ENM 509, Spring 2016
Homework 3
Due date: April 14, 2016
1. Consider the standard deterministic Economic Order Quantity problem with a fixed
ordering cost k TL, holding cost h TL/item/year and demand rate items/year. The unit
variable cost is v TL/item.
a) Show that when the optimal order quantity is used, the optimal cost is given by:
C (Q * )  v  2kh
b) So far, we have taken the fixed ordering cost k to be given. In practice, the fixed cost
can be reduced from its initial level but this requires a technological or operational
change in the ordering processes which may be costly. Let us assume that decreasing
the fixed ordering cost by 1 TL costs  TL/year. The firm then incurs a cost of (k –
k’)/year TL in order to reduce the fixed cost to k’ TL. Find the optimal level of
investment in setup cost reduction (i.e find the optimal k’) assuming that the objective
is to minimize the sum of the total yearly replenishment cost and the investment cost.
2. The supplier of a product wants to discourage large quantity purchases. Suppose that all of
the assumptions of the basic EOQ model apply except that a reverse all-unit quantity discount
is applicable. That is, unit cost of an item is
0  Q  Qb
 v0
v
Q  Qb
v0 (1  d )
where d>0.
a) Write an expression (or expressions) for the total relevant costs per year as a function
of the order quantity Q. Introduce (and define) whatever other symbols you feel are
necessary.
b) Using graphical sketches, indicate the possible positions of the best order quantity.
c) Determine the best order quantity using the data below:
Demand rate = 50,000 units/year, fixed order cost =$10, unit cost of an item=$1/unit
d=0.005, Qb = 1500 units, interest rate = 0.2/year
3. The Andy Dandy Candy Company has asked you to recommend an EOQ quantity for their
jelly bean product. The only costs involved are the cost of ordering and the cost of holding
inventory, which is equal to the average amount of capital times the interest rate. Backorders
are not allowed and there are no other costs. When a shipment arrives, the company sells the
jelly beans at a certain price until half of the shipment is sold at which time the company
raises its price. Assume that the prices charged at both parts of a cycle are known, which
implies that there are two known, continuous, constant rates of demand D1 and D2, D1 being
the rate at the first part of a cycle with D1 >D2. As a consultant, you are requested to present
the following:
a) A graphical representation of the inventory process. (What is the fraction of time spent at
rate D1?)
b) A total cost equation, defining the terms
c) Obtain the optimal order quantity.
d) Find the solution for k=$20, vr=$2, D1=30, D2=15
e) Compare the above solution with the case when a single price is charged throughout the
whole cycle with a demand rate of D1=D2=30. Compare the profits in both cases.
4. Consider the following stochastic version of the EOQ problem. When you order Q units, the
received quantity in each cycle is Q(1+x) (rather than Q) where x is a quantity that models the
delivery errors. Assume that x is a continuous random variable distributed uniformly in the
interval (-α, α) (where 0< α < 1).
a. Find the optimal order quantity minimizing the expected costs per unit time and
compare it to the regular EOQ quantity.
b. Find the optimal expected length of a cycle.
c. Explain the effect of α on the optimal ordering quantity.