ENM 509, Spring 2016
Homework 4
Due date: April 28, 2016
1. Kıtır Gevrek, a cereal manufacturer, has dedicated a plant to a major retail chain. Sales
at this retail chain average about 20000 boxes a month and production at the plant keeps
pace with this average demand. Each box of cereal costs Kıtır 3 TL and is sold to the
retailer at a wholesale price of 5 TL. Both Kıtır and the retailer use an annual holding
cost rate of 20%. For each order placed, the retailer incurs an ordering cost of 200 TL
per order. Kıtır incurs the cost of transportation and loading that totals 1000 TL per
order shipped.
a. What is the optimal order quantity that minimizes the costs of the retailer? What is the
resulting annual cost for the retailer and for Kıtır as a result of this quantity?
b. Find the optimal order quantity for a centralized supply chain that aims to minimize the
total cost for both parties. Compare the order size and cost with part (a).
For the following problems, consider the standard newsvendor setting with financial parameters
r, w, and s:
2. Demand has a continuous uniform distribution in the interval (0,1)
a. Find the optimal ordering quantity.
b. Find an expression for the optimal expected profit.
c. Find an expression for the probability that the profit is negative for a given
ordering quantity. (Hint: the profit can only be negative if demand is small
with respect to the ordering quantity).
3. Demand is exponentially distributed with mean .
a. Find the optimal ordering quantity.
b. Show that the optimal profit (using the optimal order quantity) is equal to:
(r-w)  – co Q*
4. Demand has lognormal distribution with parameters  and . Note that D has a
lognormal distribution if ln(D) has a normal distribution with  and standard deviation
. Also note that E[D] =exp(+2/2) and Var[D]= E[D]2 exp(2-1). Find the optimal
ordering quantity in terms of the standard normal random variable Z.
5. Determine the optimal order quantity in each case.
a. Demand has a Bernoulli distribution with parameter p, 0<p<1, (P(D=0)=1-p and
P(D=1)=p).
b. Demand has a discrete uniform distribution on the integers from 1 to 9.
c. Demand is a Poisson random variable with mean l.
6. Consider a supplier-retailer supply chain operating under a wholesale-price contract
with parameters r, w, and c (with s=0). Demand has a continuous uniform distribution
in the interval (0,a).
a. Show that the optimal expected profit of the centralized (integrated) supply chain is:
*int=(1/2) (r-m)2 a / r.
b. Assume that the supplier has wholesale-price setting power. Find the wholesale
price w that maximizes his profit.
c. Find the expected optimal profit of the supplier and the retailer at the optimal (from
the supplier’s point of view) wholesale price.
d. Find the supply chain efficiency (total decentralized profit/total centralized profit)
under the optimal wholesale price.