Question

A leading company in Delhi is planning to rent houses and open spaces. The houses are in three categories namely, having three bedrooms, two bedrooms and single bedroom homes. A market survey conducted by a team indicates that a maximum of 650 three bedroom homes, 500 two bedroom homes and 300 single bedroom homes can be rented. Also, the number of three bedroom homes should be at least 60% of the number of two bedroom and single bedroom homes. Open space is proportionate to the number of home units at the rates of at least 10 sq.ft, 15 sq.ft and 18 sq.ft for three bedroom, two bedroom and single bedroom homes respectively. However, land availability limits open space to no more than 10000 sq.ft. The monthly rental income is estimated at Rs. 45000, Rs. 56250 and Rs. 90000 for single bedroom, two bedroom and three bedroom homes respectively. The open space rents for Rs. 7500/sq.ft. Formulate the above as an LPP so as to get maximal revenue. (4) b) Convert the following problem to standard form explaining the various steps.

Answer 1

Let us assume, the number of three bedrooms, two bedrooms and single bedrooms that are to be rented denoted by x₁, x₂ and x₃ respectively.

Here we need to find out the number of bedrooms that are to be rented with an objective to maximize revenue and also satisfying the constraints in this regard.

Objective function:

Max. (Revenue) = (90000x₁ + 56250x₂ + 45000x₃) + (75000x₁ + 112500x₂ + 135000x₃)

Subject to Constraints:

x₁ <= 650 .................................... (1)

x₂ <= 500 .................................... (2)

x₃ <= 300 .................................... (3)

x₁ >= 0.6 (x₂ + x₃) ....................... (4)

10x₁ + 15x₂ + 18x₃ <= 10000 .... (5)

Notes:

Objective function
Points	Explanation	Equation
1	Total revenue that can be earned from the sale of flats. Where, flats with three bedrooms, two bedrooms and single bedroom can be sold at a price of Rs. 90000, Rs. 56250 and Rs. 45000 per flat respectively.	90000x1 + 56250x2 + 45000x3
2	Revenue from the sale of open space attached to each flat. Where each square feet of open space is priced at Rs. 7500.	75000x1 + 112500x2 + 135000x3
Constraint terms
Points	Explanation	Equation
1	The maximum number of three bedroom flats that can be rented should not exceed 650 flats.	x1 <= 650
2	The maximum number of two bedroom flats that can be rented should not exceed 500 flats.	x2 <= 500
3	The maximum number of single bedroom flats that can be rented should not exceed 300 flats.	x3 <= 300
4	The number of three bedroom flats to be rented should be atleast (should be equal or remain over) 60 % of the total flats with two bedrooms and single bed room.	x1 >= 0.6 (x2 + x3)
5	The maximum open space due to land availability is restricted to 10000 square feet. Where flats with three bedrooms, two bedrooms and single bedroom require an open space of 10 sq. ft, 15 sq. ft. and 18 sq. ft. per flat respectively.	10x1 + 15x2 + 18x3 <= 10000