18.5.2 Performance of Some Methods Using Unconstrained Problems As a first numerical performance study, the following four methods were implemented (Elwakeil and Arora, 1996a): covering method, acceptance-rejection method (A-R), con- trolled random search (CRS), and simulated annealing (SA). The numerical tests were performed on 29 unconstrained problems available in the literature. The problems had one to six design variables and only explicit bounds on them. Global solutions for the problems were known. Based on the results, it was concluded that the covering methods were not practical because of their inefficiency for problems with n > 2. The methods required very large com- putational effort. Also, it was difficult to generate a good estimate for the Lipschitz constant that is needed in the algorithm. Both A-R and CRS methods performed better than simulated annealing and the covering method. The fact that the A-R method does not include any stop- ping criterion makes it undesirable for practical applications. The method worked efficiently on test problems because it was stopped upon finding the known global optimum point. The CRS method contains a stopping criterion and is more efficient compared with other methods. An attempt to treat general constraints explicitly in the CRS method was not successful because constraint violations could not be corrected in reasonable computational effort. 18.5.3 Performance of Stochastic Zooming and Domain Elimination Methods In another study, the stochastic zooming method (ZOOM) and the domain elimination (DE) method were also implemented (in addition to CRS and SA), and their performance was eval- uated using 10 mathematical programming test problems (Elwakeil and Arora, 1996a). The test problems included constrained as well as unconstrained problems. Even though most 586 INTRODUCTION TO OPTIMUM DESIGN TABLE 18-1 Characteristics of Global Optimization Methods Method Can solve General Tries to Phases Needs discrete constraints? find all gradients? problems? x*? Covering (D) No No Yes G 1 Zooming (D) Yes 1 Yes No L 1 Generalized No No No G Yes descent (D) Tunneling (D) No Yes No L + G1 Multistart (S) Yes 1 Yes Yes L + G1 Clustering (S) Yes 1 Yes Yes L + G1 Controlled Yes No No L + GNo random (S) Simulated Yes No No G No annealing (S) Acceptance- Yes 1 Yes No G No rejection (S) Stochastic No No No G No integration (S) Genetic (S) Yes No No G No Stochastic Yes 1 Yes No L + G1 zooming (S) Domain Yes 1 Yes Yes L + G1 elimination (S) D: deterministic methods; S: stochastic methods; G: global phase; L: local phase. 1 Depends on the local minimization procedure used. engineering application problems are constrained, it is beneficial to test performance of the algorithms on the unconstrained problems as well. The CRS method could be used only for unconstrained problems. It is noted, however, that the problems classified as unconstrained still include simple bounds on the design variables. The sequential quadratic programming (SQP) method was used in all local searches performed in the ZOOM and DE methods. For ZOOM, the percent reduction required from one local minimum to the next was set arbi- trarily to 15 percent [i.e., g = 0.85 in Eq. (18.4)] for all the test problems. The 10 test problems used in the study had the following characteristics: 4 problems had no constraints, the number of design variables varied from 2 to 15, the total number of general constraints varied from 2 to 29, 2 problems had equality constraints, all problems had 2 or more local minima, 2 problems had 2 global minima and 1 had 4, 1 had global minimum as 0, and 4 had negative global minimum values. To compare performance of different algorithms, each of the test problems was solved five times and averages for the following evaluation criteria were recorded: number of random starting points, number of local searches performed, number of iterations used during the local search, number of local minima found by the method, cost function value of the best local minimum (the global minimum), total number of calls for function evaluations, and CPU time used. Because a random point generator with a random seed was used, the performance of the algorithms changed each time they were executed. The seed is automatically chosen based on the wall clock time. The results differed in the number of local minima found as well as for the other evaluation criteria. DE found the global solution for 9 out of 10 problems, whereas ZOOM found a global minimum for 7 out of 10 problems. In general, DE found more local minima than ZOOM did. This is attributed to the latter requiring a reduction in the cost function value after each local minimum was found. As noted earlier, ZOOM is designed to “tunnel” under some minima with relatively close cost function values. In terms of the number of function evaluations and CPU time, DE was cheaper than ZOOM. This was because the latter performed more local iterations for a particular search without finding a feasible solution. On the other hand, the number of iterations during a local search performed in DE was smaller since it could find a solution in most cases. The CPU time needed by CRS was considerably smaller than that for other methods even with a larger number of function evaluations. This was due to the use of a local search pro- cedure that did not require gradients or line search. However, the method is applicable to only unconstrained problems. Simulated annealing (SA) failed to locate the global minimum for six problems. For the successful problems, the CPU time required was three to four times that for DE. The tests also showed that there was a drastic increase in the computational effort for the problems as the number of design variables increased. Therefore, that implementation of SA was consid- ered inefficient and unreliable compared with that of both DE and ZOOM. It is noted that the SA may be more suitable for problems with discrete variables only. 18.5.4 Global Optimization of Structural Design Problems The DE and ZOOM methods have also been applied to structural design problems to find global solutions for them (Elwakeil and Arora, 1996b). In this section, we summarize and discuss results of that study which used the following 6 structures: a 10-bar cantilever truss, a 200-bar truss, a 1-bay–2-story frame, a 2-bay–6-story-frame, a 10-member cantilever frame, and a 200-member frame. These structures have been used previously in the literature to test various algorithms for local minimization (Haug and Arora, 1979). A variety of constraints were imposed on the structures: constraints and other requirements given in the Specifica- tion of the American Institute of Steel Construction (AISC, 1989), Aluminum Association Global Optimization Concepts and Methods for Optimum Design 587 Specifications (AA, 1986), displacement constraints, and constraints on the natural frequency of the structure. Some of the structures were subjected to multiple loading cases. For all prob- lems, the weight of the structure was minimized. Using these 6 structures, 28 test problems were devised by varying the cross-sectional shape of members to hollow circular tubes or I sections, and changing the material from steel to aluminum. The number of design variables varied from 4 to 116, the number of stress constraints varied from 10 to 600, the number of deflection constraints varied from 8 to 675, and the number of local buckling constraints for the members varied from 0 to 72. The total number of general inequality constraints varied from 19 to 1276. These test problems can be considered to be large compared with the ones used in the previous section. Detailed results using DE and ZOOM can be found in Elwakeil (1995). Each problem was solved five times with a different seed for the random number generator. The five runs were then combined and all the optimum solutions found were stored. It was observed that all the six structures tested possessed many local minima. ZOOM found only one local minimum for each problem (except two problems). For most of the problems, the global minimum was found with the first random starting point. Therefore, other local minima were not found since they had a higher cost function value. DE found many local minima for most of the problems except for one problem that turned out to be infeasible. The method did not find all the local minima in one run because of the imposed limit on the number of random starting designs. From the recorded CPU times, it was diffi- cult to draw a general conclusion about the relative efficiency of the two methods because for some problems one method was more efficient and for the remaining the second method was more efficient. However, each of the methods can be useful depending on the require- ments. If only the global minimum is sought, then ZOOM can be used. If all or most of the local minima are wanted, then DE should be used. The zooming method can be used to deter- mine lower-cost practical designs by appropriately selecting the parameter g in Eq. (18.4). Some problems showed only a small difference between weights for the best and the worst local minima. This indicates a flat feasible domain perhaps with small variations in the weight which results in multiple global minima. One of the problems was infeasible because of an unreasonable requirement for the natural frequency to be no less than 22Hz. However, when the constraint was gradually relaxed, a solution was found at a value of 17Hz. It is clear that the designer’s experience and knowledge about the problems, and the design requirements can affect performance of the global optimization algorithms. For example, by setting a correct limit on the number of local minima desired, the computational effort of the domain elimination method can be reduced substantially. For the zooming method, the com- putational effort will be reduced if the parameter g in Eq. (18.4) is selected judiciously. In this regards, it may be possible to develop a strategy to automatically adjust the value of g dynamically during local searches. This will avoid the infeasible problems which constitute a major computational effort in the zooming method. Also, a realistic value for F, the target value for the global minimum cost function, would improve efficiency of the method. Exercises for Chapter 18* Calculate a global minimum point for the following problems. 18.1 (Branin and Hoo, 1972) minimize fxxxxxxxx () =- + Ê Ë ˆ ¯ ++-+ () 421 1 3 44 1 2 1 4 1 2 12 2 2 2 2 . 588 INTRODUCTION TO OPTIMUM DESIGN subject to 18.2 (Lucidi and Piccion, 1989) minimize subject to 18.3 (Walster et al., 1984) minimize subject to where the coefficients (a i , b i ) (i = 1 to 11) are given as follows: (0.1975, 4), (0.1947, 2), (0.1735, 1), (0.16, 0.5), (0.0844, 0.25), (0.0627, 0.1667), (0.0456, 0.125), (0.0342, 0.1), (0.0323, 0.0833), (0.0235, 0.0714), (0.0246, 0.0625). 18.4 (Evtushenko, 1974) minimize subject to 18.5 minimize subject to 1 6 1 2 10 0 12 xx+-£. f xxx xx () =+ 23 2 121 3 2 2 01 1 6 ££ =xi i ; to fx i i i x () =- + Ê Ë ˆ ¯ È Î Í ˘ ˚ ˙ = Â 1 6 2 5 1 6 2 sin p -£ £ =22 14xi i ; to fax bbx bbxx i ii ii i x () =- - ++ È Î Í ˘ ˚ ˙ = Â 1 2 2 2 34 1 11 -££ =10 10 1 5xi i ; to f n xx xx iin i n x () = () +- () + () () [] +- () Ï Ì Ó ¸ ˝ ˛ + = - Â p pp10 1 1 10 1 2 1 2 2 1 2 1 1 sin sin -£ £22 2 x -£ £33 1 x Global Optimization Concepts and Methods for Optimum Design 589 18.6 (Hock and Schittkowski, 1981) minimize subject to 18.7 (Hock and Schittkowski, 1981) minimize subject to 18.8 (Hock and Schittkowski, 1981) minimize subject to xx 21 2 125 0 -≥ xx 12 700 0-≥ + ++ Ê Ë ˆ ¯ -cxx cxx cxx cxx xx cxx 41 3 2 2 51 3 2 3 612 2 712 3 12 81 3 2 2 8673 2000 . exp +-+-+ + () +bxx bx cx cx x cxx 71 4 282 2 12 3 22 4 231 2 2 2 28 106 1. f bx bx bx bx bxx bx xx () =- + + - + - +75 196 11 21 3 31 4 42 512 621 2 . xx 15 10-= xxx 13 2 4 10-+-= xxx 12 2 3 3 30++-= f xxxxxxxxx () =- () +- () +- () +- () 1 2 2 2 2 2 3 2 3 2 4 2 4 2 5 2 0 1 100 0 0 25 6 0 0 5 12 3 ., ,.££ ££ ££xx x uii i =+- () [] =25 50 0 01 1 99 23 ln . ; to f i x ux ii x x () =- + - - () Ê Ë ˆ ¯ 100 1 1 2 3 exp ff i i xx () = () = Â 2 1 99 xx 12 0, ≥ 1 2 1 5 10 0 12 xx+-£. 590 INTRODUCTION TO OPTIMUM DESIGN where the parameters (b i , c i ) (i = 1 to 8) are given as (3.8112E+00, 3.4604E-03), (2.0567E-03, 1.3514E-05), (1.0345E-05, 5.2375E-06), (6.8306E+00, 6.3000E-08), (3.0234E-02, 7.0000E-10), (1.2814E-03, 3.4050E-04), (2.2660E-07, 1.6638E-06), (2.5645E-01, 3.5256E-05). 18.9 (Hock and Schittkowski, 1981) minimize subject to 18.10 (Hock and Schittkowski, 1981) minimize subject to xxx 13 14 15 105 0++- ≥ xxx 10 11 12 85 0++-≥ xxx 789 70 0++-≥ xxx 456 50 0++-≥ xxx 123 60 0++-≥ 071314 33 3 £-+£ = + xx j jj ; to 071414 33 32 £-+£ = +- xx j jj ; to 071314 31 32 £-+£ = +- xx j jj ; to f x Ex x Ex x Ex kkkkkk k x () = () ++- () ++- () () ++++++ = Â 23 10 4 17 10 4 22 15 4 31 31 3 32 32 2 33 33 2 0 4 15 14££ =xi i ; to xxxx 1 2 2 2 3 2 4 2 40 0+++-= xxxx 1234 25 0-≥ fxxxxxxx () =++ () + 14 1 2 3 3 075065 12 ££ ££xx, xx 2 2 1 50 5 55 0- () () ≥ Global Optimization Concepts and Methods for Optimum Design 591 and the bounds are (k = 1 to 4): 8.0 £ x 1 £ 21.0, 43.0 £ x 2 £ 57.0, 3.0 £ x 3 £ 16.0, 0.0 £ x 3k+1 £ 90.0, 0.0 £ x 3k+2 £ 120.0, 0.0 £ x 3k+3 £ 60.0. Find all the local minimum points for the following problems and determine a global minimum point. 18.11 Exercise 18.1 18.12 Exercise 18.2 18.13 Exercise 18.3 18.14 Exercise 18.4 18.15 Exercise 18.5 18.16 Exercise 18.6 18.17 Exercise 18.7 18.18 Exercise 18.8 18.19 Exercise 18.9 18.20 Exercise 18.10 592 INTRODUCTION TO OPTIMUM DESIGN Appendix A Economic Analysis 593 The main body of this textbook describes promising analytical and numerical techniques for engineering design optimization. This appendix departs from the main theme and contains an introduction to engineering decision making based on economic considerations. More detailed treatment of the subject can be found in texts by Grant and coworkers (1982) and Blank and Tarquin (1983). A.1 Time Value of Money Engineering systems are designed to perform specific tasks. Usually many alternative designs can perform the same task. The question is, which one of the alternatives is the best? Several factors such as precedents, social environment, aesthetic, economic, and psychological values can influence the final selection. This appendix considers only the economic factors influencing the selection of an alternative. Economic problems are an integral part of engineering because engineers are sensitive to the direct cost of a design. They must anticipate maintenance and operating costs. Future economic conditions must also be taken into account in the decision-making process. We shall discuss ways to measure the value of money to enable comparisons of alternative designs. The following notation is used: n = number of interest periods, e.g., months, years. i = return per dollar per period; note that i is not the annual interest rate. This shall be further explained in examples. P = value (or sum) of money at the present time, in dollars. S n = final sum after n periods or n payments from the present date, in dollars. R = a series of consecutive, equal, end-of-period amounts of money—payment or receipt; e.g., dollars per month, dollars per year, and so on. It is important to understand the notation and the meaning of the symbols to correctly inter- pret and solve the examples and the exercises. For example, i must be interpreted as the rate of return per dollar per period and not the annual interest rate, and R is the end-of-period amount and not at the beginning of the period. It is important to note that we shall quote the annual interest rate in examples and exercises; using that one can calculate i. A.1.1 Cash Flow Diagrams A cash flow diagram is a pictorial representation of cash receipts and disbursements. These diagrams are helpful in solving problems of economic analysis. Once a correct cash flow diagram for the problem has been drawn, it is a simple matter of using proper interest for- mulas to perform calculations. In this section, we introduce the idea of a cash flow diagram. Figure A-1 gives a cash flow diagram from two points of view—the lender’s and the bor- rower’s. In the diagram, a person has borrowed a sum of $20,000 and promises to pay it back in 1 year, with simple interest paid every month. The annual interest rate is 12 percent. There- fore, $200 is paid as interest at the end of every month and $20,000 principal is paid at the end of the 12th month. Note that vertical lines with arrows pointing downward imply dis- bursements and with arrows pointing upward imply receipts. Also, disbursements are shown below and receipts above the horizontal line. A.1.2 Basic Economic Formulas Consider an investment of P dollars that returns i dollars per dollar per period. The return at the end of the first period is iP, and the original investment increases to (1 + i)P. This sum is reinvested and returns i(1 + i)P at the end of the next period so that the original amount is worth (1 + i) 2 P, and so on. If the process is continued for n periods, an original investment P will increase to the final sum S n , given by (A.1) where spcaf (i, n) is called the single payment compound amount factor. SiP inP n n =+ () = () [] 1 spcaf , 594 Appendix A Economic Analysis First period (first month) $200 interest received every month $20,200 principal and interest received 0 123456789101112 4 5 6 7 8 9 10 11 12 $20,000 loan $20,000 loan received 0 123 $200 interest payment every month $20,200 final payment of principal and interest (A) (B) FIGURE A-1 Cash flow diagrams. (A) Lender’s cash flow. (B) Borrower’s cash flow. A future payment S n made at the end of the nth period has an equivalent present worth P, which can be calculated by inverting Eq. (A.l) as (A.2) where sppwf (i, n) is called the single payment present worth factor. Note from Eq. (A.1) that sppwf (i, n) is the reciprocal of spcaf (i, n). PiS inS n nn =+ () = () [] - 1 sppwf , Appendix A Economic Analysis 595 EXAMPLE A.1 Use of Single Payment Compound Amount Factor Consider an investment of $1000 at an annual interest rate of 9 percent compounded monthly. Calculate the final sum at the end of 2 and 4 years. Solution. For the given annual interest rate, the rate of return per dollar per month is i = 0.09/12 = 0.0075. The final sum on an investment of $1000 at the end of 2 years (n = 24) using the single payment compound amount factor of Eq. (A.1) will be and at the end of 4 years (n = 48) it will become S 48 48 1 0 0075 1000 1 43141 1000 1431 41 =+ ()() = ()() = . .$. S 24 24 0 0075 24 1000 1 0 0075 1000 1 19641 1000 1196 41 = ()() =+ ()() = ()() = spcaf ., $. EXAMPLE A.2 Use of Single Payment Present Worth Factor Consider the case of a person who wants to borrow some money from the bank but can pay back only $10,000 at the end of 2 years. How much can the bank lend if the prevailing annual interest rate is 12 percent compounded monthly? Solution. Using the given rate of interest, the rate of return per dollar per period for this example is i = 0.12/12 = 0.01. Using the single payment present worth factor of Eq. (A.2), the present worth of $10,000 paid at the end of 2 years (n = 24) is given as P = [sppwf (0.01, 24)](10,000): Thus, the bank can lend only $7876.66 at the present time. P =+ ()() = () = - 1 0 01 10 000 0 787566 10 000 7876 66 24 ., .,$. [...]... (0.15, 12) = $848, 602. 09 since uspwf (0.15, 12) = 5.42062 and sppwf (0.15, 12) = 0.18 691 Calculating the annual cost of B, we get ACB = 90 0, 000 crf (0.15, 20) + 40, 000 - 60, 000 sfdf (0.15, 20) = $183, 199 .64 since crf (0.15, 20) = 0.1 597 6 and sfdf (0.15, 20) = 0.0 097 615 Now, we can calculate the present worth of B using a 12-year life span, as PWB = 183, 199 .64 uspwf (0.15, 12) = $99 3, 055.42 Therefore,... textbooks are available and should be consulted (Hohn, 196 4; Franklin, 196 8; Cooper and Steinberg, 197 0; Stewart, 197 3; Bell, 197 5; Strang, 197 6; Jennings, 197 7; Deif, 198 2; Gere and Weaver, 198 3) In addition, most software libraries have subroutines for linear algebra operations which should be directly utilized After reviewing the basic vector and matrix notations, special matrices, determinants,... other matrix (or scalar) results in a zero matrix B.2.2 Vector A matrix of order 1 ¥ n is called a row matrix, or simply row vector Similarly, a matrix of order n ¥ 1 is called a column matrix, or simply column vector A vector with n elements is called an n-component vector, or an n-vector In this text, all vectors are considered to be column vectors and denoted by a lower-case letter in boldface B.2.3... compounded monthly to buy a car and promises to pay it back in 30 monthly installments What is the monthly installment? How much money would be needed to pay off the loan after the 14th installment? A .9 604 A person wants to borrow $10,000 for one year The current interest rate is 9 percent compounded monthly What is the monthly installment to pay back the loan? A couple borrows $200,000 to buy a house... 120, 000 sppwf (0.15, 60) = 600, 000 + 333, 257.30 + 89, 715.43 + 16, 768.46 + 3134.14 + 585. 79 - 27.37 = $1, 043, 433.7 PWB = 90 0, 000 + 40, 000 uspwf (0.15, 60) - 60, 000 sppwf (0.15, 20) + 90 0, 000 uspwf (0.15, 20) - 60, 000 sppwf (0.15, 40) + 90 0, 000 uspwf (0.15, 40) - 60, 000 sppwf (0.15, 60) = 90 0, 000 + 266, 605.84 + 51, 324.23 + 3135 .93 - 13. 69 = $1, 221, 052.30 Therefore, with this procedure... month A.6 On February 1, 195 0 a person went to the bank and promised to pay the bank $20,000 on February 1, 195 1 Based on that promise, how much did the bank lend him at the beginning of every month? The first payment occurred on February 1, 195 0 and the last payment on January 1, 195 1 Assume the interest rate at that time to be 6 percent compounded monthly A.7 A person decides to deposit $50 per month... present worth method of comparison (created by G Jackson) Appendix A Economic Analysis 6 09 Appendix B Vector and Matrix Algebra Matrix and vector notation is compact and useful in describing many numerical methods and derivations Matrix and vector algebra is a basic tool needed in developing methods for the optimum design of systems The solution of linear optimization problems (linear programming) involves... said to be equivalent to another matrix B written as A ~ B if A can be transformed into B by means of one or more elementary row and/or column operations If only row (column) operations are used, we say A is row (column) equivalent to B B.2.8 Scalar Product–Dot Product of Vectors A special case of matrix multiplication of particular interest is the multiplication of a row vector by a column vector If... m ) A = aA = a (AI ( n ) ) = A(aI ( n ) ) = AS( n ) (B.17) B.2.10 Partitioning of Matrices It is often useful to divide vectors and matrices into a smaller group of elements This can be done by partitioning the matrix into smaller rectangular arrays called submatrices and a vector into subvectors For example, consider a matrix A as 1 -6 4 3˘ È 2 Í 2 3 8 -1 -3˙ ˙ A=Í Í 1 -6 2 3 8˙ Í ˙ 5 -2 7˚ ( 4¥5)... Equation (B.24) is called the cofactor expansion for |A| by the ith row; Eq (B.25) is called the cofactor expansion for |A| by the jth column Equations (B.24) and (B.25) can be used to prove Properties 2, 5, 6, and 7 directly It is important to note that Eq (B.24) or (B.25) is difficult to use to calculate the determinant of A These equations require calculation of the cofactors of the elements aij, which . x uii i =+- () [] =25 50 0 01 1 99 23 ln . ; to f i x ux ii x x () =- + - - () Ê Ë ˆ ¯ 100 1 1 2 3 exp ff i i xx () = () = Â 2 1 99 xx 12 0, ≥ 1 2 1 5 10 0 12 xx+-£. 590 INTRODUCTION TO OPTIMUM DESIGN where the. Methods for Optimum Design 5 89 18.6 (Hock and Schittkowski, 198 1) minimize subject to 18.7 (Hock and Schittkowski, 198 1) minimize subject to 18.8 (Hock and Schittkowski, 198 1) minimize subject to xx 21 2 125 0 -≥ xx 12 700. Hoo, 197 2) minimize fxxxxxxxx () =- + Ê Ë ˆ ¯ ++-+ () 421 1 3 44 1 2 1 4 1 2 12 2 2 2 2 . 588 INTRODUCTION TO OPTIMUM DESIGN subject to 18.2 (Lucidi and Piccion, 198 9) minimize subject to 18.3