Mathematical Methods Unit 4



Further Mathematics Unit 4Graphs and relations module sample modelling or problem-solving task – Aviation business costingsThe modelling or problem-solving task is to be of 2 - 3 hours duration over a period of 1 week.IntroductionIt is a common business practice to regularly review strategic direction and costings, while also taking into consideration economic forecasts in order to evaluate the viability of the company. This task examines a number of aspects of undertaking a problem or this type within a provided scenario. Students could research information relevant to the context, in this case aviation, to determine aspects of the business that would be critical for its viability by referring to websites such as: ; ; 1Using domestic airports as a starting point, examine the fees that are charged for ‘checked in’ and ‘excess’ baggage. Include in your examination the ‘online’ and ‘at terminal’ charges. Create graphical displays that allow baggage costs to be compared. The comparison could be for one airport on its own or between different airports. For a given number of luggage pieces, compare charges for single weight luggage items to that of a grouped weight of luggage items. Vary the total number of luggage pieces and investigate the change in costs. Include a comparison with online charges. Choose a domestic airport and its luggage costs. Explore the use of a linear and non-linear function to model the costs and discuss the strengths and weaknesses of such a model. (As an alternative, airport parking costs could be examined. Include in your investigation the different types of parking (eg. short term, long term, valet, etc.) and the costs for the length of time the vehicle is parked). Part 2A break-even analysis is undertaken for a chosen flight (international or domestic) with consideration given to the revenue and costs involved. Revenue could be based on the number of passengers and cost of a ticket. Costs could include a fixed amount (staff wages, airport taxes, navigation services and fuel) and a cost for food per person. For a chosen flight and a specific seat capacity, define appropriate revenue and cost equations and calculate the break-even point. Many airlines calculate their break-even point based on a certain percentage (75% – 80%) of the seat capacity being sold. By considering different percentage values of total seats sold, investigate the change that occurs in ticket prices and ultimately the profit margin. Investigate the difference made to ticket prices if the chosen percentage is increased (or decreased) by a small margin (eg. 1 – 3 %).When selling tickets for flights, airlines will charge lower prices for early purchases and gradually increase prices as time draws closer to the actual flight. Consider a tiered (piecewise) pricing structure for tickets and investigate its impact on profits.Consider the use of different sized planes based on pre-sold ticket sales. Develop a cost equation for a plane size below a chosen number of pre-sold tickets and another for a plane size above this number of pre-sold tickets. Investigate profit margins and break-even points for your chosen scenario.Part 3The viability of the company will be influenced by the size of its plane fleet. The number of planes purchased (domestic and international) will depend on a number of conditions and constraints. a.Create a series of constraints that will affect the number of planes that will be purchased. You might consider constraints linked to minimum staff and crew, minimum fuel, minimum number of planes, relationship between the number of international jets compared to domestic jets purchased. b.Explore an objective function linked to the minimum number of planes that should be purchased or an objective function linked to minimising the maintenance time/costs. c.Investigate the conditions under which a maximum profit could be calculated (assuming all original constraints did not ‘contain’ the feasible region.d.By varying the initial constraints, investigate changes that can/will occur in the solution initially found to the created objective function of part b. Areas of studyThe following content from the areas of study is addressed through this task.Area of studyContent dot pointsModule – Graphs & relations : Construction & interpretation of graphsModule – Graphs & relations : Linear programming 1, 2, 3, 4 1, 2, 3, 4, 5, 6OutcomesThe following outcomes, key knowledge and key skills are addressed through this task.OutcomeKey knowledge dot pointKey skill dot point11, 2, 3, 4, 5, 6, 7, 8, 91, 3, 4, 5, 6, 7, 8, 921, 2, 3, 41, 2, 3, 431, 2, 3, 4, 52, 3, 6, 8, 9, 10, 11 ................
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