Linear Programming Problem Setup

[nikki__10234]

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...Z_q}}

where Zi=C₁ⁱx₁ + C₂ⁱx₂+...+C_nⁱx_n

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+...+C_n¹x_n

Z₂=C₁²x₁ + C₂²x₂+...+C_n²x_n

Z_q=C₁^qx₁ + C₂^qx₂+...+C_n^qx_n

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 < Z_q

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.

 

[Ray Vickson]

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...Z_q}}

where Zi=C₁ⁱx₁ + C₂ⁱx₂+...+C_nⁱx_n

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+...+C_n¹x_n

Z₂=C₁²x₁ + C₂²x₂+...+C_n²x_n

Z_q=C₁^qx₁ + C₂^qx₂+...+C_n^qx_n

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 < Z_q

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

 

[nikki__10234]

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

Z ≤ c¹₁x₁+...+c¹_nx_n

Z ≤ c²₁x₁+...+c²_nx_n

...

Z ≤ c^q₁x₁+...+c^q_nx_n

 

[Ray Vickson]

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

Z ≤ c¹₁x₁+...+c¹_nx_n

Z ≤ c²₁x₁+...+c²_nx_n

...

Z ≤ c^q₁x₁+...+c^q_nx_n

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 Z_min = -∞.

RGV