We present a general framework for modeling routing problems based on formulating them as a traditional location problem called the Capacitated Concentrator Location Problem. We apply this framework to two classical routing problems: the Capacitated Vehicle Routing Problem and Inventory-Routing Problem. In the former case, the heuristic is proven to be asymptotically optimal for any distribution of customer demands and locations. Computational experiments show that the heuristic performs well for both problems and in most cases outperforms all published heuristics on a set of standard test problems.

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We consider the famous bin packing problem where a set of items must be stored in bins of equal capacity. In the classical version, the objective is to minimize the number of bins used. Motivated by several optimization problems that occur in the context of the storage of items, we study a more general cost structure where the cost of a bin is a concave function of the number of items in the bin. The objective is to store the items in such a way that total cost is minimized. Such cost functions can greatly alter the way the items should be assigned to the bins. We show that some of the best heuristics developed for the classical bin packing problem can perform poorly under the general cost structure. On the other hand, the so-called next-fit increasing heuristic has an absolute worst-case bound of no more than 1.75 and an asymptotic worst-case bound of 1.691 for any concave and monotone cost function. Our analysis also provides a new worst-case bound for the well studied next-tit decreasing heuristic when the objective is to minimize the number of bins used.

Because of its effectiveness in responding to uncertainties, process flexibility has gained significant attention in several industries, in particular, in the automotive industry. Evidently, it is often too expensive to achieve a high degree of flexibility, for example, full flexibility, and as a result, sparse or partial flexibility is implemented instead. One set of sparse flexibility designs is the 2-flexibility designs. A flexibility design is a 2-flexibility design if each plant produces exactly two products and demand for each product can be satisfied from exactly two plants.Of course, there are many ways to implement sparse designs, and the challenge is to identify an effective one. An important concept analyzed in the literature and applied in practice by various companies is the concept of the long chain. The first to observe the power of the long chain were Jordan and Graves (1995), who, through empirical analysis, showed that the long-chain design can provide almost as much benefit as full flexibility. In particular, Jordan and Graves (1995) found that for randomly generated demand, the expected amount of demand that can be satisfied by a long-chain design is very close to that of a full flexibility design.Though the analysis and results can be extended to more general settings, for simplicity, we focus on balanced manufacturing systems, that is, manufacturing systems with an equal number of plants and products, and each plant has a equal capacity. Given a balanced manufacturing system, a flexibility design A is represented by the arc set of a directed bipartite graph, where an arc from plant node i to product node j implies that plant i is capable of producing product j. For example, if A is a dedicated design, then A has exactly n arcs such that each plant node is incident to one arc and each product node is incident to one arc. By contrast, if A is a full flexibility design, then A has arcs connecting every plant node to all product nodes.Because A is represented by a bipartite graph, applying standard graph theory notation, we define an undirected cycle in A to be a set of arcs that forms a cycle when the arc directions are ignored. A flexibility design A is a long chain if its arcs form exactly one undirected cycle containing all plant and product nodes (see Fig. 13.1 for an example). A closed chain is defined as an induced subgraph in A that forms an undirected cycle, while an open chain is an induced subgraph in A that forms an undirected line (one arc less than an undirected cycle). Figure 13.1 presents an example of an open and a closed chain. It can be seen that any 2-flexibility design, where each product/plant node is incident to two arcs, is the union of a number of closed chains.The results presented in this chapter are motivated by a few observations made in the literature regarding the effectiveness of the long-chain flexibility design. The first is an observation that has been made in (Graves 2008 and Hopp et al. 2004) regarding the performance of the long chain for a balanced syste...

We consider the Capacitated Traveling Salesman Problem with Pickups and Deliveries (CTSPPD). This problem is characterized by a set of n pickup points and a set of n delivery points. A single product is available at the pickup points which must be brought to the delivery points. A vehicle of limited capacity is available to perform this task. The problem is to determine the tour the vehicle should follow so that the total distance traveled is minimized, each load at a pickup point is picked up, each delivery point receives its shipment and the vehicle capacity is not violated. We present two polynomial-time approximation algorithms for this problem and analyze their worst-case bounds.

The logic of logistics: theory, algorithms, and applications for logistics management I Julien Bramel, David Simchi-Levi.p. cm. -(Springer series in operations research) Includes bibliographical references and index.

The Vehicle Routing Problem with Time Windows (VRPTW) is one of the most important problems in distribution and transportation. A classical and recently popular technique that has proven effective for solving these problems is based on formulating them as a set covering problem. The method starts by solving its linear programming relaxation, via column generation, and then uses a branch and bound strategy to find an integer solution to the set covering problem: a solution to the VRPTW. An empirically observed property is that the optimal solution value of the set covering problem is very close to its linear programming relaxation which makes the branch and bound step extremely efficient. In this paper we explain this behavior by demonstrating that for any distribution of service times, time windows, customer loads, and locations, the relative gap between fractional and integer solutions of the set covering problem becomes arbitrarily small as the number of customers increases.

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