Fuzzy Probabilities and Fuzzy Sets for Web Planning

ISBN-13: 9783642055966 / Angielski / Miękka / 2010 / 190 str.

James J. Buckley

Opis

1.1 Introduction This book is written in five major divisions. The first part is the introduc- tory chapters consisting of Chapters 1-3. In part two, Chapters 4-10, we use fuzzy probabilities to model a fuzzy queuing system . We switch to employ- ing fuzzy arrival rates and fuzzy service rates to model the fuzzy queuing system in part three in Chapters 11 and 12. Optimization models comprise part four in Chapters 13-17. The final part has a brief summary and sug- gestions for future research in Chapter 18, and a summary of our numerical methods for calculating fuzzy probabilities, values of objective functions in fuzzy optimization, etc., is in Chapter 19. First we need to be familiar with fuzzy sets. All you need to know about fuzzy sets for this book comprises Chapter 2. Two other items relating to fuzzy sets, needed in Chapters 13-17, are also in Chapter 2: (1) how we plan to handle the maximum/minimum of a fuzzy set; and (2) how we will rank a finite collection of fuzzy numbers from smallest to largest.

Spis treści

1 Introduction.- 1.1 Introduction.- 1.2 Fuzzy Probabilities.- 1.3 Fuzzy Arrival/Service Rates.- 1.4 Optimization Models.- 1.5 Notation.- 1.6 References.- 2 Fuzzy Sets.- 2.1 Introduction.- 2.2 Fuzzy Sets.- 2.2.1 Fuzzy Numbers.- 2.2.2 Alpha-Cuts.- 2.2.3 Inequalities.- 2.2.4 Discrete Fuzzy Sets.- 2.3 Fuzzy Arithmetic.- 2.3.1 Extension Principle.- 2.3.2 Interval Arithmetic.- 2.3.3 Fuzzy Arithmetic.- 2.4 Fuzzy Functions.- 2.4.1 Extension Principle.- 2.4.2 Alpha-Cuts and Interval Arithmetic.- 2.4.3 Differences.- 2.5 Finding the Min/Max of a Fuzzy Number.- 2.6 Ordering/Ranking Fuzzy Numbers.- 2.7 References.- 3 Fuzzy Probabilities/Arrival Rates.- 3.1 Introduction.- 3.2 Fuzzy Probabilities from Confidence Intervals.- 3.3 Fuzzy Arrival/Service Rates.- 3.3.1 Fuzzy Arrival Rate.- 3.3.2 Fuzzy Service Rate.- 3.4 Fuzzy Numbers from Expert Opinion.- 3.5 Restricted Fuzzy Arithmetic.- 3.5.1 Probabilities.- 3.5.2 Restricted Arithmetic: General.- 3.5.3 Restricted Fuzzy Arithmetic: Book.- 3.6 Computations.- 3.6.1 First Problem.- 3.6.2 Second Problem.- 3.6.3 Another Fuzzy Computation.- 3.7 Figures.- 3.8 References.- 4 Fuzzy Markov Chains.- 4.1 Introduction.- 4.2 Fuzzy Regular Markov Chains.- 4.3 Fuzzy Absorbing Markov Chains.- 4.4 Other Fuzzy Markov Chains.- 4.5 References.- 5 Fuzzy Queuing Theory.- 5.1 Introduction.- 5.2 Queuing Theory.- 5.3 Fuzzy Queuing Theory.- 6 Computations: One Sever.- 6.1 Introduction.- 6.2 Calculations.- 6.3 References.- 7 Example: One Sever.- 7.1 Introduction.- 7.2 Computations.- 7.3 References.- 8 Computations: Two Servers.- 8.1 Introduction.- 8.2 Calculations.- 9 Example: Two Servers.- 9.1 Introduction.- 9.2 Computations.- 9.3 References.- 10 Computations: Three or More Servers.- 10.1 References.- 11 Fuzzy Arrival/Service Rates.- 11.1 Introduction.- 11.2 Fuzzy Steady State Probabilities.- 11.3 Fuzzy System Parameters.- 11.4 References.- 12 Example: Fuzzy Arrival/Service Rates.- 12.1 Introduction.- 12.2 One Server.- 12.3 Two Servers.- 12.4 Three or More Servers.- 12.5 References.- 13 Optimization: Without Revenue/Costs.- 13.1 Introduction.- 13.2 Fuzzy Probabilities.- 13.2.1 Minimize % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa0aaaeaaca % WGsbaaaaaa!36DC!$$\overline R $$.- 13.2.2 Minimize % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa0aaaeaaca % WGsbaaaaaa!36DC!$$\overline R $$ and Maximize % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa0aaaeaaca % WGvbaaaaaa!36DF!$$\overline U $$.- 13.2.3 Ranking the Fuzzy Sets.- 13.3 Fuzzy Arrival/Service Rates.- 13.3.1 Minimize % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa0aaaeaaca % WGsbaaaaaa!36DC!$$\overline R $$.- 13.3.2 Minimize % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa0aaaeaaca % WGsbaaaaaa!36DC!$$\overline R $$ and Maximize % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa0aaaeaaca % WGvbaaaaaa!36DF!$$\overline U $$.- 13.3.3 Ranking the Fuzzy Sets.- 13.4 References.- 14 Optimization: With Revenue/Costs.- 14.1 Introduction.- 14.2 Fuzzy Probabilities.- 14.2.1 Ranking the Fuzzy Sets.- 14.3 Fuzzy Arrival/Service Rates.- 14.3.1 Ranking the Fuzzy Sets.- 14.4 References.- 15 Burstiness.- 15.1 Introduction.- 15.2 Fuzzy Probabilities.- 15.2.1 Ranking the Fuzzy Numbers.- 15.3 Fuzzy Arrival/Service Rates.- 15.3.1 Ranking the Fuzzy Numbers.- 15.4 References.- 16 Long Tailed Distributions.- 16.1 Introduction.- 16.2 Fuzzy Probabilities.- 16.2.1 Ranking the Fuzzy Numbers.- 16.3 Fuzzy Arrival/Service Rates.- 16.3.1 Ranking the Fuzzy Numbers.- 16.4 References.- 17 Putting It All Together.- 17.1 Introduction.- 17.2 Fuzzy Probabilities.- 17.3 Fuzzy Arrival/Service Rates.- 18 Summary and Future Research.- 18.1 Introduction.- 18.2 Fuzzy Probabilities.- 18.3 Fuzzy arrival/Service Rate.- 18.4 Future Research.- 18.4.1 CD-ROM.- 18.4.2 Simulation.- 18.5 References.- 19 Computational Algorithms.- 19.1 Introduction.- 19.2 Computations: Fuzzy Probabilities.- 19.2.1 Premium Solver Problem.- 19.2.2 Genetic Algorithm.- 19.2.3 Optimization Models.- 19.3 Computations: Fuzzy Arrivals and Service Rates.- 19.4 References.- List of Figures.- List of Tables.

Blurb

This book presents important applications of soft computing and fuzziness to the growing field of web planning. A new method of using fuzzy numbers to model uncertain probabilities and how these can be used to model a fuzzy queuing system is demonstrated, as well as a method of modeling fuzzy queuing systems employing fuzzy arrival rates and fuzzy service rates. All the computations needed to get to the fuzzy numbers for system performance are described starting for the one server case to more than three servers. A variety of optimization models are discussed with applications to the average response times, server utilization, server and queue costs, as well as to phenomena identified with web sites such as "burstiness" and "long tailed distributions".