NAME: 3 MT263 problems Hand in your solutions, with this coversheet, for feedback to the MT263 Homework drop box in the foyer of the Bedford Building (beside the door of room 1-33) by 11am on Wednesday 23rd October. I will keep a record of how many questions you made a decent attempt on. I can provide you with additional feedback if you want it. Many students don't collect their feedback so if you don't want additional feedback, please tick the relevant box below. Otherwise, if you make a decent attempt of N questions, you can request additional feedback on | ~] questions (note the use of the floor function here). Please indicate which questions you'd like feedback on in the relevant box below (if you don't indicate what feedback you want, I'll assume you don't want any). · · Put a tick in this box if you don't want additional written feedback on your solutions. · I attempted questions. Please provide me with feedback on questions YOUR FEEDBACK TO THE LECTURER (e.g., what was hard, interesting, fun, etc. this week): FEEDBACK ON YOUR SOLUTIONS FROM THE MARKER: 1. Consider maximise subject to -x1 + X2 -2x1 - 22 ?-1 -x1 - 2x2 ?-2 x2 ? 1 x1, X2 > 0. (a) Write down the auxiliary problem. (b) Find a feasible dictionary for the auxiliary problem. (c) Use this to find a feasible dictionary for the original problem. (If at some stage you have a choice between x1 and x2 for entering variable, choose x1.) (d) Solve the original problem.
(For this question, you should obtain an optimal value of -3.) 2. Write down a flow chart that describes how to apply the 2-phase simplex method. 3. Shown below are dictionaries from an implementation of phase 1 of the two phase simplex method, and the objective function z of the original LPP. For each of the dictionaries, write down the next single step of the method. The variable x0 was introduced when forming the auxiliary dictionary. (a) z=x1- 2x2 (b) z= x1- 2x2 (c) z= x1- 2x2 (d) z= x1- 2x2 - 13 x1=2-x3-2x4-20 X2=2-x3+2x4+3x5+2x0 w= - x0 Co=2-x1- 2x2 C3=2-x1-2x2+3x4 w =- 2 +x1 + 2x2 w= - 10 C3 =- 2-x1-2x2+ x0 04 =- 3-x1+2x2+ 0 Co=1+x1-2x2 + 2x4 - 25 C3=3+x1+ 2x2 + 9x4 - 9x5 w =- 3 -x1- 6x4 - 3x6 4. Prove the following results from class. (a) The following gives a feasible solution of the auxiliary problem: @1 Cn = 0, and xo = max {| b1 |, . . . , |bm|}. (b) The original problem has a feasible solution the auxiliary problem has a feasible solution with x0 = 0. (c) The original problem has a feasible solution => optimal value of auxiliary problem is 0. 5. Solve the following LPPs. (a) Maximise subject to 2x1 + 3x2 ? 6 2x1 + 5x2 + x2 ?1, x2 ? -4 ? 0 (b) (For this question, you should obtain an optimal value of