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Linear Programming Optimal Solutions and Models Without Unique Optimal Solutions

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Finding the Optimal Point - Review X2 1000 900 800 700 600 500 400 300 200 100 0 100200 300400500600 700800 X1 OPTIMAL POINT Move the objective function line parallel to itself until it touches the last point of the feasible region.

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Minimization Objective Function X2 1000 900 800 700 600 500 400 300 200 100 0 100200 300400500600 700800 X1 OPTIMAL POINT

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Different Objective Function X2 1000 900 800 700 600 500 400 300 200 100 0 100200 300400500600 700800 X1 OPTIMAL POINT

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Another Objective Function X2 1000 900 800 700 600 500 400 300 200 100 0 100200 300400500600 700800 X1 OPTIMAL POINT

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Still Another Objective Function X2 1000 900 800 700 600 500 400 300 200 100 0 100200 300400500600 700800 X1 OPTIMAL POINT

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Extreme 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 model has an optimal solution, then extreme point an extreme point will be optimal.

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Models With No Solutions Infeasibility X2 1000 900 800 700 600 500 400 300 200 100 0 100200 300400500600 700800 X1. Max 8X 1 + 5X 2 s.t.2X 1 + 1X 2 ≤ 1000 3X 1 + 4X 2 ≤ 2400 X 1 - X 2 ≤ 350 X 1, X 2 ≥ 0 X 1 ≥ 800 No points in common. No points satisfy all constraints simultaneously. No Solutions! Problem isINFEASIBLE.

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Infeasibility infeasibleA problem is infeasible when there are no solutions that satisfy all the constraints. Infeasibility can occur from –Input Error –Misformulation –Simply an inconsistent set of contraints SolveExcel – When Solve is clicked:

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Unbounded Feasible Region Models With An “Unbounded” Solution X2 1000 900 800 700 600 500 400 300 200 100 0 100200 300400500600 700800 X1 Max 8X 1 + 5X 2 s.t. X 1 - X 2 ≤ 350 X 1 ≥ 200 X 2 ≥ 200 Can increase indefinitely Unbounded Solution

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Unbounded Feasible Region Models With An Unbounded Feasible Region – Optimal Solution X2 1000 900 800 700 600 500 400 300 200 100 0 100200 300400500600 700800 X1 Min 8X 1 + 5X 2 s.t. X 1 - X 2 ≤ 350 X 1 ≥ 200 X 2 ≥ 200 OPTIMAL POINT

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Unboundedness An unbounded feasible region extends to infinity in some direction. If the solution is unbounded, the feasible region must be unbounded. If the feasible region is unbounded, the solution may or may not be unbounded. An unbounded solution means you left out some constraints – you cannot make an “infinite” profit. SolveExcel – When Solve is clicked Means the problem isunbounded

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Multiple Optimal Solutions s.t.2X 1 + 1X 2 ≤ 1000 3X 1 + 4X 2 ≤ 2400 1X 1 - 1X 2 ≤ 350 X 1, X 2 ≥ 0 2X 1 + 1X 2 ≤ 1000 3X1 + 4X2 ≤ 2400 1X1 - 1X2 ≤ 350 X2 1000 900 800 700 600 500 400 300 200 100 0 100200 300400500600 700800 X1 MAX 8X1 + 4X2 Optimal Extreme Point All points on the boundary between optimal extreme points are also optimal

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Multiple Optimal Solutions canWhen 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— MINIn the previous example if the objective function had been MIN 8X 1 + 4X 2, 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 X 1 or X 1 = 3X 2, etc.)

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Alternate Optimal Solution To find the second optimal solution: 1.Observe that an Allowable Increase or Allowable Decrease for the objective function coefficient of some variable X j is 0 2.Add a constraint that sets the value function cell to the optimal value from the first optimal solution. 3.Change objective function to If the Allowable Increase = 0, change objective to maximize X j If the Allowable Decrease = 0, change objective to minimize X j

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Multiple Optimal Solutions in Excel Excel – Identification of multiple solutions Sensitivity Report If an Allowable Decrease or an Allowable Increase of an Objective Function Coefficient is 0.

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Alternate Optimal Solution in Excel

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Generating 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).

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Review When a linear programming model is solved it: –Has a unique optimal solution –Has multiple optimal solutions –Is Infeasible –Is unbounded Identification of each –By graph –By Excel If a linear program has an optimal solution, then an extreme point is optimal.

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