When is the assignment bound tight for the asymmetric traveling-salesman problem?

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When is the Assignment Bound Tight for the Asymmetric Traveling-Salesman Problem

We consider the probabilistic relationship between the value of a random asymmetric traveling salesman problem AT SP (M) and the value of its assignment relaxation AP (M). We assume here that the costs are given by an n × n matrix M whose entries are independently and identically distributed. We focus on the relationship between P r(AT SP (M) = AP (M)) and the probability pn that any particular entry is zero. If npn → ∞ with n then we prove that AT SP (M) = AP (M) with probability 1-o(1). This is shown to be best possible in the sense that if np(n) → c, c> 0 and constant, then P r(AT SP (M) = AP (M)) < 1 − φ(c) for some positive function φ. Finally, if npn → 0 then P r(AT SP (M) = AP (M)) → 0.

When is the assignment bound tight for the asymmetric traveling-salesman problem?

Abstract: "We consider the probabilistic relationship between the value of a random asymmetric traveling salesman problem ATSP(M) and the value of its assignment relaxation AP(M). We assume here that the costs are given by an n x n matrix M whose entries are independently and identically distributed. We focus on the relationship between Pr(ATSP(M) = AP(M)) and the probability p[subscript n] that any particular entry is zero. If np[subscript n] -> [infinity] with n then we prove that ATSP(M) = AP(M) with probability 1-o(1). This is shown to be best possible in the sense that if np(n) -> c, c > 0 and constant, then Pr(ATSP(M) = AP(M)) 0 then Pr(ATSP(M) = AP(M)) -> 0.

Combinação de abordagens GLSP e ATSP para o problema de dimensionamento e sequenciamento de lotes de produção de suplementos para nutrição animal

In this paper we study the combination of GLSP (General Lot Sizing and Scheduling Problem) and ATSP (Asymmetric Travelling Salesman Problem) approaches with sub-tour elimination and patching to a lot sizing and sequencing problem in the animal nutrition industry. This problem consists of deciding the lots size for each product as well the production sequence of the lots, while meeting demand without backlogs and minimizing production and inventory costs. The coordination of these decisions is a challenge for production scheduling in this industry as the setup times are sequence dependent. The ATSP approaches are compared with relax-and-fix approaches applied to the GLSP (General Lot-sizing and Scheduling Problem) formulated in previous research, using real data from an animal nutrition plant in Sao Paulo state. Portuguese: Neste artigo estudamos a combinação de abordagens GLSP (General Lot Sizing and Scheduling Problem) e ATSP (Asymmetric Travelling Salesman Problem) para o problema de dimensionamento e sequenciamento de lotes na indústria de nutrição animal. Este problema consiste em determinar o tamanho de cada lote de produção para cada produto, assim como a sequência de produção destes lotes, de forma a satisfazer a demanda sem atrasos e minimizar os custos de produção e estoques. Uma dificuldade para a programação da produção nesta indústria é integrar estas decisões, pois os tempos de preparação da linha de produção são dependentes da sequência produtiva e não obedecem a desigualdade triangular. A abordagem proposta é comparada com abordagens relax-and-fix para o modelo GLSP (General Lot-sizing and Scheduling Problem) estudadas em trabalhos anteriores, utilizando dados reais de um estudo de caso de uma fábrica de nutrição animal localizada no interior de São Paulo