Title: Inventory Systems for Independent Demand
1Inventory 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
2Inventory
- Definition--The stock of any item or resource
used in an organization - Raw materials
- Finished products
- Component parts
- Supplies
- Work in process ....
3INV. MANAGEMENT OBJECTIVES
- MAXIMUM CUSTOMER SERVICE
- MINIMUM INV. INVESTMENT
- MAXIMUM EFFICIENCY (MIN. COST)
- MAXIMUM PROFIT
- HIGHEST RETURN ON INVESTMENT
- TO USE INVENTORY FOR STRATEGIC ADVANTAGE
4TRADE-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
5Purposes 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.
63 QUESTIONS OF INVENTORY SYSTEM
- WHAT TO ORDER?
- HOW MANY TO ORDER?
- WHEN TO ORDER?
- TIMING IS A CRITICAL ISSUE!
7Inventory Costs
- Holding (or carrying) costs
- Setup (or production change) costs
- Ordering costs
- Shortage costs ....
8SEVERAL 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
9Independent vs. Dependent Demand
Independent Demand (Demand not related to other
items)
Dependent Demand (Derived)
E(1)
....
10Classifying Independent Demand Systems
- Fixed-Order Quantity
- Fixed-Time Period
- Hybrid Systems - Both Together - Order
Periodically unless we reach the reorder point.
11Fixed-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Demand occurs Unit withdrawn from inventory or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
....
12Fixed-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 ....
13Fixed-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) ....
14EOQ 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
....
15Cost 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)
....
16Basic 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
....
17Deriving the EOQ - Calculus Method
- Using calculus, we take the derivative of the
total cost function and set the derivative
(slope) equal to zero ....
18Deriving the EOQ - Graphical Method
- Recognizing TAHC TAOC at Optimal
- TAHC HQ/2 TAOC SD/Q
- HQ/2 SD/Q, Solving for Q Yields
19EOQ 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
15
Determine the economic order quantity and the
reorder point.
....
20Solution
Why do we round up?
....
21In-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) ....
22Solution
....
23ACHIEVING 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
24ADDING 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
25STRATEGIC 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?
26THE 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.
27EXAMPLES 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
28FORECASTING 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
29STATISTICAL 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
30DETERMINING 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
31E(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
32LETS 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.
33Fixed-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
....
34Fixed-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
....
35Fixed-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
....
36FORECASTING 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
37Fixed-Time Period Model
....
38Determining the Value of Z
....
39Determining the Value of ?TL
- The standard deviation of a sequence of random
events equals the square root of the sum of the
variances ....
40Example--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?
....
41Solution
No Interpolation, always choose the Highest Z
value
E(Z) Z 1.00 -0.90 0.92 -0.80
....
42Solution (continued)
To satisfy 96 percent of demand order 580 units
at this review period.
....
43Hybrid 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
....
44Miscellaneous 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.
....
45Miscellaneous SystemsBin Systems
Two-Bin System
Order One Bin of Inventory
Full
Empty
One-Bin System
Order Enough to Refill Bin
Periodic Check
....
46ABC 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 ....
47VILFREDO 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
48ALLOCATE 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
49CONSIDER 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
50ITEM 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
51Inventory 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?
52INVENTORY 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
53MP 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.
54LETS 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
55FORECASTS
- 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
56E(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
57THE NEWSBOY PROBLEM IS COMMON
- NEWSPAPERS
- GREETING CARDS
- CALENDARS
- TAX GUIDES
- OVERBOOKING AIRLINE SEATS
- OTHER APPLICATIONS ?????
58CUSTOMER 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
59QUEUEING 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
60A ONE SERVER QUEUING SYSTEMPoisson Arrivals and
Poisson Service Rates
61CONSIDER 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
62HOW TO IMPROVE SERVICE?