# Question

Consider the following project network (as described in Sec. 10.8), where the number over each node is the time required for the corresponding activity. Consider the problem of finding the longest path (the largest total time) through this network from start to finish, since the longest path is the critical path.

(a) What are the stages and states for the dynamic programming formulation of this problem?

(b) Use dynamic programming to solve this problem. However, instead of using the usual tables, show your work graphically. In particular, fill in the values of the various fn*(sn) under the corresponding nodes, and show the resulting optimal arc to traverse out of each node by drawing an arrowhead near the beginning of the arc. Then identify the optimal path (the longest path) by following these arrowheads from the Start node to the Finish node. If there is more than one optimal path, identify them all.

(c) Use dynamic programming to solve this problem by constructing the usual tables for n = 4, n = 3, n = 2, and n = 1.

(a) What are the stages and states for the dynamic programming formulation of this problem?

(b) Use dynamic programming to solve this problem. However, instead of using the usual tables, show your work graphically. In particular, fill in the values of the various fn*(sn) under the corresponding nodes, and show the resulting optimal arc to traverse out of each node by drawing an arrowhead near the beginning of the arc. Then identify the optimal path (the longest path) by following these arrowheads from the Start node to the Finish node. If there is more than one optimal path, identify them all.

(c) Use dynamic programming to solve this problem by constructing the usual tables for n = 4, n = 3, n = 2, and n = 1.

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