Page 1 IIT, Bombay Module 5 Design for Reliability and Quality Page 2 IIT, Bombay Module 5 Design for Reliability and Quality IIT, Bombay Lecture 5 Design for Optimization Page 3 IIT, Bombay Module 5 Design for Reliability and Quality IIT, Bombay Lecture 5 Design for Optimization IIT, Bombay Instructional Objectives By the end of this lecture, the students are expected to learn, 1. What is an optimization? 2. Different optimization tools utilized to optimize single and multi-variable problems. Defining Optimum Design In principle, an optimum design means the best or the most suitable of all the feasible conceptual designs. The optimization is the process of maximizing of a desired quantity or the minimizing of an undesired one. For example, optimization is often used by the mechanical engineers to achieve either a minimum manufacturing cost or a maximum component life. The aerospace engineers may wish to minimize the overall weight of an aircraft. The production engineers would like to design the optimum schedules of various machining operations to minimize the idle time of machines and overall job completion time, and so on. Tools for Design Optimization No single optimization method is available for solving all optimization problems in an unique efficient manner. Several optimization methods have been developed till date for solving different types of optimization problems. The optimization methods are generally classified under different groups such as (a) single variable optimization, (b) multi-variable optimization, (c) constrained optimization, (d) specialized optimization, and (e) non-traditional optimization. We would concentrate only single- and multi-variable optimization methods in this lecture and the interested readers can undertake further studies in appropriate references to understand the more advanced optimization techniques. Single Variable Optimization Methods This methods deal with the optimization of a single variable. Figure 5.5.1 depicts various types of extremes that can occur in an objective function, f(x) curve, where x is the design variable. It can be observed from the curve that both the points A and C are mathematical minima. The point A, which is the larger of the two minima, is called a local minimum, while the point C is the Page 4 IIT, Bombay Module 5 Design for Reliability and Quality IIT, Bombay Lecture 5 Design for Optimization IIT, Bombay Instructional Objectives By the end of this lecture, the students are expected to learn, 1. What is an optimization? 2. Different optimization tools utilized to optimize single and multi-variable problems. Defining Optimum Design In principle, an optimum design means the best or the most suitable of all the feasible conceptual designs. The optimization is the process of maximizing of a desired quantity or the minimizing of an undesired one. For example, optimization is often used by the mechanical engineers to achieve either a minimum manufacturing cost or a maximum component life. The aerospace engineers may wish to minimize the overall weight of an aircraft. The production engineers would like to design the optimum schedules of various machining operations to minimize the idle time of machines and overall job completion time, and so on. Tools for Design Optimization No single optimization method is available for solving all optimization problems in an unique efficient manner. Several optimization methods have been developed till date for solving different types of optimization problems. The optimization methods are generally classified under different groups such as (a) single variable optimization, (b) multi-variable optimization, (c) constrained optimization, (d) specialized optimization, and (e) non-traditional optimization. We would concentrate only single- and multi-variable optimization methods in this lecture and the interested readers can undertake further studies in appropriate references to understand the more advanced optimization techniques. Single Variable Optimization Methods This methods deal with the optimization of a single variable. Figure 5.5.1 depicts various types of extremes that can occur in an objective function, f(x) curve, where x is the design variable. It can be observed from the curve that both the points A and C are mathematical minima. The point A, which is the larger of the two minima, is called a local minimum, while the point C is the IIT, Bombay global minimum. The point D is referred to a point of inflection. To check whether the objective function is local minimum or local maximum or the inflection point, the optimality criteria is as shown below. Suppose at point x * , n x ) x ( f ; 0 x ) x ( f 2 2 = ? ? = ? ? (1) If n is odd, x * is an inflection point while if n is even, x * is a local optimum. In the case, x * is a local minima, there are two possibilities. If the second derivative is positive, x * is a local minimum. However, if the second derivative is negative, x * is a local maximum. Figure 5.5.1 Different types of extermes in objective function curve [1]. The Single variable optimization is classified into two categories â€“ direct and gradient based search methods. Direct search methods The direct search methods use only the objective function values to locate the minimum point. The typical direct search methods include uniform search, uniform dichotomous search, sequential dichotomous search, Fibonacci search and golden section search methods. Uniform search In the uniform search method, the trial points are spaced equally over the allowable range of values. Each point is evaluated in turn in an exhaustive search. For example, the designer wants f(x) Page 5 IIT, Bombay Module 5 Design for Reliability and Quality IIT, Bombay Lecture 5 Design for Optimization IIT, Bombay Instructional Objectives By the end of this lecture, the students are expected to learn, 1. What is an optimization? 2. Different optimization tools utilized to optimize single and multi-variable problems. Defining Optimum Design In principle, an optimum design means the best or the most suitable of all the feasible conceptual designs. The optimization is the process of maximizing of a desired quantity or the minimizing of an undesired one. For example, optimization is often used by the mechanical engineers to achieve either a minimum manufacturing cost or a maximum component life. The aerospace engineers may wish to minimize the overall weight of an aircraft. The production engineers would like to design the optimum schedules of various machining operations to minimize the idle time of machines and overall job completion time, and so on. Tools for Design Optimization No single optimization method is available for solving all optimization problems in an unique efficient manner. Several optimization methods have been developed till date for solving different types of optimization problems. The optimization methods are generally classified under different groups such as (a) single variable optimization, (b) multi-variable optimization, (c) constrained optimization, (d) specialized optimization, and (e) non-traditional optimization. We would concentrate only single- and multi-variable optimization methods in this lecture and the interested readers can undertake further studies in appropriate references to understand the more advanced optimization techniques. Single Variable Optimization Methods This methods deal with the optimization of a single variable. Figure 5.5.1 depicts various types of extremes that can occur in an objective function, f(x) curve, where x is the design variable. It can be observed from the curve that both the points A and C are mathematical minima. The point A, which is the larger of the two minima, is called a local minimum, while the point C is the IIT, Bombay global minimum. The point D is referred to a point of inflection. To check whether the objective function is local minimum or local maximum or the inflection point, the optimality criteria is as shown below. Suppose at point x * , n x ) x ( f ; 0 x ) x ( f 2 2 = ? ? = ? ? (1) If n is odd, x * is an inflection point while if n is even, x * is a local optimum. In the case, x * is a local minima, there are two possibilities. If the second derivative is positive, x * is a local minimum. However, if the second derivative is negative, x * is a local maximum. Figure 5.5.1 Different types of extermes in objective function curve [1]. The Single variable optimization is classified into two categories â€“ direct and gradient based search methods. Direct search methods The direct search methods use only the objective function values to locate the minimum point. The typical direct search methods include uniform search, uniform dichotomous search, sequential dichotomous search, Fibonacci search and golden section search methods. Uniform search In the uniform search method, the trial points are spaced equally over the allowable range of values. Each point is evaluated in turn in an exhaustive search. For example, the designer wants f(x) IIT, Bombay to optimize the yield of a chemical reaction by varying the concentration of a catalyst, x and x lies over the range 0 to 10. Four experiments are available, and the same are distributed at equivalent spacing over the range 10 L = (Figure 5.5.2). This divides L into intervals each of width L/n+1, where n is the number of experiments. From inspection of the results at the experimental points, we can conclude that the optimum will not lie in the ranges 2 x < or 6 x > . Therefore, we know the optimum will lie in between the range 6 x 2 < < . So, the range of values that require further search is reduced to 40% of the total range with only four experiments. Figure 5.5.2 Example of the uniform search method [1]. Uniform dichotomous search In the uniform dichotomous search method, the experiments are performed in pairs to establish whether the function is increasing or decreasing. Since the search procedure is uniform, the experiments will be spaced evenly over the entire range of values. Using the same example as in uniform search method, we place the first pair of experiments at 33 . 3 x = and the other pair at 67 . 6 x = . There will be in total n/2 numbers of pairs for n number of experiments. Hence, the range L has to be divided into 1 ) 2 n ( + intervals each of width )} 1 ) 2 / n {( L + . In this case, the pairs will be 36 . 3 , 30 . 3 x = and .70 6 , 64 . 6 x = . Since b a y y < , as observed in Figure 5.5.3, the region 33 . 3 x 0 < < is excluded from the search region. Furthermore, since d c y y > , the region 0 . 10 x 67 . 6 < < can also be excluded from the region of search. Hence, the optimum would lie in the interval 67 . 6 x 33 . 3 = = . Thus, the range of values that require further search is reduced toRead More

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