CODEWITHSUNDEEP
constraintslinear-programmingminimizationmaximizemixed-integer-programming
Renaldo
Renaldo

Reputation: 11

Integer Programming - If then statement

I am new to linear/integer programming and I am having a hard time formulating constraints for a specific if-then statement in a fixed charge problem. Suppose that there are five manufacturers of t-shirts, and a customer wishes to purchase 400 t-shirts while minimizing costs.

Producer Variable cost/t-shirt Delivery Availability
A 3 40 200
B 3.5 30 100
C 4.10 Free delivery 100
D 4.1 30 200
E 3.2 30 First 100 t-shirts
E 2.90 20 101st-150th t-shirt

Producer E has an availability of 150 t-shirts. The first 100 t-shirts bought from producer E have a variable cost of $3.20 and a delivery fee of $30. If the customer orders more than 100 t-shirts from producer E, she can buy them at a variable cost of $2.90 and an additional fee of $20.

How can I create constraints from this if-then statement:

I want constraint Xe2 <= 0 to exist when Xe1 < 100.

Thanks in advance!

Upvotes: 1

Views: 512

Answers (2)

AirSquid
AirSquid

Reputation: 11938

This is a pretty standard "discount pricing model" where a discount price is awarded at a certain quantity. You can introduce 1 binary variable to the model to do this. This sounds like a H/W assignment.... :)

Let:

Xe1 be the qty bought from E at price 1
Xe2 be the qty bought from E at price 2
Ye be the decision to buy at the second price point y ∈ {0, 1}
Me2 be the qty available at the 2nd price (or a reasonable upper bound)

Then:

Xe2 <= Ye * Me2          # purchase at 2nd price point held to zero, unless Ye==1
Xe1 >= 100 * Ye          # Ye can only be 1 if 100 are purchased at Xe1

There are some nuances here with the shipping. It isn't clear from the problem description if you bought 101 shirts from E if the shipping cost would be 30 + 20 (which would be odd) or the whole cost of shipping drops to 20. You could use the same indicator variable Ye to work something out for that as well.

Upvotes: 0

joni
joni

Reputation: 7157

Since Xe1 and Xe2 are both non-negative integer variables, your constraint is the same as Xe1 <= 99 ⇒ Xe2 = 0. This can be formulated as follows:

                b1 <= b2
99*b1 + 100*(1-b1) <= Xe1 <= 99*b1 + M*(1-b1)
             1-b2  <= Xe2 <= M'*(1-b2)

where b1,b2 are binary helper variables and M and M' are upper bounds for Xe1 and Xe2, i.e. M = 101 and M' = 400.

Upvotes: 0

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