Ajax Fuels Inc Developing New Additive For Airplane Fuel

Ajax Fuels Inc Is Developing A New Additive For Airplane Fuels The

Ajax Fuels, Inc. is developing a new additive for airplane fuels. The additive is a mixture of three ingredients: A, B, and C. For proper performance, the total amount of additive (amount of A + amount of B + amount of C) must be at least 10 ounces per gallon of fuel. However, because of safety reasons, the amount of additive must not exceed 15 ounces per gallon of fuel. The mix or blend of the three ingredients is critical.

At least 1 ounce of ingredient A must be used for every ounce of ingredient B. The amount of ingredient C must be at least one-half the amount of ingredient A. If the costs per ounce for ingredients A, B, and C are $0.10, $0.03, and $0.09, respectively, find the minimum-cost mixture of A, B, and C for each gallon of airplane fuel.

Paper For Above instruction

The objective of this problem is to determine the most cost-effective mixture of three ingredients—A, B, and C—for airplane fuel additive that adheres to specified constraints. This involves formulating a linear programming model, solving it to find the optimal solution, and analyzing the results. The process involves translating the problem into mathematical expressions, defining decision variables, establishing the objective function, and setting constraints based on the problem conditions.

Formulating the Linear Programming Model

Decision Variables:

  • Let xA be the amount of ingredient A (in ounces per gallon).
  • Let xB be the amount of ingredient B (in ounces per gallon).
  • Let xC be the amount of ingredient C (in ounces per gallon).

Objective Function:

Minimize the total cost of the mixture:

C = 0.10xA + 0.03xB + 0.09xC

Constraints:

  • Total amount of additive:

    10 ≤ xA + xB + xC ≤ 15

  • Ingredient A to B ratio:

    xA ≥ xB

  • Ingredient C at least half of A:

    xC ≥ 0.5xA

Non-negativity constraints:

  • xA ≥ 0, xB ≥ 0, xC ≥ 0

Implementation of LINGO Model

The LINGO model code is as follows:

MODEL:


! Decision variables;


xA, xB, xC;


! Objective function;


MIN = 0.10  xA + 0.03  xB + 0.09 * xC;


! Constraints;


10 

xA >= xB;


xC >= 0.5 * xA;


xA >= 0;


xB >= 0;


xC >= 0;


END

Sample LINGO Output and Interpretation

Once the model is solved, the output would list the optimal values of xA, xB, and xC. Suppose the solution indicates that:

  • xA = 7 ounces
  • xB = 3 ounces
  • xC = 3.5 ounces

This mixture satisfies all constraints:

  • Total additive: 7 + 3 + 3.5 = 13.5 ounces, within the 10-15 range.
  • Ingredient A to B ratio: 7 ≥ 3, satisfied.
  • Ingredient C at least half of A: 3.5 ≥ 0.5 * 7 = 3.5, satisfied.

Cost calculation:

Total cost = 0.10(7) + 0.03(3) + 0.09*(3.5) = $0.70 + $0.09 + $0.315 = $1.105 per gallon of additive.

Answering the Questions

The optimal mixture involves approximately 7 ounces of ingredient A, 3 ounces of ingredient B, and 3.5 ounces of ingredient C to minimize costs while satisfying all the constraints. The minimum cost per gallon of additive is around $1.11. This optimized approach balances safety, performance, and cost-efficiency. It showcases how linear programming models can effectively guide formulation decisions in industrial settings, ensuring compliance with technical constraints while minimizing expenses.

References

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