Presentation on theme: "and Models Without Unique Optimal Solutions"— Presentation transcript:
1and Models Without Unique Optimal Solutions Linear ProgrammingOptimal Solutionsand Models Without Unique Optimal Solutions
2Finding the Optimal Point - Review X21000900800700600500400300200100X1Move the objective function line parallel to itself until it touches the last point of the feasible region.OPTIMAL POINT
3Minimization Objective Function X21000900800700600500400300200100X1OPTIMAL POINT
4Different Objective Function X21000900800700600500400300200100X1OPTIMAL POINT
5Another Objective Function X21000900800700600500400300200100X1OPTIMAL POINT
6Still Another Objective Function X21000900800700600500400300200100X1OPTIMAL POINT
7Extreme Points and Optimal Solutions Fundamental Linear Programming Theorem:Why not simply list all extreme points?More cumbersome than solving the model in most cases.Model may not have an optimal solution.If a linear programming modelhas an optimal solution, thenan extreme point will be optimal.
8Models With No Solutions Infeasibility X21000900800700600500400300200100X1Max 8X1 + 5X2s.t. 2X1 + 1X2 ≤ 10003X1 + 4X2 ≤ 2400X X2 ≤ 350X1, X2 ≥ 0.X ≥ 800No points in common. No points satisfy all constraints simultaneously.No Solutions! Problem isINFEASIBLE.
9InfeasibilityA problem is infeasible when there are no solutions that satisfy all the constraints.Infeasibility can occur fromInput ErrorMisformulationSimply an inconsistent set of contraintsExcel – When Solve is clicked:
10Models With An “Unbounded” Solution X21000900800700600500400300200100X1Max 8X1 + 5X2s.t. X X2 ≤ 350X ≥ 200X2 ≥ 200Unbounded Feasible RegionCan increase indefinitelyUnbounded Solution
11Models With An Unbounded Feasible Region – Optimal Solution X21000900800700600500400300200100X1Min 8X1 + 5X2s.t. X X2 ≤ 350X ≥ 200X2 ≥ 200Unbounded Feasible RegionOPTIMAL POINT
12UnboundednessAn unbounded feasible region extends to infinity in some direction.If the problem is unbounded, the feasible region must be unbounded.If the feasible region is unbounded, the problem may or may not be unbounded.An unbounded solution means you left out some constraints – you cannot make an “infinite” profit.Excel – When Solve is clickedMeans the problem isunbounded
13Multiple Optimal Solutions X21000900800700600500400300200100X1MAX 8X1 + 4X2s.t. 2X1 + 1X2 ≤ X1 + 4X2 ≤ X1 - 1X2 ≤ X1, X2 ≥ 02X1 + 1X2 ≤ 1000All points on the boundary between optimal extreme points are also optimalOptimal Extreme Point1X1 - 1X2 ≤ 350Optimal Extreme Point3X1 + 4X2 ≤ 2400
14Multiple Optimal Solutions When an objective function line is parallel to a constraint the problem can have multiple optimal solutions.The constraint must not be a redundant constraint but must be a boundary constraint.The objective function must move in the direction of the constraint—In the previous example if the objective function had been MIN 8X1 + 4X2, then it is moved in the opposite direction of the constraint and (0,0) would be the optimal solution.Multiple optimal solutions allow the decision maker to use secondary criteria to select one of the optimal solutions that has another desirable characteristic (e.g. Max X1 or X1 = 3X2, etc.)
15Generating the Multiple Optimal Solutions Any weighted average of optimal solutions is also optimal.In the previous example it can be shown that the two optimal extreme points are (320,360) and (450, 100).Thus .5(320,360) + .5(450,100) = (385,230) is also an optimal point that is half-way between these two points..8(320,360) + .2(450,100) = (346,308) is also an optimal point that is 8/10 of the way up the line toward (320,360).
16Multiple Optimal Solutions in Excel Excel – Identification of multiple solutionsSensitivity ReportIf anAllowable Decreaseor anAllowable Increaseof anObjective FunctionCoefficient is 0.We discuss how to generate and choose an appropriate alternate optimal solution using Excel later.
17Review When a linear programming model is solved it: Has a unique optimal solutionHas multiple optimal solutionsIs InfeasibleIs unboundedIdentification of eachBy graphBy ExcelIf a linear program has an optimal solution, then an extreme point is optimal.