Operations Research Calculations Handbook

by Blumenfeld; Dennis

• ISBN13: 9780849321276
• ISBN10: 0849321271
• Format: Hardcover
• Copyright: 2001-06-27
• Publisher: CRC Press

Summary

The field of Operations Research (OR) covers a wide range of mathematical topics. Because it is so broad, results and formulas relevant to the field are widely scattered in different texts and journals and can be hard to find. As the field continues to grow, OR practitioners and students need a convenient, one-stop source for the results relevant to their work.The Operations Research Calculations Handbook meets that need. It contains more than 300 results in a single, concise volume. Organized by topic and listed in a convenient summary format, it allows readers to have frequently used results at their fingertips. Although based on the author's experience in the manufacturing industry, many of the results are basic to system modeling. They carry over easily to applications in other areas of operations research and management science.While modern software packages are useful for obtaining numerical results, formulas continue to play a significant role in systems modeling. They allow one to draw general conclusions about system behavior, reveal the underlying system model, and help provide an understanding of system performance. Whether you are a student, professor, or seasoned professional, the Operations Research Calculations Handbook offers not only a handy reference that will save time, but also a tool that will help build the intuitive understanding you need to apply systems models with confidence.

Table of Contents

Introduction	1	(2)
Means and Variances	3	(12)
Mean (Expectation) and Variance of a Random Variable	3	(2)
Covariance and Correlation Coefficient	5	(1)
Mean and Variance of a Sum of Random Variables	6	(1)
Mean and Variance of a Product of Two Random Variables	7	(1)
Mean and Variance of a Quotient of Two Random Variables	7	(1)
Conditional Mean and Variance for Jointly Distributed Random Variables	8	(1)
Conditional Mean of a Constrained Random Variable	9	(1)
Mean and Variance of the Sum of a Random Number of Random Variables	10	(1)
Mean of a Function of a Random Variable	11	(1)
Approximations for the Mean and Variance of a Function of a Random Variable	11	(1)
Mean and Variance of the Maximum of Exponentially Distributed Random Variables	12	(1)
Mean and Variance of the Maximum of Normally Distributed Random Variables	13	(2)
Discrete Probability Distributions	15	(14)
Bernoulli Distribution	15	(1)
Binomial Distribution	16	(2)
Geometric Distribution	18	(2)
Negative Binomial Distribution	20	(3)
Poisson Distribution	23	(1)
Hypergeometric Distribution	24	(3)
Multinomial Distribution	27	(2)
Continuous Probability Distributions	29	(28)
Uniform Distribution	29	(1)
Exponential Distribution	30	(2)
Erlang Distribution	32	(3)
Gamma Distribution	35	(3)
Beta Distribution	38	(2)
Normal Distribution	40	(5)
Sum of Normally Distributed Random Variables	41	(1)
Standard Normal Distribution	41	(1)
Partial Moments for the Normal Distribution	42	(1)
Approximations for the Cumulative Normal Distribution Function	43	(2)
Lognormal Distribution	45	(2)
Weibull Distribution	47	(2)
Logistic Distribution	49	(1)
Gumbel (Extreme Value) Distribution	50	(2)
Pareto Distribution	52	(2)
Triangular Distribution	54	(3)
Probability Relationships	57	(16)
Distribution of the Sum of Independent Random Variables	57	(1)
Distribution of the Maximum and Minimum of Random Variables	57	(3)
Example for the Uniform Distribution	58	(1)
Example for the Exponential Distribution	59	(1)
Change of Variable in a Probability Distribution	60	(2)
Conditional Probability Distribution for a Constrained Random Variable	62	(1)
Combination of Poisson and Gamma Distributions	63	(2)
Bayes' Formula	65	(1)
Central Limit Theorem	65	(2)
Probability Generating Function (z-Transform)	67	(1)
Moment Generating Function	68	(2)
Characteristic Function	70	(2)
Laplace Transform	72	(1)
Stochastic Processes	73	(6)
Poisson Process and Exponential Distribution	73	(2)
Properties of the Poisson Process	73	(1)
``Lack of Memory'' Property of the Exponential Distribution	74	(1)
Competing Exponentials	74	(1)
Superposition of Independent Poisson Processes	74	(1)
Splitting of a Poisson Process	74	(1)
Arrivals from a Poisson Process in a Fixed Interval	75	(1)
Renewal Process Results	75	(2)
Mean and Variance of Number of Arrivals in a Renewal Process	75	(1)
Distribution of First Interval in a Renewal Process	76	(1)
Markov Chain Results	77	(2)
Discrete-Time Markov Chains	77	(1)
Continuous-Time Markov Chains	77	(2)
Queueing Theory Results	79	(12)
Notation for Queue Types	79	(1)
Definitions of Queueing System Variables	79	(1)
Little's Law and General Queueing System Relationships	80	(1)
Extension of Little's Law	81	(1)
Formulas for Average Queue Length Lq	82	(1)
Formulas for Average Time in Queue Wq	83	(1)
References for Formulas for Average Queue Length and Time in Queue	84	(1)
Pollaczek-Khintchine Formula for Average Time in Queue Wq	85	(1)
Additional Formulas for Average Time in Queue Wq	85	(1)
Heavy Traffic Approximation for Distribution of Time in Queue	86	(1)
Queue Departure Process	87	(1)
Distribution Results for Number of Customers in M/M/1 Queue	88	(1)
Distribution Results for Time in M/M/1 Queue	89	(1)
Other Formulas in Queueing Theory	90	(1)
Production Systems Modeling	91	(8)
Definitions and Notation for Workstations	91	(1)
Basic Relationships between Workstation Parameters	91	(1)
Distribution of the Time to Produce a Fixed Lot Size at a Workstation	92	(2)
Throughput of a Serial Production Line with Failures	94	(1)
Throughput of a Two-Station Serial Production Line with Variable Processing Times	95	(1)
Two Stations without Buffer	95	(1)
Two Stations with Buffer	96	(1)
Throughput of an N-Station Serial Production Line with Variable Processing Times	96	(3)
Inventory Control	99	(24)
Economic Order Quantity (EOQ)	99	(2)
Economic Production Quantity (EPQ)	101	(3)
``Newsboy Problem'': Optimal Inventory to Meet Uncertain Demand in a Single Period	104	(2)
Inventory Replenishment Policies	106	(2)
(s, Q) Policy: Estimates of Reorder Point (s) and Order Quantity (Q)	108	(3)
(s, S) Policy: Estimates of Reorder Point (s) and Order-Up-To Level (S)	111	(2)
(T, S) Policy: Estimates of Review Period (T) and Order-Up-To Level (S)	113	(3)
(T, s, S) Policy: Estimates of Review Period (T), Reorder Point (s), and Order-Up-To Level (S)	116	(3)
Summary of Results for Inventory Policies	119	(1)
Inventory in a Production/Distribution System	120	(2)
A Note on Cumulative Plots	122	(1)
Distance Formulas for Logistics Analysis	123	(8)
``Traveling Salesman Problem'' Tour Distance: Shortest Path through a Set of Points in a Region	123	(1)
Distribution of Distance between Two Random Points in a Circle	124	(3)
Average Rectangular Grid Distance between Two Random Points in a Circle	127	(1)
Great Circle Distance	128	(3)
Linear Programming Formulations	131	(20)
General Formulation	131	(1)
Terminology	132	(1)
Example of Feasible Region	133	(1)
Alternative Formulations	134	(1)
Minimization vs. Maximization	134	(1)
Equality Constraints	134	(1)
Reversed Inequality Constraints	135	(1)
Diet Problem	135	(1)
Duality	136	(1)
Special Cases of Linear Programming Problems	137	(7)
Transportation Problem	137	(4)
Transshipment Problem	141	(1)
Assignment Problem	142	(2)
Integer Linear Programming Formulations	144	(3)
Knapsack Problem	144	(2)
Traveling Salesman Problem	146	(1)
Solution Methods	147	(4)
Simplex Method	147	(1)
Interior-Point Methods	148	(1)
Network Flow Methods	149	(1)
Cutting Planes	149	(1)
Branch and Bound	149	(2)
Mathematical Functions	151	(4)
Gamma Function	151	(1)
Beta Function	151	(1)
Unit Impulse Function	152	(1)
Modified Bessel Functions	152	(1)
Stirling's Formula	153	(2)
Calculus Results	155	(18)
Basic Rules for Differentiation	155	(2)
Integration by Parts	157	(1)
Fundamental Theorem of Calculus	157	(1)
Taylor Series	158	(1)
Maclaurin Series	158	(1)
L'Hopital's Rule	158	(1)
Lagrange Multipliers	159	(2)
Differentiation under the Integral Sign (Leibnitz's Rule)	161	(1)
Change of Variable in an Integral	162	(1)
Change of Variables in a Double Integral	162	(3)
Changing the Order of Integration in a Double Integral	165	(1)
Changing the Order of Summation in a Double Sum	166	(1)
Numerical Integration	167	(6)
Matrices	173	(4)
Rules for Matrix Calculations	173	(1)
Inverses of Matrices	174	(1)
Series of Matrices	175	(2)
Combinatorics	177	(2)
Summations	179	(2)
Finite Sums	179	(1)
Infinite Sums	179	(2)
Interest Formulas	181	(2)
References	183	(8)
Index	191