Problem 2 20 points dairy cows in most countries

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Problem 2. [20 points] 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 r d 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 m c 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 p d,c pounds in demand month d per cow calving in month c . Formulate a linear program to determine a minimum total-cost calving schedule. Clearly define in words what your decision variables represent. In the LP, label each constraint with a keyword for what the constraint corresponds to. You do not need to solve the (I)LP. Clarification: “at a loss of $ b ” means $ b per pound. Problem 3. [20 points total] A construction contractor has undertaken a job with 7 major tasks. Some of the tasks can begin at any time, but others have predecessors that must be completed first. The following table shows those predecessor task numbers, together with
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