Linear Programming - Optimization Problem 1. Brake Parts, Ltd., a specialist manufacturer makes two types of vintage car brake drums, for Dodge and Cadillac cars. It buys rough castings which are then machined, ground and then polished. Castings for Dodge brake drums cost $5 each and a completed brake drum sells for $7. Castings for Cadillac brake drums cost $8 each and a completed brake drum sells for $11. The time taken (in hours) by each part on the three processes are as follows: Dodge drum Cadillac drum machining 3 1 grinding 2 2 polishing 1 2 Each day there are a total of 90 hours of machining time available, 80 hours of grinding time and 70 hours of polishing time. How many of each type of brake drum should be produced per day to maximise the profit? To answer this question, you are to do the following: a) Formulate the above problem as a linear program with the aim of maximizing the profit to the company. Clearly identify the variables in the problem. b) Graph your linear program and hence identify the point where the optimal solution lies. c) State the solution to the problem. d) What effect would these actions have: If you can increase the capacity of the three processes (machining, grinding, polishing), which process should be given first priority? If the grinding time availability is increased from 80hrs to 90hrs, how will this increase impact the optimal profit? If the current profit contribution for Dodge is changed from $2 to $2.75, and the profit contribution for Cadillac is fixed at $3, will the current optimal solution remain the same? If the current profit contribution for Dodge is changed from $2 to $3.05, and the profit contribution for Cadillac is fixed at $3, what is the new optimal solution and profit? NOTE: consider each option independently. Problem 2. A financial advisor has recently been made redundant by a major bank. He is considering setting up business on his own and, fortunately, has identified two types of part time work that he would be able to do. These are acting as an independent financial advisor and lecturing at a number of local community colleges. He estimates that the profit per hour worked (once travelling and other expenses are taken into account) would be $17 for giving financial advice and $20 for part-time lecturing. There are a number of constraints that must be observed. For example, he is prepared to take on up to 30 hours paid work per week but at least 5 hours of this must be independent financial advising while no more than 15 hours should be lecturing. In addition, he will rent a computer by the hour at a local small business centre until he becomes established. For each hour of giving financial advice he will need to spend 0.75 hours on the computer while each hour of lecturing will need 1 hour of computer time. He can rent the computer for a maximum of 25 hours per week. a) Formulate the above problem as a linear program with the aim of maximizing the profit to the financial advisor. Clearly identify the variables in the problem. b) Graph your linear program and hence identify the point where the optimal solution lies. c) State the solution to the problem and the profit per week which is expected. d) What effect would these actions have: Doing more than 30 paid hours per week; Agreeing to consider more than 15 lecturing hours per week; Renting extra computer time? NOTE: consider each option independently