ECE580 Final Exam December 14, Please leave fractions as fractions, I do not want the decimal equivalents.

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1 ECE580 Final Exam December 14, Name: Score: /100 This exam is closed-book. Calculators may NOT be used. Please leave fractions as fractions, I do not want the decimal equivalents. Cell phones and all other electronic devices must be turned off and placed under your desk. You must show ALL of your work and provide complete justifications for your answers to obtain full credit. Please write on only one side of each page. Extra paper is available from the instructor. Please check your work carefully

2 ECE580 Final Exam December 14, Section I: Simplex Method: Answer the following questions. 1. (10 points) Hint: Read the questions below BEFORE you read this problem statement! Abe s Quarry needs to deliver 50 tons of gravel to customer the state highway department, 40 tons of slate shingles to Best Roofing Contractors, and 70 tons of curb blocks to Colossal Construction. Abe has two trucks: the Peterbuilt truck can haul 20 tons per load and the Volvo can hold 15 tons per load. The Peterbuilt costs $3 per mile driven and the Volvo costs $2 per mile driven. Assume that the number of miles driven is 30 times the number of hours required for the trip. Loads cannot mix two orders. It takes 3 hours round trip (including loading and unloading) to deliver a load to the state highway department, 5 hours to deliver to Best Roofing, and 4 hours to deliver to Colossal Construction. They have 3 drivers, one who works full time (8 hours) and two who work mornings (4 hours) only. Overtime labor costs 1.5 times normal hourly wage. (a) Which variables do we need to consider if we want to minimize the time (in days) that it takes to deliver all three orders, assuming no overtime is permitted? (b) Which variables do we need to consider if we want to minimize the cost of delivering all three orders?

3 ECE580 Final Exam December 14, (20 points) Consider the following linear programming problem. min x 1 + 3x 2 subject to x 1 2 x 1 + x 2 4 x 1 = 2x 2 (1) (a) Express this linear programming problem in standard form. Indicate the type(s) of any new variable(s) that you introduce. (b) Write the associated matrix and identify the columns you will use as your initial basis. (Identify the type of any new variables you add.)

4 ECE580 Final Exam December 14, (20 points) Consider this LP problem min 6λ 1 + 5λ 2 subject to λ 1 + 2λ 2 3 3λ 1 + 4λ 2 7 λ 1, 0 λ 2 0 (2) (a) Find this problem s dual. (b) If the stated problem has optimal cost c what is the optimal cost for its dual problem?

5 ECE580 Final Exam December 14, Section II: Answer the following multiple choice questions. Circle all answers that apply. (If no answer applies, don t circle any of them.) 4. (10 points) For parts (a) through (c), consider the linear programming problem min f(x) = x 1 2x 2 subject to x 2 + x 3 = 4 x 1 + x 2 x 4 = 2. x 0 (3) (a) Which of the vectors provide a feasible solution? i. x A = [ ] T ii. x B = [ ] T iii. x C = [ ] T iv. x D = [ ] T (b) Which of the vectors are feasible and basic? i. x A = [ ] T ii. x B = [ ] T iii. x C = [ ] T iv. x D = [ ] T (c) Which of the vectors are feasible but not basic? i. x A = [ ] T ii. x B = [ ] T iii. x C = [ ] T iv. x D = [ ] T (d) When does one use linear programming? i. When the cost function is linear and the constraints are linear or nonlinear. ii. When the cost function is nonlinear and the constraints are linear. iii. When the cost function is linear and the constraints are nonlinear. iv. When the constraints are linear and the cost is linear or nonlinear.

6 ECE580 Final Exam December 14, Section III: Karmarkar s Algorithm 5. (10 points) Describe how Karmarkar s algorithm differs from the simplex algorithms. 6. (10 points) State Karmarkar s canonical form.

7 ECE580 Final Exam December 14, (5 points) In Karmakar s algorithm, the point a 0 = [ 1 n, 1 n, 1 n (a) above the simplex (b) on the simplex (c) below the simplex. ] T lies 8. (5 points) Karmarkar s algorithm incorporates (a) only the primal problem. (b) only the dual problem. (c) both the primal and the dual. (d) neither. 9. (5 points) Give the formula for the direction vector obtained in Karmarkar s algorithm and explain what the variables in it represent. 10. (5 points) In the augmented matrix B k = [ ADk e T ], what is e T and why is it there?

8 ECE580 Final Exam December 14, Symmetric Form of Duality Primal Dual min c T x max λ T b subject to Ax b subject to λ T A c T x 0 λ 0 Asymmetric Form of Duality Primal Dual min c T x max λ T b subject to Ax = b subject to λ T A c T x 0

ECE580 Exam 1 October 4, Please do not write on the back of the exam pages. Extra paper is available from the instructor.

ECE580 Exam 1 October 4, Please do not write on the back of the exam pages. Extra paper is available from the instructor. ECE580 Exam 1 October 4, 2012 1 Name: Solution Score: /100 You must show ALL of your work for full credit. This exam is closed-book. Calculators may NOT be used. Please leave fractions as fractions, etc.