1. Integer Programming
Key characteristic of an Integer Program (IP) or Mixed Integer
Linear Program (MILP):
One or more of the decision variable must be integer.
In some cases a problem that requires an Integer solution can be
solved as a Linear program.
When?
When rounding off a real number to an integer number has no
“effect” on your solution.

2. Integer Programming LP OPTIMUM FOUND AT STEP 2
! Homework 7-2) !
! x1 - model 1 ovens
! x2 - model 2 ovens
max 12 x x2
st x <= 300
x2 <= 100
2.3 x x2 <= 500
end
free x1
LP OPTIMUM FOUND AT STEP
OBJECTIVE FUNCTION VALUE 1)
VARIABLE VALUE REDUCED COST X X
You cannot manufacture model 1 ovens. Simply
rounding off to 147 model 1 ovens is an acceptable solution.

3. Integer Programming - Example
Classic Knapsack Problem:
You want to maximize the value of items you can pack into a single suitcase (or knapsack). However, you are limited to a weight of 50 lbs.
xi = 0 if not chosen
= 1 if put in knapsack
Item
Value ($)
Weight (lbs) A 15 18 B 20 10 C 21 D 13 11 E 12

4. Integer Programming - Example
Classic Knapsack Problem: If integer constraint relaxed -
LP OPTIMUM FOUND AT STEP
OBJECTIVE FUNCTION VALUE 1)
VARIABLE VALUE REDUCED COST XA XB XC XD XE

5. Integer Programming - Example
Classic Knapsack Problem: With integer constraint -
OBJECTIVE FUNCTION VALUE 1)
VARIABLE VALUE REDUCED COST XA XB XC XD XE

6. Integer Programming - Example
Classic Knapsack Problem:
How would you modify this formulation if you realized that you could not select item A without selecting item D?
Add constraint: xA <= xD
How would you modify if you could select at most 3 items?
Add constraint: xA + xB + xC + xD + xE <= 3

7. Integer Programming - Example
Other examples of formulating problems: see handouts

8. Solving IPs and MIPs Approaches: Solve to Optimality
Branch and Bound – lesson 6
Implicit Enumeration – lesson 6
Cutting Plane Method – lesson 7
Good and sometimes Optimal Solutions
Heuristics (problem specific) – lesson 7
Meta-Heuristics – lesson 8 & 9

9. Solving IPs – Branch and Bound
See Branch and Bound On-Line Example

10. Solving IPs – Implicit Enumeration
Similar to Binary IP Branch and Bound
General Idea:
Fixed variables – those for which a value has been fixed.
Free Variable – variables which whose values are unspecified.
Completion – when all variables have been assigned a value.
Upper Bound (minimization problem) – Best feasible solution found thus far.
Lower Bound (minimization problem) – Optimal solution for a relaxed problem at a given node (e.g. some variables fixed, some free).

11. Solving IPs – Implicit Enumeration
Example Sequencing Problem:
Sequence a series of jobs to minimize the maximum lateness (Lmax) for a set of jobs to be processed on a single machine. Each job belongs to a given part family. Jobs are denoted with an (i,j) subscript indicating the jth job from family i.
Lmax = Max{Lij}
Lij = Cij – dij
Cij is the completion
time of job ij.
Family (i)
Job (j)
Processing Time (pij)
Due Date (dij) 1 3 5 2 7 4 6 9 10

12. Solving IPs – Implicit Enumeration
Example Sequencing Problem cont:
Also, when starting the processing of a new family, a family setup time is incurred. All jobs are ready to be scheduled at time 0.
Family (i)
Setup TIme 1 2 3
Optimality Condition: All jobs within a family must be sequenced in earliest due date order (EDD).

13. Solving IPs – Implicit Enumeration
Step 1 – Find an initial Upper bound
What is a good upper bound?
Step 2 – Perform implicit enumeration.
Start building partial sequences and fathom nodes if lower bound for partial sequence exceeds upper bound.
Update upper bound whenever a better value is found for a completion.
What is a good lower bounding scheme?

14. Solving IPs – Implicit Enumeration
Insert Hand Slides For Example Problem

15. Solving IPs – Using Lindo
Statements – INT and GIN
INT – forces binary solution (1 or 0) for decision variable.
GIN – forces non-negative integer (0,1,2,3,4…) for decision variable.
Knapsack problem:
max 15xa + 20xb + 18xc + 13xd + 12xe st 18xa + 10xb + 21xc + 11xd + 11xe <= 50 end int xa int xb int xc int xd int xe