2. Linear Programming Models: Graphical Method © 2007 Pearson Education from the companion CD - Chapter 2 of the book: Balakrishnan, Render, and Stair, “Managerial Decision Modeling with Spreadsheets”, 2nd ed., Prentice-Hall, 2007 http://www.stmartin.edu/ Rev. 2.3 by M. Miccio on January 20, 2015

3. Fundamental Theorem of Linear Programming 2 (stated here in two variables) A linear expression ax + by, defined over a closed bounded convex set S whose sides are line segments, takes on its maximum value at a vertex of S and its minimum value at a vertex of S. If S is unbounded, there may or may not be an optimum value, but if there is, it occurs at a vertex.

4. LP The graphical method 3 Using a graphical presentation we can represent all the constraints, the objective function, and the three types of feasible points. from the companion CD of the book: Lawrence and Pasternack, “Applied Management Science: Modeling, Spreadsheet Analysis, and Communication for Decision Making”, 2nd Edition, © 2002 John Wiley & Sons Inc.

5. Example LP Model Formulation: The Product Mix Problem Decision: How much to make of > 2 products? Objective: Maximize profit Constraints: Limited resources 4

6. Example: Flair Furniture Co. Two products: Chairs and Tables Decision: How many of each to make this month? Objective: Maximize profit 5

7. Flair Furniture Co. Data Tables (per table) Chairs (per chair) Hours Available Profit Contribution $7$5 Carpentry3 hrs4 hrs2400 Painting2 hrs1 hr1000 Other Limitations: Make no more than 450 chairs Make at least 100 tables 6

8. Decision Variables: T = Num. of tables to make C = Num. of chairs to make Objective Function: Maximize Profit Maximize $7 T + $5 C 7

9. Constraints: Have 2400 hours of carpentry time available 3 T + 4 C < 2400 (hours) Have 1000 hours of painting time available 2 T + 1 C < 1000 (hours) 8

10. More Constraints: Make no more than 450 chairs C < 450 (num. chairs) Make at least 100 tables T > 100 (num. tables) Nonnegativity: Cannot make a negative number of chairs or tables T > 0 C > 0 9

11. Model Summary z = 7T + 5Cmax!(profit) Subject to the constraints: 3T + 4C < 2400 (carpentry hrs) 2T + 1C < 1000 (painting hrs) C < 450 (max # chairs) T > 100 (min # tables) T, C > 0 (nonnegativity) 10

12. Graphical Method Graphing an LP model helps provide insight into LP models and their solutions:  A straight line is plotted in place of each disequation  A convex and bounded set (hopefully) is generated  An ideal line, that is a family of parallel lines, is drawn to represent the objective function  The optimum is found at the interception of the ideal line with a vertex  While this can only be done in two dimensions, the same properties apply to all LP models and solutions. 11 LP-2D.avi

13. Carpentry Constraint Line 3T + 4C = 2400 Intercepts (T = 0, C = 600) (T = 800, C = 0) 0 800 T C 600 0 Feasible < 2400 hrs Infeasible > 2400 hrs 3T + 4C = 2400 12

14. Painting Constraint Line 2T + 1C = 1000 Intercepts (T = 0, C = 1000) (T = 500, C = 0) 0 500 800 T C 1000 600 0 2T + 1C = 1000 13

15. 0 100 500 800 T C 1000 600 450 0 Max Chair Line C = 450 Min Table Line T = 100 Feasible Region 14

16. 0 100 200 300 400 500 T C 500 400 300 200 100 0 Objective Function Line z = 7T + 5C Profit 7T + 5C = $2,100 7T + 5C = $4,040 Optimal Point (T = 320, C = 360) 7T + 5C = $2,800 15

17. 0 100 200 300 400 500 T C 500 400 300 200 100 0 Additional Constraint Need at least 75 more chairs than tables C > T + 75 or C – T > 75 T = 320 C = 360 No longer feasible New optimal point T = 300, C = 375 16

18. LP-2D conclusion Optimal Solution: The corner point with the best objective function value is optimal 17 Extreme Point theorem : Any LP problem with a nonempty bounded feasible region has an optimal solution; moreover, an optimal solution can always be found at an (or at least one) Corner Point (extreme point) of the problem's feasible region.

19. LP-2D with MatLab® 18 LINEAR PROGRAMMING WITH MATLAB course by Edward Neuman Department of Mathematics Southern Illinois University at Carbondale

20. Function drawfr 19 function drawfr(c, A, rel, b) % Graphs of the feasible region and the line level % of the LP problem with two legitimate variables % % min (max)z = c*x % Subject to Ax = b), % x >= 0 % Enter a sequence of instructions like these into the COMMAND WINDOW: %c=[1 2]; %A=[-1 3; 1 1; 1 -1; 1 3; 2 1]; %rel=' >'; %b=[10; 6; 2; 6; 4]; % NB: % b must be a COLUMN vector % components of b vector can % indifferently be 0 It is possible to read the vertex coordinates in the Figure window by activating in the menu bar: Tools >>> Data Cursor

21. Function drawfr Cautions in its use 20 function drawfr(c, A, rel, b) % min (max)z = c*x % Subject to Ax = b), % x >= 0  Negative value of a resource drawfr accepts one or more negative resource in the vector b  Unbounded Feasible Region drawfr is currently unable of drawing an unbounded region  Equality constraints A constraint of the type a i1 x 1 + a i2 x 2 +.....+ a ij.x j +.....+ a in x n = b i must be transformed in 2 constraints of the type a i1 x 1 + a i2 x 2 +.....+ a ij.x j +.....+ a in x n ≤ b i ’ a i1 x 1 + a i2 x 2 +.....+ a ij.x j +.....+ a in x n ≥ b i ’’ with b i ’ ≈< b i ≈< b i ’’

22. LP-2D aids from Internet 21 http://www.authorstream.com/Presentation/bsndev-242949-linear- programming-entertainment-ppt-powerpoint/ http://www.authorstream.com/Presentation/bsndev-242950-linear- programming-example-2-entertainment-ppt-powerpoint/

23. LP-2D aids from a movie 22 LP-2D.avi