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).

Questions: a) Which European country did not ship out all its supply? b) Did any U.S. distributor receive shipment from more than one plant in South and Central America? Explain. c) Did any plant in South and Central America receive shipments from more than European country? Explain.

Questions: a) Which Companies had more than one route assigned to them? Explain. b) Did any Companies have no routes assigned to them? Explain.

The following information relates to a project (task times are in weeks). 1) Calculate the Expected Time and Variance for each activity.

Given the following monthly demand for surf broads at Island Waves Surf Shop: Month #Surfboards Forcast

The following table contains the measurements of the exact weight 0f 5-lb bags of?

2. Each year, a local school district contracts with a private bus company for the transportation of students in the primary grades to school. The district's annual payment is equal to $1 times the number of "kid-miles" the bus company carries. (For example, transporting 10 kids two miles each amounts to 20 kid-miles, or transporting 5 kids 4 miles each also equals 20 kid-miles.) The school district has four schools and draws students from four distinct geographic neighborhoods-North, East, West and South. The district's planning department has come up the following figures on the distance from a particular neighborhood to a particular school (distance is in miles): The capacities for Schools 1,2,3 and 4 are 324, 386, 255, and 95, respectively. The number of students in each district which are to be transported to school is 252 in North, 138 in East, 403 in West, and 196 in South. a. The district's objective is to minimize the cost of transporting students to school while satisfying the school capacity and neighborhood constraints. Formulate the linear programming problem. b. Find the cost-minimizing solution using EXCEL's Solver. Hand in copies of the answer report and the sensitivity report. c. Suppose it costs $1 to add another unit school capacity at School 1. Is it desirable to add another unit of capacity at this school? Explain with reference to the sensitivity report. d. Explain the value of School 1's shadow price with reference to the changing pattern of student transportation if School 1 had one more unit of capacity available.

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. A firm has three types of wood products (X₁, X2, X3) each requires (or potentially requires) four inputs: A B, C and D. The profit contribution of each output is $20, $27 and $34.5, respectively. a. The company's objective is to choose the product mix that maximizes its profit. Write out the objective function and the constraints. b. Find the profit-maximizing solution using EXCEL's Solver. Hand in copies of the answer report and the sensitivity report. c. If you were to give it a two-dimensional interpretation (that is, think of a two product model on a graph), how would you interpret the "allowable increase" for the X3 coefficient? d. Would it be worth for the firm to acquire one more unit of input A if it cost $10 to do so? Explain with reference to the shadow price.

1) Assume that your widget manufacturing company has a total annual demand of N widgets per year evenly distributed across the year. Each widget cost $b dollars in material and manufacturing costs to make. Every time you do a production run to make some widgets, you incur a set-up cost of P dollars. Any widgets awaiting sale must be stored and thus incur an average storage fee of c dollars per widget per year. Let x be the size of each production run (i.e. x is the number of widgets per production run). a) Write a cost function C(x) and explain each term in the equation and how it was determined. b) Write down any constraints on the allowable values of x. c) Determine a formula for the value of x that minimizes total annual cost. Show all of your work. d) Prove that your formula actually corresponds to the global minimum cost. e) Write down a formula for the number of production runs per year as a function of x.