Moshe B. Rosenwein

An interactive optimization system for bulk-cargo ship scheduling

This article considers the efficient scheduling of a fleet of ships engaged in pickup and delivery of bulk cargoes. Our optimization system begins by generating a menu of candidate schedules for each ship. This menu can contain all feasible solutions, which guarantees we will find an optimal solution or can be heuristically limited to contain only those schedules likely to be in an optimal solution. The problem of choosing from this menu an optimal schedule for the fleet is formulated as a set-packing problem and solved with a dual algorithm. Computational experience is presented based on real data obtained from the Military Sealift Command of the U. S. Navy. Run times for this data were reasonable and solutions were generated with the potential of saving up to about

Strong linear programming relaxations for the orienteering problem

30 million per year over the manual system currently in place. We also describe a color-graphics interface developed to facilitate interaction with the optimization system.

AN APPLICATION OF CLUSTER ANALYSIS TO THE PROBLEM OF LOCATING ITEMS WITHIN A WAREHOUSE

Abstract Consider a set of nodes, each with an associated profit, and a set of arcs, each with an associated length. The objective of the orienteering problem is to find the path beginning at a specified origin and terminating at a specified destination that maximizes total profit subject to 1) a constraint on the length of the path, and 2) the condition that no node is visited more than once. The problem may be formulated as a 0–1 integer program, and it has been shown to be NP-hard. Prior research has focused on heuristics that obtain lower bounds on the optimal objective function value. Some recent research has proposed a branch-and-bound algorithm that solves a linear programming (LP) relaxation of the 0–1 model at each node and obtains an optimal orienteering path. Our research is concerned with tightening the LP relaxation by adding constraints and valid inequalities. We propose a procedure to obtain upper bounds by solving three successive linear programs. We test our procedure on datasets in existing literature and demonstrate that the average deviation between our upper bounds and the best known lower bounds (feasible solutions) is less than five percent. The quality of upper bound obtained is significantly enhanced over the bound obtained from solving the basic LP relaxation.

Technical Note-An Improved Dual Based Algorithm for the Generalized Assignment Problem

Abstract The order picking function in a warehouse involves selection of items from storage areas in order to satisfy customer demand. It has been identified as the most costly warehouse operations activity. Cluster analysis identifies groups of items that are frequently ordered together. Items within a cluster may be located near one another, and efficient picking tours may, thus, be formed. We formulate the clustering problem as a p-median 0-1 integer program which may efficiently be solved to optimality.

Topological network design for SONET ring architecture

The generalized assignment problem GAP determines the minimum cost assignment of n jobs to m agents such that each job is assigned to exactly one agent, subject to an agents capacity. Existing solution algorithms have not solved problems with more than 100 decision variables. This paper designs an optimization algorithm for the GAP that effectively solves problems with up to 500 variables. Compared with existing procedures, this algorithm requires fewer enumeration nodes and shorter running times. Improved performance stems from: an enhanced Lagrangian dual ascent procedure that solves a Lagrangian dual at each enumeration node; adding a surrogate constraint to the Lagrangian relaxed model: and an elaborate branch-and-bound scheme. An empirical investigation of various problem structures, not considered in existing literature, is also presented.

An application-oriented guide for designing Lagrangean dual ascent algorithms

Service restoration and survivability have become increasingly important in telecommunications network planning with the introduction of fiber-optic high-speed networks. Synchronous optical network (SONET) technology promotes the use of interconnected rings in designing reliable networks. We describe a heuristic approach for designing networks comprised of interconnected rings. Our approach is particularly attractive for relatively sparse networks in which the set of all cycles (constituting the potential rings) can be determined at a reasonable computational effort. Most networks fall into this category. Given a set of nodes, with demand among all possible node-pairs, and a set of available links that connect the nodes, the problem is to select an optimal subset of rings, utilizing only allowable links, such that each node is included in at least one ring and each ring is connected to at least one other ring at two or more nodes. Such a multiple ring network ensures instantaneous restoration of service in case of a single link or node failure. We first generate a large set of candidate rings and approximate the cost of each ring based on the nodes that are served by the ring and based on the demands. We then apply a set covering algorithm that selects a (minimum cost) subset of the candidate rings such that each node is included on at least one ring. Finally, we select a few additional rings in order to achieve the required connectivity among the rings. We present computational results for realistic-size (e.g., 500 nodes) telecommunication networks.

An application of lagrangean decomposition to the resource‐constrained minimum weighted arborescence problem

Abstract Lagrangean relaxation is an effective method for providing bounds in integer programming. An efficient solution of the Lagrangean dual often requires a specialized algorithm that must be tailored for each application model. Lagrangean dual ascent is a generic name that describes a broad class of algorithms. Lagrangean dual ascent is often preferred to subgradient optimization in solving a Lagrangean dual since an ascent procedure guarantees monotonic bound improvement. Also, if embedded within a branch and bound scheme, Lagrangean dual ascent may adjust only a few multipliers at each node, thus conserving computer storage and enhancing processing efficiency. This paper describes a framework to construct a Lagrangean dual ascent procedure and adapts it for the generalized assignment problem and the constrained arborescence problem.

A constrained Steiner tree problem

The resource-constrained minimum weighted arborescence problem, a 0-1 integer programming model with application in hierarchical distribution network design, is introduced. Since the model is NP-hard, an enumeration method is required to solve it to optimality. Lagrangean decomposition, a special form of Lagrangean relaxation, is applied to the model. Both analytically and empirically, Lagrangean decomposition is shown to improve on bounds obtained by a conventional Lagrangean relaxation. An enumeration algorithm, that embeds a specialized Lagrangean dual ascent scheme to solve a Lagrangean decomposition dual, is designed, and problems with up to 1000 0-1 variables are solved.

A due date assignment algorithm for multiproduct manufacturing facilities

Abstract The Steiner tree problem on a graph involves finding a minimum cost tree which connects a designated subset of the nodes in the graph. Variants of the basic Steiner tree model can arise in the design of telecommunication networks where customers must be connected to a switching center. In this paper, we consider the constrained Steiner tree problem which is the Steiner tree problem on graph with one additional side constraint that imposes a budget on the total amount of a resource (e.g., employee maintenance time) required by the arcs in the solution. We discuss various problem formulations, decomposition based solution approaches, and computational experience with the proposed methods.

An improved bounding procedure for the constrained assignment problem

Abstract The order acceptance process is an important interface between a manufacturer and its customers. This paper considers a complex manufacturing facility, capable of simultaneously producing a large variety of products, e.g., printed wiring boards for telecommunications and electronic systems. A requested due date is submitted with each customer order. A batch of orders is accumulated by the manufacturer for a certain period, e.g., one week or one day. For each order, production control must either confirm the requested due date or propose an alternate due date. (Some high-priority orders may be confirmed shortly upon receipt.) We develop a heuristic, the Due Date Assignment Algorithm, to solve the order acceptance problem. Its objective is to minimize the sum of weighted (positive) deviations of the assigned due dates from the requested dates. The heuristic first generates a menu of candidate schedules for each order. It then applies a Lagrangean relaxation scheme to an integer programming formulation of the problem. Finally, an interchange procedure is applied, if necessary, to obtain primal feasibility. Computational results revealed significant improvements over the often-used policy of assigning a due date to each single order upon its arrival.