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optimization modeling with lingo

optimization modeling with lingo
fifth edition
Lingo system  INC.
1415 North Dayton Street
Chicago, Illinois 60622
Technical Support: (312) 988-9421
http://www.lindo.com
e-mail: tech@lindo.com

Contents
Contents.............................................................................................................................. iii
Preface ............................................................................................................................... xiii
Acknowledgments............................................................................................................. xiii
Ch 1 What Is Optimization? ................................................................................................. 1
1.1 Introduction ................................................................................................................... 1
1.2 A Simple Product Mix Problem ..................................................................................... 1
1.2.1 Graphical Analysis.................................................................................................. 2
1.3 Linearity ........................................................................................................................ 5
1.4 Analysis of LP Solutions................................................................................................ 6
1.5 Sensitivity Analysis, Reduced Costs, and Dual Prices .................................................. 7
1.5.1 Reduced Costs....................................................................................................... 8
1.5.2 Dual Prices............................................................................................................. 8
1.6 Unbounded Formulations.............................................................................................. 9
1.7 Infeasible Formulations............................................................................................... 10
1.8 Multiple Optimal Solutions and Degeneracy................................................................ 11
1.8.1 The “Snake Eyes” Condition................................................................................. 13
1.8.2 Degeneracy and Redundant Constraints.............................................................. 16
1.9 Nonlinear Models and Global Optimization ................................................................. 17
1.10 Problems................................................................................................................... 18
Ch 2 Solving Math Programs with LINGO ........................................................................ 21
2.1 Introduction ................................................................................................................. 21
2.2 LINGO for Windows .................................................................................................... 21
2.2.1 File Menu.............................................................................................................. 21
2.2.2 Edit Menu ............................................................................................................. 23
2.2.3 LINGO Menu ........................................................................................................ 25
2.2.4 Windows Menu..................................................................................................... 26
2.2.5 Help Menu............................................................................................................ 27
2.2.6 Summary.............................................................................................................. 27
2.3 Getting Started on a Small Problem............................................................................ 28
2.4 Integer Programming with LINGO............................................................................... 28
2.4.1 Warning for Integer Programs .............................................................................. 30
2.5 Solving an Optimization Model.................................................................................... 30
2.6 Problems..................................................................................................................... 31
Ch 3 Analyzing Solutions .................................................................................................. 33
3.1 Economic Analysis of Solution Reports....................................................................... 33
3.2 Economic Relationship Between Dual Prices and Reduced Costs ............................. 33
3.2.1 The Costing Out Operation: An Illustration ........................................................... 34
3.2.2 Dual Prices, LaGrange Multipliers, KKT Conditions, and Activity Costing............. 35
3.3 Range of Validity of Reduced Costs and Dual Prices.................................................. 36
3.3.1 Predicting the Effect of Simultaneous Changes in Parameters—The 100% Rule. 41
3.4 Sensitivity Analysis of the Constraint Coefficients....................................................... 42
3.5 The Dual LP Problem, or the Landlord and the Renter ............................................... 43
3.6 Problems..................................................................................................................... 45
Ch 4 The Model Formulation Process............................................................................... 51
4.1 The Overall Process ................................................................................................... 51
4.2 Approaches to Model Formulation .............................................................................. 52
4.3 The Template Approach.............................................................................................. 52
4.3.1 Product Mix Problems .......................................................................................... 52
4.3.2 Covering, Staffing, and Cutting Stock Problems................................................... 52
4.3.3 Blending Problems ............................................................................................... 52
4.3.4 Multiperiod Planning Problems ............................................................................. 53
4.3.5 Network, Distribution, and PERT/CPM Models..................................................... 53
4.3.6 Multiperiod Planning Problems with Random Elements........................................ 53
4.3.7 Financial Portfolio Models .................................................................................... 53
4.3.8 Game Theory Models ........................................................................................... 54
4.4 Constructive Approach to Model Formulation ............................................................. 54
4.4.1 Example ............................................................................................................... 55
4.4.2 Formulating Our Example Problem ...................................................................... 55
4.5 Choosing Costs Correctly ........................................................................................... 56
4.5.1 Sunk vs. Variable Costs ....................................................................................... 56
4.5.2 Joint Products....................................................................................................... 58
4.6 Common Errors in Formulating Models....................................................................... 59
4.7 The Nonsimultaneity Error .......................................................................................... 62
4.8 Problems..................................................................................................................... 62
Ch 5 The Sets View of the World....................................................................................... 65
5.1 Introduction ................................................................................................................. 65
5.1.1 Why Use Sets?..................................................................................................... 65
5.1.2 What Are Sets? .................................................................................................... 65
5.1.3 Types of Sets ....................................................................................................... 66
5.2 The SETS Section of a Model..................................................................................... 66
5.2.1 Defining Primitive Sets ......................................................................................... 66
5.2.2 Defining Derived Sets........................................................................................... 67
5.2.3 Summary.............................................................................................................. 68
5.3 The DATA Section ...................................................................................................... 69
5.4 Set Looping Functions ................................................................................................ 71
5.4.1 @SUM Set Looping Function ............................................................................... 72
5.4.2 @MIN and @MAX Set Looping Functions............................................................ 73
5.4.3 @FOR Set Looping Function................................................................................ 74
5.4.4 Nested Set Looping Functions.............................................................................. 75
5.5 Set Based Modeling Examples ................................................................................... 75
5.5.1 Primitive Set Example .......................................................................................... 76
5.5.2 Dense Derived Set Example................................................................................. 79
5.5.3 Sparse Derived Set Example - Explicit List........................................................... 81
5.5.4 A Sparse Derived Set Using a Membership Filter................................................. 86
5.6 Domain Functions for Variables .................................................................................. 90
5.7 Spreadsheets and LINGO........................................................................................... 90
5.8 Summary .................................................................................................................... 94
5.9 Problems..................................................................................................................... 94
Ch 6 Product Mix Problems............................................................................................... 95
6.1 Introduction ................................................................................................................. 95
Table of Contents v
6.2 Example...................................................................................................................... 96
6.3 Process Selection Product Mix Problems.................................................................... 99
6.4 Problems................................................................................................................... 104
Ch 7 Covering, Staffing & Cutting Stock Models ........................................................... 107
7.1 Introduction ............................................................................................................... 107
7.1.1 Staffing Problems ............................................................................................... 108
7.1.2 Example: Northeast Tollway Staffing Problems.................................................. 108
7.1.3 Additional Staff Scheduling Features.................................................................. 110
7.2 Cutting Stock and Pattern Selection.......................................................................... 111
7.2.1 Example: Cooldot Cutting Stock Problem........................................................... 112
7.2.2 Formulation and Solution of Cooldot .................................................................. 113
7.2.3 Generalizations of the Cutting Stock Problem .................................................... 117
7.2.4 Two-Dimensional Cutting Stock Problems.......................................................... 119
7.3 Crew Scheduling Problems....................................................................................... 119
7.3.1 Example: Sayre-Priors Crew Scheduling............................................................ 120
7.3.2 Solving the Sayre/Priors Crew Scheduling Problem........................................... 122
7.4 A Generic Covering/Partitioning Model ..................................................................... 125
7.5 Problems................................................................................................................... 126
Ch 8 Networks, Distribution and PERT/CPM.................................................................. 137
8.1 What’s Special About Network Models ..................................................................... 137
8.1.1 Special Cases .................................................................................................... 140
8.2 PERT/CPM Networks and LP ................................................................................... 140
8.3 Activity-on-Arc vs. Activity-on-Node Network Diagrams ............................................ 145
8.4 Crashing of Project Networks.................................................................................... 146
8.4.1 The Cost and Value of Crashing......................................................................... 147
8.4.2 The Cost of Crashing an Activity ........................................................................ 147
8.4.3 The Value of Crashing a Project......................................................................... 147
8.4.4 Formulation of the Crashing Problem................................................................. 148
8.5 Resource Constraints in Project Scheduling ............................................................. 151
8.6 Path Formulations..................................................................................................... 154
8.6.1 Example ............................................................................................................. 154
8.7 Path Formulations of Undirected Networks............................................................... 155
8.7.1 Example ............................................................................................................. 156
8.8 Double Entry Bookkeeping: A Network Model of the Firm......................................... 158
8.9 Extensions of Network LP Models............................................................................. 159
8.9.1 Multicommodity Network Flows .......................................................................... 160
8.9.2 Reducing the Size of Multicommodity Problems................................................. 161
8.9.3 Multicommodity Flow Example ........................................................................... 161
8.9.4 Fleet Routing and Assignment............................................................................ 164
8.9.5 Fleet Assignment................................................................................................ 168
8.9.6 Leontief Flow Models.......................................................................................... 173
8.9.7 Activity/Resource Diagrams................................................................................ 175
8.9.8 Spanning Trees .................................................................................................. 177
8.9.9 Steiner Trees...................................................................................................... 179
8.10 Nonlinear Networks................................................................................................. 183
8.11 Equilibrium Network Flows...................................................................................... 186
8.12 Problems................................................................................................................. 188
Table of Contents vi
Ch 9 Multi-period Planning Problems............................................................................. 197
9.1 Introduction ............................................................................................................... 197
9.2 A Dynamic Production Problem ................................................................................ 199
9.2.1 Formulation ........................................................................................................ 199
9.2.2 Constraints ......................................................................................................... 200
9.2.3 Representing Absolute Values ........................................................................... 202
9.3 Multi-period Financial Models ................................................................................... 203
9.3.1 Example: Cash Flow Matching ........................................................................... 203
9.4 Financial Planning Models with Tax Considerations ................................................. 207
9.4.1 Formulation and Solution of the WSDM Problem ............................................... 208
9.4.2 Interpretation of the Dual Prices ......................................................................... 210
9.5 Present Value vs. LP Analysis .................................................................................. 211
9.6 Accounting for Income Taxes.................................................................................... 212
9.7 End Effects ............................................................................................................... 215
9.7.1 Perishability/Shelf Life Constraints ..................................................................... 215
9.7.2 Startup and Shutdown Costs.............................................................................. 215
9.8 Non-optimality of Cyclic Solutions to Cyclic Problems............................................... 216
9.9 Problems................................................................................................................... 221
Ch 10 Blending of Input Materials................................................................................... 225
10.1 Introduction ............................................................................................................. 225
10.2 The Structure of Blending Problems ....................................................................... 226
10.2.1 Example: The Pittsburgh Steel Company Blending Problem............................ 227
10.2.2 Formulation and Solution of the Pittsburgh Steel Blending Problem................. 228
10.3 A Blending Problem within a Product Mix Problem ................................................. 230
10.3.1 Formulation ...................................................................................................... 231
10.3.2 Representing Two-sided Constraints................................................................ 232
10.4 Proper Choice of Alternate Interpretations of Quality Requirements ....................... 236
10.5 How to Compute Blended Quality ........................................................................... 238
10.5.1 Example ........................................................................................................... 238
10.5.2 Generalized Mean ............................................................................................ 239
10.6 Interpretation of Dual Prices for Blending Constraints............................................. 240
10.7 Fractional or Hyperbolic Programming.................................................................... 241
10.8 Multi-Level Blending: Pooling Problems.................................................................. 242
10.9 Problems................................................................................................................. 247
Ch 11 Formulating and Solving Integer Programs ........................................................ 261
11.1 Introduction ............................................................................................................. 261
11.1.1 Types of Variables............................................................................................ 261
11.2 Exploiting the IP Capability: Standard Applications................................................. 262
11.2.1 Binary Representation of General Integer Variables......................................... 262
11.2.2 Minimum Batch Size Constraints...................................................................... 262
11.2.3 Fixed Charge Problems.................................................................................... 263
11.2.4 The Simple Plant Location Problem.................................................................. 263
11.2.5 The Capacitated Plant Location Problem (CPL) ............................................... 265
11.2.6 Representing General Cost Curves with Economies of Scale .......................... 268
11.2.7 Alternate Representation of Nonlinear Cost Curves ......................................... 270
11.2.8 Converting to Separable Functions................................................................... 271
11.3 Outline of Integer Programming Methods................................................................ 272
Table of Contents vii
11.4 Computational Difficulty of Integer Programs .......................................................... 274
11.4.1 NP-Complete Problems.................................................................................... 274
11.5 Problems with Naturally Integer Solutions and the Prayer Algorithm....................... 275
11.5.1 Network LPs Revisited ..................................................................................... 275
11.5.2 Integral Leontief Constraints............................................................................. 275
11.5.3 Example: A One-Period MRP Problem............................................................. 276
11.5.4 Transformations to Naturally Integer Formulations ........................................... 278
11.6 The Assignment Problem and Related Sequencing and Routing Problems............ 280
11.6.1 Example: The Assignment Problem.................................................................. 280
11.6.2 The Traveling Salesperson Problem................................................................. 282
11.6.3 Capacitated Multiple TSP/Vehicle Routing Problems ....................................... 287
11.6.4 Minimum Spanning Tree .................................................................................. 291
11.6.5 The Linear Ordering Problem ........................................................................... 292
11.6.6 Quadratic Assignment Problem ........................................................................ 294
11.7 Problems of Grouping, Matching, Covering, Partitioning, and Packing ................... 298
11.7.1 Formulation as an Assignment Problem........................................................... 299
11.7.2 Formulation as a Packing Problem................................................................... 301
11.8 Linearizing Products of Variables............................................................................ 303
11.8.1 Example: Bundling of Products......................................................................... 304
11.9 Representing Logical Conditions............................................................................. 307
11.9.1 Simplifying Difficult Integer Programs ............................................................... 307
11.10 Problems............................................................................................................... 311
Ch 12 Decision making Under Uncertainty and Stochastic Programs......................... 321
12.1 Introduction ............................................................................................................. 321
12.2 Identifying Sources of Uncertainty........................................................................... 321
12.3 The Scenario Approach .......................................................................................... 322
12.4 A More Complicated Two-Period Planning Problem ............................................... 324
12.4.1 The Warm Winter Solution................................................................................ 326
12.4.2 The Cold Winter Solution.................................................................................. 326
12.4.3 The Unconditional Solution............................................................................... 327
12.5 Expected Value of Perfect Information (EVPI) ........................................................ 330
12.6 Expected Value of Modeling Uncertainty................................................................. 331
12.6.1 Certainty Equivalence....................................................................................... 331
12.7 Risk Aversion.......................................................................................................... 332
12.7.1 Downside Risk.................................................................................................. 333
12.7.2 Example ........................................................................................................... 334
12.8 Choosing Scenarios................................................................................................ 336
12.8.1 Matching Scenario Statistics to Targets............................................................ 337
12.8.2 Generating a Set of Scenarios with a Specified Covariance Structure ............. 338
12.8.3 Generating a Suitable Z Matrix ......................................................................... 339
12.8.4 Example ........................................................................................................... 340
12.8.5 Converting Multi-Stage Problems to Two-Stage Problems............................... 341
12.9 Decisions Under Uncertainty with More than Two Periods...................................... 341
12.9.1 Dynamic Programming and Financial Option Models ....................................... 342
12.9.2 Binomial Tree Models of Interest Rates............................................................ 343
12.9.3 Binomial Tree Models of Foreign Exchange Rates........................................... 347
12.10 Decisions Under Uncertainty with an Infinite Number of Periods .......................... 350
Table of Contents viii
12.10.1 Example: Cash Balance Management............................................................ 351
12.11 Chance-Constrained Programs............................................................................. 355
12.12 Problems............................................................................................................... 355
Ch 13 Portfolio Optimization ........................................................................................... 357
13.1 Introduction ............................................................................................................. 357
13.2 The Markowitz Mean/Variance Portfolio Model ....................................................... 357
13.2.1 Example ........................................................................................................... 358
13.3 Efficient Frontier and Parametric Analysis............................................................... 361
13.3.1 Portfolios with a Risk-Free Asset...................................................................... 361
13.3.2 The Sharpe Ratio ............................................................................................. 364
13.4 Important Variations of the Portfolio Model ............................................................. 365
13.4.1 Portfolios with Transaction Costs ..................................................................... 366
13.4.2 Example ........................................................................................................... 366
13.4.3 Portfolios with Taxes ........................................................................................ 368
13.4.4 Factors Model for Simplifying the Covariance Structure ................................... 370
13.4.5 Example of the Factor Model............................................................................ 371
13.4.6 Scenario Model for Representing Uncertainty .................................................. 372
13.4.7 Example: Scenario Model for Representing Uncertainty................................... 373
13.5 Measures of Risk other than Variance .................................................................... 375
13.5.1 Maximizing the Minimum Return ...................................................................... 376
13.5.2 Value at Risk .................................................................................................... 378
13.5.3 Example of VAR............................................................................................... 378
13.6 Scenario Model and Minimizing Downside Risk...................................................... 380
13.6.1 Semi-variance and Downside Risk ................................................................... 381
13.6.2 Downside Risk and MAD.................................................................................. 383
13.6.3 Scenarios Based Directly Upon a Covariance Matrix........................................ 383
13.7 Hedging, Matching and Program Trading................................................................ 385
13.7.1 Portfolio Hedging.............................................................................................. 385
13.7.2 Portfolio Matching, Tracking, and Program Trading.......................................... 385
13.8 Methods for Constructing Benchmark Portfolios ..................................................... 386
13.8.1 Scenario Approach to Benchmark Portfolios .................................................... 389
13.8.2 Efficient Benchmark Portfolios.......................................................................... 391
13.8.3 Efficient Formulation of Portfolio Problems....................................................... 392
13.9 Cholesky Factorization for Quadratic Programs...................................................... 394
13.10 Problems............................................................................................................... 396
Ch 14 Multiple Criteria and Goal Programming ............................................................. 397
14.1 Introduction ............................................................................................................. 397
14.1.1 Alternate Optima and Multicriteria .................................................................... 398
14.2 Approaches to Multi-criteria Problems .................................................................... 398
14.2.1 Pareto Optimal Solutions and Multiple Criteria ................................................. 398
14.2.2 Utility Function Approach.................................................................................. 398
14.2.3 Trade-off Curves............................................................................................... 399
14.2.4 Example: Ad Lib Marketing............................................................................... 399
14.3 Goal Programming and Soft Constraints................................................................. 402
14.3.1 Example: Secondary Criterion to Choose Among Alternate Optima ................. 403
14.3.2 Preemptive/Lexico Goal Programming ............................................................. 405
14.4 Minimizing the Maximum Hurt, or Unordered Lexico Minimization .......................... 408
Table of Contents ix
14.4.1 Example ........................................................................................................... 409
14.4.2 Finding a Unique Solution Minimizing the Maximum ........................................ 409
14.5 Identifying Points on the Efficient Frontier ............................................................... 414
14.5.1 Efficient Points, More-is-Better Case................................................................ 414
14.5.2 Efficient Points, Less-is-Better Case................................................................. 416
14.5.3 Efficient Points, the Mixed Case ....................................................................... 418
14.6 Comparing Performance with Data Envelopment Analysis ..................................... 419
14.7 Problems................................................................................................................. 424
Ch 15 Economic Equilibria and Pricing.......................................................................... 427
15.1 What is an Equilibrium? .......................................................................................... 427
15.2 A Simple Simultaneous Price/Production Decision ................................................. 428
15.3 Representing Supply & Demand Curves in LPs...................................................... 429
15.4 Auctions as Economic Equilibria ............................................................................. 433
15.5 Multi-Product Pricing Problems............................................................................... 437
15.6 Transportation Equilibria ......................................................................................... 441
15.6.1 User Equilibrium vs. Social Optimum................................................................ 444
15.7 General Equilibrium Models of An Economy........................................................... 446
15.8 Equilibria in Networks as Optimization Problems.................................................... 448
15.9 Problems................................................................................................................. 450
Ch 16 Game Theory and Cost Allocation ....................................................................... 453
16.1 Introduction ............................................................................................................. 453
16.2 Two-Person Games ................................................................................................ 453
16.2.1 The Minimax Strategy....................................................................................... 454
16.3 Two-Person Non-Constant Sum Games................................................................. 456
16.3.1 Prisoner’s Dilemma .......................................................................................... 457
16.3.2 Choosing a Strategy ......................................................................................... 458
16.3.3 Bimatrix Games with Several Solutions ............................................................ 461
16.4 Nonconstant-Sum Games Involving Two or More Players ...................................... 463
16.4.1 Shapley Value .................................................................................................. 465
16.5 Problems................................................................................................................. 465
Ch 17 Inventory, Production, and Supply Chain Management ..................................... 469
17.1 Introduction ............................................................................................................. 469
17.2 One Period News Vendor Problem......................................................................... 469
17.2.1 Analysis of the Decision ................................................................................... 470
17.3 Multi-Stage News Vendor ....................................................................................... 472
17.3.1 Ordering with a Backup Option......................................................................... 475
17.3.2 Safety Lotsize................................................................................................... 477
17.3.3 Multiproduct Inventories with Substitution......................................................... 478
17.4 Economic Order Quantity........................................................................................ 482
17.5 The Q,r Model......................................................................................................... 483
17.5.1 Distribution of Lead Time Demand ................................................................... 483
17.5.2 Cost Analysis of Q,r.......................................................................................... 483
17.6 Base Stock Inventory Policy.................................................................................... 488
17.6.1 Base Stock — Periodic Review ........................................................................ 489
17.6.2 Policy................................................................................................................ 489
17.6.3 Analysis............................................................................................................ 489
17.6.4 Base Stock — Continuous Review ................................................................... 491
Table of Contents x
17.7 Multi-Echelon Base Stock, the METRIC Model ....................................................... 491
17.8 DC With Holdback Inventory/Capacity .................................................................... 495
17.9 Multiproduct, Constrained Dynamic Lot Size Problems........................................... 497
17.9.1 Input Data......................................................................................................... 498
17.9.2 Example ........................................................................................................... 499
17.9.3 Extensions........................................................................................................ 505
17.10 Problems............................................................................................................... 505
Ch 18 Design & Implementation of Service and Queuing Systems.............................. 507
18.1 Introduction ............................................................................................................. 507
18.2 Forecasting Demand for Services........................................................................... 507
18.3 Waiting Line or Queuing Theory.............................................................................. 508
18.3.1 Arrival Process ................................................................................................. 509
18.3.2 Queue Discipline .............................................................................................. 510
18.3.3 Service Process ............................................................................................... 510
18.3.4 Performance Measures for Service Systems.................................................... 510
18.3.5 Stationarity ....................................................................................................... 511
18.3.6 A Handy Little Formula ..................................................................................... 511
18.3.7 Example ........................................................................................................... 511
18.4 Solved Queuing Models.......................................................................................... 512
18.4.1 Number of Outbound WATS lines via Erlang Loss Model................................. 512
18.4.2 Evaluating Service Centralization via the Erlang C Model ................................ 513
18.4.3 A Mixed Service/Inventory System via the M/G/∞ Model.................................. 515
18.4.4 Optimal Number of Repairmen via the Finite Source Model. ............................ 515
18.4.5 Selection of a Processor Type via the M/G/1 Model ......................................... 517
18.4.6 Multiple Server Systems with General Distribution, M/G/c & G/G/c .................. 518
18.5 Critical Assumptions and Their Validity ................................................................... 520
18.6 Networks of Queues ............................................................................................... 520
18.7 Designer Queues .................................................................................................... 522
18.7.1 Example: Positive but Finite Waiting Space System......................................... 522
18.7.2 Constant Service Time. Infinite Source. No Limit on Line Length ..................... 525
18.7.3 Example Effect of Service Time Distribution..................................................... 525
18.8 Problems................................................................................................................. 528
Ch 19 Design & Implementation of Optimization-Based Decision Support Systems . 531
19.1 General Structure of the Modeling Process............................................................. 531
19.1.1 Developing the Model: Detail and Maintenance................................................ 532
19.2 Verification and Validation....................................................................................... 532
19.2.1 Appropriate Level of Detail and Validation........................................................ 532
19.2.2 When Your Model & the RW Disagree, Bet on the RW .................................... 533
19.2.3 Should We Behave Non-Optimally? ................................................................. 534
19.3 Separation of Data and System Structure............................................................... 534
19.3.1 System Structure.............................................................................................. 535
19.4 Marketing the Model ............................................................................................... 535
19.4.1 Reports............................................................................................................. 535
19.5 Reducing Model Size .............................................................................................. 538
19.5.1 Reduction by Aggregation ................................................................................ 539
19.5.2 Reducing the Number of Nonzeroes................................................................. 542
19.5.3 Reducing the Number of Nonzeroes in Covering Problems.............................. 542
Table of Contents xi
19.6 On-the-Fly Column Generation ............................................................................... 544
19.6.1 Example of Column Generation Applied to a Cutting Stock Problem................ 545
19.6.2 Column Generation and Integer Programming ................................................. 549
19.6.3 Row Generation ............................................................................................... 549
19.7 Problems................................................................................................................. 549
References....................................................................................................................... 551
INDEX ............................................................................................................................... 561




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2006-4-25 10:15#1
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