Large-Scale Price Optimization via Network Flow
This paper deals with price optimization, which is to find the best pricing strategy that maximizes revenue or profit, on the basis of demand forecasting models. Though recent advances in regression technologies have made it possible to reveal price-demand relationship of a large number of products, most existing price optimization methods, such as mixed integer programming formulation, cannot handle tens or hundreds of products because of their high computational costs. To cope with this problem, this paper proposes a novel approach based on network flow algorithms. We reveal a connection between supermodularity of the revenue and cross elasticity of demand. On the basis of this connection, we propose an efficient algorithm that employs network flow algorithms. The proposed algorithm can handle hundreds or thousands of products, and returns an exact optimal solution under an assumption regarding cross elasticity of demand. Even if the assumption does not hold, the proposed algorithm can efficiently find approximate solutions as good as other state-of-the-art methods, as empirical results show.
This paper considers the problem of maximizing the seller's revenue by finding the optimal pricing strategy, given certain demand models. In particular, a binary quadratic program, first proposed by Ito and Fujimaki, is applied to formulate this price optimization problem. The authors show that the gross profit function (mapping the prices of all items to the revenue) is supermodular if the demand model satisfies the substitute-goods property. In this case the objective function of the binary quadratic programming problem is submodular, which can be reduced to a minimum s-t cut (or max-flow) problem, which is known to be solvable (e.g. Dinic’s algorithm as it is used by the authors in their experiments).   In the case that the substitute-goods property fails to hold, the authors provide an approximation algorithm by constructing an approximation function which bounds the objective function from below. Specifically, the objective function of the binary quadratic program is decomposed into a supermodular and submodular terms, and the supermodular part is replaced by another function which is a lower bound. Then, the authors provide a proximal gradient method to compute this lower bound, with the number of iterations bounded by O(1/sqrt(epsilon)). The authors claim that in practice the convergence can be observed in about 10 iterations.   Finally, the authors provide many experiments results about their proximal gradient method based algorithm. They show that their algorithm is considerably faster than the previous algorithms, and produces better pricing strategy (Table 2). In addition, they show that (in Figure 1) the approximated value of the objective function is close to the exact optimal value (by deriving upper bound of it).
For the first part with substitute goods property, unfortunately, I do not see much novelty in it. The binary quadratic programming formulation was already proposed in Ito and Fujimak’s work, while the reduction from the unconstrained binary quadratic programming problem to the minimum cut is “well known” (as also remarked by the authors).   For the second part, the idea of providing a lower bound on the objective function and using proximal gradient method is interesting, but its theoretical part is not well-developed. In particular, there is no theoretical analysis regarding “how well” does the lower bound approximate the exact optimal value. Although empirical results given in Figure 1 address this issue, but I was expecting at least an asymptotic bound to be given. Nevertheless, the empirical results given in the last section seems to be convincing.   ---- Post discussion comments ----  I have read all the reviews and the corresponding authors’ responses. My sense is that the paper is perhaps somewhat stronger than my initial assessment, even if it is only a modest contribution on the theory side.   For the first part of the paper about the reduction to min-cut, after authors’ feedback 1) and 2), I can now see some novelties. I initially had the impression that they are simply applying existing techniques (the last reviewer has the same impression and raises the same issue). I think the authors should focus more on Proposition 2 which reveals the relationship between supermodularity of gross profit function and the substitute-goods property, because I believe this contribution is even more significant. (I am actually convinced that Substitute-Goods Property is important and practical.)   The second part of the paper should be the main part, and I think that the theory part could be better developed, and some other reviewers have a similar opinion. The authors claimed that the approximation gap is bounded by (1^T A^+ 1) / 2 in their responses, but I think this is a weak bound, at least in a theoretical sense. The entries of matrix A^+ are related to functions f_{ij} defined in Equation (1), which can be very large, and I think it is not surprising that this piece was omitted.   I think this is a reasonable paper. I would say it is definitely borderline, but it would be OK to accept.