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is work presents a Variable Neighborhood Search (VNS) approach for solving the Return-To-Player (RTP) optimization problem. A large number of software companies in the gaming industry seek to solve the RTP optimization problem in order to develop modern virtual casino gambling machines. ese slot machines have a number of reels (e.g., three or more) that spin once a button is pushed. Each slot machine is required to have an RTP in a particular range according to the legislation of each country. By using a VNS framework that guides two local search operators, we show how to control the distribution of the symbols in the reels in order to achieve the desired RTP. In this study, optimization refers only to base game, the core of slot machine games, and not in bonus games, since a bonus game is triggered once two, three, or more specific symbols occur in the gaming monitor. Although other researchers have tried to solve the RTP problem in the past, this is the first time that a VNS methodology is proposed for this problem in the literature with good computational results.
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Research Article
Slot Machine RTP Optimization Using Variable
Neighborhood Search
Pantelis-Arsenios Kamanas , Angelo Sifaleras , and Nikolaos Samaras
Department of Applied Informatics, School of Information Sciences, University of Macedonia, 156 Egnatia Str.,
essaloniki 54636, Greece
Correspondence should be addressed to Angelo Sifaleras; sifalera@uom.gr
Received 11 August 2020; Accepted 27 April 2021; Published 5 May 2021
Academic Editor: Ramon Sancibrian
Copyright ©2021 Pantelis-Arsenios Kamanas et al. is is an open access article distributed under the Creative Commons
Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is
properly cited.
is work presents a Variable Neighborhood Search (VNS) approach for solving the Return-To-Player (RTP) optimization
problem. A large number of software companies in the gaming industry seek to solve the RTP optimization problem in order to
develop modern virtual casino gambling machines. ese slot machines have a number of reels (e.g., three or more) that spin once
a button is pushed. Each slot machine is required to have an RTP in a particular range according to the legislation of each country.
By using a VNS framework that guides two local search operators, we show how to control the distribution of the symbols in the
reels in order to achieve the desired RTP. In this study, optimization refers only to base game, the core of slot machine games, and
not in bonus games, since a bonus game is triggered once two, three, or more specific symbols occur in the gaming monitor.
Although other researchers have tried to solve the RTP problem in the past, this is the first time that a VNS methodology is
proposed for this problem in the literature with good computational results.
1. Introduction
Slot machines, also known as fruit machines, are the most
popular gambling games in casinos. ey are either elec-
tronic or electromechanical devices that consist of a number
of reels (n) that spin independently and a screen with a
number of rows (m). Each reel contains several symbols in
specific arrangements and quantities. Also, it has a number
of positions at which they stop, and this number is called
stops [1]. e stops are being shown on the display of a slot
machine, while the reels are spinning in front of the player.
Slot machines as mechanical devices had cylinders as
spinning reels holding the symbols that were printed on a
strip. Nowadays, slot machines are mostly electronic, and
they can be found either in traditional casinos or in online
casinos as software. However, in virtual slots, once the reels
start spinning, a Pseudo-Random Number Generator
(PRNG) is used to get the stops.
e display of a virtual slot machine shows the (m×n)
window of the game, game rules, coin values, line bet, payout
schedule, and more options. e player inserts cash or a
paper ticket with a barcode in “ticket-in, ticket-out” ma-
chine. Once the player presses a button, in case the slot
machine has the minimum amount of bet, the reels of the
base game start spinning. After that, the reels produce the
outcome of the game, which consists of all symbols that
remained visible through the window on the display and the
payment of the combination in pay table of these symbols
that are greater than zero. If the visible window contains
symbols that trigger a bonus game (e.g., free spins in base
game or to another feature game), then the player has a big
chance to win much more money than he/she bets.
e gambling industry is one of the most profitable
industries globally. With more than US$ 450 billion dollars
estimated value in 2020 and with a projected value of more
than US$ 640 billion by 2027, the gambling market is
definitely one of the largest and most profitable in the world
[2]. It is also worth mentioning that these figures are only
related to the regulated markets, as the grey and black
markets contribute with hundreds of billions of dollars to the
Hindawi
Mathematical Problems in Engineering
Volume 2021, Article ID 8784065, 8 pages
https://doi.org/10.1155/2021/8784065
total market value. ere are hundreds of companies of
different types of services that compose the gambling in-
dustry: from casinos and software developers to casino game
developers and payment solutions. e growth of remote
gaming, with online and mobile gambling market size, is
expected to grow by more than 10% by 2027, ensuring that
the gambling industry will become even more profitable in
the years to come. Hundreds of thousands of dollars are
spent every year from several companies in order to par-
ticipate in exhibitions around the world, with the most
famous and attractive being held in Las Vegas, Paris,
London, South Africa, and Singapore. A list of the most well-
known conferences and trade shows with a Gaming, Sports
Betting, and Affiliate theme may be found online [3], for
example, Paris Games Week, iGB Live!, and ICE Totally
Gaming.
Return-To-Player (RTP) is one of the two most im-
portant parameters for a player to choose if he/she is going to
play the game, but it is also the main gambling game pa-
rameter which is under government regulation [4]. It is the
win as percentage of the bet and is the phrase that casinos
and casino game makers use to describe the long-term
theoretical expected payback percentage from all wagers on a
slot machine, video poker machine, or Video Lottery Ter-
minal (VLT). If a slot machine player is going to play a
hundred rounds at 1$ on a slot game with a theoretical slot
RTP of 93%, with all other things being equal, he/she would
expect to get a return of 93$ in wins from his staked 100$. A
player wins back most of his wager. e remaining 7% of his
bet, in this game, is the house edge, the money that casino
earns off each bet. e formula for the computation of the
RTP and also some necessary notations can be found with
the formulae below:
RTP Coins Out
Coins In ×100%,(1)
or
RTP Total Prize
Cycle ×Paylines ×100%.(2)
Total Prize: the amount of money that a window in-
stance wins
Cycle: how many times the reels spun in a Monte Carlo
simulation
Pay Lines: the number of lines that the game has
enabled
Coins In: total amount of coins bet
Coins Out: total amount of coins won
In this paper, Monte Carlo simulation with 100,000 and
1,000,000 separate runs will be used for the RTP calculations.
In simulations, formula (2) is used, where the cycle repre-
sents the number of runs. In addition, there is another type
of simulation called “Full Cycle.” is latter type of simu-
lation is used in the design phase (i.e., mathematical) of a slot
machine game. It is an exhaustive cycle through all possible
stop positions. e Full Cycle simulation allows an exact
match to the calculations in the spreadsheet but is not ca-
pable of calculating coinciding statistics across a series of
feature games.
e structure of this work is as follows: In Section 2, we
discuss similar approaches for the RTP optimization
problem, focusing on recent research work and the methods
they utilize. Next, we introduce our methodology in Section
3. Section 4 presents the results of our methods on several
games. Finally, we draw our conclusions and provide some
research directions for future work in Section 5.
2. Related Work
Although the RTP optimization problem is one of the most
important parts of the slot machine game design, there are
not many scientific articles that present a heuristic or
metaheuristic approach to solve this problem. Only three
papers have been published which present a metaheuristic
approach to solve the RTP optimization problem and are
briefly presented in Table 1.
Balabanov et al. [4] presented a solution for the slot
machine RTP optimization problem, where the symbols
distribution of the base game is being controlled and, also,
some other features like free spins frequency or bonus game
frequency. e fitness function in this work was the absolute
difference between the desired RTP and the obtained RTP.
For Monte Carlo simulations, 1,000,000 or 100,000 separate
runs were used in order to calculate the obtained RTP. ey
used Genetic Algorithms (GAs), with a randomized initial
population and appropriate crossover and mutation oper-
ators. e target RTP of the example that the authors
presented was 99%, starting from 91%. So, the algorithm
increases the RTP in a small range in case the game already
exists (a quality starting RTP). However, in most cases, the
mathematicians of slot games usually try to develop a new
slot machine game (from scratch) that has initial reels with
randomized symbol distribution and the RTP might be
equal, for example, to 630% (big random value). So, it is not a
trivial task for such algorithms to handle the case of large
reductions and the huge difference between the target RTP
and the initial RTP. e decrease of the RTP in slot machine
game is the hardest problem for the mathematicians, since
they have to solve the RTP optimization problem with hand
adjustments and this usually requires a long time. Fur-
thermore, the algorithm’s behavior in the case where special
symbols appear in reels (e.g., wild, scatter) is also unknown.
e same authors in [5] also proposed a Differential
Evolution (DE) algorithm, since the problem of optimal
symbols distribution on the reels is discrete and its con-
vergence was found to be faster than GAs. Additionally, they
also used Monte Carlo simulations with 1,000,000 separate
runs over ten trials for better accuracy in base game. eir
model of slot machine game contained only simple symbols
and not scatter, wild, or bonus games (scatter represents
symbols that pay everywhere on the screen and substitute all
other symbols). e authors proposed a multicriteria opti-
mization approach as future work, for the symbol fre-
quencies, the free spins frequency, the game volatility, and
the bonus game frequency.
2Mathematical Problems in Engineering
Keremedchiev et al. [6] proposed a Genetic Algorithm
methodology for the RTP optimization problem, in bonus
and free games frequency as parallel computing evolutionary
optimization. e fitness function of base game that was
used was computed with Full Cycle method instead of
Monte Carlo simulations. In current algorithm, the model of
slot machine game does contain as features bonus game,
simple symbols that pay on each pay line, scatter symbols,
and wild and bonus symbols that trigger the bonus game.
is experiment, as it refers in it, showed that the use of
Genetic Algorithm with exact numerical calculation of RTP
as fitness function was very efficient and improved the slot
game parameters by better adjustment of RTP, bonus game,
and free games frequency. Finally, the biggest drawback was
that the exact RTP computation was time-consuming and
slowed down the optimization process. e authors also
proposed a Discrete Differential Evolution (DDE) imple-
mentation as a further research.
3. Research Methodology
Variable Neighborhood Search (VNS) constitutes a meta-
heuristic solution method which is systematically changing a
neighborhood structure. VNS was initially proposed by
Mladenovi´
c et al. [7, 8]. Nowadays, a large number of
successful VNS applications have been reported in a wide
range of areas [9–13]. In this work, a Variable Neighborhood
Descent (VND) heuristic schema was applied in base game’s
RTP and it takes all symbols into account. e VND met-
aheuristic features a strong local search phase (intensifica-
tion) utilizing a number of different neighborhood
structures but lacks a diversification phase (shaking method)
as compared to other VNS variants. e VND variant has
been applied in several research works in the literature
[14, 15].
e proposed solution method is analytically described
in the following subsections and an overview is depicted in
Figure 1.
3.1. Solution Representation. A slot machine consists of n
reels where the symbols are represented by a string and are
equally distributed. e solution representation is a two-
dimensional array with the distribution of the symbols on
each reel. For example, in case of a five-reel (number of
columns) slot machine game with 48 symbols per reel
(number of rows), the solution array (denoted as “Reels”)
would have the dimension of 48 ×5.
ree inputs of the proposed algorithm are (i) the
maximum number of iterations as a termination criterion
(also a maximum CPU solution time can be used), (ii) the
target (desired) RTP of the user (RTPdes), and (iii) the
allowed deviation denoted (R) from the target RTP. Usually,
the computation of the precise RTP is quite difficult and, due
to this reason, game developers are satisfied if the RTP
belongs in a specific range. Hence, if
RTPcurr RTPdes R, RTPdes +R
􏼂 􏼃, then the algorithm
stops. Assume, for example that if we are given R0.03,
RTPdes 0.83, and RTPcurr 0.80, then the algorithm stops.
An extra feature of the algorithm is that it allows the user to
declare a minimum number of each symbol per reel. us, a
fourth input of the proposed algorithm is an additional two-
dimensional array containing the minimum number of
symbols per reel and is denoted as “minNSR.” e number
of columns of this array is equal to the number of reels, while
its number of rows is equal to the number of distinct
symbols. us, min NSR(i, j)denotes the minimum re-
quired number of occurrences of the isymbol in the j-reel.
3.2. Neighborhood Structures. e proposed VND meth-
odology utilizes two neighborhood structures denoted as
“RMS” and “RMC.” Both these local search operators aim to
reduce the RTP as follows:
(i) RMS: initially, the symbols are sorted in ascending
order based on their corresponding payments. e
Replace Maximum Symbol (RMS) local search op-
erator replaces symboliwith the biggest payment by
symboli1that has the next biggest payment in pay
table. e players of a slot machine game may win a
monetary award in the case where a combination of
at least two identical symbols occurs in the pay lines.
e symbol that has the biggest payment in these
combinations is considered the most expensive
Table 1: Key literature contributions on RTP optimization.
Reference Approach Simulation Special symbols Runs Games tested
is work VNS Monte Carlo Yes 100,000 & 1,000,000 3
3
[4] GA(s) Monte Carlo No 100,000 or 1,000,000 1
[5] DE Monte Carlo No 1,000,000 1
[6] DDE and GA(s) Full Cycle Yes All possible 1
RTP calculation
RTP calculation - target?
End
Yes
No
RTP optimization
RMS RMC
Optimization
process
Figure 1: Solution process.
Mathematical Problems in Engineering 3
symbol. However, the replacement by the RMS
operator is permitted only after checking the input
array min NSR (minimum number of symbols per
reel requested by the game developer); otherwise, the
next most expensive symboli1is considered for
potential replacement.
(ii) RMC: the RTP increases when the placement of the
symbols in the reels leads to minimum winning
combinations of symbols (i.e., combinations of two
identical symbols only) with a high frequency. e
Replace Minimum Combination (RMC) local search
operator replaces symboliwhich participates in a
minimum winning combination with symboljthat
does not participate in such winning combinations
(e.g., it participates in winning combinations of three
or more identical symbols). Such symbols may
participate in combinations with a little larger
payment, but the frequency of their combinations is
much lower.
3.3. VND Metaheuristic Algorithm. e VND solution ap-
proach is depicted in Algorithm 1. Also, Monte Carlo
simulation (RunSimulation) is used for the calculation of the
current RTPcurr with 100,000 or 1,000,000 separate runs. e
two neighborhood structures RMS and RMC are also
denoted as N1and N2, respectively. Moreover, the number
of different neighborhood structures is denoted as lmax and
in this work it is equal to two.
e proposed VND is initialized with a randomized
initial symbol distribution per reel (the list of distinct
symbols is given by the game developer) and terminates once
a maximum number of iterations is performed rather than
not finding an improvement. is is due to the fact that, in
each iteration, a Monte Carlo simulation might return an
approximate value of the RTP and not the exact RTP that
would result after a (time-consuming) Full Cycle RTP
computation. us, in some cases, although the two local
search operators do not make any improvement, a better
RTP value can be calculated by the Monte Carlo simulations.
Moreover, each time the neighborhood is changed, the
search is performed in a different (next) reel. us, the VND
starts the optimization using the first reel (Reelcurr 1) and
afterward sequentially applies the local search operators in
each one of the nreels.
3.4. Illustrative Examples. Without loss of generality, let us
assume a hypothetical slot machine game with nreels, with
each one of them containing 12 equally distributed symbols.
e symbols are represented as string variables in an array
(R) and the pay table is stored as integer values in another
array (P).
P
0 2 4 16 64
0 0 8 32 128
0 0 8 32 128
0 0 16 64 512
.(3)
Next, an example of the minNSR array is presented,
where for simplicity we show only one of the columns
corresponding only to the first reel. Each index represents
how many of sym0isymbols should remain in current reel
and not get replaced by other, for example, minNSR[2] � 3
meaning that sym02 cannot occur less than three times in
the first reel. is way, it will not be possible to replace
sym02 with sym03.
min NSRT2 3 1 1
􏼂 􏼃.(4)
e RMC operator changes the symbols of the first reel
as follows:
sym01
sym02
sym02
sym02
sym01
sym01
sym03
sym02
sym01
sym03
sym01
sym04
sym04
sym02
sym02
sym02
sym02
sym01
sym01
sym03
sym02
sym01
sym03
sym01
sym04
sym04
.(5)
e i-row of the pay table contains the payments of each
symbol sym0i(arrays start from one) for combinations of
one, two, three, four, and five, respectively, for example,
sym03’s payment takes place in the third row of pay table
P[3]. e RMC local search operator replaces sym01 with
sym02.
If the RMS operator was applied, then sym04, that is, the
most expensive symbol according to pay table (combination
of five pays 512), will be replaced by sym03 which is the next
most expensive but cheaper than this symbol. So, in the first
reel sym04 will be replaced by sym03 as follows:
sym01
sym02
sym02
sym02
sym01
sym01
sym03
sym02
sym01
sym03
sym01
sym04
sym04
sym01
sym02
sym02
sym02
sym01
sym01
sym03
sym02
sym01
sym03
sym01
sym04
sym03
.(6)
4Mathematical Problems in Engineering
4. Experimental Results
All the computational experiments have been done using a
Dell Inspiron 5558 laptop with an Intel Core i5-5200U CPU
at 2.20 GHz, 8 GB DDR3 RAM at 1600 MHz, and running
64-bit Microsoft Windows 10 Professional. e experiments
have been done using three commercial slot machine games
developed by the Zeusplay (https://zeusplay.com) casino
game developer company.
Le Mystere Du Prince [16], a five-reel slot machine
game with wild and scatter as special symbols
Amun’s Book [17], a five-reel slot machine game that
holds one special symbol called book and acts as a wild
and scatter
Wild Charger [18], a five-reel slot machine game that
holds one wild and acts as expanding and sticky at the
same time (symbol developed by Zeusplay)
All the above games have ten pay lines (winning lines)
and they are all paid from left to right. e smaller winning is
two monetary units for a combination of two symbols for
each game. e details (e.g., pay table, symbols) of the first
slot machine game are presented below:
P
0 2 5 25 100
0 0 5 25 100
0 0 5 25 100
0 0 5 25 100
0 0 10 50 125
0 0 15 75 250
0 0 15 75 250
0 0 20 100 400
0 2 25 125 750
0 2 25 125 750
0 20 50 200 5000
0 10 250 2500 9000
.(7)
In this game, special symbols wild and scatter are in-
cluded. e distinct symbols of the game are
S
sym01
sym02
sym03
sym04
sym05
sym06
sym07
sym08
sym09
sym10
sym11
scatter
wild
.(8)
Each S[i]refers to payments of P[i]element for one, two,
three, four, and five combinations of this symbol. e only
differences with the other two games, in base game, are the
pay table and the number of distinct symbols. Amun’s Book
and Wild Charger consist of ten and eight symbols, re-
spectively. e minNSR array for these two games has been
set the same as Le Mystere Du Prince’s.
minNSR
11111
.. .. .. .. ..
.. .. .. .. ..
11111
.(9)
According to the above minNSR table, at least one
symbol on each reel must exist. In the tables below, all the
necessary information about the optimization is presented,
such as the number of separate runs with Monte Carlo
simulation, the target RTP, the initial RTP, and the RTP
that was finally obtained. e allowed deviation for the RTP
Data: RTPdes,iterationmax , R, minNSR,Reels, lmax
Result: Reels
Reelcurr 1;
RTPcurr RunSimulation;
Difference RTPcurr RTPdes;
repeat
l1;
repeat
Select ReelsNl(x)such that RTP(Reels)<RTP(Reels);
RTPcurr RunSimulation;
NeighborhoodChange (Reels,Reels, l);
Reelcurr (Reelcurr +1)mod(n);
until llmax;
until stopping criterion;
A
LGORITHM
1:
VND.
Mathematical Problems in Engineering 5
(Rvariable) was set equal to 3% for the first game, meaning
that if 77% RTPcurr 83%, then the algorithm terminates
since a good solution was found. e output is a set of reels
with the modified symbols and the best RTP that has been
found. For the second game and the third game, the Rvalue
was set equal to 2% and 1%, respectively. ese three games
are presented in order to show that the algorithm can
indeed optimize games with either large or small difference
between the initial RTP and the target RTP. e RTP of the
first game was decreased from 715% to 80%, while the RTP
of the second game was decreased from 57% to 40%, and
the RTP of the third game was decreased from 107% to
53%, approximately. e results per game are presented in
Tables 24, respectively.
e required solution time and number of iterations are
presented in Table 5.
e RTP reduction through the whole process of opti-
mization for each iteration of each game is presented in
Figures 2–7. It can be noticed that Monte Carlo simulation
with 1,000,000 separate runs is more precise and needs less
iterations to find one acceptable solution rather than the
version with 100,000 separate runs. at happens since the
larger the sample (runs) is, the more exact the results will be.
A larger sample always leads to more reliable results.
Also, it can be noticed that Amun’s Book RTP needs 110
iterations to finish for 100,000 runs and 149 iterations for
1,000,000. e difference with the other two games which
makes these big changes in RTP is due to the fact that the
wild symbol is also a scatter, so once the wild symbol is
modified to another simple symbol, the RTP rapidly
decreases.
Creating an exciting as well as attractive slot game,
consists of a variety of elements, depending on the target
group. ere are several types of players, some of which
prefer to risk on a game that might not return much when in
base game, but they have a bonus feature that is more likely
to return big wins. In order to create such games, a com-
bination of acceptable pay tables, hit rates, and bonus feature
triggering frequency must be taken in consideration. e
most common logic is to have a risky base game, with a low
RTP but with recurring small wins, so that the player does
not immediately lose his/her money, which triggers a bonus
feature more frequently and gives the player the confidence
that he/she is able to win big prizes.
Table 2: Le Mystere Du Prince.
Runs RTP initial (%) RTP target (%) RTP obtained (%)
100,000 718.374 77–83 82.025
1,000,000 712.661 77–83 78,996
Table 3: Wild Charger.
Runs RTP initial (%) RTP target (%) RTP obtained (%)
100,000 56.105 38–42 41.277
1,000,000 57.226 38–42 38.758
Table 4: Amun’s Book.
Runs RTP initial (%) RTP target (%) RTP obtained (%)
100,000 111.005 52–54 52.015
1,000,000 104.852 52–54 52.803
Table 5: Results for the three games.
Game Simulation runs Time (secs) Iterations
Le Mystere Du Prince 100,000 353 2456
Le Mystere Du Prince 1,000,000 786 501
Wild Charger 100,000 2 2062
Wild Charger 1,000,000 15 350
Amun’s Book 100,000 35 110
Amun’s Book 1,000,000 223 149
700
600
500
400
300
200
RTP %
RTP %
100
0 500 1000 1500
Iteration
2000 2500
Figure 2: Le Mystere Du Prince: 100,000 runs.
6Mathematical Problems in Engineering
5. Conclusions and Future Work
e experiments showed that the use of VNS was very ef-
ficient in developing slot games from scratch, where the
symbol distribution is not uniform. As shown in the figures
of the previous section, the required number of iterations of
the proposed method depends on the initial difference be-
tween the initial RTP and target RTP. Also, in cases where a
symbol contains two types of special symbols (e.g., such as in
the second game where the “scatter” symbol is also a “wild”
symbol), a large RTP decrease can be achieved by converting
this symbol to another.
As the complexity of games increases, the required so-
lution time increases too, and this fact shows the need for
speeding up the whole process. As future research, an
implementation of much more complex games RTP can be
110
100
90
80
70
60
RTP %
RTP %
50
0 204060
Iteration
80 100
Figure 4: Amun’s Book: 100,000 runs.
100
90
80
70
60
RTP %
RTP %
0204060
Iteration
80 100 120 140
Figure 5: Amun’s Book: 1,000,000 runs.
Figure 7: Wild Charger: 1,000,000 runs.
56
54
52
50
48
46
44
42
RTP %
RTP %
0 500 1000 1500
Iteration
2000
Figure 6: Wild Charger: 100,000 runs.
700
600
500
400
300
200
RTP %
RTP %
100
0 100 200 300
Iteration
400 500
Figure 3: Le Mystere Du Prince: 1,000,000 runs.
Mathematical Problems in Engineering 7
proposed using parallel CPU and GPU computing tech-
niques to decrease the computational time for finding an
acceptable (RTP) solution. Furthermore, the proposed al-
gorithm is a single-criterion approach for the RTP opti-
mization. Another research idea is to develop a multicriteria
method for optimizing RTP, volatility, and hit rate (i.e., how
frequently a player is expected to stop on a winning
combination).
Data Availability
e proposed methodology is applied to three commercial
casino games, which can be found online at the company’s
website (public URLs are included in the references).
Conflicts of Interest
e authors declare that there are no conflicts of interest
regarding the publication of this paper.
Acknowledgments
is work has been funded by the company Zeusplay
(Research Committee of the University of Macedonia (Grant
no. 81359)).
References
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8Mathematical Problems in Engineering
... The gambling industry plays a significant role in modern life [1,2]. It is one of the most profitable businesses worldwide with prognosed value more than USD 640 billion by 2027 [3]. In the recent years, there are more and more opportunities for everyone seeking such kind of joy. ...
... An exact formula does not exist-it should be interpreted just as the higher the number, the more volatile the win size. A game has different settings for the RTP level [3,[10][11][12], which often varies between 88% and 96%, but usually is higher than 91-92%. Every RTP level is associated with a different set of reels. ...
Article
Full-text available
Nowadays, entertainment is one of the biggest industries, which continues to expand. In this study, the problem of estimating the consolation prize as a fraction of the jackpot is dealt with, which is an important issue for each casino and gambling club. Solving the problem leads to the computation of multidimensional integrals. For that purpose, modifications of the most powerful stochastic quasi-Monte Carlo approaches are employed, in particular lattice and digital sequences, Halton and Sobol sequences, and Latin hypercube sampling. They show significant improvements to the classical Monte Carlo methods. After accurate computation of the arisen integrals, it is shown how to calculate the expectation of the real consolation prize, taking into account the distribution of time, when different numbers of players are betting. Moreover, a solution to the problem with higher dimensions is also proposed. All the suggestions are verified by computational experiments with real data. Besides gambling, the results obtained in this study have various applications in numerous areas, including finance, ecology and many others.
Book
Full-text available
This book constitutes the refereed post-conference proceedings of the 6th International Conference on Variable Neighborhood Search, ICVNS 2018, held in Sithonia, Greece, in October 2018. ICVNS 2018 received 49 submissions of which 23 full papers were carefully reviewed and selected. VNS is a metaheuristic based on systematic changes in the neighborhood structure within a search for solving optimization problems and related tasks. The main goal of ICVNS 2018 was to provide a stimulating environment in which researchers coming from various scientific fields could share and discuss their knowledge, expertise, and ideas related to the VNS metaheuristic and its applications.
Article
Full-text available
Variable neighborhood search (VNS) algorithm is proposed for scheduling identical parallel machine. The objective is to study the effect of adding a new neighborhood structure and changing the order of the neighborhood structures on minimizing the makespan. To enhance the quality of the final solution, a machine based encoding method and five neighborhood structures are used in VNS. Two initial solutionmethodswhichwere used in two versions of improved VNS (IVNS) are employed, namely, longest processing time (LPT) initial solution, denoted as HIVNS, and random initial solution, denoted as RIVNS. The proposed versions are compared with LPT, simulated annealing (SA), genetic algorithm (GA), modified variable neighborhood search (MVNS), and improved variable neighborhood search (IVNS) algorithms from the literature. Computational results show that changing the order of neighborhood structures and adding a new neighborhood structure can yield a better solution in terms of average makespan.
Conference Paper
Full-text available
Slot machines are casino gambling machines with three or more reels which spin when a button is pushed. The machine pays are based on patterns of symbols visible on the front of the machine when it stops. Most of the modern slots consist of a base game, free games and bonus games. The base game is the core of the playing process. Player’s money are usually taken as bet in the base game and no bet is taken during free games or bonus games. Each slot machine has a parameter called return to player (RTP). RTP is the average amount of money which a player will get back, in average, after each spin of the reels. The total RTP (measured in percents) can be in the range between 75% and 98%. Its components are: base game RTP, free games RTP, bonus games RTP. The base game also controls how often free games will be activated and how often bonus games will be played. In this paper an evolutionary optimization algorithm for optimization of slot machine base game RTP by rearrangement of the symbols in the reels, is proposed. The problem itself is a combinatorial problem and the fitness function used checks all possible slot machine winning screens.
Conference Paper
Full-text available
It is possible to solve slot machine RTP optimization problem by using evolutionary algorithms. In practice this optimization is done by hand adjustment of the symbols placed on the game reels. By arranging symbols positions, it is possible to achieve optimal return to player percentage (RTP). Equalization of the prizes distribution, generated by different win combinations, can be optimized also. In this paper a DE based RTP optimization and prizes equalization is proposed. DE is used in its discrete variation, because the problem of optimal symbols distribution on the reels is in the discrete domain. DE is selected as an alternative to genetic algorithms (GA) because of its faster convergence. The convergence is a key factor in such optimizations, because each fitness value is calculated based on intensive Monte-Carlo simulations. The scope of this paper is focused on the symbols distribution placed on the machine reels in such a way that two common goals to be satisfied - desired RTP and keeping relatively equal levels of the prizes (prizes expressed as amount of money won from combinations with each particular symbol), with relatively good symbol diversity on the reels.
Conference Paper
Full-text available
Slot machine RTP optimization problem is usually solved by hand adjustment of the symbols placed on the game reels. By controlling the symbols distribution, it is possible to achieve the desired return to player percent (RTP). Some other parameters can also be adjusted (for example, the free spins frequency or the bonus game frequency). In this paper RTP optimization automation, based on genetic algorithms, is proposed.
Article
Full-text available
The multi-product dynamic lot sizing problem with product returns and recovery is an important problem that appears in reverse logistics and is known to be NP-hard. In this paper we propose an efficient variable neighborhood descent heuristic algorithm for solving this problem. Furthermore, we present a new benchmark set with the largest instances in the literature. The computational results, demonstrate that our approach outperforms the state-of-the-art Gurobi optimizer.
Article
Full-text available
A heuristic algorithm based on VNS is proposed to solve the Max 3-cut and Max 3-section problems. By establishing a neighborhood structure of the Max 3-cut problem, we propose a local search algorithm and a variable neighborhood global search algorithm with two stochastic search steps to obtain the global solution. We give some numerical results and comparisons with the well-known 0.836-approximate algorithm. Numerical results show that the proposed heuristic algorithm can obtain efficiently the high-quality solutions and has the better numerical performance than the 0.836-approximate algorithm for the NP-Hard Max 3-cut and Max 3-section problems.
Article
Full-text available
The leather nesting problem is a cutting and packing optimization problem that consists in finding the best layout for a set of irregular pieces within a natural leather hide with an irregular surface and contour. In this paper, we address a real application of this problem related to the production of car seats in the automotive industry. The high quality requirements imposed on these products combined with the heterogeneity of the leather hides make the problem very complex to solve in practice. Very few results are reported in the literature for the leather nesting problem. Furthermore, the majority of the approaches impose some additional constraints to the layouts related to the particular application that is considered. In this paper, we propose a variable neighborhood search algorithm for the general leather nesting problem. We describe a fast constructive heuristic that generates a feasible layout guided by the information provided by inner-fit polygons, and we introduce new neighborhood structures based on the sequence by which the pieces are placed on the hide and on the quality of the fitness of the pieces in the current layout. To evaluate the performance of our approaches, we conducted an extensive set of computational experiments on real instances. The results of these experiments are reported at the end of the paper.
Chapter
Local search heuristic that explores several neighborhood structures in a deterministic way is called variable neighborhood descent (VND). Its success is based on the simple fact that different neighborhood structures do not usually have the same local minimum. Thus, the local optima trap problem may be resolved by deterministic change of neighborhoods. VND may be seen as a local search routine and therefore could be used within other metaheuristics. In this chapter, we discuss typical problems that arise in developing VND heuristic: what neighborhood structures could be used, what would be their order, what rule of their change during the search would be used, etc. Comparative analysis of usual sequential VND variants is performed in solving traveling salesman problem. © Springer International Publishing AG, part of Springer Nature 2018. All rights reserved.
Chapter
The Variable Neighborhood Search (VNS) metaheuristic was invented by Nenad Mladnović and Pierre Hansen and developped at GERAD (Group for Research in Decision Analysis, Montral) since 1997.