Inventory Management

1
Inventory Management

• Chapter 12

2
Independent and Dependent Demand

• Dependent Demand
• Used in the production of a final or finished
  product.
• Is derived from the number of finished units to
  be produced.
• E.g. For Each Car 4-wheels. Wheels are the
  dependent demand.
• Independent Demand
• Sold to someone.
• Precise determination of quantity impossible due
  to randomness.
• Forecasting the number of units that can be sold.

Focus on management of Independent Demand Items.
3
Types of Inventory

• Raw material
• Components
• Work-in-progress
• Finished goods
• Distribution Inventory
• Maintenance/repair/operating supply (MRO)

4
Functions of Inventory

• To meet anticipated demand.
• To smooth production requirements.
• To decouple operations.
• To protect against stockouts.
• To take advantage of economic lot size.
• To hedge against price increases or to take
  advantage of quantity discounts.

5
Objectives of Inventory Control

• Inventory Management has two main concerns
• Level of Customer Service.
• Cost of ordering and carrying inventories.
• Achieve satisfactory levels of customer service
  while keeping inventory costs within reasonable
  bounds. 2 fundamental decisions
• --Timing of Orders
• --Size of Orders
• Measures of effective Inventory Management
• Customer Satisfaction.
• Inventory Turnover COGS / Average Inventory.
• Days of Inventory expected number of days of
  sales that can be supplied from existing
  inventory.

i.e. when to order and how much to order.
6
Characteristics of Inventory Systems

• Demand
• -- Constant Vs Variable
• -- Known Vs Random
• Lead time - Known Vs Random
• Review Time - Continuous Vs Periodic
• Excess demand - Backordering Vs Lost Sales
• Changing Inventory

7
Periodic Vs Continuous Review Systems

• Periodic Review System
• Inventory level monitored at constant
  intervals.
• Decisions
• To order or not.
• How much to order?
• Realize economies in processing and shipping.
• Risk of stockout between review periods.
• Time and cost of physical count.
• Continuous Review System
• Inventory level monitored continuously.
• Decisions
• When to order?
• How much to order?
• Shortages can be avoided.
• Optimal order quantity can be determined.
• Added cost of record keeping.

? More appropriate for valuable items
8
Relevant Inventory Costs

• Item price (Cost of an item).
• Holding costs
• Variable costs dependent upon the amount of
  inventory held (e.g. capital opportunity
  costs, storage insurance, risk of
  obsolescence).
• Expressed either as a percentage of unit price or
  as a dollar amount per unit.
• Ordering setup costs
• Fixed cost of placing an order (e.g. clerical
  accounting physical handling) or setting up
  production (e.g. lost production to change tools
  clean equipment).
• Expressed as a fixed dollar amount per order
  regardless of order size.
• Shortage costs
• Result when demand exceeds the supply of
  inventory on-hand. (stock-out)
• Lost profit, expediting back ordering expenses.
• Are sometimes difficult to measure, they may be
  subjectively estimated.

9
Why Control Inventory

• Cost Vs Service
• Under-stock Frequent stock-outs
• Lost sales, loss of customer goodwill
• Low level of customer service
• Over-stock Excess inventory
• Costs of ordering and carrying inventory increase
• Objective Establish an inventory control system
  to find a balance between cost and service
• When to order?
• How much to order?

10
Ordering Quantity Approaches

• Lot-for-lot
• Order exactly what is needed.
• Fixed order quantity
• Order a predetermined amount each time.
• Min-max system
• When inventory falls to a set minimum level,
  order up to the predetermined maximum level.
• Order enough for n periods
• The order quantity is determined by total demand
  for the item for the next n-periods.
• Periodic review
• At specified intervals, order up to a
  predetermined target level.

11
Classification System

• Divides on-hand inventory into 3 classes
• A class, B class, C class (very --gt moderate --gt
  least important).
• Basis is usually annual volume
• volume Annual demand x Unit cost
• Class A 15 -20 of items but 70- 80 dollar
  usage.
• Class B 30 -35 of items but 15 -20 dollar
  usage.
• Class C 50 -60 of items but 5 -10 dollar
  usage.
• Policies based on ABC analysis
• Develop class A suppliers more
• Give tighter physical control of A items
• Forecast A items more carefully

12
Example
Classify the inventory items as A,B and C based
on annual dollar value.
72 of total value and 17 of items Class A.
25 of value and 33 of items Class B.
3 of total value and 50 of items Class C.
13
Economic-Order Quantity Models

• Basic Economic Order Quantity (EOQ).
• Economic Production Quantity (EPQ)
• (EOQ with non-instantaneous delivery).
• Quantity Discount Model.

14
EOQ Model Assumptions

• Only one product is involved.
• Demand is known constant - no safety stock is
  required.
• Lead time is known constant.
• No quantity discounts are available.
• Ordering (or setup) costs are constant.
• All demand is satisfied (no shortages).
• The order quantity arrives in a single shipment.

15
Inventory Level (Cycle)
Q
Quantity on hand
ROP
Place Order
Receive Order
Receive Order
Lead Time
16
EOQ Model
Total annual costs Annual ordering costs
Annual holding costs
17
EOQ Total Cost Equations
Minimize the TC by ordering the EOQ
D Annual Demand. H Annual Inventory
Holding/Carrying Cost per Unit. S Ordering/Setup
Cost per order.
Length of an Order Cycle (time between orders)
(Q/D) years.
18
Example

• A local distributor for a national tire company
  expects to sell approximately 9,600 belted radial
  tires for a certain size and tread design next
  year. Annual carrying cost is 16 per tire, and
  ordering cost is 75. The distributor operates
  288 days a year.
• A. What is the EOQ?
• B. How many times per year does the store order?
• C. What is the length of an order cycle?
• D. What is the total annual cost if the EOQ
  quantity is ordered?
• Pidding Manufacturing assembles security
  monitors. It purchases 3,600 black-and-white
  cathode ray tubes a year at 65 each. Ordering
  cost is 31 per order, and annual carrying cost
  per unit is 20 of the purchase price. Compute
  the optimal order quantity and the total annual
  cost of ordering and carrying the inventory.

19
Economic Production Quantity Model
Assumptions Same as the EOQ except inventory
arrives in increments is drawn down as it
arrives.
20
EPQ Equations

• Adjusted total cost
• Maximum inventory
• Adjusted order quantity
• Cycle Time Q/d.
• Run Time (production phase of the cycle) Q/p.
• where, d demand (usage rate) p production
  rate S ordering / setup cost
• D Annual demand and H Annual Holding cost.

21
Example

• A toy manufacturer uses 48,000 rubber wheels per
  year for its popular dump truck series. The firm
  makes its own wheels, which it can produce at a
  rate of 800 per day. The toy trucks are assembled
  uniformly over the entire year. Carrying cost is
  1 per wheel a year. Setup cost for a production
  run of wheels is 45. The firm operates 240 days
  per year. Determine the
• A. Optimal run size.
• B. Minimum total annual cost for carrying and
  setup.
• C. Cycle time for the optimal run size.
• D. Run time.

22
Quantity Discount Model

• Same as the EOQ, except
• Unit price depends upon the quantity ordered.
• Adjusted total cost equation

23
Total Costs with Purchasing Cost (PD)
24
Quantity Discount Representation
Order Price Quantity per Box 1 to 44 2.00 45 to
69 1.70 70 or more 1.40
TC_at_ 2.00 each
Total Cost
TC_at_ 1.70 each
TC_at_ 1.40 each
PD _at_ 2.00 each
PD _at_ 1.70 each
PD _at_ 1.40 each
0
Quantity
45
70
25
Quantity Discount Model

• The objective is to identify an order quantity
  that will represent the lowest total cost for the
  entire set of curves.
• 2 General Cases
• Carrying Cost is constant.
• Single EOQ.
• Same for all total cost curves.
• Carrying Cost is a percentage of the Price.
• Different EOQ for different prices.
• Lower carrying cost means larger EOQ.
• As unit price decreases, each curve EOQ will be
  to the right of the next higher curves EOQ.

26
Quantity Discount Procedure

• Calculate the EOQ at the lowest price.
• Determine whether the EOQ is feasible at that
  price
• Will the vendor sell that quantity at that price.
• If yes, Stop if no, Continue.
• Check the feasibility of EOQ at the next higher
  price
• Continue until you identify a feasible EOQ.
• Calculate the total costs (including purchase
  price) for the feasible EOQ model.
• Calculate the total costs of buying at the
  minimum quantity allowed for each of the cheaper
  unit prices.
• Compare the total cost of each option choose
  the lowest cost alternative.

27
Examples

• The maintenance department of a large hospital
  uses about 816 cases of liquid cleaner annually.
  Ordering costs are 12, carrying costs are 4 per
  case a year, and the new price schedule indicates
  that orders of less than 50 cases will cost 20
  per case, 50 to 79 cases will cost 18 per case,
  80 to 99 cases will cost 17 per case, and larger
  orders will cost 16 per case. Determine the
  optimal order quantity and the total cost.
• Surge Electric uses 4,000 toggle switches a year.
  Switches are priced as follows 1 to 499, 90
  cents each 500 to 999, 85 cents each and 1,000
  or more, 80 cents each. It costs approximately
  30 to prepare an order and receive it, and
  carrying costs are 40 percent of purchase price
  per unit on an annual basis. Determine the
  optimal order quantity and the total annual cost.

28
When to Order?
ROP d(LT)
where LT Lead time (in days or weeks) d
Daily or weekly demand rate
29
Example

• An office supply store sells floppy disk sets
  at a fairly constant rate of 6,000 per year. The
  accounting dept. states that it costs 8 to place
  an order and annual holding cost are 20 of the
  purchase price 3 per unit. It takes 4 days to
  receive an order. Assuming a 300-day year, find
• a) Optimal order size and ROP.
• b) Annual ordering cost, annual carrying cost.
• c) How many orders are given a year and what is
  the time between the orders?

30
What if Demand is Uncertain?
Quantity on hand
ROP
Time
31
Uncertain Demand (Lead Time)

• Safety Stock Models
• Use the same order quantity (EOQ) based on
  expected (average) annual demand.
• Determine ROP to satisfy a target Service Level
• Probability that demand will not exceed supply
  during lead time (Lead time service level).
• Percent of annual demand immediately satisfied
  (Annual service level or fill-rate).
• Equals 1- stock-out risk
• Safety Stock Stock that is held in excess of
  expected demand due to variable demand rate
  and/or lead time.

32
Adding Safety Stock

• Demand variability.
• Lead time variability.
• Order-cycle service level
• From a managerial standpoint, determine the
  acceptable probability that demand during lead
  time wont exceed on-hand inventory.
• Risk of a stockout 1 (service level).

33
Adjusted Reorder Point Equation
R reorder point d average daily demand LT
lead time in days z number of standard
deviations associated with desired service
level s standard deviation of demand during
lead time (Assumes that any variability in
demand rate or lead time can be adequately
described by a normal distribution)
34
Example

• Suppose that the manager of a construction
  supply house determined from historical records
  that demand for sand during lead time averages 50
  tons. In addition, suppose the manager determined
  that demand during lead time could be described
  by a normal distribution that has a mean of 50
  tons and a standard deviation of 5 tons. Answer
  these questions, assuming that the manager is
  willing to accept a stockout risk of no more than
  3 percent.
• A. What value of z is appropriate?
• B. How much safety stock should be held?
• C. What reorder point should be used?

35
Reorder Point -Continued

• When data on lead time demand is not readily
  available, cannot use the standard formula.
• Use the daily or weekly demand and the length of
  the lead time to generate lead time demand.
• If only demand is variable, then use
  ,
• and the ROP is

36

• If only lead time is variable, then use
  , and the ROP is
• If both demand and lead times variable, then

37
Example

• A restaurant uses an average of 50 jars of a
  special sauce each week. Weekly usage of sauce
  has a standard deviation of 3 jars. The manager
  is willing to accept no more than a 10 percent
  risk of stockout during lead time, which is two
  weeks. Assume the distribution of usage is
  normal.
• A. Which of the above formula is appropriate for
  this situation? Why?
• B. Determine the value of z.
• C. Determine the ROP.

38
Shortage and Service Level

• E.g. Suppose the standard deviation of lead time
  demand is known to be 20 units and lead time
  demand is approximately Normal.
• For a lead time service level of 90 percent,
  determine the expected number of units short for
  any order cycle.
• What lead time service level would an expected
  shortage of 2 units imply?

39
Given the following information, determine the
expected number of units short per year.
D1,000 Q250 E (n)2.5.
Given a lead time service level of 90, D1,000,
Q250, and sdLT16, determine (a) the annual
service level, and (b) the amount of cycle safety
stock that would provide an annual service level
of .98 (Given E (z) 0.048 for 90 lead time
service level).
40
Fixed-Order Interval Model

• Order groupings can produce savings in ordering
  and shipping costs.
• Can have variations in demand, lead time, or in
  both.
• Our focus is only on demand variability, with
  constant lead times.

OI Order Interval (length of time between
orders) Imax Maximum amount of inventory (also
called order-up-to-level point) Expected
demand during protection interval Safety stock
E.g. Given the following information, determine
the amount to order.
41
Single-Period Models

• Used for order perishables.
• Analysis focus on two costs Shortage and Excess.

• Goal is to identify the order quantity, or
  stocking level, that will minimize the long-run
  total excess and shortage cost.

• 2 kinds of problems
• Demand can be approximated using a continuous
  distribution.
• Demand can be approximated using a discrete
  distribution.

42
Continuous Stocking Levels
E.g. Sweet cider is delivered weekly to Cindys
Cider Bar. Demand varies uniformly between 300
liters and 500 liters per week. Cindy pays 20
cents per liter for the cider and charges 80
cents per liter for it. Unsold cider has no
salvage value and cannot be carried over into the
next week due to spoilage. Find the optimal
stocking level and its stockout risk for that
quantity.
E.g. Cindys Cider Bar also sells a blend of
cherry juice and apple cider. Demand for the
blend is approximately Normal, with a mean of 200
liters per week and a standard deviation of 10
liters per week. Cs60 cents per liter, and Ce20
cents per liter. Find the optimal stocking level
for the apple cherry blend.
Discrete Stocking Levels
E.g. Historical records on the use of spare
parts for several large hydraulic presses are to
serve as an estimate of usage for spares of a
newly installed press. Stockout costs involve
downtime expenses and special ordering costs.
These average 4,200 per unit short. Spares cost
800 each, and unused parts have zero salvage.
Determine the optimal stocking level.
43
Examples

• A large bakery buys flour in 25-pound bags. The
  bakery uses an average of 4,860 bags a year.
  Preparing an order and receiving a shipment of
  flour involves a cost of 4 per order. Annual
  carrying costs are 30 per bag.
• A. Determine the economic order quantity.
• B. What is the average number of bags on hand?
• C. Compute the total cost of ordering and
  carrying flour.
• D. If ordering cost were to increase by 1 per
  order, how much would that affect the minimum
  total annual ordering and carrying cost?

44
Examples

• A large law firm uses an average of 40 packages
  of copier paper a day. Each package contains 500
  sheets. The firm operates 260 days a year.
  Storage and handling costs for the paper are 1 a
  year per pack, and it costs approximately 6 to
  order and receive a shipment of paper.
• What order size would minimize total annual
  ordering and carrying costs?
• Compute the total annual cost using your order
  size from part a.
• Except for rounding, are annual ordering and
  carrying costs always equal at the EOQ?
• The office manager is currently using an order
  size of 400 packages. The partners of the firm
  expect the office to be managed in a
  cost-efficient manner. Would you recommend that
  the office manager use the optimal order size
  instead of 400 packages? Justify your answer.

45
Examples

• A chemical form produces sodium bisulphate in
  100-kg bags. Demand for this product is 20 tons
  per day. The capacity for producing the product
  is 50 tons per day. Setup costs 100, and storage
  and handling costs are 50 per ton per year. The
  firm operates 200 days a year. (Note 1 ton
  1,000 kg)
• How many bags per run are optimal?
• What would the average inventory be for this lot
  size?
• Determine the approximate length of a production
  run, in days.
• About how many runs per year would there be?
• How much could the company save annually if the
  setup cost could be reduced to 25 per run?

46
Examples

• A company is about to begin production of a new
  product. The manager of the department that will
  produce one of the components for the product
  wants to know how often the machine to be used to
  produce the item will be available for other
  work. The machine will produce the item at a rate
  of 200 units a day. Eighty units will be used
  daily in assembling the final product. Assembly
  will take place five days a week, 50 weeks a
  year. The manager estimates that it will take
  almost a full day to get the machine ready for a
  production run, at a cost of 300. Inventory
  holding costs will be 10 per unit a year.
• What run quantity should be used to minimize
  total annual costs?
• What is the length of a production run in days?
• During production, at what rate will inventory
  build up?
• If the manager wants to run another job between
  runs of this item, and needs a minimum of 10 days
  per cycle for the other work, will there be
  enough time.

47
Examples

• A mail-order company uses 18,000 boxes a year.
  Carrying costs are 20 cents per box per year, and
  ordering costs are 32 per order. The following
  quantity discount is available. Determine
• The optimal order quantity.
• The number of orders per year.

48
Examples

• A jewelry firm buys semi-precious stones to make
  bracelets and rings. The supplier quotes a price
  of 8 per stone and quantities of 600 stones or
  more, 9 per stone for orders of 400 to 599
  stones, and 10 per stone for lesser quantities.
  The jewelry firm operates 200 days per year.
  Usage rate is 25 stones per day, and ordering
  cost is 48 per order.
• A. If carrying cost are 2 per year for each
  stone, find the order quantity that will
  minimize total annual cost.
• B. If annual carrying cost are 30 percent of
  unit cost, what is the optimal order size?
• C. If lead time is six working days, at what
  point should the company reorder?

49
Examples

• The housekeeping department of a motel uses
  approximately 400 washcloths per day. The actual
  amount tends to vary with the number of guests on
  any given night. Usage can be approximated by a
  normal distribution that has a mean of 400 and a
  standard deviation of 9 washcloths per day. A
  linen supply company delivers towels and
  washcloths with a lead time of three days. If the
  motel policy is to maintain a stockout risk of 2
  percent, what is the minimum number of washcloths
  that must be on hand at reorder point, and how
  much of that amount can be considered safety
  stock?
• The motel in the preceding example uses
  approximately 600 bars of soap each day, and this
  tends not to vary by more than a few bars either
  way. Lead time for soap delivery is normally
  distributed with a mean of six days and a
  standard deviation of two days. A service level
  of 90 percent is desired. Find the ROP.

50
Examples

• A distributor of large appliances needs to
  determine the order quantities and reorder points
  for the various products it carries. The
  following data refers to a specific refrigerator
  in its product line
• Cost to place an order 100
• Holding Cost 20 percent of product cost per
  year.
• Cost of refrigerator 500 each.
• Annual demand 500 refrigerators.
• Standard deviation during lead time 10
  refrigerators.
• Lead time 7 days.
• Consider an even daily demand and a 365-day
  year.
• A. What is the economic order quantity?
• B. If the distributor wants a 97 service
  probability, what reorder point R should be used?
  What is the corresponding safety stock?
• C. If the current reorder point is 26
  refrigerators, what is the possibility of
  stock-out?

51
Examples

• A local service station is open 7 days a week,
  365 days per year. Sales of 10W40 grade premium
  oil average 20 cans per day. Inventory holding
  costs are 0.50 pre can per year. Ordering costs
  are 10 per order. Lead time is two weeks.
  Backorders are not practical -- the motorist
  drives away.
• A. Based on these data, choose the appropriate
  inventory model and calculate the economic order
  quantity and reorder point. ( Demand is
  deterministic).
• B. The boss is concerned about this model
  because demand really varies. The standard
  deviation of demand was determined from a data
  sample to be 6.15 cans per day. The manager
  wants a 99.5 service probability. Determine
  the new reorder point? Use Qopt from Part-A.

52
Examples

• A small copy centre uses five boxes of copy
  paper a day. Each box contains 10 packages of 500
  sheets. Experience suggests that usage can be
  well approximated by a Normal distribution with a
  mean of five boxes per day and a standard
  deviation of one-half box per day. Two days are
  required to fill an order for paper. Ordering
  cost is 10 per order, and annual holding cost is
  10 per box per year.
• Determine the economic order quantity, assuming
  250 work days a year.
• If the copy center reorders when the supply on
  hand is 12 boxes, compute the risk of a stockout.
• If fixed interval of seven days, instead of ROP,
  is used for reordering, what risk does the copy
  center incur that it will run out of stationery
  before this order arrives if it orders 36 boxes
  when the amount on hand is 12 boxes?

53
Examples

• Regional Supermarket is open 360 days per year.
  Daily use of cash register tape averages 10
  rolls. Usage appears Normally distributed with a
  standard deviation of 2 rolls per day. The cost
  of ordering tape is 5 per order, and carrying
  costs are 40 cents per roll a year. Lead time is
  three days.
• What is the EOQ?
• What ROP will provide a lead time service level
  of 96 percent?
• What is the expected number of units short per
  cycle with 96 percent service level? Per year?
• What is the annual service level?

54
Examples

• A depot operates 250 days a year. Daily demand
  for diesel fuel at the depot is Normal with an
  average of 250 liters and a standard deviation of
  14 liters. Holding cost for the fuel is 0.30 per
  liter per year, and it costs 10 in
  administrative time to submit an order for more
  fuel. It takes one day to receive a delivery of
  diesel fuel.
• Calculate the EOQ.
• Determine the amount of safety stock that would
  be needed if the manager wants.
• An annual service level of 99.5 percent.
• The expected amount of fuel short per order cycle
  to be no more than 5 liters.

55
Examples

• A drugstore uses fixed-order-interval model for
  many of the items it stocks. The manager wants a
  service level of 0.98. Determine the order size
  that will be consistent with this service level
  for the items in the following table for an order
  interval of 14 days and a lead time of 2 days.