maths

IEOR 4601 Homework 1: Due Wednesday, February 6

1. A coffee shop gets a daily allocation of 100 bagels.

The bagels can be either sold

individually at $0.90 each or can be used later in the day for sandwiches. Each bagel

sold as a sandwich provides a revenue of $1.50 independent of the other ingredients.

a) Suppose that demand for bagel sandwiches is estimated to be Poisson with param-

eter 100. How many bagels would you reserve for sandwiches?

b) Compare the expected revenue of the solution of part a) to the expected revenue

of the heuristic that does not reserve capacity for sandwiches assuming that the

demand for individual bagels is Poisson with parameter 150?

c) Answer part a) if the demand for bagel sandwiches is normal with mean 100 and

standard deviation 20.

2. Problem 2 in the textbook (page 173)

3. Problem 3 in the textbook (page 173)

[Hint: For problem 2 and 3, you might want to consider the NORMINV() function in

Excel.]

4. Suppose capacity is 120 seats and there are four fares. The demand distributions for

the different fares are given in the the following table.

Class

Fare

Demand Distribution

1

$200

Poisson(25)

2

$175

Poisson(30)

3

$165

Poisson(29)

4

$130

Poisson(30)

Determine the optimal protection levels. [Hints: The sum of independent Poisson ran-

dom variables is Poisson with the obvious choice of parameter to make the means match.

If D is Poisson with parameter λ then P (D = k + 1) = P (D = k)λ/(k + 1) for any non-

negative integer k. You might want to investigate the POISSON() function in Excel.]

5. Consider a parking lot in a community near Manhattan. The parking lot has 100 parking

spaces. The parking lot attracts both commuters and daily parkers. The parking lot

manager knows that he can fill the lot with commuters at a monthly fee of $180 each. The

parking lot manager has conducted a study and has found that the expected monthly

revenue from x parking spaces dedicated to daily parkers is approximated well by the

quadratic function R(x) = 300x − 1.5x2 over the range x ∈ {0, 1, . . . , 100}.

Note:

Assume for the purpose of the analysis that parking slots rented to commuters cannot

be used for daily parkers even if some commuter do not always use their slots.

a) What would the expected monthly revenue of the parking lot be if all the capacity

is allocated to commuters?

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b) What would the expected monthly revenue of the parking lot be if all the capacity

is allocated to daily parkers?

c) How many units should the parking manager allocate to daily parkers and how

many to commuters?

d) What is the expected revenue under the optimal allocation policy?

6. A fashion retailer has decided to remove a certain item of clothing from the racks in

one week to make room for a new item. There are currently 80 units of the item and

the current sale price is $150 per unit. Consider the following three strategies assuming

that any units remaining at the end of the week can be sold to a jobber at $30 per unit.

a) Keep the current price. Find the expected revenue under this strategy under the

assumption that demand at the current price is Poisson with parameter 50.

b) Lower the price to $90 per unit. Find the expected revenue under this strategy

under the assumption that demand at $90 is Poisson with parameter 120.

c) Keep the price at $150 but e-mail a 40% discount coupon for the item to a popula-

tion of price sensitive customers that would not buy the item at $150. The coupon

is valid only for the first day and does not affect the demand for the item at $150.

Compute the expected revenue under this strategy assuming that the you can con-

trol the number of coupons e-mailed so that demand from the coupon population

is Poisson with parameter x for values of x in the set {0, 5, 10, 15, 20, 25, 30, 35}. In

your calculations assume that demand from coupon holders arrives before demand

from customers willing to pay the full price. Assume also that you cannot deny

capacity to a coupon holder as long as capacity is available (so capacity cannot be

protected for customers willing to pay the full price). What value of x would you

select? You can assume, as in parts a) and b) that any leftover units are sold to

the jobber at $30 per unit.

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