Chapter 2 Linear Programming Models: Graphical and Computer Methods Jason C. H. Chen, Ph.D. Professor of MIS School of Business Administration Gonzaga University Spokane, WA 99223 chen@jepson.gonzaga.edu

Steps in Developing a Linear Programming (LP) Model Formulation Solution Interpretation and Sensitivity Analysis

Properties of LP Models Seek to minimize or maximize Include “constraints” or limitations There must be alternatives available 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 (per table) Chairs (per chair) Hours Available Profit Contribution $7 $5 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)

Make no more than 450 chairs C < 450 (num. chairs) 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) 3T + 4C < 2400 (carpentry hrs) 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)

Using Excel’s Solver for LP Recall the Flair Furniture Example: Max 7T + 5C (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) Go to file 2-1.xls

Flair Furniture Flair Furniture T C Tables Chairs Number of units   Tables Chairs Number of units Profit 7 5 =SUMPRODUCT(B6:C6,$B$5:$C$5) Constraints: Carpentry hours 3 4 =SUMPRODUCT(B8:C8,$B$5:$C$5) <= 2400 Painting hours 2 1 =SUMPRODUCT(B9:C9,$B$5:$C$5) 1000 Maximum chairs =SUMPRODUCT(B10:C10,$B$5:$C$5) 450 Minimum tables =SUMPRODUCT(B11:C11,$B$5:$C$5) >= 100 LHS Sign RHS Flair Furniture T C   Tables Chairs Number of units 320.0 360.0 Profit $7 $5 $4,040.00 Constraints: Carpentry hours 3 4 2400.0 <= 2400 Painting hours 2 1 1000.0 1000 Maximum chairs 450 Minimum tables >= 100 LHS Sign RHS

Microsoft Excel 10.0 Answer Report Worksheet: [2-1.xls]Flair Furniture   Target Cell (Max) Cell Name Original Value Final Value $D$6 Profit $0.00 $4,040.00 Adjustable Cells $B$5 Number of units Tables 0.0 320.0 $C$5 Number of units Chairs 360.0 Constraints Cell Value Formula Status Slack $D$8 Carpentry hours 2400.0 $D$8<=$F$8 Binding $D$9 Painting hours 1000.0 $D$9<=$F$9 $D$10 Maximum chairs $D$10<=$F$10 Not Binding 90.0 $D$11 Minimum tables $D$11>=$F$11 220.0

Add a new constraint C - T > 75 A new constraint specified by the marketing department. Specifically, they needed to ensure theat the number of chairs made this month is at least 75 more than the number of tables made. The constraint is expressed as: C - T > 75

Revised Model for Flair Furniture Max 7T + 5C (profit) Subject to the constraints: 3T + 4C < 2400 (carpentry hrs) 2T + 1C < 1000 (painting hrs) C < 450 (max # chairs) T > 100 (min # tables) - 1T + 1C > 75 T, C > 0 (nonnegativity) Go to file 2-2.xls

Flair Furniture - Modified Problem C   Tables Chairs Number of units 300.0 375.0 Profit $7 $5 $3,975.00 Constraints: Carpentry hours 3 4 2400.0 <= 2400 Painting hours 2 1 975.0 1000 Maximum chairs 450 Minimum tables >= 100 Tables vs Chairs -1 75.0 75 LHS Sign RHS Flair Furniture T C   Tables Chairs Number of units 320.0 360.0 Profit $7 $5 $4,040.00 Constraints: Carpentry hours 3 4 2400.0 <= 2400 Painting hours 2 1 1000.0 1000 Maximum chairs 450 Minimum tables >= 100 LHS Sign RHS

End of Chapter 2 No Graphical Solution will be discussed

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

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

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

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

Additional Constraint 500 400 300 200 100 Additional Constraint Need at least 75 more chairs than tables C > T + 75 Or C – T > 75 New optimal point T = 300, C = 375 T = 320 C = 360 No longer feasible 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 - 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 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 6 Max 2T + 2C Subject to: T + C < 10 T < 5 C < 6 T, C > 0 2T + 2C = 20 All points on Red segment are optimal 0 5 10 T

Special Situation in LP Unbounded Solutions – when nothing prevents the solution from becoming infinitely large C 2 1 Direction of solution Max 2T + 2C Subject to: 2T + 3C > 6 T, C > 0 0 1 2 3 T