Many uses of linear programming, mixed integer (linear) programming in this course

Linear programming Mixed integer linear programming

Game theory

Dominated strategiesMinimax strategies

Correlated equilibriumOptimal mixed strategies to

commit to

Nash equilibriumOptimal 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

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. DataTables

(per table)Chairs

(per chair)Hours

AvailableProfit Contribution $7 $5

Carpentry Req 3 hrs 4 hrs 2400Painting Req 2 hrs 1 hr 1000

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

tablesT > 0C > 0

Model SummaryMaximize 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.

CarpentryConstraint Line

3T + 4C = 2400

Intercepts

(T = 0, C = 600)

(T = 800, C = 0)

0 800 T

C

600

0

Feasible< 2400 hrs

Infeasible> 2400 hrs

3T + 4C = 2400

PaintingConstraint Line

2T + 1C = 1000

Intercepts

(T = 0, C = 1000)

(T = 500, C = 0)

0 500 800 T

C1000

600

0

2T + 1C = 1000

0 100 500 800 T

C1000

600450

0

Max Chair Line

C = 450

Min Table Line

T = 100

FeasibleRegion

0 100 200 300 400 500 T

C

500

400

300

200

100

0

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 = $2,1007T + 5C = $4,040

Optimal Point(T = 320, C =

360)

7T + 5C = $2,800

0 100 200 300 400 500 T

C

500

400

300

200

100

0

Additional Constraint

Need at least 75 more chairs than tables

C > T + 75

Or

C – T > 75

T = 320C = 360

No longer feasible

New optimal pointT = 300, C = 375

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: exampleIf 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;

• We make reproductions of two paintings

• 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: examplemaximize 30x

+ 20y

subject to4x + 2y ≤ 16

x + 2y ≤ 8

x + y ≤ 5

x ≥ 0

y ≥ 0

• We make reproductions of two paintings

• 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

Solving the linear program graphically

maximize 30x + 20y

subject to4x + 2y ≤ 16

x + 2y ≤ 8

x + y ≤ 5

x ≥ 0

y ≥ 0

2

0

4

6

8

2 4 6 8

optimal solution: x=3, y=2

Think of dotted lines as various values for 30x+20y

Modified LPmaximize 30x +

20y

subject to4x + 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) programmaximize 30x + 20y

subject to

4x + 2y ≤ 15

x + 2y ≤ 8

x + y ≤ 5

x ≥ 0, integer

y ≥ 0, integer

2

0

4

6

8

2 4 6 8

optimal LP solution: x=2.5,

y=2.5 (objective 125)

optimal IP solution: x=2,

y=3 (objective 120)

Mixed integer (linear) programmaximize 30x + 20y

subject to

4x + 2y ≤ 15

x + 2y ≤ 8

x + y ≤ 5

x ≥ 0

y ≥ 0, integer

2

0

4

6

8

2 4 6 8

optimal LP solution: x=2.5,

y=2.5 (objective 125)

optimal IP solution: x=2,

y=3 (objective 120)

optimal MIP solution: x=2.75,

y=2 (objective 122.5)

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: 4#customers: 1

willing to walk: 2

location: 7#customers: 3

willing to walk: 3

location: 9#customers: 4

willing to walk: 3

location: 1#customers: 2

willing to walk: 4

location: 15#customers: 3

willing to walk: 2