Consider the following network, where each number along a link represents the actual distance between the pair of nodes connected by that link. The objective is to find the shortest path from the origin to the destination. (a) For the forward recursion formulation of this problem, how many stages are there, and what arethe states for each stage? (origin) ( 6 f₂(A) = 11 5 B 7 8 6 f₂(C) = 13 f3(D) = 6 6 (Ε f3(E) = 7 (destination) (b) Use forward recursion to solve this problem. However, instead of using the usual tables, show your work graphically. In particular, start with the given network, where the answers already are given for fi (Sn) for four of the nodes; then solve for and fill in f₂ (B) and fi (0). Draw an arrowhead that shows the optimal link to traverse out of each of the latter two nodes. Finally, identify the optimal path by following the arrows from node O onward to node T. (c) Use forward recursion to solve this problem by manually constructing the usual tables for n = 3, n = 2, and n = 1. Some answers are already given, finish the tables by filling in the blank cells.

Consider the following model: (see picture) (c) State the corresponding dual model  (d) Solve the dual problem using Simplex algorithm

Generate the dual simplex iterations for the following problems (using TORA for convenience), and trace the path of the algorithm on the graphical solution space. (a) Minimize z = 2x, + 3xz subject to 2x, + 2x2 = 3 X1 + 2x, 2 1 X1, X2 2 0 (b) Minimize z = 5x, + 6x2 subject to X1 + x2 2 20 4x1 + x, 2 40 X1, X2 2 0 (c) Minimize z = 4x1 + 2x2 subject to X1 + x2 = 10 Зх, — х, 20 X1, X2 2 0 D 4 FIGURE 4.3 Solution space for Problem 435

Over at Duffin House, Marjory Duffin is trying to decide on the size of the print runs for the best-selling new fantasy novel Larry Plotter and the Simplex Method. The following model shows a calculation of the total cost to produce a million copies of the novel, based on an analysis of setup and storage costs c(n) = 0.0008n² - 74n+ 2,000,000 dollars, where n is the print run size (the number of books printed in each run). (a) What would be the effect on cost if the run size was increased from 20,000 to 30,000? A run of size 20,000 would cost $ and a run of 30,000 would cost $ dollars. $30,000 per month would result in a savings of Thus, increasing the run size from $20,000 to (b) Would increasing the run size in further steps of 10,000 result in the same changes in the total cost? Explain. O The cost decreases at a faster and faster rate and then begins to increase. Going from 30,000 to 40,000 decreases the cost by about the same a going from 20,000 to 30,000. O The cost decreases at…