Algorithms for two-dimensional cutting stock and strip packing problems using dynamic programming and column generation

We investigate several two-dimensional guillotine cutting stock problems and their variants in which orthogonal rotations are allowed. We first present two dynamic programming based algorithms for the Rectangular Knapsack (RK) problem and its variants in which the patterns must be staged. The first algorithm solves the recurrence formula proposed by Beasley; the second algorithm - for staged patterns - also uses a recurrence formula. We show that if the items are not so small compared to the dimensions of the bin, then these algorithms require polynomial time. Using these algorithms we solved all instances of the RK problem found at the OR-LIBRARY, including one for which no optimal solution was known. We also consider the Two-dimensional Cutting Stock problem. We present a column generation based algorithm for this problem that uses the first algorithm above mentioned to generate the columns. We propose two strategies to tackle the residual instances. We also investigate a variant of this problem where the bins have different sizes. At last, we study the Two-dimensional Strip Packing problem. We also present a column generation based algorithm for this problem that uses the second algorithm above mentioned where staged patterns are imposed. In this case we solve instances for two-, three- and four-staged patterns. We report on some computational experiments with the various algorithms we propose in this paper. The results indicate that these algorithms seem to be suitable for solving real-world instances. We give a detailed description (a pseudo-code) of all the algorithms presented here, so that the reader may easily implement these algorithms. (c) 2007 Elsevier B.V. All rights reserved.

Cutpoint selection for discretizing a continuous covariate for generalized estimating equations

We consider consider the problem of dichotomizing a continuous covariate when performing a regression analysis based on a generalized estimation approach. The problem involves estimation of the cutpoint for the covariate and testing the hypothesis that the binary covariate constructed from the continuous covariate has a significant impact on the outcome. Due to the multiple testing used to find the optimal cutpoint, we need to make an adjustment to the usual significance test to preserve the type-I error rates. We illustrate the techniques on one data set of patients given unrelated hematopoietic stem cell transplantation. Here the question is whether the CD34 cell dose given to patient affects the outcome of the transplant and what is the smallest cell dose which is needed for good outcomes. (C) 2010 Elsevier BM. All rights reserved.

An indecomposable Banach space of continuous functions which has small density

Using the method of forcing we construct a model for ZFC where CH does not hold and where there exists a connected compact topological space K of weight omega(1) < 2(omega) such that every operator on the Banach space of continuous functions on K is multiplication by a continuous function plus a weakly compact operator. In particular, the Banach space of continuous functions on K is indecomposable.

On Approximate KKT Condition and its Extension to Continuous Variational Inequalities

In this work, we introduce a necessary sequential Approximate-Karush-Kuhn-Tucker (AKKT) condition for a point to be a solution of a continuous variational inequality, and we prove its relation with the Approximate Gradient Projection condition (AGP) of Garciga-Otero and Svaiter. We also prove that a slight variation of the AKKT condition is sufficient for a convex problem, either for variational inequalities or optimization. Sequential necessary conditions are more suitable to iterative methods than usual punctual conditions relying on constraint qualifications. The AKKT property holds at a solution independently of the fulfillment of a constraint qualification, but when a weak one holds, we can guarantee the validity of the KKT conditions.

A multicentre randomized controlled trial of an empowerment-inspired intervention for adolescents starting continuous subcutaneous insulin infusion : a study protocol

Background. Continuous subcutaneous insulin infusion (CSII) treatment among children with type 1 diabetes is increasing in Sweden. However, studies evaluating glycaemic control in children using CSII show inconsistent results. Omitting bolus insulin doses using CSII may cause reduced glycaemic control among adolescents. The distribution of responsibility for diabetes self-management between children and parents is often unclear and needs clarification. There is much published support for continued parental involvement and shared diabetes management during adolescence. Guided Self-Determination (GSD) is an empowerment-based, person-centred, reflection and problem solving method intended to guide the patient to become self-sufficient and develop life skills for managing difficulties in diabetes self-management. This method has been adapted for adolescents and parents as Guided Self-Determination-Young (GSD-Y). This study aims to evaluate the effect of an intervention with GSD-Y in groups of adolescents starting on insulin pumps and their parents on diabetes-related family conflicts, perceived health and quality of life (QoL), and metabolic control. Here, we describe the protocol and plans for study enrolment. Methods. This study is designed as a randomized, controlled, prospective, multicentre study. Eighty patients between 12-18 years of age who are planning to start CSII will be included. All adolescents and their parents will receive standard insulin pump training. The education intervention will be conducted when CSII is to be started and at four appointments in the first 4 months after starting CSII. The primary outcome is haemoglobin A1c levels. Secondary outcomes are perceived health and QoL, frequency of blood glucose self-monitoring and bolus doses, and usage of carbohydrate counting. The following instruments will be used to evaluate perceived health and QoL: Disabkids, 'Check your health', the Diabetes Family Conflict Scale and the Swedish Diabetes Empowerment Scale. Outcomes will be evaluated within and between groups by comparing data at baseline, and at 6 and 12 months after starting treatment. Results and discussion. In this study, we will assess the effect of starting an insulin pump together with the model of Guided Self-Determination to determine whether this approach leads to retention of improved glycaemic control, QoL, responsibility distribution and reduced diabetes-related conflicts in the family. Trial registration: Current controlled trials: ISRCTN22444034

Mother’s experiences and perceptions of a continuous caring model with fathers after caesarean section : a qualitative study

Background: In Chile, mothers and newborns are separated after caesarean sections. The caesarean section rate in Chile is approximately 40%. Once separated, newborns will miss out on the benefits of early contact unless a suitable model of early newborn contact after caesarean section is initiated. Aim: To describe mothers experiences and perceptions of a continuous parental model of newborn care after caesarean section during mother-infant separation. Methods: A questionnaire with 4 open ended questions to gather data on the experiences and perceptions of 95 mothers in the obstetric service of Sótero Del Rio Hospital in Chile between 2009 and 2012. Data were analyzed using qualitative content analysis. Results: One theme family friendly practice after caesarean section and four categories. Mothers described the benefits of this model of caring. The fathers presence was important to mother and baby. Mothers were reassured that the baby was not left alone with staff. It was important for the mothers to see that the father could love the baby as much as the mother. This model of care helped create ties between the father and newborn during the period of mother-infant separation and later with the mother. Conclusions: Family friendly practice after caesarean section was an important health care intervention for the whole family. This model could be stratified in the Chilean context in the case of complicated births and all caesarean sections. Clinical Implications: In the Chilean context, there is the potential to increase the number of parents who get to hold their baby immediately after birth and for as long as they like. When the mother and infant are separated after birth, parents can be informed about the benefits of this caring model. Further research using randomized control trials may support biological advantages.