Chapter 2Linear Programming Models-Graphical and .ppt

1、Chapter 2 Linear Programming Models: Graphical and Computer Methods, 2007 Pearson Education,Steps in Developing a Linear Programming (LP) Model,FormulationSolutionInterpretation and Sensitivity Analysis,Properties of LP Models,Seek to minimize or maximize Include “constraints” or limitations There m

2、ust 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 profitConstraints: Limited resources,Example: Flair Furniture Co.,Two products: Chairs and TablesDecision: How many of each to mak

3、e this month?Objective: Maximize profit,Flair Furniture Co. Data,Other Limitations:Make no more than 450 chairsMake at least 100 tables,Decision Variables:T = Num. of tables to makeC = Num. of chairs to makeObjective Function: Maximize ProfitMaximize $7 T + $5 C,Constraints:,Have 2400 hours of carpe

4、ntry time available3 T + 4 C 2400 (hours) Have 1000 hours of painting time available2 T + 1 C 1000 (hours),More Constraints: Make no more than 450 chairsC 100 (num. tables)Nonnegativity: Cannot make a negative number of chairs or tablesT 0C 0,Model Summary,Max 7T + 5C (profit) Subject to the constra

5、ints:3T + 4C 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 3T + 4C = 2400Inte

6、rcepts (T = 0, C = 600) (T = 800, C = 0),0 800 T,C6000,Feasible 2400 hrs,Infeasible 2400 hrs,3T + 4C = 2400,Painting Constraint Line 2T + 1C = 1000Intercepts (T = 0, C = 1000) (T = 500, C = 0),0 500 800 T,C 10006000,2T + 1C = 1000,0 100 500 800 T,C 1000600 4500,Max Chair Line C = 450Min Table Line T

7、 = 100,Feasible Region,0 100 200 300 400 500 T,C5004003002001000,Objective Function Line 7T + 5C = Profit,7T + 5C = $2,100,7T + 5C = $4,040,Optimal Point (T = 320, C = 360),7T + 5C = $2,800,0 100 200 300 400 500 T,C5004003002001000,Additional Constraint Need at least 75 more chairs than tables C T +

8、 75 Or C T 75,T = 320 C = 360 No longer feasible,New optimal point T = 300, C = 375,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 o

9、bjective function value is optimal,Special Situation in LP,Redundant Constraints - do not affect the feasible regionExample: x 10x 12The second constraint is redundant because it is less restrictive.,Special Situation in LP,Infeasibility when no feasible solution exists (there is no feasible region)

10、Example: x 15,Special Situation in LP,Alternate Optimal Solutions when there is more than one optimal solution,Max 2T + 2C Subject to:T + C 0,0 5 10 T,C 1060,2T + 2C = 20,All points on Red segment are optimal,Special Situation in LP,Unbounded Solutions when nothing prevents the solution from becoming infinitely large,Max 2T + 2C Subject to:2T + 3C 6T, C 0,0 1 2 3 T,C210,Direction of solution,Using Excels Solver for LP,Recall the Flair Furniture Example:Max 7T + 5C (profit) Subject to the constraints:3T + 4C 100 (min # tables)T, C 0 (nonnegativity) Go to file 2-1.xls,