Chapter 2 Linear Programming Models Graphical and Computer

Chapter 2 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 > 2 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 3 hrs 4 hrs 2400 Painting 2 hrs 1 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 3 T + 4 C < 2400 (hours) • Have 1000 hours of painting time available 2 T + 1 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 7 T + 5 C (profit) Subject to the constraints: 3 T + 4 C < 2400 (carpentry hrs) 2 T + 1 C < 1000 (painting hrs) T C < 450 (max # chairs) > 100 (min # tables) T, C > 0 (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 3 T + 4 C = 2400 Infeasible > 2400 hrs 600 3 T Intercepts (T = 0, C = 600) (T = 800, C = 0) + 4 C = Feasible < 2400 hrs 24 00 0 0 800 T

C= +1 2 T + 1 C = 1000 2 T Painting Constraint Line C 1000 600 00 10 Intercepts (T = 0, C = 1000) (T = 500, C = 0) 0 0 500 800 T

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

7 T C C 4, 0 =$ 500 40 7 T + 5 C = Profit +5 Objective Function Line 7 T 400 Optimal Point (T = 320, C = 360) C +5 C 00 +5 2, 8 =$ 7 T 300 =$ 00 2, 1 200 100 0 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 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 1. Redundant Constraints - do not affect the feasible region Example: x < 10 x < 12 The second constraint is redundant because it is less restrictive.

Special Situation in LP 2. Infeasibility – when no feasible solution exists (there is no feasible region) Example: x < 10 x > 15

Special Situation in LP 3. Alternate Optimal Solutions – when there is more than one optimal solution C 10 2 T Max 2 T + 2 C All points on Red segment are optimal 2 C = 20 T + C < 10 T < 5 C< 6 T, C > 0 + Subject to: 6 0 0 5 10 T

Special Situation in LP 4. Unbounded Solutions – when nothing prevents the solution from becoming infinitely large Max 2 T + 2 C Subject to: 2 T + 3 C > 6 T, C > 0 n o i t on c re luti i D so of C 2 1 0 0 1 2 3 T

Using Excel’s Solver for LP Recall the Flair Furniture Example: Max 7 T + 5 C (profit) Subject to the constraints: 3 T + 4 C < 2400 (carpentry hrs) 2 T + 1 C < 1000 (painting hrs) C < 450 (max # chairs) T > 100 (min # tables) T, C > 0 (nonnegativity) Go to file 2 -1. xls