Linear Programming (4 Problems)

? Chapter 4 Problem 20 - Diet Mix Problem

Anna Broderick is the dietitian for the State University football team and she is attempting to determine a nutritious lunch menu for the team. She has set the followng nutritional guidlens for each lunch serving:
Between 1500 and 2000 caleries
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 follos, with the nutritional contrib ution per pound and the cost as given:

The dietitian wants to select a menu to meet the nutritonal guidelines while minimizing the total coat 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) were limited to no more than a half pound, what effect would this have on the solution?

Linear programming model showing the objective function and all 8 constraints, with each constraint labeled.

5 Points - A copy and paste of the computer solution window.

? Chapter 4 Problem 24 - Transportation Problem

Brooks City has three consolidated high schools, each with a capacity of 1,300 students. The school board has partitioned the city into five busing districts - north, south, east, west, and central - each with different high school student populations. the three schools are located in the central, west, and south districts. Some students must be bused outside of theri district, and the school board wants to minimize the total bus distance traveled by these students. The average distances from each district to the three schools and the total student population in each district are as follows.

Distance (Miles)
District Central School West School South School Student Population
North 7 12 14 750
South 11 10 0 350
East 8 18 12 800
West 8 0 10 700
Central 0 8 11 600

This is a problem that can be solved using the standard linear programming function in both QM for Windows and Excel, or the special transportation function feature in Excel QM and QM for Windows. To use these features, take a look at the Computer Solution of a Transportation Problem section in chapter 6. Otherwise, you can input the standard linear programming model in Excel or QM for Windows and go from there.

Regardless, I will still require a linear programming model for this problem showing all of the constraints (labeled) and the objective function.

Decision variables - With transportation problems, each path from a source to a destination is a decision variable since you need to determine how much of a resource to move down this path. You have 5 sources and 3 destinations, giving you 5x3 = 15 possible paths.

You have an unbalanced transportation problem where demand (the school) is greater than the supply (students).

7 Points - Linear programming model showing the objective function and all 8 constraints, with each constraint labeled.

3 Points - A copy and paste of the computer solution window.

? Chapter 4 Problem 42 -Blend Problem

A refinery blends four petroleum components into three grades of gasoline-regular, premium, and diesel. The maximum quantities available of each component and the csot per barrel are as follows:

To ensure that each gasoline grade retains certain essential characteristics, the refinery has put limits on thee perncetages of the components in each blend. The limits as well as the selling prices for the various grades are as follows:

The refinery wants to produce at least 3,000 barrels of each grade of gasoline. Management wishes to determine the optimal mix of the four components that will maximize profit.
a. Formulate a linear programming model for this problem.
b. Solve the model by using the computer

Hints are below
First, let's determine the decision variables. The objective here is to determine how much of each component to put into how much of each gasoline grade. Your decision variables should be similar to the decision variables in the textbook blend example. (You will have a total of 12 decision variables.)

Second, we need to determine the objective function. Your objective function will be fashioned in the same manner as the objective function in the textbook blend example. You will have to account for revenue in each barrel of grade you make, but also account for cost in each barrel of component you use. Your objective function will require you to combine the revenue and cost terms together.

Finally, we'll spend some time on the constraints. You will have resource constraints for each component, just like the 3 resource constraints in the example. Then, you will have 6 component specification constraints which are fashioned similarly like those in the example. Please note however that regular has 3 constraints, premium has 1 constraint, and diesel has 2 constraints. This means 6 ratio constraints which you will need to convert to linear inequalities to comply with standard form. You will then finish with requirement constraints for each grade of gasoline.

To recap, you will have 12 decision variables, and 13 constraints.

Linear programming model showing the objective function and all 8 constraints, with each constraint labeled.

A copy and paste of the computer solution window.

? Chapter 4 Problem 54 - Product Distribution Problem

A ship has two cargo holds one fore and one aft. The fore cargo hold has a weight capacity of 70,000 pounds and a volume capacity of 30,000 cubic feet. The aft hold has a weight capacity of 90,000 pounds and a volume capacity of 40,000 cubic feet. The shipowner has contracted to carry loads of packaged beef and grain. The total weight of the available beef is 85,000 pounds; the total weight of the available grain is 100,000 pounds. The volume per mass of the beef is 0.2 cubic foot per pound,and the volume per mass of the grain is 0.4 cubic foot per pound. The profit for shipping beef is $0.35 per pound, and the profit for shipping grain in $0.12 per pound. The shipowner is free to accept all or part of the avalable cargo; he wants to know how much meat and grain to accept to maximize profit.

a. Formulate a linear programming model for this problem
b. Solve the model by using the computer

Although you are dealing with two products, beef and grain, you can place all or part of each in the fore and aft cargo bays. While the objective is to decide how much beef and grain to accept to maximize profit, you still need to determine how they will be stored on the ship. Therefore, you will be actually deciding on how much of each product to store in each cargo bay (4 decision variables). There are 6 explicit constraints in this problem.

- linear programming model showing the objective function and all 6 constraints, with each labeled.

- A copy and paste of the computer solution window.