Improved approximation algorithms for inventory problems

We give new approximation algorithms for the submodular joint replenishment problem and the inventory routing problem, using an iterative rounding approach. In both problems, we are given a set of N items and a discrete time horizon of T days in which given demands for the items must be satisfied. Ordering a set of items incurs a cost according to a set function, with properties depending on the problem under consideration. Demand for an item at time t can be satisfied by an order on any day prior to t, but a holding cost is charged for storing the items during the intermediate period; the goal is to minimize the sum of the ordering and holding cost. Our approximation factor for both problems is [Formula Presented]; this improves exponentially on the previous best results

Supply Chain Management with Online Customer Selection

We consider new online variants of supply chain management models, where in addition to production decisions, one also has to actively decide on which customers to serve. Specifically, customers arrive sequentially during a selection phase, and one has to decide whether to accept or reject each customer upon arrival. If a customer is rejected, then a lost-sales cost is incurred. Once the selection decisions are all made, one has to satisfy all the accepted customers with minimum possible production cost. The goal is to minimize the total cost of lost sales and production. A key feature of the model is that customers arrive in an online manner, and the decision maker does not require any information about future arrivals. We provide two novel algorithms for online customer selection problems, which are based on repeatedly solving offline subproblems that ignore previously made decisions. For many important settings, our algorithms achieve small constant competitive ratio guarantees. That is, for any sequence of arriving customers, the cost incurred by the online algorithm is within a fixed constant factor of the cost incurred by the respective optimal solution that has full knowledge upfront on the sequence of arriving customers. Finally, we provide a computational study on the performance of these algorithms when applied to the economic lot sizing and joint replenishment problems with online customer selection.National Science Foundation (U.S.) (CMMI-0846554)United States. Air Force Office of Scientific Research (FA9550-11-1-0150)United States. Air Force Office of Scientific Research (FA9550-08-1-0369

Combinatorial Optimization

This report summarizes the meeting on Combinatorial Optimization where new and promising developments in the field were discussed. Th

The submodular joint replenishment problem

The joint replenishment problem is a fundamental model in supply chain management theory that has applications in inventory management, logistics, and maintenance scheduling. In this problem, there are multiple item types, each having a given time-dependent sequence of demands that need to be satisfied. In order to satisfy demand, orders of the item types must be placed in advance of the due dates for each demand. Every time an order of item types is placed, there is an associated joint setup cost depending on the subset of item types ordered. This ordering cost can be due to machine, transportation, or labor costs, for example. In addition, there is a cost to holding inventory for demand that has yet to be served. The overall goal is to minimize the total ordering costs plus inventory holding costs. In this paper, the cost of an order, also known as a joint setup cost, is a monotonically increasing, submodular function over the item types. For this general problem, we show that a greedy approach provides an approximation guarantee that is logarithmic in the number of demands. Then we consider three special cases of submodular functions which we call the laminar, tree, and cardinality cases, each of which can model real world scenarios that previously have not been captured. For each of these cases, we provide a constant factor approximation algorithm. Specifically, we show that the laminar case can be solved optimally in polynomial time via a dynamic programming approach. For the tree and cardinality cases, we provide two different linear programming based approximation algorithms that provide guarantees of three and five, respectively.National Science Foundation (U.S.) (CAREER Grant CMMI-0846554)United States. Air Force Office of Scientific Research (Award FA9550-11-1-0150)SMA GrantSolomon Buchsbaum AT&T Research Fun

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

Efficient Algorithms for Solving Facility Problems with Disruptions

This study investigates facility location problems in the presence of facility disruptions. Two types of problems are investigated. Firstly, we study a facility location problem considering random disruptions. Secondly, we study a facility fortification problem considering disruptions caused by random failures and intelligent attacks.We first study a reliable facility location problem in which facilities are faced with the risk of random disruptions. In the literature, reliable facility location models and solution methods have been proposed under different assumptions of the disruption distribution. In most of these models, the disruption distribution is assumed to be completely known, that is, the disruptions are known to be uncorrelated or to follow a certain distribution. In practice, we may have only limited information about the distribution. In this work, we propose a robust reliable facility location model that considers the worst-case distribution with incomplete information. Because the model imposes fewer distributional assumptions, it includes several important reliable facility location problems as special cases. We propose an effective cutting plane algorithm based on the supermodularity of the problem. For the case in which the distribution is completely known, we develop a heuristic algorithm called multi-start tabu search to solve very large instances.In the second part of the work, we study an r-interdiction median problem with fortification that simultaneously considers two types of disruption risks: random disruptions that happen probabilistically and disruptions caused by intentional attacks. The problem is to determine the allocation of limited facility fortification resources to an existing network. The problem is modeled as a bi-level programming model that generalizes the r-interdiction median problem with probabilistic fortification. The lower level problem, that is, the interdiction problem, is a challenging high-degree non-linear model. In the literature, only the enumeration method is applied to solve a special case of the problem. By exploring the special structure property of the problem, we propose an exact cutting plane method for the problem. For the fortification problem, an effective logic based Benders decomposition algorithm is proposed

Gradient-Bounded Dynamic Programming for Submodular and Concave Extensible Value Functions with Probabilistic Performance Guarantees

We consider stochastic dynamic programming problems with high-dimensional, discrete state-spaces and finite, discrete-time horizons that prohibit direct computation of the value function from a given Bellman equation for all states and time steps due to the "curse of dimensionality". For the case where the value function of the dynamic program is concave extensible and submodular in its state-space, we present a new algorithm that computes deterministic upper and stochastic lower bounds of the value function in the realm of dual dynamic programming. We show that the proposed algorithm terminates after a finite number of iterations. Furthermore, we derive probabilistic guarantees on the value accumulated under the associated policy for a single realisation of the dynamic program and for the expectation of this value. Finally, we demonstrate the efficacy of our approach on a high-dimensional numerical example from delivery slot pricing in attended home delivery.Comment: 12 pages, 4 figures. arXiv admin note: substantial text overlap with arXiv:2005.1121