4: Suppose that you have been put in charge of managing the clothing supply for a small, newly established space colony. The colony has n colonists each of whom needs at least two pairs of shoes, three jumpsuits, and a hat. You can make each of these products from some combination of cotton, thread, and glue, all of which you need to import. You can also import already made clothing for a fixed cost. For each item that you produce in the factory, you also incur a cost of d per item (a pair of shoes, jumpsuit, and hat each count as an 'item') for the use of electricity in your factory. Write down (but do not solve) a linear program to minimize the cost of acquiring all of the clothing which your colony requires. You may assume that you are allowed to make and import fractional quantities of each of these goods. (20 pts)

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?

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.

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.

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.

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

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.

5. A drug company sells three different formulations of vitamin complex and mineral complex. The first formulation consists entirely of vitamin complex and sells for $1 per unit. The second formulation consists of 3/4 of a unit of vitamin complex and 1/4 of a unit of mineral complex and sells for $2 per unit. The third formulation consists of 1/2 of a unit of each of the complexes and sells for $3 per unit. If the company has 100 units of vitamin complex and 75 units of mineral complex available, how many units of each formulation should the company produce as to maximize profit? Write down the corresponding LP, and solve it by using Theorem 22 and 23.