Question: Consider the (partial) branch and bound tree developed for the following integer programming model. (IP) Min z = 6x: +7x: + 4Xg subject to 4x4

Consider the (partial) branch and bound tree
Consider the (partial) branch and bound tree developed for the following integer programming model. (IP) Min z = 6x: +7x: + 4Xg subject to 4x4 + 6x2 + 9x9 245 8x: +6x2 + 3x3 2 46 *25*3, X20 and integer Node 1 4,65 293 S4 *125 1.58 * =2.77 y 31 X2 2 2 X 52 X 23 16 216 X-5 X =21 X2 12.13 X S2 Node 3 Node 4 Node 5 () Node 2 a) (2.5 pts) Provide the best lower bound for the optimal value of the IP. b) (2.5 pts) Provide the best upper bound for the optimal value of the IP. c) (5 pts) Consider Node 2. What can you say about this node? d) (5 pts) Consider Nodes 3 and 4. What can you say about these nodes? e) (5 pts) Is it true that X3 2 3 in any optimal solution? Why? ) (5 pts) Show that any feasible solution of the IP satisfies the following inequality, x2 + x3 + x: 28 Consider the (partial) branch and bound tree developed for the following integer programming model. (IP) Min z = 6x: +7x: + 4Xg subject to 4x4 + 6x2 + 9x9 245 8x: +6x2 + 3x3 2 46 *25*3, X20 and integer Node 1 4,65 293 S4 *125 1.58 * =2.77 y 31 X2 2 2 X 52 X 23 16 216 X-5 X =21 X2 12.13 X S2 Node 3 Node 4 Node 5 () Node 2 a) (2.5 pts) Provide the best lower bound for the optimal value of the IP. b) (2.5 pts) Provide the best upper bound for the optimal value of the IP. c) (5 pts) Consider Node 2. What can you say about this node? d) (5 pts) Consider Nodes 3 and 4. What can you say about these nodes? e) (5 pts) Is it true that X3 2 3 in any optimal solution? Why? ) (5 pts) Show that any feasible solution of the IP satisfies the following inequality, x2 + x3 + x: 28

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