From experience, Mr Subbu has found that the low bid on a construction job can be
regarded as a random variable having the uniform density
f(x)= 0.75C 2C/3 <X< 2C
= 0 otherwise
Where C is his own estimate of the cost of the job. What percentage should Harris
add to his cost estimate when submitting bids to maximize his expected profit.
To find the percentage that Harris should add to his cost estimate, we need to determine the expected profit for different bids. The expected profit is given by the formula:
Expected Profit = (Revenue - Cost) * Probability
In this case, the revenue for each bid is the low bid received, and the cost is Harris's own estimate. The probability of winning a bid is determined by the density function, which is uniform in this case.
To maximize the expected profit, we need to find the bid that maximizes the difference between revenue and cost. Let's start by writing down the expected profit formula for a bid x:
Expected Profit(x) = (x - C) * Probability(x)
Now, let's calculate the probability function for the bid x:
Probability(x) = integral of f(t) dt from 2C/3 to x
Since f(x) is a uniform density, the probability function can be calculated as:
Probability(x) = (x - 2C/3)/(2C - 2C/3)
Now, let's substitute this into the expected profit formula and simplify:
Expected Profit(x) = (x - C) * (x - 2C/3)/(2C - 2C/3)
Next, let's differentiate the expected profit formula with respect to x and set it equal to zero to find the bid that maximizes the expected profit:
d(Expected Profit(x))/dx = (2C - 2C/3) - (x - C)*1/(2C - 2C/3) = 0
Simplifying the above equation, we get:
2C - 2C/3 - (x - C)/(2C - 2C/3) = 0
Solving for x, we find:
x = 2C - 2C/3
Finally, to find the percentage that should be added to the cost estimate, we calculate (x - C)/C * 100:
Percentage to Add = [(2C - 2C/3) - C]/C * 100
= (4C/3 - C)/C * 100
= (C/3)/C * 100
= 33.33%
Therefore, Harris should add approximately 33.33% to his cost estimate when submitting bids to maximize his expected profit.