(BQ) Part 2 book “Quantitative analysis for management” has contents: Transportation and assignment models, integer programming, goal programming, and nonlinear programming, network models, project management, simulation modeling, Markov analysis,… and other contents.
9 CHAPTER Transportation and Assignment Models LEARNING OBJECTIVES After completing this chapter, students will be able to: Structure LP problems for the transportation, transshipment, and assignment models Use the northwest corner and stepping-stone methods Solve facility location and other application problems with transportation models Solve assignment problems with the Hungarian (matrix reduction) method CHAPTER OUTLINE 9.1 Introduction 9.2 9.3 The Transportation Problem The Assignment Problem 9.4 9.5 The Transshipment Problem The Transportation Algorithm 9.6 9.7 Special Situations with the Transportation Algorithm Facility Location Analysis 9.8 9.9 The Assignment Algorithm Special Situations with the Assignment Algorithm Summary • Glossary • Solved Problems • Self-Test • Discussion Questions and Problems • Internet Homework Problems • Case Study: Andrew–Carter, Inc • Case Study: Old Oregon Wood Store • Internet Case Studies • Bibliography Appendix 9.1: Using QM for Windows 341 342 CHAPTER • TRANSPORTATION AND ASSIGNMENT MODELS 9.1 Introduction In this chapter we explore three special types of linear programming problems—the transportation problem (first introduced in Chapter 8), the assignment problem, and the transshipment problem All these may be modeled as network flow problems, with the use of nodes (points) and arcs (lines) Additional network models will be discussed in Chapter 11 This first part of this chapter will explain these problems, provide network representations for them, and provide linear programming models for them The solutions will be found using standard linear programming software The transportation and assignment problems have a special structure that enables them to be solved with very efficient algorithms The latter part of the chapter will present the special algorithms for solving them 9.2 The Transportation Problem The transportation problem deals with the distribution of goods from several points of supply (origins or sources) to a number of points of demand (destinations) Usually we are given a capacity (supply) of goods at each source, a requirement (demand) for goods at each destination, and the shipping cost per unit from each source to each destination An example is shown in Figure 9.1 The objective of such a problem is to schedule shipments so that total transportation costs are minimized At times, production costs are included also Transportation models can also be used when a firm is trying to decide where to locate a new facility Before opening a new warehouse, factory, or sales office, it is good practice to consider a number of alternative sites Good financial decisions concerning the facility location also attempt to minimize total transportation and production costs for the entire system Linear Program for the Transportation Example The Executive Furniture Corporation is faced with the transportation problem shown in Figure 9.1 The company would like to minimize the transportation costs while meeting the demand at each destination and not exceeding the supply at each source In formulating this as a linear FIGURE 9.1 Network Representation of a Transportation problem, with Costs, Demands, and Supplies Source Destination Supply 100 Demand $5 Des Moines (Source 1) Albuquerque (Destination 1) 300 Boston (Destination 2) 200 Cleveland (Destination 3) 200 $4 $3 $8 300 Evansville (Source 2) $4 $3 $9 $7 300 Fort Lauderdale (Source 3) $5 9.2 THE TRANSPORTATION PROBLEM 343 program, there are three supply constraints (one for each source) and three demand constraints (one for each destination) The decisions to be made are the number of units to ship on each route, so there is one decision variable for each arc (arrow) in the network Let Xij = number of units shipped from source i to destination j where i = 1, 2, 3, with = Des Moines, = Evansville, and = Fort Lauderdale j = 1, 2, 3, with = Albuquerque, = Boston, and = Cleveland The LP formulation is Minimize total cost = 5X11 + 4X12 + 3X13 + 8X21 + 4X22 + 3X23 + 9X31 + 7X32 + 5X33 subject to X11 + X12 + X13 … 100 (Des Moines supply) X21 + X22 + X23 … 300 (Evansville supply) X31 + X32 + X33 … 300 (Fort Lauderdale supply) X11 + X21 + X31 = 300 (Albuquerque demand) X12 + X22 + X32 = 200 (Boston demand) X13 + X23 + X33 = 200 (Cleveland demand) Xij Ú for all i and j The solution to this LP problem could be found using Solver in Excel 2010 by putting these constraints into a spreadsheet, as discussed in Chapter However, the special structure of this problem allows for an easier and more intuitive format, as shown in Program 9.1 Solver is still used, but since all the constraint coefficients are or 0, the left-hand side of each constraint is simply the sum of the variables from a particular source or to a particular destination In Program 9.1 these are cells E10:E12 and B13:D13 A General LP Model for Transportation Problems The number of variables and constraints for a typical transportation problem can be found from the number of sources and destinations In this example, there were sources and destinations The LP had * = variables and + = constraints In general, for a transportation problem with m sources and n destination, the number of variables is mn, and the number of constraints is m + n For example, if there are (i.e., m = 5) constraints and (i.e., n = 8) variables, the linear program would have 5(8) = 40 variables and + = 13 constraints The use of the double subscripts on the variables makes the general form of the linear program for a transportation problem with m sources and n destinations easy to express Let xij = number of units shipped from source i to destination j cij = cost one unit from source i to destination j si = supply at source i dj = demand at destination j The linear programming model is n m Minimize cost = g g cijxij j=1 i=1 subject to n g xij … si i = 1, 2, , m g xij = dj j = 1, 2, , n j=1 m i=1 xij Ú for all i and j 344 CHAPTER • TRANSPORTATION AND ASSIGNMENT MODELS PROGRAM 9.1 Executive Furniture Corporation Solution in Excel 2010 Solver Parameter Inputs and Selections Set Objective: B16 By Changing cells: B10:D12 To: Min Subject to the Constraints: E10:E12 X4 (d) Y– = 500>X6 M6-8 (a) Y– = 30X4 - (b) Y– = 60X2 + 24 (c) Y– = 24>X5 (d) Y– = 250>X6 M6-10 X ϭ is point of inflection M6-12 Q ϭ 2,400, TR ϭ 1,440,000 M6-14 P ϭ 5.48 Module M7-18 (b) 14X1 + 4X2 … 3,360; 10X1 + 12X2 … 9,600 (d) S1 = 3,360, S2 = 9,600 (e) X2 (f) S2 (g) 800 units of X2 (h) 1,200,000 M7-20 X1 = 2, X2 = 6, S1 = 0, S2 = 0, P ϭ $36 M7-22 X1 = 14, X2 = 33, C ϭ $221 M7-24 Unbounded M7-26 Degeneracy; X1 = 27, X2 = 5, X3 = 0, P ϭ $177 M7-28 (a) Min C = 9X1 + 15X2 X1 + 2X2 Ú 30 X1 + 4X2 Ú 40 (b) X1 = 0, X2 = 20, C ϭ $300 M7-30 coffee tables, bookcases, profit ϭ 96 M7-34 (a) 7.5 to infinity (b) Negative infinity to $40 (c) $20 (d) $0 M7-36 (a) 18 Model 102, Model H23 (b) S1 = slack time for soldering S2 = slack time for inspection (c) Yes—shadow price is $4 (d) No—shadow price is less than $1.75 M7-38 (a) Negative infinity to $6 for phosphate; $5 to infinity for potassium (b) Basis won’t change; but X1, X2, and S2 will change M7-40 max P = 50 U1 + 4U2 12U1 + 1U2 … 120 20U1 + 3U2 … 250 APPENDIX H: SOLUTIONS TO SELF-TESTS Appendix H: Solutions to Self-Tests Chapter 1 10 11 12 13 14 15 c d b b c c d c d a a quantitative analysis defining the problem schematic model algorithm Chapter 2 10 11 12 13 14 15 c b a d b c a c b d b a a b a Chapter 3 10 11 12 13 14 15 16 b c c a c b a c a d b c c a c b Chapter b c d 10 11 12 b b c b c a b b c Chapter 5 10 11 12 13 14 15 b a d c b b d b d b a d b c b 13 14 a a Chapter 8 a b d d c e d c Chapter 9 10 11 12 b d b b b a b b b a b a Chapter Chapter 10 10 11 12 13 14 10 11 e e c c a b d c b a a d d d Chapter 7 10 11 12 b a b c a b c c b c a a a b a a a b b b d b e Chapter 11 10 11 12 13 14 15 c e b c b a d a b b a d shortest route maximal flow minimal spanning tree 639 640 APPENDICES (b) yes, yes, yes, yes, no, yes, yes, no, no, no Chapter 12 10 11 12 13 14 15 16 17 18 e c a d b c b a b b a a Critical path (or critical) program evaluation and review technique linear programming model optimistic, most likely, pessimistic slack monitor and control Chapter 13 10 11 12 13 14 a a b e c b c d b d c first-come, first-served negative exponentially distributed simulation Chapter 15 10 11 12 b a c c b a a a b matrix of transition probabilities collectively exhaustive, mutually exclusive vector of state probabilities Chapter 16 10 b c d a c b c d b b Module 1 a d b b c b b b Chapter 14 Module 2 10 11 12 13 14 15 10 11 12 13 14 b b a b a a d a b b d d c e (a) no, yes, no, no, no, yes, yes, yes, no, yes c b e c b a c e a a c c b b Module 3 c d b a b b c Module 4 b a c b b b a Module 5 c a b c b a e d Module 6 a d a b c d d Module 7 10 11 12 13 14 15 16 17 a d d a a d a d b a a b c c d a b Index A ABC analysis, 225 Abe Software, 462 Absorbing states, 582–586, 600 Acceptance sampling tables, 603 Accounting data, 13–14 Accounts receivable application, 582–585 Activities cost to date for, 478 defining, 461–462 Activity-based-costing (ABC) method, 474 Activity difference, 478 Activity-on-arc (AOA), 462–463 Activity-on-node (AON), 462 Activity time estimates, 463–464 Adaptive forecasting, 181 Adaptive smoothing, 181 Additive time-series models, 160 Additivity, 250 Airbus Industries simulation, 534 Airlines schedules maximizing profit, 407 Alabama Airlines, 570–571 Algorithms, Alternate optimal solutions, 276 Alternatives, 70 Ambulances in Chile evaluate and improve performance metrics, 511 American Airlines (AA) setting crew schedules, 258 American Express financial advisors, 479 American Meteorological Society (AMS), 65 Analysis ToolPak, 122 Andrew-Carter, Inc (A-C), 391 Annual carrying costs and production run model, 207–208 Annual holding costs, calculating with safety stock, 218–219 Annual ordering costs, 208 Annual setup costs, 208 ANOVA table, 149 AON networks, 463 ARCO p-charts, 611–612 Arcs, 342, 430 Area of feasible solutions, 256 Arena, 560 Arnold’s Muffler Shop exponential distribution, 51 multichannel queuing model, 512–514 single-channel queuing model, 507 Arrivals, 501–502 Artemis, 484 Aspen Technology, 283 Assignable variations, 605 Assignment algorithm balanced assignment problem, 371 final assignment, 369–370 Hungarian method, 366–369 maximization assignment problems, 371–372 opportunity cost table, 367–369 special situations, 371–372 testing for optimal assignment, 368 unbalanced assignment problems, 371 Assignment problem, 344–346, 365, 371–372 Assumptions, simplifying, 13 Athens Olympic Games Organizing Committee (ATHOC), 15 Attributes, 610–612 AT&T solving network problems, 442 Available-to-promise production scheduling, 311 Average queue length, 514 Average waiting time, 514 Averaging techniques exponential smoothing, 165–169 moving averages, 161–165 AVX-Kyocera statistical process control, 607 B Baan, 232 Backwards stepwise procedure, 133 Bad decisions, 70 Balanced assignment problem, 371 Balking, 502 Bank of America pecuniary corruption statistics, 604 Bayes, Thomas, 31 Bayes’ theorem calculating revised probabilities, 87–89 derivation of, 66 estimating probability values, 87–90 general form of, 31 probabilities and, 29–31 Bell Laboratories, 603 Bernoulli process, 38 Best level of service, 500 Beta probability distribution, 464 Bias, 159 Bill of materials (BOM), 226 Binary variables modeling, 402–406 regression models, 131–132 Binder’s Beverage, 455 Binding constraints, 263 Binomial distribution, 38–41 Binomial formula and problem solving, 39 Binomial probabilities, 624–628 Binomial tables and problem solving, 40–41 Blake Electronics, 111–112 Boeing Corporation simulation, 534 Box filling example, 606–607 Brass Department Store quantity discount model, 212–213 Break-even point (BEP), Brier, 42 British Airways (BA) program evaluation and review technique/critical path method (PERT/CPM), 462 Brownian motion, 574 Brown Manufacturing production run model, 208–209 Budgeting process, 474–477 Business games, 558–559 Business system simulation, 534 C Café du Donut marginal analysis, 222 CALEB Technologies, 407 Calling population, 501–502 Canadian Men’s Curling Championships, 42, 586 Capable-to-promise production scheduling system, 311 Capital budgeting 0-1 (binary) variables, 402–404 Carrying costs, 207–208, 219 Causal models, 154–155 Causation, 136 C-charts, 610, 613 Centered moving averages (CMA), 173–174 Centers for Disease Control and Prevention, 505 Central limit theorem, 605 Central planning engine (CPE), 401 Chase Manhattan Bank, 339 Chicago Tribune newspaper marginal analysis with normal distribution, 223–225 Closed path, 352–353 Coefficient of correlation, 121 Coefficient of determination, 120 Coefficient of realism, 73 Collectively exhaustive events, 24–27, 35, 578 Collectively exhaustive states, 574 Collinear, 133 Complete enumeration, Complex queuing models, 519 Components and material structure tree, 226 Computer languages and simulation, 535 Computers quantitative analysis role, 9–11 simulation, 519 simulation role, 560 Computer software and regression, 122–123 Conditional probabilities and decision trees, 83 Conditional probability, 27–29 Conditional values, 71 Conflicting viewpoints in defining problems, 12 Constant service time model, 514–516 Constraints, 250 binding and nonbinding, 263 dual price, 283 graphical representation, 253–257 redundant, 275–276 right-hand-side values, 282–285 solution points that satisfy, 254–255 642 INDEX Consumer market survey, 155 Continental Airlines CrewSolver system, 407 Continuous distribution and exponential distribution, 50 Continuous random variables, 33–34, 37–38 Control charts, 603 attributes, 610–613 c-charts, 613 defects, 613 QM for Windows, 619 R-chart, 605 variables, 605–610 ξ-chart (x-bar chart), 605 Controllable inputs, 543 Controllable variables, Coopers and Lybrand, 611 Corner point method, 260–262, 271–272 Corporate operating system simulation, 559 Correlation, 136 Cost analysis simulation, 557 Cost data, 474 Costs fixed, 7, 19 single-channel queuing model, 508–510 variable, waiting lines, 500–501 Crashing, 479–483 Crash time, 479 CrewSolver system, 407 Criterion of realism, 73–74 Critical path, 464–469 Critical path method (CPM), 460, 478–483 crashing, 479–483 start of, 461 Crystal Ball, 560 CSX Transportation, Inc optimization models, Cumulative probability and relation between intervals, 538 Cumulative probability distribution, 537, 543 Curling champions, probability assessments of, 42 Current state to future state, 576 Customer Equity Loyalty Management (CELM), 582 D Daily unloading rate variable, 550 Data Analysis add-in, 122 Decision analysis utility theory, 90–95 Decision making, 70–71, 89–90 automating process, decision trees, 81 Decision-making environments, 71–72 Decision making group, 155 Decision making under certainty, 71 Decision making under risk, 72, 76–80 Decision making under uncertainty, 72–75 Decision nodes, 81 Decision points, 81 Decisions good and bad, 70 opportunity cost, 367–368 Decision table, 71 Decision theory, 70–71 Decision trees alternatives, 84 analyzing problems, 81 conditional probabilities, 83 decision making, 81 expected monetary value (EMV), 84 expected value of sample information (EVSI), 85–86 lottery ticket, 90 possible outcomes and alternatives, 82–84 posterior probabilities, 83 QM for Windows, 114 sample information efficiency, 86 sensitivity analysis, 86 sequential decisions, 82–84 state-of-nature nodes, 81, 83 Decision variables, 4, 396 Decomposition method, 154, 175–177 Decoupling function, 197 Defects and control charts, 613 Degeneracy, 359–362 Degenerate solution, 352 Delphi method, 24, 155 Delta ground crew and smooth takeoff, 468 Demand fluctuating, 217 inventory, 199, 215 irregular, 197 less than or greater than supply, 359 single time period, 221–225 Department of Commerce finite population model, 517–518 Department of Corrections of Virginia, 412 Department of Health and Rehabilitative Services (HRS), 479 Dependent demand, 226–230 Dependent events, 27, 28–29 Dependent selections and 0-1 (binary) variables, 404 Dependent variables, 116, 129 Deseasonalized data, 175–176 Destinations, 342–343 Deterministic assumptions, 276 Deterministic inventory models, 543 Deterministic models, 8–9 Deviational variables, 408–409 Dice rolling, 24–25, 29–30 Diet problems, 324–325 Digital Equipment Corporation (DEC) spanning tree analysis, 435 Disaster response research, Discrete probability distribution, 35–36, 52–54 Discrete random variables, 33–35 Disney World forecasting, 179 Drawing cards, 25–26 Drexel Corp., 314–315 Dual price, 283 Dummy column or row, 371 Dummy destinations, 358–359 Dummy sources, 358–359 Dummy variables, 131–132 DuPont, 461 Dynamic Car-Planning (DCP) system, E Earliest finish time, 466 Earliest possible time, 476 Earliest start time, 466, 475 Econometric models, 559 Economic order quantity (EOQ), 200–203, 205 without instantaneous receipt assumption, 206–209 Economic systems and simulation, 559 Efficiency-based funding, 440 Empirical rule and normal distribution, 48 Employee scheduling applications, 318–319 Enterprise resource planning (ERP) systems, 232 Enumeration and integer programming problems, 397 Equally likely, 74 Equilibrium conditions, 579–581, 584 Equilibrium probabilities, 579 Equilibrium share, 579 Equilibrium states, 581 Errors, 158 Events collectively exhaustive, 24–27, 35, 578 dependent and independent, 27, 28–29 mutually exclusive, 24–27, 35, 578 statistically dependent, 28–29 statistically independent, 27–28 union of, 26 Excel absorbing states, 600 add-ins, 10, 560 Analysis ToolPak, 67 basic statistics, 66 Data Analysis add-in, 122 developing regression model, 122–123 F distribution, 49 forecasting, 162–164 fundamental matrix, 600 Goal Seek, 11 integer programming model, 401–402 linear programming (LP) problems, 264–269 linear regression equation, 134–135 Markov analysis, 599–600 mean, variance, and standard deviation, 37 multiple regression models, 129 nonlinear relationship, 134 predicting future market shares, 599 regression calculations, 122 Solver, 10–11 statistical function, 66–67 sum of squares error, 122 SUMPRODUCT function, 266 Excel 2007, activating add-ins, 635 regression analysis, 150 Excel 2010, activating add-ins, 635 regression line, 170 Solver add-in, 264–269 Excel QM, 9, 80, 635 Assignment module, 370 box filling example, 607 c-chart, 613 decision theory problems, 80 decomposition method, 176, 177 economic order quantity (EOQ), 203 exponential smoothing, 166 forecasting, 162–164 installing, 635 linear programming (LP) problems, 302–305 moving average forecast, 163 p-chart, 612 preparing spreadsheet for Solver, 264–267 production run models, 209 program crashing, 483 program evaluation and review technique/critical path method (PERT/CPM), 471 quantity discount problems, 213 regression analysis, 150 regression calculations, 122 safety stock and reorder point, 219–220 simulation module, 543 solving transportation problems, 364 technical support, 635 trend-adjusted exponential smoothing, 168–169 trend analysis, 171 Excel spreadsheets, integer programming problems, 398–399 simulation, 541–542 Expected activity time, 464 Expected demand, 540 Expected monetary value (EMV), 76–77, 79, 84 Expected opportunity loss (EOL), 78 Expected value of perfect information (EVPI), 77–78 Expected value of probability distribution, 35 Expected value of sample information (EVSI), 85–86 Expected value with perfect information (EV wPI), 77–78 Expenses, Explanatory variable, 116 Exponential distribution, 50–53 Exponential smoothing, 154, 165–169 ExtendSim, 560 Extreme point, 260 F Facility location analysis, 363–364 Facility location supply-chain reliability, 373 Factories, locating, 363–364 Factory capacity constraints, 353–354 Family Planning Research Center (Nigeria), 494–496 Fast automatic restoration (FASTAR), 442 Favorable market (FM), 87 F distribution, 48–50, 125–127, 630–631 INDEX Feasible region corner points, 260 Feasible solution, 256–257, 351 Federal Aviation Administration (FAA) simulation, 549 Fifth Avenue Industries, 312–314 Financial applications, 319–324 Financial investment 0-1 (binary) variables, 405–406 Finite population model, 516–518 Finnair, 582 First-in, first-out (FIFO) rule, 503 First in, first served (FIFS), 503 Fixed-charge problem example, 404–405 Fixed costs, 7, 19 Flair Furniture Company entering problem data, 265–266 linear programming (LP) problems, 252–253 Flight safety and probability analysis, 32 FLORIDA system, 479 Flow, 438 Flowchart, 546 Flow diagram, 546 Ford and decision theory, 74 Forecasting decomposition method, 175–177 Disney World, 179 Excel and Excel QM, 162–164 exponential smoothing, 165–169 inventory, 196 monthly sales, 190 moving averages, 161–165 QM for Windows, 191–193 time series, 156–157, 169–171 with trend and seasonal components, 175–177 Forecasts bias, 159 causal models, 154–155 combinations of weights, 162 errors, 158 mean absolute deviation (MAD), 158 mean absolute percent error (MAPE), 159 mean squared error (MSE), 158 measures of accuracy, 158–159 monitoring and controlling, 179–181 naïve model, 158 qualitative models, 155 scatter diagrams, 156–157 time-series models, 154 tracking signals, 180–181 types of, 154–155 Formulas and regression calculations, 146–147 Fortune 100 firm inventory policy for service vehicles, 210 Forward pass, 466 Forward stepwise procedure, 133 4-month moving average, 161 FREQUENCY function, 543 F test, 136, 149 Fundamental matrix, 582–586, 600 Future state from current state, 576 G Gantt charts, 461, 484 Garbage in, garbage out, Garcia-Golding Recycling, Inc constant service time model, 515–516 General Electric, 603 General Foundry, 474–477, 480 Geographic information system (GIS), 400 The Glass Slipper, 190 Global optimum, 412 Goal programming, 396, 406–411 Goals hierarchy of importance, 408 multiple, 406–411 ranking with priority levels, 409–410 satisfices, 408 weighted, 410–411 Goal Seek, 11 Goodman Shipping, 322–324 Greater-than-or-equal-to constraint, 262–263 Greenberg Motors, Inc., 314–318 Gross material requirements, 227–229 H Hanshin Expressway traffic-control system, 436 Harrison Electric Company integer programming, 396–398 Harry’s Auto Tire Monte Carlo simulation, 536–541 Harvard Project Manager, 484 Hewlett-Packard printer inventory model to reduce costs, 198 High Note Sound Company, 278, 281–282 Highway Corridor Analytical Program (HCAP), 400 Hill Construction, 494 Hinsdale Company safety stock, 216–218 Holding costs, 197, 199–201, 207–208, 215 Holiday Meal Turkey Ranch minimization problems, 270–273 Hong Kong Bank of Commerce and Industry, 318–319 Hungarian method, 366–369 Hurricane landfall location forecasts mean absolute deviation (MAD), 156 Hurwicz criterion, 73–74 643 single-period inventory models, 221–225 stockouts, 196 usage curve, 199 Inventory analysis and simulation, 543–549 Inventory control, 196–197 Inventory costs, 197–198 economic order quantity (EOQ), 200–202 Inventory models deterministic, 543 single-period, 221–225 Inventory planning and control system, 196 Inventory problem, 543 Irregular supply and demand, 197 ISO 9000 certified, 603 Isocost line approach minimization problems, 272 Isoprofit line method, 257–262 J Jackson Memorial Hospital’s operating rooms simulation, 555 JD Edwards, 232 Joint probability, 27–30 Jury of executive opinion, 155 Just-in-time inventory (JIT), 230–231 I K IBM Systems and Technology Group, 401 Immediate predecessors, 462, 471 Improved solution, 354–358 Improvement index, 352, 354 Improvement indices and transportation algorithm, 356 Independent events, 27–28 Independent variables, 116, 129–130, 133 Indicator variables, 131–132 Industrial dynamics, 559 Infeasible solution, 256–257 Ingredient blending applications, 324–327 Initial solution and degeneracy, 360–361 Input data, 4–6, 13–14 Instantaneous inventory receipt assumption, 206–209 Integer programming, 396–398 limiting number of alternatives, 404 mixed-integer programming problems, 396, 400–402 objective function measured in one dimension, 407 variables required integer values, 396 zero-one integer programming problems, 396 Integer programming problems, 324 enumeration, 397 mathematical statement, 403 rounding off, 397 Integer values, 396 International City Trust (ICT), 320–322 International Organization for Standardization (ISO), 603 Intersection, 26 Intervals and cumulative probability, 538 Inventory, 196 ABC analysis, 225 annual ordering cost, 208 annual setup cost, 208 average dollar value, 204 controlling levels, 196 cost factors, 199 decisions, 197–199 demand, 199, 215 dependent demand, 226–230 economic order quantity (EOQ), 199–205 forecasting, 196 how much to order, 197–205 just-in-time inventory (JIT), 230–231 lead time, 205, 215 optimal production quantity, 208 purchase cost, 203–205 quantity discount models, 210–213 reorder point (ROP), 205–206 safety stock, 213–220 Kanban, 230–231 Kenan Systems Corporation, 545 Kendall notation, 503–504, 506 L Labor planning, 318–319 stored in inventory, 197 Laplace, 74 Last in, first served (LIFS), 503 Latest finish time, 466, 467 Latest start time, 466, 467, 475–476 Law of addition for events not mutually exclusive, 26–27 Lead time, 205, 215, 217 Lead time variable, 546 Least-cost method, 362 Least-cost solution, 352–358 Least-squares regression, 118, 170 Less-than-or-equal to constraint, 262–263 Limited queue length, 502 Linear constraints, 412–413 Linear objective function, 414 Linear programming (LP), 250–251 assignment problem, 344–346 constraints describing network, 482–483 crash time constraints, 482 defining decision variables, 480–481 goal programming, 396 integer programming, 396–402 maximal-flow problem, 438–439 non-linear programming, 396 objective function, 407, 481 project completion constraint, 482 project crashing, 480–483 shortest-route problem, 441, 443–444 transportation problem, 342–343 transshipment problem, 346–348 Linear programming (LP) models employee scheduling applications, 318–319 financial applications, 319–324 ingredient blending applications, 324–327 manufacturing applications, 312–317 marketing applications, 308–311 transportation applications, 327–330 Linear programming (LP) problems alternate optimal solutions, 276 alternative courses of action, 250 conditions of certainty, 250 644 INDEX corner point method, 260–262 deterministic assumptions, 276 divisibility assumption, 250–251 Excel, 264–269 feasible region, 256–257 formulating, 251–253 graphical solution, 253–263 isoprofit line method, 257–260 no feasible solution, 274 objective function, 250 optimal solution, 257–260 product mix problem, 251–252 redundancy, 275–276 requirements, 250–251 sensitivity analysis, 276–285 slack, 262–263 solution points satisfying constraints simultaneously, 256 solving minimization problems, 270–273 special cases, 274–276 surplus, 262–263 unboundedness, 275 Linear trends, 169–170 Line test, 368 Little’s Flow Equations, 519 Liver transplants in United States, 25 LMS, 560 Local area network (LAN), 435 Local optimum, 412 London Stock Exchange, 479 Los Alamos Scientific Laboratory, 535 Low Knock Oil Company, 326–327 Lucent Technologies inventory requirements planning system, 212 M Machine operations and Markov analysis, 578–579 MacProject, 484 Maintenance policy simulation model, 553–557 Management Sciences Associates (MSA), 309–312 Management system simulation, 534 Manufacturing applications production mix, 312–314 production scheduling, 314–318 Mapka Institute of Technology, 410 Marginal analysis, 221–225 Marginal loss (ML), 221 Marginal probability, 27, 28 Marginal profit (MP), 221 Marketing applications, 309–312 Marketing research, 309–312 Market shares, 575–578 Market values equilibrium share, 579 Markov analysis, 574 absorbing states, 582–586 accounts receivable application, 582–586 assumptions of, 574 equilibrium conditions, 579–581 fundamental matrix, 582–586 machine operations, 578–579 matrix of transition probabilities, 574, 576–577 predicting future market shares, 577–578 reducing market costs, 582 sport of curling, 586 states, 574–576 system starting in initial state or condition, 574 vector of state probabilities, 575 Martin-Pullin Bicycle Corp (MPBC) inventory plan, 245 Material cost quantity discounts, 211 Material requirements planning (MRP), 226–230, 232 Material structure tree, 226–227 Mathematical models, 4, 7–9, 13, 534 Mathematical programming, 250 Mathematics of probability, 22–23 Matrix of transition probabilities, 574, 576–578, 583–584 Matrix reduction, 366 Maximal-flow problem, 433–439 Maximal-flow technique, 430, 433–439 Maximax criterion, 72–73 Maximin criterion, 73 Maximization assignment problems, 371–372 Maximization transportation problems, 362 Mean, 36, 76 Poisson distribution, 53 standard normal distribution, 42–44 Mean absolute deviation (MAD), 156, 158 Mean absolute percent error (MAPE), 159 Mean squared error (MSE), 125, 148, 158, 170 Media selection, 308–309 Mexicana Wire Winding, Inc., 300–301 Microsoft Project, 484 Milestones, 484 Military games, 558 Minimal-spanning tree technique, 430–433 Minimax regret, 74–75 Minimization problems, 271–272 MINVERSE function, 600 Mitigation, Mixed-integer programming problems, 396, 400–402 MMULT function, 599 Model for Evaluating Technology Alternatives (META), 543 Modeling real world, 0-1 (binary) variables, 402–406 Models, 3–4, 6–9, 13 Modified-distribution (MODI) method, 350 Monitoring solutions, Monte Carlo simulation, 535–541, 546 random numbers, 558–559 Montgomery County (Maryland) Public Health Service, 505 Monthly sales, forecasting, 190 MOSDIM, 560 Most likely time, 464 Moving averages, 154, 161–165 Multiattribute utility model (MAU), 94 Multichannel queuing model, 511–514 Multichannel system, 503 Multicollinearity, 133, 136 Multiphase system, 503 Multiple goals, 409 Multiple regression model, 121, 128–131 multicollinearity, 136 with trend and seasonal components, 177–178 Multiplicative time-series models, 160 Multiplicative-time-series seasonal index, 172 Mutually exclusive events, 24–27, 35, 578 Mutually exclusive states, 574 N Naive model, 158 NASA, 425 National Academy of Sciences, 94 National Broadcasting Company (NBC) linear, integer, and goal programming selling advertising slots, 271 National Hurricane Center (NHC), 156 National Weather Service, 156 Natural variations, 603–605 Negative exponential distribution, 50–52 Negative exponential probability distribution, 503 Net material requirements plan, 227–229 Network flow problems, 342 Network problems maximal-flow problem, 433–439 minimal-spanning tree technique, 430–433 shortest-route problem, 439–444 Networks arcs, 430 backward pass, 467 flow, 438 forward pass, 466 maximum amount of material flowing, 433–439 nodes, 430–433 program evaluation and review technique/critical path method (PERT/CPM), 460, 462 shortest distance from one location to another, 439–444 New England Foundry, Inc., 530–531 Nodes, 342, 430–433 Nonbinding constraints, 263 Nonlinear constraints, 413–414 Nonlinear objective function, 412–414 Nonlinear programming (NLP), 396, 411–414 Nonlinear regression, 133–136 Nonnegativity constraints, 253 Normal cost, 479 Normal curve, 622–623 Normal distribution, 41–48, 66 marginal analysis, 223–225 safety stock, 216 Normal time, 479 NORMDIST function, 66 NORMINV function, 543 Nortel costing projects, 474 North Carolina improving pupil transportation, 440 North-South Airline, 145–146 Northwest corner rule, 350–352 n-period moving average, 161 Numerical formatting, 633 O Oakton River bridge, 425–426 Objective function, 408 coefficient changes, 278–280 linear programming (LP), 481 Objective probability, 23–24 Oil spills and operations research, Old Oregon Wood Store, 392–393 Olympic Games, 15 Open Plan, 484 Operating characteristics, 506, 519 Operational gaming, 558–559 Operations research, 3–4 Opinion polls, 24 Opportunity costs, 366–368 Opportunity cost table, 367–369 Opportunity loss, 74–75 Opportunity loss table, 75, 78 Optimal assignment line test, 368 Optimality analysis, 277 Optimal production quantity, 208 Optimal solutions, multiple, 362 Optimistic criterion, 72–73 Optimistic time, 464 OptSolver system, 407 Oracle, 232 Ordering costs, 197, 200–201, 208 Organizations, best level of service, 500 Origins, 342 Outlier analysis, 604 P Parallel activity, 471 Parameters, Parametric programming, 277 Parents and material structure tree, 226 Partitioning matrix of transition probabilities, 584 Payoff/cost table, 216 Payoff table, 71 P-charts, 610–612 People, assigning projects to, 344–346, 365–370 People Soft, 232 Perfect information, 77–78 PERT charts, 484 PERT/Cost, 474–478 PERT/CPM charts and subprojects, 484 PERTMASTER, 462 PERT networks, 462–463 Pessimistic criterion, 73 Pessimistic time, 464 Physical models, 3, 534 Pilot plants, INDEX Pittsburgh Pirates, 12 Plutonium, 94 Poisson distribution, 52–54, 502 c-charts, 613 values for use in, 629 Polls, queuing, 515 POM-QM for Windows, 9, 632–635 Portfolio selection, 319–322 Port of Baltimore exponential smoothing, 165–166 Port of New Orleans simulation, 550–552 Posterior probabilities, 29–31, 83 Postoptimality analysis, 5, 277 Predicting future market shares, 577–578 Predictor variable, 116 Preparedness, Presently known probabilities, 574 Present value, 426 Preventive maintenance simulation, 557 Primavera Project Planner, 484 Prior probabilities, 30, 87–88 Prison expenditures in Virginia goal programming model, 412 Pritsker Corp., 25 Probabilistic models, Probabilities, 22 assessments of curling champions, 42 Bayesian analysis, 87–90 Bayes’ theorem and, 29–31 binomial distribution, 38–41 classical or logical method, 23–24 collectively exhaustive events and, 24–27 conditional, 27–29, 83 decision trees, 81–86 equilibrium share, 579 exponential distribution, 50–52 F distribution, 48–50 independent events, 28 joint, 27–30 marginal, 27–28 mathematics of, 22–23 mutually exclusive events and, 24–27 normal distribution, 41–48 objective, 23–24 Poisson distribution, 52–54 posterior, 29–31, 83, 87–89 presently known, 574 prior, 30, 87–88 random variables, 33–34 relative frequency of demand, 23 revision, 30, 32–33, 87–89 rules of, 22–23 simple, 27 sports and, 42 statistically dependent events, 28–29 statistically independent events, 27–28 subjective, 23, 24 table of normal curve areas, 43–44 types of, 23–24 Probability analysis and flight safety, 32 Probability density function, 37 Probability distributions, 13, 34, 543, 546 central tendency, 35 continuous random variables, 37–38 discrete random variable, 34–35 expected value, 35 Kendall notation, 503 mean, 36 Monte Carlo simulation, 536–537 variables, 536–537 variance, 35 Probability function, 37 Problems, 12–14 quantitative analysis, solutions to, 636–638 unbalanced, 358 Problem solving, 39–41 Process control system, 605 Processes assignable variations, 605 average, 609 dispersion, 609 natural variations, 604–605 states, 574 variability, 603–605, 609 Process Logistics Advanced Technical Optimization (PLATO) project, 15 Procomp reorder point for chips, 205–206 Production mix, 312–314 Production/operations management (POM), 9, 632 Production process setup cost, 207 Production run model, 206–209 Production scheduling, 314–318 Product mix problem, 251–252 Product quality, 602 Profit contribution, 251–252 Profit models, 7–8 Program crashing, 483 Program evaluation and review technique/critical path method (PERT/CPM) activity time estimates, 463–464 beta probability distribution, 464 critical path, 464–469 defining project and activities, 461–462 drawing network, 462–463 expected activity time, 464 general foundry example, 461–462 immediate predecessors, 462 information provided by, 471 most likely time, 464 networks, 460 optimistic time, 464 pessimistic time, 464 probability of project completion, 469–470 project management, 471–473 projects in smaller activities or tasks, 460 questions answered by, 460–461 sensitivity analysis, 471–473 variance of activity completion time, 464 Program evaluation and review technique (PERT), 460–461 Programming, 250 Project costs, 474–478 Project crashing, 479–483 Project management, 484 QM for Windows, 497–498 sensitivity analysis, 471–473 software development, 479 Projects assigning people, 344–346, 365–370 defining, 461–462 identifying activities, 460 probability of completion, 469–470 standard deviation, 470 weekly budget, 475 Project variance, computing, 469 ProModel, 560 Proof 5, 560 Proportionality, 250 Purchase cost, 203–205, 211 Puyallup Mall, 426–427 Q QM for Windows assignment module, 393 control charts, 619 decision models, 113 decision trees, 114 decomposition method, 176, 177 file extension, 633 forecasting, 191–193 goal programming module, 411 integer programming model, 401–402 integer programming problems, 398–399 inventory control, 246–247 linear programming (LP) problems, 263–264 Markov analysis, 597–598 maximal-flow problem, 438 minimal spanning tree problem, 432 minimization problems, 272 Monte Carlo simulation, 541 645 objective function coefficients changes, 279–280 program crashing, 483 project management, 495–496 Quality Control module, 619 queuing problems, 532 regression calculations, 122 regression models, 148–149 right-hand-side values changes, 283–285 transportation module, 393 Quadratic programming problem, 412 Qualitative factors, Qualitative models, 155 Quality, 602 Quality control (QC), 602–603 Quantitative analysis, 2–5 computers and spreadsheet models role, 9–11 developing model, 7–9 implementing results, 5–6 lack of commitment, 15–16 possible problems in, 12–14 real-world, resistance to change, 15 Quantitative analysis/quantitative methods (QA/QM), 632 Quantitative causal models and regression analysis, 155 Quantitative models, 13 Quantity discount models, 210–213 Quantity discounts, 197, 211 Queue discipline, 502–503 Queuing models, 505, 519 Queuing polls, 515 Queuing problem simulation, 550–552 Queuing system, 501–504, 506, 519 Queuing theory, 500, 511 R RAND function, 541 Random arrivals, 502, 513 Random numbers, 537–541, 544, 558–559 Random variables, 33–34 Range charts, 609–610 Ranking goals with priority levels, 409–410 Raw data, R-charts, 605, 607, 610 Real time network routing (RTNR), 442 Recovery, Red Brand Canners, 339–340 Red Top Cab Company c-charts, 613 Redundancy, 275–276 Regression calculations in formulas, 146–147 computer software, 122–123 least squares, 170 multiple regression model, 128–131 nonlinear, 133–136 relationship among variables, 119 with trend and seasonal components, 177–178 variance (ANOVA) table, 127–128 Regression analysis, 116, 118 cautions and pitfalls, 136 quantitative causal models, 155 Regression equations, 559 Regression models, 136 assumptions of, 124–125 binary variables, 131–132 building, 132–133 coefficient of correlation, 121 coefficient of determination, 120 dependent variable, 116 dummy variables, 131–132 errors assumptions, 124–125 estimating variance, 125 independent variable, 116 linear models, 133 measuring fit of, 119–121 nonlinear regression, 133–136 scatter diagrams, 116 significant, 126 646 INDEX simple linear regression, 117–119 statistical hypothesis test, 125–127 statistically significant, 132 stepwise regression, 133 testing for significance, 125–128 variables, 132–133 Regret, 74–75 Remington Rand, 461 Reneging, 502 Reorder point (ROP), 205–206, 214–215, 217 Residual, 122 Resistance to change, 15 Resource leveling, 484 Resources changes in, 282–285 constraints, 252–253 most effective use of, 250 slack, 262–263 storing, 197 Response, Response variable, 116 Results, 5–6, 14 Revision probability, 30–33, 87–89 @Risk, 560 Risk avoider utility curve, 92 Risk mathematical model categories, 8–9 Risk seeker utility curve, 93 RiskSim, 560 Routes, unacceptable or prohibited, 362 Rules of probability, 22 Running sum of the forecast errors (RSFE), 180 Ryder Systems, Inc., 479 S Safety stock, 215–219 Sales force composite, 155 San Miguel Corporation warehousing questions, 358 SAP, 232 Satisfices, 408 Scale models, 3–4 Scatter diagrams, 116, 156–157 Schank Marketing Research, 425 Schematic models, Seasonal indexes, 172–173 Seasonal variations, 171–174 Self-tests solutions, 639–640 Sensitivity analysis, 5, 14, 79–80, 205 decision trees, 86 input parameters values, 277 linear programming (LP) problems, 276–285 objective function coefficient changes, 278–280 project management, 471–473 resources or right-hand-side values changes, 282–285 technological coefficients changes, 280–282 what-if? questions, 277 Sequential decisions, 82–84 Service cost, 500–501, 509–510 Service facility, 503 Service level, 216 Service processes, 513 Service quality, 602 Service time distribution, 503 Setup cost, 207–208 Shipping problem, 327–330 Shortages, 197 Shortest-route problem, 439–444 Shortest-route technique, 430, 440 Significant regression model, 126 Simkin’s Hardware store, 543–548, 548–549 Simple linear regression, 117–119 Simple moving averages, 162 Simple probability, 27 Simplex algorithm, 251 SIMUL8, 560 Simulated demand, 540 Simulation, 519, 534 advantages and disadvantages, 535–536 business system, 534 collecting data, 549 computer languages, 535 computers role in, 560 controllable inputs, 543 corporate operating system, 559 cost analysis, 557 cumulative probability distribution, 543 defining problem, 543 econometric models, 559 economic systems, 559 Federal Aviation Administration (FAA), 549 flowchart, 546 history of, 535 inventory analysis, 543–549 lead time variable, 546 maintenance problems, 553–557 management system, 534 mathematical model, 534 Monte Carlo simulation, 536–543, 546 operational gaming, 558–559 physical models, 534 preventive maintenance, 557 probability distribution, 543, 546 queuing problem, 550–552 random numbers, 539 results differing, 540 systems simulation, 559 uncontrollable inputs, 543 urban government, 559 validation, 559 variables, 536, 546 verification, 559 Simulation model maintenance policy, 553–557 Simulation software tools, 560 Single-channel queuing model, 506–511 Single-channel system, 503 Single-period inventory models, 221–225 Single-phase system, 503 Sink maximal-flow technique, 433–439 Six Sigma, 603 Ski lift slowing down to get shorter lines, 505 Slack, 262–263 Slack time, 467, 469, 471–472 Smoothing constant, 165–169 Software packages and project management, 479, 484 Solutions affect of, degenerate, 352 developing, 5–6, 14 enumerating outcomes, 366 hard-to-understand mathematics, 14 implications of, improved, 354–358 integer programming, 398 multiple, 362 only one answer limiting, 14 outdates, 13 to problems, 636–638 to self-tests, 639–640 sensitivity of, stating problems as, 12–13 testing, 5–6, 14, 352–355 Solver add-in, 10–11, 264–269 changing cells, 413 minimization problems, 272 objective function, 280, 413 preparing spreadsheet for, 264–267 solving method, 413 transportation problems, 343 transshipment problem, 348 usage, 267–269 Sources, 342, 343, 433–439 Southwestern University (SWU) food and beverages at football games, 19 forecasting attendance at football games, 189 stadium construction, 494, 495 traffic problems, 456 SPC charts, 607 Special Projects Office of the U.S Navy, 461 Sport of curling and Markov analysis, 586 Sports and probability, 42 Spreadsheets, decision variables, 264, 266 entering problem data, 264–266 left-hand-side (LHS) of constraints formula, 265, 266 preparing for Solver, 264–267 quantitative analysis role, 9–11 value of objective function formula, 265, 266 Standard deviation, 36, 42–44, 217, 470 Standard deviation of the regression, 125 Standard error of the estimate, 125, 148 Standard gamble, 91 Standardized normal distribution function, 45 Standard normal curve, 622–623 Standard normal distribution, 42–44 Standard normal probability table and Haynes Construction Company example, 44, 46–47 Standard normal table, 42–44, 46 Starting Right Corporation, 110–111 State-of-nature nodes, 81, 83–84 State-of-nature points, 81 State probabilities, 574–576 calculating, 577–578 current period or next period, 580 equilibrium, 579–581 States, 574–576 accounts receivable application, 582–583 matrix of transition probabilities, 583 steady state probability, 581 States of nature, 70 Statewide Development Corporation, 571–572 Statistical dependence and joint probability, 30 Statistically dependent events, 28–29 Statistically independent events, 27–28 Statistical process control (SPC), 602, 603–605, 607 Steady state, 519 Steady state probabilities, 579, 581 Stepping-stone method, 352–358 Stepwise regression, 133 Stock level, optimum, 216 Stockout cost, 215 Stockouts, 196, 197, 213–215 Storing resources, 197 Subjective probability, 23, 24 Subprojects, 484 Successor activity, 471, 472 Sugar cane, moving in Cuba, 353 Sumco economic order quantity (EOQ), 202–203 Sum of squares due to regression (SSR), 119–120 Sum of squares error, 122 Sum of the squares error (SSE), 119, 170 Sum of the squares residual, 122 Sum of the squares total (SST), 119 SUMPRODUCT function, 266 Sun-Times marginal analysis with normal distribution, 224–225 Super cola example and ξ-chart (x-bar chart), 608–609 Supply-chain disruption problem, 373 Supply-chain optimization (SCO), 401 Supply-chain reliability, 373 Surplus, 262–263 Swift & Company, 283, 311 Synchronous optical network (SONET), 442 Systems simulation, 559 states, 574 T Taco Bell’s restaurant operation simulation, 560 TAURUS project, 479 Technological coefficients changes, 280–282 Technology, 435 Testing solutions, 14 Thermal Neutron Analysis device, 32 Thompson Lumber Company, 70–71 Three grocery stores transition probabilities, vector of state probabilities, 575–576 Three Hills Power Company simulation, 553–557 INDEX Three Rivers Shipping Company waiting lines, 501 Timeline, 484 Time series, 160–161 Time-series forecasting, 156–157, 171–172 Time-series forecasting models, 160–179 Time-series models, 154 Top Speed Bicycle Co., 327–330 Total cost, 211 Total expected cost, 501 Total opportunity costs, 368 Total quality management (TQM), 602 Tracking signals, 180–181 TransAlta Utilities (TAU), 121 Transient state, 519 Transportation algorithm changing shipping route, 355 cost-effectiveness, 353–354 degeneracy in transportation problems, 359–362 feasible solution, 351 improved solution, 355–358 improvement indices, 354, 356 initial solution, 350–352 least-cost solution, 352–358 maximization transportation problems, 362 maximum shipped on new route, 355 multiple optimal solutions, 362 northwest corner rule, 350–352 optimal shipping assignments, 357 path, 354–355 route with negative index, 355 special situations, 358–362 stepping-stone method, 352–358 summary of steps, 358 testing solution, 352–355 transportation problems, 348–358 unacceptable or prohibited routes, 362 unbalanced transportation problems, 358–359 Transportation applications, 327–330 Transportation method, 363–364 Transportation models, 350 Transportation problems cost of shipping assignment, 351 degeneracy, 359–362 demand constraints, 343 destinations, 342, 343 dummy destinations or sources, 358–359 general linear programming (LP), 343 initial shipping assignments, 351 intermediate points, 347 least-cost method, 362 linear programming (LP) for, 342–343 maximization, 362 minimizing costs, 342–343 multiple optimal solutions, 362 number of variables and constraints, 343 optimal solution, 348–358 other transportation methods, 362 sources, 342, 343 stepping-stone method, 352–358 supply constraints, 343 transportation algorithm, 348–358 transshipment point, 346 unacceptable or prohibited routes, 362 unbalanced, 358–359 Vogel’s approximation method, 362 Transportation table, 352, 354 Transshipment point, 346 Transshipment problem, 346–348, 438 Trend-adjusted exponential smoothing, 166–169 Trend analysis, 171 Trend line of deseasonalized data, 175–176 Trend projections, 154, 169–171 Trends, linear, 169–170 Trial and error method, Truck loading problem, 322–324 Tuberculosis drug allocation in Manila, 410 Tupperware International forecasting, 159 Two decision variables inventory problem, 543 Two probabilistic components inventory problem, 543 Two rules of probability, 23 U ULAM, 25 Unacceptable or prohibited routes, 362 Unbalanced assignment problems, 371 Unbalanced transportation problems, 358–359 Unboundedness, 275 Uncontrollable inputs, 543 Unfavorable market (UM), 87 Union of two events, 26 Unisys Corp experiment in health care services, 611 United Airlines, 479 United Network of Organ Sharing (UNOS), 25 University of Alberta, 121 University of Maryland, College Park, 505 UPS optimization, 321 Urban government simulation, 559 U.S Department of Agriculture, 12 U.S Department of Energy (DOE), 94 U.S Postal Service (USPS), 400, 545 Utility, 90–95 Utility curve, 91–93 Utility theory, 90–95 Utilization factor, 507 V Validation simulation, 559 Valid models, Variability in processes, 603–605 Variable costs, Variables, collinear, 133 control charts, 605–610 controllable, 647 cumulative probability distribution, 537 investigating relationship between, 116 Monte Carlo simulation, 536 multicollinearity, 133 probability distributions, 536–537 regression models, 132–133 relationship among, 119 simulation, 536 Variance (ANOVA) table, 127–128 Variance of activity completion time, 464 Variances, 35 discrete probability distribution, 36 Poisson distribution, 53 testing hypotheses about, 48–50 Variations due to assignable causes, 603 Vector of state probabilities, 575–576 Vehicle Routing Problem (VRP), 400 Venn diagram, 26 Verification, 559 VLOOKUP function, 541 Vogel’s approximation method, 362 VOLCANO (Volume, Location, and Aircraft Network Optimizer), 321 von Neumann midsquare method, 539 W Waiting costs, 501, 509–510 Waiting lines, 500–503 Warehouses, locating, 363–364 Weekly budget, 475 Weighted average, 73–74 Weighted goals and goal programming, 410–411 Weighted moving averages, 161–162 Weights, combinations and forecasts, 162 Westover Wire Works, 300 What if? questions, 308, 559 Whole Food Nutrition Center, 324–325 Winter Olympics (2002), 586 Winter Park Hotel, 531 Work breakdown structure, 460 Work package, 474 WTVX, 65 X ξ-chart (x-bar chart), 605–609 XLSim, 560 Z Zara inventory management system, 200 0-1 (binary) variables, 402–406 Zero-one integer programming problems, 396 Zero opportunity, 368 Z standard random variable, 43 ... + 10X 22 + 11X23 + 9X31 + 12X 32 + 7X33 subject to X11 + X 12 X21 + X 22 X31 + X 32 X11 + X21 + X13 … + X23 … + X33 … + X31 = X 12 + X 22 + X 32 = X13 + X23 + X33 = xij = or for all i and j The solution... 8X21 + 4X 22 + 3X23 + 9X31 + 7X 32 + 5X33 subject to X11 + X 12 + X13 … 100 (Des Moines supply) X21 + X 22 + X23 … 300 (Evansville supply) X31 + X 32 + X33 … 300 (Fort Lauderdale supply) X11 + X21 + X31... 1, 2, 3, with = Adams, = Brown, and = Cooper j = 1, 2, 3, with = Project 1, = Project 2, and = Project The LP formulation is Minimize total cost = 11X11 + 14X 12 + 6X13 + 8X21 + 10X 22 + 11X23