Scotsburn Dairy's plant in Truro, Nova Scotia, makes over $300$ types and sizes of ice cream and frozen yogurt products. The process is in two stages. In the first stage, a type of mix such as premium white is made in one of the two large pasteurizing vats $($with $5,400$ litre and $7,500$ litre capacities$),$ homogenized, cooled, and stored in one of eight storage tanks. In the second stage, the mix is fed into one of the three lines where flavour$($s$)/$fruit$($s$)$ are added and the mix is packaged in containers $($tubs$)$ of various sizes. All three lines feed into the spiral freezer and then the tubs are stored in the on$-$site warehouse. For simplicity, we assume that all the products can be aggregated and that the volume of the product determines its production speed and cost. The plant manager has received the following aggregate demand forecasts $($in units, a unit equals $10,000$ litres$)$ for the next $12$ months: A major consideration is product changeover time. This is significant as a mix is changed and as a tub size is changed. The manager estimates that out of

weeks per month, on average one week is spent for changeovers. Although production speed of a line in the second stage varies by product $($and not as a linear function of products' tub size$),$ the spiral freezer's speed is based on volume of ice cream going through it $(10,000$ litres per hour$).$ For some products, the line is a bottleneck, but for most products, the spiral freezer is$.$ For simplicity, we assume that the production capacity of the plant is determined by the capacity of the spiral freezer $($one unit or $10,000$ litres per hour$).$ The plant manager can use the following labour schemes: $40$ hours a week $(1$ shift; $10$ hours a day Monday$-$Thursday$)$; $50$ hours a week $($

hours of overtime on Fridays$)$; and $80$ hours a week $(2$ shifts; $10$ hours a day each Monday$-$Thursday$).$ The initial inventory of products is $210$ units $($i$.$e$.,2,100,000$ litres$).$ The manager would like to have safety stocks at the end of each month equal to the next month's forecast demand. Also, the ending inventory next December should be at least $170$ units. Scotsburn has on$-$site warehouse space for $6,500$ pallets of products. Each pallet's load differs by tub size. For example, $550$ two$-$litre tubs can be placed on a pallet. For simplicity, we will use this as a measure of on$-$site warehouse capacity $($i$.$e$.,$

litres or $715$ units$)$ The production cost of the $40-$hour$-$a$-$week labour scheme $($requiring three freezer operators, nine freezer packers, one freezer bypass, five palletizers, one quality control lab supervisor, one lab technician, two wash$-$up staff, four mixers, one production supervisor, one maintenance manager, and four maintenance workers$)$ is

per month. Overtime production cost is

times regular time production. A temp worker's production cost is the same as a permanent worker's. The average hiring cost of $32$ temp workers required to run the plant is

each, charged to the first month of employment. Value of one litre of product is

$,$ cost of capital is $11$ percent per year, each pallet holds $1,100$ litres, and warehousing cost is

per pallet per month. From these, inventory holding cost per unit $(10,000$ litres$)$ per month is

$.$ Shortage$/$back order is highly undesirable. Assume a cost of

per unit per month back ordered. Question What is the minimum cost aggregate production plan for Scotsburn Dairy? $($Hint: Use the second shift in May and June, supplemented by overtime in several months.$)$

can you provide numerical solutions?