What feasible set of lot sizes for each family in each week will minimize total setup and holding cost? A feasible set of lot sizes will meet the demand during the next three weeks within 40-hour weeks. Assume that a week's demand for a family cannot be split between two weeks.

The master scheduling problem at Scotsburn is to determine the production quantity for the products during each week for the next 13 weeks subject to the limitation of production hours imposed by the aggregate production plan. Because some products have ''joint'' setup, it is more efficient to group them as a family if possible. For example, some products are made from the same base mix (e.g., no sugar) and some use the same tub size (e.g., two litres). Changing base mix on a line requires washing all the equipment, and changing the tub size requires physical adjustment to the filling and packaging equipment.

Either action requires approximately a half-hour, and changing base mix also results in approximately $300 of wasted mix. Exceptions include multiflavour products (e.g., Neapolitan ice cream) which require tight coordination of flavours, and products containing a particular allergen (e.g., peanuts). In these cases, all sizes of these products should be included in their own family. Within a family, a flavour or fruit change requires approximately five minutes.

Scots burn produces tens of families of ice cream and frozen yogurt on three lines. For simplicity, we assume two families, one line, and only three weeks. Suppose that the setup time for each family is one hour, cost of labour involved in changeover is $300 per hour, and cost of mix loss is $300. Holding cost rate is 30 percent per year, cost of one litre of ice cream is $1, and there are 50 workweeks per year. The demand (in production hours) for each family per week (after adjusting for initial inventories and safety stocks) during the next three weeks are as follows:

The aggregate production plan has specified 40-hour weeks (four days of 10 hours each) during these three weeks.

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Family Week 1 Week 2 Week 3 F1 11 12 18 F2 13 16 24