Scotsburn Dairy: The MPS Problem

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.

Scotsburn 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:

Family

F$1$

F$2$

Week $1$

$11$

$13$

Week $2$

$12$

$16$

Week $3$

$18$

$24$

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

Question

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.