Optimization For Engineers By Kalyanmoy Deb - Download as PDF File .pdf), Text File .txt) VI Optimization for Engineering Design: Algorithms and Examples. Optimization for. Engineering Design. Algorithms and Examples. SECOND EDITION. KALYANMOY DEB. Department of Mechanical Engineering. Indian Institute. Obtain the benefits of reviewing practice for your lifestyle. Reserve Optimization For Engineering Design: Algorithms And Examples, 2nd Ed By Kalyanmoy Deb.

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Kalyanmoy Deb and Mayank Goyal. Mechanical in an optimal engineering design is to formulate the optimization algorithms to solve engineering design. Optimization for engineering design: Algorithms and examples. New . Deb, K., Bandaru, S., Greiner, D., Gaspar-Cunha, A., and Tutum, C. C. (). An. PDF | On Jan 1, , Abbas M. Abd and others published Engineering Optimization. K. Sasikumar · P. P. Mujumdar Optimization for Engineering Design: Algorithms and Examples. Article. Jan Kalyan Deb. View.

References Davidor, Y. Analogous crossover. Google Scholar Dawkins, R. The Blind Watchmaker. New York: Penguin Books. Dawkins, R.

Citations per year

The Selfish Gene. New York: Oxford University Press.

Optimal scheduling of casting sequence using genetic algorithms. Google Scholar Deb, K. Multi-objective optimization using evolutionary algorithms.

Chichester, UK: Wiley. An efficient constraint handling method for genetic algorithms. Computer Methods in Applied Mechanics and Engineering, 2—4 , — An introduction to genetic algorithms. Sadhana, 24 4 , — Car suspension design for comfort using geneticalgorithms. In Thomas Back Ed. Optimization for engineering design: Algorithms and examples. Delhi: Prentice-Hall. Genetic algorithms in optimal optical filter design. Balagurusamy and B. Sushila Eds.

Complex Systems, 9 — An investigation of niche and species formation in genetic function optimization,Proceedings of the Third International Conference on Genetic Algorithms, pp. Google Scholar Demie, M. Optimization of characteristics of the elasto-damping elements of a passenger car by means of a modified Nelder-Mead method.

International Journal of Vehicle Design, 10 2 , — Google Scholar Duffin, R. Geometric Programming. New York: Wiley. Macro-evolutionary Dynamics: Species,niches,and adaptive peaks. New York: McGraw-Hill. Google Scholar Fonseca, C. Genetic algorithms for multi-objective optimization: Formulation, discussion and generalization. Forrest Ed. Google Scholar Gen, M. Genetic Algorithms and Engineering Design. Google Scholar Goldberg, D. Genetic algorithms in search,optimization,and machine learning.

New York: Addison-Wesley.

A comparison of selection schemes used in genetic algorithms, Foundationsof Genetic Algorithms, edited by G. Rawlins, pp. Genetic algorithms with sharing for multimodal function optimization. Non-stationary function optimization using genetic algorithms with dominance and diploidy. Grefenstette Ed. New Jersey: Lawrence Erlbaum Associates.

Google Scholar Homaifar, A. Constrained optimization via genetic algorithms. Simulation62 4 , — Adaptation in natural and artificial systems. Ann Arbor: University of Michigan Press. Genetic algorithms- search and optimization algorithms that mimic natural evolution and genetics-are potential optimization algorithms and have been applied to many engineering design problems in the recent past. Due to their population approach and parallel processing, these algorithms have been able to obtain global optimal solutions in complex optimization problems.

Simulated annealing method mimics the cooling phenomenon of molten metals. Due to its inherent stochastic approach and availability of a convergence proof, this technique has also been used in many engineering design problems. Chapter 6 also discusses the issue of finding the global optimal solution in a multioptimal problem, where the problem contains a number of local and global optimal solutions and the objective is to find the global optimal solution.

To compare the power of various algorithms, one of the traditional constrained optimization techniques is compared with both the nontraditional optimization techniques in a multioptimal problem.

Preface Xl Some algorithms in Chapter 4 use linear programming methods, which are usually taught in operations research and transportation engineering related courses. Sometimes, linear programming methods are also taught in first or second-year undergraduate engineering courses.

Thus, a detailed discussion of linear programming methods is avoided in this book. Instead, a brief analysis of the simplex search technique of the linear programming method is given in Appendix A. The algorithms are presented in a step-by-step format so that they can be easily understood and coded in a computer language. The working principle of each algorithm is also illustrated by showing hand calculations up to a few iterations of the algorithms on a numerical optimization problem.

The hand calculations provide a better insight into the working of the optimization algorithms. Moreover, in order to compare the efficiency of different algorithms, as far as possible, the same numerical example is chosen for each algorithm. Most of the chapters contain at least one working computer code, implementing optimization algorithms presented in the chapter.

They demonstrate the ease and simplicity with which other optimization algorithms can also be coded. The computer codes presented in the text can be available by sending an e-mail to the author at deb iitk. The primary objective of this book is to introduce different optimization algorithms to students and design engineers, and provide them with a few computer codes for easy understanding. The mathematical treatment of the algorithms is. An elementary knowledge of matrix algebra and calculus would be sufficient for understanding most of the materials presented in the book.

Instructors may find this text useful in explaining optimization algorithms and solving numerical examples in the class, although occasional reference to a more theoretical treatment on optimization may be helpful.

The best way to utilize this book is to begin with Chapter 1. This chapter helps the reader to correlate the design problems to the optimization problems already discussed. Xll Preface Thereafter, subsequent chapters may be read one at a time. To have a better understanding of the algorithms, the reader must follow the steps of the solved exercise problems as they are outlined in the given algorithm.

Then, the progress of each algorithm may be understood by referring to the accompanying figure. Any comments and suggestions for improving the text would be always welcome. Goldberg of the University of illinois at Urbana- Champaign. On a lunch table, he once made me understand that probably the most effective way of communicating one's ideas is through books.

That discussion certainly motivated me in taking up this project. Professor Ghosh looked at the handout and encouraged me to revise it in the form of a textbook.

Although it took me about an year-and-half to execute that revision, I have enjoyed every bit of my experience. Most of the algorithms presented in this text are collected from various books and research papers related to engineering design optimization. My sincere thanks and appreciation are due to all authors of those books and papers. I have been particularly influenced by the concise and algorithmic approach adopted in the book entitled 'Engineering Optimization-Methods and Applications' by G.

Reklaitis, A. Ravindran, and K. Many algorithms presented here are modified abstractions from that book. I am also grateful to Professor Brahma Deo and Dr. Partha Chakroborty for their valuable comments which significantly improved the contents of this book. My special thanks are due to two of my students N. Srinivas and Ram Bhusan Agrawal for helping me in drawing some of the diagrams and checking some exercise problems.

The computer expertise provided by P. Discussions with Professors David Blank and M. Kapoor on different issues of optimization were also helpful. I am thankful to my colleagues and staff of the CAD Project for their constant support. It would. Subhransu Roy who generously provided me with his text-writing and graph plotting softwares.

India Ltd, and the discussions I had with many design engineers were valuable in writing some of the chapters. The financial assistance provided by the Continuing Education Centre at the Indian Institute of Technology Kanpur to partially compensate for the preparation of the manuscript is gratefully acknowledged.

I also wish to thank the Publishers, Prentice-Hall of India, for the meticulous care they took in processing the book. This book could not have been complete without the loving support and encouragement of my wife, Debjani. Her help in typing a significant portion of the manuscript, in proof-reading, and in preparing the diagrams has always kept me on schedule.

Finally, I take this opportunity to express my gratitude to my parents and my loving affection to my brothers. Algorithms and Examples design is usually more informed about different factors governing that design than anyone else. Thus, as far as the formulation of the optimal problem is concerned, the designer can acquire it with some practice.

However, every designer should know a few aspects of the formulation procedure which would help him or her to choose a proper optimization algorithm for the chosen optimal design problem.

This requires a knowledge about the working principles of different optimization methods. In subsequent chapters, we discuss various optimization methods which would hopefully provide some of that knowledge. In this chapter, we demonstrate the optimal problem formulation procedures of four different engineering optimal design problems. In such an activity, the feasibility of each design solution is first investigated.

Thereafter, an estimate ofthe underlying objective cost, profit, etc. This naive method is often followed because of the time and resource limitations. But, in many cases this method is followed simply because of the lack of knowledge of the existing optimization procedures. But whatever may be the reason, the purpose of either achieving a quality product or of achieving a competing product is never guaranteed to be fulfilled with the above naive method.

Optimization algorithms described in this book provide systematic and efficient ways of creating and comparing new design solutions in order to achieve an optimal design. Since an optimization algorithm requires comparison of a number of design solutions, it is usually time-consuming and computationally expensive. Thus, the optimization procedure must only be used in those problems where there is a definite need of achieving a quality product or a competitive product.

It is expected that the design solution obtained through an optimization procedure is better than other solutions in terms of the chosen objective-cost, efficiency, safety, or others. We begin our discussion with the formulation procedure by mentioning that it is almost impossible to apply a single formulation procedure for all engineering design problems.

Since the objective in a design problem and the associated design parameters vary from product to product, different techniques need to be used in different problems.

Algorithms and Examples 1. A design problem -usually involves many design parameters, of which some are highly sensitive to the proper working of the design. These parameters are called design variables in the parlance of optimization procedures. Other not so important design parameters usually remain fixed or vary in relation to the design variables.

There is no rigid guideline to choose a priori the parameters which may be important in a problem, because one parameter may be more important with respect to minimizing the overall cost of the design, while it may be insignificant with respect to maximizing the life of the product. Thus, the choice of the important parameters in an optimization problem largely depends on the user. However, it is important to understand that the efficiency and speed of optimization algorithms depend, to a large extent, on the number of chosen design variables.

In subsequent chapters, we shall discuss certain algorithms which work very efficiently when the number of design variables is small, but do not work that well for a large number of design variables.

Thus, by selectively choosing the design variables, the efficacy of the optimization process can be increased. The first thumb rule of the formulation of an optimization problem is to choose as few design variables as possible. The outcome of that optimization procedure may indicate whether to include more design variables in a revised formulation or to replace some previously considered design variables with new design variables.

The constraints represent some functional relationships among the design variables and other design parameters satisfying certain physical phenomenon and certain resource limitations. Some of these considerations require that the design remain in static or dynamic equilibrium. In many mechanical and civil engineering problems, the constraints are formulated to satisfy stress and deflection limitations. Often, a component needs to be designed in such a way that it can be placed inside a fixed housing, thereby restricting the size of the component.

The nature and number of constraints to be included in the formulation depend on the user. In many algorithms discussed in this book, it is not necessary to have an 5 Introduction explicit mathematical expression of a constraint; but an algorithm or a mechanism to calculate the constraint is mandatory. For example, a mechanical engineering component design problem may involve a constraint to restrain the maximum stress developed anywhere in the component to the strength of the material.

In an irregular-shaped component, there may not exist an exact mathematical expression for the maximum stress developed in the component. A finite element simulation software may be necessary to compute the maximum stress.

But the simulation procedure and the necessary input to the simulator and the output from the simulator must be understood at this step. There are usually two types of constraints that emerge from most considerations.

Either the constraints are of an inequality type or of an equality type. Inequality constraints state that the functional relationships among design variables are either greater than, smaller than, or equal to, a resource value. For example, the stress a x developed anywhere in a component must be smaller than or equal to the allowable strength Sallowable of the material. Mathematically, a x:: Some constraints may be of greater-than-equal- to type: Fortunately, one type of inequality constraints can be transformed into the other type by multiplying both sides by -1 or by interchanging the left and right sides.

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For example, the former constraint can be transformed into a greater-than-equal-to type by either -a x -Sallowable or Sallowable a x. Equality constraints state that the functional relationships should exactly match a resource value. Equality constraints are usually more difficult to handle and, therefore, need to be avoided wherever possible. If the functional relationships of equality constraints are simpler, it may be possible to reduce the number of design variables by using the equality constraints.

In such a case, the equality constraints reduce the complexity of the problem, thereby making it easier for the optimization algorithms to solve the problem.

In Chapter 4, we 6 Optimization for Engineering Design: Algorithms and Examples discuss a number of algorithms which are specially designed to handle equality constraints. Fortunately, in many engineering design optimization problems, it may be possible to relax an equality constraint by including two inequality constraints.

The above deflection equality constraint can be replaced by two constraints: The exact deflection requirement of 5 mm is relaxed by allowing it to deflect anywhere between 4 mm to 6 mm.

Although this formulation is inexact as far as the original requirement is concerned, this flexibility allows a smoother operation of the optimization algorithms. Thus, the second thumb rule in the formulation of optimization problems is that the number of complex equality constraints should be kept as low as possible. The common engineering objectives involve minimization of overall cost of manufacturing, or minimization of overall weight of a component, or maximization of net profit earned, or maximization total life of a product, or others.

Although most of the above objectives can be quantified expressed in a mathematical form , there are some objectives that may not be quantified easily. For example, the aesthetic aspect of a design, ride characteristics of a car suspension design, and reliability of a design are important objectives that one may be interested in maximizing in a design, but the exact mathematical formulation may not be possible. In such a case, usually an approximating mathematical expression is used.

Moreover, in any real-world optimization problem, there could be more than one objective that the designer may want to optimize simultaneously. Even though a few multiobjective optimization algorithms exist in the literature Chankong and Haimes, , they are complex and computationally expensive.

Thus, in most optimal design problem, multiple objectives are avoided. Instead, the designer chooses the most important objective as the objective function of the optimization problem, and the other objectives are included as constraints by restricting their values within a certain range.

For example, consider an optimal truss structure 7 Introduction design problem. The designer may be interested in minimizing the overall weight of the structure and simultaneously be concerned in minimizing the deflection of a specific point in the truss. In the optimal problem formulation, the designer may like to use the weight of the truss as a function of the cross-sections of the members as the objective function and have a constraint with the deflection of the concerned point to be less than a specified limit.

In general, the objective function is not required to be expressed in a mathematical form. A simulation package may be required to evaluate the objective function. But whatever may be the way to evaluate the objective function, it must be clearly understood.

The objective function can be of two types. Either the objective function is to be maximized or it has to be minimized.

Unfortunately, the optimization algorithms are usually written either for minimization problems or for maximization problems. Although in some algorithms, some minor structural changes would enable to perform either minimization or maximization, this requires extensive knowledge of the algorithm. Moreover, if an optimization software is used for the simulation, the modified software needs to be compiled before it can be used for the simulation.

Fortunately, the duality principle helps by allowing the same algorithm to be used for minimization or maximization with a minor change in the objective function instead of a change in the entire algorithm.

Optimization For Engineers By Kalyanmoy Deb

If the algorithm is developed for solving a minimization problem, it can also be used to solve a maximization problem by simply multiplying the objective function by -1 and vice versa.

Thus, the optimum solution remains the same. But once we obtain the optimum solution by minimizing the function F x , we need to calculate the optimal function value of the original function f x by multiplying F x by Certain optimization algorithms do not require this information. In these problems, the constraints completely surround the feasible region. Other problems 8 Optimization for Engineering Design: Algorithms and Examples f x 0.

The maximum point of f x is the same as the minimum point of F x. One way to remedy this situation is to make a guess about the optimal solution and set the minimum and maximum bounds so that the optimal solution lies within these two bounds. On the other hand, if any design variable corresponding to the optimal solution is found to lie on or near the minimum or the maximum bound, the chosen bound may not be correct.

The chosen bound may be readjusted and the optimization algorithm may be simulated again. Although this strategy may seem to work only with linear problems, it has been found useful in many real-world engineering optimization problems.

After the above four tasks are completed, the optimization problem can be mathematically written in a special format, known as non! Denoting the design variables 9 Introduction as a column vector! Minimize f x subject to 9j X?: Note that the above formulation can represent a formulation for maximization problems by using the duality principle and can represent a formulation for problems with the lesser-than-equal- to type inequality constraints by using the techniques described earlier.

However, the optimization algorithm used to solve the above NLP problem depends on the type of the objective function and constraints. It is important to note that the constraints must be written in a way so that the right-side of the inequality or equality sign is zero.

It is worth mentioning here that all the above four tasks are not independent of each other. While formulating the objective function, the designer may decide to include or delete some constraints. In many problems, while the constraints are being formulated, it is necessary to add some artificial design variables, which make the overall formulation easier.

The update of the design variables, the constraints, the objective function, and the variable bounds may continue for a few iterations until the designer is finally satisfied with a reasonable formulation.

Certain possible iterations are shown in Figure 1. We may mention here that this update also depends on the knowledge of the optimization algorithms to be used to solve the problem.

But this requires some practice of using optimization algorithms before such input may be incorporated into the formulation procedure.

Nevertheless, after the optimization problem is formulated, an optimization algorithm is chosen and an optimal solution of the NLP problem is obtained. We now illustrate the above four steps of the formulation procedure in four engineering optimal design problems.

To illustrate the formulation procedure, we formulate four different optimization problems.

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The first is a truss structure design problem, the second is a chemical reactor design problem, the third problem is an interesting transport scheduling problem, and the fourth problem involves the optimal design of a car suspension system. The formulation procedures show different considerations often adopted in formulating engineering optimization problems.

There are two different types of optimization problems in a truss structure design. Firstly, the topology of the truss structure the connectivity of the elements in a truss could be an optimization problem. In this problem, the objective is to find the optimal connectivity of truss elements so as to achieve the minimal cost of materials and construction. Secondly, once the optimal layout of the truss is known, the determination of every element cross-section is another optimization problem.

In this problem, the objective is to find the optimal cross-sections of all elements in order to achieve a minimum cost of materials and construction. Although both these problems attempt to achieve the same objective, the search space and the optimization algorithm required to solve each problem are different.

Here, we discuss the latter problem only. However, there exist certain algorithms which can be used to solve both the above problems simultaneously.

We discuss more about these algorithms in Chapter 6. Consider the seven-bar truss structure shown in Figure 1. The loading is also shown in the figure. Once the connectivity of the truss is given, the cross-sectional area and the material properties of the members are the design parameters.

Let us choose the cross-sectional area of members as the design variables for this problem. There are seven design variables, each specifying the cross-section of a member AI to A 7. The next task is to formulate the constraints. For the given load, we can compute the axial force generated in each element Table 1.

The positive force signifies tensile load and the negative force signifies compressive load acting on the member. Thereafter, Table 1. Algorithms and Examples Pcsco: Realizing that each of these members is connected by pin joints, we can write the Euler buckling conditions for the axial load in members AB and BD Shigley, By using Castigliano's theorem Timoshenko, , we obtain the deflection constraint: Once the constraints are formulated, the next task is to formulate the objective function.

In this problem, we are interested in minimizing the weight of the truss structure. Since we have assumed the same material for all members, the minimization of the total volume of material will yield the same optimal solution as the minimization of the total weight. Thus, we write the objective function as The fourth task is to set some lower and upper bounds for the four cross-sectional areas.

We may choose to make all four areas lie 2 between 10 and mm. Then, while moving downward, the nitrogen and hydrogen present in the feed gas undergo reaction to form ammonia in the presence of a catalyst placed in the reactor.

The production of ammonia depends on the temperature of the feed gas, the temperature at the top of the reactor, the partial pressures of the reactants nitrogen and hydrogen , and the reactor length. The optimal design problem requires to achieve of the optimal reactor length yielding maximum economic returns profits from the reactor operation corresponding to various top temperatures.

First, the decrease in the feed gas temperature must be according to the heat loss to the reaction gas: Secondly, the change in the reaction gas temperature must be according to the heat gain from the feed gas and heat generated in the reaction: The parameter S2 denotes the cross-sectional area of the catalyst.

We use the following parameter values: Note that all the above three differential equations 1. In order to solve these equations, we use the following boundary conditions: The three constraints Equations 1.

We choose to keep the reactor length X as the design variable. The objective of the reactor design problem is to achieve as much profit as possible in the production of ammonia. As in the previous example, this problem also requires numerical solution of coupled differential equations. Thus, the NLP problem is as follows: The solid lines Station B Figure 1. The problem is to determine schedules for the routes such that the transit system provides the best level of service LOS to its passengers, within the resources available.

One of the good measures of the LOS is the amount of time -passengers wait during their ou: On any transit network, passengers wait either to board the vehicle at the station of origin or they wait at a transfer station at which they transfer from one vehicle to another. For example, a passenger wishing to travel from station 18 Optimization for Engineering Design: Algorithms and Examples A to station B in the network shown in Figure 1. Thus, the optimization problem involves finding a schedule of vehicles on all routes arrival and departure times such that the total waiting time for the passengers is minimum.

The design variables in this problem are the arrival time af and departure time df for the k-th vehicle at i-th route. Thus, if in a problem, there are a total of M routes and each route has K vehicles, the total number of design variables is 2MK. In addition, there are a few more artificial variables which we shall discuss later.

The constraints in this problem appear from different service related limitations. Some of these constraints are formulated in the following: Minimum stopping time: A vehicle cannot start as soon as it stops; it has to wait at the stop for a certain period of time, or d7 - a7 2: Smin for all i and k. Maximum stopping time: A vehicle cannot stop for more than a certain period of time even if it means increasing the total transfer time on the network, or d7 - a Maximum allowable transfer time: No passenger on the transit network should have to wait more than a certain period of time T at any transfer station.

This can be enforced by checking all possible differences between departure and arrival times and limiting those values to T. Mathematically, this is difficult to achieve. We sim- plify the formulation of this constraint by introducing a new set of variables btl between the k-th vehicle of the i-th route and the l-th vehicle of the j-th route.

These variables can take either a zero or a one. A value of zero means that the transfer of passengers between those two vehicles is not feasible. A value of one means otherwise. Consider the arrival and departure times of vehicles in two different routes at a particular station, as shown in Figure 1. J Figure 1.Utkarsh is currently reading it Jan 22, In many industrial design activities, optimization is achieved indirectly by comparing a few chosen design solutions and accepting the best solution.

There are usually two types of constraints that emerge from most considerations. Algorithms and Examples where the forces F 1 to F 6 are calculated as follows: Google Scholar Powell, D. In addition, there are a few more artificial variables which we shall discuss later.

There exist a number of other search and optimization algorithms which are comparatively new and are becoming popular in engineering design optimization problems in the recent past. The proposed technique is compared with binarycoded genetic algorithms, Augmented Lagrange multiplier method, Branch and Bound method and Hooke and Jeeves pattern search method. The production of ammonia depends on the temperature of the feed gas, the temperature at the top of the reactor, the partial pressures of the reactants nitrogen and hydrogen , and the reactor length.

The objective is to minimize the foUowing function:

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