportation systems was elaborated in [
22
]. The main variables of the model that were
investigated were the transportation prices and tanker truck fleet. The Nash and Stackelberg
equilibrium models were applied to formulate and solve the problem.
A coalition game theory model was elaborated in [
23
] for route planning in a shared
transportation system. The authors studied the problem of automated vehicle collection.
The subsidy problem in railways was investigated by applying a game model [
24
].
The players were railway operators, shippers and local governments. The game model was
implemented for a railway express line in China.
Game theory was used to study inter-city transportation pricing [
25
]. The authors
applied the game model with the aim to maximize profit. The authors studied a rail and
bus company.
A Stackelberg game model was used to study railway transportation companies and
customers regarding costs and prices [26].
A game theory model to investigate an urban public transport integration policy
is given in [
27
]. The problem is staged for a two-player game and multi-player game,
providing services in separate and the same districts.
Sustainability 2024,16, 9199 6 of 31
A review of game theory that was applied to investigate urban traffic congestion
management systems is undertaken in [
28
]. Here, the authors discuss different applications
of game models.
In [
29
], a game theory approach was adopted to optimize public transport traffic. The
two players are the passenger and authorities. Strategies were determined for each of
the players. To provide the Nash equilibrium, a set of conditions had to be satisfied: the
frequency of transport and the payoff functions of players.
Game theory and the Golden Template algorithm were integrated with the purpose to
investigate urban mobility [
30
]. The relevant intersections were chosen as players. Two
game models were studied, one for public transport operators, and the other on the number
of passengers. The model permits us to plan multimodal transportation routes in urban
conditions. The methodology was tested for public transport in Bucharest.
In [
31
], game theory was used to establish the best line in urban transport. The payoff
matrix was formed by means of a survey conducted among residents for route service
preference. The function of risk of expected losses was chosen as the element of the payoff
matrix. The linear programming method was applied to make a decision.
The sharing of transportation between a distribution center and service outlets in a
city was investigated based on game theory [32]. The model aimed to reduce the costs for
transportation. The multiagent game model was elaborated on to realize transportation
sharing between multiple enterprises.
Game theory was applied in the area of supply chains [
33
]. The game model was
formed based on Nash competition and Stackelberg competition. The Nash competition
was described between the retailers of the supply chains and between the manufacturers of
the supply chains. The Stackelberg competition was presented between the manufacturer
and the retailer.
A cooperative game theory approach was used to make a decision about the selection
of a transportation network in a multi-level supply chain [
34
]. Linear transportation
programming was used to make models for multiple supply chains. The Shapley value
was applied to determine the total value of the coalition and establish the optimal mode in
terms of transportation.
Intermodal rail–road transport was studied using Stackelberg game theory in order to
reduce both the subsidy and carbon emissions [35].
It can be concluded that previous literature has elaborated on different game the-
ory models and has thus contributed to the development of transport planning. The
criteria used for making a decision based on the game model were as follows: trans-
portation costs [
8
,
32
], link costs [
9
], total costs (profit) [
10
,
16
,
17
,
19
,
25
], subsidy [
24
,
35
],
utility function [
15
], number of passengers [
30
], function of risk of expected losses [
31
],
and pricing [20].
The authors investigated cooperative and non-cooperative games. In cooperative
games the players make coalitions and have a common interest. The non-cooperative
games present the competition between the players. The linear programming method is
applied to make a decision in these cases [9,12,17,20,21,31,33].
In [
36
], the authors investigated the transport time, costs, transport quality, the service
in transport, transport tools, and the social benefits as criteria by which to assess different
routes for multimodal transportation. A fuzzy analytic hierarchy process (FAHP) and
an artificial neural network (ANN) were applied for route selection. The transport time,
transport price, congestion time, accident risk and noise, and CO
2
emissions were chosen
for route selection in [
37
]. The authors used the AHP method to establish the weights
of criteria, and the PROMETHEE method for ranking routes. The criteria—travel time,
transportation cost, and CO
2
emissions generated by all transportation modes—were
applied in a multi-objective optimization model for transport mode selection [38].
The criteria of transportation costs, transportation time, security risk, operational
risk, infrastructure risk, macro risk, policy, legality risks, and environmental risk were
used in [
39
] for route selection in multimodal transport network. The author proposed
Sustainability 2024,16, 9199 7 of 31
a hybrid multi-criteria approach integrating the analytic hierarchy process (AHP), data
envelopment analysis (DEA), and the technique for order preference by similarity to ideal
solution (TOPSIS). The case of route selection between Thailand and Vietnam was studied.
Five main groups of criteria, eco-sociological, urban planning, constructional, technical,
and transport and economic were introduced in [
40
,
41
] with purpose of evaluating a
road route. Sub-criteria were set for each main group of criteria. The PROMETHEE
method was used to rank the alternative routes. The weights of the criteria were calculated
through experts’ assessment. The case of optimal railroad route between Rijeka and Zagreb
was studied.
In [
42
], the authors developed a multi-objective optimization model for optimizing
transportation routes. The criteria that were chosen were minimizing the total transporta-
tion time, transportation costs and container usage costs. The example of transportation in
Panzhihua City, Sichuan Province was studied.
In [
43
], four main groups of criteria—technology, economy, society and
environment—were
studied for railway route selection. TOPSIS, the entropy weight method and Grey cor-
relation analysis was applied to make a decision. The case study was investigated for
China’s railway.
On the basis of this research, it can be concluded that the most important factors for route
selection are transport price [36–43], transport time [36–39,42], and CO2emissions [38,39,42].
Some authors have focused efforts to integrate game theory and multi-criteria decision
making (MCDM) methods to support decision making in different areas of research.
A hybrid methodology comprising stepwise weight assessment ratio analysis (SWARA),
weighted aggregated sum product assessment (WASPAS) and game theory is elaborated
in [
44
]. The authors applied the SWARA method to assess the criteria weights, and the
WASPAS method to evaluate and select the best Nash equilibriums to make a decision. The
methodology was tested for supply chain management.
In [
45
], the authors elaborate an integrated model comprising the analytic network
process (ANP) method, entropy weight, game theory, the decision-making trial and evalu-
ation laboratory (DEMATEL) method and evidence theory. Supplier management of an
uncertain case was studied. The weights of criteria were determined using a combination
of game theory and the DEMATEL method.
The analytical hierarchy process (AHP) method and game theory were integrated to
select the most suitable network, [
46
]. Non-cooperative game theory was applied. Both
of the players in the game (for network and for user) participated without any form of
collaboration. The payoff matrix of the game was formed by taking into account the weight
estimation of the networks given by AHP method.
The concept of a combination of fuzzy AHP and game theory is presented in [
47
], in
which the risk factors associated with a tunnel project were assessed. A case study for the
Resalat tunnel in Iran is presented. The fuzzy AHP was used to analyze qualitative decisions
into quantitative scores. Both non-cooperative and cooperative games were investigated.
An integration between the TOPSIS method and cooperative games is presented in [
48
].
The authors investigated airlines. The authors evaluated the impacts of alliances and
mergers on both airlines and passengers using Shapley values and the Nash equilibrium.
The TOPSIS method was applied to assess the factors affecting airline mergers.
In [
49
], a multi-agent MCDM is represented as an evolutionary game. The criteria
and alternatives were considered as strategies of the agents (players). A case study for the
Indian tea industry was considered.
A hybrid game theory technique and multi-criteria decision-making methods are
discussed in [
50
]. A total of 56 papers from the year 2008 to 2020 have been analyzed to
provide different applications studied from previous research. The authors found that the
most used multi-criteria methods which have been integrated with game theory are AHP
and TOPSIS. The authors concluded that the integration between MCDM and game theory
has been increasingly used to support decision-making.
Sustainability 2024,16, 9199 8 of 31
In summary, multi-criteria methods can be said to be a powerful tool when integrated
with game theory to support the making of sustainable decisions. The authors applied AHP,
ANP, TOPSIS, SWARA, WASPAS, DEMATEL methods for multi-criteria analysis. However,
all of these methods use expert evaluation of the criteria, which makes them subjective.
This paper proposes the integration of the SIMUS method of multi-criteria analysis
with game theory for the first time. This is a non-subjective MCDM method.
The difference between this study and the literature described above is the develop-
ment of a new integrated approach. The combination of multi-criteria analysis based on the
SIMUS method and game theory has not yet been presented in the literature. This paper
deals with a new approach which permits us to assess the common benefits of quantitative
and qualitative criteria when choosing strategies for two players with conflicting interests
in non-cooperative games. These benefits are considered as the total utility, which is used
to construct the payoff matrix in the game model.
3. Materials and Methods
The methodology of the new integrated SIMUS–game theory approach for decision
making consists of the following steps, as in Figure 1:
Sustainability 2024, 16, x FOR PEER REVIEW 9 of 32
Figure 1. Scheme of the methodology.
3.1. Step 1: Determination of the Strategies for Players A and B—Determination of the Criteria
for Assessment of the Strategies
The players have antagonistic interests, i.e., one player’s gain is another player’s loss.
The objective is to determine the strategies of both players, so that each of them has a
profit.
Initial decision matrices 𝐴 and 𝐵 are constructed for each of the players.
Matrix A and matrix B contain the criteria values for each strategy of both players.
Figure 1. Scheme of the methodology.
Sustainability 2024,16, 9199 9 of 31
Step 1: Determination of the strategies for players A and B and determination of the
criteria to evaluate these strategies.
Step 2. Application of the SIMUS method for determining the benefits of the strategies,
taking into account the criteria.
Step 3: Model formation in game theory. Determination of the strategies for both players.
Finally, to verify the results of the new approach we propose the following four criteria
for decision making: Laplace’s criterion, minimax and maximin criteria, Savage’s criterion
and Hurwitz’s criterion [51].
3.1. Step 1: Determination of the Strategies for Players A and B—Determination of the Criteria for
Assessment of the Strategies
The players have antagonistic interests, i.e., one player’s gain is another player’s loss.
The objective is to determine the strategies of both players, so that each of them has a profit.
Initial decision matrices
Amxn1
and
Bmxn2
are constructed for each of the players.
Matrix Aand matrix Bcontain the criteria values for each strategy of both players.
Amxn1=
a11 a12 . . . a1i. . . a1n1
a21 a22 . . . a2i. . . a2n1
. . . . . . . . . . . . . . . ..
ak1ak2. . . aki . . . akn1
. . . . . . . . . . . . . . . . . .
am1am2. . . ami . . . amn1
Bmxn2=
b11 b12 . . . b1i. . . b1n2
b21 b22 . . . b2i. . . b2n2
. . . . . . . . . . . . . . . ..
bk1bk2. . . bki . . . bkn2
. . . . . . . . . . . . . . . . . .
bm1bm2. . . bmi . . . bmn2
,
(1)
where
k=
1,
. . .
,
m
is the number of criteria,
i=
1,
. . .
,
n1
is the number of strategies for
player A, and j=1, . . . , n2is the number of strategies for player B.
3.2. Step 2: Application of the SIMUS Method for Solving the Benefits of the Strategies
SIMUS is a multi-criteria method that uses linear programming, weighted sum, and
outranking [
4
]. Its main advantage is that it does not use criteria weights, i.e., it does not
use a subjective approach.
The SIMUS method includes the following operations:
•Determination of the initial decision matrix.
•Calculation of the normalized decision matrix.
•
Formulation and solving of the linear optimization models for each criterion as an
objective function.
•Determination of the efficient results matrix.
•Calculation of the normalized efficient results matrix.
•Calculation of the scores of alternatives.
The initial decision matrix determined by means of the SIMUS method is formed with
alternatives in columns and criteria in rows using the matrices
Amxn1
and
Bmxn2
of players
A and B, respectively, that were composed in the previous step.
The initial decision matrix is normalized. For this purpose, different methods of nor-
malization can be used. The ranking of the alternatives is not affected by the normalization
method. This research proposes the use of the total sum in row method.
The thresholds for each row are determined. This determination depends on the type
of optimization. The threshold is equal to the maximum normalized value of the row
when the optimization is of the maximum of the criterion. The threshold is equal to the
minimum normalized value of the row when the optimization is of the minimum of the
Sustainability 2024,16, 9199 10 of 31
criterion. The thresholds are necessary for the formulation of the restrictive conditions of
linear optimization models.
The simplex algorithm is applied to solve the linear optimization models. The un-
knowns are the scores of each alternative. The number of criteria determine the number of
optimization models. The first linear optimization model is formed by taking into account
the first criterion as the objective function. In this case the restrictive conditions are formed
by the other rows. This procedure is applied consistently to all criteria considered as an
objective function. When the optimization is of a maximum, the operator of the restrictive
condition is “
≤
”;when the optimization is of a minimum the operator of the restrictive
condition is “
≥
”. The results of the optimizations are recorded in the efficient results matrix
(ERM). The elements of the ERM are the optimal values of the scores of each alternative for
each optimization model. This matrix is a new decision matrix.
The normalization procedure is applied again to ERM using the sum of the row method.
The SIMUS procedure uses normalized efficient results matrix (NERM) to determine the
ranking of the alternatives. For this purpose, the sum of all elements in each column (SC)
and the Participation Factor (PF) are consistently determined.
The sum of all elements in each column (
SCAi
;
SCBj
) is determined for each player as
follows:
SCAi=∑m
k=1
xmi
∑n1
i=1xmi +∑n2
j=1ymj
,i=1, . . . , n1;j=1, . . . , n2(2)
SCBj=∑m
k=1
xmj
∑n1
i=1xmi +∑n2
j=1ymj
,i=1, . . . , n1;j=1, . . . , n2; (3)
where
xmi
represents the scores of each alternative for player A;
ymj
represents the scores of
each alternative for player B;
SCAi
represents the sum of all elements in each strategy for
player A;
SCBj
represents the sum of all elements in each strategy for player B; and where
i=1, . . . , n1;j=1, . . . , n2;k=1, . . . , m.
The number of participations of each alternative in each column of NERM represents
the participation factor (PF). The number of participations of each strategy
i
of player
A
are
PFAiand the number of participations of each strategy jof player B are PFBj.
The normalized values of the participation factor (NPF) are calculated dividing each
value of PF ( PFAi;PFBjby the total number of criteria m.
The scores of the strategies are determined as follows:
ai=PFAi
m·SCAi(4)
bj=PFBj
m·SCBj(5)
where
ai
is the score of strategy
i
of player
A
;
i=
1,
. . .
,
n1
is the number of strategies for
player A;
bj
is the score of strategy
j
of player
B
; and
j=
1,
. . .
,
n2
is the number of strategies
for player B.
The maximal values of all scores shows the best alternative.
3.3. Step 3: Model Formation in Game Theory—Determination of the Strategies for Both Players
In this step, a payoff matrix is formed and a mathematical representation of the game
theory problem for both players is undertaken. The probabilities when applying the
strategies for each player are determined using the dual simplex method.
The payoff matrix (pi jn1x n2represents the game, as in Table 1.
The strategies for player A (
i=
1,
. . .
,
n1
) are given by rows and the strategies for
player B (
j=
1,
. . .
,
n2
) are given by the columns. For example,
pi1
is the profit of player A
for his ith strategy and the first strategy for player B.
Sustainability 2024,16, 9199 11 of 31
Table 1. Payoff matrix for game theory.
Player Probability
B1B2BjBn2
y1y2yjyn2
A1x1p11 =a1−b1p12 =a1−b2. . . p1j=a1−bj. . . p1n2=a1−bn2
A2x2p21 =a2−b1p22 =a2−b2. . . p2j=a2−bj. . . p2n2=a2−bn2
... ... ... ... ... ... ...
Aixipi1=ai−b1pi2=ai−b2. . . pij =ai−b2pin2=an1−bn2
... ... ... ... ... ... ...
An1xn1pn11 =an1−b1pn12 =an1−b2. . . pn1j=a2−bj. . . pn1n2=an1−bn2
The elements of the payoff matrix are formed using the results of the scores of the
alternatives of both players received by the SIMUS method. The payoff matrix is made up
of rows expressing the payoff of player A. As the SIMUS scores express the benefits of each
strategy for both players, and as they have antagonistic interests, the profit of player A is
reduced by the profit of player B. For example,
p21
is the profit of player A for his second
strategy and the first strategy for player B, i.e., the value of
p21
, is calculated by reducing
the benefit of the second strategy,
a2
, for player A by the benefit of the first strategy,
b1
, for
B. The reduction of the benefits of all strategies of player A when player B uses his first
strategy is found by subtracting the benefits of this strategy by the benefits of all of the
strategies of player A. This is determined in order to account for the competition between
the players. It is possible that some of the elements in the matrix are negative. This means
that, in this case, there are losses. If all values in the payoff matrix are negative, then the
problem is solved against player B. In this case the payoff matrix is determined by reducing
the benefits of all of the strategies of player B by subtracting the benefits of player A.
Game theory indicates a two-person zero-sum game wherein the gain of one player
is equal to the loss for the other. The game is of a mixed strategy when both players have
several optimal strategies. In this case, dual linear optimization is applied to determine
the strategies of both players. This makes it possible to determine the optimal strategies
for both players simultaneously. The linear optimization model for player A involves
maximizing their benefits, just as the linear optimization model for player B involves
minimizing their losses. The value of the game is the same for both optimization problems.
For player A, the optimal mixed strategies are as follows:
max
xinmin∑n1
i=1pi1xi,∑n1
i=1pi2xi, . . . , ∑n1
i=1pin2xio, (6)
x1+x2+. . . +xn1=1, (7)
0≤xi≤1, i=1, 2, . . . , n1(8)
where xirepresents the probability of player A using strategy Ai.
For player B, the optimal mixed strategies are as follows:
min
yjnmax∑n2
j=1p1jyj,∑n2
j=1a2jyj, . . . , ∑n2
j=1amj yj,o, (9)
y1+y2+. . . +yn2=1, (10)
0≤yj≤1, j=1, 2, . . . , n2(11)
where yjrepresents the respective probabilities for strategy Bjfor player B.
To be able to solve the indicated mathematical models, transformations are made in
order to apply a dual simplex method.
For the model presented by Equation (6) the following transformation is made:
ζ=min∑n1
i=1pi1xi,∑n1
i=1pi2xi, . . . , ∑n1
i=1pin2xi(12)
Sustainability 2024,16, 9199 12 of 31
For the model presented by Equation (9) the following transformation is made:
ω=max∑n2
j=1p1jyj,∑n2
j=1p2jyj, . . . , ∑n2
j=1yj,(13)
Based on Equations (12) and (13), the problem for both players can be written as follows:
Maximize ζ=ϑ, (14)
where ϑrepresents the value of the game.
Subject to the following:
p11x1+p12x2+. . . +pn11xn1≥ϑ
p12x1+p22x2+. . . +pn12xn1≥ϑ
. . .
p1n2x1+p2n2x2+. . . +pn1n2xn1≥ϑ
(15)
and Equations (7) and (8).
Minimize ω=ϑ, (16)
Subject to the following:
p11y1+p12 y2+. . . +p1nyn2≤ϑ
p21y1+p22 y2+. . . +p2nyn2≤ϑ
. . .
pn1y1+pn12y2+. . . +pn1n2yn2≤ϑ
(17)
and Equations (10) and (11).
Using additional transformations models (14)–(17) are represented by dual linear
optimization.
The problem of player A is transformed as follows:
Minimize ζ=X1+X2+. . . +Xn1(18)
Subject to the following:
p11X1+p21X2+. . . +pn11 Xn1≥1
p12xX1+p22 X2+. . . +pn12 Xn1≥1
. . .
p1n2X1+p2n2X2+. . . +pn1n2Xn1≥1
(19)
Xi≥0, i=1, 2, . . . , n2(20)
Xi=xi
ϑ;ϑ=1
ζ(21)
The problem of player B is transformed as follows:
Maximize ω=(Y1+Y2+. . . +Yn)(22)
Subject to the following:
p11Y1+p12Y2+. . . +p1nyn2≤1
p21Y1+p22Y2+. . . +p2nyn2≤1
. . .
pn1Y1+pn12Y2+. . . +pn1n2yn2≤1
(23)
Yj≥0, j=1, 2, . . . , n2(24)
Sustainability 2024,16, 9199 13 of 31
Yj=yi
ϑ;ϑ=1
ω(25)
The results for the game model are given by the solution of the transformed models.
4. Results
The elaborated new integrated SIMUS–game theory approach was applied to the
decision making of strategies of railway and road operators for the carriage of containers.
Both transport operators serve the in the direction from Sofia to Varna and have antagonistic
interests, i.e., the rail operator’s gain is the road operator’s loss.
The limitations of this research are as follows:
•The players are competitors.
•The players have antagonistic interests.
•The transport is in 40-foot containers.
•The player benefits are determined by preset criteria.
•The strategies of both players are about the choice of the route between two points.
4.1. Step 1: Determination of the Strategies for Railway and Road Operator—Determination of the
Criteria for Assessment of the Strategies
Let us assume that the railway operator is player A, and that the road operator is
player B.
Player A has two strategies for the carriage of container block trains in the
Sofia–Varna
direction, which differ from each other on the route between Sofia and Varna. The first
strategy (A1) is carriage along the
Sofia–Gorna
Oryahovitsa–Varna route; the second strat-
egy (A2) is carriage along the Sofia–Karlovo–Karnobat–Varna route. A1 is served by a fully
double-tracked railway line. A2 is, for the most part, a single-track railway, with some
double-track sections. Figure 2illustrates the strategies for both operators.
Sustainability 2024, 16, x FOR PEER REVIEW 14 of 32
Player A has two strategies for the carriage of container block trains in the Sofia–
Varna direction, which differ from each other on the route between Sofia and Varna. The
first strategy (A1) is carriage along the Sofia–Gorna Oryahovitsa–Varna route; the second
strategy (A2) is carriage along the Sofia–Karlovo–Karnobat–Varna route. A1 is served by
a fully double-tracked railway line. A2 is, for the most part, a single-track railway, with
some double-track sections. Figure 2 illustrates the strategies for both operators.
It is assumed that the container block trains have 20 wagons with 40-foot containers
and a gross mass of 20 t. The gross train weight is 1086 t. The container trucks each have
load capacity of 24 tons and a gross weight of 36 tons.
Player B has three strategies for carriage by container trucks, called road trains in the
Sofia–Varna direction which differ by route. The first strategy (B1) is carriage by road
trains along the Sofia–Veliko Tarnovo–Varna route; the second strategy (B2) is carriage by
road train along the Sofia–Plovdiv–Burgas–Varna route; the third strategy (B3) is carriage
by road trains along the Sofia–Plovdiv–Karnobat–Shumen–Varna route.
Strategy B1 has about 280 km of secondary road. This road crosses northern Bulgaria
and is well maintained throughout the year. The road passes through many populated
areas. The speed of movement is limited, which makes maneuvering difficult.
Strategy B2 mainly involves motorway. There is 125 km of secondary road which is
located in the eastern part of the country.
Strategy B3, compared with road B2, has approximately the same length of secondary
road. This road passes through a mountainous section and has many sharp turns.
Figure 2. Scheme of the strategies for the railway and road operator.
This study uses the criteria for the assessment of container carriages determined in
[52], as follows:
C1—carbon dioxide emissions during transport, g/UTI;
C2—operational costs and fees for loading and unloading operations, EUR/UTI;
C3—charges for the use of railway and road infrastructure, EUR/UTI;
Figure 2. Scheme of the strategies for the railway and road operator.
Sustainability 2024,16, 9199 14 of 31
It is assumed that the container block trains have 20 wagons with 40-foot containers
and a gross mass of 20 t. The gross train weight is 1086 t. The container trucks each have
load capacity of 24 tons and a gross weight of 36 tons.
Player B has three strategies for carriage by container trucks, called road trains in the
Sofia–Varna direction which differ by route. The first strategy (B1) is carriage by road trains
along the Sofia–Veliko Tarnovo–Varna route; the second strategy (B2) is carriage by road
train along the Sofia–Plovdiv–Burgas–Varna route; the third strategy (B3) is carriage by
road trains along the Sofia–Plovdiv–Karnobat–Shumen–Varna route.
Strategy B1 has about 280 km of secondary road. This road crosses northern Bulgaria
and is well maintained throughout the year. The road passes through many populated
areas. The speed of movement is limited, which makes maneuvering difficult.
Strategy B2 mainly involves motorway. There is 125 km of secondary road which is
located in the eastern part of the country.
Strategy B3, compared with road B2, has approximately the same length of secondary
road. This road passes through a mountainous section and has many sharp turns.
This study uses the criteria for the assessment of container carriages determined in [
52
],
as follows:
C1—carbon dioxide emissions during transport, g/UTI;
C2—operational costs and fees for loading and unloading operations, EUR/UTI;
C3—charges for the use of railway and road infrastructure, EUR/UTI;
C4—duration of transportation, h;
C5—light on the route, km;
C6—infrastructure factor, coef. This describes the transport infrastructure and can have
values between 1 and 2. C6 = 2 in the case of a two-track railway line or a motorway, while
C6 = 1 in the case of a single-track railway line or a first-class/second-class road. When the
infrastructure consists of mixed sections, the infrastructure factor is determined by taking
into account its percentage, i.e., C6 = 1 + k, where k is a parameter that indicates the relative
share of double-track sections or motorway for the route.
C7—security and safety, coef. This can have values of 0 or 1. When C7 = 1 this indicates a
more secure and safe transport.
The criteria included in the study cover ecology (criterion C1—carbon dioxide emis-
sions during transport), economic criteria related to operating costs and infrastructure
fees (C2 and C3), technology (C4), infrastructure (C5 and C6), and security and safety
(C7). It can be summarized that various aspects of achieving transport sustainability
are covered—ecology, technology, infrastructure, costs and fees for transport, security
and safety.
Based on the literature review it can be concluded that the criteria that refer to costs,
transport time and carbon dioxide emissions during transport were used by most authors
when they explored route selection. This study also applies criteria related to the type of
infrastructure and security and safety. Considering the type of infrastructure is important,
as it affects the speed and capacity of the transport.
In the first step of the methodology, initial decision matrices
A7×2
and
B7×3
are
constructed for each player. Matrix Aand matrix Bcontain the criteria values for each
strategy of both players.
In order to have commensurability between the container truck and the container train
composition, the data regarding the railway operator in the matrix represent one transport
unit, i.e., one wagon (one container).
A7×2=
238, 257.00 248, 250.00
152.00 147.00
96.00 94.00
9.05 8.57
543.00 514.00
2.00 1.00
2.00 2.00
;B7×3=
442, 887.00 519, 180.00 545, 931.00
190.00 227.00 189.00
51.00 61.00 65.00
7.14 8.12 8.62
447.00 524.00 551.00
1.53 1.75 1.82
1.00 1.00 1.00
(26)
Sustainability 2024,16, 9199 15 of 31
where
k=
1,
. . .
, 7 represents the number of criteria,
i=
1, 2 represents the number of
strategies for player A, and j=1, 2, 3 represents the number of strategies for player B.
In this study were used data obtained from [52].
The values of carbon dioxide emissions during transport (C1) were determined while
taking into account the consumption of electricity for the movement of the trains and the
consumption of fuel for the movement of the trucks. The consumption of electricity for train
movement was determined according to data provided by the Bulgarian State Railways
Holding. The amount of carbon dioxide (CO
2
) during train movement was determined
while taking into account the amount of CO
2
t/MWh released by the power plants during
the production of electricity. This research used a value of 0.460 t/MWh amount of CO
2
,
according to data from the National Statistics Institute of Bulgaria.
The data from the real test of the European Truck Challenge 2014 were used to calculate
the amount of carbon dioxide released during the movement of a truck on the considered
routes. The value of 49.54 g CO
2
/ton.km were used. It is assumed that the trucks are Euro
6 standard.
The operational costs (C2) for a freight train were determined cost rates per locomotive
kilometer and wagon kilometer. These rates include the costs for movement, depreciation
costs, salary and social security costs. The study used a rate of 0.6 EUR/loc.km and rate
of 0.1 EUR/wag.km for a freight wagon, according to Bulgarian State Railways Holding.
The operating costs included the costs for electricity, depreciation costs, salary and social
security costs, costs for maintenance and repair of rolling stock. The operating costs for a
freight train with a composition of 20 wagons and 2 locomotives are 3.2 EUR/train.km.
The operational costs (C2) for a truck include fuel consumption, driver wages, taxes
and insurance, the costs of tires, lubricants, maintenance, and other types of costs. Fuel
consumption is close to 60% of a transport company’s operating costs. The data from the
European Truck Challenge 2014 were used to determine the value of operational costs
based on average rate for variable costs of 0.467 EUR/km, and average rate for fixed costs
of 0.223 EUR/km, for a total of 0.69 EUR/km. In this situation, fuel consumption is 60% of
the operating costs, and the other groups of operating costs make up the remaining 40%.
The charges (C3) for the use of railway infrastructure were determined according
to the tariff for infrastructure fees, determined by the National Railway Infrastructure
Company of Bulgaria. The charges (C3) for the use of road infrastructure were determined
by tariff for toll road charges.
The duration of the transportation by rail and by road (C4) were determined while
taking into account the timetable of the trains and the permitted speed for trucks on the
road infrastructure.
The light on the route by rail and by road (C5) was determined according to data on
the sections in the transport infrastructure.
The infrastructure factor (C6) was determined regarding the type of transport infras-
tructure by sections of the transport network.
The value of the criterion security and safety (C7) is either 0 or 1. These values are
determined based on analysis of freight train and truck accidents for the studied sections.
In recent years, there have been no incidents with container trains on both railway routes.
There are accidents with trucks on highways and road infrastructure. In general, rail
transport is the safer mode of transport because it runs on a separate infrastructure. The
trucks move on motorways and on road networks on which private cars also move.
4.2. Step 2: Application of the SIMUS Method for Determining the Benefits of the Strategies for
Both Transport Operators
The next step of the methodology includes determination of the scores of both players
by applying the SIMUS method.
Table 2represents the initial decision matrix. The left-hand side (LHS) of the initial
decision matrix presents the values of the criteria for the alternatives of both players. The last
Sustainability 2024,16, 9199 16 of 31
column shows the type of optimization for each criterion. Table 3represents the normalized
decision matrix. The sum of the rows is used.
Table 2. Initial decision matrix.
Criteria
Player A Player B
Type
Alternatives Alternatives
Symbol Dimension A1 A2 B 1 B2 B3
C1 CO2, g/UTI 238,257.00 248,250.00 442,887.00 519,180.00 545,931.00 min
C2 EUR/UTI 152.00 147.00 190.00 227.00 189.00 min
C3 EUR/UTI 96.00 94.00 51.00 61.00 65.00 min
C4 h 9.05 8.57 7.14 8.12 8.62 min
C5 km 543.00 514.00 447.00 524.00 551.00 min
C6 - 2.00 1.00 1.53 1.75 1.82 max
C7 - 2.00 2.00 1.00 1.00 1.00 max
Table 3. Normalized decision matrix.
Criteria
Player A Player B
Type Operator RHS Values
Alternatives Alternatives
A1 A2 B1 B2 B3
Symbol
Dimension
C1 CO2,
g/UTI 0.12 0.12 0.22 0.26 0.27 Min ≥0.12
C2 EUR/UTI 0.17 0.16 0.21 0.25 0.21 Min ≥0.16
C3 EUR/UTI 0.26 0.26 0.14 0.17 0.18 Min ≥0.14
C4 h 0.22 0.21 0.17 0.20 0.21 Min ≥0.17
C5 km 0.21 0.20 0.17 0.20 0.21 Min ≥0.17
C6 - 0.25 0.12 0.19 0.22 0.22 Max ≤0.25
C7 - 0.29 0.29 0.14 0.14 0.14 Max ≤0.29
The elements of the ERM represent the scores of each alternative.
The procedure includes successively constructing and solving linear optimizations
targeting each of the criteria. The first optimization is drawn up with the first criterion
considered as an optimization function. It is removed from the decision matrix. The
restrictive conditions are considered by the other rows of the normalized matrix. This
procedure is repeated successively with the other criteria. The results of all optimization
models are structured in an efficient results matrix (ERM), as in Table 4.
Table 4. Efficient results matrix (ERM). Preliminary analysis.
Efficient Results Matrix Preliminary Analysis
Objective
Player A Player B
Objective
Function Values
RHS
Values Type Objective
Satisfied? Comment
Alternatives Alternatives
A1 A2 B1 B2 B3
Z1 0.97 0.00 0.00 0.00 0.00 0.12 0.12 Min Yes Satisfied 100%
Z2 0.68 0.00 0.00 0.00 0.14 0.14 0.16 Min No The objective function
value is lower than the
RHS value
Z3 0.00 0.00 1.00 0.00 0.00 0.14 0.14 Min Yes Satisfied 100%
Z4 0.00 0.00 0.00 0.85 0.00 0.17 0.17 Min Yes Satisfied 100%
Z5 0.00 0.53 0.36 0.00 0.00 0.17 0.17 Min Yes Satisfied 100%
Z6 0.00 0.00 0.00 0.00 2.00 0.45 0.25 Max No The objective function is
greater than the RHS value
Z7 0.00 2.00 0.00 0.00 0.00 0.57 0.29 Max No The objective function is
greater than the RHS value
Z1–Z7 are the objectives equivalent to criteria C1–C7.
Sustainability 2024,16, 9199 17 of 31
The second part of Table 4represents the values of the objective function for each of
the optimization models, the right-hand side (RHS) values and the preliminary analysis
of optimization. The preliminary analysis makes a comparison between the RHS values,
and the results for objective functions obtained through the SIMUS method. The RHS
presents the defined values for each objective for the criteria. The objective is satisfied 100%
in the case of equality of the objective function values and RHS. Objective Z2, operating
costs, which have to be minimized, shows that they are a little less (0.14) than the defined
value (0.16). This could be due to the small amount of freight for railways, or the high
frequencies of carriage of container trucks. Objectives Z6 and Z7 refer respectively to the
state of infrastructure, and to security and safety, which must be maximum. The computed
values of the objective function (0.45) and (0.57) are greater than the RHS values (0.25) and
(0.29) in the original table. It can be concluded that the actual state of the infrastructure and
security are higher than expected. This is positive for freight transport, but for transport
company finances it indicates a strain on operating costs.
Table 5shows the normalized efficient results matrix (NERM), and the determination
of the scores of alternatives.
Table 5. Normalized efficient results matrix (NERM).
Objective
Player A Player B
Alternatives Alternatives
A1 A2 B1 B2 B3
Z1 1.00 0.00 0.00 0.00 0.00
Z2 0.83 0.00 0.00 0.00 0.17
Z3 0.00 0.00 1.00 0.00 0.00
Z4 0.00 0.00 0.00 1.00 0.00
Z5 0.00 0.59 0.41 0.00 0.00
Z6 0.00 0.00 0.00 0.00 1.00
Z7 0.00 1.00 0.00 0.00 0.00
SC (SCAi;SCBj)1.83 1.59 1.41 1.00 1.17
PF (PFAi;PFBj)2 2 2 1 2
NPF 0.29 0.29 0.29 0.14 0.29
SC ×NPF a1a2b1b2b3
0.52 0.46 0.40 0.14 0.33
SC—sum of values in the column; PF—participation factor; NPF—norm. participation factor; SC
×
NPF—final
result. Number of targets = 7; ranking: A1–A2–B1–B5–B4.
4.3. Step 3: Model Formation in Game Theory
The next part of the methodology consists of the formation of the game theory model.
The elements of the payoff matrix are structured based on the results of the scores of
alternatives for both players, determined using the SIMUS method.
The payoff matrix is made up of rows for player A. The values of the matrix are
calculated by reducing the profit of player A with the profit of player B for each row of
payoff matrix. Table 6shows the payoff matrix.
Table 6. Payoff matrix of SIMUS–Game Theory Approach.
Player
Probability
B1B2B3
y1y2y3
A1x1p11 =a1−b1=
0.52 −0.40 =0.12 p12 =a1−b2=
0.52 −0.14 =0.38 p12 =a1−bn3=
0.52 −0.33 =0.19
A2x2p21 =a2−b1=
0.46 −0.40 =0.06 p12 =a2−b2=
0.46 −0.14 =0.32 p13 =a2−bn2=
0.46 −0.33 =0.13
The mathematical representation of the game model is as follows:
For player A can be written as follows:
Maximize ζ=ϑ, (27)
Sustainability 2024,16, 9199 18 of 31
Subject to the following:
0.12·x1+0.06·x2≥ϑ
0.38·x1+0.32·x2≥ϑ
0.19·x1+0.13·x2≥ϑ
x1+x2=1
(28)
For player A can be written as follows:
Minimize ζ=X1+X2, (29)
Subject to the following:
0.12·X1+0.06·X2≥1
0.38·X1+0.32·X2≥1
0.19·X1+0.13·X2≥1
X1+X2=1
Xi≥0, i=1, 2
X1=x1
ϑ;X2=x2
ϑ;ϑ=1
ζ
(30)
The problem of player B can be written as follows:
Minimize ω=ϑ, (31)
Subject to the following:
0.12y1+0.38y2+0.19y3≤ϑ
0.06y1+0.32y2+0.13y3≤ϑ
y1+y2+y3=1,
0≤yj≤1, j=1, 2, 3
(32)
The Game model for player B is as follows:
Maximize ω=(Y1+Y2+Y3)(33)
Subject to the following:
0.12Y1+0.38Y2+0.19Y3≤1
0.06Y1+0.32Y2+0.13Y3≤1
Yj≥0, j=1, 2, 3.
Y1=y1
ϑ;Y2=y2
ϑ;Y3=y3
ϑ;ϑ=1
ω
(34)
The game models were solved by means of duallinear programming. The results are given
in Table 7. For player A, the optimal strategy is A1, while for player B, the optimal strategy is
B1. Player A will apply their first strategy with probability 1, player B will also apply their first
strategy with probability 1. The value of the game is 0.12. This means that both players with
antagonistic interests will have a profit if they implement their optimal strategies.
Table 7. Results of the game models of SIMUS–Game Theory Approach.
Player A Player B
Probabilities Value of the Game Probabilities Value of the Game
x1x2ζy1y2y3ω
8.33 0.00 8.33 8.33 0.00 0.00 8.33
X1=x1
ϑX2=x2
ϑϑ=1
ζY1=y1
ϑY2=y2
ϑY3=y3
ϑϑ=1
ω
1.00 0.00 0.12 1.00 0.00 0.00 0.12
Sustainability 2024,16, 9199 19 of 31
5. Discussion
5.1. Verification of the Results
The new integrated SIMUS–game theory approach for decision making elaborated herein
permits us to determine the strategies of both players when they have antagonistic interests.
The results were verified by applying the following four criteria for decision making:
Laplace’s criterion, minimax and maximin criteria, Savage’s criterion and Hurwitz’s crite-
rion [
51
]. These criteria serve to determine the optimal strategies for each of the players.
The payoff matrix given in Table 6is used to calculate the values of the criteria.
The goal of player A is to choose a strategy with maximum utility. The value of
Laplace’s criterion is determined as follows:
Li=∑n2
j=1pij
n2,i=1, . . . , n1, (35)
where
i=
1,
. . .
,
n1
is the number of strategies for player A and
pij
represents the elements
of the payoff matrix.
The maximum value of Laplace’s criterion determines the optimal alternative for
player A, as follows:
Lopt =max
iLi. (36)
The goal of player B is to reduce their losses, as both players have antagonistic interests.
In this case, the value of Laplace’s criterion is determined as follows:
Lj=∑n1
i=1pij
n1,i=1, . . . , n1, (37)
where
j=
1,
. . .
,
n2
is the number of strategies for player B and
pij
represents the elements
of the payoff matrix.
The maximin criterion expresses the choice of the worst option among the best options.
The optimal alternative for player A is determined by the maximin criterion, as follows:
Lopt =min
jLj,j=1, . . . , n2. (38)
The maximin criterion expresses the choice of the worst option among the best options.
The optimal alternative for player A is determined by the maximin criterion, as follows:
max
imin
jpij ,i=1, . . . , n1; j =1, . . . .n2(39)
The minimax criterion expresses the choice of the best option among the worst options.
The optimal alternative for player B is determined by the minimax criterion, as follows:
min
jmax
ipij ,i=1, . . . , n1; j =1, . . . .n2. (40)
Savage’s criterion is determined based on the newly formed matrix with the following
elements:
rij =ri j −min
jpij , when pij represents the costs (41)
rij =max
ipij −pij , when pij represents the profits (42)
The newly formed matrix represents a loss. The minimax criterion is applied to make a
decision. For this purpose, the maximum costs by rows for each strategy of the new-formed
matrix are determined. The optimal strategy is that for which the maximum cost is the
smallest. The minimax criterion for both players is determined as follows:
Sustainability 2024,16, 9199 20 of 31
min
imax
jrij for player A (43)
min
jmax
irij , for player B (44)
Hurwitz’s criterion uses the minimum and the maximum values of the rows to make
a decision. The optimal strategy is determined as follows:
Hopt =max
iαmax
jpij +(1−α)min
jpij ,when pij represents the profits (45)
Hopt =min
jαmin
jpij +(1−α)max
jpij , when pij represents the costs (46)
where αis a coefficient of optimism, 0 ≤α≤1. Generally, α=0.5.
The strategy with the highest profit is the aim for player A. Table 8represents the
values of the criteria for verification for player A.
Table 8. Parameters of criteria for verification of player A.
Decision Matrix
Laplace’s
Criterion
Maximin
Criterion
rij
Savage’s
Criterion
Max Min
Hurwitz’s
Criterion
Player A Player B
B1 B2 B3 B1 B2 B3
A1 0.12 0.23 0.38 0.19 0.12 0 0 0 0 0.38 0.12 0.25
A2 0.06 0.17 0.32 0.13 0.06 0.06 0.06 0.06 0.06 0.32 0.06 0.19
Player B aims to implement the strategy with the least loss. The decision matrix for
player B is structured as a transposed matrix of player A, because the profit of player A is a
loss for player B. Table 9represents the values of the criteria for verification for player B.
Both operators have opposite interests.
Table 9. Parameters of criteria for verification for player B.
Decision Matrix
Laplace’s
Criterion
Minimax
Criterion
rij Savage’s
Criterion Max Min Hurwitz’s
Criterion
Player B Player A
A1 A2 A1 A2
B1 0.12 0.09 0.06 0.12 0 0 0 0.12 0.06 0.09
B2 0.38 0.35 0.32 0.06 0.26 0.26 0.06 0.38 0.32 0.35
B3 0.19 0.16 0.13 0.12 0.07 0.07 0.07 0.19 0.13 0.16
The results show that the optimal strategy for player A is A1 since it has a maximum
value for Laplace’s criterion, maximin criterion and Hurwitz’s criterion, and minimum value
for Savage’s criterion. The optimal strategy for player B is strategy B1 since it has a minimum
value for Laplace’s criterion, minimax criterion, Savage’s criterion and Hurwitz’s criterion.
Figures 3and 4illustrate the results for all criteria for verification.
Sustainability 2024, 16, x FOR PEER REVIEW 21 of 32
𝐻=max
𝛼 max
𝑝+(1−𝛼) min
𝑝 , when 𝑝 represents the profits, (45)
𝐻=min
𝛼min
𝑝+(1−𝛼) max
𝑝,when 𝑝 represents the costs (46)
where 𝛼 is a coefficient of optimism, 0 ≤𝛼≤1. Generally, 𝛼= 0.5.
The strategy with the highest profit is the aim for player A. Table 8 represents the
values of the criteria for verification for player A.
Player B aims to implement the strategy with the least loss. The decision matrix for
player B is structured as a transposed matrix of player A, because the profit of player A is
a loss for player B. Table 9 represents the values of the criteria for verification for player B.
Both operators have opposite interests.
Table 8. Parameters of criteria for verification of player A.
Decision Matrix Laplace’s
Criterion
Maximin
Criterion 𝒓𝒊𝒋 Savage’s
Criterion Max Min
Hurwitz’s
Criterion
Player A Player B
B1 B2 B3 B1 B2 B3
A1 0.12 0.23 0.38 0.19 0.12 0 0 0 0 0.38 0.12 0.25
A2 0.06 0.17 0.32 0.13 0.06 0.06 0.06 0.06 0.06 0.32 0.06 0.19
Table 9. Parameters of criteria for verification for player B.
Decision Matrix Laplace’s
Criterion
Minimax
Criterion 𝒓𝒊𝒋 Savage’s
Criterion Max Min Hurwitz’s
Criterion
Player B Player A
A1 A2 A1 A2
B1 0.12 0.09 0.06 0.12 0 0 0 0.12 0.06 0.09
B2 0.38 0.35 0.32 0.06 0.26 0.26 0.06 0.38 0.32 0.35
B3 0.19 0.16 0.13 0.12 0.07 0.07 0.07 0.19 0.13 0.16
The results show that the optimal strategy for player A is A1 since it has a maximum
value for Laplace’s criterion, maximin criterion and Hurwi’s criterion, and minimum
value for Savage’s criterion. The optimal strategy for player B is strategy B1 since it has a
minimum value for Laplace’s criterion, minimax criterion, Savage’s criterion and Hur-
wi’s criterion.
Figures 3 and 4 illustrate the results for all criteria for verification.
It can be concluded that the results obtained by using the criteria of verification give
the same choice of strategies for both players as those are obtained through game theory.
Figure 3. Criteria for verification. Player A.
Figure 3. Criteria for verification. Player A.
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Sustainability 2024, 16, x FOR PEER REVIEW 22 of 32
Figure 4. Criteria for verification. Player B.
The verification of the results confirms the hypothesis of this research about the pos-
sibility of decision making by means of the new integrated SIMUS–game theory approach
elaborated on herein.
5.2. Calculation of the Weights of Criteria
The SIMUS method allows one to determine the weights of criteria based on the re-
sults in the normalized ERM matrix. It can be used only for analysis. Further procedures
with the weights are not carried out. The number of columns in the NERM matrix is cal-
culated as a sum of strategies for player A and player B (𝑛+ 𝑛) and the number of rows
correspond to the number of criteria. The maximum value of each row max
𝑁𝐸𝑅𝑀 is
determined.
The weights are determined as follows:
𝑤=
∑
, (47)
0≤𝑤≤1, (48)
∑𝑤
=1, (49)
where 𝑘=1,…,𝑚 is the number of criteria,
𝑞=1,…,𝑛+ 𝑛 is the total number of
strategies for player A and player B, 𝑛 is the number of strategies for player A, and 𝑛
is the number of strategies for player B.
Table 10 presents the NERM matrix and the weights of the criteria. The criteria with
the greatest impact are those of carbon dioxide emissions (C1), infrastructure charges for
the use of railway and road infrastructure (C3), duration of transportation (C4), infrastruc-
ture factor (C6), and security and safety (C7), as in Figure 5.
The determined values of weights could serve a decision maker only for analysis and
not for consequent actions.
Table 10. Weights of objectives obtained using the SIMUS method.
Objective
Player A Player B 𝐦𝐚𝐱
𝒒𝑵𝑬𝑹𝑴𝒌𝒒 𝒘𝒌
Alternatives Alternatives
A1 A2 B1 B2 B3
Z1 1.00 0.00 0.00 0.00 0.00 1.00 0.16
Z2 0.83 0.00 0.00 0.00 0.17 0.83 0.13
Z3 0.00 0.00 1.00 0.00 0.00 1.00 0.16
Z4 0.00 0.00 0.00 1.00 0.00 1.00 0.16
Z5 0.00 0.59 0.41 0.00 0.00 0.59 0.09
Z6 0.00 0.00 0.00 0.00 1.00 1.00 0.16
Z7 0.00 1.00 0.00 0.00 0.00 1.00 0.16
Total
6.42 1.00
Figure 4. Criteria for verification. Player B.
It can be concluded that the results obtained by using the criteria of verification give
the same choice of strategies for both players as those are obtained through game theory.
The verification of the results confirms the hypothesis of this research about the possi-
bility of decision making by means of the new integrated SIMUS–game theory approach
elaborated on herein.
5.2. Calculation of the Weights of Criteria
The SIMUS method allows one to determine the weights of criteria based on the results
in the normalized ERM matrix. It can be used only for analysis. Further procedures with the
weights are not carried out. The number of columns in the NERM matrix is calculated as a
sum of strategies for player A and player B
(n1+n2)
and the number of rows correspond
to the number of criteria. The maximum value of each row max
qNERMkq is determined.
The weights are determined as follows:
wk=
max
qNERMkq
∑m
k=1NERMkq
, (47)
0≤wk≤1, (48)
∑m
k=1wk=1, (49)
where
k=
1,
. . .
,
m
is the number of criteria,
q=
1,
. . .
,
n1+n2
is the total number of
strategies for player A and player B,
n1
is the number of strategies for player A, and
n2
is
the number of strategies for player B.
Table 10 presents the NERM matrix and the weights of the criteria. The criteria with
the greatest impact are those of carbon dioxide emissions (C1), infrastructure charges for the
use of railway and road infrastructure (C3), duration of transportation (C4), infrastructure
factor (C6), and security and safety (C7), as in Figure 5.
Table 10. Weights of objectives obtained using the SIMUS method.
Objective
Player A Player B
max
qNERMkq wk
Alternatives Alternatives
A1 A2 B1 B2 B3
Z1 1.00 0.00 0.00 0.00 0.00 1.00 0.16
Z2 0.83 0.00 0.00 0.00 0.17 0.83 0.13
Z3 0.00 0.00 1.00 0.00 0.00 1.00 0.16
Z4 0.00 0.00 0.00 1.00 0.00 1.00 0.16
Z5 0.00 0.59 0.41 0.00 0.00 0.59 0.09
Z6 0.00 0.00 0.00 0.00 1.00 1.00 0.16
Z7 0.00 1.00 0.00 0.00 0.00 1.00 0.16
Total 6.42 1.00
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Figure 5. Weights of criteria determined using the SIMUS method.
5.3. Comparison with the Game Cost-Based Model
In order to demonstrate the advantage of the new integrated SIMUS–game theory
methodology, a comparison of the obtained results is made with an approach for applying
a game theory model based only on costs. For this purpose, the data for operational costs
and fees for loading and unloading operations (C2) presented in Table 2 were used. As the
costs of player B regarding all of his strategies are greater than those of player A, the payoff
matrix was formed for player B. This is shown in Table 11. The results for the game cost-
based model is represented in Table 12.
Table 11. Payoff matrix of game model by costs.
Player
Costs
Probability
𝑨
𝟏
𝑨
𝟐
EUR/UTI 152 147
𝒙𝟏 𝒙𝟐
𝐵 190 𝑦 190−152=38 170−147=43
𝐵 227 𝑦 227−152=75 227−147=80
𝐵 189 𝑦 189−152=37 189−147=42
Table 12. Results of game models by costs.
Player A Player B
Probabilities Value of the Game Probabilities Value of the Game
𝑥 𝑥 𝜁 𝑦 𝑦 𝑦 𝜔
0.00 0.024 0.024 8.33 0.00 0.024 0.024
𝑋=𝑥
𝜗 𝑋=𝑥
𝜗 𝜗=1
𝜁
𝑌=𝑦
𝜗 𝑌=𝑦
𝜗 𝑌=𝑦
𝜗 𝜗=1
𝜔
0.00 1.00 42 0.00 0.00 1.00 42
It can be seen that the solution in this case shows that the optimal strategy for player
A is the second one, while for player B the optimal strategy is the third one. The results
are different compared with those of the integrated SIMUS–game theory approach. This
is due to the fact that only costs are used, and not a set of criteria with their total benefit.
When the game theory is used by itself, the strategies of both players are determined
only in terms of the revenue or cost criteria. The ranking of the alternatives of both players
is performed, and their strategies are considered jointly. The best alternative among all of
the strategies of both players is thus determined.
5.4. Sensitivity Analysis Using SIMUS
The SIMUS method utilizes linear programming. This makes it possible to define, for
each optimization, the limits of the values of the criteria at which the solution is preserved.
Figure 5. Weights of criteria determined using the SIMUS method.
The determined values of weights could serve a decision maker only for analysis and
not for consequent actions.
5.3. Comparison with the Game Cost-Based Model
In order to demonstrate the advantage of the new integrated SIMUS–game theory
methodology, a comparison of the obtained results is made with an approach for applying
a game theory model based only on costs. For this purpose, the data for operational costs
and fees for loading and unloading operations (C2) presented in Table 2were used. As
the costs of player B regarding all of his strategies are greater than those of player A, the
payoff matrix was formed for player B. This is shown in Table 11. The results for the game
cost-based model is represented in Table 12.
Table 11. Payoff matrix of game model by costs.
Player
Costs
Probability
A1A2
EUR/UTI 152 147
x1x2
B1190 y1190 −152 =38 170 −147 =43
B2227 y2227 −152 =75 227 −147 =80
B3189 y3189 −152 =37 189 −147 =42
Table 12. Results of game models by costs.
Player A Player B
Probabilities Value of the Game Probabilities Value of the Game
x1x2ζy1y2y3ω
0.00 0.024 0.024 8.33 0.00 0.024 0.024
X1=x1
ϑX2=x2
ϑϑ=1
ζY1=y1
ϑY2=y2
ϑY3=y3
ϑϑ=1
ω
0.00 1.00 42 0.00 0.00 1.00 42
It can be seen that the solution in this case shows that the optimal strategy for player
A is the second one, while for player B the optimal strategy is the third one. The results are
different compared with those of the integrated SIMUS–game theory approach. This is due
to the fact that only costs are used, and not a set of criteria with their total benefit.
When the game theory is used by itself, the strategies of both players are determined
only in terms of the revenue or cost criteria. The ranking of the alternatives of both players
is performed, and their strategies are considered jointly. The best alternative among all of
the strategies of both players is thus determined.
Sustainability 2024,16, 9199 23 of 31
5.4. Sensitivity Analysis Using SIMUS
The SIMUS method utilizes linear programming. This makes it possible to define, for
each optimization, the limits of the values of the criteria at which the solution is preserved.
For each objective we performed a sensitivity analysis. The allowable range of varia-
tion of criteria were thus determined. Table 13 shows the upper limit (U) and the lower
limit (L) for each of the criteria determined by SIMUS.
Table 13. Sensitivity analysis using SIMUS.
Objectives Player A Player B
A1 A2 B1 B2 B3
C1 U ∞ ∞ ∞ ∞ ∞
238,257.00 248,250.00 442,887.00 519,180.00 545,931.00
L 0 230,420 297,821 355,818 296,254
C2 U 152 ∞ ∞ ∞ 195
152.00 147.00 190.00 227.00 189.00
L 82 146 153 180 154
C3 U ∞ ∞ 52 ∞ ∞
96.00 94.00 51.00 61.00 65.00
L 62 59 0 60 63
C4 U ∞ ∞ 7.26 ∞ ∞
9.05 8.57 7.14 8.12 8.62
L 5 5 1 8 8
C5 U ∞516 466 ∞ ∞
543.00 514.00 447.00 524.00 551.00
L 541 420 428 511 528
C6 U 2 2 2 2 2
2.00 1.00 1.53 1.75 1.82
L 1 1 1 1 1
C7 U 2 2 2 2 2
2.00 2.00 1.00 1.00 1.00
L 1 1 1 1 1
It can be seen in Table 13 that some values are unlimited or equal to zero. These values
are only theoretical.
The following conditions have to be taken into account:
•
The limits for the criterion “carbon dioxide emissions” (C1) depends on the electricity
consumption for train movement and fuel consumption for truck movement. It is
also related to the permissible speed for movement in the sections, the type of cargo
transported, the weight of the cargo, and the restrictions in the infrastructure.
•
The limits of the criterion “operational costs” (C2) depend on the cost rates for fixed
and variable costs, depreciation of the rolling stock, operating personnel, and the
characteristics of the type of transport.
•
The limits of the criterion “infrastructure charges” (C3) depend on the state’s policy
and the state of the transport infrastructure.
•
The limits of the criterion “duration of transportation” (C4) depend on the restrictions
on the speed of movement in the sections of the transport infrastructure, the state of
the infrastructure, and the traffic load.
•
The value of the criterion “light on the route” (C5) cannot be changed. It cannot be
reduced or increased as no changes are planned in the studied routes. The upper and
the lower limits are only theoretical.
•The upper limit for both of the criteria C6 and C7 is 2. The lower limit is 1.
Sustainability 2024,16, 9199 24 of 31
The obtained values for the criteria limits could also be used for analysis in cases when
the criteria values are not precisely set. If some or all of the criteria are changed within the
established limits, the obtained decision for the scores of the alternatives does not change.
5.5. Comparison with the SIMUS Ranking Model
The results of the newly integrated approach can be compared with the results obtained
using the SIMUS method. For this purpose, Tables 5and 7are considered. The last
row in Table 5shows the ranking of the strategies of the players when all strategies are
considered together.
It should be noted that the SIMUS method aims to rank the alternatives and not
to determine the optimal strategies for each of the players. It can be seen that the most
appropriate route selection strategy is A1, i.e., travel by carriage via the Sofia–Gorna
Oryahovitsa–Varna railway route, as in Table 5. This strategy is also optimal for the
railway operator according to the results obtained through the new approach. If the
ranking obtained by SIMUS is analyzed, it can be seen that, out of all strategies of the road
operator, strategy B1 has the highest rating, i.e., carriage by container trucks along the
Sofia–Veliko Tarnovo–Varna route. When analyzed separately, the rail and road operator
rankings show which strategies are most appropriate for each of them. This procedure
can also be used to verify the results obtained through the new integrated approach. The
independent application of the SIMUS method is not appropriate in case of antagonistic
interests between two transport operators. Although the ranking is based on the overall
utility of the quantitative and qualitative criteria used, this method does not show what
the cost of the game is and what is the probability that each of the players will use each
of their strategies under antagonistic interests. This is achieved with the newly integrated
SIMUS–game theory approach proposed in this research.
5.6. Comparison of the Results
A comparison of the three procedures—individual application of the SIMUS method,
game theory using a costs-based model and the integrated SIMUS–game theory approach—
is shown in Figure 6. The most suitable alternative among all investigated alternatives of
both players is determined.
Sustainability 2024, 16, x FOR PEER REVIEW 25 of 32
5.5. Comparison with the SIMUS Ranking Model
The results of the newly integrated approach can be compared with the results ob-
tained using the SIMUS method. For this purpose, Tables 5 and 7 are considered. The last
row in Table 5 shows the ranking of the strategies of the players when all strategies are
considered together.
It should be noted that the SIMUS method aims to rank the alternatives and not to
determine the optimal strategies for each of the players. It can be seen that the most ap-
propriate route selection strategy is A1, i.e., travel by carriage via the Sofia–Gorna Or-
yahovitsa–Varna railway route, as in Table 5. This strategy is also optimal for the railway
operator according to the results obtained through the new approach. If the ranking ob-
tained by SIMUS is analyzed, it can be seen that, out of all strategies of the road operator,
strategy B1 has the highest rating, i.e., carriage by container trucks along the Sofia–Veliko
Tarnovo–Varna route. When analyzed separately, the rail and road operator rankings
show which strategies are most appropriate for each of them. This procedure can also be
used to verify the results obtained through the new integrated approach. The independent
application of the SIMUS method is not appropriate in case of antagonistic interests be-
tween two transport operators. Although the ranking is based on the overall utility of the
quantitative and qualitative criteria used, this method does not show what the cost of the
game is and what is the probability that each of the players will use each of their strategies
under antagonistic interests. This is achieved with the newly integrated SIMUS–game the-
ory approach proposed in this research.
5.6. Comparison of the Results
A comparison of the three procedures—individual application of the SIMUS method,
game theory using a costs-based model and the integrated SIMUS–game theory ap-
proach—is shown in Figure 6. The most suitable alternative among all investigated alter-
natives of both players is determined.
The results obtained through the game theory cost-based model show that the best
alternative for the railway operator is the use of container block trains on the Sofia–Kar-
lovo–Karnobat–Varna route (A2). This transport service offers minimal operating costs
and transport time, but the carbon dioxide emissions are larger compared with railway
route A1. The proposed strategy for the road operator is carriage by container trucks on
the Sofia–Plovdiv–Karnobat–Shumen–Varna route (B3). This alternative offers compara-
ble costs with route B1, but time travel and carbon dioxide emissions are greater.
It can be seen in Figure 6 that the SIMUS ranking is A1 > A2 > B1 > B3 > B2. The most
appropriate strategy in this case is A1 (container block trains on the Sofia–Gorna Or-
yahovitsa–Varna route).
Figure 6. Comparison of the results using the SIMUS method, SIMUS–game theory and game theory
by costs.
Figure 6. Comparison of the results using the SIMUS method, SIMUS–game theory and game theory
by costs.
The results obtained through the game theory cost-based model show that the best
alternative for the railway operator is the use of container block trains on the Sofia–Karlovo–
Karnobat–Varna route (A2). This transport service offers minimal operating costs and
transport time, but the carbon dioxide emissions are larger compared with railway route
A1. The proposed strategy for the road operator is carriage by container trucks on the
Sustainability 2024,16, 9199 25 of 31
Sofia–Plovdiv–Karnobat–Shumen–Varna route (B3). This alternative offers comparable
costs with route B1, but time travel and carbon dioxide emissions are greater.
It can be seen in Figure 6that the SIMUS ranking is A1 > A2 > B1 > B3 > B2. The
most appropriate strategy in this case is A1 (container block trains on the Sofia–Gorna
Oryahovitsa–Varna route).
The newly integrated SIMUS–game theory approach determines the strategies of
each of the players, in this case A1 for player A and B1 for player B; each of these will
implement their strategy with a probability equal to 1. This means that, for the railway
operator, the optimal strategy is carriage by container block trains on the Sofia–Gorna
Oryahovitsa–Varna route, while, for the road operator, the optimal strategy is carriage by
container trucks on the Sofia–Veliko Tarnovo–Varna route. For the railway operator this
result shows minimal carbon dioxide emissions, comparable operating costs and travel
times with the alternative rail transport strategy (Sofia–Karlovo–Karnobat–Varna). For the
road operator the result shows carriage with minimal carbon dioxide emissions, operating
costs close to the minimum and the shortest transport time. These results demonstrate the
overall benefits of the investigated criteria. It can be said that the results obtained through
the new integrated SIMUS–game theory approach are better when compared with those
obtained though the game theory cost-based model regarding operating costs, time during
and carbon dioxide emissions.
5.7. Concept of Application in the Case of Games with Imperfect Information: Fuzzy SIMUS–Game
Theory Approach
There are many cases when the information regarding the criteria values of the two
players cannot be precisely determined. In this situation, the studied system is in a state of
uncertainty. In these cases, the proposed new integrated approach SIMUS–game theory
can be extended by applying a fuzzy SIMUS method. The application is also for non-
cooperative games. The fuzzy SIMUS method is elaborated in [
53
]. The steps of the
integrated fuzzy SIMUS–game theory approach are as follows:
•
For each player, three initial decision matrices with the values of the criteria for each
alternative are set—lower (L), medium (M), and upper (U).
•
An average decision matrix is formed. Its elements are calculated as an average value
by using the lower, medium, and upper values.
•Determination of the normalized average, upper and lower matrices.
•
Determination of the objective function per criterion for both lower and upper matrices.
•
Fuzzy linear optimization with linear membership function. For each objective, the op-
timization is formed and calculated sequentially based on the average
decision matrix
.
•The results of fuzzy linear models are recorded in fuzzy efficient results matrix.
•The SIMUS procedure ranking is then applied.
•
The payoff matrix for game model is formed as in the SIMUS–game theory approach.
The applicability of the new fuzzy SIMUS–game theory approach is shown below
for the case study from point 4 under uncertainty. Tables 14 and 15 represent the upper
and lower matrices. It can be assumed that the matrix shown in Table 2is a medium
matrix. Criteria C5, C6 and C7 are set with constant values in the matrices. The criterion
C5 presents the light of the route, which is constant for each route, criterion C6 presents the
infrastructure factor, and criterion C7 presents the security. The changes in carbon dioxide
values can be explained by changes in load weight, electricity consumption, and changes in
speed of movement. The changes in operating costs and tariffs can be explained by changes
in prices and government policy.
The SIMUS method was calculated separately for each of the given matrices. Table 15
shows the results for objective functions for both upper and lower matrices (
Zi,U,Zi,L)
and
the threshold values (
RHSj,U,RHSj,L
. These results were used in fuzzy linear models
for membership functions. For objectives Z5, Z6 and Z7 we applied the SIMUS linear
procedure by using the average matrix. This is because only one value is set for criteria C5,
C6 and C7.
Sustainability 2024,16, 9199 26 of 31
Table 14. Upper decision matrix.
Criteria
Player A Player B
Type
Alternatives Alternatives
Symbol Dimension A1 A2 B1 B2 B3
C1 CO2, g/UTI 262,083 273,075 487,176 571,098 600,524 min
C2 EUR/UTI 167 162 209 250 208 min
C3 EUR/UTI 101 99 54 64 68 min
C4 h 9.2 9.15 7.25 8.3 9 min
C5 km 543 514 447 524 551 min
C6 - 2 1 1.53 1.75 1.82 max
C7 - 2 2 1 1 1 max
Table 15. Lower decision matrix.
Criteria Player A Player B
Type
Alternatives Alternatives
Symbol Dimension A1 A2 B1 B2 B3
C1 CO2, g/UTI 214,431 223,425 398,598 467,262 491,338 min
C2 EUR/UTI 137 132 171 204 170 min
C3 EUR/UTI 91 89 48 58 62 min
C4 h 9 8.55 7.1 8.1 8.55 min
C5 km 543 514 447 524 551 min
C6 - 2 1 1.53 1.75 1.82 max
C7 - 2 2 1 1 1 max
Fuzzy linear models were formed for each criterion. For this purpose, the average
decision matrix was used. The results of the SIMUS method for normalized lower and
upper matrices are applied to build the optimization models. The fuzzy linear models are
solved by linear membership function.
The efficient results fuzzy matrix (ERFM) presents the results of fuzzy linear models.
The first part of Table 16 shows the results of FERM and the values of objectives. The rows
represent the scores of the alternatives obtained by optimization models. The second part
of Table 17 represents the results of normalized ERFM and the SIMUS scores.
Table 16. Results of the SIMUS method for normalized lower and upper matrices.
Objective RHSj,UZi,URHSj,LZi,LZi,U
Zi,U−Zi,L
Zi,L
Zi,U−Zi,L
RHSj,U
RHSj,U−RHSj,L
RHSj,L
RHSj,U−RHSj,L
Z1 0.133 0.133 0.126 0.127 147.916 146.92 490,849.91 490,848.91
Z2 0.102 0.095 0.105 0.097 367.193 368.19 333.00 332.00
Z3 0.116 0.116 0.117 0.119 71.182 70.18 71.18 70.18
Z4 0.112 0.119 0.113 0.121 0.059 1.06 0.07 1.07
Z5 0.104 0.099 0.104 0.097 - - - -
Z6 0.167 0.169 0.167 0.169 - - - -
Z7 0.120 0.120 0.124 0.126 - - - -
The results of fuzzy SIMUS are applied to form the payoff matrix, which is made up of
rows for player A. The values of the matrix are calculated by reducing the profit of player A
with the profit of player B for each row of payoff matrix. Table 18 shows the payoff matrix.
The game models were formed using the payoff matrix. Table 19 shows the results.
Sustainability 2024,16, 9199 27 of 31
Table 17. Normalized efficient results fuzzy matrix (NERFM). Results of fuzzy SIMUS score.
Efficient Results Fuzzy Matrix Normalized Efficient Results Matrix
Objective
Player A Player B Player A Player B
Alternatives Alternatives Alternatives Alternatives
A1 A2 B1 B2 B3 A1 A2 B1 B2 B3
Z1 0.000 0.870 0.000 0.000 0.000 0.000 1.000 0.000 0.000 0.000
Z2 0.823 0.000 0.000 0.000 0.000 1.000 0.000 0.000 0.000 0.000
Z3 0.823 0.000 0.000 0.000 0.000 1.000 0.000 0.000 0.000 0.000
Z4 0.823 0.000 0.000 0.000 0.000 1.000 0.000 0.000 0.000 0.000
Z5 0 0.53 0.36 0 0 0.000 0.596 0.404 0.000 0.000
Z6 0 0 0 0 2 0.000 0.000 0.000 0.000 1.000
Z7 0 2 0 0 0 0.000 1.000 0.000 0.000 0.000
SC (SCAi;SCBj) - - - - - 3.000 2.596 0.404 0.000 1.000
PF (PFAi;PFBj) - - - - - 3 4 1 0 1
NPF - - - - - 0.429 0.571 0.143 0.000 0.143
SC ×NPF -----a1a2b1b2b3
1.286 1.483 0.058 0.000 0.143
Table 18. Payoff matrix of Fuzzy SIMUS–Game Theory Approach.
Player Probability B1B2B3
y1y2y3
A1x1p11 =a1−b1=1.228 p12 =a1−b2=1.286 p12 =a1−bn3=1.143
A2x2p21 =a2−b1=1.054 p12 =a2−b2=1.112 p13 =a2−bn2=0.969
Table 19. Results of the game models of Fuzzy SIMUS–Game Theory Approach.
Player A Player B
Probabilities Value of the Game Probabilities Value of the Game
x1x2ζy1y2y3ω
0.875 0.000 0.875 0.00 0.00 0.875 0.746
X1=x1
ϑX2=x2
ϑϑ=1
ζY1=y1
ϑY2=y2
ϑY3=y3
ϑϑ=1
ω
1.00 0.00 1.14 0.00 0.00 1.00 1.14
It can be seen that the optimal strategy for player A is A1, the optimal strategy for
player B is B3. The value of the game is 1.14. This means that both players with antagonistic
interests will have a profit if they implement their optimal strategies. It can be seen that
the results are different from those in the case of certainty obtained by the SIMUS–game
theory approach.
The change in results can be explained by the changes in the values of the criteria.
Some upper limits of criteria values determined by the linear optimization models using
the SIMUS method are exceeded in the upper decision matrix, Table 12.
6. Conclusions
The contributions of the conducted research can be summarized in the following aspects:
(1)
Regarding decision theory—A new integrated approach to determine the optimal
strategies of two players in non-cooperative games is developed, in which the payoff
matrix of the game model consists of the benefits of quantitative and qualitative
criteria for evaluating the strategies determined using the SIMUS multicriteria analysis
method. This approach can be applied in various fields of research. A concept has
Sustainability 2024,16, 9199 28 of 31
been developed for the case of uncertainty, where the fuzzy SIMUS–game theory
approach is applied.
(2)
For society—The proposed approach can be used in various fields of research. The
article examines a case for the Bulgarian transport network for choosing a route for
the transport of containers between Sofia and Varna by competing railway and road
operators. Applying the proposed approach allows the two transport operators to
choose their optimal route strategies so that they can both benefit. This includes not
only operating costs, but also environmental, technological, infrastructural criteria, as
well as security and safety of transportation.
(3)
Regarding the transport technique—The study examines two types of transport, rail
and road, which are in competition with antagonistic interests, and the corresponding
transport technique, container trains and container trucks. The application of the
elaborated methodology allows both types of transport to be distributed along efficient
routes, taking into account the benefit of a set of criteria applied in the study. This
approach allows both competing operators to carry out transportation along a route
that will bring them equal benefit. The use of different types of transport on parallel
lines between the starting point and the final point allows for adequate use of the
transport infrastructure and transport technique.
(4) Regarding the researched example of the Bulgarian transport network—The strategies
for railway and road operator for the carriage of containers between Sofia and Varna
were obtained based on the new integrated SIMUS–game theory approach. It was
determined that the proposed strategies for both operators determined by the new
integrated approach are better when compared with those obtained by the game
theory cost-based model concerning operating costs, time travel and carbon dioxide
emissions. For the railway operator, the proposed route offers minimal carbon dioxide
emissions and operating costs and travel times that are close to the minimum values.
For the road operator the result shows that carriage has minimal carbon dioxide
emissions, operating costs that are close to minimum, and the shortest transport time.
This paper creates new scientific knowledge regarding decision making using the
SIMUS method and non-cooperative games in a new approach decision making.
The advantages of the new integrated SIMUS–game theory approach can be summa-
rized as follows:
•
It allows for strategies of competing players to be determined taking into account the
benefits of criteria affecting the transport process.
•
The integration of the game theory method with the SIMUS method permits us to
take into account different criteria that influence decision making about the optimal
strategies for each of the players. In this case, the payment matrix for game theory
is compiled taking into account the benefits of all criteria determined through the
SIMUS method.
•
In this research, the SIMUS method is chosen among numerous MCDM to be integrated
with game theory because it uses linear optimization, the ranking is based on benefits,
criteria weights are not used, expert evaluations are not used, and for this reason there
is no subjectivity in decision making.
•
The combination of the SIMUS method and game theory allow one to assess the
weights of criteria in the formation of the common benefit. Thus, it can assess which
criteria have the greatest influence on the total benefit of all criteria in determining
the strategies of the competing players. The calculation of the weights of the criteria
can be used only for analysis by the decision maker after performing the optimization
procedures. The criteria weights are not used in subsequent operations.
•
The individual application of the SIMUS method does not allow one to determine
what the probability is for each of the players to apply their most appropriate strategy
in the case of antagonistic interests. It ranks only alternatives based on benefits. The
combination of the SIMUS method and game theory makes it possible to determine
Sustainability 2024,16, 9199 29 of 31
the probabilities of the application of the most suitable strategies for both players with
antagonistic interests, taking into account the benefits of a set of criteria.
The new methodology was applied for sustainable decision making for a railway and
a road operator, with antagonistic interests, for the carriage of containers on the Bulgarian
transport network. Seven criteria were applied to assess the benefits for both players. The
strategies for both operators were investigated. The results were verified using four criteria
for decision making.
The new integrated SIMUS–game theory methodology can be applied in different
areas of research, when the strategies for two players in non-cooperatives games have to
be established. The procedure helps the decision maker to establish those strategies for
the players that will bring them benefits. The benefits are established through the overall
impact of various criteria.
In future work the new integrated approach will be extended to the case of cooperative
games. This is of interest because many non-antagonistic conflicts allow participants to
cooperate to maximize their profits.
Funding: This work was supported by the Bulgarian National Science Fund—BNSF—of the Ministry
of Education and Science of Bulgaria [project number No.KP-06-H77/11 of 14.12.2023 “Modeling and
development of a complex system for environmental and energy efficiency of urban transport”].
Institutional Review Board Statement: Not applicable.
Informed Consent Statement: Not applicable.
Data Availability Statement: The original contributions presented in the study are included in the
article, further inquiries can be directed to the corresponding author.
Conflicts of Interest: The author declares no conflicts of interest.
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