Strayer MAT 540 Week 8 Assignment and Chapter 4 Problems #14, 19, 20, 22, 36 & 43

Assignment 1. Linear Programming Case Study

Your instructor will assign a linear programming project for this assignment according to the following specifications.

It will be a problem with at least three (3) constraints and at least two (2) decision variables. The problem will be bounded and feasible. It will also have a single optimum solution (in other words, it won’t have alternate optimal solutions). The problem will also include a component that involves sensitivity analysis and the use of the shadow price.

You will be turning in two (2) deliverables, a short writeup of the project and the spreadsheet showing your work.

Writeup.

Your writeup should introduce your solution to the project by describing the problem. Correctly identify what type of problem this is. For example, you should note if the problem is a maximization or minimization problem, as well as identify the resources that constrain the solution. Identify each variable and explain the criteria involved in setting up the model. This should be encapsulated in one (1) or two (2) succinct paragraphs.

After the introductory paragraph, write out the L.P. model for the problem. Include the objective function and all constraints, including any non-negativity constraints. Then, you should present the optimal solution, based on your work in Excel. Explain what the results mean.

Finally, write a paragraph addressing the part of the problem pertaining to sensitivity analysis and shadow price.

Excel.

As previously noted, please set up your problem in Excel and find the solution using Solver. Clearly label the cells in your spreadsheet. You will turn in the entire spreadsheet, showing the setup of the model, and the results.

Click here to view the grading rubric for this assignment.

The Rayhoon Restaurant:
Brayden and Behrad were roommates. They decided to open a restaurant in the small town that they were living. They bought an old home for their new restaurant which they named Rayhoon.
Not knowing the taste of their customers, they decided to serve only two full course meals each night, one with pork and the other with chicken.
They estimated that they would sell a maximum of 60 meals each night. Each chicken dinner requires 15 minutes to prepare, and each pork dinner takes twice as long. There is a total of 20 hours of kitchen staff labor available each day. They thought that they would sell at least 3 chicken dinners for every two pork dinners. They also believe that at least 10 percent of their customers will order pork dinners. The profit from each chicken dinner will be $12, and the profit from a pork dinner will be $16.
Formulate a linear programming model for B & B, that will help them estimate the number of meals they should prepare each night and solve this model by computer.
A. They are considering investing in some advertising to increase the maximum number of meals they serve. They estimate that if they spend $30 per day on a newspaper ad, it will increase the maximum number of meals they serve per day from 60 to 70. Should they make the investment?
B. They are concerned about the reliability of their kitchen staff. They estimate that on some evenings they could have a staff reduction of as much as 5 hours. How would this affect their profit level?
C. The final question they would like to explore is raising the price of the chicken dinner. Brayden has suggested a price increase that will increase profit for the chicken dinner to $14. Would this be acceptable, and how much additional profit would be realized?

MAT540

Week 8 Homework

Chapter 4

14. Grafton Metalworks Company produces metal alloys from six different ores it mines. The company has an order from a customer to produce an alloy that contains four metals according to the following specifications: at least 21% of metal A, no more than 12% of metal B, no more than 7% of metal C and between 30% and 65% of metal D. The proportion of the four metals in each of the six ores and the level of impurities in each ore are provided in the following table:

Ore	Metal (%)	Impurities (%)	Cost/Ton
A	B	C	D
1	19	15	12	14	40	27
2	43	10	25	7	15	25
3	17	0	0	53	30	32
4	20	12	0	18	50	22
5	0	24	10	31	35	20
6	12	18	16	25	29	24

When the metals are processed and refined, the impurities are removed.

The company wants to know the amount of each ore to use per ton of the alloy that will minimize the cost per ton of the alloy.

a. Formulate a linear programming model for this problem.

b. Solve the model by using the computer.

19. As a result of a recently passed bill, a congressman’s district has been allocated $4 million for programs and projects. It is up to the congressman to decide how to distribute the money. The congressman has decided to allocate the money to four ongoing programs because of their importance to his district – a job training program, a parks project, a sanitation project, and a mobile library. However, the congressman wants to distribute the money in a manner that will please the most voters, or, in other words, gain him the most votes in the upcoming election. His staff’s estimates of the number of votes gained per dollar spent for the various programs are as follows.

Program	Votes/ Dollar
Job training	0.02
Parks	0.09
Sanitation	0.06
Mobile library	0.04

In order also to satisfy several local influential citizens who financed his election, he is obligated to observe the following guidelines:

· None of the programs can receive more than 40% of the total allocation.

· The amount allocated to parks cannot exceed the total allocated to both the sanitation project and the mobile library

· The amount allocated to job training must at least equal the amount spent on the sanitation project.

Any money not spent in the district will be returned to the government; therefore, the congressman wants to spend it all. The congressman wants to know the amount to allocate to each program to maximize his votes.

a. Formulate a linear programming model for this problem.

b. Solve the model by using the computer.

20. Anna Broderick is the dietician for the State University football team, and she is attempting to determine a nutritious lunch menu for the team. She has set the following nutritional guidelines for each lunch serving:

· Between 1,500 and 2,000 calories

· At least 5 mg of iron

· At least 20 but no more than 60 g of fat

· At least 30 g of protein

· At least 40 g of carbohydrates

· No more than 30 mg of cholesterol

She selects the menu from seven basic food items, as follows, with the nutritional contributions per pound and the cost as given:

	Calories (per lb.)	Iron (mg/lb.)	Protein (g/lb.)	Carbo-hydrates (g/lb.)	Fat (g/lb.)	Chol-esterol (mg/lb.)	Cost $/lb.
Chicken	520	4.4	17	0	30	180	0.80
Fish	500	3.3	85	0	5	90	3.70
Ground beef	860	0.3	82	0	75	350	2.30
Dried beans	600	3.4	10	30	3	0	0.90
Lettuce	50	0.5	6	0	0	0	0.75
Potatoes	460	2.2	10	70	0	0	0.40
Milk (2%)	240	0.2	16	22	10	20	0.83

The dietician wants to select a menu to meet the nutritional guidelines while minimizing the total cost per serving.

a. Formulate a linear programming model for this problem.

b. Solve the model by using the computer

c. If a serving of each of the food items (other than milk) was limited to no more than a half pound, what effect would this have on the solution?

22. The Cabin Creek Coal (CCC) Company operates three mines in Kentucky and West Virginia, and it supplies coal to four utility power plants along the East Coast. The cost of shipping coal from each mine to each plant, the capacity at each of the three mines and the demand at each plant are shown in the following table:

	Plant
Mine	1	2	3	4	Mine Capacity (tons)
1	$ 7	$ 9	$10	$12	220
2	9	7	8	12	170
3	11	14	5	7	280
Demand (tons)	110	160	90	180

The cost of mining and processing coal is $62 per ton at mine 1, $67 per ton at mine 2, and $75 per ton at mine 3. The percentage of ash and sulfur content per ton of coal at each mine is as follows:

Mine	% Ash	% Sulfur
1	9	6
2	5	4
3	4	3

Each plant has different cleaning equipment. Plant 1 requires that the coal it receives have no more than 6% ash and 5% sulfur; plant 2 coal can have no more than 5% ash and sulfur combined; plant 3 can have no more than 5% ash and 7% sulfur; and plant 4 can have no more than 6% ash and sulfur combined. CCC wabts to determine the amount of coal to produce at each mine and ship to its customers that will minimize its total cost.

a. Formulate a linear programming model for this problem.

b. Solve this model by using the computer.

36. Joe Henderson runs a small metal parts shop. The shop contains three machines – a drill press, a lathe, and a grinder. Joe has three operators, each certified to work on all three machines. However, each operator performs better on some machines than on others. The shop has contracted to do a big job that requires all three machines. The times required by the various operators to perform the required operations on each machine are summarized as follows:

Operator	Drill Press (min)	Lathe (min)	Grinder (min)
1	23	18	35
2	41	30	28
3	25	36	18

Joe Henderson wants to assign one operator to each machine so that the topal operating time for all three operators is minimized.

a. Formulate a linear programming model for this problem.

b. Solve the model by using the computer

c. Joe’s brother, Fred, has asked him to hire his wife, Kelly, who is a machine operator. Kelly can perform each of the three required machine operations in 20 minutes. Should Joe hire his sister-in-law?

43. The Cash and Carry Building Supply Company has received the following order for boards in three lengths:

Length	Order (quantity)
7 ft.	700
9 ft.	1,200
10 ft.	300

The company has 25-foot standard-length boards in stock. Therefore, the standard-length boards must be cut into the lengths necessary to meet order requirements. Naturally, the company wishes to minimize the number of standard-length boards used.

a. Formulate a linear programming model for this problem.

b. Solve the model by using the computer

c. When a board is cut in a specific pattern, the amount of board left over is referred to as “trim-loss.” Reformulate the linear programming model for this problem, assuming that the objective is to minimize trim loss rather than to minimize the total number of boards used, and solve the model. How does this affect the solution?