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Raúl Poler · Josefa Mula

ManuelDíaz-Madroñero

Operations

Research

Problems

Statements and Solutions

Operations Research Problems

Rau¥l Poler

Josefa Mula

Manuel D¥az-Madronòero

Operations Research

Problems

Statements and Solutions

123

Rau¥l Poler

Josefa Mula

Manuel D

¥az-Madronòero

Research Centre on Production

Management and Engineering (CIGIP)

Department of Business Management

Universitat Polite

`cnica de Vale`ncia Alcoy Spain ISBN 978-1-4471-5576-8 ISBN 978-1-4471-5577-5 (eBook)

DOI 10.1007/978-1-4471-5577-5

Springer London Heidelberg New York Dordrecht

Library of Congress Control Number: 2013949477

©Springer-Verlag London 2014

Translators: Helen Warbuton and Antonio Maravilla Burgos (HyA Translations)

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Preface

Operations Research Problemsis an advanced textbook: designed primarily to meet the demands of a operations research and management science course taught at postgraduate M. Eng. or MBA level. This book includes an overview of each topic considered with worked examples in the text, problems, solutions to prob- lems, end-of-chapter references and Index. The objective of this book is to provide a valuable compendium of problems as a reference for undergraduate and postgraduate students, faculty, researchers and practitioners of operations research and management science. These problems can serve as a basis for the development or study of assignments and exams. Also, they can be useful as a guide for the rst stage of the model formulation, i.e. the denition of a problem. The book is divided into 10 chapters that address the following topics: Linear programming, integer programming, nonlinear pro- gramming, network modelling, inventory theory, queueing theory, decision theory, games theory, dynamic programming and markov processes. Readers are going to nd a considerable number of statements of operations research applications for management decision-making. The solutions of these problems are provided in a concise way although all topics start with a more developed overview. The pro- posed problems are based on the research experience of the authors in real-world companies so much as on the teaching experience of the authors in order to develop exam problems for industrial engineering and business administration studies.

The chapters have been arranged as follows:

Linear Programming

This book chapter provides some linear programming applications. After reading this chapter, the reader should be capable of: understanding the nature of resources optimisation problems and their formulation by linear programming; knowing the resolution graphic model for linear programming problems with two decision variables; comprehending the Simplex Method"s logic nature and of applying it to solve linear programming models; formulating the dual problem of a linear v programming model; economically interpreting the solutions of a dual problem; understanding the sensitivity analysis concept and of using it for decision making.

Integer Programming

Integer linear programming models are employed in a large number of problems with intrinsically integer variables. After reading this chapter, the reader should be capable of knowing and modelling different integer linear programming prototype problems and formulating these models with binary variables.

Non-Linear Programming

The objective of this book chapter is to help learn the formulation of nonlinear programming models and of presenting some of their applications in the industrial engineering and management domain. After reading this chapter, the reader should be capable of formulating different prototype nonlinear programming problems, and of modelling multivariant and multimodel functions with inequality con- straints by the Kuhn-Tucker conditions.

Network Modelling

The purpose of this book chapter is to help learn the formulation of network modelling models and to show some of their applications in the industrial engi- neering and management area. Therefore, management problems are modelled using graphs, while models are solved with shortest path, maximal ow and minimal spanning tree problems. After reading this chapter, readers should be able to model and solve different prototype shortest path, maximal ow and minimal spanning tree problems and to model minimal cost ow problems.

Inventory Theory

The objective of this chapter is to help learn to formulate and solve deterministic models based on the Inventory Theory in the independent demand context and to show some of their applications in the industrial engineering and management area. Thus, management problems are modelled by analytical EOQ formulation along with some of its variants. After reading this chapter, readers should be able to model and solve different inventory problems by means of the basic EOQ model viPreface and its variants with discounts for volume, delivery times other than zero and backorders.

Queueing Theory

The objective of this book chapter is to help to learn how to formulate and solve analytical Queueing Theory models and to show some of their applications in the industrial engineering and management domain. Therefore, some management problems are modelled by the analytical formulation of various steady-state queueing systems. After reading this chapter, readers should be able to quantita- tively and qualitatively characterise a queue by a mathematical analysis, and determine the suitable levels of certain queueing system parameters which balance the social waiting cost with the cost associated with the resources consumed.

Decision Theory

With this chapter, readers should be able to understand the nature of risk decision problems and their modelling by means of decision trees, to discover the solution method by calculating the EMV, to apply Bayes Theorem, to calculate revised probabilities according to new information and to interpret solutions by calculating the limit values of the costs of further information.

Games Theory

The purpose of this chapter is to provide mechanisms to understand the cooper- ative and non-cooperative decision problems; identify the players from a games problem and its feasible strategies; calculate the losses and gain associated with each combination of strategies and solve zero-sum games played by two people by applying Mini-Max for pure strategies, the algebraic method for mixed strategies with 292 matrices, the Graph Method for 29M and the general linear pro- gramming method for any dimension.

Dynamic Programing

After reading this chapter, readers should be able to comprehend the nature of multiphase decision problems that can be modelled by dynamic programming; dene the stages of the problem, its input stages and the decisions that can be

Prefacevii

made; dene the transition function between the input state and the output stage according to the decision made in each stage; construct the recursive function of a dynamic programming model; calculate the optimal costs in each stage, as well as the optimal decisions and obtain the optimal solutions that provide dynamic programming for decision making.

Markov Processes

The objective of this book chapter is to help to understand the nature of stochastic systems that can be modelled by a Markov chain; calculate the one-step transition probabilities between the various system states; know the different calculation formulae of the several step transition probabilities, stationary probabilities and the mean rst passage times and calculate the mean system operation costs. Operations Research Problemsis related to operations management, production planning and quantitative methods. Readers would be able to identify different operations management problems in order to improve the decision-making process concerning to them. Therefore, it is not only useful for academic purposes but also for industrial practitioners. We want to acknowledge the valuable help of Lu QiDa Lian, China, ?for the development of this book. viiiPreface

Contents

1 Linear Programming................................. 1

1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Bonuses and Merits. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.3 Production Planning in a Textile Firm. . . . . . . . . . . . . . . . . 7

1.4 Portfolio of Investments. . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.5 Transferring Currencies . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.6 Production Planning in a Metallurgical Company . . . . . . . . . 14

1.7 Production Planning in a Cosmetics Firm . . . . . . . . . . . . . . 16

1.8 Product Mix of Aluminium Windows . . . . . . . . . . . . . . . . . 19

1.9 Production Planning in the Automobile Industry. . . . . . . . . . 21

1.10 Portfolio of Investments. . . . . . . . . . . . . . . . . . . . . . . . . . . 22

1.11 Investment Funds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

1.12 Renting Warehouses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

1.13 Production Planning in a Tubes Factory . . . . . . . . . . . . . . . 27

1.14 Production Planning of a Wires Manufacturer . . . . . . . . . . . 29

1.15 Mixed Investments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

1.16 Production Planning in a Carpentry Firm. . . . . . . . . . . . . . . 31

1.17 Product Mix. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

1.18 Transport Planning in a Olive Production Firm . . . . . . . . . . 34

1.19 Programming Weekly Production in a Metallurigcal

Company. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

1.20 Production Planning in a Bricks Factory . . . . . . . . . . . . . . . 40

1.21 Sensitivity Analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

1.22 Crop Planning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

2 Integer Programming................................. 49

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

2.2 Postmen/Women"s Shifts . . . . . . . . . . . . . . . . . . . . . . . . . . 50

2.3 Distribution of Air-Conditioning Units . . . . . . . . . . . . . . . . 54

2.4 Contracting Carpenters . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

2.5 Planning Production and Inventories . . . . . . . . . . . . . . . . . . 60

2.6 Tarmacking Shifts. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

ix

2.7 Transport Planning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

2.8 Production Planning of Automobile Seats . . . . . . . . . . . . . . 66

2.9 Production Planning in the Shoe Industry . . . . . . . . . . . . . . 67

2.10 Allocating Orders to Machines. . . . . . . . . . . . . . . . . . . . . . 70

2.11 Opening LED TV Production Plants . . . . . . . . . . . . . . . . . . 72

2.12 Planning Surveys . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

2.13 Production Planning in a Toys Firms . . . . . . . . . . . . . . . . . 76

2.14 Allocation in a Lawyersí Office . . . . . . . . . . . . . . . . . . . . . 77

2.15 Production Planning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

2.16 Planning University Schedules . . . . . . . . . . . . . . . . . . . . . . 80

2.17 The Shortest Path . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

2.18 Planning Sales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

3 Non-Linear Programming.............................. 87

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

3.2 Selecting Investments . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

3.3 Location of a Stationery Store . . . . . . . . . . . . . . . . . . . . . . 92

3.4 Placing a Handrail . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

3.5 Planning the Construction of Homes. . . . . . . . . . . . . . . . . . 96

3.6 Production Planning in a Drinks Firm . . . . . . . . . . . . . . . . . 97

3.7 Production Planning in a Firm in the Automobile Sector. . . . 99

3.8 Planning Materials in the Chemical Industry . . . . . . . . . . . . 102

3.9 Production Planning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

3.10 Maximising Utility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

3.11 Designing Boxes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

3.12 Designing and Planning a Project to Launch

a Scientific Balloon. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

3.13 Production Planning in an Oil-Packing Firm . . . . . . . . . . . . 109

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

4 Network Modelling................................... 115

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

4.2 Planning Storage in a Library. . . . . . . . . . . . . . . . . . . . . . . 117

4.3 Organising Tables for a Charity Gala . . . . . . . . . . . . . . . . . 119

4.4 Planning Loads on Trucks . . . . . . . . . . . . . . . . . . . . . . . . . 121

4.5 Designing a Communications Network . . . . . . . . . . . . . . . . 121

4.6 Assigning Cases in a Lawyerís Office. . . . . . . . . . . . . . . . . 122

4.7 Project Planning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

4.8 Production Planning in a Plastics Firms. . . . . . . . . . . . . . . . 125

4.9 Overbooking on Airlines . . . . . . . . . . . . . . . . . . . . . . . . . . 126

4.10 Production Sequencing in a Firm of the Metal Sector . . . . . . 127

4.11 Fibre Optics Network Planning. . . . . . . . . . . . . . . . . . . . . . 129

4.12 Assigning Subject Matters to Teachers . . . . . . . . . . . . . . . . 130

xContents

4.13 Water Network Planning . . . . . . . . . . . . . . . . . . . . . . . . . . 132

4.14 Production Planning in a Firm of the Textile Sector . . . . . . . 133

4.15 Project Planning and Designing an Air Conditioning

Network. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

4.16 Machinery Transport Planning . . . . . . . . . . . . . . . . . . . . . . 135

4.17 Emergency Routes Should a Fire Break Out . . . . . . . . . . . . 136

4.18 Planning Customers Orders . . . . . . . . . . . . . . . . . . . . . . . . 138

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

5 Inventory Theory.................................... 141

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

5.2 Minced Meat Inventory . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

5.3 Soft Drinks Orders in a Cafeteria . . . . . . . . . . . . . . . . . . . . 146

5.4 Orders Management in the Footwear Industry . . . . . . . . . . . 148

5.5 Components Order Management in an Industrial Firm . . . . . 151

5.6 Training Planning for Drivers. . . . . . . . . . . . . . . . . . . . . . . 153

5.7 Inventory Management Without Stockouts. . . . . . . . . . . . . . 154

5.8 Inventory Management of Two Products . . . . . . . . . . . . . . . 155

5.9 Inventory Management of Cold Water Dispensers . . . . . . . . 158

5.10 Orders Management in the Footwear Industry . . . . . . . . . . . 160

5.11 Orders Management of X-ray Plaques in a Hospital . . . . . . . 160

5.12 Orders Management with Discount . . . . . . . . . . . . . . . . . . . 161

5.13 Paper Inventory Management. . . . . . . . . . . . . . . . . . . . . . . 162

5.14 Inventory Planning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

5.15 Inventory Management in a Textile Firm. . . . . . . . . . . . . . . 166

5.16 Inventory Management in a Supermarket. . . . . . . . . . . . . . . 167

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

6 Queueing Theory.................................... 171

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

6.2 System with Three Stations . . . . . . . . . . . . . . . . . . . . . . . . 175

6.3 Planning Bank Cashiers. . . . . . . . . . . . . . . . . . . . . . . . . . . 177

6.4 Hospital Pharmacy Planning. . . . . . . . . . . . . . . . . . . . . . . . 179

6.5 Luggage Control in Airports . . . . . . . . . . . . . . . . . . . . . . . 180

6.6 System with a Processing Centre . . . . . . . . . . . . . . . . . . . . 181

6.7 Processing Jobs in a Calculation Centre. . . . . . . . . . . . . . . . 181

6.8 Customer Counselling Service . . . . . . . . . . . . . . . . . . . . . . 183

6.9 Acquiring Machinery for an Assembly Line. . . . . . . . . . . . . 185

6.10 Receiving Orders Via a Telephone

Communications Service . . . . . . . . . . . . . . . . . . . . . . . . . . 186

6.11 Customer Services in an Agency . . . . . . . . . . . . . . . . . . . . 188

6.12 Orders from Cars in a Fast Food Restaurant . . . . . . . . . . . . 190

6.13 Buying a Photocopier . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192

6.14 A Printer Shared in a Computer Network . . . . . . . . . . . . . . 193

Contentsxi

6.15 Income from Temporary Work. . . . . . . . . . . . . . . . . . . . . . 193

6.16 Fast Food Restaurants . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194

6.17 Loading Vans to Deliver Orders. . . . . . . . . . . . . . . . . . . . . 196

6.18 Contracting Shop Assisitants in an Ice-Cream Parlour. . . . . . 197

6.19 Traffic in Routers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198

6.20 A Taxi Rank at an Airport. . . . . . . . . . . . . . . . . . . . . . . . . 199

6.21 Attending Telephone Calls. . . . . . . . . . . . . . . . . . . . . . . . . 200

6.22 Warehouse Management in a Firm

in the Automobile Sector. . . . . . . . . . . . . . . . . . . . . . . . . . 201 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203

7 Decision Theory..................................... 205

7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205

7.2 Selecting Suppliers to Manufacture Solar Panels . . . . . . . . . 210

7.3 Selling a Car . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216

7.4 Football Bets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218

7.5 Public Tender for Maritime Measuring Instruments . . . . . . . 223

7.6 Providing Personal Loans . . . . . . . . . . . . . . . . . . . . . . . . . 226

7.7 Spying in Genetic Research . . . . . . . . . . . . . . . . . . . . . . . . 228

7.8 Choosing an Elliptical Trainer Bike . . . . . . . . . . . . . . . . . . 231

7.9 Acquiring a New Machine. . . . . . . . . . . . . . . . . . . . . . . . . 236

7.10 Wine Tasting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239

7.11 Maintaining Machinery . . . . . . . . . . . . . . . . . . . . . . . . . . . 243

7.12 Antidoping Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246

7.13 An Urban Development Purchasing Problem . . . . . . . . . . . . 248

7.14 Purchasing a Forklift Truck . . . . . . . . . . . . . . . . . . . . . . . . 252

7.15 Planning to Commercialise a New Irrigation System. . . . . . . 255

7.16 Maximising Profits in the Film Industry . . . . . . . . . . . . . . . 258

7.17 Planning Routes for a Vending Machine Firm . . . . . . . . . . . 261

7.18 Replacing Petrol Cars with Electric Ones . . . . . . . . . . . . . . 265

7.19 A Football Playerís Injury . . . . . . . . . . . . . . . . . . . . . . . . . 268

7.20 Medical Decisions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271

7.21 Developing New Textile Products. . . . . . . . . . . . . . . . . . . . 273

7.22 Locating Production Plants in the Automobile Industry. . . . . 275

7.23 Selecting Antidumping Purchases and Measures. . . . . . . . . . 278

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280

8 Games Theory...................................... 281

8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281

8.2 Encounter in a Building. . . . . . . . . . . . . . . . . . . . . . . . . . . 287

8.3 Products Improvement Strategy Between

Competing Firms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289

8.4 Strategies to Set Oil Prices . . . . . . . . . . . . . . . . . . . . . . . . 290

8.5 Playing Spoof. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292

8.6 Investments Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294

xiiContents

8.7 Strategy to Select Routes. . . . . . . . . . . . . . . . . . . . . . . . . . 295

8.8 TV Ads Selection Strategy. . . . . . . . . . . . . . . . . . . . . . . . . 296

8.9 Advertising Campaign Design . . . . . . . . . . . . . . . . . . . . . . 297

8.10 Playing Bluff . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298

8.11 Security Measures to Avoid Snipers . . . . . . . . . . . . . . . . . . 300

8.12 A Game Between Friends . . . . . . . . . . . . . . . . . . . . . . . . . 301

8.13 Strategies to Improve the Market Share

of Transport Companies. . . . . . . . . . . . . . . . . . . . . . . . . . . 303

8.14 Negotiation Between the Government and Transport

Associations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304

8.15 Selecting Locations to Open Fast-Food Restaurants . . . . . . . 305

8.16 Strategies for Davis Cup Tennis Heats . . . . . . . . . . . . . . . . 306

8.17 Developing a High-Tech Product . . . . . . . . . . . . . . . . . . . . 307

8.18 Electoral Strategies in Local Elections . . . . . . . . . . . . . . . . 309

8.19 Strategies to Develop 3D Television Sets . . . . . . . . . . . . . . 311

8.20 Tennis Playing Strategies. . . . . . . . . . . . . . . . . . . . . . . . . . 312

8.21 An Interactive TV Services Market. . . . . . . . . . . . . . . . . . . 313

8.22 A Security Firmís Service Strategy to Avoid Holdups. . . . . . 315

8.23 Strategies to Improve the Market Share of Airlines. . . . . . . . 316

8.24 Paying and Avoiding Tax . . . . . . . . . . . . . . . . . . . . . . . . . 317

8.25 Electoral Campaign Strategies . . . . . . . . . . . . . . . . . . . . . . 319

8.26 World Robots Championship . . . . . . . . . . . . . . . . . . . . . . . 321

8.27 Strategies to Increase the Market Share

of Travelling Vendors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324

9 Dynamic Programming................................ 325

9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325

9.2 Commercialisation of Products Sold Under Licence . . . . . . . 329

9.3 Investing in an Advertising Campaign. . . . . . . . . . . . . . . . . 331

9.4 Planning Routes to Refill Products in Vending Machines . . . 333

9.5 Planning an Electoral Campaign. . . . . . . . . . . . . . . . . . . . . 335

9.6 Seeking a Study Strategy. . . . . . . . . . . . . . . . . . . . . . . . . . 336

9.7 Production Planning in a Textile Firm. . . . . . . . . . . . . . . . . 338

9.8 Frozen Cakes Production Planning . . . . . . . . . . . . . . . . . . . 340

9.9 Managing Hire Cars in Travel Agencies . . . . . . . . . . . . . . . 341

9.10 Fruit and Vegetable Production Planning. . . . . . . . . . . . . . . 342

9.11 Production Planning in a Recycled Glass Firm. . . . . . . . . . . 344

9.12 Selecting Slot Machines . . . . . . . . . . . . . . . . . . . . . . . . . . 345

9.13 Planning an Electrical Expansion Project. . . . . . . . . . . . . . . 346

9.14 Transport Planning in a Firm of the Automobile Sector. . . . . 349

9.15 Inventories Planning in a Chemistís . . . . . . . . . . . . . . . . . . 350

9.16 Truck Load Planning. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352

9.17 Orders Planning of Imported Products. . . . . . . . . . . . . . . . . 353

9.18 Selecting Investment Strategies. . . . . . . . . . . . . . . . . . . . . . 354

Contentsxiii

9.19 Replacing a Car . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 356

9.20 Operation Programming of Electric Generators. . . . . . . . . . . 358

9.21 Summer Cakes Production Planning . . . . . . . . . . . . . . . . . . 360

9.22 Selecting Investments . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362

9.23 Marketing Mix in a Mobile Phone Company . . . . . . . . . . . . 364

9.24 Location of Production Plants . . . . . . . . . . . . . . . . . . . . . . 366

9.25 Reducing Time to Develop Software. . . . . . . . . . . . . . . . . . 368

9.26 Planning Holidays. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 369

9.27 Personnel Planning in Industry with Seasonal Demand . . . . . 371

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374

10 Markov Processes.................................... 375

10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375

10.2 How Social Classes Evolve . . . . . . . . . . . . . . . . . . . . . . . . 378

10.3 Replacing a Car . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 380

10.4 Forecasting Evaluation Tasks for Subject Matters. . . . . . . . . 382

10.5 Market Research in Ice-Cream Parlours. . . . . . . . . . . . . . . . 384

10.6 Evolution of Breakfast Cereal Purchases . . . . . . . . . . . . . . . 386

10.7 Car Insurance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 388

10.8 Market Research into Car Purchases . . . . . . . . . . . . . . . . . . 389

10.9 Experiment Done with Mice . . . . . . . . . . . . . . . . . . . . . . . 391

10.10 Evolution of Financial Products . . . . . . . . . . . . . . . . . . . . . 393

10.11 Market Research into Cars for Firms. . . . . . . . . . . . . . . . . . 394

10.12 Natural Gas Repairs Technician . . . . . . . . . . . . . . . . . . . . . 396

10.13 Weather Forecasts. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 398

10.14 Students Evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399

10.15 How a Squirrel Population Evolves. . . . . . . . . . . . . . . . . . . 400

10.16 How Trees in a Nature Reserve Evolve. . . . . . . . . . . . . . . . 401

10.17 Study into a Radioactive Element. . . . . . . . . . . . . . . . . . . . 402

10.18 Files Proceedings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403

10.19 Controlling Machinery. . . . . . . . . . . . . . . . . . . . . . . . . . . . 405

10.20 Analysing the Effects of a New Virus. . . . . . . . . . . . . . . . . 407

10.21 Market Research into Cars for Fleets of Vehicles . . . . . . . . . 409

10.22 Industrial Machinery Maintenance . . . . . . . . . . . . . . . . . . . 410

10.23 Distribution of Operators in Workshop Areas. . . . . . . . . . . . 413

10.24 Behaviour of Cells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414

10.25 How the Staff in a Consultancy Firm Evolve. . . . . . . . . . . . 416

10.26 How the Effects of a New Drug Evolve . . . . . . . . . . . . . . . 417

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419 About the Authors....................................... 421 Index................................................ 423 xivContents

Synopsis

Operations Research Problemsprovide readers a valuable compendium of problems as a reference for undergraduate and graduate students, faculty, researchers and practitioners of operations research and management science. These problems can serve as a basis for the development or study of assignments and exams. Also, they can be useful as a guide for the rst stage of the model formulation, i.e. the denition of a problem. The chapters address the following topics:

Linear programming,

Integer programming,

Nonlinear programming,

Network modelling,

Inventory theory,

Queueing theory,

Tree decision,

Games theory,

Dynamic programming and

Markov processes.

The objective ofOperations Research Problemsis to present a considerable number of statements of operations research applications for management deci- sion-making. The solutions of these problems are provided in a concise way although all topics start with a more developed resolution. The proposed problems are based on the research experience of the authors in real-world companies so much as on the teaching experience of the authors in order to develop exam problems for industrial engineering and business administration studies. xv

Chapter 1

Linear Programming

AbstractThis chapter commences with an introduction to linear programming. This is followed by a varied set of linear programming problems with their cor- responding solutions. This chapter aims to help learn the formulation, resolution and interpretation of linear programming models and to show some of their applications in the industrial engineering and management area. Hence new variants of basic product mix models and mixtures applied to Industrial Organi- zation Engineering and its management are proposed. Resolution methods are also reviewed: graphic method and Simplex Method. Creating and solving dual prob- lems are also considered. Finally, special attention is paid to the economic inter- pretation of the results of some of the problems presented through their sensitive analysis.

1.1 Introduction

Operations research employs the scientific method as a basis to deal with decision- making problems by designing and solving mathematical models. One of the most studied and developed is linear programming, which seeks to optimise a linear objective function that is subject to some constraints which are also linear. Thequotesdbs_dbs21.pdfusesText_27

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