The Green Apple\'s problem. Green Apple produces both interior and exterior pain

Question

The Green Apple's problem. Green Apple produces both interior and exterior paints from two raw materials, M1 and M2 with the maximum daily availability 24 tons and 6 tons, respectively. The number of tons of raw materials per ton of exterior paint are 6 and 1 tons for M1 and M2, respectively and leading to the profit of 5000 dollars per ton of exterior paint. Similarily, the number of tons of raw materials per ton of interior paint are 4 and 2 tons for M1 and M2, respectively and leading to the profit of 4000 dollars per ton of interior paint.

A market survey indicates that the daily demand for interior paint cannot exceed that of exterior paint by more than 1 ton. Also the maximum daily demand of interior paint is 2 tons. Green Apple wants to determine the best (optimum) product mix of interior and exterior paints that maximizes the total daily profit.

Formulate a complete linear program model for this problem. Explain the meaning of decision variables, objective function, and all constraints in terms of the daily usage of raw materials by both paints including thier availability and corresponding profits.

Explanation / Answer

The required table is as below:

Particulars

Tons of material per ton of

Maximum tons of material available

Material M1

6

4

24

Material M2

1

2

6

Profit/ton ($1,000)

5

4

Decision variable: This is the finding element in the model. Decision variables are as below:

Production of exterior paint per day = X ton

Production of interior paint per day = Y ton

Objective function: The objective is to increase profit at the highest level. It is expressed as below:

Maximize Z = 5X + 4Y

Constraints: These are the limitations within which the production should be done. Constraints are expressed as below:

      6X + 4Y 24

                   X + 2Y 6

                  -X + Y 1

                          Y 2

Therefore, the required linear program model is as below:

Maximize Z = 5X + 4Y

Subject to, 6X + 4Y 24

                   X + 2Y 6

                  -X + Y 1

                          Y 2

Where, X, Y 0

Particulars

Tons of material per ton of

Maximum tons of material available

Material M1

6

4

24

Material M2

1

2

6

Profit/ton ($1,000)

5

4

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