In this paper we introduce a new operation for linear programming (LP), called \emph{LP complementation}, which resembles many properties of LP duality. Given a maximisation (resp.~minimisation) LP $P$, we define its ...
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In this paper we introduce a new operation for linear programming (LP), called \emph{LP complementation}, which resembles many properties of LP duality. Given a maximisation (resp.~minimisation) LP $P$, we define its \emph{complement} $Q$ as a specific minimisation (resp.~maximisation) LP which has the \emph{same} objective function as $P$. Our central result is the LP complementation theorem, that establishes the following relationship between the optimal value $\Opt(P)$ of $P$ and the optimal value $\Opt(Q)$ of its complement: $\frac{1}{\Opt(P)}+\frac{1}{\Opt(Q)}=1$. The LP complementation operation can be applied if and only if $P$ has an optimum value greater than 1. As it turns out, LP complementation has far-reaching implications. To illustrate this, we first apply LP complementation to \emph{hypergraphs}. For any hypergraph $H$, we review the four classical LPs, namely \emph{covering} $K(H)$, \emph{packing} $P(H)$, \emph{matching} $M(H)$, and \emph{transversal} $T(H)$. For every hypergraph $H=(V,E)$, we call $\HComplement{H}=(V,\{V\setminus e : e\in E\})$ the \emph{complement} of $H$. For each of the above four LPs, we relate the optimal values of the LP for the dual hypergraph $\HDual{H}$ to that of the complement hypergraph $\HComplement{H}$ (e.g.~$\frac{1}{ \Opt( K(\HDual{H}) ) }+\frac{1}{\Opt( K(\HComplement{H}) ) } = 1$). As a second application of LP complementation, we consider \emph{fractional graph theory}.We prove that the LP for the \Define{fractional in-dominating number} of a digraph $D$ is the complement of the LP for the \Define{fractional total out-dominating number} of the digraph complement $\DComplement{D}$ of $D$. Furthermore we apply the hypergraph complementation theorem to matroids. We establish that the fractional matching number of a matroid coincide with its edge *** result can then be applied to graphic matroids, yielding a formula for the edge toughness of a ***, we derive an alternative proof of the relations
Engineers face significant challenges when determining what geometry provides the most profitable and safe stope for extraction. Several techniques and optimization algorithms have been developed in recent years, but ...
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To address the issue of load recovery after blackout in distribution networks, a linear programming method for fault recovery based on dynamic switching technology is proposed. Firstly, an analytical expression elucid...
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ISBN:
(数字)9798350386127
ISBN:
(纸本)9798350386134
To address the issue of load recovery after blackout in distribution networks, a linear programming method for fault recovery based on dynamic switching technology is proposed. Firstly, an analytical expression elucidating the functional relationship between the adjacency matrix of the network topology and node connectivity is proposed. Based on this, a dynamic model of the network structure is developed to enhance the network reconfiguration strategy. Secondly, considering the dynamic characteristics of non-black start units, a load recovery power adequacy constraint is constructed, and the distribution network fault recovery model is established, which was transformed into a mixed integer linear programming form. Finally, the validity of the islanding strategy is tested based on the optimal scheduling and active and reactive power balance. The effectiveness of this method is validated through a simulation using a 43-bus distribution system example. The results show that the method can effectively coordinates network topology flexibility with distributed resources, hence improving the distribution network's resilience.
The production planning of Fast-Moving Consumer Goods (FMCG) industries is a challenging task due to uncertainty and risk. This study proposes a novel algorithm that combines linear programming and Monte Carlo simulat...
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This paper mainly discusses the influence among various kinds of catalyst combinations and temperature on the rate of ethanol conversion and the selectivity shown in C4 olefin, the catalyst combination and temperature...
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This research aims to provide the best strategic supply chain network design decisions for e-groceries to accommodate growing demand at the minimum cost. The decisions include supply chain network configuration, facil...
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This paper focuses on developing a path planning algorithm for a wheeled mobile robot with actuator constraints. The proposed motion planning approach is divided into two stages: path planning, followed by path follow...
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This paper evaluates the economic impact of short-term and long-term energy storage capacity on power system operation cost. First, the unit commitment (UC) model with short-term and long-term energy storage comprisin...
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The central path method is a crucial technique for solving a wide range of optimization problems. The method relies on an equation solving step, which hits its limit for very large instances in practise. This paper pr...
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We are developing an approach that is an alternative to the Zimmermann approach to solving a multiple objective linear programming problem. We use fuzzy equivalence relations to solve the problem, where fuzzy sets are...
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