Assignment A1.4 (A Recycling Center): (approx. 25 points) A recycling center is collecting four types of solid waste materials and treats them so that they can be amalgamated into a salable product. (Treating and amalgamating are separate processes). Three different grades of this product can be made (see the first column of Table 1), depending upon the mixture of the materials used. Although there is some flexibility in the mix for each grade, certain quality standards specify the minimum or maximum amount allowed for the proportion of a material in the product grade. (This proportion is the weight of the material expressed as a percentage of the total weight for the product grade). For each of the two higher grades, a fixed percentage is specified for one of the materials. These specifications are given in Table 1 along with the cost of amalgamation and the selling price for each grade. The recycling center collects its solid waste materials from regular sources and is usually able to maintain a steady rate for treating them. Table 2 gives an overview of the quantities available for collection and treatment each week as well as the cost of treatment for each type of material. The recycling center is solely owned by Green World, an organization devoted to dealing with environmental issues. Green World has raised contributions and grants, amounting to 180,000 RMB per week which should be exclusively used to cover the entire treatment cost for the solid waste materials. The management of the recycling center is instructed to divide this money among the materials in such a way that at least half of the amount available of each material is actually collected and treated. These additional restrictions are listed in Table 2. Within the restrictions specified in Tables 1 and 2, the management wants to determine the amount of each product grade to produce and the exact mix of materials to be used for each grade. The objective is to maximize the net weekly profit (total sales income minus total amalgamation Amalgamation Grade Specification Cost per kg Selling Price per kg A 45.00 105.00 Material 1: Not more than 30% of total Material 2: Not less than 40% of total Material 3: Not more than 50% of total Material 4: Exactly 20% of total Material 1: Not more than 50% of total Material 2: Not less than 10% of total Material 4: Exactly 10% of total Material 1: Not more than 70% of total B 32.00 88.00 23.00 60.00 Table 1: Product Data (Prices in RMB). Material Kg per Week Available Additional Restrictions 1 Treatment Cost per kg 24,00 48,00 32,00 40,00 3,000 2,000 4,000 1,000 2 3 4 For each material at least half of the kg per week available should be collected and treated 180,000 RMB per week should be used to treat these materials Table 2: Solid Waste Materials Data for the Recycling Center (Prices in RMB). costs), exclusive of the fixed treatment cost of 180,000 RMB per week that is being covered by gifts and grants. a) Formulate this problem as a constrained optimization problem. b) Discuss whether the problem can be formulated or rewritten as linear optimization problem. Assignment A1.4 (A Recycling Center): (approx. 25 points) A recycling center is collecting four types of solid waste materials and treats them so that they can be amalgamated into a salable product. (Treating and amalgamating are separate processes). Three different grades of this product can be made (see the first column of Table 1), depending upon the mixture of the materials used. Although there is some flexibility in the mix for each grade, certain quality standards specify the minimum or maximum amount allowed for the proportion of a material in the product grade. (This proportion is the weight of the material expressed as a percentage of the total weight for the product grade). For each of the two higher grades, a fixed percentage is specified for one of the materials. These specifications are given in Table 1 along with the cost of amalgamation and the selling price for each grade. The recycling center collects its solid waste materials from regular sources and is usually able to maintain a steady rate for treating them. Table 2 gives an overview of the quantities available for collection and treatment each week as well as the cost of treatment for each type of material. The recycling center is solely owned by Green World, an organization devoted to dealing with environmental issues. Green World has raised contributions and grants, amounting to 180,000 RMB per week which should be exclusively used to cover the entire treatment cost for the solid waste materials. The management of the recycling center is instructed to divide this money among the materials in such a way that at least half of the amount available of each material is actually collected and treated. These additional restrictions are listed in Table 2. Within the restrictions specified in Tables 1 and 2, the management wants to determine the amount of each product grade to produce and the exact mix of materials to be used for each grade. The objective is to maximize the net weekly profit (total sales income minus total amalgamation Amalgamation Grade Specification Cost per kg Selling Price per kg A 45.00 105.00 Material 1: Not more than 30% of total Material 2: Not less than 40% of total Material 3: Not more than 50% of total Material 4: Exactly 20% of total Material 1: Not more than 50% of total Material 2: Not less than 10% of total Material 4: Exactly 10% of total Material 1: Not more than 70% of total B 32.00 88.00 23.00 60.00 Table 1: Product Data (Prices in RMB). Material Kg per Week Available Additional Restrictions 1 Treatment Cost per kg 24,00 48,00 32,00 40,00 3,000 2,000 4,000 1,000 2 3 4 For each material at least half of the kg per week available should be collected and treated 180,000 RMB per week should be used to treat these materials Table 2: Solid Waste Materials Data for the Recycling Center (Prices in RMB). costs), exclusive of the fixed treatment cost of 180,000 RMB per week that is being covered by gifts and grants. a) Formulate this problem as a constrained optimization problem. b) Discuss whether the problem can be formulated or rewritten as linear optimization