Suppose you are going to develop five new nutritious food to sell in Hong Kong. You would like to know which food and how many of them should be produced in order to give the maximum profit for your production. The following table shows the developing cost (set-up cost), units of material required to produce each unit of food, and revenue of each kind of nutritious food. Developing cost Materials required Revenue per unit 1 $40,000 40 units S60 $35,000 20 units $40 Nutritious Food 3 $45,000 50 units $75 4 $60,000 70 units $120 5 $35,000 60 units $100 (a) (7 points) We only have 500,000 units of materials. The production amount should be an integer. Formulate an IP model with binary variables for this problem. (b) (8 points) We have the following restrictions. The total developing cost must be no more than $200,000. At most four products can be produced. We can produce Product 4 only if we have produced either Product 1 or 2. We can produce Product 5 only if we have produced either Product 2 or 3. Suppose now the amount of each kind of food produced must be an integral multiple of 0.5 (Assume selling half unit of a product receives half amount of profit per unit). Reformulate the IP model with binary variables for this problem. (C) (10 points) Suppose we DO NOT consider the developing cost and revenue in the above table and the restrictions. We have a new table showing the profit we can gain from each production level of the nutritious food. At most 6 thousand units of food can be produced. We should determine the production level of each food before we start the production. Formulate a BIP model for this problem. Profit from Nutritious Food (in million) 3 Production level (in thousand) 1 2 5 0 1 SO -SI S2 SO -S0.5 SI $2.5 SO -S1.S SI S3 SO SI $1.5 SO SO.S SI S2 2 3 Suppose you are going to develop five new nutritious food to sell in Hong Kong. You would like to know which food and how many of them should be produced in order to give the maximum profit for your production. The following table shows the developing cost (set-up cost), units of material required to produce each unit of food, and revenue of each kind of nutritious food. Developing cost Materials required Revenue per unit 1 $40,000 40 units S60 $35,000 20 units $40 Nutritious Food 3 $45,000 50 units $75 4 $60,000 70 units $120 5 $35,000 60 units $100 (a) (7 points) We only have 500,000 units of materials. The production amount should be an integer. Formulate an IP model with binary variables for this problem. (b) (8 points) We have the following restrictions. The total developing cost must be no more than $200,000. At most four products can be produced. We can produce Product 4 only if we have produced either Product 1 or 2. We can produce Product 5 only if we have produced either Product 2 or 3. Suppose now the amount of each kind of food produced must be an integral multiple of 0.5 (Assume selling half unit of a product receives half amount of profit per unit). Reformulate the IP model with binary variables for this problem. (C) (10 points) Suppose we DO NOT consider the developing cost and revenue in the above table and the restrictions. We have a new table showing the profit we can gain from each production level of the nutritious food. At most 6 thousand units of food can be produced. We should determine the production level of each food before we start the production. Formulate a BIP model for this problem. Profit from Nutritious Food (in million) 3 Production level (in thousand) 1 2 5 0 1 SO -SI S2 SO -S0.5 SI $2.5 SO -S1.S SI S3 SO SI $1.5 SO SO.S SI S2 2 3
