Dairy cows in most countries calve on a regular annual basis. Their milk output

Question

Dairy cows in most countries calve on a regular annual basis. Their milk output varies over the year accordingly, with a peak reached a few months after calving, followed by a decline to almost zero in the tenth month. Knowing these facts, farmers in an agricultural cooperative are trying to plan calving months c = 1, 2, . . . , 12 to make it easier to meet contractual seasonal milk demands rd pounds in months d = 1, . . . , 12. Government regulations do not allow milk to be discarded. Any milk produced beyond these demands must be sold on the bulk market at a loss of $b. The annual cost mc of maintaining a cow calving in month c varies significantly over the year because low-cost grazing is available only in certain seasons. From scientific studies the farmers can estimate the yield pd,c pounds in demand month d per cow calving in month c.

Formulate a linear program to determine a minimum total-cost calving schedule.

Explanation / Answer

Let Xc = Number of cows calved in month c

Mc = Cost of maintaining a cow hat will calf in month c

Yd = Excess milk produced in month d

b = Loss $b

Rd = Seasonal milk demands (pounds) in months

P dc = Yield in demand month d per cow calving in month c

The LPP model will be as shown below:

Minimize Z =  Xc * Mc +  Yd * b

Subject to

Xc, P dc, Yd, Rd >= 0

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