There are 25 rooms in a hotel. Each room can accommodate at the most three people. For each room, the single occupancy charge is Rs. 2000 per day, the double occupancy charge is Rs. 3000 per day, and the triple occupancy charge is Rs. 3500 per day.
If there are 55 people staying in the hotel today, what is the maximum possible revenue from room occupancy charges today?
C
Step-by-Step Solution
Key idea: This is a linear optimization problem with discrete constraints. We need to maximize revenue by allocating people to rooms such that the revenue per person is maximized, while respecting the total room and total people constraints.
Step 1: Define variables and the objective function.
Let be the number of single, double, and triple occupancy rooms used.
Total rooms:
Total people:
Revenue
Step 2: Express Revenue in terms of fewer variables.
From the people constraint, .
Substitute into the Revenue equation:
Step 3: Identify constraints on and .
Since , we have .
Substitute into the room constraint:
Step 4: Minimize the penalty term .
To maximize , we must minimize subject to and .
From , we get .
Substitute this into the penalty term:
Penalty .
To minimize this, we need the smallest possible value for .
Substitute into :
.
Step 5: Calculate the maximum revenue.
The minimum valid is 5.
If , then .
Then .
Maximum Revenue .
Answer: Rs. 77500