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Linear Programming Problem Setup

  • #1
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

  • #2
Ray Vickson
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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
 
  • #3
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
 
  • #4
Ray Vickson
Science Advisor
Homework Helper
Dearly Missed
10,706
1,728
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
 

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