Mixed integer formulations using natural variables for single machine scheduling around a common due date

34 pages, 10 figuresWhile almost all existing works which optimally solve just-in-time scheduling problems propose dedicated algorithmic approaches, we propose in this work mixed integer formulations. We consider a single machine scheduling problem that aims at minimizing the weighted sum of earliness tardiness penalties around a common due-date. Using natural variables, we provide one compact formulation for the unrestrictive case and, for the general case, a non-compact formulation based on non-overlapping inequalities. We show that the separation problem related to the latter formulation is solved polynomially. In this formulation, solutions are only encoded by extreme points. We establish a theoretical framework to show the validity of such a formulation using non-overlapping inequalities, which could be used for other scheduling problems. A Branch-and-Cut algorithm together with an experimental analysis are proposed to assess the practical relevance of this mixed integer programming based methods

Dominance inequalities for scheduling around an unrestrictive common due date

The problem considered in this work consists in scheduling a set of tasks on a single machine, around an unrestrictive common due date to minimize the weighted sum of earliness and tardiness. This problem can be formulated as a compact mixed integer program (MIP). In this article, we focus on neighborhood-based dominance properties, where the neighborhood is associated to insert and swap operations. We derive from these properties a local search procedure providing a very good heuristic solution. The main contribution of this work stands in an exact solving context: we derive constraints eliminating the non locally optimal solutions with respect to the insert and swap operations. We propose linear inequalities translating these constraints to strengthen the MIP compact formulation. These inequalities, called dominance inequalities, are different from standard reinforcement inequalities. We provide a numerical analysis which shows that adding these inequalities significantly reduces the computation time required for solving the scheduling problem using a standard solver.Comment: 30 pages, 7 figures and 4 table

Effects and cost of different strategies to eliminate hepatitis C virus transmission in Pakistan: a modelling analysis

Background The WHO elimination strategy for hepatitis C virus advocates scaling up screening and treatment to reduce global hepatitis C incidence by 80% by 2030, but little is known about how this reduction could be achieved and the costs of doing so. We aimed to evaluate the effects and cost of different strategies to scale up screening and treatment of hepatitis C in Pakistan and determine what is required to meet WHO elimination targets for incidence. Methods We adapted a previous model of hepatitis C virus transmission, treatment, and disease progression for Pakistan, calibrating using available data to incorporate a detailed cascade of care for hepatitis C with cost data on diagnostics and hepatitis C treatment. We modelled the effect on various outcomes and costs of alternative scenarios for scaling up screening and hepatitis C treatment in 2018–30. We calibrated the model to country-level demographic data for 1960–2015 (including population growth) and to hepatitis C seroprevalence data from a national survey in 2007–08, surveys among people who inject drugs (PWID), and hepatitis C seroprevalence trends among blood donors. The cascade of care in our model begins with diagnosis of hepatitis C infection through antibody screening and RNA confirmation. Diagnosed individuals are then referred to care and started on treatment, which can result in a sustained virological response (effective cure). We report the median and 95% uncertainty interval (UI) from 1151 modelled runs. Findings One-time screening of 90% of the 2018 population by 2030, with 80% referral to treatment, was projected to lead to 13·8 million (95% UI 13·4–14·1) individuals being screened and 350 000 (315 000–385 000) treatments started annually, decreasing hepatitis C incidence by 26·5% (22·5–30·7) over 2018–30. Prioritised screening of high prevalence groups (PWID and adults aged ≥30 years) and rescreening (annually for PWID, otherwise every 10 years) are likely to increase the number screened and treated by 46·8% and decrease incidence by 50·8% (95% UI 46·1–55·0). Decreasing hepatitis C incidence by 80% is estimated to require a doubling of the primary screening rate, increasing referral to 90%, rescreening the general population every 5 years, and re-engaging those lost to follow-up every 5 years. This approach could cost US$8·1 billion, reducing to$3·9 billion with lowest costs for diagnostic tests and drugs, including health-care savings, and implementing a simplified treatment algorithm. Interpretation Pakistan will need to invest about 9·0% of its yearly health expenditure to enable sufficient scale up in screening and treatment to achieve the WHO hepatitis C elimination target of an 80% reduction in incidence by 2030. Funding UNITAID

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