Convexification of Queueing Formulas by Mixed-Integer Second-Order Cone Programming: An Application to a Discrete Location Problem with Congestion

Mixed-Integer Second-Order Cone Programs (MISOCPs) form a nice class of mixed-inter convex programs, which can be solved very efficiently due to the recent advances in optimization solvers. Our paper bridges the gap between modeling a class of optimization problems and using MISOCP solvers. It is shown how various performance metrics of M/G/1 queues can be molded by different MISOCPs. To motivate our method practically, it is first applied to a challenging stochastic location problem with congestion, which is broadly used to design socially optimal service networks. Four different MISOCPs are developed and compared on sets of benchmark test problems. The new formulations efficiently solve large-size test problems, which cannot be solved by the best existing method. Then, the general applicability of our method is shown for similar optimization problems that use queue-theoretic performance measures to address customer satisfaction and service quality

Location, inventory and testing decisions in closed-loop supply chains: a multimedia company

Our partnering firm is a Chinese manufacturer of multimedia products that needs guidance developing its imminent Closed-Loop Supply Chain (CLSC). To study this problem, we take into account location, inventory, and testing decisions in a CLSC setting with stochastic demands of new and time-sensitive returned products. Our analysis pays particular attention to the different roles assigned to the reverse Distribution Centers (DCs) and how each option affects the optimal CLSC design. The roles considered are collection and consolidation, additional testing tasks, and direct shipments with no reverse DCs. The problem concerning our partnering firm is formulated as a scenario-based chance-constrained mixed-integer program and it is reformulated to a conic quadratic mixed-integer program that can be solved efficiently via commercial optimization packages. The completeness of the model proposed allows us to develop a decision support tool for the firm and to offer several useful managerial insights. These insights are inferred from our computational experiments using data from the Chinese firm and a second data set based on the U.S. geography. Particularly interesting insights are related to how changes in the reverse flows can impact the forward supply chain and the inventory dynamics concerning the joint DCs.This research is partially supported by the National Natural Science Foundation of China under grants 71771135, 71371106 and 71332005

A capacitated facility location model with bidirectional flows

Supply chains with returned products are receiving increasing attention in the operations management community. The present paper studies a capacitated facility location model with bidirectional flows and a marginal value of time for returned products. The distribution system consists of a single supplier that provides one new product to a set of distribution centers (DCs), which then ships to the final retailers. While at the retailers' site, products can be shipped back to the supplier for reprocessing. Each DC is capacitated and handles stocks of new and/or returned products. The model is a nonlinear mixed-integer program that optimizes DC location and allocation between retailers and DCs. We show that it can be converted to a conic quadratic program that can be efficiently solved. Some valid inequalities are added to the program to improve computational efficiency. We conclude by reporting numerical experiments that reveal some interesting properties of the model

Combined Location-Inventory Optimization of Deteriorating Products Supply Chain Based on CQMIP under Stochastic Environment

The design and optimization of combined location-inventory model for deteriorating products are a main focus in supply chain management. There were many combined location-inventory design models in this field, but these models are under the assumptions of adequate capacity facilities, invariable lead time, unique product, and uncorrelated retailer’s demands. These assumptions have a big gap in the practical situation. In this paper, we design a combined location-inventory model for deteriorating products under capacitated facilities, stochastic lead time, multiple products, and correlated retailers’ stochastic demands assumptions. These constraints are near to actual supply chain circumstance. The problem is modeled as conic quadratic mix-integer programming (CQMIP) to minimize the total expected cost. We explain how to formulate these problems as conic quadratic mixed-integer problems, and in order to obtain better computational results we use extended cover cuts. Simultaneously we compare our method with the previous Lagrange methods; the result is that the new CQMIP method can get better solution

Conic Reformulations for Kullback-Leibler Divergence Constrained Distributionally Robust Optimization and Applications

In this paper, we consider a distributionally robust optimization (DRO) model in which the ambiguity set is defined as the set of distributions whose Kullback-Leibler (KL) divergence to an empirical distribution is bounded. Utilizing the fact that KL divergence is an exponential cone representable function, we obtain the robust counterpart of the KL divergence constrained DRO problem as a dual exponential cone constrained program under mild assumptions on the underlying optimization problem. The resulting conic reformulation of the original optimization problem can be directly solved by a commercial conic programming solver. We specialize our generic formulation to two classical optimization problems, namely, the Newsvendor Problem and the Uncapacitated Facility Location Problem. Our computational study in an out-of-sample analysis shows that the solutions obtained via the DRO approach yield significantly better performance in terms of the dispersion of the cost realizations while the central tendency deteriorates only slightly compared to the solutions obtained by stochastic programming