2450 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 57, NO. 10, OCTOBER 2012 Choice Prediction With Semidefinite Optimization When Utilities are Correlated Vinit Kumar Mishra, Karthik Natarajan, Member, IEEE, Hua Tao, and Chung-Piaw Teo Abstract—We consider the problem of making choice prediction by optimizing the expectation of maximum utility in discrete choice situations, and propose a discrete choice model which generates choice probabilities through a semidefinite program. The choice model, termed as the cross moment model (CMM) is parsimonious in that it uses only the mean and covariance information of the util- ities. It is encouraging that CMM generates reasonable choice es- timates when utilities are correlated even though no distributional assumptions on random utilities are made. We present a few examples in route choice setting and random walk to test the quality of choice prediction using CMM. By capturing correlations among utilities, CMM avoids some of the common behavioral limitations, such as the Independence of Irrel- evant Alternatives and the Invariate Proportion of Substitution, present in several discrete choice models. Being a convex opti- mization problem, it obviates the use of exhaustive simulation in computing choice probabilities for which the multinomial probit (MNP) model is often criticized. CMM can be easily used in design problems such as product-line selection and assortment planning. We exploit CMM to solve a flexible packaging design problem faced by online retailers and warehouses. We use the data provided by a local service part supplier for this design problem and compare the results with Multinomial Logit and MNP models. We find that CMM not only captures utility correlations in this problem and provides good designs but also has computational advantages over MNP which uses simulation. Index Terms—Cross moment model (CMM), multinomial probit (MNP). I. INTRODUCTION C ONSIDER a set of alternatives and a set of consumers where each consumer is indexed with at- tributes . The utility that the consumer associates with alterna- tive is . From the point of view of a researcher, this utility is random. Under the standard additive random utility model, the utility of alternative is specified as Manuscript received April 01, 2010; revised February 07, 2011; accepted September 19, 2011. Date of publication August 02, 2012; date of current ver- sion September 21, 2012. Recommended by Associate Editor C.-H. Chen. V. K. Mishra, H. Tao, and C.-P. Teo are with the Department of Deci- sion Sciences, NUS Business School, National University of Singapore, Singapore, 117591 (e-mail: vinit_mishra@nus.edu.sg; bizteocp@nus.edu.sg; jenny.sendero@gmail.com). K. Natarajan is with the Department of Engineering Systems and De- sign, Singapore University of Technology and Design, Singapore (e-mail: natarajan_karthik@sutd.edu.sg). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAC.2012.2211175 The deterministic component of the utility is the part which the researcher can describe through various observable alternative attributes such as price, cost, etc., and consumer at- tributes such as age, income, etc. All the unobserved factors (id- iosyncracies) are captured through the random component of the utility . Each consumer chooses the alternative with the highest utility with the researcher interested in estimating the probability of a consumer choosing an alternative. The prob- ability that a consumer with attributes chooses alternative (choice probability) is Based on the assumptions on the joint distribution of the random error terms, varying choice models can be generated that are consistent with the utility maximizing behavior of consumers. For ease of exposition, we focus on customers with fixed attributes and suppress the dependence of the results on . When are i.i.d. type-1 extreme value distributed random variables, the choice model is the Multinomial Logit (MNL) model, popularized by Luce [20] and McFadden [22], among others. Under MNL, the cumulative distribution function of each error term is assumed to be with the choice probability equal to This choice model however suffers from the Independence of Irrelevant Alternatives (IIA) property, namely, the ratio of choice probabilities for any two alternatives is unaffected by the presence of other alternatives. When alternatives share many common attributes, MNL tends to exaggerate the market shares. There are plenty of examples in practice where the utility evaluation is not independent across alternatives. In transportation choice problems such as airline networks (Bront et al. [5]), slight variation in features are used to distinguish alternatives. In general, in these problem settings, different resources are combined to provide for the configuration of different alternatives (e.g., each resource corresponds to a single-leg flight, and an alternative is defined as an itinerary and fare-class combination). The sharing of common resources can result in high correlations in utilities among the alternatives. In these circumstances, a model which ignores the correlation among the utilities associated with the alternatives can give inaccurate choice probabilities. 0018-9286/$31.00 © 2012 IEEE