Robust Timing of Markdowns

My student Mike Dziecichowicz, former student Daniela Caro and I recently completed a paper on the robust timing of markdowns in revenue management. We apply robust optimization to the arrival rates of the demand processes, in an approach that does not require the knowledge of the underlying probability functions and instead incorporates range forecasts on those rates, as well as captures the degree of the manager's aversion to risk through intuitive budget-of-uncertainty functions. These budget functions bound the cumulative deviation of the arrival rates from their nominal values over the lengths of time for which a product is offered at a given price.

A key issue is that using lengths of time as decision variables - a necessity for problems on timing markdowns but a departure from the traditional robust optimization framework - introduces non-convexities in the problem formulation when budget functions are strictly concave. Concavity is a common assumption in the literature (it reflects that the longer a product is on sale, the more the various sources of uncertainty tend to cancel each other out, in the spirit of the law of large numbers, so that the marginal increase in time of the protection level decreases) and therefore must be incorporated in a tractable manner.

Specifically, we make the following contributions:

• We model uncertainty on the arrival rates of the demand processes through range forecasts and capture the manager's risk aversion through a "budget of uncertainty" function (of time on sale at a specific price point), which limits the cumulative deviation of the arrival rates from their mean and is determined by the decision-maker.
• In the nominal case and in the case where the budget of uncertainty function is linear in time, we provide closed-form solutions for the optimal sale time in the single-product case.
• In the case where the budget of uncertainty is concave and increasing, again for a single product, we derive a mixed-integer problem (MIP) that approximates the robust non-convex formulation.
• We develop a policy about the optimal time to put products on sale, which depends on both the number of items unsold and on the time-to-go, and use our robust optimization model to determine its parameters.
• We extend our analysis to the case of multiple products. In particular, we present the idea of constraint aggregation to maintain the performance of robust optimization for that problem structure.
• We provide numerical experiments to test the performance of the robust optimization approaches described in this paper.

The full paper can be downloaded here. Mike, a former NSF IGERT fellow, is a doctoral candidate who will be graduating next month (and is looking for a job - you can learn more about him here); Daniela received both her Bachelor and her Master's degrees from our department in 2008 and 2009, respectively (the Master's as a Presidential Scholar) and now works as a corporate pricing specialist at MillerCoors in Chicago. Feel free to email us any comments!

This research was funded through NSF Grant CMMI-0540143.

Comments

[Ilya Kipnis]

O_o...I'm hoping he has more luck than I am. 6 months and counting here and still nothing >_<.

In the meantime, if Daniela can write papers with you, got a bone to toss me, Prof. Thiele? I'd think my background would be relatively similar to Dani's =X...

(It doesn't have to be in finance either...just SOMETHING to pass the time that I can put on the resume =X)

(Oh, btw, I need recommendations for UNC/Duke/UMich :P. Hopefully you haven't forgotten (I know you're busy).

[Aurelie C. Thiele]

Hi Ilya,
I'll send something your way asap!
Also, I haven't forgotten about your recommendation letters - things here have been hectic, but they're at the top of my to-do list. I'm expecting to have submitted everything before Thanksgiving.
I'll be in touch soon.