Jeremy F. Shapiro

A Survey of Lagrangean Techniques for Discrete Optimization

Publisher Summary This chapter proposes Lagrangean techniques for discrete optimization problems. A simple method for trying to solve zero–one integer programming (IP) problems is discussed. This method is used as a starting point for discussing many of the developments since then. The behavior of Lagrangean techniques in analyzing and solving zero–one IP problems is typical of their use on other discrete optimization problems. The chapter discusses a number of questions about this method and its relevance for optimizing the original IP problem. The goal of Lagrangean techniques is to try to establish sufficient optimality conditions: Lagrangean techniques are useful in computing zero–one solutions to IP problems with soft constraints or in parametric analysis of an IP problem over a family of right hand sides. Parametric analysis of discrete optimization problems is also discussed. The use of Lagrangean techniques as a distinct approach to discrete optimization has proven theoretically and computationally important for three reasons. First, dual problems derived from more complex discrete optimization problems can be represented as linear programming (LP) problems, but ones of immense size, which cannot be explicitly constructed and then solved by the simplex algorithm. Second, reason for considering the application of Lagrangean techniques to dual problems, in addition to the simplex algorithm, is that the simplex algorithm is exact and the dual problems are relaxation approximations. Lagrangean techniques as a distinct approach to discrete optimization problems emphasize the need they satisfy for exploiting special structures, which arise in various models.

LARGE SCALE OPTIMIZATION OF BEAM WEIGHTS UNDER DOSE-VOLUME RESTRICTIONS

The problem of choosing weights for beams in a multifield plan which maximizes tumor dose under conditions that recognize the volume dependence of organ tolerance to radiation is considered, and its solution described. Structures are modelled as collections of discrete points, and the weighting problem described as a combinatorial linear program (LP). The combinatorial LP is solved as a mixed 0/1 integer program with appropriate restrictions on normal tissue dose. The method is illustrated through the assignment of weights to a set of 10 beams incident on a pelvic target. Dose-volume restrictions are placed on surrounding bowel, bladder, and rectum, and a limit placed on tumor dose inhomogeneity. Different tolerance restrictions are examined, so that the sensitivity of the target dose to changes in the normal tissue constraints may be explored. It is shown that the distributions obtained satisfy the posed constraints. The technique permits formal solution of the optimization problem, in a time short enough to meet the needs of treatment planners.

On the connections among activity-based costing, mathematical programming models for analyzing strategic decisions, and the resource-based view of the firm

This paper examines connections between data-driven models for analyzing a firms strategic plans, which use activity-based costing and mathematical programming, and the resource-based view of the firm. After brief reviews of the three disciplines, extensions of activity-based costing methods to mathematical programming models for strategic resource planning are discussed. Applications of these models to supply chain planning in a multi-national food manufacturer, a specialty chemicals company, and a wholesaling/retailing company are presented. The paper concludes by using concepts from the resource-based view of the firm to interpret optimal solutions from mathematical programming models. Extensions to strategic planning under uncertainty using stochastic programming are also discussed briefly.

An Adaptive Group Theoretic Algorithm for Integer Programming Problems

Group theory is used to integrate a wide variety of integer programming methods into a common computational process. Included are group optimization algorithms, Lagrangian methods, the cutting plane method, and the method of surrogate constraints. These methods are controlled by a supervisor which performs four main functions: set-up, directed search, subproblem analysis, and prognosis. Some computational experience is given. One appendix contains an algorithm for dynamically solving unconstrained group problems. A second appendix gives an algorithm for solving zero-one group problems.

Group Theoretic Algorithms for the Integer Programming Problem II: Extension to a General Algorithm

The main result of this paper is a group theoretic algorithm GTIP2 for the integer programming problem. This algorithm is an extension of an algorithm from an earlier paper part I. The algorithm in part I solves a group optimization problem derived from a given integer programming problem. The optimal solution to the group problem thereby obtained is an optimal solution to the integer programming problem if it is feasible. Unfortunately, an optimal solution to the group problem may yield an infeasible integer solution. The algorithm GTIP2 of this paper is an extension of the method of part I when it fails. In particular, GTIP2 employs a search procedure to find an optimal solution to the integer programming problem. The extent of the search is bounded by procedures derived from a variety of relevant group problems that are solved by the algorithm of part I. There is a discussion of the class of problems for which GTIP2 is primarily intended and the relation of GTIP2 to other algorithms is indicated. A numerical example and some partial computational results are included.

Computational experience with a group theoretic integer programming algorithm

This paper gives specific computational details and experience with a group theoretic integer programming algorithm. Included among the subroutines are a matrix reduction scheme for obtaining group representations, network algorithms for solving group optimization problems, and a branch and bound search for finding optimal integer programming solutions. The innovative subroutines are shown to be efficient to compute and effective in finding good integer programming solutions and providing strong lower bounds for the branch and bound search.

Chapter 8 Mathematical programming models and methods for production planning and scheduling

Publisher Summary This chapter focuses on the mathematical programming decomposition methods that allow large-scale models to be broken down into manageable sub-models, and then systematically reassembled. These methods show considerable promise for time critical scheduling applications, especially when the methods have been adapted for and implemented on parallel computers. The mathematical programming models fall into several categories: linear programming, network optimization, mixed integer programming, nonlinear programming, dynamic programming, multiple criteria optimization, and stochastic programming. The linear programming model assumes that all transformation activities are linear and additive. Network optimization models determine monthly production plans for tools and tool/machine combinations. Nonlinear mixed integer programming models for capacity expansion planning of electric utilities have received considerable attention. The chapter presents stochastic programming with recourse model that explicitly treats uncertainties regarding demand, fuel costs, and environmental restrictions.

Bottom-Up Vs. Top-Down Approaches to Supply Chain Modeling

The term “supply chain management” crystallizes concepts about integrated planning proposed by operations research practitioners, logistics experts, and strategists over the past 40 years (e.g., Hanssman (1959), LaLonde et al (1970), Porter (1985)). Integrated planning refers to functional coordination within the firm, between the firm and its suppliers, and between the firm and its customers. It also refers to intertemporal coordination of supply chain decisions as they relate to the firm’s operational, tactical and strategic plans.

Modeling and IT Perspectives on Supply Chain Integration

The essence of supply chain management is integrated planning of activities across the firms supply chain, including those of its suppliers and customers. Integrated planning refers to functional, geographical and inter-temporal coordination of managerial decisions. It requires and invokes modeling systems and processes that support data-driven or fact-based planning. A major challenge is to understand how to seamlessly imbed modeling systems in the firms information technology. This paper examines five perspectives on these connections.

Sensitivity Analysis in Integer Programming

This paper uses an IP duality Theory recently developed by the authors and others to derive sensitivity analysis tests for IP problems. Results are obtained for cost, right hand side and matrix coefficient variation.