An optimization framework for solving capacitated multi-level lot-sizing problems with backlogging

This paper proposes two new mixed integer programming models for capacitated multi-level lot-sizing problems with backlogging, whose linear programming relaxations provide good lower bounds on the optimal solution value. We show that both of these strong formulations yield the same lower bounds. In addition to these theoretical results, we propose a new, effective optimization framework that achieves high quality solutions in reasonable computational time. Computational results show that the proposed optimization framework is superior to other well-known approaches on several important performance dimensions

On the equivalence of strong formulations for capacitated multi-level lot sizing problems with setup times

Several mixed integer programming formulations have been proposed for modeling capacitated multi-level lot sizing problems with setup times. These formulations include the so-called facility location formulation, the shortest route formulation, and the inventory and lot sizing formulation with (l,S) inequalities. In this paper, we demonstrate the equivalence of these formulations when the integrality requirement is relaxed for any subset of binary setup decision variables. This equivalence has significant implications for decomposition-based methods since same optimal solution values are obtained no matter which formulation is used. In particular, we discuss the relax-and-fix method, a decomposition-based heuristic used for the efficient solution of hard lot sizing problems. Computational tests allow us to compare the effectiveness of different formulations using benchmark problems. The choice of formulation directly affects the required computational effort, and our results therefore provide guidelines on choosing an effective formulation during the development of heuristic-based solution procedures

Modelling the Demand for Long-term Care to Optimise Local Level Planning

Long-term care (LTC) includes the range of health, social and voluntary support services provided to those with chronic illness, physical or mental disability. LTC has been widely studied in the literature, in particular due to concerns surrounding how future demographic shifts may impact the LTC system’s ability to cater to increasing amounts of patients not withstanding what the future cost impact might be. With that said, few studies have attempted to model demand at the local level for the purposes of informing local service delivery and organisation. Many developing countries with mature and developed systems of LTC in place are under pressure to reduce health care spend, whilst delivering greater value for money. We suggest that the lack of local studies in LTC stems from the lack of a strong case for the benefits of demand modelling at the local level in combination with low quantity and incomplete social care data. We propose a mathematical model to show how savings may be generated under different models of commitment with third party providers. Secondly, we propose a hybrid-fuzzy demand model to generate estimates of demand in the short to medium term that can be used to inform contract design based on local area needs – such an approach we argue is more suited to problems in which historic activity is incomplete or limited. Our results show that commitment models can be of great use to local health care planners with respect to lowering their care costs, at the same time our formulation had wider generic applicability to procurement type problems where commitment size in addition to the timing of commitments needs to be determined