Chapter Linear Programming Models: Graphical and Computer Methods © 2007 Pearson Education Steps in Developing a Linear Programming (LP) Model 1) Formulation 2) Solution 3) Interpretation and Sensitivity Analysis Properties of LP Models 1) Seek to minimize or maximize 2) Include “constraints” or limitations 3) There must be alternatives available 4) All equations are linear Example LP Model Formulation: The Product Mix Problem Decision: How much to make of > products? Objective: Maximize profit Constraints: Limited resources Example: Flair Furniture Co Two products: Chairs and Tables Decision: How many of each to make this month? Objective: Maximize profit Flair Furniture Co Data Tables Chairs (per table) (per chair) Profit Contribution $7 $5 Hours Available Carpentry hrs hrs 2400 Painting hrs hr 1000 Other Limitations: • Make no more than 450 chairs • Make at least 100 tables Decision Variables: T = Num of tables to make C = Num of chairs to make Objective Function: Maximize Profit Maximize $7 T + $5 C Constraints: • Have 2400 hours of carpentry time available T + C < 2400 (hours) • Have 1000 hours of painting time available T + C < 1000 (hours) More Constraints: • Make no more than 450 chairs C < 450 (num chairs) • Make at least 100 tables T > 100 (num tables) Nonnegativity: Cannot make a negative number of chairs or tables T>0 C>0 Model Summary Max 7T + 5C (profit) Subject to the constraints: 3T + 4C < 2400 (carpentry hrs) 2T + 1C < 1000 (painting hrs) T C < 450 (max # chairs) > 100 (min # tables) T, C > (nonnegativity) Graphical Solution • Graphing an LP model helps provide insight into LP models and their solutions • While this can only be done in two dimensions, the same properties apply to all LP models and solutions Carpentry Constraint Line C 3T + 4C = 2400 Infeasible > 2400 hrs 600 3T Intercepts (T = 0, C = 600) (T = 800, C = 0) + 4C = Feasible < 2400 hrs 24 00 0 800 T C 1000 1C 2T + 1C = 1000 + 2T Painting Constraint Line =1 600 000 Intercepts (T = 0, C = 1000) (T = 500, C = 0) 0 500 800 T Max Chair Line C 1000 C = 450 Min Table Line 600 450 T = 100 Feasible Region 100 500 800 T + 7T C =$ 4,0 500 7T + 5C = Profit 5C Objective Function Line 7T Optimal Point (T = 320, C = 360) 400 C +5 C +5 00 2,8 =$ 7T 300 =$ 00 2,1 200 100 0 100 200 300 400 500 T C Additional Constraint Need at least 75 more chairs than tables New optimal point T = 300, C = 375 500 400 T = 320 C = 360 No longer feasible C > T + 75 Or C – T > 75 300 200 100 0 100 200 300 400 500 T LP Characteristics • Feasible Region: The set of points that satisfies all constraints • Corner Point Property: An optimal solution must lie at one or more corner points • Optimal Solution: The corner point with the best objective function value is optimal Special Situation in LP Redundant Constraints - not affect the feasible region Example: x < 10 x < 12 The second constraint is redundant because it is less restrictive Special Situation in LP Infeasibility – when no feasible solution exists (there is no feasible region) Example: x < 10 x > 15 Special Situation in LP Alternate Optimal Solutions – when there is more than one optimal solution C 10 2T Max 2T + 2C All points on Red segment are optimal 2C = 20 T + C < 10 T < C< T, C > + Subject to: 0 10 T Special Situation in LP Unbounded Solutions – when nothing prevents the solution from becoming infinitely large Max 2T + 2C Subject to: 2T + 3C > T, C > on n i t c ri e lutio D so of C 0 T Using Excel’s Solver for LP Recall the Flair Furniture Example: Max 7T + 5C (profit) Subject to the constraints: 3T + 4C < 2400 2T + 1C < 1000 C < 450 T > 100 T, C > (carpentry hrs) (painting hrs) (max # chairs) (min # tables) (nonnegativity) Go to file 2-1.xls ... Product Mix Problem Decision: How much to make of > products? Objective: Maximize profit Constraints: Limited resources Example: Flair Furniture Co Two products: Chairs and Tables Decision: How many... Painting hrs hr 1000 Other Limitations: • Make no more than 450 chairs • Make at least 100 tables Decision Variables: T = Num of tables to make C = Num of chairs to make Objective Function: Maximize... An optimal solution must lie at one or more corner points • Optimal Solution: The corner point with the best objective function value is optimal Special Situation in LP Redundant Constraints