1. Integer Programming Integer Linear Programming A ll-Integer Linear Program AA ll variables must be integers Mixed-Integer Linear Program SSome, but not all variables must be integers 0-1 Integer Linear Program AAll variables must be 0 or 1, also known as binary variables

2. Integer Programming Integer Programming – All Integers Northern Airlines is a small regional airline. Management is now considering expanding the company by buying additional aircraft. One of the main decisions is whether to buy large or small aircraft to use in the expansion. The table below gives data on the large and small aircraft that may be purchased. As noted in the table, management does not want to buy more than 2 small aircraft, while the number of large aircraft to be purchased is not limited. How many aircraft of each type should be purchased in order to maximize annual profit? SmallLargeCapital Available Annual profit$1 million$5 million Purchase cost$5 million$50 million$100 million Maximum purchase quantity2No maximum

3. Integer Programming Define Variables - Northern Airlines Let: S = # of Small Aircraft L = # of Large Aircraft

4. Integer Programming General Form - Northern Airlines Max 1S + 5L s.t. 5S + 50L <= 100 S <= L S, L >= 0 & Integer

5. Integer Programming Northern Airlines – Graph Solution Budget Small AC LP Relaxation (2, 1.8)

6. Integer Programming Integer Linear Programming All-Integer Linear Program AAll variables must be integers M ixed-Integer Linear Program SS ome, but not all variables must be integers 0-1 Integer Linear Program AAll variables must be 0 or 1, also known as binary variables

7. Integer Programming General Form - Northern Airlines Max 1S + 5L s.t. 5S + 50L <= 100 S <= L S, L >= 0 & L Integer

8. Integer Programming Northern Airlines – Graph Solution Budget Small AC

9. Integer Programming General Form - Northern Airlines Max 1S + 5L s.t. 5S + 50L <= 100 S <= L S, L >= 0 & S Integer

10. Integer Programming Northern Airlines – Graph Solution Budget Small AC

11. Integer Programming Integer Linear Programming All-Integer Linear Program AAll variables must be integers Mixed-Integer Linear Program SSome, but not all variables must be integers 0-1 Integer Linear Program AAll variables must be 0 or 1, also known as binary variables

12. Integer Programming 0-1 Integer Linear Program (Binary Integer Programming) Assists in selection process 11 corresponding to undertaking 00 corresponding to not undertaking

13. Integer Programming 0-1 Integer Linear Program (Binary Integer Programming) Allows for modeling flexibility through: MMultiple choice constraints k out of n alternatives constraint MMutually exclusive constraints CConditional & co-requisite constraint