Abstract
We study a dynamic pricing problem, where a firm offers a product to be sold over a fixed time horizon. The firm has a given initial inventory level, but there is uncertainty about the demand for the product in each time period. The objective of the firm is to determine a dynamic pricing strategy that maximizes revenue throughout the entire selling season. We develop a tractable optimization model that directly uses demand data, therefore creating a practical decision tool. We show computationally that regret‐based objectives can perform well when compared to average revenue maximization and to a Bayesian approach. The modeling approach proposed in this study could be particularly useful for risk‐averse managers with limited access to historical data or information about the true demand distribution. Finally, we provide theoretical performance guarantees for this sampling‐based solution.
Get full access to this article
View all access options for this article.
References
1.
Adida
E.
Perakis
G.
. 2006 . A robust optimization approach to dynamic pricing and inventory control with no backorders . Math. Program. 107 (1 ): 97 –129 .
2.
Araman
V. F.
Caldentey
R.
. 2009 . Dynamic pricing for non‐perishable products with demand learning . Oper. Res. 57 (5 ): 1169 –1188 .
3.
Aviv
Y.
Pazgal
A.
. 2002 . Pricing of short life‐cycle products through active learning . Working paper.
4.
Ball
M. O.
Queyranne
M.
. 2009 . Toward robust revenue management: Competitive analysis of online booking . Oper. Res. 57 (4 ): 950 –963 .
5.
Ben‐Tal
A.
Nemirovski
A.
. 2000 . Robust solutions of linear programming problems contaminated with uncertain data . Math. Program. 88 : 411 –424 .
6.
Ben‐Tal
A.
Goryashko
A.
Guslitzer
E.
Nemirovski
A.
. 2004 . Adjustable robust solutions of uncertain linear programs . Math. Program. 99 (2 ): 351 –376 .
7.
Ben‐Tal
A.
Golany
B.
Nemirovski
A.
Vial
J.
. 2005 . Retailer‐supplier exible commitments contracts: A robust optimization approach . Manuf. Serv. Oper. Manag. 7 (3 ): 248 –271 .
8.
Bertsimas
D.
Sim
M.
. 2004 . The price of robustness . Oper. Res. 52 (1 ): 35 –53 .
9.
Bertsimas
D.
Thiele
A.
. 2006 . A robust optimization approach to supply chain management . Oper. Res. 54 (1 ): 150 –168 .
10.
Bertsimas
D.
Vayanos
P.
. 2017 . Data‐driven learning in dynamic pricing using adaptive optimization . Working paper.
11.
Bertsimas
D.
Iancu
D.
Parrilo
P.
. 2010 . Optimality of affine policies in multi‐stage robust optimization . Math. Oper. Res. 35 (2 ): 363 –394 .
12.
Bertsimas
D.
Iancu
D. A.
Parrilo
P. A.
. 2011 . A hierarchy of near‐optimal policies for multistage adaptive optimization . IEEE Trans. Autom. Control 56 (12 ): 2809 –2824 .
13.
Calafiore
G.
2009 . On the expected probability of constraint violation in sampled convex programs . J. Optim. Theory Appl. 143 (2 ): 405 –412 .
14.
Calafiore
G.
Campi
M.
. 2005 . Uncertain convex programs: Randomized solutions and confidence levels . Math. Program. 102 : 25 –46 .
15.
Calafiore
G.
Campi
M.
. 2006 . The scenario approach to robust control design . IEEE Trans. Autom. Control 51 (5 ): 742 –753 .
16.
Campi
M.
Garatti
S.
. 2008 . The exact feasibility of randomized solutions of uncertain convex programs . SIAM J. Optim. 19 (3 ): 1211 –1230 .
17.
Caramanis
C.
2006 . Adaptable optimization: Theory and algorithms . Ph.D dissertation, Department of Electrical Engineering and Computer Science, MIT .
18.
Elmaghraby
W.
Keskinocak
P.
. 2003 . Dynamic pricing: Research overview, current practices and future directions . Management Sci. 49 : 1287 –1309 .
19.
Eren
S. S.
Maglaras
C.
. 2010 . Monopoly pricing with limited demand information . J. Revenue Pricing Manage. 9 : 23 –48 .
20.
Farias
V. F.
Van Roy
B.
. 2010 . Dynamic pricing with a prior on market response . Oper. Res. 58 (1 ): 16 –29 .
21.
Gallego
G.
Moon
I.
. 1993 . The distribution free newsboy problem: Review and extensions . J. Oper. Res. Soc. 44 (8 ): 5 .
22.
Gaur
V.
Kesavan
S.
Raman
A.
Fisher
M.
. 2007 . Estimating demand uncertainty using judgmental forecasts . Manuf. Serv. Oper. Manag. 9 (4 ): 480 –491 .
23.
Kleywegt
A.
Shapiro
A.
Homem‐de Mello
T.
. 2001 . The sample average approximation method for stochastic discrete optimization . SIAM J. Optim. 12 (2 ): 479 –502 .
24.
Lim
A.
Shanthikumar
J.
Watewai
T.
. 2009 . Robust multi‐product pricing and a new class of risk‐sensitive control problems . Working paper.
25.
Lim
A. E.
Shanthikumar
J. G.
Vahn
G. Y.
. 2012 . Robust portfolio choice with learning in the framework of regret: Single‐period case . Management Sci. 58 (9 ): 1732 –1746 .
26.
Lin
K. Y.
2006 . Dynamic pricing with real‐time demand learning . Eur. J. Oper. Res. 174 (1 ): 522 –538 .
27.
Lobel
R.
2012 . Pricing and incentive design in applications of green technology subsidies and revenue management . Ph.D. dissertation, Operations Research Center, MIT .
28.
Luenberger
D. G.
1969 . Optimization by Vector Space Methods . John Wiley & Sons , New York .
29.
Pagnoncelli
B.
Ahmed
S.
Shapiro
A.
. 2009 . Computational study of a chance‐constrained portfolio problem . J. Optim. Theory Appl. 142 (2 ): 399 –416 .
30.
Perakis
G.
Roels
G.
. 2010 . Robust controls for network revenue management . Manuf. Serv. Oper. Manag. 12 (1 ): 56 –76 .
31.
Rusmevichientong
P.
Van Roy
B.
Glynn
P. W.
. 2006 . A non‐parametric approach to multi‐product pricing . Oper. Res. 54 (1 ): 89 –98 .
32.
Talluri
K. T.
Van Ryzin
G. J.
. 2006 . The Theory and Practice of Revenue Management , vol. 68. Springer Science & Business Media , New York, NY .
33.
Vayanos
P.
Kuhn
D.
Rustem
B.
. 2012 . A constraint sampling approach for multi‐stage robust optimization . Automatica 48 (3 ): 459 –471 , ISSN 0005‐1098.