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 Rounding may fail

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

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 Trillion

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

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

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

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

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

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

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

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 Sensitivity Analysis Relaxation Rounding

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!