Chapter 12: Inventory Control

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Chapter 12 Inventory Control

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Purposes of Inventory

• 1. To maintain independence of operations
• 2. To meet variation in product demand
• 3. To allow flexibility in production scheduling
• 4. To provide a safeguard for variation in raw
  material delivery time
• 5. To take advantage of economic purchase-order
  size

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Inventory Costs

• Holding (or carrying) costs
• Costs for capital, storage, handling,
  shrinkage, insurance, etc
• Setup (or production change) costs
• Costs for arranging specific equipment setups,
  etc
• Ordering costs
• Costs of someone placing an order, etc
• Shortage costs
• Costs of canceling an order, etc

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Independent vs. Dependent Demand
Finished product
E(1)
Component parts
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Inventory Systems

• Single-Period Inventory Model
• One time purchasing decision (Example vendor
  selling t-shirts at a football game)
• Seeks to balance the costs of inventory overstock
  and under stock
• Multi-Period Inventory Models
• Fixed-Order Quantity Models
• Event triggered (Example running out of stock)
• Fixed-Time Period Models
• Time triggered (Example Monthly sales call by
  sales representative)

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The Newsvendor Model
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Wetsuit example

• The too much/too little problem
• Order too much and inventory is left over at the
  end of the season
• Order too little and sales are lost.
• Example Selling Wetsuits

• Economics
• Each suit sells for p 180
• Seller charges c 110 per suit
• Discounted suits sell for v 90

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Too much and too little costs

• Co overage cost (i.e. order one too many ---
  demand
• The cost of ordering one more unit than what you
  would have ordered had you known demand if you
  have left over inventory the increase in profit
  you would have enjoyed had you ordered one fewer
  unit.
• For the example Co Cost Salvage value c
  v 110 90 20
• Cu underage cost (i.e. order one too few
  demand order amount)
• The cost of ordering one fewer unit than what you
  would have ordered had you known demand if you
  had lost sales (i.e., you under ordered), Cu is
  the increase in profit you would have enjoyed had
  you ordered one more unit.
• For the example Cu Price Cost p c 180
  110 70

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Newsvendor expected profit maximizing order
quantity

• To maximize expected profit order Q units so that
  the expected loss on the Qth unit equals the
  expected gain on the Qth unit
• Rearrange terms in the above equation -
• The ratio Cu / (Co Cu) is called the critical
  ratio (CR).
• We shall assume demand is distributed as the
  normal distribution with mean m and standard
  deviation s
• Find the Q that satisfies the above equality use
  NORMSINV(CR) with the critical ratio as the
  probability argument.
• (Q-m)/s z-score for the CR so
• Q m z s

Note where F(Q) Probability Demand
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Finding the examples expected profit maximizing
order quantity

• Inputs
• Empirical distribution function table p 180 c
  110 v 90 Cu 180-110 70 Co 110-90
  20
• Evaluate the critical ratio
• NORMSINV(.7778) 0.765
• Other Inputs mean m 3192 standard deviation
  s 1181
• Convert into an order quantity
• Q m z s
• 3192 0.765 1181
• 4095

Find an order quantity Q such that there is a
77.78 prob that demand is Q or lower.
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Single Period Model Example

• A college basketball team is playing in a
  tournament game this weekend. Based on past
  experience they sell on average 2,400 tournament
  shirts with a standard deviation of 350. They
  make 10 on every shirt sold at the game, but
  lose 5 on every shirt not sold. How many shirts
  should be ordered for the game?
• Cu 10 and Co 5 P 10 / (10 5)
  .667
• Z.667 .432 (use NORMSINV(.667) therefore we
  need 2,400 .432(350) 2,551 shirts

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Hotel/Airline Overbooking

• The forecast for the number of customers that DO
  NOT SHOW UP at a hotel with 118 rooms is Normally
  Dist with mean of 10 and standard deviation of 5
• Rooms sell for 159 per night
• The cost of denying a room to the customer with a
  confirmed reservation is 350 in ill-will and
  penalties.
• Let X be number of people who do not show up X
  follows a probability distribution!
• How many rooms ( Y ) should be overbooked (sold
  in excess of capacity)?

• Newsvendor setup
• Single decision when the number of no-shows in
  uncertain.
• Underage cost if X Y (insufficient number of
  rooms overbooked).
• For example, overbook 10 rooms and 15 people do
  not show up lose revenue on 5 rooms
• Overage cost if X overbooked).
• Overbook 10 rooms and 5 do not show up pay
  penalty on 5 rooms

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Overbooking solution

• Underage cost
• if X Y then we could have sold X-Y more rooms
• to be conservative, we could have sold those
  rooms at the low rate, Cu rL 159
• Overage cost
• if X
• and incur an overage cost Co 350 on each
  bumped customer.
• Optimal overbooking level
• Critical ratio

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Optimal overbooking level

• Suppose distribution of no-shows is normally
  distributed with a mean of 10 and standard
  deviation of 5
• Critical ratio is
• z NORMSINV(.3124) -0.4891
• Y m z s 10 -.4891 5 7.6
• Overbook by 7.6 or 8
• Hotel should allow up to 1188 reservations.

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Multi-Period ModelsFixed-Order Quantity Model
Model Assumptions (Part 1)

• Demand for the product is constant and uniform
  throughout the period
• Lead time (time from ordering to receipt) is
  constant
• Price per unit of product is constant

• Inventory holding cost is based on average
  inventory
• Ordering or setup costs are constant
• All demands for the product will be satisfied (No
  back orders are allowed)

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Basic Fixed-Order Quantity Model and Reorder
Point Behavior
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Cost Minimization Goal
By adding the item, holding, and ordering costs
together, we determine the total cost curve,
which in turn is used to find the Qopt inventory
order point that minimizes total costs
C O S T
Holding Costs
Ordering Costs
Order Quantity (Q)
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Deriving the EOQ

• Using calculus, we take the first derivative of
  the total cost function with respect to Q, and
  set the derivative (slope) equal to zero, solving
  for the optimized (cost minimized) value of Qopt

We also need a reorder point to tell us when to
place an order
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Basic Fixed-Order Quantity (EOQ) Model Formula
TCTotal annual cost D Demand C Cost per unit Q
Order quantity S Cost of placing an order or
setup cost R Reorder point L Lead time HAnnual
holding and storage cost per unit of inventory
Total Annual Cost
Annual Purchase Cost
Annual Ordering Cost
Annual Holding Cost


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EOQ Class Problem 1

• Dickens Electronics stocks and sells a particular
  brand of PC. It costs the firm 450 each time it
  places and order with the manufacturer. The cost
  of carrying one PC in inventory for a year is
  170. The store manager estimates that total
  annual demand for computers will be 1200 units
  with a constant demand rate throughout the year.
  Orders are received two days after placement
  from a local warehouse maintained by the
  manufacturer. The store policy is to never have
  stockouts. The store is open for business every
  day of the year. Determine the following
• Optimal order quantity per order.
• Minimum total annual inventory costs (i.e.
  carrying plus ordering ignore item costs).
• The optimum number of orders per year (D/Q)

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EOQ Problem 2

• The Western Jeans Company purchases denim from
  Cumberland textile Mills. The Western Jeans
  Company uses 35,000 yards of denim per year to
  make jeans. The cost of ordering denim from the
  Textile Mills is 500 per order. It costs
  Western 0.35 per yard annually to hold a yard of
  denim in inventory. Determine the following
• a. Optimal order quantity per order.
• b. Minimum total annual inventory costs (i.e.
  carrying plus ordering).
• c. The optimum number of orders per year.
•       

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Problem 3

• A store specializing in selling wrapping paper is
  analyzing their inventory system. Currently the
  demand for paper is 100 rolls per week, where the
  company operates 50 weeks per year.. Assume that
  demand is constant throughout the year. The
  company estimates it costs 20 to place an order
  and each roll of wrapping paper costs 5.00 and
  the company estimates the yearly cost of holding
  one roll of paper to be 50 of its cost.
• If the company currently orders 200 rolls every
  other week (i.e., 25 times per year), what are
  its current holding and ordering costs (per
  year)?

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Problem 3

• The company is considering implementing an EOQ
  model. If they do this, what would be the new
  order size (round-up to the next highest
  integer)? What is the new cost? How much money
  in ordering and holding costs would be saved each
  relative to their current procedure as specified
  in part a)?

• The vendor says that if they order only twice per
  year (i.e., order 2500 rolls per order), they can
  save 10 cents on each roll of paper i.e., each
  roll would now cost only 4.90. Should they take
  this deal (i.e., compare with part bs answer)
  Hint For c. calculate the item, holding, and
  ordering costs in your analysis.

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Safety Stocks
Suppose that we assume orders occur at a fixed
review period and that demand is probabilistic
and we want a buffer stock to ensure that we
dont run out
Probability of stockout (1.0 - 0.85 0.15)
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Safety Stock Formula
Reorder Point Average demand Safety stock
Reorder Point Demand during Lead Time Safety
Stock Demand during lead time daily demand L
dL Safety stock Zservicelevel sL Where
sL square root of Ls2, where s is the
standard deviation of demand for one day
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Problem 4

• A large manufacturer of VCRs sells 700,000 VCRs
  per year. Each VCR costs 100 and each time the
  firm places an order for VCRs the ordering charge
  is 500. The accounting department has
  determined that the cost of carrying a VCR for
  one year is 40 of the VCR cost. If we assume
  350 working days per year, a lead-time of 4 days,
  and a standard deviation of lead time of 20 per
  day, answer the following questions.
• How many VCRs should the company order each time
  it places an order?
• If the company seeks to achieve a 99 service
  level (i.e. a 1 chance of being out of stock
  during lead time), what will be the reorder
  point? How much lower will be the reorder point
  if the company only seeks a 90 service level?

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Fixed-Time Period Model with Safety Stock Formula
q Average demand Safety stock Inventory
currently on hand
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Multi-Period Models Fixed-Time Period Model
Determining the Value of sTL

• The standard deviation of a sequence of random
  events equals the square root of the sum of the
  variances

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Example of the Fixed-Time Period Model
Given the information below, how many units
should be ordered?
Average daily demand for a product is 20 units.
The review period is 30 days, and lead time is 10
days. Management has set a policy of satisfying
96 percent of demand from items in stock. At the
beginning of the review period there are 200
units in inventory. The daily demand standard
deviation is 4 units.
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Example of the Fixed-Time Period Model Solution
(Part 1)
The value for z is found by using the Excel
NORMSINV function.
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ABC Classification System

• Items kept in inventory are not of equal
  importance in terms of
• dollars invested
• profit potential
• sales or usage volume
• stock-out penalties

So, identify inventory items based on percentage
of total dollar value, where A items are
roughly top 15 , B items as next 35 , and the
lower 65 are the C items