operations research calculations handbook - 2001

[...]... 2001 CRC Press LLC Chapter 14 Matrices 14.1 Rules for Matrix Calculations 14.2 Inverses of Matrices 14.3 Series of Matrices Chapter 15 Combinatorics Chapter 16 Summations 16.1 Finite Sums 16.2 Infinite Sums Chapter 17 Interest Formulas References 11 2001 CRC Press LLC 1 Introduction The objective of this handbook is to provide a concise collection of analytical results and formulas that arise in operations. .. properties of matrices and Chapter 15 gives the standard formulas for combinatorial calculations Chapter 16 lists some common summation results Chapter 17 gives basic interest formulas important in investment calculations For each result or formula in this handbook, references are given for derivations and additional details 12 2001 CRC Press LLC 2127_C02_frame Page 3 Friday, December 20, 2002 9:55 AM 2... methodologies in optimization In keeping with the intent of this handbook to focus on analytical results and formulas, this chapter presents the mathematical formulations of basic linear programming problems and gives references for the solution methods Chapters 12–17 contain basic results in mathematics that are relevant to operations research Chapter 12 lists some common mathematical functions that... LLC 1 Introduction The objective of this handbook is to provide a concise collection of analytical results and formulas that arise in operations research applications The first few chapters are devoted to results on the stochastic modeling aspects of operations research Chapter 2 covers a range of formulas that involve the mean and variance of random variables Chapters 3 and 4 list the main properties... applying the above formulas for the mean and variance in the two-variable case and the correlation of Y and max(X1, X2) The results for the three-variable case are approximate, since max(X1, X2) is not normally distributed This procedure for approximate results can be extended to any finite number of normal random variables (Clark, 1961) 24 2001 CRC Press LLC 3 3.1 Discrete Probability Distributions BERNOULLI... for P(x) to be a probability distribution 29 2001 CRC Press LLC The probability that the geometric random variable X is less than or equal to a non-negative integer k is given by Pr{ X ≤ k} = k ∑ P( x) = 1 − (1 − p) k +1 x =0 The probability that X is greater than k is given by Pr{X > k} = (1 – p)k+1 The geometric distribution has the property that, for non-negative integers k and m, the conditional... denotes the cumulative distribution function of X If X takes non-negative values only (0 ≤ X < ∞), then the mean of X is given by ∞ E[ X ] = ∫ x f ( x)dx (2.6) 0 ∞ = ∫ [1 – F( x)]dx (2.7) 0 (Çinlar, 1975; Mood, Graybill, and Boes, 1974) For any random variable X, the variance is given by { Var[ X ] = E ( X − E[ X ]) [ ] 2 } = E X 2 − ( E[ X ]) 2001 CRC Press LLC 2 (2.8) (2.9) 14 2127_C02_frame Page 5 Friday,... between a and b, is given by 19 2001 CRC Press LLC b E[ X | a < X < b] = ∫ xf ( x )dx a Pr{a < X < b} (2.35) b = ∫ xf ( x )dx a b (2.36) ∫ f ( x )dx a b = ∫ xf ( x )dx a F(b) − F( a) (2.37) (Stirzaker, 1994) 2.8 MEAN AND VARIANCE OF THE SUM OF A RANDOM NUMBER OF RANDOM VARIABLES Let X1, X2, …, XN be N independent and identically distributed random variables, where N is a non-negative integer random variable... other analytical results that frequently arise in probability Chapters 6 and 7 present formulas that arise in stochastic processes and queueing theory The next three chapters cover applications of operations research in the areas of stochastic modeling Chapter 8 presents some results in production systems modeling and Chapter 9 covers the basic formulas in inventory control Chapter 10 gives distance... distribution P(x) (x = 0, 1) is used to characterize a random experiment with two possible outcomes The outcomes are generally 25 2001 CRC Press LLC FIGURE 3.1 Example of the Bernoulli distribution referred to as “success” (x = 1) and “failure” (x = 0), with probabilities p and 1-p, respectively Bernoulli Trials Repeated random experiments that are independent and have two possible outcomes with constant . index. ISBN 0-8 49 3-2 12 7-1 (alk. paper) 1. Operations research Handbooks, manuals, etc. 2. Mathematical. analysis—Handbooks, manuals, etc. I. Title. T57.6 B57 2001 658.4 ′ 034—dc21 20010 25425 2 2001. America 2 3 4 5 6 7 8 9 0 Printed on acid-free paper Library of Congress Cataloging-in-Publication Data Blumenfeld, Dennis. Operations research calculations handbook / by Dennis Blumenfeld. p