Interactive Operations Research with Maple, 2000 Methods and Models

Interactive Operations Research with Maple: Methods and Models has two ob­ jectives: to provide an accelerated introduction to the computer algebra system Maple and, more importantly, to demonstrate Maple's usefulness in modeling and solving a wide range of operations research (OR) problems. This book is written in a format that makes it suitable for a one-semester course in operations research, management science, or quantitative methods. A nwnber of students in the departments of operations research, management science, oper­ ations management, industrial and systems engineering, applied mathematics and advanced MBA students who are specializing in quantitative methods or opera­ tions management will find this text useful. Experienced researchers and practi­ tioners of operations research who wish to acquire a quick overview of how Maple can be useful in solving OR problems will find this an excellent reference. Maple's mathematical knowledge base now includes calculus, linear algebra, ordinary and partial differential equations, nwnber theory, logic, graph theory, combinatorics, statistics and transform methods. Although Maple's main strength lies in its ability to perform symbolic manipulations, it also has a substantial knowledge of a large nwnber of nwnerical methods and can plot many different types of attractive-looking two-dimensional and three-dimensional graphs. After almost two decades of continuous improvement of its mathematical capabilities, Maple can now boast a user base of more than 300,000 academics, researchers and students in different areas of mathematics, science and engineering.

1 Introduction to Operations Research.- 1.1 A Brief History.- 1.2 The Method of OR.- 1.3 About This Book.- 2 A Quick Tour of Maple.- 2.1 Introduction.- 2.2 Symbolics.- 2.2.1 Expanding, Factoring and Simplifications.- 2.2.2 Function Definitions Using “->” and unapply ().- 2.2.3 Lists, Sets and Arrays.- 2.2.4 Additional Commands to Manipulate Expressions.- 2.2.5 Solving Equations Analytically.- 2.3 Numerics.- 2.3.1 Solving Equations Numerically.- 2.3.2 Statistical Computations Using the stats Package.- 2.3.3 The simplex Package.- 2.4 Graphics.- 2.4.1 Two-dimensional Plots.- 2.4.2 Three-Dimensional Plots.- 2.5 Other Useful Commands and Packages.- 2.5.1 The piecewise () Command.- 2.5.2 The interface () Command.- 2.5.3 The infolevel[] Command.- 2.5.4 Exporting to C, FORTRAN and LaTeX.- 2.5.5 Programming in Maple.- 2.5.6 The finance Package.- 2.6 Summary.- 2.7 Exercises.- 3 Maple and Mathematical Foundations of Operations Research.- 3.1 Introduction.- 3.2 Algebra.- 3.2.1 Solution of a Single Equation or Inequality.- 3.2.2 Solution of a System of Nonlinear Equations.- 3.3 Calculus.- 3.3.1 Limits.- 3.3.2 Derivatives.- 3.3.3 Integrals.- 3.3.4 Finite Sums and Infinite Series.- 3.4 Linear Algebra.- 3.4.1 Matrix Operations.- 3.4.2 Solution of Simultaneous Linear Equations.- 3.4.3 Eigenvalues, Eigenvectors and Diagonalization.- 3.4.4 Least Squares Fitting to Data.- 3.4.5 Special Matrices.- 3.4.6 Positive Definiteness.- 3.5 Differential Equations.- 3.5.1 Ordinary Differential Equations.- 3.5.2 Partial Differential Equations.- 3.5.3 Difference (Recurrence) Equations.- 3.6 Transform Methods.- 3.6.1 Laplace Transforms.- 3.6.2 Generating Functions.- 3.7 Probability.- 3.7.1 Continuous and Discrete Random Variables.- 3.7.2 Expectation.- 3.7.3 Jointly Distributed Random Variables.- 3.8 Summary.- 3.9 Exercises.- 4 Linear Programming.- 4.1 Introduction.- 4.2 Graphical Solution.- 4.3 The Simplex Method.- 4.3.1 Manual Solution of the Production Problem.- 4.3.2 Solution Using Maple’s simplex Package.- 4.3.3 A Problem with Mixed Constraints.- 4.4 Special Cases and Difficulties.- 4.4.1 Infeasibility.- 4.4.2 Unbounded Solutions.- 4.4.3 Degeneracy.- 4.5 Other Examples.- 4.5.1 The Transportation Problem.- 4.5.2 Two-Person Zero-Stun Games.- 4.5.3 A Linear Program with Randomly Generated Data.- 4.6 Sensitivity Analysis and Duality.- 4.6.1 Changes in the RHS Values and Dual Prices.- 4.6.2 Changes in the Objective Function Coefficients.- 4.6.3 Addition of a New Decision Variable.- 4.6.4 Duality.- 4.7 Integer Linear Programming.- 4.8 Summary.- 4.9 Exercises.- 5 Nonlinear Programming.- 5.1 Introduction.- 5.2 Convexity of Sets and Functions.- 5.2.1 Positive and Negative Definite Matrices.- 5.2.2 Convexity of a Function and Definiteness of its Hessian.- 5.2.3 Examples of Definiteness/Convexity.- 5.2.4 Cholesky Factorization.- 5.3 Unconstrained Optimization.- 5.4 Inequality and Equality Constrained Optimization.- 5.4.1 Geometric Interpretation of Kuhn-Tucker Conditions.- 5.4.2 Constraint Qualification.- 5.4.3 An Equivalent Formulation Using the Lagrangian.- 5.4.4 Economic Interpretation of the Lagrange Multipliers.- 5.5 Lagrangian Duality.- 5.6 Summary.- 5.7 Exercises.- 6 Dynamic Programming.- 6.1 Introduction.- 6.2 Stagecoach Problem.- 6.3 Models with a Linear System and Quadratic Cost.- 6.3.1 The Infinite-Stage Problem.- 6.4 Continuous-Time Dynamic Programming.- 6.4.1 A Problem with Linear System and Quadratic Cost.- 6.4.2 A Problem with Quadratic and Linear Costs.- 6.5 A Constrained Work Force Planning Model.- 6.6 A Gambling Model with Myopic Optimal Policy.- 6.7 Optimal Stopping Problems.- 6.7.1 The Infinite-Stage Problem.- 6.8 Summary.- 6.9 Exercises.- 7 Stochastic Processes.- 7.1 Introduction.- 7.2 Exponential Distribution and Poisson Process.- 7.2.1 Memoryless Property of the Exponential Distribution.- 7.2.2 Hazard Rate Function.- 7.2.3 The Erlang Random Variable.- 7.2.4 Poisson Process.- 7.2.5 Interarrival Times of the Poisson Process.- 7.2.6 Density of the Erlang—An Equivalent Derivation.- 7.2.7 Nonhomogeneous Poisson Process.- 7.2.8 Compound Poisson Process.- 7.3 Renewal Theory.- 7.3.1 Preliminaries.- 7.3.2 Distribution of N(t).- 7.3.3 Renewal Function and Renewal Density.- 7.3.4 Renewal Equation and the Key Renewal Theorem.- 7.3.5 Renewal Reward Process.- 7.4 Discrete-Time Markov Chains.- 7.4.1 Chapman-Kolmogorov Equations.- 7.4.2 Limiting Probabilities and the Stationary Distribution.- 7.4.3 Classification of States.- 7.4.4 Imbedded Markov Chain Technique.- 7.4.5 Transient Behavior of Markov Chains.- 7.5 Continuous-Time Markov Chains.- 7.5.1 Kolmogorov Differential Equations.- 7.5.2 Limiting Probabilities.- 7.6 Summary.- 7.7 Exercises.- 8 Inventory Models.- 8.1 Introduction.- 8.2 Classification of Inventory Models.- 8.3 Costs Associated with Inventory Models.- 8.3.1 Procurement Cost.- 8.3.2 Holding (or Carrying) Cost.- 8.3.3 Penalty Cost.- 8.4 Deterministic Inventory Models.- 8.4.1 The Basic EOQ Model.- 8.4.2 The EOQ Model with Planned Backorders.- 8.4.3 Analysis of Implied Backorder Costs.- 8.4.4 Quantity Discounts.- 8.5 Probabilistic Inventory Models.- 8.5.1 The Continuous-Review Model: Approximate Formulation.- 8.5.2 The Continuous-Review Model: Exact Formulation.- 8.5.3 One-Period (Newsvendor) Model with Random Demand.- 8.5.4 Dynamic Inventory Models.- 8.5.5 Diversification Under Yield Randomness.- 8.6 Summary.- 8.7 Exercises.- 9 Queueing Systems.- 9.1 Introduction.- 9.2 Markovian Queueing Systems.- 9.2.1 Birth and Death Processes.- 9.2.2 M/M/1 Queueing System.- 9.2.3 Finite Capacity Markovian Queue: M/M/1/K.- 9.2.4 Multiserver Queue M/M/c.- 9.2.5 Markovian Bulk Arrival System: MX/M/1.- 9.3 Transient Solutions.- 9.3.1 Finite Capacity M/M/1/K Queue with K= 1.- 9.3.2 Ample Server System M/M/?.- 9.4 Queueing Networks.- 9.4.1 Serial Queue with Blocking.- 9.4.2 Jackson Networks.- 9.5 Optimization of Queueing Systems.- 9.5.1 A Transportation Queueing Process.- 9.5.2 An M/M/1 System with Controlled Arrivals.- 9.5.3 Optimal Dynamic Service Rate Control.- 9.6 Summary.- 9.7 Exercises.- 10 Simulation.- 10.1 Introduction.- 10.2 Generating (Pseudo-) Random Numbers.- 10.2.1 Mixed-Congruential Method.- 10.2.2 Maple’s Uniform Random Number Generator.- 10.2.3 Kolmogorov-Smirnov Test for Uniformity.- 10.3 Generating Random Variates from Other Distributions.- 10.3.1 Exponential Random Variates.- 10.3.2 Maple’s Random Variate Generators.- 10.4 Monte Carlo Simulation.- 10.4.1 Numerical Evaluation of Definite Integrals.- 10.4.2 Simulation of a Static (Single-Period) Problem.- 10.5 Dynamic Simulation Models.- 10.5.1 Simulation of an Inventory System with Random Yield.- 10.5.2 Simulation of a Non-Markovian Queue.- 10.6 Optimization by Random Search.- 10.7 Summary.- 10.8 Exercises.- References.