definition of linear programming

Question : Old McDonald's Farms owns a 3600-acre field. They want to plant Iceberg Lettuce and romaine lettuce. The U.S Department of Agriculture recommends. No more than 2000 acres of Iceberg lettuce and no more than 2200 acres of. Romaine is planted in order to reduce loss due to disease. Current contracts call for at least 1200 acres of each is planted. Old McDonald's Farms earns a profit of $200 per acre for Iceberg lettuce and $250 per acre for Romaine lettuce. As the farm foreperson, you are to deter..

Answer : Let's plot this on a graph paper. Let x axis represent Iceberg lettuce planted in acres, Let y axis represent Romaine lettuce planted in acres. These are the constraints: x + y <= 3600 (you want to make use of the whole field) 1200 <= x <= 2000 (must be more than 1200 but less than 2000), similarly, 1200 <= y <= 2200 x >= 0, y>=0 Now you can plot these lines on a graph and shade accordingly. x=0, y=0, y=-x+3600 is a triagle with formed by (0,0), (3600,0), (0,3600). Area for x and y can also be shaded. Now profit, p = 200x + 250y Let p = 0, then y = -4x/5 So, you need to shift this line until it hits the top most point of the shaded area for the maximum profit, and if you shift it to the lowest point of the shaded area, it will be the minimum profit. By calculation, the maximum point is when you plant maximum amount of Romaine x = 2200, then y = 1400 as x + y = 3600. So profit (maximum) = 200(1400) + 250(2200) = 830000 Bonus = 3% of profits = 3% x 830000 = ....   More from Yahoo Answers

Question : I'm having quite a lot of trouble with this question. a) State what it means for a subset C of R^n to be a convex. b) The feasible region A for a linear programming problem is given by the constraints 2x_1+3x_2+4x_3 less than12 2x_1+x_2+3x_3 less than 9 5x_1-5x_2+x_3 = 10 x_1, x_2, x_3 greater than 0 Show, using your definition from part (a), that the region A is convex. so for (a), could I say, the subset C of R^n is convex if for every pair of points within the R^n and every poin..

Answer : a) Here is the definition: a subset C of R is convex if for every pair of points within C all points on the connecting straight line segment that joins them also belongs to C, or symbolically: if X' and X" are n-dimensional points in C, then every linear combination X = k'X' + k"X" /k' 0, k" 0, k' + k" = 1/ also belongs to C ("convex linear combination" describing the connecting segment). b) The following statements are easy to prove using the above definition: - every hyperplane (a set of solutions of a linear equation or inequality) in R is a convex subset; - the intersection of convex subsets of R is also convex. The feasible region A, being an intersection of hyperplanes and semi-spaces is convex...   More from Yahoo Answers