IP Modeling Techniques III: Non-Binary Variables & Piecewise Linear Functions, Slides of Discrete Structures and Graph Theory

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IP modeling techniques III

In this handout,

Modeling techniques:

 Making choices with non-binary variables

 Piecewise linear functions

Making choices with non-binary variables

• Recall the furniture manufacturer problem.
• Extra requirement : From the 3 possible products (tables, chairs, desks), at most two should be chosen to be produced. That is, at most two of xt , xc , xd can be non-zero.
• How to achieve this in the model?
• Introduce new binary variables. For i=t,c,d,
• To enforce the requirement, need the following constraint: y (^) t + yc + yd ≤ 2
• Need also to relate xi ’s and yi ’s. Add constraints: xi ≤ Myi for i=t,c,d and large positive M

0 if product icannot be produced (x 0)

y^1 if product icanbe produced (x 0)

i i i

• How to include piecewise linear cost functions in an

objective function of IP?

• Idea : Introduce a new variable for each cost segment.

For i=1,2,3,

y i = number of items produced at cost ci

Then the total number of items is x = y 1 +y 2 +y 3 +y 4.

We need constraints

0 ≤ y 1 ≤ 5, 0 ≤ y 2 ≤ 4, 0 ≤ y 3 ≤ 7, 0 ≤ y 4 ≤ 10 , (*)

and the production cost in the objective function is

11y 1 + 8y 2 + 5y 3 + 7y 4

• What is the shortcoming of this model?

• We should require that
  • y 2 >0 implies that y1 =5 (1)
  • y 3 >0 implies that y2 =4 (2)
  • y 4 >0 implies that y3 =7 (3)
• Introduce new variables to translate these requirements into linear constraints. For i=1,2,3,4,
• Proper constraints relating wi and yi will provide that requirements (1)-(3) are satisfied. y 2 ≤ 4w 1 and 5w 1 ≤ y1 provide (1) y 3 ≤ 7w 2 and 4w 2 ≤ y2 provide (2) y 4 ≤ 10w 3 and 7w 3 ≤ y3 provide (3)

0 otherwise

w 1 if yi is at its upper bound

i