This paper addresses a combinatorial optimization problem (COP), namely a variant of the (standard) matrix chain product (MCP) problem where the matrices are square and either dense (i.e. full) or lower/upper triangular. Given a matrix chain of length n, we first present a dynamic programming algorithm (DPA) adapted from the well known standard algorithm and having the same O(n 3) complexity. We then design and analyse two optimal O(n) greedy algorithms leading in general to different optimal solutions i.e. chain parenthesizations. Afterwards, we establish a comparison between these two algorithms based on the parallel computing of the matrix chain product through intra and inter-subchains coarse grain parallelism. Finally, an experimental study illustrates the theoretical parallel performances of the designed algorithms.

We consider a system consisting of two not necessarily identicalexponential servers having a common Poisson arrival process. Uponarrival, customers inspect the first queue and join it if it isshorter than some threshold n. Otherwise, they join the secondqueue. This model was dealt with, among others, by Altman et al. [Stochastic Models 20 (2004) 149–172].We first derive an explicitexpression for the Laplace-Stieltjes transform of the distributionunderlying the arrival (renewal) process to the second queue. Second,we observe that given that the second serveris busy, the two queue lengths are independent.Third, we develop two computational schemes for thestationary distribution of the two-dimensional Markov process underlying thismodel, one with a complexity of$O(n \log\delta^{-1})$, the other with a complexity of $O(\log n\log^2\delta^{-1})$, where δ is the tolerance criterion.

This paper is motivated by operating self service transport systemsthat flourish nowadays. In cities where such systems have been setup with bikes, trucks travel to maintain a suitable number of bikesper station.It is natural to study a version of the C-delivery TSP defined byChalasani and Motwani in which, unlike their definition, C is partof the input: each vertex v of a graph G=(V,E) has a certainamount xv of a commodity and wishes to have an amount equal toyv (we assume that $\sum_{v\in V}x_v=\sum_{v\in V}y_v$ and allquantities are assumed to be integers); given a vehicle of capacityC, find a minimal route that balances all vertices, that is,that allows to have an amount yv of the commodity on each vertexv.This paper presents among other things complexity results, lowerbounds, approximation algorithms, and a polynomial algorithm whenG is a tree.

Data envelopment analysis (DEA) has been proven as an excellent data-oriented efficiency analysis method for comparing decision making units (DMUs) with multiple inputs and multiple outputs. In conventional DEA, it is assumed that the status of each measure is clearly known as either input or output. However, in some situations, a performance measure can play input role for some DMUs and output role for others. Cook and Zhu [Eur. J. Oper. Res. 180 (2007) 692–699] referred to these variables as flexible measures. The paper proposes an alternative model in which each flexible measure is treated as either input or output variable to maximize the technical efficiency of the DMU under evaluation. The main focus of this paper is on the impact that the flexible measures has on the definition of the PPS and the assessment of technical efficiency. An example in UK higher education intuitions shows applicability of the proposed approach.