P re-positioning emergency inventory in selected facilities is commonly adopted to prepare for potential disaster threat. In this study, we simultaneously optimize the decisions of facility location, emergency inventory pre-positioning, and relief delivery operations within a single-commodity disaster relief network. A min-max robust model is proposed to capture the uncertainties in both the left-and right-hand-side parameters in the constraints. The former corresponds to the proportions of the pre-positioned inventories usable after a disaster attack, while the latter represents the demands of the inventories and the road capacities in the disaster-affected areas. We study how to solve the robust model efficiently and analyze a special case that minimizes the deprivation cost. The application of the model is illustrated by a case study of the 2010 earthquake attack at Yushu County in Qinghai Province of PR China. The advantage of the min-max robust model is demonstrated through comparison with the deterministic model and the two-stage stochastic model for the same problem. Experiment variants also show that the robust model outperforms the other two approaches for instances with significantly larger scales.

With the low-carbon revolution, many firms face the continuous challenge for reducing carbon emissions in their supply chain activities. Built upon the strategic safety stock placement model, in this paper, we model the carbon emissions for each stage of a multi-echelon supply chain as a function of the service time guaranteed by this stage to its immediate downstream stages. We establish that there is a negative correlation between service time and carbon emissions at each stage. Based on this, we develop two models to study the trade-off between service time and carbon emissions for safety stock placement in multi-echelon supply chains by considering carbon cap and carbon tax, respectively. Both of them are solved using iterative piecewise linear approximations. We implement these two models to study how carbon cap, carbon emission cost rate and guaranteed service time affect the optimal safety stock placement using a real chain.

Most existing facility location models assume that the facility cost is either a fixed setup cost or made up of a fixed setup and a problem-specific concave or submodular cost term. This structural property plays a critical role in developing fast branch-and-price, Lagrangian relaxation, constant ratio approximation, and conic integer programming reformulation approaches for these NP-hard problems. Many practical considerations and complicating factors, however, can make the facility cost no longer concave or submodular. By removing this restrictive assumption, we study a new location model that considers general nonlinear costs to operate facilities in the facility location framework. The general model does not even admit any approximation algorithms unless P = NP because it takes the unsplittable hard-capacitated metric facility location problem as a special case. We first reformulate this general model as a set-partitioning model and then propose a branch-and-price approach. Although the corresponding pricing problem is NP-hard, we effectively analyze its structural properties and design an algorithm to solve it efficiently. The numerical results obtained from two implementation examples of the general model demonstrate the effectiveness of the solution approach, reveal the managerial implications, and validate the importance to study the general framework.

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