1 / 16

# Previously in Chapter 4 - PowerPoint PPT Presentation

Previously in Chapter 4. Assignment Problems Network Flow Problems Sequential Decision Problems Vehicle Routing Problems Transportation Problems Staffing Problems Production Problems. Agenda. Quiz Hardness Modeling with Binary Variables Issues with binary/integer variables

I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.
Download Presentation

## PowerPoint Slideshow about ' Previously in Chapter 4' - olesia

An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript

• Assignment Problems

• Network Flow Problems

• Sequential Decision Problems

• Vehicle Routing Problems

• Transportation Problems

• Staffing Problems

• Production Problems

• Quiz

• Hardness

• Modeling with Binary Variables

• Issues with binary/integer variables

• Rounding may fail

• 24 hour take-home

• Posted noon Monday

• Due by noon on Tuesday

• Coverage: through today’s lecture

• LP with n variables

• can be solved in √n matrix operations

• 2n possibilities for n binary variables

• No really faster way knownfor some cases (NP hard problems)

• fame + \$1m Clay prize for proving it

• Piecewise linear functions

• If statements

• Discontinuous functions

• Set Covering

• Versions of the assignment problem

• n items

• item i has weight wi, value vi

• maximize the value in the knapsack

• s.t. weight limit B is not exceeded

max x1v1+…+xnvn

s.t. x1w1+…+xnwn ≤ B

xi binary

xi = 1 if item i in the knapsack

NP hard problem

• Operating coal plant

• \$3000 penalty (per day) if emissions > b

(emissions always < 88kg/day)

• \$3000 penalty (per day) if emissions > b

(emissions always < 88kg/day)

• emissions p

• f binary

• p ≤ 88 + (b-88)f

• penalty: (1-f)3000

• unintended option?

• Transportation Problem

• Fixed cost of \$1000 for any shipment

(quantity shipped always less than 100)

• xij quantity shipped from i to j

• fij binary (1 if xij > 0)

• xij ≥ 0, xij ≤ 100 fij

• fixed cost of 1000 fij

Solution to Maximal Covering Problem w/ 10 facilities

Dc=300

Set covering – Find min. # needed to cover all demands

Max covering – Cover max # DEMANDS w/ fixed # facilities

P-center – Cover all demand nodes w/ fixed # facilities in smallest possible distance

Slide courtesy of Prof. Daskin

0 ≤x and

If x≤b, then y=c, else y=d

• create binary 0/1 variable z

• add the constraints

(b-x)/b ≤ z (if x≤b, then z=1)

z≤1+(b-x)/b (if x>b, then z=0)

y=cz+d(1-z) (if z=1, then y=c else y=d)

• Sensitivity Analysis

• Relaxation

• Rounding

Example courtesy of Prof. Daskin

Note that none of the points you would get to by rounding(9,9) (10,9), (9,8), (10,8)

is feasible!

Solution