Rafael Diaz

1. A Simulation Optimization Approach to Solve Stochastic Inventory Problems with Autocorrelated Demand Rafael Diaz Old Dominion University   Virginia Modeling, Analysis, & Simulation Center 1030 University  Blvd. Suffolk, VA 23435  Rafael Diaz - Old Dominion University

2. Content • Introduction and Motivation • Inventory Model • Simulation Optimization Method • Results • Conclusions and Future Work Rafael Diaz - Old Dominion University

3. Introduction • A simulation-based optimization technique: • Simulating Annealing • Pattern Search • Ranking and Selection used to approximate solutions to stochastic inventory models that consider autocorrelated demands. • Failing to capture the probabilistic properties of input processes that exhibit autocorrelation generates errors that can be characterized: • Regression analysis. Rafael Diaz - Old Dominion University

4. Motivation • Stochastic Inventory System • Rewards and inconvenience • Decisions and rules are designed to rationalize, coordinate, and control • Stochastic demand • Positively / Negatively autocorrelated. • Examples: • Electronic retailing, consumer goods, and grocery shops i.e. Erkip 1990, Lee et al. 2001. • Other authors: Ray (1980, 1981), Lau and Wang (1987), Fotopoulos et al (1988), Marmorstein and Zinn (1993), Charnes et al. (1995), and Urban (2000, 2005). • The difficulty: • Multivariate times series integration Rafael Diaz - Old Dominion University

5. Goal The purpose of this research: • Design a simulation-based optimization method to approximate solutions in terms of inventory policy. • Characterize the error generated by stochastic modeling techniques that ignore serial correlation components in the demand • lost-sale case inventory system, competitive markets. • Analyze the impact of ignoring autocorrelation components. • Average Total Costs • Stockouts Rafael Diaz - Old Dominion University

6. 1. Inventory Model and Autocorrelated Demands The lost sale case Rafael Diaz - Old Dominion University

7. The Inventory Problem • Control system • Continuous review • Stochastic demand • Serially-correlated • Inventory • System • Lost-sale • Minimizing Costs • Ordering • Penalty • Holding Rafael Diaz - Old Dominion University

8. Inventory X(i)‏ S z1 z2 s 0 1 2 i Assumptions and Inventory sequence xi yi Hold i Order zi Demand ξi i+1 Objective Function Ordering Decision (s, S) Constraints Penalty and Holding Decision ; ; Rafael Diaz - Old Dominion University

9. P_01 P_12 P_23 P_33 P_00 3 0 1 2 Autocorrelation Error Stochastic Demand Correlation boundaries π 1 π2 π 0 π3 3 0 1 2 Representing Discrete Markovian-modulated demand and Autoregressive AR(1) Autoregressive - AR(1)‏ • Given Transition Probabilities Autocorrelated Case Correlation-free Case Examples: electronics retail, grocery food Industry, and general Promotions

10. 2. Simulation-Based Optimization Rafael Diaz - Old Dominion University

11. Simulation-Based Optimization • Continuous: • Stochastic Approximation (gradient based methods)‏ • RSM. • Discrete: • Statistical selection • Random search • Metaheuristics • In this study: • Simulated Annealing • Pattern Search • R&S Optimization Procedure Input Simulation Model Output Rafael Diaz - Old Dominion University

12. Simulation-Based Optimization • Simulated Annealing (SA) • Large decision space. • Mathematically proven. • Generate, evaluate, and pre-select candidate solutions. • Pattern Search (PS) • 80’s and 90’s • Systematic exploring. • Additional neighbors. • Ranking and Selection (R&S) • Improve estimation MOP due to stochastic nature. • Evaluate incumbent and neighbors. • Related work • Simulated Tampering • SA and R&S (Ahmed & Alkhamis, 2002) • PS and R&S (Sriver & Chrissis, 2004) Rafael Diaz - Old Dominion University

13. Simulation Optimization Procedure Start Arbitrary Inventory Policy Inventory policy constrains Generate stochastic demand Continuous Demand Multivariate input parameters Inventory model Inventory input parameters: holding, ordering, and penalty cost Simulating Annealing parameters Pattern Search parameters R&S parameters Evaluation and selection False Termination Stopping criteria True End Rafael Diaz - Old Dominion University

14. Obtained by SA Obtained by PS Obtained by R&S Combining SA and PS &RS Decision Space Cost T1 T2 S T3 Sample Path T4 s Rafael Diaz - Old Dominion University

15. Numerical Experimentation Input Type Description 1. Inventory model Demand distribution 1.1.2. Continuous demand modeled as AR(1) process 1.2.1 Ordering = 1 1.2.2 Holding = 2.5 1.2.3 Penalty = 19 Costs 1.3 Maximum / minimum inventory level allowed in the system (s = 500; S = 8,000). Constraints 2.SAPSR&S algorithm 2.1.1 Maximum temperature (based on acceptation 98%)‏ 2.1.2 Temperature Gradient 2.1.3 Length of the stage (20,000 periods) 2.1.4 Stopping criteria Simulated Annealing 2.2.1 Step Size 2.2.2 Number of neighbors to explore per iteration = 3^2 Pattern Search 2.3.1 Indifference zone value 5%. 2.3.2 h based on the indifference value and the number of neighbor to explore 3.619 2.3.3 Initial number of replications R&S Rafael Diaz - Old Dominion University

16. 3. Results Rafael Diaz - Old Dominion University

17. Results – Ignoring Autocorrelation • Assuming IID demands • Ignoring autocorrelated demands Figure 1 Rafael Diaz - Old Dominion University

18. Results – Acknowledging Autocorrelation • Assuming IID demands • Considering autocorrelated demands Figure 2 Figure 3 Rafael Diaz - Old Dominion University

19. Total Costs and Stockouts Figure 5 Rafael Diaz - Old Dominion University Figure 6

20. Error Characterization Figure 6 Rafael Diaz - Old Dominion University

21. Analyzing the difference - ANOVA • H0 = The autocorrelated demand does not change the average total cost of the inventory system. • HA = The autocorrelated demand does change the average total cost of the inventory system • Significance level - 0.95 Rafael Diaz - Old Dominion University

22. Improving candidate solutions • Using PS combined with R&S improves candidate solutions. • The enhancement was defined and measured by: • The number of times that a candidate solution improved upon the evaluation process. • It enhances between 40-60% candidate solution proposal Figure 7 Rafael Diaz - Old Dominion University

23. Conclusions and Future Work Rafael Diaz - Old Dominion University

24. Summary • Ignoring φleads to errors. • Substantial and significant. • It shows a better performance. • Autocorrelation is high, • Stockouts are controlled. • Lower costs • It demonstrated 40%-60% improvement • evaluating and selecting candidate solutions. Rafael Diaz - Old Dominion University

25. Future work • Study other types of stochastic dependent demand. • Compare with other inventory models. • Study more complex supply chain problems. • Study the effect of autocorrelation demand in other settings, i.e. scheduling. Rafael Diaz - Old Dominion University

26. Thank you!!!! Rafael Diaz - Old Dominion University

27. Questions????? Rafael Diaz - Old Dominion University

28. Appendices Rafael Diaz - Old Dominion University

29. Autocorrelation • It refers to the correlation of a time series with its own past and future values. • +: positive/negative departures from the mean tend to be followed by positive/negative departures from the mean. • -: a tendency for positive departures to follow negative departures, and vice versa. • Tools: • the time series plot, • the lagged scatterplot, and • the autocorrelation function. Rafael Diaz - Old Dominion University

30. Mathematical formulation • Stochastic formulation • Continuous case • Discrete case • Service Level Rafael Diaz - Old Dominion University

31. Autocorrelation Error Stochastic Demand Correlation boundaries Generating AR(1) demands Autoregressive - AR(1)‏ . Examples: electronics retail, grocery food industry 1. Given values of the parameters , , and 2. Generate from the normal distribution with a given mean and variance 3. Set 4. Set and go to 2 Rafael Diaz - Old Dominion University