# American Institute of Mathematical Sciences

doi: 10.3934/jimo.2020064

## 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

Received  June 2019 Revised  September 2019 Published  March 2020

Fund Project: 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

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.

Citation: Hideaki Takagi. Extension of Littlewood's rule to the multi-period static revenue management model with standby customers. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2020064
##### References:

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##### References:
Two-period static revenue management model with standby customers
Domain $\{ D _0 > y _0 ^* , D _0 + D _1 > y _1 ^* \}$ for the two-period model
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].
 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].
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
 $t$ 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
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
 (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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