7. A company manufactures three products (A, B and C) using three machines (X, Y and Z). The time required (in minutes) to produce one unit of product on these machines is as shown below: Product A B C X 1.1 1.8 2.3 Machine Y 0.5 0.9 1.6 Z 3.3 2.7 4.3 For example producing one unit of product A on machine X requires 1.1 minutes. Units of A can also be produced on machines Y and Z: each unit of A produced on machine Y requires 0.5 minutes, each unit of A produced on machine Z requires 3.3 minutes. In the forthcoming production period it is estimated that there will be (at most) 1500 minutes available on machine X, 2700 minutes available on machine Y and 2400 minutes available on machine Z. These machines are very reliable, with only 2% of available time being lost due to machine breakdown. Technological constraints mean that for every 5 units of A produced at most 8 units of B can be produced. The company would like (if possible) to meet forecast demand, which for the forthcoming production period is 3950 units of A, 4400 units of B and 970 units of C. Should it not be possible to meet forecast demand then the most important product is A, with the second most important product being B and the third most important product being C. (a) Formulate this problem as a weighted goal program with linear constraints. Note here that you should not attempt to solve this problem numerically or attempt to simplify any linear equations that you give. (20 marks) (b) How does your formulation change if you use a sequential goal programming approach? Note here that you should not attempt to solve this problem numerically or attempt to simplify any linear equations that you give. 7. A company manufactures three products (A, B and C) using three machines (X, Y and Z). The time required (in minutes) to produce one unit of product on these machines is as shown below: Product A B C X 1.1 1.8 2.3 Machine Y 0.5 0.9 1.6 Z 3.3 2.7 4.3 For example producing one unit of product A on machine X requires 1.1 minutes. Units of A can also be produced on machines Y and Z: each unit of A produced on machine Y requires 0.5 minutes, each unit of A produced on machine Z requires 3.3 minutes. In the forthcoming production period it is estimated that there will be (at most) 1500 minutes available on machine X, 2700 minutes available on machine Y and 2400 minutes available on machine Z. These machines are very reliable, with only 2% of available time being lost due to machine breakdown. Technological constraints mean that for every 5 units of A produced at most 8 units of B can be produced. The company would like (if possible) to meet forecast demand, which for the forthcoming production period is 3950 units of A, 4400 units of B and 970 units of C. Should it not be possible to meet forecast demand then the most important product is A, with the second most important product being B and the third most important product being C. (a) Formulate this problem as a weighted goal program with linear constraints. Note here that you should not attempt to solve this problem numerically or attempt to simplify any linear equations that you give. (20 marks) (b) How does your formulation change if you use a sequential goal programming approach? Note here that you should not attempt to solve this problem numerically or attempt to simplify any linear equations that you give