Question

A farmer has a 1000-acre farm on which carrots, onions, and tomatoes could be planted. He has 1600 tons of fertilizer, 900 tons of insecticide-A, and 800 tons of insecticide-B.

Each acre of carrots requires two tons of fertilizer, one ton of insecticide-A and 0.5 tons of insecticide-B. Each acre of onions requires one ton of fertilizer, 0.25 tons of insecticide-A and 0.75 tons of insecticide-B. Each acre of tomatoes requires 1.5 tons of fertilizer, one ton of insecticide-A and one ton of insecticide-B.

The farmer has determined that planted acres of tomatoes must be at least half the number of acres s/he allocates for planting carrots The expected profit figures per acre of carrot, onion, and tomatoes in AED are 25,000, 30,000, and 40,000 respectively. The farmer would like to determine how many acres should he plant of Carrots, onions, and tomatoes in order to maximize his total profit.

Requirement:

Provide below the algebraic LP model for the above case.
Your model should include complete definition of the decision variables, objective function, and constraint.

Answer 1

The decision that farmer has to take is how many acres of land to be planted with Carrot, Onions, and Tomatoes.

Let them be our decision variables

X- No.of acres in which Carrots are planted

Y- No.of acres in which Onions are planted

Z- No.of acres in which Tomatoes are planted

The farmer would like to get the most economic benefit out of his hard work and hence would like to maximize his profit.

As per given expected profits per acre of Carrot, Onions, and Tomatoes, the following would be profit for the farmer which to be maximized.

MAX= X* 25000 + Y* 30000 + Z* 40,000 ......Objective function

However, the farmer has only a maximum of 1000 acres of land.

X + Y + Z <= 1000 .....Constraint on availability of land

The framer has decided that planted acres of tomatoes should be at least half the number of acres s/he allocates for planting carrots.

Z >= X/2 ..... Constraint as per a decision by the farmer

He also has a limited amount of supply for Fertilizer, Insecticide-A, and Insecticide- B

2 tons of fertilizer per acre is required for carrots, 1 ton of fertilizer per acre is required for Onions, 1.5 tons of fertilizer per acre is required for Tomatoes. The total consumption should be less than the available amount. Hence,

2*X + Y + 1.5 <= 1600 ...Constraint on availability of Fertilizer

1 ton of Insecticide-A per acre is required for carrots, 0.25 ton of Insecticide-A per acre is required for Onions, 1 ton of Insecticide-A per acre is required for Tomatoes. The total consumption should be less than the available amount. Hence,

X + 0.25 *Y + Z <= 900 ...Constraint on availability of Insecticide-A

0.5 ton of Insecticide-B per acre is required for carrots, 0.75 ton of Insecticide-B per acre is required for Onions, 1 ton of Insecticide-B per acre is required for Tomatoes. The total consumption should be less than the available amount. Hence,

0.5*X + 0.75*Y + Z <=800 ...Constraint on availability of Insectide-B

Also, X, Y, Z must be non-negative values.

X, Y, Z >=0 ..... Constraint on nature of variable