Many uses of linear programming, mixed
integer (linear) programming in this course
Game theory
Linear programming
Mixed integer linear
programming
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,
Winner determination in
expressive
auctions, exchanges, … with
marketplaces
partially acceptable bids
Mechanism
design
Automatically designing
optimal mechanisms that use
randomization
Winner determination in:
auctions, exchanges, …
without partially
acceptable bids; Kemeny,
Slater, other voting rules;
kidney exchange
Automatically designing
optimal mechanisms that
do not use randomization
Brief introduction to linear
and mixed integer
programming
Properties of Linear
Programming (LP) Models
1) Seek to minimize or maximize
2) Include “constraints” or limitations
3) There must be alternatives available
4) All equations are linear
Example LP Model Formulation:
The Product Mix Problem
Decision: How much to make of two
products?
Objective: Maximize profit
Constraints: Limited resources
Example
Two products: Chairs and Tables
Decision: How many of each to make this
month?
Objective: Maximize profit
Flair Furniture Co. Data
Tables
Chairs
(per table)
(per chair)
Profit
$7
Contribution
Carpentry Req 3 hrs
Painting Req
2 hrs
4 hrs
Hours
Available
2400
1 hr
1000
$5
Other Limitations:
• Make no more than 450 chairs
• Make at least 100 tables
Decision Variables:
T = Num. of tables to make
C = Num. of chairs to make
Objective Function: Maximize Profit
Maximize $7 T + $5 C
Constraints:
• Have 2400 hours of carpentry time
available
3 T + 4 C < 2400 (hours)
• Have 1000 hours of painting time
available
2 T + 1 C < 1000 (hours)
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
Model Summary
Maximize 7T + 5C
(profit)
Subject to the constraints:
3T + 4C < 2400 (carpentry hrs)
2T + 1C < 1000 (painting hrs)
C < 450 (max # chairs)
T
> 100 (min # tables)
T, C > 0
(nonnegativity)
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.
Carpentry
Constraint Line
C
3T + 4C = 2400
Infeasible
> 2400 hrs
600
Intercepts
(T = 0, C = 600)
(T = 800, C = 0)
Feasible
< 2400 hrs
0
0
800 T
C
1000
Painting
Constraint Line
2T + 1C = 1000
600
Intercepts
(T = 0, C = 1000)
(T = 500, C = 0)
0
0
500
800 T
Max Chair Line
C
1000
C = 450
Min Table Line
600
450
T = 100
Feasible
0
Region
0 100
500
800 T
C
Objective
Function Line
7T + 5C = Profit
Think of line of
solutions at a
particular profit.
Anywhere on
that line yields
the same profit
500
Optimal Point
(T = 320, C =
360)
400
300
200
100
0
0
100
200
300
400
500 T
C
Additional
Constraint
Need at least 75
more chairs than
tables
C > T + 75
New optimal point
T = 300, C = 375
500
400
T = 320
C = 360
No longer
feasible
300
Or
C – T > 75
200
100
0
0
100
200
300
400
500 T
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
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
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
Linear programs: example
• We make reproductions of
two paintings
maximize 30x
+ 20y
subject to
4x + 2y ≤ 16
x + 2y ≤ 8
• Painting 1 sells for $30, painting 2
sells for $20
x+y≤5
• Painting 1 requires 4 units of blue, 1
x≥0
green, 1 red
• Painting 2 requires 2 blue, 2 green, 1
y≥0
red
• We have 16 units blue, 8 green, 5 red
Solving the linear program graphically
maximize 30x +
20y
subject to
8
Think of dotted lines as various
values for 30x+20y
6
4x + 2y ≤ 16
x + 2y ≤ 8
4
optimal solution:
x=3, y=2
x+y≤5
x≥0
y≥0
2
0
2
4
6
8
Modified LP
maximize 30x +
20y
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 = 7.5 + 5 =
12.5
Half paintings? –
interested in integer
answers
Integer (linear) program
maximize 30x + 20y
8
subject to
4x + 2y ≤ 15
6
x + 2y ≤ 8
optimal IP
solution: x=2,
y=3
(objective 120)
optimal LP
solution: x=2.5,
y=2.5
(objective 125)
4
x+y≤5
x ≥ 0, integer
2
y ≥ 0, integer
0
2
4
6
8
Mixed integer (linear) program
maximize 30x + 20y
subject to
4x + 2y ≤ 15
8
6
x + 2y ≤ 8
x+y≤5
x≥0
4
optimal IP
solution: x=2,
y=3
(objective 120)
optimal LP
solution: x=2.5,
y=2.5
(objective 125)
optimal MIP
solution: x=2.75,
y=2
(objective 122.5)
2
y ≥ 0, integer
0
2
4
6
8
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)
Practice modeling hard problems
as LP or (M)IP problems
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?
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
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: 15
location: 1
location: 4
location: 7
location: 9
#customers: 3
#customers: 2
#customers: 1
#customers: 3
#customers: 4
willing to walk: 4 willing to walk: 2 willing to walk: 3 willing to walk: 3 willing to walk: 2