a. Consider the task facing the plant manager of a company that manufactures special additives used in mineral processing plants. She must determine the proper amounts of each raw material to purchase for the production of a certain product. Three raw materials are available. Each litre of the finished product must have a combustion point of at least 105C. In addition, the gamma content (which causes hydrocarbon pollution) cannot exceed 6 percent of the volume. The zeta content (which cleans the internal moving parts mineral processing plant) must be at least 12 percent by volume. Each raw material has varying degrees of these characteristics. A B C Raw material characteristics Combustion (C) Gamma content (%) Zeta content (%) 95 85 4 3 140 10 8 20 10 Raw material A costs K6.00 per litre; raw material B, K4.00: and raw material C. K5.00. The plant manager wishes to minimize the cost of raw materials per litre of product. Refer to the POM for Windows output below showing the solution to this problem. Variable Value Reduced Cost Original Val Lower Bound Upper Bound 2632 0 6 A 4.1818 Infinity 5.1667 B .4211 0 4 - Infinity 3158 0 5 3.6 15 Constraint Dual Value Slack/Surplus Original Val Lower Bound Upper Bound Combustion -.0246 0 105 87 109.1667 0 .5263 6 5.4737 Infinity Gamma content Zeta content - 1754 0 12 9.2727 17.3333 Fraction - 1579 0 1 9296 1.2083 i. For minimum cost, how much of each raw material should be used by volume? (3 Points) ii. How much cost per volume will be incurred by producing the output? (2 Points) iii. Is there any Gamma content left over? If so, how much? If not, what is its shadow price (dual value)? Explain what this value means. (3 Points) iv. Interpret the meaning of the lower bound to Zeta content in the Ranging analysis. That is, explain how the solution would change if the amount of Zeta content fell below that lower value. (2 Points) a. Consider the task facing the plant manager of a company that manufactures special additives used in mineral processing plants. She must determine the proper amounts of each raw material to purchase for the production of a certain product. Three raw materials are available. Each litre of the finished product must have a combustion point of at least 105C. In addition, the gamma content (which causes hydrocarbon pollution) cannot exceed 6 percent of the volume. The zeta content (which cleans the internal moving parts mineral processing plant) must be at least 12 percent by volume. Each raw material has varying degrees of these characteristics. A B C Raw material characteristics Combustion (C) Gamma content (%) Zeta content (%) 95 85 4 3 140 10 8 20 10 Raw material A costs K6.00 per litre; raw material B, K4.00: and raw material C. K5.00. The plant manager wishes to minimize the cost of raw materials per litre of product. Refer to the POM for Windows output below showing the solution to this problem. Variable Value Reduced Cost Original Val Lower Bound Upper Bound 2632 0 6 A 4.1818 Infinity 5.1667 B .4211 0 4 - Infinity 3158 0 5 3.6 15 Constraint Dual Value Slack/Surplus Original Val Lower Bound Upper Bound Combustion -.0246 0 105 87 109.1667 0 .5263 6 5.4737 Infinity Gamma content Zeta content - 1754 0 12 9.2727 17.3333 Fraction - 1579 0 1 9296 1.2083 i. For minimum cost, how much of each raw material should be used by volume? (3 Points) ii. How much cost per volume will be incurred by producing the output? (2 Points) iii. Is there any Gamma content left over? If so, how much? If not, what is its shadow price (dual value)? Explain what this value means. (3 Points) iv. Interpret the meaning of the lower bound to Zeta content in the Ranging analysis. That is, explain how the solution would change if the amount of Zeta content fell below that lower value. (2 Points)