Engineering Optimization: Theory and Practice, Fourth Edition

4.4 Decomposition Principle 203 μj,i≥ 0 , i= 1 , 2 ,... , sj, j= 1 , 2 ,... , p (4.32) Since the extreme points X(k) 1 ,X(k) 2 , ...

204 Linear Programming II: Additional Topics and Extensions FertilizerAshould not contain more than 60% of ammonia andBshould co ...

4.4 Decomposition Principle 205 subject to A 1 X 1 +A 2 X 2 ≤b 0 B 1 X 1 ≤b 1 B 2 X 2 ≤b 2 X 1 ≥ 0 , X 2 ≥ 0 (E 5 ) where X 1 = ...

206 Linear Programming II: Additional Topics and Extensions X( 12 )= ointp S= { 0 0 } X( 22 )= ointp T= { 1000 2000 } X( 32 )= o ...

4.5 Sensitivity or Postoptimality Analysis 207 μ 11 +μ 12 +μ 13 = 1 μ 21 +μ 22 +μ 23 = 1 with μ 11 ≥ 0 ,μ 12 ≥ 0 ,μ 13 ≥ 0 ,μ 21 ...

208 Linear Programming II: Additional Topics and Extensions 4.5.1 Changes in the Right-Hand-Side Constantsbi Suppose that we hav ...

4.5 Sensitivity or Postoptimality Analysis 209 that is, xi= ∑m j= 1 βijbj, i= 1 , 2 ,... , m (4.38) Finally, the change in the ...

210 Linear Programming II: Additional Topics and Extensions SOLUTION Letx 1 , x 2 , x 3 , andx 4 denote the number of units of p ...

4.5 Sensitivity or Postoptimality Analysis 211 Result of pivot operation: x 3 53 0 1 73 154 − 151 0 8003 x 2 301 1 0 − 301 − 150 ...

212 Linear Programming II: Additional Topics and Extensions If the variables are not renumbered, Eq. (4.36) will be applicable f ...

4.5 Sensitivity or Postoptimality Analysis 213 If thecjare changed tocj+ cj, the original optimal solution remains optimal, pro ...

214 Linear Programming II: Additional Topics and Extensions Example 4.7 Find the effect of changingc 3 from − 3 0 to−24 in Examp ...

4.5 Sensitivity or Postoptimality Analysis 215 cost coefficients corresponding to the new variablesxn+kbe denoted byai,n+k, i= 1 ...

216 Linear Programming II: Additional Topics and Extensions the procedure outlined in the preceding section. The second possibil ...

4.5 Sensitivity or Postoptimality Analysis 217 SinceA 1 is changed, we have c 1 =c 1 −πTA 1 = − 45 −(−^223 −^23 ) { 6 10 } =^173 ...

218 Linear Programming II: Additional Topics and Extensions 4.5.5 Addition of Constraints Suppose that we have solved a LP probl ...

4.5 Sensitivity or Postoptimality Analysis 219 Thus Eq. (E 1 ) can be expressed as 2 x 1 + 5 (^403 − 301 x 1 + 301 x 4 + 1501 x ...

220 Linear Programming II: Additional Topics and Extensions 4.6 Transportation Problem This section deals with an important clas ...

4.6 Transportation Problem 221 The problem stated in Eqs. (4.52) to (4.56) was originally formulated and solved by Hitchcock in ...

222 Linear Programming II: Additional Topics and Extensions Figure 4.2 Transportation array. 4.Select a variable to leave from t ...