Multi-Period Capacitated Lot Sizing with Variable Batch Sizes

In the production planning for strong seasonal demand products, it is uneconomical to configure the supply chain for throughputs equivalent to the demand peaks. Instead, a holistic approach to supply chain optimisation is adopted where forward demand forecasts drive the production planning process. There is a considerable amount of research literature available for the various types of lot-sizing models involved in production planning. It is generally assumed in these models that a continuous production system is employed for the relatively high volume, low value products. However, these models are not directly applicable for a large number of specialised products operating in batch mode for increased flexibility on multi-purpose equipments. This research addresses the batch sizing operation within a lot-sizing model in order to derive a simultaneous batch sizing and production planning optimisation. The degrees of freedom in this combined problem are the monthly batchsizes of each product, integer number of batches of each product produced each month, amount of overtime working and outsourcing required in each month as well as the time-varying inventory positions across the chain are manipulated. Values for these are selected to balance the trade-offs in batch costs (each batch produced incurs a fixed charge associated with set-up and cleaning), stock costs (these are proportional to the product batchsizes and the amounts of inventory carried) as well as the overtime and outsourcing costs. In greater detail, the multi-item lot-sizing problem has been extended to incorporate the batch-sizing complexity, where the production costs are aggregated to the batch level, with an additional discrete dispersion cost proportional to the batchsize. The operational requirements for overtime work and outsourcing are presented as vector constraints of each product. Through linearisation of the inherent non-linear product of the integer number of batches and its corresponding batchsize, a mixed-integer linear programming (MILP) model is formulated with a minimum cost objective function. For a large number of products with consumer paints, this single-stage deterministic model is intractable due to the large number of variables involved, despite effective reformulations for tighter linear programming relaxations. Methods have been developed in the research to disaggregate the vector constraints from the batch sizing and production planning model for each product. These implicitly or explicitly utilise decomposition algorithms to reduce computational complexity and solution times. They are able to solve the applied industrial problem.