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Extension of Littlewood's rule to the multi-period static revenue management model with standby customers
Professor Emeritus, University of Tsukuba, Tsukuba Science City, Ibaraki 305-8573, Japan |
Classical Littlewood's rule (1972) for the two-period static revenue management of a single perishable resource is extended to a generic $ T $-period model with monotonically increasing fixed fares, ending with standby customers with a special fare. The expected revenue in the entire period is expressed explicitly in terms of multiple definite integrals involving the distribution function of the demand in each period. The exact optimal protection level in each period is calculated successively, resulting in the maximized total expected revenue. The Brumelle-McGill's theorem for the optimal booking limits in the $ T $-period model is also extended to a similar model with standby customers. We show some numerical examples with comments on the effects of accepting standby customers on the optimal booking limits and the increase in the expected revenue.
References:
[1] |
P. P. Belobaba, Air Travel Demand and Airline Seat Inventory Management, Ph.D thesis, Massachusetts Institute of Technology, 1987. |
[2] |
S. L. Brumelle and J. I. McGill,
Airline seat allocation with multiple nested fare classes, Operations Research, 41 (1993), 127-137.
doi: 10.1287/opre.41.1.127. |
[3] |
R. E. Curry,
Optimal airline seat allocation with fare classes nested by origins and
destinations, Transportation Science, 24 (1990), 169-243.
doi: 10.1287/trsc.24.3.193. |
[4] |
M. Z. F. Li and T. H. Oum,
A note on the single leg, multifare seat allocation problem, Transportation Science, 36 (2002), 271-354.
doi: 10.1287/trsc.36.3.349.7830. |
[5] |
K. Littlewood,
Forecasting and control of passenger bookings, J. Revenue Pricing Manag., 4 (2005), 111-123.
|
[6] |
M. Müller-Bungart, Revenue Management with Flexible Products: Models and Methods for the Broadcasting Industry, Springer-Verlag Berlin Heidelberg, 2007. |
[7] |
S. Netessine and R. Shumsky,
Introduction to the theory and practice of yield management, INFORMS Transactions on Education, 3 (2002), 34-44.
doi: 10.1287/ited.3.1.34. |
[8] |
R. Phillips, Pricing and Revenue Optimization, Stanford University Press, 2005.
![]() |
[9] |
L. W. Robinson,
Optimal and approximate control policies for airline booking with sequential
nonmonotonic fare classes, Operations Research, 43 (1995), 252-263.
doi: 10.1287/opre.43.2.252. |
[10] |
H. Takagi, Explicit calculation of optimal booking limits for the static revenue management with standby customers, in Conference Proceedings, Joint International Conference of Service Science and Innovation and Serviceology, ICSSI2018 and ICServe2018, Taichung, Taiwan, (2018), 119–126. |
[11] |
K. T. Talluri, Revenue management, in The Oxford Handbook of Pricing Management (eds. Ö. Özer and R. Phillips), Oxford University Press, (2012), 655–678.
doi: 10.1093/oxfordhb/9780199543175.013.0026. |
[12] |
K. T. Talluri and G. J. van Ryzin, The Theory and Practice of Revenue Management, International Series in Operations Research & Management Science, vol. 68, Kluwer Academic Publishers, Boston, MA, 2004.
doi: 10.1007/b139000. |
[13] |
G. J. van Ryzin and K. T. Talluri, An introduction to revenue management, Tutorials in Operations Research, (2005), 142–194.
doi: 10.1287/educ.1053.0019. |
[14] |
D. Walczak, E. A. Boyd and R. Cramer, Revenue management, in Quantitative Problem Solving Methods in the Airline Industry (eds. C. Barnhart and B. Smith), Springer, Boston, MA, (2012), 101–161.
doi: 10.1007/978-1-4614-1608-1_3. |
[15] |
R. D. Wollmer,
An airline seat management model for a single leg route when lower fare
classes book first, Operations Research, 40 (1992), 26-37.
doi: 10.1287/opre.40.1.26. |
show all references
References:
[1] |
P. P. Belobaba, Air Travel Demand and Airline Seat Inventory Management, Ph.D thesis, Massachusetts Institute of Technology, 1987. |
[2] |
S. L. Brumelle and J. I. McGill,
Airline seat allocation with multiple nested fare classes, Operations Research, 41 (1993), 127-137.
doi: 10.1287/opre.41.1.127. |
[3] |
R. E. Curry,
Optimal airline seat allocation with fare classes nested by origins and
destinations, Transportation Science, 24 (1990), 169-243.
doi: 10.1287/trsc.24.3.193. |
[4] |
M. Z. F. Li and T. H. Oum,
A note on the single leg, multifare seat allocation problem, Transportation Science, 36 (2002), 271-354.
doi: 10.1287/trsc.36.3.349.7830. |
[5] |
K. Littlewood,
Forecasting and control of passenger bookings, J. Revenue Pricing Manag., 4 (2005), 111-123.
|
[6] |
M. Müller-Bungart, Revenue Management with Flexible Products: Models and Methods for the Broadcasting Industry, Springer-Verlag Berlin Heidelberg, 2007. |
[7] |
S. Netessine and R. Shumsky,
Introduction to the theory and practice of yield management, INFORMS Transactions on Education, 3 (2002), 34-44.
doi: 10.1287/ited.3.1.34. |
[8] |
R. Phillips, Pricing and Revenue Optimization, Stanford University Press, 2005.
![]() |
[9] |
L. W. Robinson,
Optimal and approximate control policies for airline booking with sequential
nonmonotonic fare classes, Operations Research, 43 (1995), 252-263.
doi: 10.1287/opre.43.2.252. |
[10] |
H. Takagi, Explicit calculation of optimal booking limits for the static revenue management with standby customers, in Conference Proceedings, Joint International Conference of Service Science and Innovation and Serviceology, ICSSI2018 and ICServe2018, Taichung, Taiwan, (2018), 119–126. |
[11] |
K. T. Talluri, Revenue management, in The Oxford Handbook of Pricing Management (eds. Ö. Özer and R. Phillips), Oxford University Press, (2012), 655–678.
doi: 10.1093/oxfordhb/9780199543175.013.0026. |
[12] |
K. T. Talluri and G. J. van Ryzin, The Theory and Practice of Revenue Management, International Series in Operations Research & Management Science, vol. 68, Kluwer Academic Publishers, Boston, MA, 2004.
doi: 10.1007/b139000. |
[13] |
G. J. van Ryzin and K. T. Talluri, An introduction to revenue management, Tutorials in Operations Research, (2005), 142–194.
doi: 10.1287/educ.1053.0019. |
[14] |
D. Walczak, E. A. Boyd and R. Cramer, Revenue management, in Quantitative Problem Solving Methods in the Airline Industry (eds. C. Barnhart and B. Smith), Springer, Boston, MA, (2012), 101–161.
doi: 10.1007/978-1-4614-1608-1_3. |
[15] |
R. D. Wollmer,
An airline seat management model for a single leg route when lower fare
classes book first, Operations Research, 40 (1992), 26-37.
doi: 10.1287/opre.40.1.26. |


Literature | class 1 customers | class 2 customers |
Littlewood [5] | high-yield passengers | low-yield passengers |
Müller-Bungart [6,p. 55] | high fare passengers | low fare passengers |
Netessine and Shumsky [7] | business customers | leisure customers |
Phillips [8,p. 149] | full-fare customers | discount customers |
Talluri and van Ryzin [12] |
class 1 demands | class 2 demands |
Walczak et al. [14,p. 133] | high fare demand | low fare demand |
Literature | class 1 customers | class 2 customers |
Littlewood [5] | high-yield passengers | low-yield passengers |
Müller-Bungart [6,p. 55] | high fare passengers | low fare passengers |
Netessine and Shumsky [7] | business customers | leisure customers |
Phillips [8,p. 149] | full-fare customers | discount customers |
Talluri and van Ryzin [12] |
class 1 demands | class 2 demands |
Walczak et al. [14,p. 133] | high fare demand | low fare demand |
Fare | Mean | Standard | ||||
period | deviation | |||||
0 | Variable | 10.0 | 10.0000 | 2.0 | 2.0000 | 0.2000 |
1 | 105 | 20.3 | 20.5135 | 8.6 | 8.3414 | 0.4236 |
2 | 83 | 33.4 | 33.9289 | 15.1 | 14.4936 | 0.4521 |
3 | 57 | 19.3 | 19.7139 | 9.2 | 8.7453 | 0.4767 |
4 | 39 | 29.7 | 30.1047 | 13.1 | 12.6264 | 0.4411 |
Fare | Mean | Standard | ||||
period | deviation | |||||
0 | Variable | 10.0 | 10.0000 | 2.0 | 2.0000 | 0.2000 |
1 | 105 | 20.3 | 20.5135 | 8.6 | 8.3414 | 0.4236 |
2 | 83 | 33.4 | 33.9289 | 15.1 | 14.4936 | 0.4521 |
3 | 57 | 19.3 | 19.7139 | 9.2 | 8.7453 | 0.4767 |
4 | 39 | 29.7 | 30.1047 | 13.1 | 12.6264 | 0.4411 |
(a) 2-period model with standby customers (expected total demand = 64.442). | ||||||
r0 | b1* | b2* | R(b1*,b2*) | S(b1*,b2*) | ||
150 | 98.04880 | 82.53349 | 6465.337 | 64.40167 | ||
120 | 99.30070 | 83.08922 | 6165.701 | 64.40301 | ||
106 | 101.69624 | 83.55498 | 6025.950 | 64.40352 | ||
105 | 107 | 83.61577 | 6015.975 | 64.40354 | ||
90 | 107 | 85.47684 | 5866.468 | 64.40366 | ||
83 | 107 | 86.34620 | 5796.699 | 64.40369 | ||
50 | 107 | 89.94355 | 5467.795 | 64.403745 | ||
30 | 107 | 91.58374 | 5268.461 | 64.403750 | ||
0 | 107 | 93.43602 | 4969.460 | 64.403753 | ||
(b) 3-period model with standby customers (expected total demand = 84.156). | ||||||
r0 | b1* | b2* | b3* | R(b1*,b2*,b3*) | S(b1*,b2*,b3*) | |
150 | 98.04880 | 82.53349 | 46.72013 | 7468.847 | 82.9636 | |
120 | 99.30070 | 83.08922 | 47.23458 | 7174.891 | 82.9934 | |
106 | 101.69624 | 83.55498 | 47.63984 | 7039.305 | 83.0073 | |
105 | 107 | 83.61577 | 47.68707 | 7029.778 | 83.0082 | |
90 | 107 | 85.47684 | 48.90773 | 6890.287 | 83.0211 | |
83 | 107 | 86.34620 | 49.52548 | 6825.326 | 83.0250 | |
50 | 107 | 89.94355 | 52.73081 | 6824.932 | 83.0332 | |
30 | 107 | 91.58374 | 54.77039 | 6334.970 | 83.0345 | |
0 | 107 | 93.43602 | 57.73468 | 6057.888 | 83.0352 | |
(c) 4-period model with standby customers (expected total demand = 114.261). | ||||||
r0 | b1* | b2* | b3* | b4* | R(b1*,b2*,b3*,b4*) | S(b1*,b2*,b3*,b4*)) |
150 | 98.049 | 82.533 | 46.720 | 18.50316 | 7864.765 | 95.6543 |
120 | 99.301 | 83.089 | 47.235 | 19.00966 | 7438.307 | 96.0016 |
106 | 101.696 | 83.555 | 47.640 | 19.40404 | 7312.431 | 96.2417 |
105 | 107 | 83.616 | 47.687 | 19.44909 | 7304.006 | 96.2670 |
90 | 107 | 85.477 | 48.908 | 20.57752 | 7191.889 | 96.8510 |
83 | 107 | 86.346 | 49.525 | 21.15184 | 7141.103 | 97.1230 |
50 | 107 | 89.944 | 52.731 | 24.21901 | 6914.435 | 98.3425 |
30 | 107 | 91.584 | 54.770 | 26.29302 | 6786.543 | 98.9860 |
0 | 107 | 93.436 | 57.735 | 29.54009 | 6606.416 | 99.7556 |
(a) 2-period model with standby customers (expected total demand = 64.442). | ||||||
r0 | b1* | b2* | R(b1*,b2*) | S(b1*,b2*) | ||
150 | 98.04880 | 82.53349 | 6465.337 | 64.40167 | ||
120 | 99.30070 | 83.08922 | 6165.701 | 64.40301 | ||
106 | 101.69624 | 83.55498 | 6025.950 | 64.40352 | ||
105 | 107 | 83.61577 | 6015.975 | 64.40354 | ||
90 | 107 | 85.47684 | 5866.468 | 64.40366 | ||
83 | 107 | 86.34620 | 5796.699 | 64.40369 | ||
50 | 107 | 89.94355 | 5467.795 | 64.403745 | ||
30 | 107 | 91.58374 | 5268.461 | 64.403750 | ||
0 | 107 | 93.43602 | 4969.460 | 64.403753 | ||
(b) 3-period model with standby customers (expected total demand = 84.156). | ||||||
r0 | b1* | b2* | b3* | R(b1*,b2*,b3*) | S(b1*,b2*,b3*) | |
150 | 98.04880 | 82.53349 | 46.72013 | 7468.847 | 82.9636 | |
120 | 99.30070 | 83.08922 | 47.23458 | 7174.891 | 82.9934 | |
106 | 101.69624 | 83.55498 | 47.63984 | 7039.305 | 83.0073 | |
105 | 107 | 83.61577 | 47.68707 | 7029.778 | 83.0082 | |
90 | 107 | 85.47684 | 48.90773 | 6890.287 | 83.0211 | |
83 | 107 | 86.34620 | 49.52548 | 6825.326 | 83.0250 | |
50 | 107 | 89.94355 | 52.73081 | 6824.932 | 83.0332 | |
30 | 107 | 91.58374 | 54.77039 | 6334.970 | 83.0345 | |
0 | 107 | 93.43602 | 57.73468 | 6057.888 | 83.0352 | |
(c) 4-period model with standby customers (expected total demand = 114.261). | ||||||
r0 | b1* | b2* | b3* | b4* | R(b1*,b2*,b3*,b4*) | S(b1*,b2*,b3*,b4*)) |
150 | 98.049 | 82.533 | 46.720 | 18.50316 | 7864.765 | 95.6543 |
120 | 99.301 | 83.089 | 47.235 | 19.00966 | 7438.307 | 96.0016 |
106 | 101.696 | 83.555 | 47.640 | 19.40404 | 7312.431 | 96.2417 |
105 | 107 | 83.616 | 47.687 | 19.44909 | 7304.006 | 96.2670 |
90 | 107 | 85.477 | 48.908 | 20.57752 | 7191.889 | 96.8510 |
83 | 107 | 86.346 | 49.525 | 21.15184 | 7141.103 | 97.1230 |
50 | 107 | 89.944 | 52.731 | 24.21901 | 6914.435 | 98.3425 |
30 | 107 | 91.584 | 54.770 | 26.29302 | 6786.543 | 98.9860 |
0 | 107 | 93.436 | 57.735 | 29.54009 | 6606.416 | 99.7556 |
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