Question

1. Suppose we are selling hot dogs during a basketball game. A hot dog sells for $15 each but only costs $5 each to make. If we run out of hot dogs during the game, it will be impossible to get more. On the other hand, a leftover hot dog has a value of S1 for each hot dog. Assume that we believe the fans would buy 100 hot dogs with probability 3/10, 110 hot dogs with probability 2/10, 120 hot dogs with probability 2/10, 130 hot dogs with probability 1/10, and 140 hot dogs with probability 2/10. (a) Find the expected demand. (b) If 120 hot dogs are prepared, what is the expected profit? (c) What is the corresponding expected (potential) cost?

Answer 1

Solution :-

Given data :

A hot dog sells for $15 each but only costs $5 each to make.

A leftover hot dog has a value of $1 for each hot dog.

demand and probabilities are

1. $x_1$ = 100 hot dogs with probability P($x_1$) = 3/10,

2. $x_2$ = 110 hot dogs with probability P($x_2$) = 2/10,

3. $x_3$ = 120 hot dogs with probability P($x_3$) = 2/10,

4. $x_4$ = 130 hot dogs with probability P($x_4$) = 1/10,

5. $x_5$ = 140 hot dogs with probability P($x_5$) = 2/10.

( a ) :-

Here we need to calculate the expected demand.

i.e, E[X].

$E[X] = \sum x_i*P(x_i)$

$E[X] =100*(3/10)+110*(2/10)+120*(2/10)+130*(1/10)+140(2/10)$

$E[X] =30+22+24+13+28$

$E[X] =117$

$\therefore$ Expected demand, $E[X] =117$

( b ) :-

Here we need to calculate the expected profit, If 120 hot dogs are prepared.

When a hot dog sold, will earn a profit of ( $15 - $5 ) = $10

When a hot dog unsold, will be a loss of ( $5 - $1 ) = $4

The demand is more that too will be a potential loss, If the hot dogs are finished and this will be a loss of potential profit of $10.

If the 120 hot dogs are prepared then, the profit will be also have the probability distribution according to the distribution of demand.

Consider the demand is 100 , then 20 hot dogs will be left.

and the profit = (100*10)-(20*4)

profit = 1000-80

   profit = 920

Consider the demand is 110 , then 10 hot dogs will be left.

and the profit = (110*10)-(10*4)

profit = 1100-40

profit = 1060

Then the expected profit becomes as

Expected profit =$920*(3/10)+1060*(2/10)+1200*(2/10)+1200*(1/10)+1200*(2/10)$

Expected profit = $920*0.3+1060*0.2+1200*0.2+1200*0.1+1200*0.2$

Expected profit = $276+212+240+120+240$

Expected profit = $1088

$\therefore$  Expected profit = $1088

( c ) :-

Here we need to calculate  the corresponding expected (potential) cost.

Here we know the formula,

Expected (potential) cost = Expected demand * Cost

Expected (potential) cost = 117 * $5

Expected (potential) cost = $585   

$\therefore$ Expected (potential) cost = $585