1. Terminology Tight vs. Slack constraints Sign / Nonnegativity vs. normal constraints Shadow prices Reduced costs Agenda: –Shadow prices –Reduced costs –Duality

2. Shadow Prices (Change in objective) / (Change in constraint right hand side) 0 for loose constraints Applies to normal constraints Interpretation: –Price for additional capacity

3. Reduced Cost Shadow price for x >= 0 constraints 0 for variables that are positive Change in objective when variable set to 1 Making constraint tighter –Objective becomes worse –Contribution (profit from setting variable to 1) - opportunity cost (loss from reallocating resources)

4. Recreational Vehicles Example Shadow prices: –E (engine shop) = $140/hr –B (body shop) = $420/hr Reduced cost for L (luxury car) –Contribution $1200 –Opportunity cost, needs 1hr in engine shop + 3hr in body shop = 1E+3B=$1400 –Reduced cost = 1200-1400= $ -200

5. Excel Solver -> options -> “assume linear model” Solver -> Solve -> Sensitivity Report Concepts apply to nonlinear problems too –Shadow price = “Lagrange multiplier” –Reduced cost = “Reduced gradient”

6. Dual Problem Theory is works particularly for LPs Also an LP (when original is an LP) Another way of understanding problem Normal constraints become variables Variables become constraints Max becomes min …

7. Dual Problem Dual: mincost to buy all capacity s.t.willing to sell capacity instead of produce variables are prices Original: maxprofit from running plant s.t.capacity not exceeded variables are production quantities

8. Dual Problem Dual: minprice E * 120 hr engine shop capacity + … s.t.3hr * E + 1hr *B + 2hr * SF >= $840 (car profit) … variables E, B, SF, FF, FL are prices Original: max$840 profit * S cars + … s.t.3hr * S + 2hr * F + 1hr * L <= 120hr engine shop capacity … variables S, F, L are production quantities

9. Dual Problem max x p T x s.t. Ax <= c x >= 0 equivalent to min y c T y s.t.A T y >= p y >= 0