Question

A company manufactures two products A and B. These products are processed in the same machine. It takes 10 minutes to process one unit of product A and 2 minutes for each unit of product B and the machine operates for a maximum of 35 hours in a week. Product A requires 1kg and B 0.5kg of raw material per unit, the supply of which is 600kg per week. Market constraints on product B is known to be minimum of 800 units every week. Product A costs $5 per unit and sold at $10. product B costs $5 per unit and can be sold in the market at a unit price of $8. Determine by simplex method how many items of A and B per week can be produced to maximize profit.

Answer 1

Answer:

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Step-by-step explanation:

Decision Variable: Let $x_{1}$ and $x_{2}$ be the number of products A and B respectively.

Objective function: Cost of product A per unit is $5 and the selling price is $10 per unit. Therefore Profit on one unit of product A = 10 −5 = $5.

$x_{1}$ units of product A contribute a profit of $5$x_{1}$, profit contribution from one unit of product B = 8 −6 = $2$x_{2}$ units of product B contribute a profit of $2$x_{2}$ .The objective function is given by Maximize z = 5$x_{1}$ + 2$x_{2}$

Constraints: The requirement constraint is given by

                                      10 + 2x2 ≤(35 × 60)

                                         10$x_{1}$ + 2x2 ≤2100.

The raw material constraint is given by,

                                    $x_{1}$ + 0.5x2 ≤600

Market demand for product B is 800 units every week

                                         $x_{2}$≥800

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Answer 2

A and B can be produced P₂ ≥ 800 to maximize profit.  

Given:

• Product A takes 10 minutes to process one unit.  
• Product B takes 2 minutes for each unit.
• Product A requires 1 kg and B 0.5 kg of raw material per unit.
• Supply of raw material per week = 600 kg
• Market constraints on product B are known to have a minimum of 800 units weekly.
• Product A costs $5 per unit and can be sold at $10
• Product B costs $5 per unit and can be sold at $8.

To find:

The number of items of A and B per week can be produced to maximize profit.

Solution:

We can simply solve this type of mathematical problem by using the following process.

Now,

We can assume that P₁ and P₂ are the numbers of products A and B respectively.

Case-1:

The cost of product A is $5 and the market price is $10

∴ The profit on Product A = 10 - 5 = $5

P₁ units of product A contribute a profit of $5P₁

Case-2:

Product B costs $5 per unit and can be sold at $8

∴ The profit on Product B = 8 - 5 = $3

P₂ units of product B contribute a profit of $3P₂

As we know,

Maximize Profit = 5P₁ + 3P₂

Constraints:

The required constraint is

                 10P₁ + 2P₂  ≤  (35 × 60)

                10P₁ + 2P₂  ≤  2100

The raw material constraint is

                 P₁ + 0.5P₂  ≤ 600

Market demand for the product B is 800 units per week

So,            P₂ ≥ 800

Hence, A and B can be produced P₂ ≥ 800 to maximize profit.  

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