Linear Programming

Optimal Solutions and Models Without Unique Optimal Solutions http://business.fullerton.edu/isds/jlawrence/SPRING%20 2004/B-Spring%2004/B- PowerPoint%20Excel%20HW.htm

Finding the Optimal Point - Review

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

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OPTIMAL POINT

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Minimization Objective Function

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OPTIMAL POINT

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Different Objective Function

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OPTIMAL POINT

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Another Objective Function

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OPTIMAL POINT

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Still Another Objective Function

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OPTIMAL POINT

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Extreme Points and Optimal Solutions

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Fundamental Linear Programming Theorem: If a linear programming model has an optimal solution, then an extreme point will be optimal.

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Why not simply list all extreme points?

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More cumbersome than solving the model in most cases.

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Model may not have an optimal solution.

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Models With No Solutions Infeasibility Max 8X s.t.

2X 3X 1 X 1 1 1 X 1 + 5X 2 + 1X 2 ≤ 1000 + 4X 2 ≤ 2400 - X 2 ≤ 350 X 1 , X 2 ≥ 800 ≥ 0

.

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 Max 8X 1 s.t.

+ 5X 2 X 1 - X 2 ≤ 350 X 1 ≥ 200 X 2 ≥ 200 Unbounded Feasible Region Can increase indefinitely

Unbounded Solution

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Models With An Unbounded Feasible Region – Optimal Solution

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Min

s.t.

8X 1 + 5X 2 X 1 - X 2 ≤ 350 X 1 ≥ 200 X 2 ≥ 200

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Unbounded Feasible Region

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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

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Optimal Extreme Point MAX 8X1 + 4X2 s.t.

All points on the boundary between 2X 3X 1 1X 1 1 X 1 + 1X 2 + 4X - 1X 2 , X 2 2 ≤ 1000 ≤ 2400 ≤ 350 ≥ 0 optimal extreme points are also optimal

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Optimal Extreme Point

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Multiple Optimal Solutions

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When an objective function line is parallel to a constraint the problem

can

have multiple optimal solutions.

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The constraint must not be a redundant constraint but must be a boundary constraint.

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The objective function must move in the direction of the constraint —

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In 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.)

Generating the Multiple Optimal Solutions

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Any weighted average of optimal solutions is also optimal.

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In the previous example it can be shown that the two optimal extreme points are (320,360) and (450, 100).

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Thus .5(320,360) + .5(450,100) = (385,230) is also an optimal point that is half-way between these two points.

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.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).

Multiple Optimal Solutions in Excel

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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.

Review

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When a linear programming model is solved it:

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Has a unique optimal solution

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Has multiple optimal solutions

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Is Infeasible

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Is unbounded

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Identification of each

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By graph

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By Excel

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If a linear program has an optimal solution, then an extreme point is optimal.