Math364 - Practice Midterm Exam (Oct 4, 2005) - Optimization Problems, Exams of Linear Algebra

Download Math364 - Practice Midterm Exam (Oct 4, 2005) - Optimization Problems and more Exams Linear Algebra in PDF only on Docsity! Math364 - Practice Midterm Exam (Oct 4, 2005) 1 Principles of Optimization Practice Midterm Examination • The total points (given in parentheses) add up to 105. You will be graded for 100 points. • The mid-term exam will be similar to this exam – but it will not be identical to this practice exam! 1. (15) Solve the following LP graphically. Indicate the feasible region clearly and give the coordi- nates of the optimal solution and the optimal objective function value. max z = 3x1 + 4x2 s.t. − x1 + x2 ≤ 4 x1 + 2x2 ≤ 12 9x1 + 5x2 ≤ 45 x1 urs, x2 ≥ 0 2. (20) A fertilizer distributor has received a special order from a local discount store for 5000 bags of all-purpose fertilizer with specifications 8-14-8 (i.e., it has 8% nitrogen, 14% phosphorus, and 8% potash; the rest of its contents is inert filler). Each bag weighs 25 pounds. The fertilizer will be made by mixing three finished bulk fertilizers (with specifications given below) and inert filler in appropriate quantities. Formulate an LP which the producer can use to minimize the total cost of making the 5000 bags of fertilizer. Ingredient Cost/lb. (cents) Availability (lbs.) 4-8-6 2 60,000 20-20-20 8 40,000 8-12-4 3 50,000 filler 0.05 no limit 3. (20) Alice was given a min-LP, but she wrongly used the criteria for a max-LP and obtained the optimal tableau given below. She actually pivoted x1 into the basis to obtain this tableau. Find the correct optimal tableau that she should have obtained (i.e., the optimal tableau for the min-LP). z x1 x2 x3 x4 rhs 1 0 5 2 0 8 0 1 1 1 0 4 0 0 -2 -1 1 2