Introduction to the Mathematics of Operations Research with Mathematica«, Second Edition

by Hastings; Kevin J.

• ISBN13: 9781574446128
• ISBN10: 1574446126
• eBook ISBN(s): 9781351992169
• Edition: 2nd
• Format: Hardcover
• Copyright: 2006-05-30
• Publisher: Chapman & Hall/

Summary

The breadth of information about operations research and the overwhelming size of previous sources on the subject make it a difficult topic for non-specialists to grasp. Fortunately, Introduction to the Mathematics of Operations Research with Mathematica®, Second Edition delivers a concise analysis that benefits professionals in operations research and related fields in statistics, management, applied mathematics, and finance. The second edition retains the character of the earlier version, while incorporating developments in the sphere of operations research, technology, and mathematics pedagogy. Covering the topics crucial to applied mathematics, it examines graph theory, linear programming, stochastic processes, and dynamic programming. This self-contained text includes an accompanying electronic version and a package of useful commands. The electronic version is in the form of Mathematica notebooks, enabling you to devise, edit, and execute/reexecute commands, increasing your level of comprehension and problem-solving. Mathematica sharpens the impact of this book by allowing you to conveniently carry out graph algorithms, experiment with large powers of adjacency matrices in order to check the path counting theorem and Markov chains, construct feasible regions of linear programming problems, and use the "dictionary" method to solve these problems. You can also create simulators for Markov chains, Poisson processes, and Brownian motions in Mathematica, increasing your understanding of the defining conditions of these processes. Among many other benefits, Mathematica also promotes recursive solutions for problems related to first passage times and absorption probabilities.

Table of Contents

Graph Theory and Network Analysis	1	(142)
Definitions and Examples	2	(22)
Spanning Trees	24	(23)
Undirected Spanning Trees	25	(11)
Directed Spanning Trees	36	(11)
Minimal Cost Networks	47	(25)
Undirected Graphs	47	(9)
Directed Graphs	56	(16)
Critical Path Algorithm	72	(19)
Maximal Flow Problems	91	(22)
Problem Description	91	(3)
Main Results and Algorithm	94	(8)
Examples	102	(11)
Maximum Matching Problems	113	(22)
Definitions and Problem Description	113	(5)
Matching Algorithm	118	(6)
Examples	124	(11)
Other Problems of Graph Theory	135	(8)
Graph Coloring Problem	136	(2)
Shortest Paths Problem	138	(1)
Traveling Salesman Problem	139	(4)
Linear Programming	143	(68)
Two-Variable Problems	145	(14)
Geometry of Linear Programming	159	(11)
Simplex Algorithm for the Standard Maximum Problem	170	(23)
The Simplex Algorithm	170	(6)
Special Behavior	176	(8)
Tableau Method	184	(9)
Duality and the Standard Minimum Problem	193	(18)
Further Topics in Linear Programming	211	(50)
Non-Standard Problems	213	(14)
Transportation Problem	227	(15)
Sensitivity Analysis	242	(19)
Discussion of the Problem	242	(2)
Matrix-Geometric View of the Simplex Method	244	(5)
Determining Sensitivity of Parameters	249	(12)
Markov Chains	261	(66)
Definitions and Examples	263	(11)
Simulation	268	(6)
Short-Run Distributions	274	(10)
First Passage Times	284	(8)
Classification of States	292	(11)
Limiting Probabilities	303	(15)
Main Results	303	(8)
Long-Run Discounted Cost	311	(7)
Absorption Probabilities	318	(9)
Continuous Time Processes	327	(76)
Poisson Processes	327	(13)
Definitions and Main Results	327	(5)
Examples	332	(8)
Birth and Death Processes	340	(15)
Preliminaries	340	(5)
Kolmogorov Equations	345	(10)
Renewal Processes	355	(13)
Introduction	355	(1)
Short-Run Distributions	356	(4)
Long-Run Results	360	(3)
Renewal Reward Processes	363	(5)
Queueing Theory	368	(18)
Preliminaries	368	(2)
Simple Poissonian Queues	370	(6)
M/G/1 Queue	376	(4)
G/M/1 Queue	380	(6)
Brownian Motion	386	(17)
Relation to Random Walks	386	(3)
Definition and Properties of Standard Brownian Motion	389	(5)
Brownian Motion with Drift	394	(9)
Dynamic Programming	403	(98)
The Markovian Decision Model	403	(15)
Deterministic Dynamic Programming	404	(2)
Stochastic Dynamic Programming: The Finite Horizon Problem	406	(5)
Examples	411	(7)
The Finite Horizon Problem	418	(17)
Dynamic Programming Algorithm, Stochastic Case	421	(4)
Examples	425	(10)
The Discounted Reward Problem	435	(14)
Method of Successive Approximations	435	(5)
Examples	440	(9)
Policy Improvement	449	(12)
Main Theorem and Policy Improvement Algorithm	449	(4)
Examples	453	(8)
Optimal Stopping of a Markov Chain	461	(15)
Dynamic Programming Approach	461	(7)
Linear Programming Approach	468	(8)
Extended Applications	476	(25)
American Option Problem	477	(6)
Inventory Problem	483	(11)
Conclusion	494	(7)
Appendix A - Probability Review	501	(29)
Appendix B - Answers to Selected Exercises	530	(17)
Appendix C - Glossary of Mathematica Commands	547	(11)
References	558	(4)
Index	562