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Multi - Objective Optimization Using Evolutionary Computation iii
Preface
The Project P roposal is one of the significant documents as it identifies the aims and goals of the project. It is important to the success of the project because it clearly outlines what the project team must achieve in order to classify the project to be complete. This document describes the objective and the scope of the project in detail along with the brief introduction of single-objective optimization, multi-objective optimization, and different techniques for solving multi-objective optimization problem using genetic algorithms.
Multi - Objective Optimization Using Evolutionary Computation iv
Table of Contents
CERTIFICATE OF APPRO VAL ................................ ................................................................ ........... II
PREFACE ................................ ................................ ................................................................ ................ III
TABLE OF CONTENTS ................................ ................................................................ ....................... IV
LIST OF FIGURES ................................................................................................ ............................... VII
Multi - Objective Optimization Using Evolutionary Computation vii
- SECTION LIST OF TABLES VIII
- INTRODUCTION
- 1.1 PURPOSE
- 1.2 S COPE
- 1.3 DEFINITIONS , ACRONYMS AND A BBREVIATIONS
- 1.3.1 Definitions:
- 1.3.2 Abbreviations:
- 1.4 OVERVIEW.
- 1.5 P ROJECT G ENERIC INFORMATION
- 1.5.1 Project Team
- 1.5.2 Project Deliverables
- SECTION
- OBJECTIVES OF RESEARCH
- 2.1 SOP
- 2.2 W HY E VOLUTIONARY?
- 2.3 E VOLUTIONARY C OMPUTATION :
- 2.3.1 Some Biological Terms Used in Genetic Algorithms:
- 2.3.2 Working of Genetic Algorithms:
- 2.4 MOP
- 2.4.1 Linear and Nonlinear MOP
- 2.4.2 Objectives in Multi - objective Optimization
- 2.5 DIFFERENCE BETWEEN SINGLE AND M ULTI - OBJECTIVE OPTIMIZATION
- 2.5.1 Two Goals Instead of One
- 2.5.2 Dealing with Two Search Spaces
- 2.5.3 No Artificial Fix Ups
- 2.6 APPLICATIONS OF MOEA S:
- 2.7 T ECHNIQUES FOR MOEA S :
- 2.8 PERFORMANCE M ETRICS Multi - Objective Optimization Using Evolutionary Computation v
- 2.9 B ENCHMARK T EST P ROBLEMS
- 2.9.1 Schaffer s Test Problem:
- 2.9.2 Fonseca and Fleming Test Problem
- 2.9.3 Zitzler - Deb -Thiele s (ZDT) Test Problems
- 2.10 OPTIMIZATION PROBLEMS IN M OBILE C OMMUNICATION :
- SECTION
- SECTION
- OVERALL DESCRIPTION
- 3.1 P RODUCT PERSPECTIVE
- 3.2 P RODUCT FUNCTIONS
- 3.3 USER C HARACTERISTICS
- 3.4 C ONSTRAINTS
- 3.4.1 Implementation Language Constraints
- 3.4.2 Input Parameter Constraints
- 3.5 ASSUMPTIONS AND DEPEN DENCIES
- SECTION
- SPECIFIC REQUIREMENT S
- 4.1 E XTERNAL INTERFACES
- 4.1.1 User Interface
- 4.1.2 Hardwar e Interfaces
- 4.1.3 Software Interfaces
- 4.2 M ODES OF OPERATIONS
- 4.3 PERFORMANCE R EQUIREMENTS
- 4.3.1 Time Requirements
- 4.3.2 Space Requirements
- 4.3.3 Accuracy
- 4.4 QUALITY A TTRIBUTES
- 4.4.1 Security
- 4.4.2 Availability
- 4.4.3 Maintainability
- 4.5 OTHER R EQUIREMENTS
- SECTION
- PROJECT PLAN
- 5.1 SEMESTER 6 TH (AND SUMMER VACATIONS )
- 5.2 SEMESTER 7 TH
- 5.3 SEMESTER 8 TH
- 5.4 P ROJECT D URATION Multi - Objective Optimization Using Evolutionary Computation vi
- SECTION
- CONCLUSION
- SECTION
- REFERENCES
- FIGURE 1 P REFERENCE - BASED MULTI - OBJECTIVE OPTIMIZATI ON PROBLEM [1] List of Figures
- FIGURE 2 A FLOWCHART OF THE WO RKING PRINCIPLE OF GA [1]
- FIGURE 3 GUI FOR THE SOFTWARE
- FIGURE 4 DIFFERENT T ECHNIQUES
- FIGURE 5 STOPPING C RITERIA
- FIGURE 6 O UTPUT FUNCTION
- FIGURE 7 O PTIONS FOR DISPLAYIN G G RAPH
Multi - Objective Optimization Using Evolutionary Computation 1
Section 1
Introduction
This document introduces the reader to the research of Multi-objective Optimization using Evolutionary Computation project to be carried by the final year students in Computer & Information Sciences at Pakistan Institute of Engineering and Applied Sciences (PIEAS) in the year 2007 - 2008. This section provide as an introduction to this document by defining the purpose, scope, and intended audience for this document. It also outlines the domain specific definitions, acronyms and abbreviations used in this document. Finally it presents the organizati on of the rest of the document.
1.1 Purpose
This document serves as a monitoring tool for the progress of the project and also as a signification of verifying and testing the finished system against original requirements. Both functional and nonfunctional requirements of the system have been mentioned in this document which is to be developed. This document serves as a starting point for documentation and user manual of the software to be built.
1.2 Scope
The scope of research is to design a user friendly toolbox for multi- objective optimization using evolutionary computation. Decision parameters will be optimized for conflicting objective problems. The objective of this project is to develop toolbox that is used for solving multi- objective optimization problems w ith the help of genetic algorithms. The toolbox will be first tested on the benchmark test problems. After testing the implemented algorithms on test problems, these algorithms will be implemented on the real world application for determining its optimal solution like mobile communication optimization problems.
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1.3 Definitions, Acronyms and Abbreviations
This section presents basic terms and definitions that might be useful for the reader in gaining a better understanding of the rest of the document
1.3.1 Definitions :
Some important definitions relevant to this document are listed below:
o Genetic Algorithm (GA)
A genetic algorithm (GA) is a search technique used in computing to find true or approximate solutions to optimization and search problems.
o Evolutionary Algorit hms (EA)
Evolutionary algorithms are the copy of natural evolutionary principles that represent search and optimization procedures.
o Evolutionary Computation (EC)
Evolutionary computation involves combinatorial optimization problems. It may be loosely recognized by the following criteria: iterative progress, growth or development, population based, guided random search, parallel processing and often biologically inspired.
o Single -objective Optimization
If an optimization problem involves only one objective function for modeling a physical system, then the method of finding the optimal solution is called Single- objective optimization. The objective function is scalar valued i.e. it can be measured by a single number.
o Multi -objective Optimization
If an optimization problem involves more then one objective function then the method of finding one or more optimal solution is called multi objective optimization. The objective function is vector valued i.e. its value is expressed by an n -tuple of numbers. It is the p rocess of optimizing two or more conflicting objectives subject to certain constraints simultaneously.
Multi - Objective Optimization Using Evolutionary Computation 4
1.4 Overview
The rest of the document introduces the reader with the components of research to be conducted. Section 2 describes the objectives of research, gives a brief overview of benchmark problems and different applications in MOEAs. Section 3 describes the product functions, user characteristics, constraints and then the assumptions and dependencies for the software to be developed. In Section 4 all the specific features of the software are illustrated, which involve input and output of the software, its external interfaces and the operations to be performed on the given optimization proble ms and other user related information. Section 5 describes the project plan.
1.5 Project Generic Information
1.5.1 Project Team
Student Name
Bushra Sadia
Project Supervisor
Dr. Muhammad Arif
Project Co- supervisor
Mr. Nauman Shamim
Project Coordinator
Mr. S M Haroon
1.5.2 Project Deliverables
The deliverables of this product include the MATLAB toolbox for solving multi - objective Optimization problems GUI for solving multi- objective Optimization problems. Complete project report that w ill include the details on the development and operation of the software.
Multi - Objective Optimization Using Evolutionary Computation 5
Section 2
Objectives of Research
Most real world search and optimization problems naturally involve multiple objectives. The problem cannot be optimized using single-objective optimization when the rest of the objectives are also important. Different solutions may produce different trade-offs among different objectives. As compared to single-objective optimization, multi-objective optimization has also been studied extensively. Many algorithms and case studies exist there involving multi-objectives. The majority of these methods avoid complexity involve in the true multi-objective optimization problem and transform multi-objective into single-objective function using some user defined parameters. In fact, multi-objective optimization is considered as an application of single objective optimization for handling multiple objectives. Thus single objective optimization is a degenerate case of multi-objective optimization and multi -objective optimization is not simply the extension of single- objective optimization. The fundamental difference between the single-objective optimization and the multi-objective optimization lies in the cardinality in the optimal set of solution.
2.1 SOP
If an optimization problem involves only one objective function for modeling a physical system, then the method of finding the optimal solution is called Single- objective optimization. The objective function is scalar valued i.e. it can be measured by a single number.
2.2 Why Evolutionary?
The classical way to solve multi-objective optimization problem is to follow the preference -based approach in which a relative preference vector is used to scalarize multiple objectives. A diagram representat ion of preference-based multi- objective optimization is shown in figure 1.
Multi - Objective Optimization Using Evolutionary Computation 7
The main advantage of genetic algorithms is their flexibility and robustness as a global searc h method. GAs are simple but even then they help to solve complex problems that other techniques might not have ability to accomplish. Genetic algorithms are generally more straightforward to apply, because no restrictions for the definition of the objecti ve function exist. These have shown better search efficiency as compared to tradi tional optimization algorithms. GA is a very effective way of quickly finding a reasonable solution to a complex problem. They are good at doing searches through a large and complex search space in a better manner, and are most effective in a search space for which little is known. If the exact solution is known, but you don t know how to achieve this solution then genetic Algorithms are the best approach to solve this problem.
2.3.1 Some Biological Terms Used in Genetic Algorithms:
As techniques used in GAs are inspired by natural genetics, it uses a lot of biological terms. In this section the correspondence between these biological terms and some common terms used in GAs are review ed as : All living organisms consist of cells. In each cell there is the same set of chromosomes. Each chromosome consists of genes which is the building block of DNA. In GAs, a chromosome is referred to as a string. One or more chromosomes combine together to form the whole genetic package called g enotype. In GAs, the total genetic package is called structure. Basically each gene encodes a trait , or physical property for example color of eyes and possible settings for a trait (e.g. blue, brown) are called alleles. In GAs, alleles are known as feature value. Each gene has its own position in the chromosome termed as locus analogous to string position in GAs. The organism formed by the interaction of genotypes is called phenotyp e which is known as parameter s et in artificial genetic systems. In the process of reproduction , first a recombination or crossover occurs and then genes from the parents form a new chromosome.
Multi - Objective Optimization Using Evolutionary Computation 8
Th e new created offspring can be mutated. Mutation i.e. the ele ments of DNA are a bit change d. These changes are mainly caused by errors in copying genes from parents. The fitness of an organism is measured by success of the organism in its life.
2.3.2 Working of Genetic Algorithms:
The working principle of GAs is very different from the most classical optimization techniques. The flowchart for the working principle is shown in figure 2. It can be divided into the following steps: Initialization Selection Reproduction Termination
Figure 2 A flowchart of the working principle of GA [ 1]
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On the basis of GA, algorithms will be implemented for MOP. After then these algorithms should be tested on following benchmark problems discussed in sub section 2.9.
2.4 MOP
A multi-objective optimization problem has a number of objective functions which are to be maximized or minimized. Any feasible solution including optimal solution must satisfy the number of constraints a specific problem has. The general form of multi - objective optimization problem is:
where,
A solution x is a vector of n decision variables: x = (x 1 , x 2 , x 3 , ., x (^) n ) T Last set of constraints are variable bound i.e. each decision variable xi should take a value within a lower x (^) i (L)^ and an upper x (^) i (U)^ bound. gi (x) and h (^) k (x) are called constraint functions. J and K are inequality and equality constraints respectively. Multi - objective opt imization is sometimes called vector optimization because a vector of objectives is optimized instead of single - objective [1].
2.4.1 Linear and Nonlinear MOP
If all objective functions and constraint functions are linear then the MOP is called the multi - objective linear problem (MOLP) and if the objective functions and the constraint functions are nonlinear then the MOP is called nonlinear multi- objective optimization problem. Solution techniques do not have convergence proofs for nonlinear problems.
2.4.2 Objectives in Multi - objective Optimization
The search space for multiple objective problems is divided into two non overlapping regions; one is optimal and the other is non optimal. The set of optimal solutions have more than one solution in case of conflicting objectives. It is very difficult to prefer one solution over the other in the presence of Pareto-optimal solution without having
( ) ( )
m j kL U i i i
Minimize Maximize f x m M subject to g x j J h x k K x x x i 1, 2,......., n ;
Multi - Objective Optimization Using Evolutionary Computation 11
any further information about the problem. In this case, all Pareto-optimal solutions have equal importance. There are two goals in multi - objective optimization: To find a set of solutions as close as possible to the Pareto - optimal front. To find the set of solution as diverse as possible [1].
2.5 Difference between Single and Multi-objective
Optimization
There are number of fundamental differences between multi-objective optimization and single - objective optimization. There are: Two goals instead of one; Dealing with two search spaces No artificial fix - ups.
2.5.1 Two Goals Instead of One
In single-objective optimization, there is only one goal which is the search space for optimum solution. While the search space may have number of optimal solution, the goal is always to find the global optimum solution. In case of multi-objective optimization, finding a number of local and global optimal solutions is the goal instead of finding one solution. In single-objective optimization algorithms, new solution can be accepted as long as the new solution has a better objective function with respect to the old ones. In multi-objective optimization, it is imp ortant goal to progress towards Pareto-optimal front. However it is also essential to maintain a diverse set of soluti on in the non - dominated front [1].
2.5.2 Dealing with Two Search Spaces
Multi -objective optimization involves two search spaces as compared to single- objective optimization. In single-objective optimization, there is only one search space call ed the decision variable space [1].
2.5.3 No Artificial Fix Ups
As most real world problems are naturally posed multi-objective optimization problems, designers had to innovate different fix ups because of the lack of suitable means of handling multi-objective problems as true multi-objective optimization
Multi - Objective Optimization Using Evolutionary Computation 13
nylon 6 semi - batch reactor H. A. Guvenir and E. Erel (1998) Inventory Clas sification D. Cvetkovic and I. Parmee (1998) Airframe design and conceptual design R. Kumar and P. I. Rockett (1998) Hierarchical learning of pattern spaces C. M. Fonseca and P. J. Fleming (1998b) Gas turbine engine design S. Mardle et al. (1998) Fis hery Modeling B. Paechter et al. (1998) Class timetabling of University T. Bagchi (1999) Multi - criterion flow - chart scheduling S. Garg and S. K. Gupta (1999) Free radical bulk polymerization reactor N. Eklund and M. Embrechts (1999) Color - Efficiency t rade - off in filtered light C. Poloni et al. (2000) Aerodynamic shape design E. Schlemmer et al. (2000) Hydroelectric generator Design P. Di Barba et al. (2000) Electrostatic micro - motor design L. Costa and P. Oliveira (2000) Laminated composite plate design A. J. Blumel et al. (2000) Autopilot controller design K. B. Matthews et al. (2000) Land use planning H. Meunier et al. (2000) Radio Network optimization X. Li et al. (2000) Medical Image reconstruction F. B. Zhou et al. (2000) Continuous cas ting process A. Petrovski and j. McCall (2001) Cancer chemotherapy M. Lahanas et al. (2001) Dose optimization in brachytherapy W. El Moudani et al. (2001) Airlines crew rostering M. Erickson et al. (2001) Groundwater quality management M. Thomopson ( 2001) Analog filter tuning N. Laumanns et al. (2001) Road train design D. Sasaki et al. (2001) Supersonic wing design Ishibuchi et al. (2001) Linguistic rule extraction I. F. Sbalzarini et al. (2001) Microchannel flow optimization H. E. Aguirre et al. (2001) Halftone image generation
After testing the implemented algorithms on test problems, these algorithms should be implemented on the real world application for determining its optimal solution like mobile communication optimization problems.
Multi - Objective Optimization Using Evolutionary Computation 14
2.7 Techn iques for MOEAs:
Techniques to be developed for MOP are: Non - Elitist Mul ti -O bjective Optimization using Evolutionary Algorithms It involves following algorithms: o Vector Evaluated Genetic Algorithms o Vector - Optimized Evolution Strategy o Weight- Based Genetic A lgorithms o Random Weighted GA o Multiple Objectives Genetic Algorithm o Non - dominated Sorting Genetic Algorithm o Niched - Pareto Genetic Algorithm o Predator - Prey Evolution Strategy Elitist Multi - Objective Optimization using Evolutionary Algorithms It involves follo wing algorithms: o Rudolph s Elitist Multi- objective Evolutionary Algorithm o Elitist Non - dominated Sorting Genetic Algorithm o Distance - Based Pareto Genetic Algorithm o Strength Pareto Evolutionary Algorithm o Thermodynamical Genetic Algorithm o Pareto - Archived Evolu tion Strategy o Multi - Objective Messy Genetic Algorithm Constrained Multi - Objective Optimization using Evolutionary Algorithms It involves following algorithms: o Jimenez - Verdegy- Gomez -Skarmeta s Method o Constrained tournament Method o Ray - Tai -Seow s Method
2.8 Perf ormance Metrics
It is very important to compare different implementations of MOEAs in term o f performance. With existence of these different MOEAs, it is important that their performance should be measured on existed test problems. The choice of performanc e metric is very important for the algorithm and the choice of test problem is also important.
