An Evaluation of Heuristic Methods for Determining the Best Table Mix in Full-Service Restaurants...
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Transcript of An Evaluation of Heuristic Methods for Determining the Best Table Mix in Full-Service Restaurants...
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
7
6
5
4
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
10
15
20
25
30
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