Technical Note—Preservation of Supermodularity in Parametric Optimization Problems with Nonlattice Structures

Abstract

This paper establishes a new preservation property of supermodularity in a class of two-dimensional parametric optimization problems, where the constraint sets may not be lattices. This property and its extensions unify several results in the literature and provide powerful tools to analyze a variety of operations models including a two-product coordinated pricing and inventory control problem with cross-price effects that we use as an illustrative example.

Dynamic Stochastic Inventory Management with Reference Price Effects

“…Unfortunately, the requirement on the feasible set is not satisfied in our problem. Interestingly, the concavity of the profit-to-go functions enabled by our transformation technique allows us to apply results developed by Chen et al (2013) on preserving concavity and supermodularity in a class of parametric optimization problems with non-lattice structures and show that a list-price policy is optimal.…”

“…By Assumption 1, it can be verified that N M is a non-negative matrix, and N N is a positive diagonal matrix. Thus, v t (θ + r − x, r) is supermodular by Remark 1 in Chen et al (2013).…”

“…We first show that π(r, q) is supermodular in r and q when p(r, q) ≥ p. Because η(z) is increasing and concave, it is straightforward to see (p − c)[B − Ap + η(z)] is component-wise concave and supermodular in (p, z) when p ≥ c. By Corollary 2 in Chen et al (2013), π(r, q) is supermodular when p(r, q) ≥ p ≥ c.…”

“…This allows us to prove the optimality of a reference-price-dependent base-stock policy. In addition, our transformation technique coupled with a recently developed result on preserving supermodularity with non-lattice structures (Chen et al 2013) allows us to prove the optimality of a list-price policy. We also study how the optimal base-stock level and target reference price depend on the current reference price, and compare long-run optimal price and base-stock level to those in the case without the reference price effect.…”

L♮-convexity and its applications in operations

“…(Chen et al 2013) Given any 2 Â n matrix A³0, if S is a nonempty closed convex sublattice, then so is the set T; moreover, if g is concave and supermodular on S, then so is f on T.…”