# Linear Programming Problem Setup

We were given this problem in a Linear Programming class and asked to define the constraints.

## Homework Statement

max Z = max (xεS) {min {Z1, Z2...Zq}}

where Zi=C1ix1 + C2ix2+...+Cnixn

## Homework Equations

Constraints need to be defined to set up the problem.

## The Attempt at a Solution

The first few Z equations would be:
Z1=C11x1 + C21x2+...+Cn1xn

Z2=C12x1 + C22x2+...+Cn2xn

Zq=C1qx1 + C2qx2+...+Cnqxn

I think the best way to ensure that the Z is at a minimum is to define the inequalities below:
Z < Z1
Z < Z2
...
Z < Zq

This ensures that we pick the minimum value of Z. But these constraints should also include some way to maximize x and I am confused as to how to include that.

## Answers and Replies

Ray Vickson
Science Advisor
Homework Helper
Dearly Missed
We were given this problem in a Linear Programming class and asked to define the constraints.

## Homework Statement

max Z = max (xεS) {min {Z1, Z2...Zq}}

where Zi=C1ix1 + C2ix2+...+Cnixn

## Homework Equations

Constraints need to be defined to set up the problem.

## The Attempt at a Solution

The first few Z equations would be:
Z1=C11x1 + C21x2+...+Cn1xn

Z2=C12x1 + C22x2+...+Cn2xn

Zq=C1qx1 + C2qx2+...+Cnqxn

I think the best way to ensure that the Z is at a minimum is to define the inequalities below:
Z < Z1
Z < Z2
...
Z < Zq

This ensures that we pick the minimum value of Z. But these constraints should also include some way to maximize x and I am confused as to how to include that.

As written, your formulation will fail, but it can be modified slightly to work properly. Come back for additional hints when you have dealt with this issue.

RGV

Would it work to write the constraints as follows instead of saying Z<Zi. This way you are telling the program to choose the maximum values of x that lead to the minimum Z.

Z ≤ c11x1+...+c1nxn

Z ≤ c21x1+...+c2nxn

...

Z ≤ cq1x1+...+cqnxn

Ray Vickson
Science Advisor
Homework Helper
Dearly Missed
Would it work to write the constraints as follows instead of saying Z<Zi. This way you are telling the program to choose the maximum values of x that lead to the minimum Z.

Z ≤ c11x1+...+c1nxn

Z ≤ c21x1+...+c2nxn

...

Z ≤ cq1x1+...+cqnxn

Why would you want to minimize Z? You were asked to maximize the minimum, not minimize it. Anyway, as written your problem would be unbounded, with Zmin = -∞.

RGV