http://directory.umm.ac.id/Journals/Journal%20of%20Operations%20Management/Vol19.Issue2.Feb2001/245.pdf

Arcs of integration: an international study of supply chain strategies

(This article originally appeared in Management Science, January 1981, Volume 27, Number 1, pp. 1-18, published by The Institute of Management Sciences.)

One of the most computationally useful ideas of the 1970s is the observation that many hard integer programming problems can be viewed as easy problems complicated by a relatively small set of side constraints. Dualizing the side constraints produces a Lagrangian problem that is easy to solve and whose optimal value is a lower bound (for minimization problems) on the optimal value of the original problem. The Lagrangian problem can thus be used in place of a linear programming relaxation to provide bounds in a branch and bound algorithm. This approach has led to dramatically improved algorithms for a number of important problems in the areas of routing, location, scheduling, assignment and set covering. This paper is a review of Lagrangian relaxation based on what has been learned in the last decade.

https://www.jstor.org/stable/info/2631139

The Lagrangian Relaxation Method for Solving Integer Programming Problems

Vendor selection criteria and methods

https://isiarticles.com/bundles/Article/pre/pdf/19101.pdf

A review of methods supporting supplier selection

A decision support system for supplier selection using an integrated analytic hierarchy process and linear programming

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

A Survey of Lagrangian Techniques for Discrete Optimization.

A computerized vendor selection system was implemented for the purchasing department of IBM's Poughkeepsie manufacturing facility. The system, called the Vendor Selection System (VSS), employs advanced mixed integer optimization techniques within a flexible but structured environment to help purchasing personnel to make better contract decisions. They can use the VSS without prior training in computer science or programming and achieve significant savings.

Improving Purchasing Productivity at IBM with a Normative Decision Support System

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.

Large scale optimization of beam weights under dose-volume restrictions.

This paper reports on the application of stochastic programming with recourse to strategic planning decisions regarding resource acquisition. A resource directed decomposition method, which simultaneously exploits stochastic programming and mixed integer programming model structures, is proposed. Computational experience with the method applied to fuel contract and plant construction decisions faced by an electric utility is presented.

Optimizing Resource Acquisition Decisions by Stochastic Programming

The current emphasis on optimizing the value chain, or integrated planning, advocated by business strategists and logistics practitioners indicates that the time is ripe for large increases in the number and scope of mathematical programming applications. Recent breakthroughs in information technology make it possible to develop advanced decision support systems based on mathematical programming models for integrated planning that operate on desk-top computers. We developed an advanced DSS operating on a personal computer that was used to evaluate strategic production and distribution planning questions in a large consumer products company.

Jeremy F. Shapiro

Papers

A Survey of Lagrangian Techniques for Discrete Optimization.

Improving Purchasing Productivity at IBM with a Normative Decision Support System

Large scale optimization of beam weights under dose-volume restrictions.

Optimizing Resource Acquisition Decisions by Stochastic Programming

Optimizing the Value Chain