=Paper=
{{Paper
|id=Vol-2864/paper40
|storemode=property
|title=Algorithm of Analysis and Conversion of Input Data of a Two-factor Multi-variative Transport Problem with Weight Coefficients
|pdfUrl=https://ceur-ws.org/Vol-2864/paper40.pdf
|volume=Vol-2864
|authors=Oleksii Chyzhmotria,Olena Chyzhmotria,Tetiana Vakaliuk
|dblpUrl=https://dblp.org/rec/conf/cmis/ChyzhmotriaCV21
}}
==Algorithm of Analysis and Conversion of Input Data of a Two-factor Multi-variative Transport Problem with Weight Coefficients==
https://ceur-ws.org/Vol-2864/paper40.pdf
Algorithm of analysis and conversion of input data of a two‐
factor multi‐variative transport problem with weight
coefficients
Oleksii Chyzhmotriaa, Olena Chyzhmotriaa and Tetiana Vakaliuka
a
Zhytomyr Polytechnic State University, Chudnivska str., 103, Zhytomyr, 10005, Ukraine
Abstract
The article is devoted to the analysis of input data of a two-factor multivariate transport
problem with weighting factors. The article aims to develop and describe an algorithm for
bringing this problem to a form suitable for the application of one of the existing methods of
solving the classical transport problem. The developed algorithm should be such that it can
be relatively easily programmed in one of the existing programming languages. The input
data for the development of the algorithm is the presence of two independent optimization
criteria; different values of weighting factors of two factors for each pair "supplier-
consumer"; a different number of options for transportation of goods for each pair "supplier-
consumer" with the corresponding values of both factors. According to the goal, the
algorithm must choose the objectively best of several options for transportation of goods for
each pair "supplier-consumer", taking into account the two-factor and the presence of
weights. The issues related to the choice of the best of the options for transportation of goods
for a single pair "supplier-consumer" taking into account the weight coefficients are
considered on the examples. An analysis of the influence of the values of the factors of one
pair "supplier-consumer" on the resulting criteria of other pairs. Developed an algorithm for
bringing the initial data to a single numerical range, calculating the resulting criteria, and
determining the best transportation option for each pair "supplier-consumer".
Keywords 1
Two-factor transport problem, transport transportation, quality criteria, weight factors, weight
coefficients, multivariate, algorithms.
1. Introduction
One of the types of transport problem is a two-factor problem, in which it is necessary to minimize
costs simultaneously for two factors. There is a common method of pairwise multiplication of the
corresponding values of factors with subsequent selection of the smallest of the obtained values.
In the case of several variants of pair wise values of two factors for the pair "supplier-consumer",
the mentioned method of pair wise multiplication of the corresponding values of factors with the
subsequent selection of the smallest of the obtained values remains valid. The selected value will be
used in the future as the best option for the existing pair "supplier-consumer".
Suppose we have two factors (C, T) and three transport options from supplier A to consumer B
with different values for each factor ((c1, c2, c3 and t1, t2, t3). For the minimization problem, it will be
enough to choose the minimum value from three pairwise products of the values of factors C and T:
r minc1 t1 ; c2 t 2 ; c3 t3 (1)
CMIS-2021: The Fourth International Workshop on Computer Modeling and Intelligent Systems, April 27, 2021, Zaporizhzhia, Ukraine
EMAIL: 4ov.ztu@gmail.com (O. V. Chyzhmotria); ch-o-g@ztu.edu.ua (O.G.Chyzhmotria); tetianavakaliuk@gmail.com (T. Vakaliuk)
ORCID: 0000-0002-5515-6550 (O. V. Chyzhmotria); 0000-0001-8597-1292 (O.G.Chyzhmotria); 0000-0001-6825-4697 (T. Vakaliuk)
© 2020 Copyright for this paper by its authors.
Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
CEUR Workshop Proceedings (CEUR-WS.org)
The value of r and will be the resulting criterion, and the corresponding transportation option - the
best of the proposed.
In real conditions, different suppliers and different consumers may have different priorities when
transporting goods: someone needs to receive the goods as quickly as possible, someone needs to
minimize the cost of transportation, someone is looking for a balanced solution, etc. Thus, there is a
need to vary the "weight" of factors for each pair "supplier-consumer" within one transport task.
Assuming the presence of weight coefficients for each of the factors (kc, kt), there is a problem of
choosing an objectively better option from the proposed for each pair of "supplier-consumer". Note
that for each variant of a certain pair "supplier-consumer" weight coefficients are constant values.
Taking into account the weight coefficients, formula 1 takes the following form:
r minc1 k c t1 kt ; c2 k c t 2 kt ; c3 k c t3 kt (2)
or after mathematical transformations:
r k c k t minc1 t1 ; c2 t 2 ; c3 t3 (3)
From formula 3 it can be seen that the resulting best option is still chosen as the minimum of three
pair wise products of the values of factors C and T. The weight coefficients only equally increase each
of the products in kc×kt times and in no way affect the choice of the best option. This method
eliminates the very essence of weight coefficients as levers of influence when choosing the best
option.
Simple pairwise multiplication of the values of the factors and their weight coefficients with the
subsequent choice of the smallest of the obtained values does not allow to choose the objectively best
of the proposed options.
2. Review of the literature
Nowadays mathematical methods solve many problems of operational planning for transportation.
Many scientists dedicate their work to the topic of transport optimization.
Chyzhmotria O. et al. [4] on an example considered the problems connected with the search of the
optimum plan of transportations simultaneously on two quality criteria. They conducted a
comparative analysis of the four transportation plans according to the condition of the example.
Burduk A., Musial K. [2; 3] solved the problem of optimization using genetic algorithms. They
described genetic algorithms, their properties, and their capabilities in solving computational
problems. To solve the problem under study, the authors used the program MATLAB.
Prifti V. et al. [6] considered a real problem of linear programming in detail by taking an example
in an Albania company. For the company under consideration the problem of minimizing
transportation costs was solved by solving 3 methods: The North West Corner Method, the Least Cost
Method, and the Vogel's Approximation Method. The calculations showed that based on the demand
from the 9 geographical sites (destinations) and the capacity offered by the two manufacturing plants
(sources), the most optimal solution turns out to be the one obtained by Vogel's method.
Sun Y. et al. [8] in their study presented a systematic review of the problem of planning route
transportation of goods in a multimodal transport network. In this study, the formulation
characteristics are divided and classified into six aspects, and optimization models in recent studies
are determined based on the respective formulation characteristics.
Gunantara N. [5] in his work considered two methods of multi-objective optimization (MOO) that
do not require complicated mathematical equations. These two methods are Pareto and scalarization.
In the article by Zhang Y. et al. [10] an optimization method for multiple batches of express freight
demands is proposed for the shippers of railway express freight to select the most suitable
transportation products to transport, considering the priority of shippers and capacity constraints. Five
transport attributes, the most common concerns of express freight shippers, including freight transit
time, transport cost, convenience, safety, and reliability, are selected as main indexes. Furthermore, a
solution algorithm is designed by considering important clients who prioritize choosing transportation
products.
Badica A. et al. [1] proposed a method for declarative modeling and optimization of freight
transportation brokering using agents and constraint logic programming.
Stoilova S. [7] in her study proposed a step-by-step approach to determining the transport plan of
passenger trains. In the first step, criteria for optimizing the transport plan were defined. In the second
stage, variants of the transport plan were formulated. In the third stage, the weight coefficients of the
criteria were determined. Multi-purpose optimization was performed in the fourth step. The impact of
changes in passenger traffic on the choice of the optimal transport plan was studied in the fifth stage.
In the article of the authors Zabolotnii S. and Mogilei S. [9] a study of existing methods for
constructing support plans for the transportation problem with several means of cargo delivery is
conducted, and the task itself is defined as multimodal. Based on the criterion of reducing the number
of numerous iterations in finding solutions to such a problem, a more perfect method of constructing
its support plans, the so-called Steiner method, is proposed. And also a general formulation of the
multimodal transport problem is implemented - its objective function (criterion) of optimization and
an admissible set of solutions are formalized.
The article aims to develop and describe an algorithm for bringing this problem to a form suitable
for the application of one of the existing methods of solving the classical transport problem. The
developed algorithm should be such that it can be relatively easily programmed in one of the existing
programming languages.
3. Results
Consider and analyze the following method of choosing the best option, taking into
account the weight coefficients kc and kt. The essence of the method will be to add the values
of the two factors C and T, multiplied by the corresponding weight coefficients.
The following example 1 should provide an answer as to the feasibility or inadmissibility
of this method. The input data will be the values of the factorsc1 = 900, c2 = 850, t1 = 1, t2 =
5.In example 1, we will deal with two options for transporting goods from point A to point B.
The weight coefficients will change in the range from 0 to 1 in steps of 0.1. The sum of the
coefficients will always be equal to 1.
The best option for transportation and the value of the resulting criterion will be sought by
the formula:
r minc1 kc t1 kt ; c2 kc t 2 kt (4)
The results of the calculations are summarized in table 1:
Table 1
Calculation of values and selection of the resulting criterion
kc kt r1 = c1×kc+t1×kt r2 = c2×kc+t2×kt r
1 0 900 850 r2
0,9 0,1 810,1 765,5 r2
0,8 0,2 720,2 681 r2
0,7 0,3 630,3 596,5 r2
0,6 0,4 540,4 512 r2
0,5 0,5 450,5 427,5 r2
0,4 0,6 360,6 343 r2
0,3 0,7 270,7 258,5 r2
0,2 0,8 180,8 174 r2
0,1 0,9 90,9 89,5 r2
0 1 1 5 r1
Let us analyze the data in Table 1, according to which it is seen that with a decrease in the
influence of factor C and a corresponding increase in the influence of factor T, the difference
between the values of the resulting criteria r1 and r2 decreases. But even when the weight of
the factor T acquires a conditional 90% (kt = 0.9), still the smaller of the two values of the
resulting criterion remains r2. And this even though under the condition of the example, the
value of t2 is five times greater than the value of t1. This result can be considered distorted
and biased, and the method of adding the values of the two factors C and T multiplied by the
corresponding weight coefficients should be considered unacceptable as such.
It should be noted that in this example we were dealing with two different numerical
ranges, one of which was hundreds of times larger than the other. From the very beginning of
the calculations, this range of values dominated and, accordingly, the corresponding factor
dominated. This had a direct impact on the value of the resulting criterion and the choice of
the best freight option. Even the use of weight coefficients could not significantly affect the
final result. The difference in the values of the two ranges was so great that the weight
coefficients could not perform their direct function in terms of influencing the choice of the
best option, for which they were generally introduced into the mathematical model. Thus, the
reason for the biased results was a large difference in the values of the numerical ranges of
the two factors. It can also be argued that even a small difference can also negatively affect
the objectivity of the result.
In this study, we consider an algorithm that will get rid of the distortion of the results due
to the difference in the values of the two numerical ranges. The essence of the algorithm will
first be represented schematically:
Figure 1: Scheme of the algorithm for bringing data to a single numerical range
At the first stage of the algorithm, it is necessary to determine the largest value for each factor.
For factor C:
cmax maxc1 , c2 ,, cm (5)
For factor T:
t max maxt1 , t 2 ,, t m (6)
When programming to find the maximum value, as in the case of finding the minimum value, use
arrays, cyclic structures with a precondition or postcondition, and branching structures.
In the second stage of the algorithm, it is necessary to calculate the least common multiple (LCM)
for the values
c max t max (7)
LCM c max , t max
GCDc max , t max
where GCD is the greatest common divisor.
To find the largest common divisor, as an option, you can use the well-known Euclidean
algorithm. The algorithm contains a loop with a premise and several branches and can be easily
programmed.
It should be noted that at the beginning of the Euclidean algorithm, all input data must be integers,
so, if necessary, it is necessary to simultaneously increase the input data by 10n times.
In the third stage, additional factors are determined for each of the factors.
For factor C:
cam LCM cmax , t max cmax (8)
For factor T:
t am LCM cmax , t max t max (9)
In the last, fourth stage, each value of each of the factors must be multiplied by the corresponding
additional factor (cam or tam, respectively).
At the end of the algorithm, you can proceed to search by formula 4 the value of the resulting
criterion and choose the best option.
Let's return to the above example 1. The input data in the example were the values of the factors
c1 = 900, c2 = 850, t1 = 1, t2 = 5.
According to the algorithm for bringing data to a single numerical range (see Fig. 1) step by step
we get:
cmax = max{900, 850} = 900, tmax = max{1, 5} = 5.
LCM(cmax,tmax) = LCM(900, 5) = 900.
cam = LCM(cmax,tmax) / cmax = 900 / 900 = 1;
tam = LCM(cmax,tmax) / tmax = 900 / 5 = 180.
c1'= c1×cam =900× 1 = 900;
c2' = c2×cam = 850× 1 = 850;
t1' = t1×tam =1× 180 = 180;
t2' = t2×tam =5× 180 = 900.
The value of the resulting criterion and the best option for transportation will be sought by formula
4. The weight coefficients will be changed in the range from 0 to 1 in steps of 0.1.
The results of the calculations are summarized in table2:
The results shown in table 2 are radically different from the results in table 1. From table 2 we see
that after reducing the initial data to a single numerical range, even the minimum effect of factor T (kt
= 0.1 or 10%) was sufficient to the resulting criterion was the criterion r1 '. This choice is logical,
because the factor T in its values differs from the minimum to the maximum 5 times, while the factor
C - only 1.06 times. Accordingly, the effect of factor C should be minimal. At the same time, it is
the T factor that should play a key role in choosing the best transportation options, as demonstrated by
the proposed algorithm.
The resulting criterion r' will thus participate in the further solution of the two-factor transport
problem.
Table 2
Adjusted calculation of values and selection of the resulting criterion
kc kt r1' = c1'×kc+t1'×kt r2' = c2'×kc+t2'×kt r'
1 0 900 850 r2'
0,9 0,1 828 855 r1'
0,8 0,2 756 860 r1'
0,7 0,3 684 865 r1'
0,6 0,4 612 870 r1'
0,5 0,5 540 875 r1'
0,4 0,6 468 880 r1'
0,3 0,7 396 885 r1'
0,2 0,8 324 890 r1'
0,1 0,9 252 895 r1'
0 1 180 900 r1'
It will be recalled that before that it was a question of transportation of cargo from supplier A to
consumer B. However, the transport task assumes the presence of m departure points A1, A2, ..., Am
and n consumers B1, B2, ..., Bn. For each pair "supplier-consumer" within the research topic, the
variability of the values of each of the two factors is allowed. Also, each pair "supplier-consumer" for
the problem may have different values of weights for these factors. Thus, there is a task of developing
an algorithm for bringing a two-factor multivariate transport problem with weight coefficients to a
unified form, suitable for the application of one of the existing methods for solving problems of the
corresponding type. Separate questions are the possible influence of factor values of one "supplier-
consumer" pair on the resulting criteria of other pairs and the probable different numerical ranges of
factor values for different "supplier-consumer" pairs.
For further analysis and development of the algorithm as example 2 consider a fragment of a two-
factor multivariate transport problem, given in the tabular form:
Table 3
A fragment of a two‐factor multivariate transport problem
Consumers
Suppliers Cargo stocks
B1 B2 B3 B4
C T
A1 ... ... 15 6 ... a1
10 9
A2 ... ... ... a2
...
C T
4 6
A3 ... 5 5 ... ... a3
2 13
3 9
Cargo needs b1 b2 b3 b4
To analyze the possible influence of factor values of one pair "supplier-consumer" on the resulting
criteria of other pairs and to address the issue of different numerical ranges of factor values for
different pairs "supplier-consumer" it will suffice to use two pairs "supplier-consumer" (in our
example pairs A1-B3 and A3-B2). The same two pairs "supplier-consumer" will develop an algorithm
for bringing a two-factor multivariate transport problem with weight coefficients to a unified form,
suitable for the application of one of the existing methods of solving problems of the corresponding
type.
To begin with, we determine the best transportation option and the value of the resulting criterion
separately for each pair A1-B3 and A3-B2. To do this, we use formula 4 and the algorithm for
bringing data to a single numerical range (see Fig. 1, formulas 5-9).
For the A1-B3 pair we have two options for cargo transportation. According to the algorithm step
by step we get:
c13max = max{15, 10} = 15, t13max = max{6, 9} = 9.
LCM(c13max,t13max) = LCM(15, 9) = 45.
c13am = LCM(c13max,t13max) / c13max = 45 / 15 = 3;
t13am = LCM(c13max,t13max) / t13max = 45 / 9 = 5.
c131'= c131×c13am =15× 3 = 45;
c132' = c132×c13am = 10× 3 = 30;
t131' = t131×t13am =6× 5 = 30;
t132' = t132×t13am =9× 5 = 45.
Weight coefficients will be changed in the range from 0 to 1 in steps of 0.1. The results of
calculations according to formula 4 are summarized in table 4:
Table 4
Calculation of values and selection of the resulting criterion for the pair A1‐B3
k13c k13t r131' r132' r13'
1 0 45 30 r132'
0,9 0,1 43,5 31,5 r132'
0,8 0,2 42 33 r132'
0,7 0,3 40,5 34,5 r132'
0,6 0,4 39 36 r132'
0,5 0,5 37,5 37,5 r131' = r132'
0,4 0,6 36 39 r131'
0,3 0,7 34,5 40,5 r131'
0,2 0,8 33 42 r131'
0,1 0,9 31,5 43,5 r131'
0 1 30 45 r131'
For the pair A3-B2 we have four options for cargo transportation. According to the algorithm step
by step we get:
c32max = max{4, 5, 2, 3} = 5, t32max = max{6,5, 13, 9} = 13.
LCM(c32max,t32max) = LCM(5, 13) = 65.
c32am = LCM(c32max,t32max) / c32max = 65 / 5 = 13;
t32am = LCM(c32max,t32max) / t32max = 65 / 13 = 5.
c321'= c321×c32am =4× 13 = 52;
c322' = c322×c32am = 5× 13 = 65;
c32 3' = c32 3 ×c32am = 2× 13 = 26;
c32 4' = c32 4 ×c32am = 3× 13 = 39;
t321' = t321×t32am =6× 5 = 30;
t322' = t322×t32am =5× 5 = 25;
t32 3' = t32 3 ×t32am = 13 × 5 = 65;
t32 4' = t32 4 ×t32am = 9 × 5 = 45.
The results of calculations according to formula 4 are summarized in table5.
The results of the calculations for the pair A3-B2, shown in table 5, indicate that each of the
options for transporting goods from supplier A3 to consumer B2 may be the best option depending on
the weight coefficients.
Table 5
Calculation of values and selection of the resulting criterion for the pair A3‐B2
k32c k32t r321' r322' r323' r324' r32'
1 0 52 65 26 39 r323'
0,9 0,1 49,8 61 29,9 39,6 r323'
0,8 0,2 47,6 57 33,8 40,2 r323'
0,7 0,3 45,4 53 37,7 40,8 r323'
0,6 0,4 43,2 49 41,6 41,4 r324'
0,5 0,5 41 45 45,5 42 r321'
0,4 0,6 38,8 41 49,4 42,6 r321'
0,3 0,7 36,6 37 53,3 43,2 r321'
0,2 0,8 34,4 33 57,2 43,8 r322'
0,1 0,9 32,2 29 61,1 44,4 r322'
0 1 30 25 65 45 r322'
For both pairs A1-B3 and A3-B2, the algorithm for calculating the value of the resulting criterion
and finding the best option yielded results. They can be considered objective, but only separately for
each couple. If we analyze and compare the calculations for both pairs, we see that the values of the
factors in these pairs were in different numerical ranges. For the pair A1-B3, the numerical range with
the largest value of 45 was obtained, and for the pair A3-B2, the numerical range with the largest
value of 65 was obtained. The case with different numerical ranges has already been considered and
described above. It has also been concluded that it is inadmissible to use the obtained values in this
form for further calculations.
To solve this problem in the study it is proposed to combine the values of each of the factors of all
options from both pairs "supplier-consumer" of the current problem. This option is perfectly
acceptable, because under the condition of the transport problem, the load is homogeneous, and the
factors together with the units of measurement are the same for all pairs of "supplier-consumer" of the
current problem. In this case, the algorithm for bringing data to a single numerical range, shown in
Fig. 1, remains unchanged. Also, we have the opportunity to identify and analyze the possible
influence of the values of the factors of one pair "supplier-consumer" on the resulting criteria of other
pairs.
For the current example, after performing the first three steps of the algorithm, we obtain:
cmax = max{15, 10, 4, 5, 2, 3} = 15,
tmax = max{6, 9, 6, 5, 13, 9} = 13;
LCM(cmax,tmax) = LCM(15, 13) = 195;
cam = LCM(cmax,tmax) / cmax = 195 / 15 = 13;
tam = LCM(cmax,tmax) / tmax = 195 / 13 = 15.
For pair A1-B3 the fourth step of the algorithm:
c131'= c131×cam = 15 × 13 = 195;
c132' = c132×cam = 10× 13 = 130;
t131' = t131×tam = 6 × 15 = 90;
t132' = t132×tam = 9 × 15 = 135.
For the pair A3-B2, the fourth step of the algorithm:
c32 1'= c32 1×cam = 4 × 13 = 52;
c322' = c32 2×cam = 5× 13 = 65;
c32 3' = c32 3 ×cam = 2× 13 = 26;
c32 4' = c32 4 ×cam = 3× 13 = 39;
t321' = t32 1×tam = 6 × 15 = 90;
t322' = t32 2×tam = 5 × 15 = 75;
t32 3' = t32 3 ×tam = 13 × 15 = 195;
t32 4' = t32 4 ×tam = 9 × 15 = 135.
The results of calculations according to formula 4 for the pair A1-B3 are summarized in table 6.
Table 6
Adjusted calculation of values and selection of the resulting criterion for the pair A1‐B3
k13c k13t r131' r132' r13'
1 0 195 130 r132'
0,9 0,1 184,5 130,5 r132'
0,8 0,2 174 131 r132'
0,7 0,3 163,5 131,5 r132'
0,6 0,4 153 132 r132'
0,5 0,5 142,5 132,5 r132'
0,4 0,6 132 133 r131'
0,3 0,7 121,5 133,5 r131'
0,2 0,8 111 134 r131'
0,1 0,9 100,5 134,5 r131'
0 1 90 135 r131'
The results of calculations according to formula 4 for the pair A3-B2 are summarized in table 7:
Table 7
Adjusted calculation of values and selection of the resulting criterion for the pair A3‐B2
k32c k32t r321' r322' r323' r324' r32'
1 0 52 65 26 39 r323'
0,9 0,1 55,8 66 42,9 48,6 r323'
0,8 0,2 59,6 67 59,8 58,2 r324'
0,7 0,3 63,4 68 76,7 67,8 r321'
0,6 0,4 67,2 69 93,6 77,4 r321'
0,5 0,5 71 70 110,5 87 r322'
0,4 0,6 74,8 71 127,4 96,6 r322'
0,3 0,7 78,6 72 144,3 106,2 r322'
0,2 0,8 82,4 73 161,2 115,8 r322'
0,1 0,9 86,2 74 178,1 125,4 r322'
0 1 90 75 195 135 r322'
Let's analyze the results.
For pair A1-B3 we compare the data of tables 4 and 6. In table 4 at weight coefficients of
0,5 / 0,5 both variants of transportation of freight have identical result: r13' = r13 1' = r13 2'.
After combining the values of each of the factors of all options from both pairs "supplier-
consumer" of the current problem at the same weight coefficients of 0.5 / 0.5, the resulting
criterion was chosen r13 2': r13' = r13 2'.
Given the invariance of the initial data in the pair A1-B3, it is possible to draw an
unambiguous conclusion about the direct impact on the result in the pair A1-B3 values of the
second pair A3-B2.
For the pair A3-B2, we compare the data from Tables 5 and 7. Here we see even greater
differences in the results: for six of the eleven pairs of weight coefficients, the best option for
transporting cargo from the four existing ones has changed. Here, too, with constant initial
data, the significant influence of the values of the pair A1-B3 is obvious.
4. Conclusions
For a two-factor multivariate transport problem with weight coefficients, the method of pair wise
multiplication of factor values and their weight coefficients with subsequent selection of the smallest
of the obtained values does not allow to choose the objectively best of the proposed options, as in this
case weight coefficients do not affect the choice of the best option.
The difference in the values of the numerical ranges of the two factors leads to biased results, as
the weight coefficients, in this case, can not fully perform their direct function to influence the choice
of the best option. Therefore, it is mandatory to bring the values of the numerical ranges of the two
factors to a single range both within a single pair "supplier-consumer" and within the entire transport
task.
The proposed algorithm for bringing the initial data to a single numerical range, calculating the
resulting criteria, and determining the best option for transportation for each pair "supplier-consumer"
has fully performed its function. The obtained resulting criteria are ready for use in the further
solution of the transport problem by one of the existing methods.
The algorithm allowed us to draw an important conclusion about the influence of the values of the
factors of a single pair "supplier-consumer" on the resulting criteria of other pairs.
The developed algorithm can be relatively easily programmed in one of the existing programming
languages. To write a program according to the given algorithm will require knowledge, skills, and
abilities to work, in particular, with one-dimensional and multidimensional arrays, cycles of different
types, branching design.
5. References
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Brokerage Using Agents and Constraints. International Conference on Engineering Applications
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9_38
[2] Burduk A., Musiał K.: Genetic Algorithm Adoption to Transport Task Optimization.
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[3] Burduk A., Musiał K.: Optimization of Chosen Transport Task by Using Generic Algorithms,
vol. 9842. (2016) 197-205. DOI: 10.1007/978-3-319-45378-1_18
[4] Chyzhmotria O., Chyzhmotria O., Bolotina V.: Analysis of the Two-Criteria Transport Task.
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