{"title": "Assortment Optimization Under the Mallows model", "book": "Advances in Neural Information Processing Systems", "page_first": 4700, "page_last": 4708, "abstract": "We consider the assortment optimization problem when customer preferences follow a mixture of Mallows distributions. The assortment optimization problem focuses on determining the revenue/profit maximizing subset of products from a large universe of products; it is an important decision that is commonly faced by retailers in determining what to offer their customers. There are two key challenges: (a) the Mallows distribution lacks a closed-form expression (and requires summing an exponential number of terms) to compute the choice probability and, hence, the expected revenue/profit per customer; and (b) finding the best subset may require an exhaustive search. Our key contributions are an efficiently computable closed-form expression for the choice probability under the Mallows model and a compact mixed integer linear program (MIP) formulation for the assortment problem.", "full_text": "Assortment Optimization Under the Mallows model\n\nAntoine D\u00e9sir\nIEOR Department\nColumbia University\n\nantoine@ieor.columbia.edu\n\nSrikanth Jagabathula\n\nIOMS Department\n\nNYU Stern School of Business\nsjagabat@stern.nyu.edu\n\nVineet Goyal\n\nIEOR Department\nColumbia University\n\nvgoyal@ieor.columbia.edu\n\nDanny Segev\n\nDepartment of Statistics\n\nUniversity of Haifa\n\nsegevd@stat.haifa.ac.il\n\nAbstract\n\nWe consider the assortment optimization problem when customer preferences\nfollow a mixture of Mallows distributions. The assortment optimization problem\nfocuses on determining the revenue/pro\ufb01t maximizing subset of products from a\nlarge universe of products; it is an important decision that is commonly faced by\nretailers in determining what to offer their customers. There are two key challenges:\n(a) the Mallows distribution lacks a closed-form expression (and requires summing\nan exponential number of terms) to compute the choice probability and, hence, the\nexpected revenue/pro\ufb01t per customer; and (b) \ufb01nding the best subset may require an\nexhaustive search. Our key contributions are an ef\ufb01ciently computable closed-form\nexpression for the choice probability under the Mallows model and a compact\nmixed integer linear program (MIP) formulation for the assortment problem.\n\n1\n\nIntroduction\n\nDetermining the subset (or assortment) of items to offer is a key decision problem that commonly\narises in several application contexts. A concrete setting is that of a retailer who carries a large\nuniverse of products U but can offer only a subset of the products in each store, online or of\ufb02ine. The\nobjective of the retailer is typically to choose the offer set that maximizes the expected revenue/pro\ufb01t1\nearned from each arriving customer. Determining the best offer set requires: (a) a demand model and\n(b) a set optimization algorithm. The demand model speci\ufb01es the expected revenue from each offer\nset, and the set optimization algorithm \ufb01nds (an approximation of) the revenue maximizing subset.\nIn determining the demand, the demand model must account for product substitution behavior,\nwhereby customers substitute to an available product (say, a dark blue shirt) when her most preferred\nproduct (say, a black one) is not offered. The substitution behavior makes the demand for each\noffered product a function of the entire offer set, increasing the complexity of the demand model.\nNevertheless, existing work has shown that demand models that incorporate substitution effects\nprovide signi\ufb01cantly more accurate predictions than those that do not. The common approach to\ncapturing substitution is through a choice model that speci\ufb01es the demand as the probability P(a|S)\nof a random customer choosing product a from offer set S. The most general and popularly studied\nclass of choice models is the rank-based class [9, 24, 12], which models customer purchase decisions\nthrough distributions over preference lists or rankings. These models assume that in each choice\ninstance, a customer samples a preference list specifying a preference ordering over a subset of the\n1As elaborated below, conversion-rate maximization can be obtained as a special case of revenue/pro\ufb01t\n\nmaximization by setting the revenue/pro\ufb01t of all the products to be equal.\n\n30th Conference on Neural Information Processing Systems (NIPS 2016), Barcelona, Spain.\n\n\fproducts, and chooses the \ufb01rst available product on her list; the chosen product could very well be\nthe no-purchase option.\nThe general rank-based model accommodates distributions with exponentially large support sizes\nand, therefore, can capture complex substitution patterns; however, the resulting estimation and\ndecision problems become computationally intractable. Therefore, existing work has focused on\nvarious parametric models over rankings. By exploiting the particular parametric structures, it has\ndesigned tractable algorithms for estimation and decision-making. The most commonly studied\nmodels in this context are the Plackett-Luce (PL) [22] model and its variants, the nested logit (NL)\nmodel and the mixture of PL models. The key reason for their popularity is that the assumptions\nmade in these models (such as the Gumbel assumption for the error terms in the PL model) are\ngeared towards obtaining closed-form expressions for the choice probabilities P(a|S). On the other\nhand, other popular models in the machine learning literature such as the Mallows model have\nlargely been ignored because computing choice probabilities under these models has been generally\nconsidered to be computationally challenging, requiring marginalization of a distribution with an\nexponentially-large support size.\nIn this paper, we focus on solving the assortment optimization problem under the Mallows model.\nThe Mallows distribution was introduced in the mid-1950\u2019s [17] and is the most popular member\nof the so-called distance-based ranking models, which are characterized by a modal ranking !\nand a concentration parameter \u2713. The probability that a ranking is sampled falls exponentially\nas e\u2713\u00b7d(,!). Different distance functions result in different models. The Mallows model uses\nthe Kendall-Tau distance, which measures the number of pairwise disagreements between the two\nrankings. Intuitively, the Mallows model assumes that consumer preferences are concentrated around\na central permutation, with the likelihood of large deviations being low.\nWe assume that the parameters of the model are given. Existing techniques in machine learning\nmay be applied to estimate the model parameters. In settings of our interest, data are in the form of\nchoice observations (item i chosen from offer set S), which are often collected as part of purchase\ntransactions. Existing techniques focus on estimating the parameters of the Mallows model when the\nobservations are complete rankings [8], partitioned preferences [14] (which include top-k/bottom-k\nitems), or a general partial-order speci\ufb01ed in the form of a collection of pairwise preferences [15].\nWhile the techniques based on complete rankings and partitioned preferences don\u2019t apply to this\ncontext, the techniques proposed in [15] can be applied to infer the model parameters.\n\nOur results. We address the two key computational challenges that arise in solving our problem:\n(a) ef\ufb01ciently computing the choice probabilities and hence, the expected revenue/pro\ufb01t, for a given\noffer set S and (b) \ufb01nding the optimal offer set S\u21e4. Our main contribution is to propose two alternate\nprocedures to to ef\ufb01ciently compute the choice probabilities P(a|S) under the Mallows model. As\nelaborated below, even computing choice probabilities is a non-trivial computational task because it\nrequires marginalizing the distribution by summing it over an exponential number of rankings. In\nfact, computing the probability of a general partial order under the Mallows model is known to be a\n#P hard problem [15, 3]. Despite this, we show that the Mallows distribution has rich combinatorial\nstructure, which we exploit to derive a closed-form expression for the choice probabilities that takes\nthe form of a discrete convolution. Using the fast Fourier transform, the choice probability expression\ncan be evaluated in O(n2 log n) time (see Theorem 3.2), where n is the number of products. In\nSection 4, we exploit the repeated insertion method (RIM) [7] for sampling rankings according to\nthe Mallows distribution to obtain a dynamic program (DP) for computing the choice probabilities\nin O(n3) time (see Theorem 4.2). The key advantage of the DP speci\ufb01cation is that the choice\nprobabilities are expressed as the unique solution to a system of linear equations. Based on this\nspeci\ufb01cation, we formulate the assortment optimization problem as a compact mixed linear integer\nprogram (MIP) with O(n) binary variables and O(n3) continuous variables and constraints. The\nMIP provides a framework to model a large class of constraints on the assortment (often called\n\u201cbusiness constraints\") that are necessarily present in practice and also extends to mixture of Mallows\nmodel. Using a simulation study, we show that the MIP provides accurate assortment decisions in a\nreasonable amount of time for practical problem sizes.\nThe exact computation approaches that we propose for computing choice probabilities are necessary\nbuilding blocks for our MIP formulation. They also provide computationally ef\ufb01cient alternatives to\ncomputing choice probabilities via Monte-Carlo simulations using the RIM sampling method. In fact,\nthe simulation approach will require exponentially many samples to obtain reliable estimates when\n\n2\n\n\fproducts have exponentially small choice probabilities. Such products commonly occur in practice\n(such as the tail products in luxury retail). They also often command high prices because of which\ndiscarding them can signi\ufb01cantly lower the revenues.\n\nLiterature review. A large number of parametric models over rankings have been extensively\nstudied in the areas of statistics, transportation, marketing, economics, and operations management\n(see [18] for a detailed survey of most of these models). Our work particularly has connections to the\nwork in machine learning and operations management. The existing work in machine learning has\nfocused on designing computationally ef\ufb01cient algorithms for estimating the model parameters from\ncommonly available observations (complete rankings, top-k/bottom-k lists, pairwise comparisons,\netc.). The developed techniques mainly consist of ef\ufb01cient algorithms for computing the likelihood of\nthe observed data [14, 11] and sampling techniques for sampling from the distributions conditioned\non observed data [15, 20]. The Plackett-Luce (PL) model, the Mallows model, and their variants have\nbeen, by far, the most studied models in this literature. On the other hand, the work in operations\nmanagement has mainly focused on designing set optimization algorithms to \ufb01nd the best subset\nef\ufb01ciently. The multinomial logit (MNL) model has been the most commonly studied model in this\nliterature. The MNL model was made popular by the work of [19] and has been shown by [25] to be\nequivalent to the PL model, introduced independently by Luce [16] and Plackett [22]. When given\nthe model parameters, the assortment optimization problem has been shown to be ef\ufb01ciently solvable\nfor the MNL model by [23], variants of the nested logit (NL) model by [6, 10], and the Markov chain\nmodel by [2]. The problem is known to be hard for most other choice models [4], so [13] studies the\nperformance of the local search algorithm for some of the assortment problems that are known to be\nhard. As mentioned, the literature in operations management has restricted itself to only those models\nfor which choice probabilities are known to be ef\ufb01ciently computed. In the context of the literature,\nour key contribution is to extend a popular model in the machine learning literature to choice contexts\nand the assortment problem.\n\n2 Model and problem statement\n\nNotation. We consider a universe U of n products. In order to distinguish products from their\ncorresponding ranks, we let U = {a1, . . . , an} denote the universe of products, under an arbitrary\nindexing. Preferences over this universe are captured by an anti-re\ufb02exive, anti-symmetric, and\ntransitive relation , which induces a total ordering (or ranking) over all products; speci\ufb01cally,\na b means that a is preferred to b. We represent preferences through rankings or permutations. A\ncomplete ranking (or simply a ranking) is a bijection : U ! [n] that maps each product a 2 U to\nits rank (a) 2 [n], where [j] denotes the set {1, 2, . . . , j} for any integer j. Lower ranks indicate\nhigher preference so that (a) < (b) if and only if a b, where denotes the preference\nrelation induced by the ranking . For simplicity of notation, we also let i denote the product ranked\nat position i. Thus, 12 \u00b7\u00b7\u00b7 n is the list of the products written by increasing order of their ranks.\nFinally, for any two integers i \uf8ff j, let [i, j] denote the set {i, i + 1, . . . , j}.\nMallows model. The Mallows model is a member of the distance-based ranking family models\n[21]. This model is described by a location parameter !, which denotes the central permutation, and\na scale parameter \u2713 2 R+, such that the probability of each permutation is given by\n\n() =\n\ne\u2713\u00b7d(,!)\n\n (\u2713)\n\n,\n\nwhere (\u2713) = P exp(\u2713 \u00b7 d(, !)) is the normalization constant, and d(\u00b7,\u00b7) is the Kendall-\nTau metric of distance between permutations de\ufb01ned as d(, !) = Pi s , or ii) ai was chosen at position s 1, and ak+1 is inserted at a position\n` \uf8ff s 1. Consequently, we have for all i \uf8ff k\n\nnPs=1\n\n\u21e1(i, s, n).\n\n\u21e1(i, s, k + 1) =\n\nk+1X`=s+1\n\npk+1,` \u00b7 \u21e1(i, s, k) +\n\npk+1,` \u00b7 \u21e1(i, s 1, k)\n= (1 k+1,s) \u00b7 \u21e1(i, s, k) + k+1,s1 \u00b7 \u21e1(i, s 1, k),\n\ns1X`=1\n\n`=1 pk,` for all k, s.\n\nwhere k,s =Ps\nCase 2 : ak+1 2 S\nak+1 2 S. Consider a product ai with i \uf8ff k. This product is chosen at position s only\nak+1 2 S\nif it was already chosen at position s and ak+1 is inserted at a position `> s . Therefore, for all\ni \uf8ff k, \u21e1(i, s, k + 1) = (1 k+1,s) \u00b7 \u21e1(i, s, k). For product ak+1, it is chosen at position s only if\nall products ai for i \uf8ff k are at positions ` s and ak+1 is inserted at position s, implying that\n\n\u21e1(k + 1, s, k + 1) = pk+1,s \u00b7Xi\uf8ffk\n\nnX`=s\n\n\u21e1(i, `, k).\n\nAlgorithm 1 summarizes this procedure.\n\n6\n\n\fAlgorithm 1 Computing choice probabilities\n1: Let S be a general offer set. Without loss of generality, we assume that a1 2 S.\n2: Let \u21e1(1, 1, 1) = 1.\n3: For k = 1, . . . , n 1,\n\n(a) For all i \uf8ff k and s = 1, . . . k + 1, let\n\n\u21e1(i, s, k + 1) = (1 k+1,s) \u00b7 \u21e1(i, s, k) + 1l[ak+1 /2 S] \u00b7 k+1,s1 \u00b7 \u21e1(i, s 1, k).\n\n(b) For s = 1, . . . , k + 1, let\n\n\u21e1(k + 1, s, k + 1) = 1l[ak+1 2 S] \u00b7 pk+1,s \u00b7Xi\uf8ffk\n\n\u21e1(i, `, k).\n\nnX`=s\n\n4: For all i 2 [n], return P(ai|S) =Pn\n\ns=1 \u21e1(i, s, n).\n\nTheorem 4.2 For any offer set S, Algorithm 1 returns the choice probabilities under a Mallows\ndistribution with location parameter ! and scale parameter \u2713.\n\nThis dynamic programming approach provides an O(n3) time algorithm for computing P(a|S) for\nall products a 2 S simultaneously. Moreover, as explained in the next section, these ideas lead to an\nalgorithm to solve the assortment optimization problem.\n\n5 Assortment optimization: integer programming formulation\n\nIn the assortment optimization problem, each product a has an exogenously \ufb01xed price ra. Moreover,\nthere is an additional product aq that represents the outside option (no-purchase), with price rq = 0\nthat is always included. The goal is to determine the subset of products that maximizes the expected\nrevenue, i.e., solve (2). Building on Algorithm 1 and introducing a binary variable for each product,\nwe formulate 2 as an MIP with O(n3) variables and constraints, of which only n variables are binary.\nWe assume for simplicity that the \ufb01rst product of S (say a1) is known. Since this product is generally\nnot known a-priori, in order to obtain an optimal solution to problem (2), we need to guess the \ufb01rst\noffered product and solve the above integer program for each of the O(n) guesses. We note that the\nMIP formulation is quite powerful and can handle a large class of constraints on the assortment (such\nas cardinality and capacity constraints) and also extends to the case of the mixture of Mallows model.\n\nTheorem 5.1 Conditional on a1 2 S, the optimal solution to 2 is given by S\u21e4 = {i 2 [n] : x\u21e4i = 1},\nwhere x\u21e4 2{ 0, 1}n is the optimal solution to the following MIP:\n\nmax\n\nx,\u21e1,y,z Xi,s\n\nri \u00b7 \u21e1(i, s, n)\n\ns.t.\u21e1 (1, 1, 1) = 1,\u21e1 (1, s, 1) = 0,\n\n\u21e1(i, s, k + 1) = (1 wk+1,s) \u00b7 \u21e1(i, s, k) + yi,s,k+1,\n\n\u21e1(k + 1, s, k + 1) = zs,k+1,\nyi,s,k \uf8ff k+1,s1 \u00b7 \u21e1(i, s 1, k 1),\n0 \uf8ff yi,s,k \uf8ff k+1,s1 \u00b7 (1 xk),\nzs,k \uf8ff pk+1,s \u00b7\n0 \uf8ff zs,k \uf8ff pk+1,s \u00b7 xk,\nx1 = 1, xq = 1, xk 2{ 0, 1}\n\n\u21e1(i, `, k 1),\n\nk1Xi=1\n\nnX`=s\n\n8s = 2, . . . , n\n8i, s,8k 2\n\n8s,8k 2\n8i, s,8k 2\n8i, s,8k 2\n8s,8k 2\n8s,8k 2\n\nWe present the proof of correctness for this formulation in Appendix C.\n\n7\n\n\f6 Numerical experiments\n\nIn this section, we examine how the MIP performs in terms of the running time. We considered the\nfollowing simulation setup. Product prices are sampled independently and uniformly at random from\nthe interval [0, 1]. The modal ranking is \ufb01xed to the identity ranking with the outside option ranked\nat the top. The outside option being ranked at the top is characteristic of applications in which the\nretailer captures a small fraction of the market and the outside option represents the (much larger) rest\nof the market. Because the outside option is always offered, we need to solve only a single instance\nof the MIP (described in Theorem 5.1). Note that in the more general setting, the number of MIPs\nthat must be solved is equal the minimum of the rank of the outside option and the rank of the highest\nrevenue item2. Because the MIPs are independent of each other, they can be solved in parallel. We\nsolved the MIPs using the Gurobi Optimizer version 6.0.0 on a computer with processor 2.4GHz\nIntel Core i5, RAM of 8GB, and operating system Mac OSX El Capitan.\n\nStrengthening of the MIP formulation. We use structural properties of the optimal solution\nto tighten some of the upper bounds involving the binary variables in the MIP formulation. In\nparticular, for all i, s, and m, we replace the constraint yi,s,m \uf8ff m+1,s1 \u00b7 (1 xm) with the\nconstraint yi,s,m \uf8ff m+1,s1 \u00b7 ui,s,m \u00b7 (1 xm), where ui,s,m is the probability that product ai\nis selected at position (s 1) after the mth step of the RIM when the offer set is S = {ai\u21e4, aq},\ni.e. when only the highest priced product is offered. Since we know that the highest price product\nis always offered in the optimal assortment, this is a valid upper bound to \u21e1(i, s 1, m 1) and,\ntherefore, a valid strengthening of the constraint. Similarly, for all s and m, we replace the constraint,\nzs,m \uf8ff \u21b5m+1,s \u00b7 xm with the constraint zs,m \uf8ff \u21b5m+1,s \u00b7 vs,m \u00b7 xm, where vs,m is equal to the\nprobability that product that product i is selected at position ` = s, . . . , n when the offer set is\nS = {aq} if ai w ai\u21e4, and S = {aq, ai\u21e4} otherwise. Again using the fact that the highest price\nproduct is always offered in the optimal assortment, we can show that this is a valid upper bound.\n\nResults and discussion. Table 1 shows the running time of the strengthened MIP formulation for\ndifferent values of e\u2713 and n. For each pair of parameters, we generated 50 different instances. We\n\ne\u2713 = 0.8\n\ne\u2713 = 0.9\n\ne\u2713 = 0.8\n\nn Average running time (s) Max running time (s)\ne\u2713 = 0.9\n5.80\n10\n28.79\n15\n189.93\n20\n25\n1,817.98\n\n4.60\n19.04\n48.08\n143.21\n\n4.72\n21.30\n105.30\n769.78\n\n5.64\n27.08\n58.09\n183.78\n\nTable 1: Running time of the strengthened MIP for various values of e\u2713 and n.\n\nnote that the strengthening improves the running time considerably. Under the initial formulation,\nthe MIP did not terminate after several hours for n = 25 whereas it was able to terminate in a few\nminutes with the additional strengthening. Our MIP obtains the optimal solution in a reasonable\namount of time for the considered parameter values. Outside of this range, i.e. when e\u2713 is too small\nor when n is too large, there are potential numerical instabilities. The strengthening we propose is\none way to improve the running time of the MIP but other numerical optimization techniques may be\napplied to improve the running time even further. Finally, we emphasize that the MIP formulation\nis necessary because of its \ufb02exibility to handle versatile business constraints (such as cardinality or\ncapacity constraints) that naturally arise in practice.\n\nExtensions and future work. Although the entire development was focused on a single Mallows\nmodel, our results extend to a \ufb01nite mixture of Mallows model. Speci\ufb01cally, for a Mallows model\nwith T mixture components, we can compute the choice probability by setting \u21e1(i, s, n) =PT\nt=1 \u21b5t \u00b7\n\u21e1t(i, s, n), where \u21e1(i, s, n) is the probability term de\ufb01ned in Section 4, \u21e1t(\u00b7,\u00b7,\u00b7) is the probability for\nmixture component t, and \u21b5t > 0 are the mixture weights. We then have P(a|S) =Pn\ns=1 \u21e1(i, s, n)\nfor the mixture model. 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