1. MA3264 Mathematical Modelling Lecture 9 Chapter 7 Discrete Optimization Modelling Continued

2. Theorem 1 p. 257-258 Question Where is objective function max? Increasing direction of objective function Level curves feasible region is blue

3. Theorem 1 p. 257-258 Question Where is objective function max? Increasing direction of objective function

4. Theorem 1 p. 257-258 Question Where is objective function max? Increasing direction of objective function

5. Example 1 page 261 How many tables and how many bookcases should a carpenter make each week to maximize profit? He realizes a profit of $25 per table and $30 per bookcase. He has 690 feet of lumber per week and 120 hours of labor per week. Each table requires 20 feet of lumber and 5 hours of labor. Each bookcase requires 30 feet of lumber and 4 hours of labor. This Carpenter Problem has different parameters from that on page 238

6. Mathematical Formulation Maximize decision variables objective function Subject to constraints explicitly stated common sense

7. Figure 7.12 p. 261 Question Is the feasible region convex?

8. Slack Variables Question SV satisfy which inequalities?

9. Reformulation Question SV satisfy which inequalities? Maximize Subject to

10. Simplex Method Remark At each extreme point at least 2 of these 4 variables equal 0. Always assume constraints But manipulate these

11. Simplex Method These pairs of independent variables depend on the extreme point Start at Move to At each step 1 independent variable and 1 dependent variable change places

12. Simplex Method Start at Move to In step 1 independent variable becomes dependent since it has the largest negative coeff. in equation for the objective variable

13. Simplex Method The variable Start at Move to the eqn. it is in restricts becomes independent since to be smallest.

14. Simplex Method Equations express dependent Start at Move to functions of as  independent at

15. Simplex Method We pivot to express dependent Start at Move to functions of as  independent at

16. Simplex Method In step 2 independent variable Start at Move to dependent since it has the largest negative becomes coeff. in equation for the objective variable

17. Simplex Method The variable Start at Move to becomes independent since the eqn. it is in restrictsto be smallest.

18. Simplex Method We pivot to express dependent Start at Move to functions of as  independent at

19. Simplex Method We stop since in the equation for objective Start at Move to variable all coeff. are nonnegative.

20. Tableau Format Study Example 1 on pages 268-271 Observe that Tableau 0 represents the 3 equations in vufoil 13 Tableau 1 represents the 3 equations in vufoil 15 Tableau 2 represents the 3 equations vufoil in 18

21. Suggested Reading Linear Programming 2: Algebraic Solutions p. 259-263 Linear Programming 3: The Simplex Method p. 263-273

22. Tutorial 9 Due Week 27–31 Oct Problem 1. Modify the Bus Stop Waiting program to write a program to compute the waiting time histogram assuming that people form a queue so that people who arrive first are the first to board a bus. Then RUN this program and compare the waiting times with those without queuing. Problem 2. Do problem 5 on page 259. Suggestion: study how to formulate a Chebyshev approximation problem into a linear program on pages 107-109.

23. Homework 3 Due Friday 31 October 7.4 Project 1 Write a computer program to perform the basic simplex algorithm. The program should print out the sequence of all tableaus from the initial to the final one. It should print out the coordinates of all of the extreme points and the values of the objective function at these points. Then use the program to solve problem 3 on p. 258.