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Unformatted text preview: IE 426 – Quiz #2 Answers 1 BranchandBoundandBoundandBranch (20) Consider the following knapsack problem: z * = max7 x 1 + 9 x 2 + 3 x 3 subject to 3 x 1 + 5 x 2 + 2 x 3 ≤ 6 x 1 , x 2 , x 3 ∈ { , 1 } 1.1 Problem (12 points) Solve the following knapsack problem with branch and bound. Show your branch and bound tree. You get no credit for just writing down an answer. Clearly label the order in which you evaluated the nodes, as your answers to the next problems rely on this order. If you are running short on time, just try to evaluate at least five nodes so you can answer questions 1.2 and 1.3. Answer: The variables are already in the greedy c j /a j order: { 1,2,3 } is the sequence we use to solve the relaxation at each node. Node 1 : Solution : x 1 = 1 , x 2 = 3 / 5 , x 3 = 0 , z * = 7 + 27 / 5 = 12 . 4 Action : Branch on x 2 Node 2: ( x 2 = 0) Solution : x 1 = 1 , x 2 = 0 , x 3 = 1 , z * = 10 Action : Solution feasible: New z L = 10 Node 3: ( x 2 = 1) Solution : x 1 = 1 / 3 , x 2 = 1 , x 3 = 0 , z * = 11 . 33 Action : Branch on x 1 Node 4: ( x 1 = 0 , x 2 = 1) Solution : x 1 = 0 , x 2 = 1 , x 3 = 1 / 2 , z * = 10 . 5 Action : Branch on x 3 Node 5: ( x 1 = 1 , x 2 = 1) Solution : INFEASIBLE. 12.4 11.3 10 10.5 INF 1 2 3 5 4 At the end of five nodes, we could actually quit, since the largest upper bound is z U = 10 . 5 and we have a lower bound z L = 10. Since the objective function can take only integer values in this case, it is true that node 4 (with the z U = 10 . 5 surely can’t yield a solution better than 10. ♦ Branchandbound is based on determining values z L and z U such that z L ≤ z * ≤ z U . The following questions will ask you to deduce values of z L and z U at various points in the branchandbound procedure. 1.2 Problem (4 points) After completing the third node of your branchandbound tree, what values of z L and z U can you deduce? Answer: z L = 10 , z U = 11 . 3333 ♦ 1.3 Problem (4 points) After completing the fifth node of your branchandbound tree, what values of z L and z U can you deduce? IE426 Quiz #2 Answers Prof. Linderoth Answer: z L = 10 , z U = 10 . 5 or z U = 10 since the objective function must be an integer value. ♦ 2 Scheduling Teachers 30 Points Each semester, fifty students must take each of the following three classes: • Advanced Operations Research (AOR), • Network Modeling (NM), • Engineering Economics (EE)....
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This note was uploaded on 02/29/2008 for the course IE 426 taught by Professor Linderoth during the Spring '08 term at Lehigh University .
 Spring '08
 Linderoth

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