1. An Evaluation of Heuristic Methods for Determining the Best Table Mix in Full-Service Restaurants Sheryl E. Kimes and Gary M. Thompson Cornell University CORNELL

2. Related Research Revenue Management: Optimal demand mix to maximize revenue Capacity Planning: Optimal supply mix to minimize cost

3. Research Problem What is the supply mix that will maximize revenue?

4. Supply Mix Problems Restaurants Airlines Performing arts centers Self-storage facilities Hotels

5. Factors Affecting Table Mix Space constraints Party characteristics Layout Table combinability

6. Problem Setting 240-seat restaurant in busy shopping center in California On a wait every night 2 two-tops, 56 4-tops and 2 6-tops Over 60% of parties are parties of 1 or 2 Mean dining time = 49.5 minutes Average check = $13.88/person

7. Party Arrival Rate by 15-minute Period

8. Party Size Mix by Day of Week.

9. Party SizeSunMonTueWedThuFriSat 140.538.343.035.740.648.342.9 245.547.047.847.547.947.446.4 348.453.452.652.949.753.750.5 451.152.652.856.753.055.754.9 554.562.360.359.566.155.455.2 669.967.061.866.359.863.561.0 768.773.583.074.158.859.862.4 862.676.289.059.592.574.872.2 964.351.461.267.859.883.271.8 1064.351.461.267.859.883.271.8 Mean Dining Duration (Minutes) by Party Size and Day of Week

10. Seat Occupancy by Day of Week and Time of Day

11. Our Approach TableMix simulation used for complete enumeration Increased demand level Experiments –Maximize revenue by day of week –Maximize revenue over the entire week

12. Daily Party Arrival Rates by 15-Minute Period

15. Complete Enumeration Evaluated every combination of table mix 13,561 possible combinations –105 within 1% of optimal –292 within 2% of optimal

16. Weekly Revenue Distribution Across all Table Mixes

17. Impact of Results Restaurant adopted one of the “near” optimal table mixes Revenue increased by 2.1% Payback expected within 14 months

18. Discussion of Results Profit impact was high Complete enumeration impractical in larger restaurants What other methods could be used and how would they perform? Was it worthwhile to reconfigure the restaurant from day to day?

19. Methods Tested Integer programming –Naïve –Time-based –Revenue management based Simulated annealing

20. Factors Considered

21. General Policies No combinability (Thompson 2002) Table assignment rules

22. NaïveIP-A

23. NaïveIP-B

24. Time IP for

25. RevMgtIP for

26. Simulated Annealing Temperature parameter decremented every 2 iterations (100 iteration limit) Ensured that we never evaluated the same mix twice. No particular tuning of the parameters (e.g. temp, cooling, DropProp, probabilities of selecting different table sizes). Two approaches –SimAnneal-S –SimAnneal-N

27. Solution Times (minutes) a On a Pentium IV 2.0 GHz personal computer. b Model solved using SAS-OR ®. c To evaluate 100 table mixes.

28. Recommended Table Mixes

29. Percentage of Optimal

30. Single Day Premiums

31. Results All methods within 2% of optimal 1.Simulated Annealing 2.NaïveIP-A 3.NaïveIP-B 4.Time IP 5.RM IP Optimizing by day of week provides a 1.1% premium

32. Discussion Naïve IP-A performed very well How well would it hold up in different operating situations?

33. Factors to be Tested

34. Results

35. Discussion Factors influencing performance –Duration difference –Party size –Restaurant size –Demand intensity Factors not influencing performance –Average check differences –Coefficient of variation

36. Summary and Conclusion An improved supply mix can help increase revenue Simulated annealing provided the best solution NaïveIP-A within 0.5% of optimal, but... Reoptimizing by day provided a 1.1% premium

37. Future Research Restaurant industry –Optimal station size Other industries –Optimal supply mix –Revenue impact of optimal supply mix