## 2020, issue 4, p. 5-14

*Received **21.10.2020;
Revised 05.12.2020; Accepted 17.12.2020*

*Published **31.12.2020;
First Online 22.01.2021*

*https://doi.org/10.34229/2707-451X.20.4.1*

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**On Linear
and Quadratic Two-Stage Transportation Problem**

Petro
Stetsyuk ^{1 *} ,
Oleksii Lykhovyd ^{1}, Anton Suprun ^{1}

^{1}* V.M. Glushkov Institute of Cybernetics of
the NAS of Ukraine, Kyiv*

^{*}* Correspondence: This email address is being protected from spambots. You need JavaScript enabled to view it.*

**Introduction.** When
formulating the classical two-stage transportation problem, it is assumed that
the product is transported from suppliers to consumers through intermediate
points. Intermediary firms and various kinds of storage facilities (warehouses)
can act as intermediate points.

The article discusses two mathematical models for two-stage transportation problem (linear programming problem and quadratic programming problem) and a fairly universal way to solve them using modern software. It uses the description of the problem in the modeling language AMPL (A Mathematical Programming Language) and depends on which of the known programs is chosen to solve the problem of linear or quadratic programming.

**The purpose of the article** **is** to
propose the use of AMPL code for solving a linear programming two-stage
transportation problem using modern software for linear programming problems,
to formulate a mathematical model of a quadratic programming two-stage
transportation problem and to investigate its properties.

**Results.** The properties
of two variants of a two-stage transportation problem are described: a linear
programming problem and a quadratic programming problem. An AMPL code for
solving a linear programming two-stage transportation problem using modern
software for linear programming problems is given. The results of the
calculation using Gurobi program for a linear programming two-stage
transportation problem, which has many solutions, are presented and analyzed. A
quadratic programming two-stage transportation problem was formulated and
conditions were found under which it has unique solution.

**Conclusions.** The developed AMPL-code
for a linear programming two-stage transportation problem and its modification
for a quadratic programming two-stage transportation problem can be used to
solve various logistics transportation problems using modern software for
solving mathematical programming problems. The developed AMPL code can be
easily adapted to take into account the lower and upper bounds for the quantity
of products transported from suppliers to intermediate points and from intermediate
points to consumers.

**Keywords:** transportation
problem, linear programming problem, AMPL modeling language, Gurobi program,
quadratic programming problem.

**Cite as: **Stetsyuk P.,
Lykhovyd O., Suprun A. On Linear and Quadratic Two-Stage Transportation
Problem. *Cybernetics and Computer Technologies*. 2020. **4**. P. 5–14.
(in Ukrainian) https://doi.org/10.34229/2707-451X.20.4.1

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3. Stetsyuk P.I., Liashko V.I., Mazutynetz G.V. Two-stage transportation problem and its AMPL-implementation. Naukovi zapysky NaUKMA. Kompiuterni nauky. 2018. 1. P. 14–20. (in Ukrainian) https://doi.org/10.18523/2617-3808.2018.14-20

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6. Gurobi Optimization, Inc., Gurobi Optimizer Reference Manual, 2014. https://www.gurobi.com/ (last accessed: 06.09.2020)

7. Grygorak M.Yu. Intellectualization of the logistics services market: concept, methodology, competence: monograph. Kyiv: Sik Grup Ukraine, 2017. 513 p. (in Ukrainian)

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*ISSN 2707-451X (Online) *

*ISSN 2707-4501 (Print) *

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