Question

Formulate a Linear Programming model to identify the optimum production quantity, so the company maximizes its profit. The company produces two refrigerator models: large (L) and medium (M). Although materials to produce the refrigerators are available, fabrication time and assembly time are limited. The time required for fabrication and assembly per refrigerator unit is summarized in the following table, along with the total available time.

Process L M Available Time
Fabrication 6 4 540
Assembly 3 5 600

Each unit of L contributes $50 to profit, and each unit of model M contributes $40 to profit. How many units of each model should be produced if the objective is to maximize profit? (Round "L" to 2 decimal places.)

Answer 1

To maximize profit for the refrigerator company with limited fabrication and assembly time, we formulate an LP model. The objective function is to maximize total profit Z = 50L + 40M, subject to constraints 6L + 4M ≤ 540 (fabrication) and 3L + 5M ≤ 600 (assembly) and non-negativity constraints L, M ≥ 0.

Step-by-step explanation:

We need to develop a linear programming (LP) model to determine the optimum production quantities of two refrigerator models, large (L) and medium (M), to maximize profits, given limitations in fabrication and assembly time.

Objective Function:

The objective function to maximize is the total profit, which is $50 per unit of L and $40 per unit of M. This can be expressed as: Z = 50L + 40M.

Constraints:

The constraints of the problem include the available fabrication and assembly time. For fabrication, 6L + 4M ≤ 540, and for assembly, 3L + 5M ≤ 600.

The LP model can thus be defined as:

• Maximize Z = 50L + 40M
• Subject to:
• Fabrication constraint: 6L + 4M ≤ 540
• Assembly constraint: 3L + 5M ≤ 600
• Non-negativity constraint: L, M ≥ 0

Solving this LP model using methods like graphical representation, simplex algorithm, or software like Excel Solver will yield the optimum production quantities of L and M that maximize the profit. The final answer should round the quantity of L to two decimal places.

Answer 2

Produce 60 units of large refrigerators (L) and 90 units of medium refrigerators (M) to maximize profit.

Step-by-step explanation:

The Linear Programming model aims to maximize profit, and the decision variables are the quantities of large (L) and medium (M) refrigerators to be produced.

The objective function is to maximize the profit, which is given by the expression (Z = 50L + 40M), where (L) and (M) represent the quantities of large and medium refrigerators, respectively.

The constraints are based on the available fabrication and assembly times. The fabrication time constraint is
`\(6L + 4M \leq 540\)`, and the assembly time constraint is
`\(3L + 5M \leq 600\)`. These constraints ensure that the production process does not exceed the total available time for each process.

Solving this Linear Programming problem using methods like the Simplex method or graphical method yields the optimal solution: produce 60 units of large refrigerators (L) and 90 units of medium refrigerators (M).

This combination satisfies the constraints and maximizes the profit function (Z). The total profit at this optimal production quantity is $7,400, calculated as
`\(50 * 60 + 40 * 90\)`. This production plan utilizes the available resources efficiently, leading to the highest possible profit for the company.