Capacitated assortment and price optimization under the nested logit model

Abstract

We study the capacitated assortment and price optimization problem, where a retailer sells categories of substitutable products subject to a capacity constraint. The goal of the retailer is to determine the subset of products as well as their selling prices so as to maximize the expected revenue. We model the customer purchase behavior using the nested logit model and formulate this problem as a non-linear binary integer program. For this NP-complete problem, we show that there exists a pseudo polynomial time approximation scheme that finds its $\epsilon$-approximate solution. We first convert the original problem into an equivalent fixed point problem. We then show that finding an $\epsilon$-approximate solution to the fixed point problem can be achieved by binary search, where a non-linear auxiliary problem is repeatedly approximated by a dynamic programing based algorithm involving an approximation to a series of multiple-choice parametric knapsack problems. For the special case when the capacity constraints are cardinal and nest-specific, we develop an algorithm that finds the optimal solution in strongly polynomial time. Moreover, our algorithm can be directly applied to find an $\epsilon$-approximate solution to the capacitated assortment optimization problem under the nested logit model, which is the first approximate algorithm that is polynomial with respect to the number of nests in the literature.

Appendices

Appendix A: Heuristic algorithm for a lower bound

In this section, we present a tractable way to obtain a lower bound for $z^*$ using a marginal-contribution maximization heuristic algorithm, which is described in Algorithm 4 and works repeatedly. In line 1, we let $\widehat{z}$ be a temporary lower bound, b be the unoccupied capacity and initialize them to zero and c, respectively. In line 2, we introduce $\varvec{S}$ as the current assortment and $\overline{\varvec{S}}$ as the set of products that have not been offered in current assortment. In lines 3–11, we check whether there exist a feasible product in $\overline{\varvec{S}}$ that has positive ratio of marginal contribution and its weight. If this is true, we add the one with maximum marginal-contribution per unit weight to the current assortment and delete the product from $\overline{\varvec{S}}$ in line 6, then we update $\widehat{z}$ and the unoccupied capacity in line 7. Otherwise, we terminate the procedure in line 9 and output $\widehat{z}$ as the lower bound for $z^*$ in line 12. Note that in each iteration, the unoccupied capacity may reduce by the weight of one product, which is more than one, or remain the same, which means terminating the procedure. So the complexity of Algorithm 4 is bounded by O(cmnd). We remark that for non-degenerate cases with $c\ge \min _{i\in M, j\in N_i} \left\{ w_{ij}\right\}$, the output $\widehat{z}$ must be greater than zero.

Appendix B: Notation

In this section, we summarize key notation used throughout the paper for ease of reading:

M:

$=\{1,2,\ldots , m\}$, the set of nests under the nested logit model.

$N_i$:

$\{1,2,\ldots , n_i\}$, the set of products under nest $i\in M$.

n:

Maximum products under each nest.

$\langle i,j\rangle$:

Product $j\in N_i$ under nest $i\in M$.

D:

$=\{1,2,\ldots , d\}$., the set of pre-determined price levels for products.

$p_{ijk}$:

Price of product $\langle i,j\rangle$ at level $k\in D$.

$w_{ij}$:

Weight of product $\langle i,j\rangle$,

c:

Capacity limit for all offered products.

U:

The ratio between the maximum and the minimum preference weights of the products.

$S_{ijk}$:

Binary variable to indicate whether product $\langle i,j\rangle$ at level $k\in D$ is offered.

$v_{ijk}$:

Preference weight for product $\langle i,j\rangle$ at level $k\in D$.

$\gamma _i$:

Scale parameter for nest i.

$\varvec{S}$:

$=(\varvec{S}_1, \varvec{S}_2,\ldots ,\varvec{S}_m)$, the set of all decision variables.

$V_i(\varvec{S}_i)$:

Total preference weights of products under nest i.

$R_i(\varvec{S}_i)$:

Expected revenue under nest i with assortment $\varvec{S}_i$.

$F(z,\varvec{S}), \mathcal {F}(z)$:

Functions defined in Eqs. (6)–(7).

$\mathcal {A}_i(z,b_i)$:

Collection of candidate assortment associated with nest i with $b_i$ unit capacity and known z.