Stage 2 General Mathematics Investigation Assignment

FOLIO TASK 1

Topic 1 - Subtopic 1.2: Linear Programming

You will be assessed on the following

• Setting up constraints and objective function
• Graphing feasible region
• Finding optimal solutions
• Considering wastage
• Dealing with optimal solution that is not achievable because only discrete values are allowed
• Effect on optimal solution/feasible region when the original parameters of a linear programming model is changed.
• Usage of technology is required in this task.

Your work needs to include clear and concise communication as well as correct Mathematical representation for Figures and Tables.

It should include a discussion which addresses Limitations, Reasonableness and Assumptions made in the LP modelling task.

Timeline:

You will have class time and homework time in Week 6 to work on this and only homework time in Week 7 and 8.

Your draft is due on _______________at 3:30pm (before the end of the school day)

From the date you receive your drafted work, you will have 1 week for final submission. The final submission date will vary between students.

Stage 2 General Mathematics

AT2: Mathematical Investigation

Introduction:

Tim runs a business called “Muscle Meals”. He is a qualified dietician and chef who provides meals for athletes. He has just won a contract to provide high energy military rations for soldiers in the Australian Army, so that they are prepared for their treacherous days during their desert training sessions.

He has come up with two possible packages: “Clusters Bombs” and “Frosty Frags”.  

Only whole packets of these rationed meals can be sold, and each packet contains the key nutrients:  Protein, Vitamins, Carbohydrates and Fats

A packet of “Cluster Bombs” contains 20 units of Protein, 50 units of Vitamins, 20 units of Carbohydrates and 10 units of Fat.

A packet of “Frosties Frags” contains 100 units of Protein, 30 units of Vitamins, 40 units of Carbohydrates and 30 units of Fat.

A soldier looking to excel during their desert training needs at least 300 units of Protein but no more than 530 units of Vitamins, 380 units of Carbohydrates and 270 units of Fat.

A packet of “Clusters Bombs” and a bag of “Frosty Frags” yield a profit of $18 and $24 respectively for Tim’s business

A soldier is responsible for purchasing suitable food supplies as part of the training exercise.

Mathematical Investigations

Task One

• Construct a table showing the values of the elements for each item and available amounts or minimum requirements for each element.
• State the objective function and constraints.
• Graph the solution to the linear programming situations showing all key points, intersections, and the feasible region.
• Optimise the objective function, explaining what the solution means.
• Explain and calculate whether it is possible for a soldier to rely on one type of breakfast alone to provide the sustenance he needs to excel during training.
• Determine the wastage or oversupply for each of the four types of nutrients.

Explain how you can tell, by looking at the graph, if a soldier is receiving more, less or the exact amount of each nutrient that is required.

  Task Two

Consider the objective function for Tim’s profit:  

Investigate the impact on the optimal solution for different values of, when

Summarise for different values of , that generate optimal solutions. Consider instances where multiple solutions may occur.

Task Three

Consider when the Vitamin constraint equation is

Explore the impact on the optimal solution and feasible region for different values of , when .

Task Four

There has been a major downturn in the supply of ingredients that has pushed up Tim’s costs. As a result, packets of “Clusters Bombs” are now effectively making a loss of $10, although the packets of “Frosty Frags” are still providing a profit of $24.

Due to this downturn Tim’s Business “Muscle Meals” has had to impose some restrictions on the amounts of “Cluster Bombs” and “Frosties” that can be purchased by the soldiers.

• No more than 10 products can be purchased
• The number of bags of Frosty Frags purchased is at least one-third the number of bags of Clusters Bombs purchased.
• At most 7 bags of “Frosty Frags” can be purchased
• At least 2 bags of “Clusters Bombs” must be purchased.