1. Mixed integer linear programming
Many uses of linear programming, mixed integer (linear) programming in this course
Linear programming
Mixed integer linear programming
Game theory
Dominated strategies
Minimax strategies
Correlated equilibrium
Optimal mixed strategies to commit to
Nash equilibrium
Optimal mixed strategies to commit to in more complex settings
Social choice, expressive marketplaces
Winner determination in auctions, exchanges, … with partially acceptable bids
Winner determination in: auctions, exchanges, … without partially acceptable bids; Kemeny, Slater, other voting rules; kidney exchange
Mechanism design
Automatically designing optimal mechanisms that use randomization
Automatically designing optimal mechanisms that do not use randomization

2. Brief introduction to linear and mixed integer programming

3. Properties of Linear Programming (LP) Models
Seek to minimize or maximize
Include “constraints” or limitations
There must be alternatives available
All equations are linear

4. Example LP Model Formulation: The Product Mix Problem
Decision: How much to make of two products?
Objective: Maximize profit
Constraints: Limited resources

5. Two products: Chairs and Tables
Example
Two products: Chairs and Tables
Decision: How many of each to make this month?
Objective: Maximize profit

6. Flair Furniture Co. Data
Tables
(per table)
Chairs
(per chair)
Hours Available
Profit Contribution $7 $5
Carpentry Req
3 hrs
4 hrs
2400
Painting Req
2 hrs
1 hr
1000
Other Limitations:
Make no more than 450 chairs
Make at least 100 tables

7. Decision Variables:
T = Num. of tables to make
C = Num. of chairs to make
Objective Function: Maximize Profit
Maximize $7 T + $5 C

8. Constraints: Have 2400 hours of carpentry time available
3 T C < (hours)
Have 1000 hours of painting time available
2 T C < (hours)

9. Make no more than 450 chairs C < 450 (num. chairs)
More Constraints:
Make no more than 450 chairs
C < 450 (num. chairs)
Make at least 100 tables
T > 100 (num. tables)
Nonnegativity:
Cannot make a negative number of chairs or tables
T > 0
C > 0

10. Model Summary Maximize 7T + 5C (profit)
Subject to the constraints:
3T + 4C < 2400 (carpentry hrs)
2T + 1C < 1000 (painting hrs)
C < (max # chairs)
T > (min # tables)
T, C > 0 (nonnegativity)

11. Graphical Solution
Graphing an LP model helps provide insight into LP models and their solutions.
While graphing can only be done in two or three dimensions, the same properties apply to all LP models and solutions.

12. Carpentry Constraint Line 3T + 4C = 2400 Intercepts (T = 0, C = 600)
Carpentry
Constraint Line
3T + 4C = 2400
Intercepts
(T = 0, C = 600)
(T = 800, C = 0)
Infeasible
> 2400 hrs
3T + 4C = 2400
Feasible
< 2400 hrs T

13. Painting Constraint Line 2T + 1C = 1000 Intercepts (T = 0, C = 1000)
600
Painting
Constraint Line
2T + 1C = 1000
Intercepts
(T = 0, C = 1000)
(T = 500, C = 0)
2T + 1C = 1000 T

14. Max Chair Line C = 450 Min Table Line T = 100 Feasible Region C 1000
600
450
Max Chair Line
C = 450
Min Table Line
T = 100
Feasible
Region T

15. Objective Function Line 7T + 5C = Profit
500
400
300
200
100
Objective Function Line
7T + 5C = Profit
Think of line of solutions at a particular profit.
Anywhere on that line yields the same profit
7T + 5C = $4,040
Optimal Point
(T = 320, C = 360)
7T + 5C = $2,800
7T + 5C = $2,100 T

16. Additional Constraint
500
400
300
200
100
Additional Constraint
Need at least 75 more chairs than tables
C > T + 75 Or
C – T > 75
New optimal point
T = 300, C = 375
T = 320
C = 360
No longer feasible T

17. LP Characteristics
Feasible Region: The set of points that satisfies all constraints
Corner Point Property: An optimal solution must lie at one or more corner points
Optimal Solution: The corner point with the best objective function value is optimal

18. In higher dimensions
In multiple dimensions, the region of interest is of higher dimension.
polytope – a polygon in higher dimension - a geometric object with flat sides
simplex – a generalization of the notion of a triangle or tetrahedron to arbitrary dimension – a convex hull
Algorithmic method – optimal value will always be on a vertex of the polytope – walking along edges to vertices of higher objective function

20. Linear programs: example
We make reproductions of two paintings
If x is the number of painting 1 and y is the number of painting 2, state this poroblem as a series of equations of requirements: Something to maximize and inequalities to satisfy;
Painting 1 sells for $30, painting 2 sells for $20
Painting 1 requires 4 units of blue, 1 green, 1 red
Painting 2 requires 2 blue, 2 green, 1 red
We have 16 units blue, 8 green, 5 red

21. Linear programs: example
We make reproductions of two paintings
maximize 30x + 20y
subject to
4x + 2y ≤ 16
x + 2y ≤ 8
x + y ≤ 5
x ≥ 0
y ≥ 0
Painting 1 sells for $30, painting 2 sells for $20
Painting 1 requires 4 units of blue, 1 green, 1 red
Painting 2 requires 2 blue, 2 green, 1 red
We have 16 units blue, 8 green, 5 red

22. Solving the linear program graphically
maximize 30x + 20y
subject to
4x + 2y ≤ 16
x + 2y ≤ 8
x + y ≤ 5
x ≥ 0
y ≥ 0 8
Think of dotted lines as various values for 30x+20y 6 4
optimal solution: x=3, y=2 2 2 4 6 8

23. Modified LP maximize 30x + 20y Optimal solution: x = 2.5, y = 2.5
subject to
4x + 2y ≤ 15
x + 2y ≤ 8
x + y ≤ 5
x ≥ 0
y ≥ 0
Optimal solution: x = 2.5, y = 2.5
Solution value = = 12.5
Half paintings? – interested in integer answers

24. Integer (linear) program
maximize 30x + 20y
subject to
4x + 2y ≤ 15
x + 2y ≤ 8
x + y ≤ 5
x ≥ 0, integer
y ≥ 0, integer 8
optimal IP solution: x=2, y=3
(objective 120) 6
optimal LP solution: x=2.5, y=2.5
(objective 125) 4 2 4 6 8 2

25. Mixed integer (linear) program
maximize 30x + 20y
subject to
4x + 2y ≤ 15
x + 2y ≤ 8
x + y ≤ 5
x ≥ 0
y ≥ 0, integer 8
optimal IP solution: x=2, y=3
(objective 120) 6
optimal LP solution: x=2.5, y=2.5
(objective 125) 4
optimal MIP solution: x=2.75, y=2
(objective 122.5) 2 4 6 8 2

26. Solving linear/integer programs
Linear programs can be solved efficiently
Simplex, ellipsoid, interior point methods…
(Mixed) integer programs are NP-hard to solve
Quite easy to model many standard NP-complete problems as integer programs (try it!)
Search type algorithms such as branch and bound
Standard packages for solving these
GNU Linear Programming Kit, CPLEX, …
LP relaxation of (M)IP: remove integrality constraints
Gives upper bound on MIP (~admissible heuristic)

27. Practice modeling hard problems as LP or (M)IP problems

28. Exercise in modeling: knapsack-type problem
We arrive in a room full of precious objects
Can carry only 30kg out of the room
Can carry only 20 liters out of the room
Want to maximize our total value
Unit of object A: 16kg, 3 liters, sells for $11
There are 3 units available
Unit of object B: 4kg, 4 liters, sells for $4
There are 4 units available
Unit of object C: 6kg, 3 liters, sells for $9
Only 1 unit available
What should we take?

29. Exercise in modeling: cell phones (set cover)
We want to have a working phone in every continent (besides Antarctica)
… but we want to have as few phones as possible
Phone A works in NA, SA, Af
Phone B works in E, Af, As
Phone C works in NA, Au, E
Phone D works in SA, As, E
Phone E works in Af, As, Au
Phone F works in NA, E

30. Exercise in modeling: hot-dog stands
We have two hot-dog stands to be placed in somewhere along the beach
We know where the people that like hot dogs are, how far they are willing to walk
Where do we put our stands to maximize #hot dogs sold? (price is fixed)
location: 1
#customers: 2
willing to walk: 4
location: 4
#customers: 1
willing to walk: 2
location: 7
#customers: 3
willing to walk: 3
location: 9
#customers: 4
willing to walk: 3
location: 15
#customers: 3
willing to walk: 2