Previously in chapter 4
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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

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Previously in Chapter 4

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Previously in chapter 4

Previously in Chapter 4

  • Assignment Problems

  • Network Flow Problems

  • Sequential Decision Problems

  • Vehicle Routing Problems

  • Transportation Problems

  • Staffing Problems

  • Production Problems


Agenda

Agenda

  • Quiz

  • Hardness

  • Modeling with Binary Variables

  • Issues with binary/integer variables

  • Rounding may fail


Previously in chapter 4

Quiz

  • 24 hour take-home

  • Posted noon Monday

  • Due by noon on Tuesday

  • Coverage: through today’s lecture


Hardness

Hardness

  • 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


Binary variables

Binary Variables

  • Piecewise linear functions

  • If statements

  • Discontinuous functions

  • Set Covering

  • Versions of the assignment problem


Knapsack problem

Knapsack Problem

  • n items

  • item i has weight wi, value vi

  • maximize the value in the knapsack

  • s.t. weight limit B is not exceeded


Knapsack problem1

Knapsack Problem

max x1v1+…+xnvn

s.t.x1w1+…+xnwn ≤ B

xi binary

xi = 1 if item i in the knapsack

NP hard problem


Penalty

Penalty

  • Operating coal plant

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

    (emissions always < 88kg/day)


Penalty1

Penalty

  • $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?


Fixed cost

Fixed Cost

  • Transportation Problem

  • Fixed cost of $1000 for any shipment

    (quantity shipped always less than 100)


Fixed cost1

Fixed Cost

  • xij quantity shipped from i to j

  • fij binary (1 if xij > 0)

  • xij ≥ 0, xij ≤ 100 fij

  • fixed cost of 1000 fij


Facility location

Facility Location

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


If statements part 2

If statements (Part 2)

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)


Other integer binary issues

Other Integer/Binary issues

  • Sensitivity Analysis

  • Relaxation

  • Rounding


Rounding may fail

Rounding may fail

Example courtesy of Prof. Daskin


Solution

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

is feasible!

Solution


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