ISyE 6203 - Transportation and Supply Chain Systems
Spring 2022
Homework 1
1. A car manufacturer faces a weekly demand of 175 units for a specific model, and you manage
the tire orders for this model. Each tire costs $42 and placing an order incurs in a fixed cost
of $300. Given the location of your warehouse, you estimate that the weekly holding costs
per tire are of $0.35 per tire.
(a) Determine the optimal number of tires for the company to order at one time, and the
time between orders in weeks. Keep in mind that a vehicle needs four tires.
(b) What is the weekly cost associated with tire orders? Specify each component of the
total.
(c) Suppose the supplier is only able to send shipments every week, every two weeks or
every four weeks. What is the best order interval under this restriction?
(d) How does the total cost change compared to the cost associated with the optimal order
quantity?
(e) Suppose the company now decides to include a spare tire with each car. Does your
answer for (c) change?
2. You manage the inventory of cars in a local dealership. You face deterministic demand of
2000 units per month, and a monthly holding cost of $15 per vehicle. You have been using
EOQ to determine the order quantity, and currently do not backorder any demand. Because
the price of used cars is increasing, customers have an increased willingness to wait for their
new vehicles; and so you start considering allowing backorders, estimating a corresponding
cost of $40 per unit per month in customer goodwill.
(a) What should be the percentage of demand met from backorders if you optimize your
reorder point with respect to holding and backorder cost, regardless of the order quan-
tity?
(b) Suppose you keep your current order quantity but optimize the reorder point with
respect to holding and backorder cost. Excluding purchasing cost (i.e. variable order
cost), by what percentage would your monthly cost change?
(c) Considering the same conditions as in (b), how would you maximum inventory change
in percentage?
(d) Suppose you re-optimize the order quantity to consider backorders. Is your new quan-
tity larger or smaller? In percentage terms, by how much does the quantity change?
3. Suppose you run a small fast food restaurant that focuses on serving Potatoes in any possible
way. The restaurant is located in San Diego, and your local farmer sells you every pound of
potatoes at $0.3 but also charges you a cost of $500 for delivery on his own truck (regardless
of the quantity order). You have a deterministic demand of 8000 pounds per week, and the
cost of storing a pound of potatoes is assumed to be $0.05 per week.
(a) What is the economic order quantity and how often should you order?
(b) If you could only have a maximum of 10000 pounds of potatoes in inventory at any
time, how would it affect the order quantity, the frequency of orders and the total cost
per pound of potatoes?
(c) You have a very good friend with an empty storage location just besides your restaurant
where you can fit 10000 pounds of potatoes. He tells you that he is happy to store your
potatoes as long as you pay a fee of $500 each time a new order comes to the location
to deliver the goods plus a storing cost of $0.05 per pound per week. Would you seek
help from your friend?
(d) Suppose the local farmer sells the potatoes at a discounted price of $0.25 per pound for
orders of 14000+ pounds (the entire order has this new price). Would you seek help
from your friend? What is the order quantity, frequency of orders and total cost per
pound of potatoes?
(e) Summer is approaching and because of higher temperatures, you can only store pota-
toes for up to two weeks. How does that affect the new order quantity, frequency of
orders and total cost per pound of potatoes?
4. You are in charge of ordering products for a small gaming store in metro Atlanta. You face
a steady monthly demand of 490 units of high-performing graphic cards, and the manufac-
turer (M1) charges you $650 per unit, plus an ordering cost of $500. Besides that, you must
pay $2800 for transportation in airplane and same-day ground delivery (the graphic cards
become available in inventory the same day you place the order). The inventory costs are
computed based on the total costs associated with the purchase, and a monthly interest rate
of 2.25% (assume a month has 4 weeks).
(a) What is the holding cost rate? Include appropriate units of measurement in your an-
swer.
(b) What is your optimal order quantity?
(c) What is the optimal monthly cost associated with this product?
(d) How often should you place an order?
Your store also sells processors, facing a steady monthly demand of 630 units. The manufac-
turer (M2) charges you $320 per unit, a $250 ordering cost and transportation via airplane
plus same-day ground shipping costs $2500; whereas the holding costs are computed just
like for the graphic cards. Both manufacturers M1 and M2 ship from California, so after
talking with the transportation company you arranged a consolidated shipment with total
cost of $3000 plus an extra administrative cost for consolidation of $500.
(e) Determine the optimal order policy for the joint order of the two products. What is the
order amount for graphic cards? What is the order amount for processors?
(f) What is the total monthly cost for both items under this setup?
(g) How often do you place a joint order?
(h) How much does this arrangement save you per month, compared to ordering the two
products independently?
Furthermore, both manufacturers also have production plants in China and you could ar-
range a consolidated shipment from there, with a cost of $4000 plus the consolidation cost
of $500.
(i) Assuming both manufacturers agree to reduce the price of the items by $5 (while still
charging the same processing fees), and that items also arrive in the day when the or-
ders are made, would you start making the deliveries from China? Justify your answer.
(j) If there was a delivery option from China with a total cost (considers consolidation)
of $2000, but where items become available 7 days after you make the orders and you
pay the same day the order is placed. Would you change towards that delivery option?
Justify your answer.
5. You have a small company that sells only one item that must be purchased from the only
manufacturer in the market. In order to manipulate orders such as to meet their goals, the
manufacturer modifies the fixed cost f it charges per order according to the demand d per
period, to always have a constant value f d.
(a) Assume f d = 450, and the holding cost per unit per period is 1. What is the optimal
order quantity?
(b) The manufacturer imposes a new rule: Each order must be a multiple of 20 units (20, 40,
60, . . .). What is the optimal number of units to order? Does it depend on the values of
f , d (given that f d = 450) or the unit cost of each item? You can use a counter example
if needed.
(c) Is there any value of x such that f d = x makes the choice between 20 and 40 units
equivalent? Justify your answer.
6. This problem will help you understand why we can assume a constant order quantity in the
EOQ model: Consider an EOQ system with demand of d units per month, and costs f , c and
h in appropriate units. Suppose that at the beginning of this month, we have no inventory.
We have decided to place two orders over the course of the month and end the month again
with no inventory. So we must place one order at the start of the month for q1 units, and
when these run out place a second order for q2 units, where q1 + q2 = d because the total
demand over one month is d. Use simple calculus to show that, assuming we place two
orders, we minimize total cost over the month by placing two equal orders, q1 = q2 = d/2,
one at the start of the month and the other halfway through the month.