1. Previously Optimization Probability Review Inventory Models Markov Decision Processes Queues

2. Friday Finish simulation Review Post –Sample problems for final –Case to discuss on Monday –Office hour schedule

3. Agenda Simulation

4. Simulation (Ch 15) Interested in quantity X (it is random) Run simulation to get realizations of X: –X 1, X 2, X 3, …, X n Evaluate output: –look at averageE[X] ≈ AVERAGE(X 1,…,X n ) –standard deviation  [X] ≈ STDEV(X 1,…,X n ) –distribution of realizations

5. Agenda Confidence intervals –for output evaluation Generating realizations

6. Random Numbers ORMM Random Variables add-in –Add RV (to define random variable) –RV_sim(name,seed) or RV_simV(name) RiskSim add-in from book (p562) –functions randnormal, randuniform, …

7. Ex. Yield Management (15.7) How many plane tickets to sell? –2 types: full fare and discount –120 seats on plane –Discount tickets sold first Full FareDiscount Fare DemandN(50,20 2 )N(100,30 2 ) Contribution (per seated passenger) $400$150 P(no-show)15%5% Fee for re-booking$0$75 Cost of overbooking$125

8. Yield Management 1.Decide ticket availability –q Discount tickets –t Total tickets 2.Demand for Discount tickets –# Reservations –Tickets left 3.Demand for Full-fare tickets –# Reservations 4.# No shows on day of flight: 5.Calculate: –# Overbooked, # Passengers, Profits/Fees

9. Stochastic Optimization Approach 1: 1.Pick q, t 2.Estimate E[Profit(q,t)] 3.Go to step 1

10. Approach 2 1.Generate n scenarios 2.Calculate Profit(x) for each –x are the decision variables 3.Use Excel to optimize –maximize average Profit(x) –average over all scenarios

11. Asset Allocation Example R i = random return of asset class i x i = % allocation to asset class i –decision variable total return = ∑ i R i x i max P(total return > 3%)