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**and Models Without Unique Optimal Solutions**

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 X1 Move the objective function line parallel to itself until it touches the last point of the feasible region. OPTIMAL POINT

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**Minimization Objective Function**

X2 1000 900 800 700 600 500 400 300 200 100 X1 OPTIMAL POINT

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**Different Objective Function**

X2 1000 900 800 700 600 500 400 300 200 100 X1 OPTIMAL POINT

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**Another Objective Function**

X2 1000 900 800 700 600 500 400 300 200 100 X1 OPTIMAL POINT

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**Still Another Objective Function**

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

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Infeasibility A 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 Excel – When Solve is clicked:

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**Models With An “Unbounded” Solution**

X2 1000 900 800 700 600 500 400 300 200 100 X1 Max 8X1 + 5X2 s.t. X X2 ≤ 350 X ≥ 200 X2 ≥ 200 Unbounded Feasible Region Can increase indefinitely Unbounded Solution

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**Models With An Unbounded Feasible Region – Optimal Solution**

X2 1000 900 800 700 600 500 400 300 200 100 X1 Min 8X1 + 5X2 s.t. X X2 ≤ 350 X ≥ 200 X2 ≥ 200 Unbounded Feasible Region OPTIMAL POINT

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Unboundedness An 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 clicked Means the problem is unbounded

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**Multiple Optimal Solutions**

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

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**Multiple 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.)

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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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**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. We discuss how to generate and choose an appropriate alternate optimal solution using Excel later.

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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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