CO-350-1025-Midterm_exam

CO-350-1025-Midterm_exam - Ma‘l'k 501. UNIVERSITY OF...

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Unformatted text preview: Ma‘l'k 501. UNIVERSITY OF WATERLOO m {(HQYM diam-L EXAMINATION SPRING TERM 2002 Surname: ——————————————‘—‘——_—' First Name: __——-—————-—-————‘————_— Name: (Signature) Id.#: . [:1 Section 1 8:30 TR LeCture seetlom El Section 2 10:30 MWF Course Number C850 350 Course Title Linear Optimization Sections 1 and 2 Instructor (1) K.Cheung (2) J .Cheriyan Date of Exam June 18, 2002 Time Period 5:00 pm. — 7:00 p.m. Number of Exam Pages 8 pages (including this cover sheet) Exam Type Closed Book INSTRUCTIONS: I. Write your name and Student Id.# in the blanks above. Put a check mark in the box next to your assigned lecture time. 2. Answer each of the questions in the space provided; use the back of the previous page for additional space. 3. You may NOT use a calculator. 1. [16 marks] Solve the following linear programming problem (P) by using the simplex method. [15] ' Use the smallest subscript rule for choosing entering variables, and show all of your work. _ . max1m1ze ~x1 + :52 (P) subject to 2'1 + :122 — 2233 S 5 —2x1+x2+ x3S2 x1 + 5173 S 3 1‘1 7 $2 , $3 2 0 2. [17 marks] (a) Define what is meant by (i) a convex set; (ii) an extreme point of a convex set. (b) Let F be the set of m satisfying — 2:81 + 2x2 + 12345671173 -— 2.234 = 2 $1 + x2 — 9876543$3 + .734 = 3 1:1 7 1'2 7 $3 7 $4 2 0 (i) Find two extreme points of F. Justify your answer. (ii) Find a point of F that is not an extreme point. Justify your answer. [7] 3. [9 marks] Consider the LP problem (P) max 2 : 8x1 + 9232 + 10:33 subject to — 3271 + x2 + 5.23 = ——10 2x1 — 3:63 2 —20 — 71131 + 5313 S —30 $1 , 172 , $3 2 0 (a) Write down an auxiliary problem (Phase 1 problem) for Clearly indicate the artificial variables and the slack variables. How many artificial variables are there? (Do NOT solve the auxiliary problem.) (b) Write down an initial feasible basis B for the auxiliary problem, and write down an initial basic feasible solution :c*. 4. [8 marks] [8] Let A be a 3 X 4 matrix, and let 6 2 [10, 20, 30, OF. Consider the LP problem max 2 = cTr subject to ACE = b a: Z 0. Suppose that the auxiliary problem (Phase 1 problem) has two artificial variables, $5 and $6. The final tableau for Phase 1, with basis {1, 2, 5}, is given below. Write down an initial tableau for Phase 2. (Do NOT solve the Phase 2 problem.) w + 334 + 25% = 0 $1 + 41173 + 5134 = 1 .232 i - 21’3 — .734 + 35%- = 1 — $3 + 3554 + $5 + %$5 = 0 5. [20 marks] (a) Define what is meant by (i) a basis of an m X n matrix A, where A has rank m; (ii) a basic solution of a system A3: = b (where A is as above); (P) max 2 2 0T1: (iii) atableau for the linear programming problem subject to At = b a: 2 0. (b) Consider the following tableau (T) with basis B = {2,3,6}, where [3 is a (real— valued) parameter. Suppose that the initial basis consists of the slack variables $4a$53$6- Z + {131 ‘— $4 —' {1:5 3 0 371 + $2 + 2134 + $5 = 0 — 3:1 + $3 + 2334 — 411:5 = 0 $1 +flm4—m5+xe=0 (i) Suppose that B =1. List all the pairs (away) such that 33;, could be the entering variable and 33,. could be the leaving variable on a simplex iteration beginning from (T). (ii) Suppose that fl = l, and that .134 is the entering variable. Find the leaving variable using the lexicographic rule. Justify your answer. (iii) Suppose that 274 is the entering variable. Find all values of 3 such that the lexicographic rule chooses $6 as the leaving variable. Justify your answer. 6. [20 marks] For each of the following statements, answer whether it is true or false. If true, give a complete explanation, and if false, give a counter—example. (a) If the feasible region of a linear programming problem is unbounded, then the linear programming problem is also unbounded. (b) Every subset of a convex set is convex. (c) The set of optimal solutions of a linear programming problem is a convex set. ((1) If we have a tableau (not necessarily feasible) such that E, S 0 for all j E N, then every'feasible solution (of the LP problem) has objective value 3 17 (recall that 1—) denotes the number on the right hand side of the z—row). (e) In a degenerate iteration of the simplex method, both the old tableau and the new tableau are degenerate. 7. [10 marks] (a) State the Fundamental Theorem of Linear Programming for LP problems in standard equality form. (b) Prove that if the set F = {:L' : A1: = b, at least one extreme point. (Note that A may not have full row rank.) (Hint: Try using part (21).) a: _>_ 0} is non—empty, then it has ...
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CO-350-1025-Midterm_exam - Ma‘l'k 501. UNIVERSITY OF...

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