Question: Hello, I need help with this question: Question 1. Linear Programming Models [9 Points] A company produces 3 types of industrial solvents (S1, S2, S3)
Hello, I need help with this question:
Question 1. Linear Programming Models [9 Points]
A company produces 3 types of industrial solvents (S1, S2, S3) by blending 3 grades of distillates (D1, D2, D3) in different proportions. The selling price per gallon for solvents S1, S2, and S3 are $5, $4, and $ 3, respectively. The cost price per gallon for distillates D1, D2, and D3 are $4, $3, and $2, respectively. The company is contractually obligated to produce at least 1,000,000 gallons of each solvent. It has 1,000,000 gallons of each distillate available in its inventory. For each solvent, regulations dictate the minimum proportion by volume of distillate D1 required, and the maximum proportion by volume of distillate D3 allowed. Solvent S1 must contain at least 50% of distillate D1, and no more than 30% of distillate D3. Solvent S2 must contain at least 40% of distillate D1, and no more than 40% of distillate D3. Solvent S3 must contain at least 10% of distillate D1, and no more than 60% of distillate D3. There are no restrictions on the proportion of distillate D2. The table below specifies these restrictions and summarizes other relevant information.
| Solvent | S1 | S2 | S3 | Cost | Available |
| Selling price | $5.00 | $4.00 | $3.00 |
|
|
| Minimum Quantity | 1,000,000 | 1,000,000 | 1,000,000 |
|
|
| Min prop of D1 | 0.5 | 0.4 | 0.1 | $4.00 | 1,000,000 |
| Distillate D2 |
|
|
| $3.00 | 1,000,000 |
| Max prop of D3 | 0.3 | 0.4 | 0.6 | $2.00 | 1,000,000 |
The company aims to maximize profits subject to the specified constraints.
Task 1.1. Formulate the problem as a linear programming model. [2 Points]
Decision variables:
Objective Function:
Subject to Constraints:
Task 1.2. Solve the LP model and present optimal solutions. [4 Points]
Maximum profit attainable: $ .
| Solvent | S1 | S2 | S3 |
| Gallons of D1 in |
|
|
|
| Gallons of D2 in |
|
|
|
| Gallons of D3 in |
|
|
|
| Distillate | Available | Used | Surplus |
| D1 | 1,000,000 |
|
|
| D2 | 1,000,000 |
|
|
| D3 | 1,000,000 |
|
|
Task 1.3. Sensitivity analysis. [3 Points]
At most how much should the company be willing to pay to increase the availability of each grade of distillate by 1 unit?
- Pay up to $ per gallon to increase the availability of distillate D1.
Reasoning:
- Pay up to $ per gallon to increase the availability of distillate D2.
Reasoning:
- Pay up to $ per gallon to increase the availability of distillate D3.
Reasoning:
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