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# solutions02 - 1 15.093J/2.098J Optimization Methods...

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1 15.093J/2.098J Optimization Methods Assignment 2 Solutions Exercise 2.1 BT, Exercise 3.17. The initial tableau in Phase I is 5 1 1 3 1 2000 x 6 =2 1 3 0 4 1 1 0 0 x 7 1 2 0 3 1 0 1 0 x 8 =1 1 4 3 0 0 0 0 1 The ﬁnal tableau in Phase I is 0 0 1 0 7 0 2 0 1 x 1 1 3 0 4 1 1 0 0 x 7 =0 0 1 0 7 0 11 0 x 3 01 / 314 / 31 / / 301 / 3 Drive the artiﬁcal variable x 7 out of the basis 0 0 0 0 0 0 1 1 1 x 1 100 17 1 2 3 0 x 2 010 7 0 1 1 0 x 3 0011 1 / / 32 / 3 1 / / 3 The initial tableau in Phase II is 7 000 3 5 x 1 17 1 x 2 010 7 0 x 3 1 / / 3 3 5 8 2 / 7 x 5 1 1 7 / 7 001 x 4 0 1 / 7 x 3 / 3 1 / 3 4 / 3100 The ﬁnal tableau in Phase II is Exercise 2.2 BT, Exercise 3.19. 10 δ 4 1 η 1 α 4010 β γ 3 (a) The current solution is optimal but the current basis is not. Thus the current solution is a degenerate optimal solution. So we have β = 0. Update the tableau using one simplex iteration 10 δ + 2 γ 0002 / 3 3 1 γη x 3 =4 η 3 3 x 4 α + 4 γ 0014 / 3 3 γ x 2 1001 / 3 3 There are multiple optimal solutions, thus δ + 2 3 γ = 0. In addition, we need to have a feasible direction, which requires γ 0. 3 The conditions could be β , δ + 2 3 γ , and γ 0. (b) The optimal cost is −∞ when we have a feasible solution in the current tableau, a nonbasic variable x i with c i < 0and u i = B 1 A i 0 . We need β 0 for problem feasiblity.

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2 The variable x 2 cannot satisfy all the conditions for the −∞ cost. For the variable x 1 , the conditions then can be expressed as follows: α 0, λ 0, and δ< 0. (c) The current solution is feasible if β 0. If the solution is not degenerate then the current solution is deﬁnitely not optimal. Thus a condition could be simply β> 0. Exercise 2.3 BT, Exercise 3.31. (a) The set of all ( b 1 ,b 2 ) is the convex hull of four points A 1 (2 , 1), A 2 (3 , 2), A 3 (1 , 1), and A 4 (1 , 3). (b) There are more than one basis corresponding to a degenerate basic feasible solution. The set of all ( b 1 2 ) is the union of six segments A 1 A 2 , A 1 A 3 , A 1 A 4 , A 2 A 3 , A 2 A 4 ,and A 3 A 4 . (c) If ( b 1 2 ) is one of the four points A i , then there is a basic feasible solution associated with it and this basic feasible solution has three bases.
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solutions02 - 1 15.093J/2.098J Optimization Methods...

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