PLEASE GIVE STEP-BY-STEP REASONING AND CLEAR WRITING. PLEASE ANSWER ALL PARTS

Problem 1 In class, we have discussed the fact that shortest path lengths satisfy Bellman's equations. Thus, one possible method of computing shortest path distances is by solving Bellman's equations directly. Part A Solve the shortest path problem from node 1 to all other nodes for the network given in Figure by using Bellman's equations directly. Note: Bellman's equations can be stated as P = min{Cik +Px}, where P = 0 for origin 1. Part B Solve the same shortest path problem by Dijkstra's algorithm. Part C Suppose I tell you that the shortest distance to each node from origin node 1 is as follows. Node 1: 0 Node 2: 8 Node 3: 7 Node 4: 16 Node 5: 13 Node 6: 14 Node 7: 19 Use the optimality conditions given in class as a check of optimality. Based on your findings, is my solution correct? Explain in terms of optimality conditions. 2 5 8 9 8 1 2 4 8 7 3 3 6 7