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1

Route planning for multiple unmanned aerial vehicles

Aguiar Antônio Lucas Sousa Aguiar, Pinto Vandilberto P., Sousa Lígia Maria Carvalho, Pinheiro José Lucas Da Silva, do Nascimento José Cleilton

Przegląd Elektrotechniczny

|

2024

|

R. 100, nr 9

219--223

EN

This study addresses efficient task assignment for collaborative systems, with a focus on route planning for multiple Unmanned Aerial Vehicles (UAVs). Using the Ant Colony Optimization algorithm and the A* algorithm for obstacle avoidance, the results show that the proposed method allows route planning with acceptable computational time, providing guidance on the optimal number of UAVs in a mission.

PL

Niniejsze badanie dotyczy efektywnego przydzielania zadań dla systemów współpracujących, ze szczególnym uwzględnieniem planowania trasy dla wielu bezzałogowych statków powietrznych (UAV). Wykorzystując algorytm optymalizacji kolonii mrówek i algorytm A* do omijania przeszkód, wyniki pokazują, że proponowana metoda umożliwia planowanie trasy w akceptowalnym czasie obliczeniowym, zapewniając wskazówki dotyczące optymalnej liczby UAV w misji.

2

Assignment of tasks to machines under data replication with a tie to Steiner systems

Wojciechowski Paweł, Kasprzak Marta

International Journal of Applied Mathematics and Computer Science

|

2024

|

Vol. 34, no. 2

263--275

EN

In the paper a problem of assignment of tasks to machines is formulated and solved, where a criterion of data replication is used and a large size of data imposes additional constraints. This problem is met in practice when dealing with large genomic files or other types of vast data. The necessity of comparing all pairs of files within a big set of DNA sequencing results, which we collected, maintained, and analyzed within a national genomic project, brought us to the proposed results. This problem resembles that of generating a particular Steiner system, and a mechanism observed there is employed in one of our algorithms. Based on the problem complexity, we propose two heuristic algorithms, which work very well even for instances with tight constraints and a heterogeneous environment defined. In addition, we propose a simplified method, nevertheless capable of finding very good solutions and surpassing the algorithms in some special cases. The methods are validated in tests on a wide set of instances, where values of parameters reflect our real-world application and where their usefulness is proven.

3

Modeling environmental and economic factors in regional energy optimization

Coban Hasan Huseyin, Lewicki Wojciech, Brelik Agnieszka

Economics and Environment

|

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

A new geometric approach to multiobjective linear programming problems

Kaci Mustapha, Radjef Sonia

Mathematica Applicanda

|

2023

|

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.

5

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

|

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.

6

Efficient exact A* algorithm for the single plant Hydro Unit Commitment problem

Heintzmann Alexandre, Artigues Christian, Bendotti Pascale, Ngueveu Sandra Ulrich, Rottner Cécile

Annals of Computer Science and Information Systems

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2023

|

Vol. 35

533--543

EN

The Hydro Unit Commitment problem (HUC) specific to hydroelectric plants is part of the electricity production planning problem, called Unit Commitment Problem (UCP). More specifically, the studied case is that of the HUC with a single plant, denoted 1-HUC. The plant is located between two reservoirs. The horizon is discretized in time periods. The plant operates at a finite number of points defined as pairs of the generated power and the corresponding water flow. Several constraints are considered. Each reservoir has an initial volume, as well as window resource constraints, defined by a minimum and maximum volume per time period. At each time period, there is an additional positive, negative or zero intake of water in the reservoirs. The case of a price-taker revenue maximization problem is considered. An efficient exact A* variant, so called HA*, is proposed to solve the 1-HUC accounting for window constraints, with a reduced search space and a dedicated optimistic heuristic. This variant is compared to a classical Resource Constrained Shortest Path Problem (RCSPP) algorithm and a Mixed Integer Linear Programming formulation solved with CPLEX. Results show that the proposed algorithm outperforms both concurrent alternatives in terms of computational time in average on a set of realistic instances, meaning that HA* exhibits a more stable behavior with a larger number of instances solved.

7

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

|

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.

8

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

Popławski Tomasz, Kurkowski Bartosz

Rynek Energii

|

2021

|

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%.

9

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.

10

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.

11

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

Lahiri Somdeb

Managerial Economics

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2021

|

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.

12

Optimum transportation of relief materials aftermath the disaster

Nautiyal A., Kumar A., Poddar A., Parajuli N.

Journal of Achievements in Materials and Manufacturing Engineering

|

2021

|

Vol. 109, nr 1

26--41

EN

Purpose: Natural disasters disrupt not only the lives of individuals but also the functioning of society. Given the unpredictability of disasters and the uncertainty associated with them, preparation is the best way to mitigate and reduce the effects of the disaster. Design/methodology/approach: The study presents a mathematical model in the form of a multi-objective linear programming problem for the relief distribution from the airports which minimizes the total operational cost as well as travel time. Further, the solution approach and analytical results have also been discussed. Findings: The main aims at the preparedness stage are to identify and build infrastructures that might function as useful operation centres during a disaster. The study also provides decisions that include the type and number of vehicles for each affected location. Research limitations/implications: Airports can function as centres for relief collection and distribution. However, relief operations carried out through airports are often subject to problems such as stockpiling. Further, various modes are available for the transport of relief supplies- air, water, and land transport modes primarily. While aircraft and helicopters are faster, their costs of operation are too high. Instead, trucks are economical but very slow as compared to aircraft. Practical implications: The choice of model depends on many factors including the availability of vehicles, availability of routes, and criticality of situations. The choices made in turn affect the costs and the time of operations. Originality/value: The model converts a disaster scenario into a demand-supply problem with the aim being to decide allocations at specified intervals of time.

13

Conditions and a computation method of the constrained regulation problem for a class of fractional-order nonlinear continuous-time systems

Si Xindong, Yang Hongli, Ivanov Ivan G.

International Journal of Applied Mathematics and Computer Science

|

2021

|

Vol. 31, no. 1

17--28

EN

The constrained regulation problem (CRP) for fractional-order nonlinear continuous-time systems is investigated. New existence conditions of a linear feedback control law for a class of fractional-order nonlinear continuous-time systems under constraints are proposed. A computation method for solving the CRP for fractional-order nonlinear systems is also presented. Using the comparison principle and positively invariant set theory, conditions guaranteeing positive invariance of a polyhedron for fractional-order nonlinear systems are established. A linear feedback controller model and the corresponding algorithm of the CRP for fractional nonlinear systems are also proposed by using the obtained conditions. The presented model of the CRP is formulated as a linear programming problem, which can be easily implemented from a computational point of view. Numerical examples illustrate the proposed method.

14

Risk management in the process of personnel allocation to jobs

Krynke Marek

System Safety : Human - Technical Facility - Environment

|

2020

|

Vol. 2, nr 1

82--99

EN

The strategy should be designed in such a way as the risk management can operate not only as a system for avoiding losses, but also risk management should allow recognizing and making use of occasions and create new opportunities for the organization. Risk management includes both an evaluation (analytical and evaluation) undertaking as well as planning and control activities aimed at minimizing (reducing) risk or maintaining it at an acceptable level. Security management can in particular be reduced to the issue of risk management, because risk is a quantitative expression of the functioning of systems in an environment where there are active sources of threats to system security. The article presents the problem of personnel allocation in hazardous conditions, emphasizing the possibilities of undertaking optimization actions in the safety management process. A mathematical model was formulated for this issue. An algorithm solving the problem of personnel allocation is presented. The proposed analysis is the starting point for determining the risk when using multi-station work.

15

Managing the effectiveness of machines during the assignment of tasks to the production

Krynke Marek

Quality Production Improvement - QPI

|

2020

|

Vol. 2, Iss. 1

188--196

EN

The allocation of production tasks to specific production resources is an important part of preparing the manufacturing process. The amount of profit and costs incurred depends on this division. The efficiency of production resources depends not only on the technologies used, but also on the tasks that will be carried out on them. Therefore, the management of machine efficiency includes both an evaluation (analytical and assessment undertaking, e.g. OEE) and planning activities aimed at maximizing the efficiency of machines by appropriately assigning production tasks to them. The article presents the problem of the allocation of production of various products to various production resources, including the efficiency of the use of machines and devices, emphasizing the possibilities of undertaking optimization actions in the cost management process. A mathematical model was formulated for this issue. An algorithm solving the problem of allocation of production tasks is presented. The solution was obtained using the Octave computing environment.

16

A linear programming methodology for approximate dynamic programming

Díaz Henry, Sala Antonio, Armesto Leopoldo

International Journal of Applied Mathematics and Computer Science

|

2020

|

Vol. 30, no. 2

363--375

EN

The linear programming (LP) approach to solve the Bellman equation in dynamic programming is a well-known option for finite state and input spaces to obtain an exact solution. However, with function approximation or continuous state spaces, refinements are necessary. This paper presents a methodology to make approximate dynamic programming via LP work in practical control applications with continuous state and input spaces. There are some guidelines on data and regressor choices needed to obtain meaningful and well-conditioned value function estimates. The work discusses the introduction of terminal ingredients and computation of lower and upper bounds of the value function. An experimental inverted-pendulum application will be used to illustrate the proposal and carry out a suitable comparative analysis with alternative options in the literature.

17

Pipe-lining Dynamic Programming Processes in Order to Synchronize Energy Production and Consumption

Bendali Fariha, Quilliot Alain, Mole Kamga Eloise, Mailfert Jean, Toussaint Hélène

Annals of Computer Science and Information Systems

|

2020

|

Vol. 21

303--306

EN

Synchronizing heterogeneous processes remains a difficult issue in Scheduling area. Related ILP models are in trouble, because of large gaps induced by rational relaxation. We propose here a pipe-line decomposition of a dynamic programming process for energy production and consumption scheduling, and describe the way related sub-processes interact in order to achieve efficient synchronization.

18

An Experimental Study on Symmetry Breaking Constraints Impact for the One Dimensional Bin-Packing Problem

Hadj Salem Khadija, Kieffer Yann

Annals of Computer Science and Information Systems

|

2020

|

Vol. 21

317--326

EN

We consider the classical One-Dimensional Bin Packing Problem (1D-BPP), an NP-hard optimization problem, where, a set of weighted items has to be packed into one or more identical capacitated bins. We give an experimental study on using symmetry breaking constraints for strengthening the classical integer linear programming proposed to optimally solve this problem. Our computational experiments are conducted on the data-set found in BPPLib and the results have confirmed the theoretical results.

19

Heuristic method of assembly scheduling of multi-option products for assembly line with intermediate buffers

Magiera Marek

Technologia i Automatyzacja Montażu

|

2019

|

nr 3

4--10

EN

The method intended to build the possibly shortest assembly schedules is presented in the paper. The method is constructed for assembly lines without parallel machines. The intermediate buffers are located between the assembly machines. A flow of products of different types is unidirectional in the assembly line. The proposed method is a heuristic because the fixed tasks are characterized by a large size and considerable computational complexity. It is a relaxation heuristic. The linear mathematical model is used in the mentioned heuristic method. The method is constructed for assembly scheduling for multioption products. Assembly of various types of products in different variants is the reply to the contemporary challenges faced by the manufacturers, who try to satisfy the demands of individual customers. The planned downtimes of machines are regarded in the method, for example downtime of machines connected with maintenance. The results of computational experiments with proposed heuristic method are presented. The described heuristic method is compared with optimal method based on the integer programming. The fixed makespans and computational fixed times using heuristic method and optimal method are compared.

PL

Przedstawiono metodę przeznaczoną do budowy najkrótszych harmonogramów montażu produktów w liniach montażowych bez maszyn równoległych. Konfiguracja linii montażowej uwzględnia obecność buforów międzyoperacyjnych pomiędzy maszynami. Przepływ równocześnie montowanych produktów różnych typów jest jednokierunkowy. Ze względu na rozmiary rozwiązywanych problemów i związaną z tym złożoność obliczeniową metoda jest heurystyką. Jest to heurystyka relaksacyjna, w której wykorzystano model zadania programowania liniowego. Metoda dotyczy produktów wielowariantowych. Produkty danego typu mogą różnić się wariantami wykonania - specyficznymi cechami, uwzględniającymi wymagania odbiorców. Metodę wyróżnia także uwzględnienie planowanych przestojów maszyn, np. przeznaczonych na konserwację. Zaprezentowano wyniki eksperymentów obliczeniowych, za pomocą których oceniono jakość opracowanej metody. Długości harmonogramów wyznaczanych za pomocą przedstawionej metody porównano z długościami harmonogramów optymalnych, znanymi dzięki zastosowaniu modelu zadania programowania całkowitoliczbowego. Porównano także czasochłonność obliczeń.

20

Analysis of the problem of staff allocation to work stations

Krynke Marek, Mielczarek Krzysztof, Vaško Alan

Quality Production Improvement - QPI

|

2019

|

Vol. 1, Iss. 1

545--550

EN

The proper organization of work is to set the workflow to the slightest effort of man and machine operation to obtain maximum results. The article presents the problem of the allocation of personnel that occurs in a real company. The mathematical model for this issue was formulated. An algorithm solving the problem of personnel allocation is presented. The proposed analysis is a starting point for determining the production capacity and load of each workstation, which is particularly important when using multi-station work and balancing the production line.