Engineering Optimization: Theory and Practice, Fourth Edition

11.2 Basic Concepts of Probability Theory 643 11.2.8 Probability Distributions There are several types of probability distributi ...

644 Stochastic Programming Figure 11.4 Standard normal density function. By the same token, the values ofzcorresponding top < ...

11.2 Basic Concepts of Probability Theory 645 Table 11.1 Standard Normal Distribution Table z f (z) φ (z) 0.0 0.398942 0.500000 ...

646 Stochastic Programming =[1−P (Z≤ 1. 667 )]+[1−P (Z≤ 1. 667 )] = 2. 0 − 2 P (Z≤ 1. 667 ) = 2. 0 − 2 ( 0. 9525 )= 0. 095 = 9 . ...

11.3 Stochastic Linear Programming 647 11.2.9 Central Limit Theorem IfX 1 , X 2 ,... , Xnarenmutuallyindependent random variable ...

648 Stochastic Programming wherecj,aij, andbiare random variables andpiare specified probabilities. Notice thatEqs. (11.65) indi ...

11.3 Stochastic Linear Programming 649 wherehiis a new random variable defined as hi= ∑n j= 1 aijxj−bi= ∑n+^1 k= 1 qikyk (11.72) ...

650 Stochastic Programming = ∑n k= 1 [ x^2 kVar (aik)+ 2 ∑n l=k+ 1 xkxlCov (aik, ail) ] +Var(bi)− 2 ∑n k= 1 xkCov (aik, bi) (11. ...

11.3 Stochastic Linear Programming 651 Machining time required per unit (min) Maximum time available Part I Part II per week (mi ...

652 Stochastic Programming As the value of the standard normal variate (si) orresponding to the probability 0.99c is 2.33 (obtai ...

11.4 Stochastic Nonlinear Programming 653 If the standard deviations ofyi,σyi, are small,f(Y)can be approximated by the first tw ...

654 Stochastic Programming By introducing the new variable θ= gj−gj σgj (11.94) and noting that ∫∞ −∞ 1 √ 2 π e−t (^2) / 2 dt= 1 ...

11.4 Stochastic Nonlinear Programming 655 Figure 11.5 Column under compressive load. Mean diameter of the section=(d, σd)=(d, 0. ...

656 Stochastic Programming the objective function can be expressed asf (Y)= 5 W+ 2 d= 5 ρlπdt+ 2 d. Since Y=        ...

11.4 Stochastic Nonlinear Programming 657 The mean values of the constraint functions are given by Eq. (11.92) as g 1 = P πdt −f ...

658 Stochastic Programming ∂g 4 ∂y 6 ∣ ∣ ∣ ∣Y=^1.^0 ∂g 5 ∂yi ∣ ∣ ∣ ∣Y= ∂g 6 ∂yi ∣ ∣ ∣ ∣Y=0 fori=1 to 6 Since the value of the st ...

11.5 Stochastic Geometric Programming 659 Thus the equivalent deterministic optimization problem can be stated as follows: Min- ...

660 Stochastic Programming whereNc is the number of active turns,Qthe number of inactive turns, andρthe weight density. Noting t ...

References and Bibliography 661 References and Bibliography 11.1 E. Parzen,Modern Probability Theory and Its Applications, Wiley ...

662 Stochastic Programming 11.22 S. S. Rao and C. P. Reddy, Mechanism design by chance constrained programming, Mechanism and Ma ...