This section provides an index to the supplements to the chapters of the text Operations Research Models and Methods.
We also provide several new chapters. All supplements are PDF documents. Click on a title to download a supplement. Supplements can be read with the free reader provided by Adobe.
Methods Of Operations Research
Download Acrobat Reader. Bard Copyright - All rights reserved. Presentations PDF Documents. PDF Supplements to the Text. Additional Exercises. Equivalent Linear Programs There are a number of problems that do not at first appear to be candidates for linear programming but, in fact, have an equivalent or approximate representation that fits the LP framework.
In these instances, the solution to the equivalent problem gives the solution to the original problem.
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Among the subjects considered in this supplement are: nonlinear objective functions, maximizing the minimum, goal programming, fractional programming, and the complementarity problem. The supplement includes exercises.
Chemical Processing We consider a somewhat larger problem than those used in Chapter 2 as a medium for the discussion of the modeling process. The example is especially informative in that it incorporates a number of features found in many real-world applications. This supplement describes the relationship between the two models, their primal and dual variables and solution algorithms.
Primal Simplex Algorithm for the Pure Minimal Cost Flow Problem Based on the primal and dual variables described in Chapter 6, we provide an algorithm to solve the minimum cost flow problem for pure network problems. Exercises are included. Generalized Network Flow Programming This chapter long supplement adapts the bounded variable primal simplex method to the generalized minimum cost flow problem.
Additional Branch and Bound Algorithms Two additional sections and associated exercises comprise this supplement. One provides an example branch and bound applied to the mixed integer programming problem. The second describes Balas' implicit enumeration procedure.
Algorithms for Constrained Optimization In this additional section, we present several procedures for solving the nonlinear programming problem. The first is the now classical penalty approach developed by Fiacco and McCormick  and is perhaps the simplest to implement.
The second is Zoutendijk's feasible direction method. Other primal approaches discussed in the literature include the gradient projection method and the generalized reduced gradient method that is a simplex-like algorithm.
Dynamic Programming Models Dynamic programming has been described as the most general of the optimization approaches because conceivably it can solve the broadest class of problems.
In many instances, this promise is unfulfilled because of the attending computational requirements. Certain problems, however, are particularly suited to the model structure and lend themselves to efficient computational procedures; in cases involving discontinuous functions or discrete variables, dynamic programming may be the only practical solution method. This supplement is a complete chapter with exercises.
Dynamic Programming Methods Although all dynamic programming DP problems have a similar structure, the computational procedures used to find solutions are often quite different.
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Unlike linear and integer programming where standard conventions are used to describe parameters and coefficients, there is no common data structure that unifies all DPs. Models are problem specific, and for the most part, are represented by recursive formulas rather than algebraic expressions.
The chapter describes the principal compuational procedures used to solve dynamic programming models. Probability Models Here we present the language of probability that is used to model situations whose outcomes cannot be predicted with certainty.
In important instances, reasonable assumptions allow single random variables to be described using one of the named discrete or continuous random variables. The chapter summarizes the formulas for obtaining probabilities and moments for these distributions.
When combinations of several random variables affect the system output, the method of simulation is often used to learn about a system. Here we provide an introduction to the subject. Time Series and Forecasting A time series is a sequence of observations of a periodic random variable.
Examples are: the monthly demand for a product, the annual freshman enrollment in a department of the university and the daily flows in a river. Time series are important for operations research because they are often the drivers of operations research decision models.
Time series analysis provides tools for selecting a model that describes the time series and using the model to forecast future events. Modeling the time series is a statistical problem because observed data is used in computational procedures to estimate the parameters of a supposed model.
Decision Analysis Consider a situation requiring a series of decisions.
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The problem is complicated however in that the results of some of the decisions are not deterministic. Rather, the results are governed by known probability distributions. Decision analysis provides a model for handling processes involving an ordered series of decisions interrupted by random events.
Many practical situations can be modeled in this way.
Solution procedures find the optimum decisions that maximize the expected net benefit. Game Theory The decision problems we often face are associated with another individual who may be an opponent.
We must make decisions knowing that the result will be governed in part by the actions of another. The part of operations research that deals with this kind of situation is called game theory.
Although clearly applicable to games as the name implies, it is appropriate for the wide variety of contexts in which one must make a decision whose result will be impacted by the actions of one or more individuals. Inventory Theory Because of practical and economic importance of industrial inventories, the subject of inventory control is a major consideration in many situations.
Questions must be constantly answered as to when and how much raw material should be ordered, when a production order should be released to the plant, what level of safety stock should be maintained at a retail outlet, or how in-process inventory is to be maintained in a production process.
These questions are amenable to quantitative analysis through the subject of inventory theory. Reliability The name reliability is given to the field of study that attempts to assign numbers to the propensity of systems to fail. In a more restrictive sense, the term "reliability" is defined to be the probability that a system performs its mission successfully. Since mission is often specified in terms of time, reliability is often defined as the probability that a system will operate satisfactorily for a given time.
Thus reliability may be a function of time.
We use probability models to compute the reliability of a complex system. Stochastic Optimization Operations research has been particularly successful in two areas of decision analysis: i optimization of problems involving many variables when the outcome of the decisions can be predicted with certainty, and ii the analysis of situations involving a few variables when the outcome of the decisions cannot be predicted with certainty.
In this chapter, we combine optimization and uncertainty and consider two decision models that explicitly incorporate the probability distributions of random variables.
Both are solved with linear programming LP. Although we will see that rather simple situations lead to large LP models, the general availability of efficient LP algorithms may make the solution of such problems practical.