# Linear Programming 5: Alternate solutions, Infeasibility, Unboundedness, & Redundancy

Welcome to this tutorial as we discuss special

cases encountered while solving some linear programming problems. The special cases we’ll discuss include

Alternative optimal solutions Infeasibility

Unboundedness, and Redundancy. Let’s begin with alternate optimal solutions.

Consider this linear programming problem and its graph. Here is an instance of the objective function

line. Since this is a maximization problem, moving

this line shows that the optimal solutions occur at these two extreme points. That is,

more than one optimal solution exists. We refer to these as Alternative or Multiple

Optimal Solutions. Generally, this situation occurs when the

objective function line is parallel to a binding constraint.

Note that the objective function and the first constraint line have the same slope. So they

are parallel. That is why the objective function line coincides

with the constraint line here. In fact, not only these 2 points are optimal

solution points. Every single point between them on the line is also optimal. There’s nothing wrong with alternative optimal

solutions. In fact, they offer the benefit of different combinations of the decision

variables that produce the same optimal profit or cost. The next special case is Infeasibility

Consider this linear programming problem and its graph. The first constraint is a “less than”

constraint and is satisfied in the region here towards the origin while the second is

a “greater than” constraint satisfied here away from the origin. As you can see, there is no region satisfying

both constraints at the same time. So we say there is no feasible solution or no feasible

region, or we have a condition referred to as “infeasibility”. The next special case is Unboundedness.

This is a situation that applies only to maximization problem. Because all these 3 constraints are “greater

than” constraints, the feasible region is this region shaded here. Since this is a maximization problem, and

the feasible region is open ended, we can move the objective function line infinitely

without reaching the end of the feasible region. As a result, this situation is called “unboundedness”.

It suggests that we can make unlimited profit which is impractical. It is usually due to

a formulation error or by assuming a resource could be unlimited in supply. The last special case we’ll discuss here

is redundancy. Suppose you’re applying to enter a contest

where applicants have to be at least 15 years old. If X represents age, then we have X ≥ 15.

Suppose also that applicants must also have a valid driver’s license.

If the minimum licensing age is 16, then applicants must be at least 16 years old and we have

X ≥ 16. So we will call this first constraint redundant

because its effect has been overshadowed by the second constraint. Now consider this minimization problem. Here’s a graph showing its constraint lines.

Since these are all “greater than” constraints, here’s the feasible region. As you can see here, constraint 3 line does

not touch the feasible region at all. Therefore it is redundant. So a redundant constraint is one that does

not affect the feasible region. In essence we could remove a redundant constraint

without changing the feasible region or optimal solution. And that concludes this video on special cases

in solving linear programming problems. Thank you for watching.

you are amazing 🙂 you explained me in 20 minutes everything that my reacher could not in 2 months, thank you

Thanks a lot you make it very easy for me .

Really amazing! So understandable!

Thank you so much for this:)

thank you…

you explained something that my professor in graduate school took 2 hrs to explain, you did it in 3:42. thanks

That's perfect clear!

So do alternative optimal solutions always exist when the slope of the objective function equals the slope of any of the constraint boundary line slopes, in both a max or min LP problem?

How do I show Alternate solutions, Infeasibility, Unboundedness, & Redundancy WITHOUT drawing a graph? I.e. algebraically

thank you my man <3 helped me alot

Sorry Joshua do you have any other videos I am possibly missing that explain solving linear programming problems using the Simplex method? i have found your way of teaching very helpful and better than anyone's

Excellent

OH THANK YOU SO MUCH!! I'm going to pass my paper tomorrow for sure. God bless

thanks Sir. can i have a problem where two equations do not intersect and there is a feasible region?

Thank you for making the videos so short and the the point!

Thanks Joshua, I was looking for this! It was really helpful.

You make it so much more simpler!

Thank u… Good job

Man. you saved my life! thanks

wow… helped me in solving

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a great video! Very precise& quite clear

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Good

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Great video!! Excellent for last-minute revision!

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simple and clear

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My college is waiting my time and money without understanding anything from them, Your video is really very helpful.

Many thanks again

good lecture. pls add degeneracy also

Thank you🙏 really awesomely explained..

Superb sir….

Is redundancy same as a degenerate solution ?

Best explaination

you taught this very easily , THANKS 🙂

In the unboundedness, the state is minimization

Thank you so much, this lesson made it easy for me to understand linear programming special cases

Thanks sir

How do we find an objective function given an optimal solution ??