Relating Optimization Problems to Systems of Inequalities and Equalities

Corley, H. W.; Dwobeng, Emma

doi:10.4236/ajor.2020.106016

Cited by 5 publications

(8 citation statements)

References 26 publications

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“…The peak of the load distribution can serve as the upper bound A. Now, if the function 𝑝(𝑥) in the Hayashi's inequality (15) is interpreted as the function 1 − 𝐹 𝑆 (𝑥) and ℎ(𝑥) in ( 15) is interpreted as the probability density distribution 𝑓 𝐿 (𝑥) of the load , the conditions for the validity of the Hayashi's inequality are fulfilled because 1 − 𝐹 𝑆 (𝑥) is non-increasing in the interval [a,b] and 0 ≤ 𝑓 𝐿 (𝑥) ≤ 𝐴 in the interval [a,b].…”

Section: An Application Of the Hayashi's Inequality For Determining T...mentioning

confidence: 99%

“…Algebraic inequalities have numerous applications in mathematics and engineering and discussion related to algebraic inequalities can be found in. [1][2][3][4][5][6][7][8][9][10][11] Algebraic inequalities have also been widely used in engineering [11][12][13][14][15][16][17] . In engineering, algebraic inequalities are commonly used for deriving bounds by using inequalities or expressing design constraints.…”

Section: Introductionmentioning

confidence: 99%

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Probabilistic interpretation of algebraic inequalities related to reliability and risk

Todinov

2023

Quality & Reliability Eng

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The paper explores the probabilistic interpretations of algebraic inequalities and presents several findings. First, the inequality of the additive ratios can be used to increase the probability of an event occurring within a set of mutually exclusive and exhaustive events. The interpretation of this inequality produced a counter‐intuitive result, that for suppliers delivering the same quantity of reliable products alongside unknown numbers of unreliable products, the probability of purchasing a reliable product from a randomly selected supplier is higher than the probability of purchasing a reliable product from the market formed by all suppliers. Next, the paper discusses how averaging the reliabilities of components from different varieties can lead to a significant overestimation of the calculated system reliability, as demonstrated by another algebraic inequality interpretation. Finally, the paper derives tight bounds for the reliability of demand in a load‐strength interference model by interpreting the Hayashi's inequality. Notably, these bounds do not depend on the shape of the load distribution and only require the strength distribution to be known in relatively small vicinities of the lower and upper bound of the load.

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Section: An Application Of the Hayashi's Inequality For Determining T...mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Probabilistic interpretation of algebraic inequalities related to reliability and risk

Todinov

2023

Quality & Reliability Eng

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“…In engineering design, design inequalities have been widely used to express design constraints guaranteeing that the design will perform its required function. More recently, in non-linear programming problems, systems of inequalities have been used to present, non-linear goal and non-linear resource constraints (Corley and Dwobeng 2020). Inequalities have also been used in reliability and risk research to characterise reliability functions (Ebeling 1997;Xie and Lai 1998;Makri and Psilakis 1996;Hill et al 2013;Berg and Kesten 1985;Kundu and Ghosh 2017;Dohmen 2006).…”

Section: Introductionmentioning

confidence: 99%

A general class of algebraic inequalities for generating new knowledge and optimising the design of systems and processes

Todinov¹

2022

Res Eng Design

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A special class of general inequalities has been identified that provides the opportunity for generating new knowledge that can be used for optimising systems and processes in diverse areas of science and technology. It is demonstrated that inequalities belonging to this class can always be interpreted meaningfully if the variables and separate terms of the inequalities represent additive quantities.The meaningful interpretation of a new algebraic inequality based on the proposed general class of inequalities led to developing a light-weight design for a supporting structure based on cantilever beams, reducing the maximum force upon impact, generating new knowledge about the deflection of elastic elements connected in parallel and series and optimising the allocation of resources to maximise the expected benefit. The interpretation of the new inequality yielded the result stating that the deflection of elastic elements connected in parallel is at least 2 n times smaller than the deflection of the same elastic elements connected in series irrespective of the individual stiffness values of the elastic elements.The interpretation of another algebraic inequality from the proposed general class led to a method for decreasing the stiffness of a mechanical assembly by cyclic permutation of the elastic elements building the assembly. The analysis showed that a decrease of stiffness exists only if asymmetry of the stiffness values in the connected elements is present.

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“…More recently, in non-linear programming problems, systems of inequalities have been used to present non-linear goal and non-linear resource constraints. 14 This paper shows that the use of algebraic inequalities in engineering is far reaching and is certainly not restricted to specifying constraints. The meaningful interpretation to generate knowledge and improve the design of systems and processes is a new dimension in the use of algebraic inequalities.…”

Section: Introductionmentioning

confidence: 99%

“…More recently, in non-linear programming problems, systems of inequalities have been used to present non-linear goal and non-linear resource constraints. 14…”

Section: Introductionmentioning

confidence: 99%

Optimised design of systems and processes using algebraic inequalities

Todinov

2021

Proceedings of the Institution of Mechanical Engineers, Part C:

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A method for optimising the design of systems and processes has been introduced that consists of interpreting the left- and the right-hand side of a correct algebraic inequality as the outputs of two alternative design configurations delivering the same required function. In this way, on the basis of an algebraic inequality, the superiority of one of the configurations is established. The proposed method opens wide opportunities for enhancing the performance of systems and processes and is very useful for design in general. The method has been demonstrated on systems and processes from diverse application domains. The meaningful interpretation of an algebraic inequality based on a single-variable sub-additive function led to developing a light-weight design for a supporting structure based on cantilever beams. The interpretation of a new algebraic inequality based on a multivariable sub-additive function led to a method for increasing the kinetic energy absorbing capacity during inelastic impact. The interpretation of a new inequality has been used for maximising the mass of deposited substance during electrolysis.

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Relating Optimization Problems to Systems of Inequalities and Equalities

Cited by 5 publications

References 26 publications

Probabilistic interpretation of algebraic inequalities related to reliability and risk

Probabilistic interpretation of algebraic inequalities related to reliability and risk

A general class of algebraic inequalities for generating new knowledge and optimising the design of systems and processes

Optimised design of systems and processes using algebraic inequalities

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