Engineering Optimization: Theory and Practice, Fourth Edition

Problems 543 subject to Fi(k) xiσi∗ ≤ 1 , i= 1 , 2 ,... , n, k= 1 , 2 ,... , q (2) ∑n i= 1 Fi(k)li xiE∗i sij≤ 1 , j= 1 , 2 ,... ...

9 Dynamic Programming 9.1 Introduction In most practical problems, decisions have to be made sequentially at different points in ...

9.2 Multistage Decision Processes 545 technique suffers from a major drawback, known as thecurse of dimensionality. How- ever, d ...

546 Dynamic Programming seen to be in series and the system has to be treated as a multistage decision problem. Finally, conside ...

9.2 Multistage Decision Processes 547 Figure 9.3 Multistage decision problem (initial value problem). wherexidenotes the vector ...

548 Dynamic Programming 9.2.3 Conversion of a Nonserial System to a Serial System According to the definition, a serial system i ...

9.3 Concept of Suboptimization and Principle of Optimality 549 Figure 9.5 Types of multistage problems: (a)initial value problem ...

550 Dynamic Programming Figure 9.6 Water tank system. Example 9.1 Explain the concept of suboptimization in the context of the d ...

9.3 Concept of Suboptimization and Principle of Optimality 551 Figure 9.7 Suboptimization (principle of optimality). suboptimiza ...

552 Dynamic Programming Consider the first subproblem by starting at the final stage,i=1. If the input to this stages 2 is speci ...

9.4 Computational Procedure in Dynamic Programming 553 the optimum value offi= ∑i k= 1 Rkfor any specified value of the inputsi+ ...

554 Dynamic Programming get the following simplified statement: f 2 ∗(s 3 ) =opt x 2 [R 2 (x 2 , s 3 )+f 1 ∗(s 2 )] (9.19) Thus ...

9.5 Example Illustrating the Calculus Method of Solution 555 Assuming that the suboptimization sequence has been carried on to i ...

556 Dynamic Programming Figure 9.10 Suboptimization of components 1, 2 ,... , ifor various settings of the input state variables ...

9.5 Example Illustrating the Calculus Method of Solution 557 Figure 9.11 Four-bar truss. Memberi pi di= (stressi)li E = Ppili xi ...

558 Dynamic Programming Figure 9.12 Example 9.2 as a four-stage decision problem. sinceδ 1 =s 2 , and x 1 ∗= 1. 5625 s 2 (E 5 ) ...

9.5 Example Illustrating the Calculus Method of Solution 559 Lets 4 be the displacement available for allocation to the first th ...

560 Dynamic Programming the minimum ofF (s 5 , x 4 ) for any specified value of, s 5 , is given by ∂F ∂x 4 = 0. 6 − ( 11. 5596 ) ...

9.6 Example Illustrating the Tabular Method of Solution 561 Table 9.1 Component 3 (Tank) Type of tank Load acting on the tank,s ...

562 Dynamic Programming Table 9.3 Component 1 (Foundation) s 1 =s 2 + Type of foundation s 2 (kgf) R 1 cost ($) Self-weight (kgf ...