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Extension of Littlewood's rule to the multi-period static revenue management model with standby customers

The author is supported by the Grant-in-Aid for Scientific Research (C) No. 17K00435 from the Japan Society for the Promotion of Science (JSPS) in 2018. This work was presented in the 2019 INFORMS Revenue Management and Pricing Conference, Graduate School of Business, Stanford University, Stanford, California, U.S.A., June 6–7, 2019

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  • 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.

    Mathematics Subject Classification: Primary: 90B50; Secondary: 91B26, 90B06.

    Citation:

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  • Figure 1.  Two-period static revenue management model with standby customers

    Figure 2.  Domain $ \{ D _0 > y _0 ^* , D _0 + D _1 > y _1 ^* \} $ for the two-period model

    Table 1.  A variety of terms used for two classes of customers in the literature on the two-period static revenue management model

    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
    $ {} ^* $Also Talluri [11,p. 663] and van Ryzin and Talluri [13].
     | Show Table
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    Table 2.  The fare and parameters of demand in each period used in the numerical example

    $ t $ th Fare Mean Standard $ \sigma _t / \mu _t $
    period $ r _t \quad $ $ \mu _t \; $ $ \sigma _t \; $ 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
     | Show Table
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    Table 3.  Optimization of booking limits in the 2-, 3-, and 4-period static revenue management models with standby customers. $ C = 107 $ is the total number of seats

    (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
     | Show Table
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    [8] R. PhillipsPricing and Revenue Optimization, Stanford University Press, 2005. 
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    [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.
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