Inventory Systems for Independent Demand

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Inventory Systems for Independent Demand
OPERATIONS MANAGEMENT CLASS

• This presentation explores
• The definition and purpose of inventory
• Inventory costs
• Independent vs. dependent demand
• Basic fixed order quantity system
• Basic fixed time period system
• Customer Service - Achieving Percent Fill Rates

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Inventory

• Definition--The stock of any item or resource
  used in an organization
• Raw materials
• Finished products
• Component parts
• Supplies
• Work in process ....

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INV. MANAGEMENT OBJECTIVES

• MAXIMUM CUSTOMER SERVICE
• MINIMUM INV. INVESTMENT
• MAXIMUM EFFICIENCY (MIN. COST)
• MAXIMUM PROFIT
• HIGHEST RETURN ON INVESTMENT
• TO USE INVENTORY FOR STRATEGIC ADVANTAGE

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TRADE-OFFS

• YOU CANT SELL FROM AN EMPTY CART
• OUR LINE OF CREDIT (BUDGET) WILL SUPPORT ONLY SO
  MUCH
• 1 IN INVENTORY COSTS ABOUT .25/YEAR
• OUT-OF-STOCKS gt LOST SALES gt LOST
  CUSTOMERS
• OUR SHAREHOLDERS WANT A 15 ROI
• ITS IMPOSSIBLE TO FORECAST PERFECTLY

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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
• TO DECOUPLE AND ECONOMIC LOT SIZE.

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3 QUESTIONS OF INVENTORY SYSTEM

• WHAT TO ORDER?
• HOW MANY TO ORDER?
• WHEN TO ORDER?
• TIMING IS A CRITICAL ISSUE!

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Inventory Costs

• Holding (or carrying) costs
• Setup (or production change) costs
• Ordering costs
• Shortage costs ....

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SEVERAL DIFFERENT SYSTEMS

• INDEPENDENT DEMAND - MUST BE FORECASTED.
• WAGONS, BICYCLES, ETC.
• FINISHED GOODS OR SERVICES
• DEPENDENT DEMAND - CAN AND SHOULD BE CALCULATED
• WHEELS, SPOKES, HANDLE BARS
• COMPONENTS FOR SERVICES
• SIMPLE BUT POWERFUL PRINCIPLE

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Independent vs. Dependent Demand
Independent Demand (Demand not related to other
items)
Dependent Demand (Derived)
E(1)
....
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Classifying Independent Demand Systems

• Fixed-Order Quantity
• Fixed-Time Period
• Hybrid Systems - Both Together - Order
  Periodically unless we reach the reorder point.

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Fixed-Order Quantity Models
Idle state Waiting for demand
Demand occurs Unit withdrawn from inventory or
back ordered
No
Is status lt Reorder point?
Compute inventory status Status On hand On
order - Back order
Yes
Issue order for Exactly Q units
....
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Fixed-Order Quantity ModelsAssumptions

• 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 ....

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Fixed-Order Quantity ModelsAssumptions

• 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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EOQ Model--Basic Fixed-Order Quantity Model
Number of units on hand
R Reorder point Ld Q Economic order
quantity L Lead time d Demand per Unit of
Time
....
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Cost Minimization Goal
TAHC H(Q/2) TAOC SD/Q TAC TAHC TAOC
C O S T
Total Cost
Holding Costs
Annual Cost of Items (DC)
Annual Ordering Costs
QOPT
Order Quantity (Q)
....
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Basic Fixed-Order Quantity Model
TC Total 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 H Annual holding and storage cost per unit
of inventory
....
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Deriving the EOQ - Calculus Method

• Using calculus, we take the derivative of the
  total cost function and set the derivative
  (slope) equal to zero ....

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Deriving the EOQ - Graphical Method

• Recognizing TAHC TAOC at Optimal
• TAHC HQ/2 TAOC SD/Q
• HQ/2 SD/Q, Solving for Q Yields

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EOQ Example
Annual Demand 1,000 units Days per year
considered in average daily demand 365 Cost to
place an order 10 Holding cost per unit per
year 2.50 Lead time 7 days Cost per unit
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Determine the economic order quantity and the
reorder point.
....
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Solution
Why do we round up?
....
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In-Class Exercise
Annual Demand 10,000 units Days per year
considered in average daily demand 365 Cost to
place an order 10 Holding cost per unit per
year 10 of cost per unit Lead time 10
days Cost per unit 15
Determine the economic order quantity and the
reorder point.
Note (Tag hidden-slide icon to project
solution) ....
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Solution
....
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ACHIEVING HIGH CUSTOMER SERVICE

• ADD BUFFER STOCK
• ORDER EARLIER
• ACHIEVE DESIRED CUSTOMER SERVICE
• PERCENT FILL RATE .99 9,900/10,000
• ARE SUPPLIED DIRECTLY FROM STOCK

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ADDING SAFETY STOCK TO ORDER EARLIER

• Fixed Order Quantity System Under Uncertainty

B ADD BUFFER STOCK IN CASE DEMAND DURING
LEADTIME gt EXPECTED, dL.

R Reorder point Q Economic order quantity L
Lead time SL Standard Error of Estimate During
Leadtime B Buffer stock set to achieve P
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STRATEGIC IMPORTANCE OF FILL (P)

• P IS SET BY TOP MANAGEMENT
• P IS STRATEGIC PERFORMANCE MEASURE
• CONTEMPORARY P ARE INCREASING
• P99.8 IS BECOMING MORE COMMON
• HOW DO WE CONTROL INVENTORIES TO ACHIEVE P?

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THE VARIANCE LAW
IF X AND Y ARE INDEPENDENT RANDOM VARIABLES AND
Z X Y THEN __ __ __ Z X Y
AND ?Z2 ?X2 ?Y2
______________ ?Z ? ?X2 ?Y2 This is
extremely important in many applications.
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EXAMPLES OF THE VARIANCE LAW
CPM with Critical Path Consisting of activities
A, C, D, F, and G Tcp TA TC TD TF TG
Mean time of CP AND ?CP2 ?A2 ?C2 ?D2
?F2 ?G2 Var of CP
______________ This is extremely important in
many applications. For example if A, C, D, F,
and G were stock returns from investments then
the average returns and variances equal those of
the above assuming they are independent of each
other. In forecasting/inventory control we see
that
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FORECASTING DEMAND DURING LT

• MEAN DEMAND dL
• Where L is length of LT
• Assume L 3, ?d SEE in forecasting,
• ?L SEE during leadtime.
• 
  ___
• ?L ?d1 ?d2 ?d3 ?d ? L
• This Assumes Demand is Normally Independently
  Distributed
• This Allows the Use of the E(Z) Concept

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STATISTICAL INVENTORY CONTROLSCIENTIFIC
INVENTORY CONTROL

• SCIENTIFICALLY SET THE REORDER POINT TO YIELD
  DESIRED PERCENT FILL (P)
• R dL Z?L Mean Buffer Stock where
• R Reorder Point
• dL Mean Demand During Leadtime
• ? L Standard Error of Forecast During LT
• Z Value Determined to Achieve P

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DETERMINING THE BEST Z-VALUE

• (1-P)D No. Units Short Per Year
• E(Z) Units Short Per Leadtime when ? L is 1.
• No. Units Short Per Year E(Z) ? LD/Q
• Equating Both
• E(Z) ? LD/Q (1-P)D rearranging
• E(Z) (1-P)Q/ ? L

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E(Z) (1-P)Q/ ? L

• THIS RELATES THE FOLLOWING
• PERCENT FILL RATE (P)
• FORECAST ACCURACY
• SAFETY STOCK
• ORDER QUANTITY
• NO. OF EXPOSURES PER YEAR
• REORDER POINT

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LETS WORK A PROBLEM

• D5000, P.99, C3/UNIT, i.20//YR
• S10, d100, ? d30, L3
• Determine the EOQ
• Reorder Point
• Safety Stock
• Annual Investment in Inventory
• Comment on the effectiveness of this solution.

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Fixed-Time Period Model
No
Is it Time to Place an Order?
Yes
Idle state Waiting for demand
Demand occurs Unit withdrawn from inventory or
back ordered
....
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Fixed-Time Period System Under Certainty
q d(TL) - I - On Order_____
______________________
Number of units on hand
dT
dT
T
lt----------T--------gt
lt-------T------------gt
Average Q dT T Time between review L
Lead time d Demand per Unit of Time
....
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Fixed-Time Period System - Under Uncertainty
q d(TL) Buffer
Stock - I - On Order____________
Number of units on hand
dT
dT
T
lt---------T---------gt
lt-------T------------gt
Average Q dT T Time between review L
Lead time d Demand per Unit of Time
....
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FORECASTING DEMAND DURING L T PERIODS

• MEAN DEMAND dL
• _____
• STD. DEV. ?LT ?d ? T L
• We Assume Demand is Normally Distributed
• This Allows the Use of the E(Z) Concept

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Fixed-Time Period Model
....
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Determining the Value of Z
....
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Determining the Value of ?TL

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

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Example--Fixed-Time Period Model
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. How many units should be
ordered?
....
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Solution
No Interpolation, always choose the Highest Z
value
E(Z) Z 1.00 -0.90 0.92 -0.80
....
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Solution (continued)
To satisfy 96 percent of demand order 580 units
at this review period.
....
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Hybrid Model - Fixed-Time Period Model
No
Time to Place an Order or is status lt Reorder
point?
Yes
Idle state Waiting for demand
Demand occurs Unit withdrawn from inventory or
back ordered
....
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Miscellaneous SystemsOptional Replenishment
System
Maximum Inventory Level, M
q M - I
Actual Inventory Level, I
M
I
Q minimum acceptable order quantity If q gt Q,
order q.
....
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Miscellaneous SystemsBin Systems
Two-Bin System
Order One Bin of Inventory
Full
Empty
One-Bin System
Order Enough to Refill Bin
Periodic Check
....
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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 ....

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VILFREDO PARETO

• FAMOUS ITALIAN MATH., ECON. AND ENGI.
• SMALL OF POP GET LARGE OF INCOME
• 20-80 RULE (SMALL/LARGE)
• 80 OF SALES FROM 20 PRODUCTS
• 80 OF PROFITS FROM 20 OF PRODUCT
• 80 SALES FROM 20 OF PEOPLE
• 95 OF PROBLEMS FROM 5 OF CAUSES
• OTHER EXAMPLES

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ALLOCATE CONTROL RESOURCES TO THE VITAL FEW

• CONSIDER ANNUAL DOLLAR USAGE
• HIGHEST ANNUAL USAGE ARE A ITEMS
• REMAINING ARE B AND C ITEMS
• MORE CLOSELY CONTROL A
• LESS CLOSELY CONTROL B AND C
• ASSURE THAT WE DO NOT RUN OUT OF STOCK OF ANY
  ITEMS
• CONSIDER AN EXAMPLE

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CONSIDER A PURCHASING DEPT

• 30 ORDERS PER YEAR IS MAX. ACTIVITY
• 80 MIL OF A ITEMS
• 15 MIL OF B ITEMS
• 5 MIL OF C ITEMS
• EACH ARE ORDERED 10 TIMES PER YEAR

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ITEM Q Q/2 HQ/2

• A 80 10 8 4
  .254 1.000
• B 15 10 1.5 .75
  .1875
• C 5 10 .5 .25
  .0625
• 
  5.0 1.25
• A 80 20 4 2
  .5
• B 15 5 3 1.5
  .375
• C 5 5 1 .5
  .125
• 
  4.0 1.00

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Inventory Accuracy and Cycle Counting

• Inventory accuracy - Bank Teller Vs. Store Room
• Do inventory records agree with physical count?
• Eliminate the Causes of Errors
• Eliminate the Errors
• Cycle Counting
• Frequent counts
• Which items?
• When?
• By whom?

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INVENTORY MANAGEMENT USINGMARGINAL ANALYSIS

• ONE-TIME MODEL
• CLASSICALLY CALLED THE NEWSBOY PROBLEM
• ITEMS ORDERED ONLY ONCE
• TOO MANY - THEN ONLY SALVAGE VALUE
• TOO LITTLE - CUSTOMERS UPSET

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MP ML

• STOCK UNTIL MP (IF SOLD)ML (IF NOT)
• UNDER CONDITIONS OF UNCERTAINTY E(MP) E(ML)
• P1(MP) P2(ML) where P1 Prob of Selling and P2
  is Prob of Not Selling
• NOW P2 1-P1
• P1(MP)(1-P1)ML, yields
• P1 ML/(MPML), that is, stock items until the
  probability of selling the last item equals P1.

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LETS WORK A PROBLEM

• FAMOUS AMOS PROBLEM 22
• MP MR - MC .69-.49 .20
• ML MC - SAL .49-.29 .20
• .20
• P1 ------------------ .5
• .20 .20
• STOCK ITEMS UNTIL P(SELLING) THE LAST ITEM .5

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FORECASTS

• QTY. P(DQ) P(DgtQ)
• 1800 .05 1.00
• 2000 .10 .95
• 2200 .20 .85
• 2400 .30 .65
• 2600 .20 .35
• 2800 .10 .15
• 3000 .05 .05
• 1.00

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E(PROFIT) E(MP) - E(ML)
P1MPdQ - (1-P1)MLdQ

• E(P)1800 1.0.201800 - 0.0.201800 360
• E(P)2000 360 .95.20200 - .05.20200
  360 38 - 2
  396
• E(P)2200 396 .85.20200 - .15.20200
  396 34 - 6 424
• E(P)2400 424 .65.20200 - .35.20200
  424 26 - 14 436
• E(P)2600 436 .35.20200 - .65.20200
  436 14 - 26 424

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THE NEWSBOY PROBLEM IS COMMON

• NEWSPAPERS
• GREETING CARDS
• CALENDARS
• TAX GUIDES
• OVERBOOKING AIRLINE SEATS
• OTHER APPLICATIONS ?????

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CUSTOMER SERVICE IN SERVICES

• QUEUEING MODEL ARE USEFUL
• CONSIDER THE ONE-PERSON SERVICE CTR.
• RANDOM ARRIVAL RATE (POISSON)
• EXPONENTIAL SERVICES TIME
• SINGLE SERVER
• UNLIMITED QUEUE LENGTH
• FCFS
• Pop ? Queue ? Server ? Back to Population

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QUEUEING STATISTICS

• L arrival rate U service rate Both in
  Units/Time
• Ns L/(U-L) Mean Number in System
• Ts 1/(U-L) Mean Time in System
• p L/U Utilization Rate
• Pn (1-L/U)(L/U)n Prob. of n people in System

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A ONE SERVER QUEUING SYSTEMPoisson Arrivals and
Poisson Service Rates
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CONSIDER A HEALTH CLINIC

• L 18 per hour, U 19 per hour
• Ns L/(U-L) 18/(19-18) 18 people
• Ts 1/(U-L) 1/(19-18) 1 hour
• p L/U 18/19 .947
• Pn (1-L/U)(L/U)n
• Prob. of 30 people in System
• P30 (1-18/19)(18/19)30

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HOW TO IMPROVE SERVICE?

• 1)
• 2)
• 3)
• 4)