=Paper=
{{Paper
|id=Vol-2928/paper12
|storemode=property
|title=Forecasting Methods in Logistics Problems
|pdfUrl=https://ceur-ws.org/Vol-2928/paper12.pdf
|volume=Vol-2928
|authors=Anna V. Zykina,,Angela A. Kanunnikova
}}
==Forecasting Methods in Logistics Problems==
https://ceur-ws.org/Vol-2928/paper12.pdf
Forecasting Methods in Logistics Problems
Anna V. Zykina Angela A. Kanunnikova
Omsk State Technical University Omsk State Technical University
Omsk, Russia, Mira Ave. 11, 644050 Omsk, Russia, Mira Ave. 11, 644050
avzykina@mail.ru marymind99@gmail.com
Abstract
The article considers the use of forecasting methods in solving problems
of delivering bottled water to customers. It also examines the method
of regression analysis of data. Moreover, a mathematical model is de-
veloped to predict the amount of water in the following months. To
determine the optimal allocation of financial resources, a multi-index
transport problem is constructed. The constructed models are imple-
mented in Python.
1 Introduction
Logistics is the science of managing material flows, as well as related information, financial, service and other
flows, to optimize the functioning of economic systems through the efficient utilization of all types of resources
[Sak2018].
The main strategic goals of logistics for this study are:
1) to ensure a continuous flow of external resources needed for the operation of the enterprise;
2) to optimize the overall cost of resources of the enterprise, as well as to manage the stocks of purchased
products;
3) to manage suppliers, namely, their selection and evaluation, and development of relationships with them;
4) to coordinate the procurement process with other functional areas of logistics,e. g. the logistics of produc-
tion, distribution, and transportation.
This study uses methods of mathematical statistics and optimization methods to achieve its goals.
In market conditions, several factors determine the relevance of companies’ use of logistics.
First of all, there is an economic factor. Every company aims to maximize profits, so the focus is on finding
ways to reduce the time and money costs that come with products transportation.
Another factor is the organizational-economic one. The market is developing, new organizational forms are
emerging, and the development of interaction between manufacturers, consumers, and transport companies is
becoming increasingly important.
The rapid development of technology, as well as the growth of the information sector of the economy, con-
tributes to the search for different methods to increase company profits and reduce its costs [Pic2020, Mas2006].
Opportunities and tools for the management of logistics processes are emerging. New theories and methods of
operation research are being developed that can help to optimize the processes.
Constant development of market relations, technical opportunities, as well as a wide range of goals and
objectives of logistics, makes this field relevant to the study and highlights new challenges to be addressed.
Copyright ⃝
c by the paper’s authors. Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
In: Sergei S. Goncharov, Yuri G. Evtushenko (eds.): Proceedings of the Workshop on Applied Mathematics and Fundamental
Computer Science 2021 , Omsk, Russia, 24-04-2021, published at http://ceur-ws.org
1
2 Problem statement
Currently, many companies propose products’ delivery services. One group of such companies are those delivering
bottled water.
In modern society, the availability of clean drinking water in the workplace is one of the requirements of sanitary
and epidemiological regulations. The number of business and industrial enterprisers, where these regulations must
be followed, is quite large. Therefore, water delivery companies are in high demand.
Since water delivery companies, like any other company, strive to minimize costs and improve the quality of
their services, their work is carried out with the use of logistics processes.
There are several main work areas for logisticians:
– resource procurement;
– production;
– end product sales.
When the subject area was studied, two tasks were identified:
– forecast of water demand;
– optimization of the bottled water delivery to customers.
A mathematical model was developed for each problem, parameters were selected, and the results were
interpreted.
The essence of modeling using economic-statistical methods is that the forecast indicator is determined
through specific models that show its functional dependence on certain factors [Chi2020, Ash2019].
Problem 1. Forecast of water demand.
The water delivery company should anticipate what demand is expected in the future. The option to prepare
excess products is not suitable in the case of bottled water. Firstly, the warehouse has a limited capacity.
Secondly, bottled water has a limited expiry date even when closed, so it is necessary to make a certain amount
of water so as not to suffer losses.
Consumer companies do not always order the same amount of water each time. This can be influenced by
various factors. Therefore, the following forecast task is being investigated: how much water the company will
order, or how much water the company should order to provide sufficient water amount for employees.
To be able to make a forecast, it is necessary to build a model which will be used to predict the amount
of water. Three types of parameters will be used for constructing the model: the number of employees, the
number of working days and the time of the year. Let us build a regression model using these parameters.
Linear regression is a standard representative of machine learning algorithms. Regression belongs to the class of
supervised learning tasks, where a particular target variable must be predicted for a given set of characteristics
of the object under observation [Lin2020].
It is necessary to predict the amount of water, so the data on the number of bottles ordered is used as the
dependent variable y. The independent variables are the number of employees in the company x1 , the number
of working days in a month x2 , and the time of the year z1 , z2 , z3 .
To build the model, monthly data for one company have been taken for two years. The method of least squares
is used to find the model parameters. Instead of the season variable, which is nominative, let us introduce three
dummy variables: winter month or not; spring month or not; summer month or not.
As a result of the calculation, the following regression equation was obtained:
y = −6, 0534 + 1, 5135x1 + 1, 7924x2 − 1, 8131z1 + 1, 1467z2 − 0, 9204z3 .
To evaluate the quality measure of the regression equation, the determination coefficient R2 was calculated. In
this case, R2 = 0.86, which means that the resulting regression equation explains the change in the resulting
variable y by 86. The remaining 14 are accounted for by unaccounted factors.
To check the regression equation for significance, an F -test is used. The ”zero” hypothesis H0 of statistical
insignificance of the regression equation is put forward. In the model, criterion Fobsrv = 28.284. Using F -criterion
table we find the critical value Fcrit (0.01; 5; 23) = 2.64. Compare the two values: Fobsrv > Fcrit . This means
that the zero hypothesis is rejected, and the equation is significant.
To check the correctness of the construction and further automate the calculation of the forecast, the model is
implemented using the Python language. The language allows one to work with big data, build various models
and present a visualization of the resulting calculations. To build the model, the Linear Regression() function
from the sklearn library is used.
2
The data for the construction are loaded from the excel-file (Figure 1).
Figure 1: Data view in the file
As a result of using the function, the following data were obtained for the model:
– intercept is free coefficient;
– slope is the slope of the linear regression, are coefficients at independent variables of the model;
– and coefficient of determination R2 (Figure 2).
As seen from F igure2, the data from manual calculations and those made with Python are the same.
Figure 2: Formation of the mathematical model in Python
Using the model, let us predict the amount of water for the next few months. To do this, data should be
entered for each of the variables. The result will be a forecast of how many bottles of water the company will
need in a given month (Figure 3).
Figure 3: Data entry and forecasting
To view the dynamics of the water orders, let us build a diagram with the data used to find the model, and
mark the point of the predicted quantity on the diagram (Figure 4).
The implementation results in a total of 87 bottles of water for 36 employees for December 2020, which has
23 workdays.
Problem 2. Bottled water delivery to customers.
As a rule, the company’s costs associated with the transportation of products are not predicted. The calcula-
tion of operating efficiency is based on the actual costs of the transportation division. This makes it impossible
to estimate costs in advance when planning transportation. Moreover, the task of determining the optimal route
for the transportation organization, together with the task of loading transport and product allocation, are the
most important in transport logistics. The task of logistics management of product flows to minimize costs
and maximize profits in grocerycompanies is a complex methodological problem, since the technological chain of
food production includes a large number of interacting participants. It is necessary to include all participants of
production, their interaction, peculiarities of organization and operating technology [Mok2020].
3
Figure 4: Schedule of changes in water delivery
Based on the forecast of water demand, the problem of delivering bottled water to customers can be solved
as a linear programming problem of transport type. Let us consider the formulation of the mathematical model
of the problem.
There are n kinds of water bottles, m shipment points, l consumer delivery points, g vehicles of the first type,
and f vehicles of the second type, which will be used to deliver products.
Set constant parameters:
−aij is the stock of the i-th type of product in the j-th point;
−bik is the demand of the k-th consumer of the i-th type of product;
−dip is thethe amount of the i-th type of product, that can be loaded in the transport of the first type;
−eit is the amount of the i-th type of product that can be loaded in the transport of the second type;
−cijkp is transport costs per unit of the product of the i-th type of the j-th supplier delivered to the k-th
consumer by transport of the first type;
−qijkt is transport costs per unit of the product of the i-th type of the j-th supplier delivered to the k-th
consumer by the second type of transport.
The following parameters are selected as variables:
−xijkp is the amount of the i-th type of product from the j-th shipment point to be delivered to the k-th
consumer by the first type of transport;
−yijkt is the amount of the i-th type of product from the j-th shipment point to be delivered to the k-th
consumer by the second type of transport.
The mathematical model of the problem has the form:
xijkp ≥ 0, yijkt ≥ 0, i = 1,¯n , j = 1,¯m , k = 1,¯ l, p = 1,¯g, t = 1,¯f , (1)
∑
l ∑
g ∑
l ∑
f
xijkp + yijkt ≤ aij , i = 1,¯n , j = 1,¯m, (2)
k=1 p=1 k=1 t=1
∑
m ∑
g ∑
m ∑
f
xijkp + yijkt = bik , i = 1,¯n, k = 1,¯ l, (3)
j=1 p=1 j=1 t=1
∑
n
xijkp ≤ dip , j = 1,¯m, k = 1,¯ l, p = 1,¯g, (4)
i=1
∑
n
yijkt ≤ eit , j = 1,¯m, k = 1,¯ l, t = 1,¯f , (5)
i=1
4
∑
n ∑
m ∑
l ∑
g ∑
n ∑
m ∑
l ∑
f
L= cijkp xijkp + qijkt yijkt → min (6)
i=1 j=1 k=1 p=1 i=1 j=1 k=1 t=1
All elements xijkp and yijkt must be either positive or equal to zero (1). This means that from the shipment
point a certain amount of a certain type of product will be transported to the consumer, or the product will not
be transported to them. The solution matrix cannot contain negative numbers.
Condition (2) describes the constraint that the amount shipped from the shipment point on all types of
transport should not exceed the total stocks of this shipment point, and condition (3) means that the number
of shipped products must meet the demand of the consumer. Conditions (4, 5) refer to the restrictions on the
capacity of transport that will be used for delivery: the amount of transported products must not exceed the
volume that the vehicle can accommodate.
Target function (6) of the problem sets the minimization of transport costs in the delivery of products.
This problem is a multi-index transport problem of linear programming. To form the cost table, a three-index
matrix is constructed.
3 Numerical experiment
When compiling a data table for further formation of the solution in this example, a different type of transport
for delivery is given for each type of product.
Figure 5 shows a table of input data.
Figure 5: The table of input data
Here:
– L1, L2, L3 are consumers;
– M1, M2 are shipment points;
– 1G, 2F are bottles of the 1st and 2nd types and the corresponding type of transport for delivery;
– M 11 , M 12 are shipments of bottles of the 1st and 2nd types from the first point;
– M 21 , M 22 are shipments of bottles of the 1st and 2nd types from the second point.
The transport variables are recorded in the cells at the intersection. In places, where the delivery of certain
products will not be made by this type of transport, lines are drawn.
Columns L and N contain the transport costs per unit of transported products. The problem has the following
target function
F = 3x1 + 4x2 + 5x3 + 7x4 + 4x5 + 6x6 + 7y1 + 2y2 + 5y3 + 5y4 + 3y5 + 2y6 → min
As a result of solving the problem, the following volumes of product delivery to the consumer and the total
costs are obtained (Figure 6).
To solve the problem, the Scipy library for the Python programming language is used. To solve linear
programming problems, Scipy offers the linprog function (included in the optimize component). The function
looks as follows: scipy.optimize.linprog(c, A ub, b ub, Ae q, b eq) and solves the problem by the simplex method.
5
Figure 6: Results
4 Conclusion
Using a regression model, a forecast of the demand for the number of bottles has been obtained, these forecast
results have been used to formulate and solve the linear programming problem of transport type. In this case not
only the need for the number of water bottles has been forecast, but also the company’s costs for water delivery.
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