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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. Quiz Hardness Modeling with Binary Variables Issues with binary/integer variables

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Previously in Chapter 4

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  1. Previously in Chapter 4 • Assignment Problems • Network Flow Problems • Sequential Decision Problems • Vehicle Routing Problems • Transportation Problems • Staffing Problems • Production Problems

  2. Agenda • Quiz • Hardness • Modeling with Binary Variables • Issues with binary/integer variables • Rounding may fail

  3. Quiz • 24 hour take-home • Posted noon Monday • Due by noon on Tuesday • Coverage: through today’s lecture

  4. 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

  5. Binary Variables • Piecewise linear functions • If statements • Discontinuous functions • Set Covering • Versions of the assignment problem

  6. 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

  7. Knapsack Problem max x1v1+…+xnvn s.t. x1w1+…+xnwn ≤ B xi binary xi = 1 if item i in the knapsack NP hard problem

  8. Penalty • Operating coal plant • $3000 penalty (per day) if emissions > b (emissions always < 88kg/day)

  9. 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?

  10. Fixed Cost • Transportation Problem • Fixed cost of $1000 for any shipment (quantity shipped always less than 100)

  11. 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

  12. 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

  13. 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)

  14. Other Integer/Binary issues • Sensitivity Analysis • Relaxation • Rounding

  15. Rounding may fail Example courtesy of Prof. Daskin

  16. 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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