hmk_2_s

# hmk_2_s - ESE304 - Introduction to Optimization (Homework...

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ESE304 - Introduction to Optimization (Homework #2) Fall Semester, 2010 M. Carchidi –––––––––––––––––––––––––––––––––––– Problem #1 (15 points) Use the simplex method to solve the following LP. max z =2 x 1 x 2 + x 3 (Objective Function) st 3 x 1 + x 2 + x 3 60 (Constraint #1) x 1 x 2 +2 x 3 10 (Constraint #2) x 1 + x 2 x 3 20 (Constraint #3) x 1 ,x 2 ,x 3 0 (Sign Restrictions) This is Problem #3 on page 149 of the text: Operations Research by Wayne L. Winston. –––––––––––––––––––––––––––––––––––– Problem #2 (15 points) Characterize all optimal solutions to the following LP. max z = 8 x 5 (Objective Function) st x 1 + x 3 +3 x 4 +2 x 5 =2 (Constraint #1) x 2 +2 x 3 +4 x 4 +5 x 5 =5 (Constraint #2) x 1 ,x 2 ,x 3 ,x 4 ,x 5 0 (Sign Restrictions) This is Problem #9 on page 154 of the text: Operations Research by Wayne L. Winston. ––––––––––––––––––––––––––––––––––––

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–––––––––––––––––––––––––––––––––––– Problem #3 (15 points) Show that the LP min w = x 1 3 x 2 (Objective Function) st x 1 2 x 2 4 (Constraint #1) x 1 + x 2 3 (Constraint #2) x 1 ,x 2 0 (Sign Restrictions) is unbounded and determine all directions of unboundedness. This is Prob- lem #6 on page 158 of the text: Operations Research by Wayne L. Winston. –––––––––––––––––––––––––––––––––––– Problem #4 (15 points) Show that if ties are broken in favor the lower numbered rows, then cycling occurs when the simplex method is used to solve the following LP. max z = 3 x 1 + x 2 6 x 3 (Objective Function) st 9 x 1 + x 2 9 x 3 2 x 4 0 (Constraint #1) x 1 + 1 3 x 2 2 x 3 1 3 x 4 0 (Constraint #2) 9 x 1 x 2 +9 x 3 +2 x 4 1 (Constraint #3) x 1 ,x 2 ,x 3 ,x 4 0 (Sign Restrictions) This is Problem #4 on page 172 of the text: Operations Research by Wayne L. Winston. –––––––––––––––––––––––––––––––––––– 2
–––––––––––––––––––––––––––––––––––– Problem #5 (15 points) Use both the big M method and the two-phase method to solve the following LP. min w = x 1 + x 2 (Objective Function) st 2 x 1 + x 2 + x 3 =4 (Constraint #1) x 1 + x 2 +2 x 3 =2 (Constraint #2) x 1 ,x 2 ,x 3 0 (Sign Restrictions) This is Problem #5 on page 178 of the text: Operations Research by Wayne L. Winston. –––––––––––––––––––––––––––––––––––– Problem #6 (15 points)

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## This note was uploaded on 03/02/2011 for the course ESE 304 taught by Professor Michaela.carchidi during the Winter '11 term at UPenn.

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hmk_2_s - ESE304 - Introduction to Optimization (Homework...

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