Engineering Optimization: Theory and Practice, Fourth Edition

Problems 583 (c)The objective function,f=(R 1 +R 2 )R 3 , is separable. (d)A nonserial system can always be converted to an equi ...

584 Dynamic Programming Figure 9.23 Possible paths fromAtoP. Figure 9.24 Three subsystems connected in series. 9.4 The altitude ...

Problems 585 Figure 9.25 Altitudes of the airplane in Example 9.4. To altitude (ft): From altitude (ft): 0 8,000 16,000 24,000 3 ...

586 Dynamic Programming Figure 9.26 Pipe network. For the segmentsBitoCjandCitoDj To nodej From nodei 1 2 3 1 8 12 19 2 9 11 13 ...

Problems 587 Thermal Station,i Return function,Ri(x) 1 2 3 Ri(0) 0 0 0 Ri(1) 2 1 3 Ri(2) 4 5 5 Ri(3) 6 6 6 Find the investment p ...

10 Integer Programming 10.1 Introduction In all the optimization techniques considered so far, the design variables are assumed ...

10.2 Graphical Representation 589 Table 10.1 Integer Programming Methods Linear programming problems Nonlinear programming probl ...

590 Integer Programming Figure 10.1 Graphical solution of the problem stated in Eqs. (10.1). by changing the constraint 3x 1 + 1 ...

10.3 Gomory’s Cutting Plane Method 591 x 2 = 412 , andf= 3412. The truncation of the fractional part of this solution gives x 1 ...

592 Integer Programming adding these additional constraints is to reduce the original feasible convex region ABCDto a new feasib ...

10.3 Gomory’s Cutting Plane Method 593 expressed, from theith equation of Table 10.2, as xi=bi− ∑n j= 1 aijyj (10.2) wherebiis a ...

594 Integer Programming Table 10.3 Optimal Solution with Gomory Constraint Basic Coefficient corresponding to: variables x 1 x 2 ...

10.3 Gomory’s Cutting Plane Method 595 be avoided by using the all-integer integer programming algorithm developed by Gomory [10 ...

596 Integer Programming SOLUTION Step 1: Solve the LP problem by neglecting the integer requirement of the variables xi, i= 1 to ...

10.3 Gomory’s Cutting Plane Method 597 betweenx 1 andx 2 , let us selectx 1 as the basic variable having the largest fractional ...

598 Integer Programming Here cj −arj = 7 12 × 36 11 = 21 11 for columny 1 = 5 12 × 36 1 =15 for columny 2. Since^2111 is minimum ...

10.3 Gomory’s Cutting Plane Method 599 Since onlyarjcorresponding to columny 2 is negative, the pivot element will be− 111 in th ...

600 Integer Programming where a+ij= { aij if aij≥ 0 0 if aij< 0 (10.11) a−ij= { 0 if aij≥ 0 aij if aij< 0 (10.12) Eq. (10. ...

10.3 Gomory’s Cutting Plane Method 601 Thus Eq. (10.13) yields ∑n j= 1 (ai+j+a−ij)yj≤βi− 1 (10.21) Since ∑n j= 1 a−ijyj≤ ∑n j= 1 ...

602 Integer Programming which can be seen to be infeasible. Hence the constraint Eq. (10.26) is added at the end of Table 10.2, ...