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Index

->,8 :,6 :=,8 ;,6

%1,369 %2,369

adaptive random search optimization using, 441

advertising policies, 127 algsubs, 217 all values, 41, 342, 368 alternating renewal process, 265 arrays, II assign, 11,369 assume, 8, 307 augment, 59

balance equations, 319 basic solution, 105 basic solution set, 105 basis,63 basis matrix, 105 birth and death process, 306, 377

bivariate normal plot of density, 25

Chapman-Kolmogorov equations, 276 charpoly,62 Cholesky factorization, 164 Cobb-Douglas production function,

179 coeff,217 collect, 13 combine,7 complementary slackness, 171 constraint qualification, 174 continuous-time Markov chain, 306

balance equations, 319 exponential matrix, 313 infinitesimal generator, 306 Kolmogorov differential equa-

tions, 308 limiting probabilities, 318

contourplot,22 convert, II convex function, 68, ISS convex set, 101, 154 convexity

464 Index

relation to Hessian, 160 cycle length, 425

D,45 definite, 67, 159 derivatives, 45 describe, 17 det, 56 diagonalization, 61 diff,45 difference equation

nonlinear, 229 differential equation, 15

analytic solution, 69 infolevel,31 numerical solution, 22 partial,77 savings account, 69 system

in optimal control, 73 Lancaster's, 72 numerical solution, 74

Van der Pol's numerical solution, 70

Digits, 277 display, 101 dsol ve, 23, 72, 76, 241, 395 dual,139 dual price, 131 duality, 136

economic interpretation, 138 in game theory, 143 main results, 140

dynamic inventory model dynamic programming formu­

lation, 361 optimal base-stock policy, 362 optimality of (s, S) policy, 366

dynamic programming backward recursion, 195 continuous time, 211 cost-to-go (value) function, 196 decision, 194 discount factor, 211, 213 forward recursion, 195

policy, 194 stage, 194 state, 193 transformation, 194

dynamic simulation inventory model

random yield, 436

eigenvalUes, 61, 159 eigenvectors, 61 eigenvectors, 62 elementary renewal theorem, 261 empirical,91 EOQ model, 335

planned backorders, 338 equation

analytic solution, 14 difference (recurrence), 79 numerical solution, 16 solution of cubic, 38 solution of quadratic, 38

equations differential-difference, 244 homogeneous linear system, 60 linear system, 58 nonlinear system, 41

eval,8 evalf,16

int, 433 evalm, 12,55 expand,6 exponential,313 exponential matrix, 313 exporting to

FORTRAN,32 LaTeX, 32

extrema, 68 extreme point, 101

factor, 6 feasible points, 105 Fibonacci sequence, 79 finance, 33 fit, 64 fsolve, 17,40, 170

futurevalue,29

G/G/l simulation, 438

gambling model, 225 geneqns,58 generating function, 81, 85,296,302

as opposed to z-transfonn, 85 numerical inversion, 87

genfunc,298,302,388 genmatrix,107 grad,156 gradient, 67 gradient vector, 156 graphics

three-dimensional, 24 two-dimensional, 20

Hamilton-Jacobi-Bellman equation, 213

hazard rate, 240 Hessian, 66, 156 hessian, 157 histogram, 18

ifactor, 16 ilp,146 implicitdiff,48 implici tplot, 22 implied backorder costs, 343 indefinite, 157 inequal, 101 inequality

solution, 39 infeasible basic solution, 106 infinite series, 54 infolevel, 113 infolevel,30 integer programming, 145 integral, 50

multiple, 53 integration

infolevel,30 interface, 29 intersect, 10

inttrans, 242, 258 inventory, 331

Index 465

classification, 332 reasons for keeping, 332

inventory cost holding, 334 penalty, 335 procurement, 334

inventory model backorder cost function, 161 backorders, 24 constrained multi-item, 41 continuous-review, 49 lost sales, 46 periodic review, 274, 277 periodic-review

stationary distribution, 279 quantity discounts, 27 random yield

simulation, 436 stochastic, 14

inverse, 56 invlaplace,81 is,68 iso-profit lines, 103 isolate, 359

Jackson network, 398 Jacobian, 67, 95

key renewal theorem, 264 Kolmogorov differential equations

backward, 310 forward, 313

Kolmogorov-Smimov test, 426 Kuhn-Tucker conditions, 171

geometric interpretation, 173

I'Hopital's rule, 296 Lagrange multipliers, 171

economic interpretation, 181 Lagrangian dual, 186 Lagrangian function, 176 laplace, 82 Laplace transfonn, 81

466 Index

numerical inversion, 83 least squares, 64 leastsquare,64 Leibniz's rule, 47, 358 limit, 44 limiting (steady-state) probabilities,

278 linalg, 55, 302 linear programming, 19

degeneracy, 119 graphical solution, 100 infeasibility, 118 mixed constraints, 116 number of iterations, 107 simplex, 104 transportation model, 121 unbounded solution, 119

linear system homogeneous, 60 solution, II

linear-quadratic control, 187,201 infinite stage, 207 relation to nonlinear program-

ming,204 linsol ve, 58 lists, to location problem, 163, 169

M(t)/M(t)/S/N dynamic service rate control, 406

M/Ek/ I,299 M/ M/l, 298, 378

controlled arrivals, 404 M/M/I/I

transient solution, 390 M/M/l/K,382 M/M/c,383 M/M/oo

transient solution, 391 M/M/oo,78 makeproc,79 map, 12 Maple

books on, 6 help facility, 5

Internet resources, xiii knowledge base, xi Release 5.1 and 6, xiii users, xi

Markov chain, 61, 272 aperiodic, 288 communicating states, 287 equivalence class, 288 ergodic, 290 first passage time probability, 288 imbedded, 293 irreducible, 288 machine maintenance, 283 Markovian property, 273 mean recurrence time, 290 null recurrent, 290 pathological case, 286 period of a state, 288 positive recurrent, 290 reachable states, 287 recurrent state, 288 transient behavior, 301 transient state, 288

matrix multiplication, 55 matrix operations, 55 matrixDE,311 Milton's sonnets, 375 minus, 280 mixed-congruential method, 424 Monte Carlo simulation, 432

numerical evaluation of definite integrals, 432

multiply, 129 MX /M/I,387 myopic solution, 227

negative definite, 157 negative semidefinite, 157 nonbasic variables, 105 nonhomogeneous Poisson process, 247 nonlinear programming, 153

duality theorem, 186 NONNEGATIVE, 126 nops,11 normal,7

odetest, 69, 72 op, 10,30 open-loop decision, 220 operations research

definition, 2 journals, 2 origins, 1 without mathematics, 37

optimal stopping, 228 infinite stage, 230

option pricing blackscholes,34

orientation in plot3d, 26

output=listprocedure, 410

partial differential equation, 77 partial fraction expansion, 82 pdsolve,77 piecewise, 27, 222, 347, 409 piecewise-quadratic cost, 189 plot,249 plot3d, 25, 282 Poisson process, 243,308

compound, 252 nonhomogeneous, 247

Pollaczek-Khinchine formula, 298 Pontryagin's principle, 408 portfolio selection, 154, 171 positive definite, 157 positive definiteness, 67 positive semidefinite, 157 principal minor, 158 principle of optimality, 195 probabilistic inventory model

continuous review with backo­rders

approximate formulation, 349 exact formulation, 354

one period (newsvendor), 358 yield randomness, 367

probability conditional, 277 unconditional, 277

programming, 32

Index 467

Pushkin, 273

quadratic form, 157 quantity discounts, 346 queueing networks, 394 queueing notation, 376 queueing system

optimization, 401

random, 420 uniform, 426

random variable beta, 231 binomial, 418

moment generating function, 55

simulation, 420 Cauchy, 94 continuous

pdf, cdf, icdf, 89 discrete

pf, dcdf, idcdf, 89 Erlang, 242

density, 246, 299 exponential, 14,52,238

memory less property, 239 geometric, 55 normal, 16, 89

mean and variance, 92 Poisson, 90 triangular, 28 uniform, 423 unit normal, 52 Weibull, 241

plot of density, 20 random variate generation, 18

exponential, 429 inverse transform method, 430 random, 430

random vector joint density, 95

random walk, 272 randvector, 129 rank,58 renewal density, 259

468 Index

renewal equation, 262 renewal function, 259 renewal process, 256 renewal reward theorem, 268

use in inventory models, 335 rgf _encode, 85 rgf _expand, 85 rgf _expand, 303 rsolve, 79, 229, 383 Runge-Kutta method, 395

saddle-point conditions, 186 select, 437 sensitivity analysis

adding a new decision variable, 135

change in objective function, 134 change in RHS values, 131

serial queue with blocking, 394 sets, 10 simplex, 19

final table, 115 simplex, 113

not useful in integer program­ming,145

simplify, 7, 370 simulation, 417

number of runs, 421 programs, 418 runlength,421

slack variable, 104 solve, 16,171,181 sprint, 315 stagecoach problem, 197 statevalf,89 static simulation

car and goats problem, 434 s ta ts, 17, 64,426

distributions, 430 statsplots,18 stochastic matrix, 272 stochastic process, 237 strong law oflarge numbers, 433 student, 248 subs,8

successive approximations, 207 sum, 170,385

tallyinto, 18 time~ependentarrivalrate, 75 transform, 18 transportation queueing process, 402 type=nurneric, 23, 70, 76

unapply,8, 10,304,355 union, 10,280

work force planning model, 219

zero-sum game solution using linear program­

ming, 124 ztrans,85