Engineering Optimization: Theory and Practice, Fourth Edition

10.8 Generalized Penalty Function Method 623 Figure 10.11 Three-bar truss. A general convergence proof of the penalty function m ...

624 Integer Programming g 3 (X)= 1 − 0. 5 x 1 − 2 x 2 1. 5 x 1 x 2 + √ 2 x 2 x 3 + 1. 319 x 1 x 3 ≥ 0 g 4 (X)= 1 + 0. 5 x 1 − 2 ...

References and Bibliography 625 Step 3: The output of the program is shown below: Optimization terminated. x = 1 1 1 1 1 Referen ...

626 Integer Programming 10.18 C. A. Trauth, Jr., and R. E. Woolsey, Integer linear programming: a study in computa- tional effic ...

Problems 627 10.2 Define the following terms: (a)Cutting plane (b)Gomory’s constraint (c)Mixed-integer programming problem (d)Ad ...

628 Integer Programming 10.3 Maximizef=^4 x 1 +^3 x 2 subject to 3 x 1 + 2 x 2 ≤ 18 x 1 , x 2 ≥ 0 ,integers 10.4 Maximizef=^3 x ...

Problems 629 2.The pipes leading out ofBor ofCshould have total capacities of either 2 or 3. 3.No pipe between any two cities mu ...

630 Integer Programming 10.14 Find the solution of Problem 10.1 using the branch-and-bound method coupled with the graphical met ...

Problems 631 10.20 Find the solution of the following problem using a graphical method based on the generalized penalty function ...

11 Stochastic Programming 11.1 Introduction Stochasticorprobabilistic programmingdeals with situations where some or all of the ...

11.2 Basic Concepts of Probability Theory 633 phenomena are chance dependent and one has to resort to probability theory to desc ...

634 Stochastic Programming the range−∞to∞. Such a quantity (likeX)is called arandom variable. We denote a random variable by a c ...

11.2 Basic Concepts of Probability Theory 635 possible value isT, then FX(x) = 0 for allx < S and FX(x) = 1 for allx>T Pro ...

636 Stochastic Programming Discrete Case. Let us assume that there arentrials in which the random variable Xis observed to take ...

11.2 Basic Concepts of Probability Theory 637 Figure 11.2 Two density functions with same mean. SOLUTION X= ∑^6 i= 0 xipX(xi) = ...

638 Stochastic Programming = 21. 3333 + 29. 3333 = 50. 6666 σX^2 =E[X^2 ]−(E[X])^2 = 05. 6666 −( 6. 6667 )^2 = 6. 2 222 or σX= 2 ...

11.2 Basic Concepts of Probability Theory 639 11.2.5 Jointly Distributed Random Variables When two or more random variables are ...

640 Stochastic Programming exclusive ways of obtaining the points lying betweenxandx+dx. Let the lower and upper limits of y bea ...

11.2 Basic Concepts of Probability Theory 641 Then the joint distribution functionFY(y) by definition, is given by, FY(y) =P(Y≤y ...

642 Stochastic Programming These results can be generalized to the case whenYis a linear function of several random variables. T ...