Integer Programming - If then statement

Question

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:

Xe1 = number of expensive t-shirts bought at producer E
Xe2 = number of cheap t-shirts bought at producer E

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

Thanks in advance!

score 0 · Answer 1 · answered Dec 13 '21 at 16:51

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.

score 0 · Answer 2 · answered Dec 13 '21 at 20:22

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.

Integer Programming - If then statement

2 Answers2