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

Related Research

• Revenue Management: Optimal demand mix to maximize revenue

• Capacity Planning: Optimal supply mix to minimize cost

Research Problem

What is the supply mix

that will maximize revenue?

Supply Mix Problems

• Restaurants

• Airlines

• Performing arts centers

• Self-storage facilities

• Hotels

Factors Affecting Table Mix

• Space constraints

• Party characteristics

• Layout

• Table combinability

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

0

10

20

30

40

50

60

70

10:4

511

:15

11:4

512

:15

12:4

513

:15

13:4

514

:15

14:4

515

:15

15:4

516

:15

16:4

517

:15

17:4

518

:15

18:4

519

:15

19:4

520

:15

20:4

521

:15

21:4

522

:15

22:4

523

:15

Ave

rag

e N

um

ber

of

Par

ties

Sunday

Weekday

Weekend

Party Arrival Rate by 15-minute Period

0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

Sun Mon Tue Wed Thu Fri Sat

% o

f A

ll P

arti

es

10+

9

8

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3

2

1

Party Size Mix by Day of Week.

Party Size Sun Mon Tue Wed Thu Fri Sat

1 40.5 38.3 43.0 35.7 40.6 48.3 42.9

2 45.5 47.0 47.8 47.5 47.9 47.4 46.4

3 48.4 53.4 52.6 52.9 49.7 53.7 50.5

4 51.1 52.6 52.8 56.7 53.0 55.7 54.9

5 54.5 62.3 60.3 59.5 66.1 55.4 55.2

6 69.9 67.0 61.8 66.3 59.8 63.5 61.0

7 68.7 73.5 83.0 74.1 58.8 59.8 62.4

8 62.6 76.2 89.0 59.5 92.5 74.8 72.2

9 64.3 51.4 61.2 67.8 59.8 83.2 71.8

10 64.3 51.4 61.2 67.8 59.8 83.2 71.8

Mean Dining Duration (Minutes) by Party Size and Day of Week

0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

11 12 13 14 15 16 17 18 19 20 21

Hour of Day

Sea

t O

ccu

pan

cy

Sunday

Monday

Tuesday

Wednesday

Thursday

Friday

Saturday

Seat Occupancy by Day of Week and Time of Day

Our Approach

• TableMix simulation used for complete enumeration

• Increased demand level

• Experiments– Maximize revenue by day of week– Maximize revenue over the entire week

0

5

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20

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35

4:30 PM 4:45 PM 5:00 PM 5:15 PM 5:30 PM 5:45 PM 6:00 PM 6:15 PM 6:30 PM 6:45 PM 7:00 PM 7:15 PM 7:30 PM 7:45 PM

Time Period

Par

ty A

rriv

al R

ate

(per

15-

min

ute

s)

Sun

Mon

Tue

Wed

Thu

Fri

Sat

Daily Party Arrival Ratesby 15-Minute Period

Complete Enumeration

• Evaluated every combination of table mix

• 13,561 possible combinations– 105 within 1% of optimal– 292 within 2% of optimal

0

200

400

600

800

1000

1200

1400

1600

1800

2000

$20,000 $22,000 $24,000 $26,000 $28,000 $30,000 $32,000 $34,000 $36,000 $38,000 $40,000 $42,000 $44,000

Revenue

Nu

mb

er o

f S

olu

tio

ns

Weekly Revenue Distribution Across all Table Mixes

Impact of Results

• Restaurant adopted one of the “near” optimal table mixes

• Revenue increased by 2.1%

• Payback expected within 14 months

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?

Methods Tested

• Integer programming– Naïve– Time-based– Revenue management based

• Simulated annealing

Factors Considered

Factor Naïve IP-A Naïve IP-B TimeIP RevMgtIP SimAnnealDemand Level x x xMean Dining Time x x x xVariance of Time xParty Value x x xParty Mix x x x x xCapacity x x x x x

General Policies

• No combinability (Thompson 2002)

• Table assignment rules

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

Solution Times (minutes)

aOn a Pentium IV 2.0 GHz personal computer.bModel solved using SAS-OR ®.cTo evaluate 100 table mixes.

Method

Solution Time per Single Daya

Solution Time per Weeka

Enum 139.00 973.00 NaiveIPAb 0.01 0.01 NaïveIPBb 0.02 0.02 TimeIPb 0.05 0.18 RevMgtIP15b 0.02 0.03 RevMgtIP5b 0.02 0.73 RevMgtIP3b 0.13 7.58 SimAnnealc 1.16 6.05

Recommended Table Mixes

Method Sun Mon Tue Wed Thu Fri Sat Whole Week

Enum 50-23-4-3 59-23-5-0 67-22-3-0 59-22-3-2 55-23-5-1 52-24-4-2 51-23-5-2 56-24-4-1 NaiveIP-A 46-22-6-3 57-20-5-2 64-20-4-1 59-21-5-1 55-23-5-1 51-23-5-2 49-24-5-2 53-22-5-2

NaïveIP-B 42-22-6-4 51-20-7-2 59-21-5-1 50-22-6-2 50-24-6-1 45-24-5-3 44-23-6-3 50-22-6-2

TimeIP 46-24-6-2 58-22-6-0 65-23-3-0 57-22-5-1 49-26-5-1 47-26-5-2 47-25-5-2 52-25-6-0

RevMgtIP15 47-20-7-3 54-24-6-0 62-21-4-1 52-19-6-3 51-25-5-1 47-26-3-3 46-25-4-3 48-23-6-2

RevMgtIP5 46-20-6-4 55-22-7-0 68-20-4-0 57-20-5-2 54-22-6-1 47-27-5-1 47-23-5-3 49-24-5-2

RevMgtIP3 45-22-5-4 54-24-6-0 67-20-3-1 57-17-7-2 51-22-7-1 50-25-4-2 47-23-5-3 50-22-6-2

SimAnneal-S* 49-22-5-3

59-23-5-0 (45)

67-22-3-0 (35)

59-22-3-2 (78)

55-23-5-1 (98)

55-26-3-1

51-23-5-2 (33)

56-24-4-1 (80)

SimAnneal-N* 50-23-4-3 (31)

59-23-5-0 (41)

67-22-3-0 (55)

59-22-3-2 (33)

57-22-5-1

52-24-4-2 (64)

51-23-5-2 (4)

56-24-4-1 (47)

Existing 2-56-2-0 2-56-2-0 2-56-2-0 2-56-2-0 2-56-2-0 2-56-2-0 2-56-2-0 2-56-2-0

Percentage of Optimal

Method Sun Mon Tue Wed Thu Fri Sat Whole Week

Single-Day Total

Enum 100.0% 100.0% 100.0% 100.0% 100.0% 100.0% 100.0% 100.00% 100.00% NaiveIP-A 99.1% 99.1% 98.9% 99.9% 100.0% 99.6% 99.9% 99.60% 99.49%

NaïveIP-B 98.4% 97.0% 97.3% 97.6% 99.3% 97.5% 98.0% 98.85% 97.86% TimeIP 99.4% 99.8% 99.7% 99.9% 99.5% 99.6% 99.6% 99.48% 99.63%

RevMgtIP15 99.4% 98.9% 98.5% 97.7% 99.9% 98.8% 99.0% 98.32% 98.89% RevMgtIP5 99.2% 99.1% 99.9% 99.6% 99.9% 99.1% 99.1% 98.89% 99.41%

RevMgtIP3 99.2% 98.9% 99.6% 98.9% 99.2% 99.8% 99.1% 98.85% 99.26% SimAnneal-S 99.9% 100.0% 100.0% 100.0% 100.0% 99.6% 100.0% 100.00% 99.93%

SimAnneal-N 100.0% 100.0% 100.0% 100.0% 99.9% 100.0% 100.0% 100.00% 99.98% Existing 77.3% 73.2% 70.3% 74.0% 75.1% 75.9% 76.2% 75.4% N/A

Single Day Premiums

Single Day Premium

Enum 1.1%NaïveIP-A 0.5%NaïveIP-B 0.1%TimeIP 1.3%RevMgtIP15 1.7%RevMgtIP5 1.6%RevMgtIP3 1.5%SimAnneal-S 1.0%SimAnneal-N 1.1%

Results

• All methods within 2% of optimal1. Simulated Annealing

2. NaïveIP-A

3. NaïveIP-B

4. Time IP

5. RM IP

• Optimizing by day of week provides a 1.1% premium

Discussion

• Naïve IP-A performed very well

• How well would it hold up in different operating situations?

Factors to be Tested

Factor Levels Description Naïve IP-A Naïve IP-B SimAnnealMeal Duration Difference 2 Low, High No Yes YesMean Party Size 2 2.5, 3.5 Yes Yes YesDemand Intensity 2 100%, 120% No No YesCoefficient of Variation 2 0.3, 0.5 No No YesAverage Check Difference 2 Low, High No No YesRestaurant Size 3 50, 200, 1000 Yes Yes Yes

6 12 96Total Experiments

Is Factor Included in Model?

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

Future Research

• Restaurant industry– Optimal station size

• Other industries– Optimal supply mix– Revenue impact of optimal supply mix