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 2 n possibilities for n binary variables No really faster way known for some cases (NP hard problems) fame + $1m Clay prize for proving it 2 10 1024 2 40 Trillion 2 100 10 30

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 w i, value v i maximize the value in the knapsack s.t. weight limit B is not exceeded

7. Knapsack Problem max x 1 v 1 +…+x n v n s.t.x 1 w 1 +…+x n w n ≤ B x i binary x i = 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 x ij quantity shipped from i to j f ij binary (1 if x ij > 0) x ij ≥ 0, x ij ≤ 100 f ij fixed cost of 1000 f ij

12. Facility Location 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 Solution to Maximal Covering Problem w/ 10 facilities D c =300 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. Solution Note that none of the points you would get to by rounding (9,9) (10,9), (9,8), (10,8) is feasible!