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1

A steepest feasible direction method for linear programming : derivation and embedding in the simplex method

Wolde Biressaw C., Larsson Torbjörn

Operations Research and Decisions

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2024

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Vol. 34, No. 2

163--182

EN

A feasible direction method for linear programming has been proposed. The method is embedded in the framework of the simplex method, even though it works with non-edge feasible directions. The direction used is the steepest in the space of all variables or an approximation thereof, and it is found by solving a strictly convex quadratic program in the space of the nonbasic variables. Further, this program guarantees the feasibility of the direction even in the case of degeneracy. To remain within the simplex framework, the direction is represented by an auxiliary, or external, nonbasic column, which is a nonnegative linear combination of original nonbasic columns. We have made an experimental evaluation of the suggested method on both nondegenerate and highly degenerate problem instances. The overall results are very promising for continued research along this line, especially concerning various computational strategies that can be applied when the method is implemented.

2

Efficiency and Fairness Optimization in Energy Management

Zalewski Grzegorz, Granat Janusz, Makowski Marek

Journal of Telecommunications and Information Technology

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2024

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nr 4

86--92

EN

The paper proposes a solution to the problem of distributing electricity originating from various sources. In the proposed model, each source has a different cost of acquisition and is characterized by varying energy efficiency factors. Additionally, in the case of renewable sources, the costs of storing energy are taken into consideration as well. This work presents a fair and cost-efficient approach to distributing the demands of energy providers. A model has been developed and verified for the purpose of corroborating the process.

3

Modeling environmental and economic factors in regional energy optimization

Coban Hasan Huseyin, Lewicki Wojciech, Brelik Agnieszka

Economics and Environment

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2024

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No. 2

art. no. 748

EN

This article addresses the contemporary environmental challenges stemming from rapid economic growth, surging energy consumption, urban expansion, and mounting waste issues. The study explores the optimisation of a regional energy system, considering not only the electric energy sector but also the fuel and thermal energy sectors for the selected geographical destination. In this study, the application of the Linprog optimisation function in MATLAB programming tool to solve Regional Energy System Optimization with renewable resources is explained. The primary objective is to develop a mathematical model that identifies the optimal energy balance structure, allowing for the partial replacement of hydrocarbon sources with bioresources and waste in heat and electricity generation, as well as in vehicle fuel consumption. The modelling approach involves linear programming and integrates two key criteria: economic (cost of energy for consumers) and environmental (carbon footprint). The novelty of this approach lies in applying life cycle analysis to assess potential environmental consequences. Results reveal optimal generation volumes based on economic and environmental considerations. When optimising solely for economic criteria, municipal solid waste, along with wind energy, emerges as the preferred source. In contrast, the simultaneous optimisation of economic and environmental parameters aligns with the economic calculation, demonstrating a balanced approach to sustainable development.

PL

W artykule omówiono współczesne wyzwania środowiskowe wynikające z szybkiego wzrostu gospodarczego, rosnącego zużycia energii, ekspansji miast i narastających problemów z odpadami. W opracowaniu podjęto próbę optymalizacji regionalnego systemu energetycznego, uwzględniając nie tylko sektor energii elektrycznej, ale także sektor paliwowy i energetyki cieplnej dla wybranej lokalizacji geograficznej. W tym opracowaniu wyjaśniono zastosowanie funkcji optymalizacyjnej Linprog w narzędziu programistycznym MATLAB do rozwiązywania problemów z optymalizacją regionalnego systemu energetycznego przy użyciu zasobów odnawialnych. Podstawowym celem jest opracowanie modelu matematycznego identyfikującego optymalną strukturę bilansu energetycznego, pozwalającą na częściowe zastąpienie źródeł węglowodorów biosurowcami i odpadami w procesie wytwarzania ciepła i energii elektrycznej, a także w zużyciu paliwa przez pojazdy. Podejście modelowe obejmuje programowanie liniowe i integruje dwa kluczowe kryteria: ekonomiczne (koszt energii dla konsumentów) i środowiskowe (ślad węglowy). Nowatorstwo tego podejścia polega na zastosowaniu analizy cyklu życia do oceny potencjalnych konsekwencji dla środowiska. Wyniki ujawniają optymalne wielkości produkcji w oparciu o względy ekonomiczne i środowiskowe. W przypadku optymalizacji wyłącznie pod kątem kryteriów ekonomicznych, preferowanym źródłem są stałe odpady komunalne oraz energia wiatrowa. Natomiast jednoczesna optymalizacja parametrów ekonomicznych i środowiskowych pokrywa się z kalkulacją ekonomiczną, wykazując zrównoważone podejście do zrównoważonego rozwoju.

4

The use of rank and optimisation methods in strategic management in higher education

Ryńca Radosław, Piórkowska Natalia

Operations Research and Decisions

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2023

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Vol. 33, No. 1

93--111

EN

The article presents proposals for a university management model supporting the process of strategic management at a university. The proposed model is based on the use of multi-criteria methods such as the 0–10 technique, object ranking, and optimisation methods – linear programming. The proposed solution integrates ranking and optimisation methods, the use of which may be helpful in the hands of managers in making management decisions. The proposed approach may also be helpful in developing a strategic scorecard, especially in the stage of formulating goals. It also enables the optimal selection of goals with the existing time constraints for the implementation of the strategy. The article presents a proposal for the use of the strategy implementation model and an example of its use. The strengths and weaknesses of the model were also indicated.

5

Linear programming models for optimal workload and batching in pick-and-pass warehousing systems

Tarczyński Grzegorz

Operations Research and Decisions

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2023

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Vol. 33, No. 3

141--158

EN

Pick-and-pass systems are a part of picker-to-parts order-picking systems and constitute a very common storage solution in cases where customer orders are usually small and need to be completed very quickly. As workers pick items in the zones connected by conveyors, their work needs to be coordinated. The paper presents MILP models that optimize the order-picking process. The first model uses information about the expected demand for items to solve the storage location problem and balance the workload across zones. The task of the next model is order-batching and sequencing – two concepts are presented that meet different assumptions. The results of the exemplary tasks solved with the use of the proposed MILP models show that the total picking time of a set of orders can be reduced by about 35-45% in comparison with random policies. The paper presents an equation for the lower bound of a makespan. Recommendations about the number of zones that guarantee the required system efficiency are also introduced.

6

A new geometric approach to multiobjective linear programming problems

Kaci Mustapha, Radjef Sonia

Mathematica Applicanda

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2023

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Vol. 51, no. 1

3--12

EN

This paper is a follow-up to a previous work where we developed a new geometric approach to sensitivity analysis. In this paper, we present a simple method to determine whether a given multiobjective linear programming problem (MOLPP) has an ideal solution (i.e. all of the objective functions are optimized simultaneously) without having to calculate the optimal value of each objective function. First, we divide the space of linear forms into a finite number of sets based on a fixed convex polygonal subset of R2 using an equivalency relationship. All the elements from a given equivalency class have the same optimal solution. Next, we characterize the equivalence classes of the quotient set using a geometric approach to sensitivity analysis. Finally, a numerical example is given to illustrate the method.

PL

W tym artykule przedstawiamy nową metodę rozwiązywania problemów programowania liniowego z wieloma celami (MOLPP), która eliminuje potrzebę obliczania optymalnej wartości każdej funkcji celu. Metoda ta jest kontynuacją naszych wcześniejszych prac dotyczących analizy wrażliwości, gdzie opracowaliśmy nowe podejście geometryczne. Pierwszym krokiem naszego podejścia jest podział przestrzeni form liniowych na skończoną liczbę zbiorów opartych na stałym wypukłym podzbiorze wielokąta R2. Dokonujemy tego za pomocą relacji równoważności, która zapewnia, że wszystkie elementy z danej klasy równoważności mają takie same rozwiązanie optymalne. Następnie charakteryzujemy klasy równoważności zbioru ilorazowego za pomocą podejścia geometrycznego do analizy wrażliwości. Ten krok jest kluczowy w identyfikacji rozwiązania idealnego dla MOLPP. Korzystając z tego podejścia, możemy określić, czy dana MOLPP ma rozwiązanie idealne, bez konieczności obliczania optymalnej wartości każdej funkcji celu. Jest to znacząca poprawa w stosunku do istniejących metod, ponieważ znacznie zmniejsza złożoność obliczeniową i czas wymagany do rozwiązania MOLPP. Aby zilustrować naszą metodę, przedstawiamy numeryczny przykład, który dowodzi jej skuteczności. Nasza metoda jest prosta, ale potężna i może być łatwo zastosowana do szerokiego zakresu MOLPP. Niniejsza praca przyczynia się do dziedziny optymalizacji poprzez przedstawienie nowego podejścia do rozwiązywania MOLPP, które jest wydajne, skuteczne i łatwe do zaimplementowania.

7

The application of linear programming for the optimal profit of PT. Naruna using the simplex method

Syifa Eka Auliya, Istiqomah Tita Nuril, Puspita Nikken Prima, Ratnasari Lucia, Khabibah Siti, Anggoro Paulus Wisnu, Bawono Baju

Management Systems in Production Engineering

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2023

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nr 2 (31)

138--143

EN

PT. Naruna is a ceramics factory located in Salatiga, Indonesia. In PT. Naruna ceramics, all products are handmade with contemporary designs and have a high artistic value in shape and color. Getting profit is the company's primary goal, but many companies still need to learn the maximum profit that can be obtained by optimizing their resources, one of which is PT. Naruna. PT. Naruna produces goods based on intuition. As a result, a lot of goods are piled up in warehouses. Meanwhile, with the development of the times, new trends and images will appear more attractive so that consumer tastes and motifs from ceramics will change. In addition, ceramic products that have gone through the combustion process cannot be recycled and must be burned. This research focuses on the production of glasses with three different types according to price. The aim of this paper is to optimize profits by determining the composition of the number of products produced. We used linear programming with a simplex method to solve our problem in PT. Naruna. Linear programming is the most appropriate method for solving problems that exist in PT. Naruna, namely by paying attention to the objective and constraint functions. The objective function is to maximize profit, so it takes the form of a linear equation with the variable X1 being the first type of glass, X2 being the second type of glass, and X3 being the third type of glass. The constraint functions used include the number of products, the number of workers, the amount of clay, and the time for production. The results show that PT. Naruna can achieve maximum profit when producing glass type 1 less than type 3 less than type 2.

8

New Algorithm Permitting the Construction of an Effective Spanning Tree

Belkacem Salima Nait, Abbasy Moncef

Foundations of Computing and Decision Sciences

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2023

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Vol. 48, No. 3

313--329

EN

In this paper, we have done a rapid and very simple algorithm that resolves the multiple objective combinatorial optimization problem. This, by determining a basic optimal solution, which is a strong spanning tree constructed, according to a well-chosen criterion. Consequently, our algorithm uses notions of Bellman’s algorithm to determine the best path of the network, and Ford Fulkerson’s algorithm to maximise the flow value. The Simplex Network Method that permits to reach the optimality conditions manipulates the two algorithms. In short, the interest of our work is the optimization of many criteria taking into account the strong spanning tree, which represents the central angular stone of the network. To illustrate that, we propose to optimize a bi-objective distribution problem.

9

Deployment of a predictive-like optimal control law on a servo drive system using linear programming approach

Horla Dariusz, Pinczewski Piotr

Archives of Electrical Engineering

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2023

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Vol. 72, nr 4

1005--1016

EN

Current drive control systems tend to push control loops to the limits of their performance. One of the ways of doing so is to use advanced optimization algorithms, usually related to model-based off-line calculations, such as genetic algorithms, the particle swarmoptimisation or the others. There is, however, a simpler way, namely to use predictive control formalism and by formulation of a simple linear programming problem which is easy to solve using powerful solvers, without excessive computational burden, what is a reliable solution, as whenever the optimization problem has a feasible solution, a global minimizer can be efficiently found. This approach has been deployed for a servo drive system operated by a real-time sampled-data controller, verified between model-in-the-loop and hardwarein- the-loop configurations, for a range of prediction horizons, as an attractive alternative to classical quadratic programming-related formulation of predictive control task.

10

List Of Pareto Optimal Solutions of a Biobjective Shortest Path Problem

Laskov Lasko, Marinov Marin

Annals of Computer Science and Information Systems

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2023

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Vol. 35

603--613

EN

Many applications in practice involve the search for a shortest path in a network by optimizing two conflicting objective functions. Such problems often are referred to as biobjective optimization problems. Their goal is to find special optimal paths that are nondominated and are also known in the specialized literature as to as Pareto optimal. While most of the existing methods aim to find the minimum complete set of Pareto optimal paths, we propose an approach that is able to generate a list of all Pareto optimal solutions in a given network.

11

A linearisation approach to solving a non-linear shelf space allocation problem with multi-oriented capping in retail store and distribution centre

Czerniachowska Kateryna, Hernes Marcin, Lutosławski Krzysztof

Operations Research and Decisions

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2022

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Vol. 32, No. 4

33--56

EN

Shelf space is one of the essential resources in logistic decisions. Order picking is the most time-consuming and labourintensive of the distribution processes in distribution centres. Current research investigates the allocation of shelf space on a rack in a distribution centre and a retail store. The retail store, as well as the distribution centre, offers a large number of shelf storage locations. In this research, multi-orientated capping as a product of the rack allocation method is investigated. Capping allows additional product items to be placed on the rack. We show the linearisation technique with the help of which the models with capping could be linearised and, therefore, an optimal solution could be obtained. The computational experiments compare the quality of results obtained by non-linear and linear models. The proposed technique does not increase the complexity of the initial non-linear problem.

12

Complex Fibonacci (c, p) : numbers

Prasad Bandhu

Mathematica Applicanda

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2022

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Vol. 50, no. 2

235--247

EN

In this paper a new complex Fibonacci Q_{p,c} matrix for complex Fibonacci (c,p)-numbers, where p is a positive integer and c is a non-zero complex number, is introduced. Thereby, we discuss various properties of Q_{p,c} matrix, coding and decoding method followed from the Q_{p,c} matrix.

PL

W artykule przedstawiono nową macierz zespoloną Fibonacciego oznaczaną Qp,c dla liczb zespolonych Fibonacciego (c, p), gdzie p jest liczbą całkowitą dodatnią, a c jest niezerową liczbą zespoloną. Omówiono różne własności macierzy Qp,c, oraz sposób kodowania i dekodowania wynikający z macierzy Qp,c.

13

A coding theoretical interpretation of Gaussian-Pell polynomials

Prasad Bandhu

Mathematica Applicanda

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2022

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Vol. 50, no. 1

129--138

EN

In this paper, we establish a new result followed from Gaussian Pell polynomials matrix, Qn(x)P(x) (cf. Serpil and Sinan (2018)) whose elements are Gaussian Pell polynomials and we develop new coding and decoding method follow from Gaussian Pell polynomials matrix, Qn(x)P(x). The correction ability of this method is 93:33%.

PL

W artykule z wykorzystaniem macierzy wielomianów Gaussa Pella, Qn(x)P(x) (v. Serpil and Sinan (2018)), opracowano nową metodę kodowania. Ta metoda wynika z własności tej macierzy. Uzyskany kod daje możliwość korekcji na poziomie 93:33% .

14

Global optimal vessel capacity for LNG production and transportation

Igboanusi Cninemerem, Amuji Harrison Obiora, Onukwube Obioma G., Chikadibia Nwosu Azuwueze

Journal of Sustainable Development of Transport and Logistics

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2022

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Vol. 7, No. 1

82--89

EN

In this paper, developed a linear programming model to determine the volume of vessels that will give an optimal return on investment. The solution to the developed model was carried out using the Interior Point algorithm with the help of the MATLAB package. The analysis observed that the production and transporting of the LNG with a vessel of capacity 178,5006m3 would give an optimal profit of 633,640 million USD. And from the results of the analysis, we observed that the decision to take the maximum modern capacity instead of lower capacities yields the highest profit.

15

Independent component analysis based on Jacobi iterative framework and L1-norm criterion

Borowicz Adam

Annals of Computer Science and Information Systems

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2022

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Vol. 30

305--312

EN

Most recently, a link between principal component analysis (PCA) based on L1-norm and independent component analysis (ICA) has been discovered. It was shown that the ICA can actually be performed by L1-PCA under the whitening assumption, inheriting the improved robustness to outliers. In this paper, a novel ICA algorithm based on Jacobi iterative framework is proposed that utilizes the non-differentiable L1-norm criterion as an objective function. We show that such function can be optimized by sequentially applying Jacobi rotations to the whitened data, wherein optimal rotation angles are found using an exhaustive search method. The experiments show that the proposed method provides a superior convergence as compared to FastICA variants. It also outperforms existing methods in terms of source extraction performance for Laplacian distributed sources. Although the proposed approach exploits the exhaustive search method, it offers a lower computational complexity than that of the optimal L1-PCA algorithm.

16

Optymalizacja kosztów transportu gazu skroplonego LNG pod względem lokalizacji magazynów na przykładzie Polski

Popławski Tomasz, Kurkowski Bartosz

Rynek Energii

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2021

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Nr 4

3--8

PL

Wielu ekspertów zarówno z Polski jak i UE jednoznacznie stwierdza, że Gazociąg Nordstream 2 zmniejsza bezpieczeństwo gazowe zarówno Polski jaki i państw z Europy środkowowschodniej. W związku z tym konieczne są różne przedsięwzięcia w celu dywersyfikacji dostaw gazu do Polski. Przy istniejącej i wciąż rozwijającej się infrastrukturze drogowej oraz kolejowej, jak również w obliczu istnienia terminali do odbioru gazu w Świnoujściu i Gdańsku, jednym z możliwych działań w tym kierunku jest budowa nowych magazynów lub rozbudowa starych w taki sposób, aby można było przewozić i przechowywać tam skroplony gaz LNG. Jednak, aby przedsięwzięcie to było opłacalne, lokalizacje tych magazynów muszą tak zostać dobrane, aby zminimalizować przyszłe koszty transportowe. W niniejszym artykule zaproponowano metodę opartą o programowanie liniowe, która pozwala na optymalną alokację magazynów LNG. Jej innowacyjnym względem istniejących metod elementem jest dynamicznie aktualizująca się waga dostawcy, w zależności od przypisanych do niego odbiorców. Dla zaprezentowanego przykładu zaproponowana metoda pozwoliła na zmniejszenie kosztów drogowych o ponad 25%.

EN

Many experts from both Poland and the EU clearly state that the Nordstream 2 gas pipeline reduces gas safety both in Poland and in Central and Eastern European countries. With the existing and still developing road and rail infrastructure, as well as the gas receiving terminals in Świnoujście and Gdańsk, one of the possible actions is the construction of new warehouses or expansion of old ones so that LNG can be transported and stored there. However, for this venture to be profitable, the locations of these warehouses must be selected to minimize future transport costs. This article proposes a method based on linear programming that allows for the optimal allocation of LNG storage facilities. Its innovative element compared to the existing methods is the dynamically updating weight of the supplier, depending on the recipients assigned to it. For the example shown, the proposed method allowed to reduce road costs by more than 25%.

17

An optimal design of non-causal recursive digital filters with zero phase shift using Chebyshev approximation and linear programming

Mouffak Adnane

Przegląd Elektrotechniczny

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2021

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R. 97, nr 9

100--105

EN

This paper presents an optimal design for the special class of Non-Causal Recursive (NR) digital filters with zero phase shift. The design is based on the Chebyshev approximation problem. It can be transformed to an equivalent linear program under linear constraints of the zero phase. The given design yields more interesting pole-zero patterns that are not necessarily restricted to the classical design of Kormylo and Jain. The proposed optimal design allows an accurate zero phase shift and better magnitude characteristics in passband and stopband.

PL

W artykule przedstawiono optymalną procedurę projektowania dla specjalnej klasy nieprzyczynowych filtrów rekurencyjnych (NR) z zerowym przesunięciem fazowym. Projekt opiera się na problemie aproksymacji Czebyszewa. Można go przekształcić do równoważnego programu liniowego przy ograniczeniach liniowych fazy zerowej. Projekt daje bardziej interesujące wzory bieguna zerowego, które niekoniecznie ograniczają się do klasycznego projektu Kormylo i Jaina. Zaproponowana optymalna procedura umożliwia dokładne zerowe przesunięcie fazowe i lepszą charakterystykę amplitudy w paśmie przepuszczania i zatrzymywania.

18

Comparative Study of AMPL, Pyomo and JuMP Optimization Modeling Languages on a Network Linear Programming Problem Example

Karbowski Andrzej, Wyskiel Krzysztof

Pomiary Automatyka Robotyka

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2021

|

R. 25, nr 3

23--30

EN

The purpose of this work is a comparative study of three languages (environments) of optimization modeling: AMPL, Pyomo and JuMP. The comparison will be based on three implementations of the shortest path problem formulated as a linear programming problem. The codes for individual models and differences between them will be presented and discussed. Various aspects will be taken into account, such as: simplicity and intuitiveness of implementation, availability of specific data structures for a LP network problems, etc.

PL

Celem pracy jest zbadanie i porównanie możliwości trzech języków (środowisk) modelowania optymalizacyjnego: AMPL, Pyomo i JuMP. Porównanie zostanie oparte na trzech implementacjach zadania najkrótszej ścieżki sformułowanego jako zadanie programowania liniowego. Przedstawione i omówione zostaną kody poszczególnych modeli oraz różnice między nimi. Pod uwagę będą brane różne aspekty, takie jak: prostota i intuicyjność implementacji, dostępność określonych struktur danych dla problemów z siecią LP itp.

19

A new geometric approach for sensitivity analysis in linear programming

Kaci Mustapha, Radjef Sonia

Mathematica Applicanda

|

2021

|

Vol. 49, no. 2

145--157

EN

The article presents a geometric method of sensitivity analysis in linear programming, which is a computationally practical way to study the behavior of an optimal solution to a linear programming problem. In this approach, we improve the implementation of the constraints, and then we formulate the problem of linear programming geometrically. In this way, we obtain a new, equivalent geometrical formulation of the problem for the sensitivity analysis using the concepts of affine geometry. It consists in entering the objective function coefficient vector in the polar coordinates and determining all angles for which the solution remains unchanged. The method is presented in detail and illustrated by a numerical example.

PL

W niniejszym artykule przedstawiamy nowe podejście geometryczne do analizy wrażliwości w programowaniu liniowym, które jest praktyczne obliczeniowo. Pozwala analizować zachowania optymalnego rozwi¡zania problemu programowania liniowego przy zmianach danych zadania. Najpierw ustalamy dopuszczalną dziedzinę (naprawiamy ograniczenia liniowe). Następnie geometrycznie formułujemy problem programowania liniowego. Nast¦pnie podajemy nowe równoważne sformułowanie geometryczne problemu analizy wrażliwości przy użyciu pojęć geometrii afinicznej. Piszemy wektor współczynników funkcji celu we współrzędnych biegunowych i wyznaczamy wszystkie kąty, dla których rozwiązanie pozostaje niezmienione. Proponowane podejście zostało szczegółowo przedstawione i zilustrowane przykładem liczbowym.

20

An example to illustrate several aspects of optimization theory in Managerial Economics

Lahiri Somdeb

Managerial Economics

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2021

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Vol. 22, no. 1

23--34

EN

We provide a single example that illustrates all aspects of linear, integer and dynamic programming, including such concepts such as value of perfect and imperfect information. Such problems, though extremely plausible and realistic are hardly ever discussed in managerial economics.