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

Hideaki Takagi

doi:10.3934/jimo.2020064

Article Contents

2021, Volume 17, Issue 4: 2181-2202. Doi: 10.3934/jimo.2020064

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

Hideaki Takagi

Professor Emeritus, University of Tsukuba, Tsukuba Science City, Ibaraki 305-8573, Japan

Received: June 2019

Revised: September 2019

Early access: March 22, 2020

Published online: March 22, 2020

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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Abstract

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.

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

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

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

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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].

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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	$ \sigma _t / \mu _t $
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

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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

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