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Hw2s - ENGRI 1101 Engineering Applications of OR Fall 2008...

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ENGRI 1101 Engineering Applications of OR Fall 2008 Homework 2 Homework 2 Solutions 1. As suggested in the problem, we will introduce nodes [ i, k ], corresponding to k months having elapsed at end of phase i . We have a node [ i, k ] for every value of k at which the i th phase could be completed by hiring some combination of crews for phases up to and including i . The resulting graph is in the picture below. The arcs represent hiring the crew of a particular size to execute a particular phase of the project (the arc costs indicate which crew is being hired). When crews of two different sizes can complete the phase in the same amount of time, the cheaper crew will be hired (otherwise, we would have, for example, two copies of the arc from [0 , 0] to [1 , 1], with costs 2 and 3 million dollars). While providing both arcs is not a mistake, it is clear that only the cheaper of the two could be a part of the shortest path. [0,0] [1,1] [4,8] [3,6] [2,4] [1,2] [2,3] [2,5] [3,7] [3,4] [3,5] [3,8] [4,12] [4,11] [4,9] [4,7] [4,6] [4,5] [4,10] Before we add costs to the arcs, let’s simplify the graph.
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