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Linear Programming Models: Graphical Method 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 © 2007 Pearson Education Fundamental Theorem of Linear Programming (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. 2 LP The graphical method 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. 3 Example LP Model Formulation: The Product Mix Problem Decision: How much to make of > 2 products? Objective: Maximize profit Constraints: Limited resources 4 Example: Flair Furniture Co. Two products: Chairs and Tables Decision: How many of each to make this month? Objective: Maximize profit 5 Flair Furniture Co. Data Tables Chairs (per table) (per chair) Profit Contribution $7 $5 Hours Available Carpentry 3 hrs 4 hrs 2400 Painting 2 hrs 1 hr 1000 Other Limitations: • Make no more than 450 chairs • Make at least 100 tables 6 Decision Variables: T = Num. of tables to make C = Num. of chairs to make Objective Function: Maximize Profit Maximize $7 T + $5 C 7 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 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 Model Summary z = 7T + 5C max! (profit) Subject to the constraints: 3T + 4C < 2400 (carpentry hrs) 2T + 1C < 1000 (painting hrs) T C < 450 (max # chairs) > 100 (min # tables) T, C > 0 (nonnegativity) 10 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. LP-2D.avi 11 Carpentry Constraint Line C 3T + 4C = 2400 Infeasible > 2400 hrs 600 Intercepts (T = 0, C = 600) (T = 800, C = 0) Feasible < 2400 hrs 0 0 800 T 12 C 1000 Painting Constraint Line 2T + 1C = 1000 600 Intercepts (T = 0, C = 1000) (T = 500, C = 0) 0 0 500 800 T 13 Max Chair Line C 1000 C = 450 Min Table Line 600 450 T = 100 Feasible 0 Region 0 100 500 800 T 14 C Objective Function Line z = 7T + 5C Profit 500 Optimal Point (T = 320, C = 360) 400 300 200 100 0 15 0 100 200 300 400 500 T New optimal point T = 300, C = 375 C Additional Constraint Need at least 75 more chairs than tables 500 400 T = 320 C = 360 No longer feasible C > T + 75 or 300 C – T > 75 200 100 0 16 0 100 200 300 400 500 T LP-2D conclusion 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. • Optimal Solution: The corner point with the best objective function value is optimal 17 LP-2D with MatLab® LINEAR PROGRAMMING WITH MATLAB course by Edward Neuman Department of Mathematics Southern Illinois University at Carbondale 18 Function drawfr 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 (or 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 or >0 It is possible to read the vertex coordinates in the Figure window by activating in the menu bar: Tools >>> Data Cursor 19 Function drawfr Cautions in its use function drawfr(c, A, rel, b) % % % min (max)z = c*x Subject to Ax <= b (or 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 ai1 x1 + ai2 x2 +.....+ aij .xj+.....+ ain xn = bi must be transformed in 2 constraints of the type ai1 x1 + ai2 x2 +.....+ aij .xj+.....+ ain xn ≤ bi’ ai1 x1 + ai2 x2 +.....+ aij .xj+.....+ ain xn ≥ bi’’ with bi’ ≈< bi ≈< bi’’ 20 LP-2D aids from Internet http://www.authorstream.com/Presentation/bsndev-242949-linearprogramming-entertainment-ppt-powerpoint/ http://www.authorstream.com/Presentation/bsndev-242950-linearprogramming-example-2-entertainment-ppt-powerpoint/ 21 LP-2D aids from a movie LP-2D.avi 22