3: Solve the Klee-Minty problem, as described in the notes/text, for n=3. Use any pivot rule (maximum coefficient pivot may be slow) (15 pts).

Exercises Ex 1: Find a maximum flow from node s to node t. The number associated with each arc indicates the capacity of the arc. Formulate this maximal flow problem as a linear programming problem and use solver to find the optimal solution.

6-9. In intermodal transportation, loaded truck trailers are shipped between railroad terminals on special flatbed carts. Figure 6.38 shows the location of the main railroad terminals in the United States and the existing railroad tracks. The objective is to decide which tracks should be "revitalized" to handle the intermodal traffic. In particular, the Los Angeles (LA) terminal must be linked directly to Chicago (CH) to accommodate expected heavy traffic. Other than that, all the remaining terminals can be linked, directly or indirectly, such that the total length (in miles) of the selected tracks is minimized. Determine the segments of the railroad tracks that must be included in the revitalization program.

6-10. Figure 6.39 gives the mileage of the feasible links connecting nine offshore natural gas wellheads with an inshore delivery point. Because wellhead 1 is the closest to shore, it is equipped with sufficient pumping and storage capacity to pump the output of the remaining eight wells to the delivery point. Determine the minimum pipeline network that links the wellheads to the delivery point.

1. The Better Widgets Company has five production plants in Vancouver, Salem, Fresno, Bakersfield and Imperial. The harpsicords are all shipped to one of six distribution centers in Charlotte, Montreal, Philadelphia, Orlando, Hartford, and Augusta. The transportation costs between plants and distribution centers are as follows: The maximum capacity of the Vancouver plant is 122; the capacity of the Salem plant is 81; the capacity of the Fresno plant is 103; the capacity of the Bakersfield plant is 89; and the capacity of Imperial plant is 68. The minimum required shipments to Charlotte, Montreal, Philadelphia, Orlando, Hartford, and Augusta are 82, 61, 77, 90, 84, and 58, respectively. a. The company's objective is to minimize the cost of transporting its product from its plants to its distribution center while satisfying the above constraints. Write out the objective function and the constraints. b. Find the cost-minimizing solution using EXCEL's Solver. Hand in copies of the answer report and the sensitivity report. c. How do you interpret the shadow prices for the capacity constraints? Would it be profitable to add another unit of capacity to the Vancouver plant if the cost of an additional unit of capacity is $51? Explain your answer with reference to the sensitivity report. d. Explain the value of Salem's shadow price with reference to the changing pattern of shipments if Salem had one more unit of capacity available. e. By how much could the cost of shipping from Salem to Augusta change by without changing your initial answer? Explain your answer with reference to the sensitivity report.

6. Prove that there are infinitely many optimal solutions for the problem in Exercise 5 above. First prove that there are two solutions at extreme points of the constraint set. Then consider the line segment between these solutions/points.

3. Show that the LP problem Minimize: g(x1, x2) = 2x1-5x2 subject to x1+x2 ≥ 2 x1-2x₂ ≤0 x1-2x1 ≤ 1 x1,x2 ≥ 20 is unbounded.

4. Solve each of the LP problems below by sketching the constraint set and applying Theorem

MATH 308 - Assignment 1 1. Find necessary and sufficient conditions for the numbers s and t to make the LP problem Maximize: f(x1, x2) = x1+x2 subject to sx1+tx2 ≤ 1 x1, x2 ≥ 0 a) have an optimal solution. b) be infeasible. c) be unbounded. Prove your answers.

2. Prove or disprove: If a canonical LP problem Maximize: subject to