Let G =(V, E) be a graph with m edges. For reals p ∈ [0, 1] and q = 1-p, let m;(G) be the minimum of qe(V;) + pe(V;) over partitions V = V;∪ V;, where e(V;) denotes the number of edges spanned by V;. We sh...Let G =(V, E) be a graph with m edges. For reals p ∈ [0, 1] and q = 1-p, let m;(G) be the minimum of qe(V;) + pe(V;) over partitions V = V;∪ V;, where e(V;) denotes the number of edges spanned by V;. We show that if m;(G) = pqm-δ, then there exists a bipartition V;, V;of G such that e(V;) ≤ p;m-δ + p(m/2);+ o(√m) and e(V;) ≤ q;m-δ + q(m/2);+ o(√m) for δ = o(m;). This is sharp for com;lete graphs up to the error term o(√m). For an integer k ≥ 2, let fk(G) denote the maximum number of edges in a k-partite subgraph of G. We prove that if fk(G) =(1-1/k)m + α,then G admits a k-partition such that each vertex class spans at most m/k;-Ω(m/k;) edges forα = Ω(m/k;). Both of the above im;rove the results of Bollob′as and Scott.展开更多
In this work, we examine the impact of crude distillation unit(CDU) model errors on the results of refinery-wide optimization for production planning or feedstock selection. We compare the swing cut + bias CDU model w...In this work, we examine the impact of crude distillation unit(CDU) model errors on the results of refinery-wide optimization for production planning or feedstock selection. We compare the swing cut + bias CDU model with a recently developed hybrid CDU model(Fu et al., 2016). The hybrid CDU model computes material and energy balances, as well as product true boiling point(TBP) curves and bulk properties(e.g., sulfur% and cetane index, and other properties). Product TBP curves are predicted with an average error of 0.5% against rigorous simulation curves. Case studies of optimal operation computed using a planning model that is based on the swing cut + bias CDU model and using a planning model that incorporates the hybrid CDU model are presented. Our results show that significant economic benefits can be obtained using accurate CDU models in refinery production planning.展开更多
基金Supported by NSFC(Grant No.11671087)New Century Programming of Fujian Province(Grant No.JA14028)
文摘Let G =(V, E) be a graph with m edges. For reals p ∈ [0, 1] and q = 1-p, let m;(G) be the minimum of qe(V;) + pe(V;) over partitions V = V;∪ V;, where e(V;) denotes the number of edges spanned by V;. We show that if m;(G) = pqm-δ, then there exists a bipartition V;, V;of G such that e(V;) ≤ p;m-δ + p(m/2);+ o(√m) and e(V;) ≤ q;m-δ + q(m/2);+ o(√m) for δ = o(m;). This is sharp for com;lete graphs up to the error term o(√m). For an integer k ≥ 2, let fk(G) denote the maximum number of edges in a k-partite subgraph of G. We prove that if fk(G) =(1-1/k)m + α,then G admits a k-partition such that each vertex class spans at most m/k;-Ω(m/k;) edges forα = Ω(m/k;). Both of the above im;rove the results of Bollob′as and Scott.
基金supported by the Ontario Research FoundationMc Master Advanced Control ConsortiumImperial Oil
文摘In this work, we examine the impact of crude distillation unit(CDU) model errors on the results of refinery-wide optimization for production planning or feedstock selection. We compare the swing cut + bias CDU model with a recently developed hybrid CDU model(Fu et al., 2016). The hybrid CDU model computes material and energy balances, as well as product true boiling point(TBP) curves and bulk properties(e.g., sulfur% and cetane index, and other properties). Product TBP curves are predicted with an average error of 0.5% against rigorous simulation curves. Case studies of optimal operation computed using a planning model that is based on the swing cut + bias CDU model and using a planning model that incorporates the hybrid CDU model are presented. Our results show that significant economic benefits can be obtained using accurate CDU models in refinery production planning.