Linear Programming

1
Linear Programming

• Optimal Solutions
• andModels Without Unique Optimal Solutions

2
Finding the Optimal Point - Review
Move the objective function line parallel to
itself until it touches the last point of the
feasible region.
3
Minimization Objective Function
4
Different Objective Function
5
Another Objective Function
6
Still Another Objective Function
7
Extreme Points andOptimal 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.
8
Models With No SolutionsInfeasibility
Max 8X1 5X2 s.t. 2X1 1X2 1000 3X1 4X2
2400 X1 - X2 350 X1, X2 0
.
No points in common.No points satisfy all
constraints simultaneously.
No Solutions!Problem is INFEASIBLE.
9
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

10
Models With An Unbounded Solution
Max 8X1 5X2 s.t. X1 - X2 350 X1
200 X2 200
Unbounded Solution
11
Models With An UnboundedFeasible Region
Optimal Solution
Min 8X1 5X2 s.t. X1 - X2 350 X1
200 X2 200
12
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
13
Multiple Optimal Solutions
s.t. 2X1 1X2 1000 3X1 4X2 2400 1X1 -
1X2 350 X1, X2 0
2X1 1X2 1000
1X1 - 1X2 350
3X1 4X2 2400
14
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.)

15
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).

16
Multiple Optimal Solutions in Excel

• Excel Identification of multiple solutions

We discuss how to generate and choose an
appropriate alternate optimal solution using
Excel later.
17
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.