Question

Domestic cows can be bought for $200 each but Hybrid cows cost $500 each. The domestic cows produce 30 gallons of milk per week and the Hybrid ones produce 50 gallons of milk per week. A gallon of milk can be sold for $3. A cow costs $100 per week to feed. If the financial constraint is to spend $8000 for cows and the capacity constraint is that total number of cows to be bought cannot exceed 20 cows. Formulate a linear programming model for this problem, and find the optimal combination of domestic and hybrid cows that maximize the total profit earned using the graphic Method

Answer 1

The following variables are defined:

Ti, i = l, ... , 28 =Time, in weeks from

the end of the pretrail period.

A1 i, j = l, ..., 26 =Mean weekly milk

yield over 7 pre-trial

weeks, for the jth

cow.

A2i, j = l, ... , 26 =Mean weekly milk

yield over last 4

weeks of pre-trial

period, for the jth

cow.

A 3 j, j = l, ... , 26 = Mean weekly fatcorrected milk over 7

pre-trial weeks, for

the jth cow.

A4 j, j = l, ... , 26 =Mean weekly fatcorrected milk over

last 4 weeks of pretrial period, for the

jth cow.

W oi = Liveweight of jth cow at the

beginning of the trial period

(in pounds) .

W ii = Liveweight of jth cow in the

ith week of the trial period

(in pounds).

f:j_ Wii = Wii - W<i-!Ji =Change in liveweight from the

previous period

(in pounds). 

0 ii = Concentrate consumed by jth cow in the ith period (pounds per week). Hii =Hay consumed by jth cow in the ith period (pounds per week). Mii =Milk yield of jth cow in the ith period (pounds per week). FOM·ii =Fat-corrected milk yield of the jth cow in the ith period (pounds per week).