Parallel machine scheduling by column generation

Parallel machine scheduling problems concern the scheduling of n jobs on m machines to minimize some function of the job completion times. If preemption is not allowed, then most problems are not only NP-hard, but also very hard from a practical point of view. In this paper, we show that strong and fast linear programming lower bounds can be computed for an important class of machine scheduling problems with additive objective functions. Characteristic of these problems is that on each machine the order of the jobs in the relevant part of the schedule is obtained through some priority rule. To that end, we formulate these parallel machine scheduling problems as a set covering problem with an exponential number of binary variables, n covering constraints, and a single side constraint. We show that the linear programming relaxation can be solved efficiently by column generation, since the pricing problem is solvable in pseudo-polynomial time. We display this approach on the problem of minimizing total weighted completion time on m identical machines. Our computational results show that the lower bound is singularly strong and that the outcome of the linear program is often integral. Moreover, they show that our branch-and-bound algorithm that uses the linear programming lower bound outperforms the previously best algorithm. Keywords and Phrases: parallel machine scheduling, set covering formulation, linear programming, column generation, dynamic programming, total weighted completion time

Column generation strategies and decomposition approaches for the two-stage stochastic multiple knapsack problem

\u3cp\u3eMany problems can be formulated by variants of knapsack problems. However, such models are deterministic, while many real-life problems include some kind of uncertainty. Therefore, it is worthwhile to develop and test knapsack models that can deal with disturbances. In this paper, we consider a two-stage stochastic multiple knapsack problem. Here, we have a multiple knapsack problem together with a set of possible disturbances. For each disturbance, or scenario, we know its probability of occurrence and the resulting reduction in the sizes of the knapsacks. For each knapsack we decide in the first stage which items we take with us, and when a disturbance occurs we are allowed to remove items from the corresponding knapsack. Our goal is to find a solution where the expected revenue is maximized. We use branch-and-price to solve this problem. We present and compare two solution approaches: the separate recovery decomposition (SRD) and the combined recovery decomposition (CRD). We prove that the LP-relaxation of the CRD is stronger than the LP-relaxation of the SRD. Furthermore, we investigate numerous column generation strategies and methods to create additional columns outside the pricing problem. These strategies reduce the solution time significantly. To the best of our knowledge, there is no other paper that investigates such strategies so thoroughly.\u3c/p\u3

Time-indexed formulations for machine scheduling problems : column generation

Time-indexed formulations for machine scheduling problems have received a great deal of attention; not only do the linear programming relaxations provide strong lower bounds, but they are good guides for approximation algorithms as well. Unfortunately, time-indexed formulations have one major disadvantage their size. Even for relatively small instances the number of constraints and the number of variables can be large. In this paper, we discuss how Dantzig-Wolfe decomposition techniques can be applied to alleviate, at least partly, the difficulties associated with the size of time-indexed formulations. In addition, we show that the application of these techniques still allows the use of cut generation techniques

Prognostic value of handgrip strength in people aged 60 years and older: A systematic review and meta-analysis

AIM: The aim of the present study was to systematically review the literature on the predictive value of handgrip strength as a marker for vulnerability. Furthermore, we aimed to update a recent systematic review on the association between handgrip strength and mortality. METHODS: Literature searches using Cochrane, PubMed and Embase databases, and searching reference lists of included studies. Eligible studies were observational longitudinal studies presenting handgrip strength at baseline as an independent variable and its association with cognition, depression, mobility, functional status, hospitalization or mortality at follow up in a general population aged 60 years and older. With respect to mortality, we updated a recent systematic review. RESULTS: We included 34 articles. Most of them involved the association between handgrip strength and cognition (n = 9), functional status (n = 12), mobility (n = 6) or mortality (n = 22), and mainly found a positive relationship, meaning that higher handgrip strength at baseline is protective for declines in these outcome measures. Statistical pooling was carried out for functional status and mortality, with a pooled ratio for functional status of 1.78 (95% CI 1.28-2.48) for categorical variables (high vs low handgrip strength) and 0.95 (95% CI 0.92-0.99) for handgrip strength as a continuous variable. The pooled hazard ratio for mortality was 1.79 (95% CI 1.26-2.55) for categorical variables and 0.96 (95% CI 0.93-0.98) for continuous variables. CONCLUSIONS: Handgrip strength has a predictive validity for decline in cognition, mobility, functional status and mortality in older community-dwelling populations.status: publishe