This paper studies the batch sizing scheduling problem with earliness and tardiness penalties which is closely related to a two-level supply chain problem.In the problem,there are K customer orders,where each customer...This paper studies the batch sizing scheduling problem with earliness and tardiness penalties which is closely related to a two-level supply chain problem.In the problem,there are K customer orders,where each customer order consisting of some unit length jobs has a due date.The jobs are processed in a common machine and then delivered to their customers in batches,where the size of each batch has upper and lower bounds and each batch may incur a fixed setup cost which can also be considered a fixed delivery cost.The goal is to find a schedule which minimizes the sum of the earliness and tardiness costs and the setup costs incurred by creating a new batch.The authors first present some structural properties of the optimal schedules for single-order problem with an additional assumption(a):The jobs are consecutively processed from time zero.Based on these properties,the authors give a polynomial-time algorithm for single-order problem with Assumption(a).Then the authors give dynamic programming algorithms for some special cases of multiple-order problem with Assumption(a).At last,the authors present some structural properties of the optimal schedules for single-order problem without Assumption(a) and give a polynomial-time algorithm for it.展开更多
An upper bound is established on the parameter Γ -(G) for a cubic graph G and two infinite families of 3-connected graphs G k, G * k are constructed to show that the bound is sharp and, moreover, the difference Γ -(...An upper bound is established on the parameter Γ -(G) for a cubic graph G and two infinite families of 3-connected graphs G k, G * k are constructed to show that the bound is sharp and, moreover, the difference Γ -(G * k)-γ s(G * k) can be arbitrarily large, where Г -(G * k) and γ s(G * k) are the upper minus domination and signed domination numbers of G * k, respectively. Thus two open problems are solved.展开更多
基金National Nature Science Foundation of China under Grant Nos.11471210and 11171207the Natural Science Foundation of Ningbo City under Grant No.2015A610168the Natural Science Foundation of Zhejiang Province of China under Grant No.LQl3A010010
文摘This paper studies the batch sizing scheduling problem with earliness and tardiness penalties which is closely related to a two-level supply chain problem.In the problem,there are K customer orders,where each customer order consisting of some unit length jobs has a due date.The jobs are processed in a common machine and then delivered to their customers in batches,where the size of each batch has upper and lower bounds and each batch may incur a fixed setup cost which can also be considered a fixed delivery cost.The goal is to find a schedule which minimizes the sum of the earliness and tardiness costs and the setup costs incurred by creating a new batch.The authors first present some structural properties of the optimal schedules for single-order problem with an additional assumption(a):The jobs are consecutively processed from time zero.Based on these properties,the authors give a polynomial-time algorithm for single-order problem with Assumption(a).Then the authors give dynamic programming algorithms for some special cases of multiple-order problem with Assumption(a).At last,the authors present some structural properties of the optimal schedules for single-order problem without Assumption(a) and give a polynomial-time algorithm for it.
文摘An upper bound is established on the parameter Γ -(G) for a cubic graph G and two infinite families of 3-connected graphs G k, G * k are constructed to show that the bound is sharp and, moreover, the difference Γ -(G * k)-γ s(G * k) can be arbitrarily large, where Г -(G * k) and γ s(G * k) are the upper minus domination and signed domination numbers of G * k, respectively. Thus two open problems are solved.
