On modelling non-linear quantity discounts in a supplier selection problem by mixed linear integer optimization

Abstract

Applying traditional integer programming techniques in order to solve real logistic problems can be an important challenge. To ensure tractability, real instances are often either simplified in scope or limited in size, given rise to solutions that may not address realistic issues. In this paper we present a novel approach to solve a multicommodity capacitated network flow problem with concave routing costs, considering also outsourcing, overload and underutilization facility costs. It is derived from a real NP production and transportation problem concerning to the processing of biological samples in a large health-care network, with consideration of volume-based price incentives—i.e. economies of scale—on the shipping costs. It is a tactical level model providing the global view of network layout and the coordinating policy among facilities with realistic assessment of long-term operations costs. The goal is to find an efficient resolution procedure in order to integrate it into a Decision Support System used by planners. With this aim, we analyse three alternative methods of linearizing the involved modified all-units discount cost function. Performance of the different modelling techniques is shown through extensive computations.

Appendices

Appendix 1: Discounts shapes and problem size for alternative formulations

A set of five MAUD shapes has been considered in our computations. The details of each of them are shown in Tables 11, 12, 13, 14 and 15, presenting on the left side the involved extended range set—i.e. used in the seminal work (Andrade-Pineda et al. 2013)—and on the right one the modified range set—i.e. used at each of the alternative formulations. Further, the sizes of the alternative MILPs arisen from addressing the reference network $\overline{G} ^{A}=(\overline{V^{481}} ,\overline{L^{2835}} ,K^{6})$—i.e. graph #A1 in our test bed, and base of Class A instances—are outlined, as an illustration of its dependence on the model applied.

Table 15 Fifth MAUD shape with extended range set of 13 intervals, and reduced range set of 7 intervals: $13Q\_7R$

Table 16 Comparison of the LOGDCC and the logarithmic approach in Li et al. (2013)

Appendix 2: Comparison of logarithmic approaches

The convenience of the modification of the logarithmic approach in Li et al. (2013) which this paper proposes is validated with the comparative results from addressing the Class A instances in our test bed—see Table 16. Particularly, the version from Li et al. (2013) ran out of memory with Instances $17 and $18, which happens not because of the size of the branch-and-bound trees—i.e. with regards to number of nodes—but rather owing to the size of the individual LPs within the tree. In a whole, the LogDCC model, as a locally ideal reformulation of the original in Li et al. (2013) exhibits improved tractability.