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# Chapter 2

## on Jan 25, 2014

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## Chapter 2Presentation Transcript

• 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 7T + 5C (profit) Subject to the constraints: 3T + 4C < 2400 (carpentry hrs) 2T + 1C < 1000 (painting hrs) C < 450 T (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 3T + 4C = 2400 Infeasible > 2400 hrs 600 3T Intercepts (T = 0, C = 600) (T = 800, C = 0) + 4C = Feasible < 2400 hrs 24 00 0 0 800 T
• C 1000 1C 2T + 1C = 1000 + 2T Painting Constraint Line 000 =1 600 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
• + 7T C 40 4,0 =\$ 7T + 5C = Profit 5C Objective Function Line 500 7T Optimal Point (T = 320, C = 360) 400 C +5 C +5 00 2 ,8 =\$ 7T 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 2T Max 2T + 2C All points on Red segment are optimal 2C = 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 2T + 2C Subject to: 2T + 3C > 6 T, C > 0 n tio on c re luti Di so of C 2 1 0 0 1 2 3 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 > 0 (carpentry hrs) (painting hrs) (max # chairs) (min # tables) (nonnegativity) Go to file 2-1.xls