1. Agenda Office Hours: Friday 2-3 Final: Friday 4pm - Sat. 4pm

2. LP Shadow Prices (Lec 4) (Change in objective) / (Change in constraint right hand side) 0 for loose constraints Interpretation: –Price for additional capacity Excel –Solver -> options -> “assume linear model” –Solver -> Solve -> Sensitivity Report

3. LP Piecewise linear (Lec 6) As a constraint: c(x) <= b Add decision variable z and replace constraint with z <= b and x s 1 + t 1 <= z x s 2 + t 2 <= z x s 2 + t 2 x s 1 + t 1

4. Diseconomy of Scale quantity revenue or profit quantity cost mathematically equivalent

5. More Optimization Assignment Problem Integer Programs / IF statements Lec 10 slides reviewed for the quiz

6. Inventory Models Newsvendor Uncertain Demand (D) Single-period Specification –q = # to have on hand –b = contribution per sale –c = cost per unsold item P(D ≤ q*) = b/(b+c) –round q* up to nearest integer Base Stock Uncertain Demand (D) Multi-period –Inventory –Lost-sales p= Service level Probability of running out P(D ≤ q*) = p Safety Stock = q- E[D] = constant √E[D]

7. Order Quantity Model (EOQ) Deterministic Demand Continuous review –Inventory –No backlogging Solution –Reorder when inventory at r = AL –Order size q* =(2AK/H) 1/2 (cycle stock, Economic Order Quantity) Specification Replenishment lead time L Order placement cost K (Independent of order size) Unit holding cost H

8. Markov Decision Processes (MDP) States i=1,…,n Possible actions in each state Reward R(i,k) of doing action k in state i Law of motion: P(j | i,k) probability of moving i  j after doing action k

9. current + future profit of doing action k in state i MDP as LP f(i) = largest expected current + future profit if currently in state i f(i) decision variables in LP min ∑ j f(i) s.t. f(i) ≥ R(i,k) + ∑ j P(j|i,k) f(j) for all i,k Tight if k is optimal action for state i