Two-stage stochastic nonlinear winner determination for logistics service procurement auctions under quantity discounts

Abstract

Quantity discount is a frequently adopted scheme that has not been explicitly investigated in logistics service procurement auctions. This paper focuses on a revised winner determination problem under quantity discounts and demand uncertainty for a fourth-party logistics (4PL) provider in a combinatorial reverse auction. To characterize our research problem, a two-stage stochastic nonlinear programming model is constructed. Inspired by the idea of sample average approximation (SAA), the nonlinear model is reformulated as a deterministic mixed integer linear programming model by using a linearization technique with superior expressions. Since the reformulation has a large number of decision variables and constraints, we integrate SAA with a dual decomposition Lagrangian relaxation technique (DDLR) to develop a solution method called SAA-DDLR. Numerical experiments are conducted to illustrate the effectiveness and applicability of our model and method. Sensitivity analysis reveals that both the 4PL and 3PLs can benefit from the quantity discount scheme. Managerial insights are drawn for the 4PL to run a cost-effective logistics system in the presence of quantity discounts.