#### Applications of Linear Programming

Instructor: Mary Pugh, Sidney Smith room 4058. Office hours Monday 2-3, Friday 1-2, or by appointment.

Prerequisites: MAT 223H or MAT 240H

Text: Bernard Kolman and Robert E. Beck, Elementary Linear Programming with Applications, 2nd edition .

Additional Materials: Eight problem sets are given on the other side of this page. Solutions are available for short-term loan in the Gerstein Science Information Centre, as are copies of all Fall term tests since 1997 (including solutions). The Association of Part-time Undergraduate Students in SS 1089 has old final exams for most courses currently offered at the St. George campus.

Marking Scheme: Two term tests, each worth 20\% of the total mark, and a two-hour final exam worth 60\% of the total mark. Calculators are not allowed during the term tests and the final exam. The times and locations of the term tests will be announced later.

Course Summary (subject to change): review of linear algebra (Chapter 0), introduction to linear programming (Chapter 1), the simplex method (Chapter 2), further topics in linear programming (Chapter 3), the transportation problem (Section 5.1). If time permits, I will present some "real-world" applications of linear programming such as game theory and data-fitting.

Recommended problems: Make sure you can do the following problems. Warning: we may cover additional material, in which case I'll add extra problems to this list.

problem set 1: p 21, #6b. (Additional instructions: (i). Solve for x, y, and z in terms of w. (ii). Solve for x, z, and w in terms of y.) p 21, #9a, p 28, #6c, #8b, p 42, #5d, #6b
problem set 2: p 57, #2, #4. p. 28, #6, #8, p 59, #10. Put these problems in canonical and in standard form.
problem set 3: p 82, #14. p 83, #16. In the preceding questions, replace instructions (b) and (c) with "draw the line z = c^T x = k, where k is the optimal value of z." p 91, #4, #8, #12. p 100, #6, #8.
Supplementary problems:
1. Prove that the set of optimal solutions of any linear programming problem is convex.
2. Prove that the set of objective values which a linear programming problem attains over its feasible region is convex.
3. Give an example of a convex set in R^2 which is not a line segment, and hwihc has (1,0) and (0,2) as its only extreme points.
4. Find all extreme points (in R^3) of the set
{(x1,x2,x3) such that x1 - x2 + x3 = -1, 3 x1 - 2 x2 + 4 x3 = 2, x1 + x2 + 3 x3 = 9, x1 >=0, x2 >= 0, x3 >= 0}.
problem set 4: p 120, #6 (solve the problem which this tableau represents), #8 (insert row labels and solve the problem). p.212, #14, #16, #19. p. 122, #22, #23. p 131, #6.
problem set 5: p. 150 #2, p. 152, #10, #20.
Supplementary problems:
1. Maximize z = -x1 + 5 x2 + 3 x3 subject to the constraints
2 x1 + x2 + x3 = 5, 3 x1 + 2 x2 >= 6, 4 x1 + 3 x2 - x3 <= 7, x1 >= 0, x2 >= 0, x3 >= 0.
2. Maximize z = -9 x1 - 4 x3 subject to the constraints
9 x1 + x2 + x3 >= 27, 3 x1 + x2 + 2 x3 = 9, x1 >= 0, x2 >= 0, x3 >= 0.
problem set 6: p 165, #2, #4. p. 166 #6, p. 183, #9. Modify this problem by replacing "x1, x2, x3 unrestricted" with "x1 >= 0, x2 >= 0, x3 >= 0". p 184, #11.
Supplementary problems pertaining to section 3.3:
Consider the standard problem: Maximize z = -76 x1 + 13 x2 - 11 x3 - 27 x4 + 2 x5, subject to constraints
11 x1 + x2 - x3 - 7 x4 + x5 <= 3, -15 x1 - x2 + 2 x3 + 23 x4 - 3 x5 <= 2, -7 x1 + 2 x2 - 2 x3 - 9 x4 + x5 <= 4, x1, x2, x3, x4, x5 >= 0.
a) insert slack variables x6, x7, x8 for the first, second, and third constraints respectively, and write the initial simplex tableau for the problem.
b) The final simplex tableau for the problem has basic variables x5, x3, x2 (in that order). Without using the simplex method to find it, write the final simplex tableau for the problem.
c) Let w_i (i=1,2,3) be the dual variable associated with the ith primal constraint. What is the optimal solution of the dual problem?
problem set 7: p. 214, #2. p 215, #4, #8. p 223, #2, #4. p 224, #6, #8.
problem set 8: p 325, #8, #10, #11, #12.