Competition under time‐varying demands and dynamic lot sizing costs

Abstract

We develop a competitive pricing model which combines the complexity of time-varying demand and cost functions and that of scale economies arising from dynamic lot sizing costs. Each firm can replenish inventory in each of the T periods into which the planning horizon is partitioned. Fixed as well as variable procurement costs are incurred for each procurement order, along with inventory carrying costs. Each firm adopts, at the beginning of the planning horizon, a (single) price to be employed throughout the horizon. Based on each period's system of demand equations, these prices determine a time series of demands for each firm, which needs to service them with an optimal corresponding dynamic lot sizing plan. We establish the existence of a price equilibrium and associated optimal dynamic lotsizing plans, under mild conditions. We also design efficient procedures to compute the equilibrium prices and dynamic lotsizing plans.

A review of multi-product pricing models

“…The relevant papers include Bernstein and Federgruen [5], Federgruen and Meissner [36], Oxenstierna [64], and Tanaka [85]. The equilibrium policy was obtained via solving first-order conditions in some of these papers.…”

“…See Lederer and Li [53], and Li and Lee [56] for other models of pricing and delivery-time competition. Federgruen and Meissner [36] may be the first to discuss a competitive pricing model that combines the complexity of time-dependent demand and cost functions with that arising from dynamic lot sizing costs. They assume that each firm i adopts one price p i to be employed throughout the horizon.…”

“…where d t ðpÞ is the demand function and b t is the uncertainty parameter, at time t. For example, assuming that each firm offers one product, Federgruen and Meissner [36] uses the demand system: Then in Bernstein and Federgruen [7], the fill rate (or service level) f was incorporated into the multiplicative demand model as both price and service competition were considered. They assumed the average sales function of the form d i ðp; f Þ ¼ w i ðf Þq i ðpÞ, where w i ðf Þ is a function of the service levels of all firms, and q i ðpÞ is a standard demand function satisfying the (LID) property.…”

Capacitated dynamic lot sizing with capacity acquisition

“…In analogy to the algorithm presented by Federgruen and Meissner (2009), who present an algorithm for a combined pricing and uncapacitated lot sizing problem, the heuristic developed here considers each possible number of setups n, n = 1, . .…”

Competition under capacitated dynamic lot-sizing with capacity acquisition

“…Ubhaya (1979) provides an algorithm to find the optimal approximation convex function of a quasi-convex function. Because the approximation process is not our focus, in analogy to Federgruen and Meissner (2009), we apply an approximation function model of the lot-sizing cost K i (C i ) as below: In order to estimate the approximation function more accurately, we calculate the constants of the approximation function based on different demand seasonality patterns and fixed setup cost levels. Assuming that firm i faces a planning horizon of T periods, and the demand behaves according to d it = β tdi , six seasonality patterns (1) Time-invariant demand functions: β t = 1; t = 1, .…”