Inventory Hecl
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Transcript of Inventory Hecl
1
Supply Chain Management
Traditional Inventory Models for Independent Demand
Prof. Prem Vrat
Vice Chancellor
UP Technical University
Lucknow
2
What is an Inventory System
Inventory is defined as the stock of any item or resource used in an organization.
An Inventory System is made up of a set of policies and controls designed to monitor the levels of inventory and designed to answer the following questions: What levels should be maintained? When stock should be replenished? and How large orders should be? i.e. what is the
optimal size of the order?
3
Cost of holding inventory
Capital cost (Interest) Storage cost Obsolescence cost Consist of about 20% of total cost in the
United States
4
Inventory issues
Demand Constant vs. variable deterministic vs. stochastic
Lead time Review time
Continuous vs. periodic Excess demand
Backorders, lost sales Inventory change
Perish, obsolescence
Inventory Decisions: When, What, and how many to order
5
Two basic types of Inventory Systems
1) continuous (fixed-order quantity) an order is placed for the same constant amount
when inventory decreases to a specified level, ie. Re-order point
2) periodic (fixed-time) an order is placed for a variable amount after a
specified period of time used in smaller retail stores, drugstores, grocery
stores and offices
6
basic inventory elements
1.1. Carrying cost, Carrying cost, Cc
• Include facility operating costs, record keeping, interest, etc.
2.2. Ordering cost, Ordering cost, Co
• Include purchase orders, shipping, handling, inspection, etc.
3.3. Shortage (stock out) cost, Shortage (stock out) cost, Cs
• Sometimes penalties involved; if customer is internal, work delays could result
Carrying Costs
Category
Cost (and Range) as a Percent of Inventory
Value
Housing costs (including rent or depreciation, operating costs, taxes, insurance)
6% (3 - 10%)
Material handling costs (equipment lease or depreciation, power, operating cost)
3% (1 - 3.5%)
Labor cost 3% (3 - 5%)
Investment costs (borrowing costs, taxes, and insurance on inventory)
11% (6 - 24%)
Pilferage, space, and obsolescence 3% (2 - 5%)
Overall carrying cost 26%
8
Inventory
- to study methods to deal with
“how much stock of items should be kept on hands that would meet customer demand”
Objectives are to determine:
a) how much to order, and
b) when to order
9
Inventory models
Here, we onlyonly study the following three different models:
1. Basic model
2. Model with “discount rate”
3. Model with “re-order points”
10
1. Basic model
The basic model is known as:
“Economic Order Quantity” (EOQ) Models
Objective is to determine the optimal order size that will minimize total inventory costs
How the objective is being achieved?
11
Quantityon hand
Q
Receive order
Placeorder
Receive order
Placeorder
Receive order
Lead time
Reorderpoint
Usage rate
Profile of Inventory Level Over Time
12
Profile of … Frequent Orders
13
Basic EOQ models
Three models to be discussed:
1. Basic EOQ model
2. EOQ model without instantaneous
receipt
3. EOQ model with shortages.
14
Basic Fixed-Order Quantity Model
This model attempts to estimate the order size (Q) and determine the point (R) at which an order should be placed.
Model assumptions:
1. Annual demand (D) for the product is known, constant and uniform throughout the period,
2. Lead time (L) is known and constant,
3. Product unit price (C) is known and constant,
4. Per unit holding or carrying cost (Cc) is known and constant,
5. Ordering or setup cost (Co) is known and constant,
15
Basic Fixed-Order Quantity Model
Model assumptions:
6 No backorders are allowed,
7. There is no interaction with other products, the inventory control system operates independent of its environment.
16
1 Economic Order Quantity (EOQ) Model
QOrder Size
L Time
Q/2
d
L
R
Daily Usage Rate
17
The Basic EOQ Model
• The optimal order size, Q, is to minimize the sum of carrying costs and ordering costs.
• Assumptions and Restrictions: - Demand is known with certainty and is relatively constant over time. - No shortages are allowed. - Lead time for the receipt of orders is constant. (will consider later) - The order quantity is received all at once and instantaniously.
How to determinethe optimal valueQ*?
18
Determine of Q
We try to Find the total cost that need to spend for keeping
inventory on hands = total ordering + stock on hands Determine its optimal solution by finding its first
derivative with respect to Q
How to get these values?1. Find out the total carrying cost2. Find out the total ordering cost3. Total cost = 1 + 24. d (Total cost) /d Q = 0, and find Q*
19
The Basic EOQ ModelWe assumed that, we will only keep half the inventory over a year then
The total carry cost/yr = Cc x (Q/2). Total order cost = Co x (D/Q)
Then , Total cost = 2QC
QDCTC co Finding optimal Q*
20
The Basic EOQ Model
• EOQ occurs where total cost curve is at minimum value and carrying cost equals ordering cost:
•Where is Q* located in our model?
c
o
co
CDCQ
QCQDCTC
2
2
*
min
(How to obtain this?)Then, *c
o
co
CDCQ
QCQDCTC
2
2
*
min
21
EOQ Derivation
Min. The Total Cost Function by finding first derivative and equating it to zero:
TC = DCo/Q + QCc/2
dTC
Solving for Q: EOQ =
We could achieve the same result by equating Holding (Carrying Cost) to Ordering Cost and solve for Q .
Reorder point: R = d.L + SS (safety stock)
dQ = (- DCo/Q ) + Cc/2 = 0
2
2DCoCc
22
The Basic EOQ Model
• Total annual inventory cost is sum of ordering and carrying cost:
2QC
QDCTC co
Figure The EOQ cost model
To order inventory
To keep inventory
Try to get this value
23
The Basic EOQ ModelExample
Consider the following:
days store 62.2 5
311*/
days 311 timecycleOrder
5000,2000,10 :yearper orders ofNumber
500,1$2
)000,2()75.0(000,2000,10)150(
2 :costinventory annual Total
yd 000,2)75.0(
)000,10)(150(22* :sizeorder Optimal
10,000yd D $150, C $0.75, C :parameters Model
*
*min
oc
QD
QD
QCQDCTC
CDCQ
optco
c
o
No. of working days/yr
*
Note: You should pay attention thatall measurement units must be the same
Consider the same example, with yearly
24
The Basic EOQ ModelEOQ Analysis with monthly time frame
$1,500 ($125)(12) cost inventory annual Total
monthper 125$2
)000,2()0625.0(000,2
)3.833()150(2*
* :costinventory monthly Total
yd 000,2)0625.0(
)3.833)(150(22* :sizeorder Optimal
monthper yd 833.3 D order,per $150 C month,per ydper $0.0625 C :parameters Model
min
oc
QCQDCTC
CDCQ
co
c
o
(unit be based on yearly)
12 months a year
Robust Model
The EOQ model is robustThe EOQ model is robust
It works even if all parameters It works even if all parameters and assumptions are not metand assumptions are not met
The total cost curve is relatively The total cost curve is relatively flat in the area of the EOQflat in the area of the EOQ
26
2 Fixed-Order Quantity Model With Usage
In this model production and usage of the item being manufactured occur simultaneously. The graph below illustrates the model.
TC = DC + DS/Q +
Production Rate, p
Usage Rate, d
R
Q
L
Build up
Usage Rate
(p-d)
No production
usage only
Max.
(p - d).Q.H 2p
27
The EOQ Model with Noninstantaneous Receipt
Figure The EOQ model with noninstantaneous order receiptAlways greater than 0why?
The order quantity is received gradually over time and inventory is drawn on
at the same time it is being replenished.
Example: Let p = production, d = demand,
28
The EOQ Model with Noninstantaneous ReceiptModel Formulation
)/1(2 :sizeorder Optimal
12
:cost inventory annual Total
demanded isinventory at which ratedaily drate) productionor ( over time received isorder heat which t ratedaily p
*
pdCDCQ
pdQC
QDCTC
c
o
co
Assuming placing an order/yr
29
The EOQ Model with finite-replenishment rate
yd 772,1150
2.3218.256,21* levelinventory Maximum
runs 43.48.256,2
000,10*
runs)n (productioyear per orders ofNumber
days 05.15150
8.256,2* length run Production
329,1$150
2.3212
)8.256,2()075(.)8.256,2()000,10()150(1
2*
*min:costinventory annual minimum Total
yd 8.256,2
1502.32175.0
)000,10)(150(2
1
2* :sizeorder Optimal
dayper yd 150 p day,per yd 32.2 10,000/311 year,per yd 10,000 D unit,per $0.75 Cc $150, Co
pdQ
QD
pQ
pdQCc
QDCoTC
pdCc
CoDQ
Let,
30
The EOQ Model with Shortages
• In the EOQ model wth shortages, the assumption that shortages cannot exist is relaxed.
• Assumed that unmet demand can be backordered with all demand eventually satisfied.
Shortage = S/Q
On hand = (Q-S)/Q t1 + t2 = S/D + (Q-S)/D = Q/D
Shortage
What we needed
Max level of inventory
Here, we allow Q being shortageshortage, so that we could borrow or replenish the stockslater
Total cost is
31
3 The EOQ Model with Shortages
• In the EOQ model wth shortages, the assumption that shortages cannot exist is relaxed.
• Assumed that unmet demand can be backordered with all demand eventually satisfied.
Shortage = S/Q
On hand = (Q-S)/Q t1 + t2 = S/D + (Q-S)/D = Q/D
Shortage
What we needed
½*base* height = ½ * (Q-S) * (Q-S)/Q = ½ * (Q-S)2 /Q
Area = ½ * (S/Q) * S = ½ * S2 /Q
32
The EOQ Model with Shortages
sc
c
s
cs
c
o
ocs
CCCQS
C
CC
C
DCQ
Q
DC
Q
SQC
Q
SCTC
** :level Shortage
2*:quantityorder Optimal
2
)(
2:costinventory Total
cost ordering total costs carrying total costs shortage Total cost Total
22
33
The EOQ Model with ShortagesExample
$1,279.20 639.60 465.16 $174.44
2.345,2)000,10)(150(
)2.345,2(2)6.705,1)(75.0(
)2.345,2(2)6.639)(2(
**2*)*(
*2:costinventory Total
yd 6.63975.02
75.02.345,2* :level Shortage
yd 2.345,22
75.0275.0
)000,10)(150(22:quantityorder Optimal
yd 10,000 D yd,per $2 C yd,per $0.75 C $150, C
2222
*
*
sco
QDC
QSQC
QSCTC
CCCQS
CCC
CDCQ
ocs
sc
c
s
cs
c
o
Let,
34
The EOQ Model with Shortages
Additional Parameters in Example
sday 19.9or year 0.064 10,000639.6
D t shortage a is re which theduring Time
days 53.2or 0.17110,000
639.6-2,345.2 t handon isinventory which during Time
ordersbetween days 0.7326.4
311
orders ofnumber
yearper days t ordersbetween Time
yd 6.705,16.6392.345,2 levelinventory Maximum
yearper orders 26.42.345,2
000,10 orders ofNumber
2
1
SD
SQ
SQ
QD
= Q/D
35
4. Model with “discount rate”
Price discounts are often offered if a predetermined number of units is ordered or when ordering materials in high volume.
How do we decide if we should order more to take advantage of the discount being offered?
36
All-Unit Quantity Discounts
Pricing schedule has specified quantity break points q0, q1, …, qr, where q0 = 0
If an order is placed that is at least as large as qi but smaller than qi+1, then each unit has an average unit cost of Ci
The unit cost generally decreases as the quantity increases, i.e., C0>C1>…>Cr
The objective for the company (a retailer for example) is to decide on a lot size that will minimize the sum of material, order, and holding costs
37
All-Unit Quantity Discount Procedure
Step 1: Calculate the EOQ for the lowest price. If it is feasible (i.e., this order quantity is in the range for that price), then stop. This is the optimal lot size. Calculate TC for this lot size.
Step 2: If the EOQ is not feasible, calculate the TC for this price and the smallest quantity for that price.
Step 3: Calculate the EOQ for the next lowest price. If it is feasible, stop and calculate the TC for that quantity and price.
Step 4: Compare the TC for Steps 2 and 3. Choose the quantity corresponding to the lowest TC.
Step 5: If the EOQ in Step 3 is not feasible, repeat Steps 2, 3, and 4 until a feasible EOQ is found.
38
All-Unit Quantity Discounts: Example
Cost/Unit
$3$2.96
$2.92
Order Quantity
5,000 10,000
Order Quantity
5,000 10,000
Total Material Cost
39
All-Unit Quantity Discount: Example
Order quantity Unit Price0-5000 $3.005001-10000 $2.96Over 10000 $2.92
q0 = 0, q1 = 5000, q2 = 10000C0 = $3.00, C1 = $2.96, C2 = $2.92D = 120000 units/year, Co = $100/lot, Cc = 0.2
40
All-Unit Quantity Discount: Example
Step 1: Calculate Q2* = Sqrt[(2DCo)/CcC2] = Sqrt[(2)(120000)(100)/(0.2)(2.92)] = 6410Not feasible (6410 < 10001)Calculate TC2 using C2 = $2.92 and q2 = 10001TC2 = (120000/10001)(100)+(10001/2)(0.2)(2.92)+ (120000)(2.92)
= $354,520Step 2: Calculate Q1* = Sqrt[(2DCo)/CcC1]=Sqrt[(2)(120000)(100)/(0.2)(2.96)] = 6367Feasible (5000<6367<10000) StopTC1 = (120000/6367)(100)+(6367/2)(0.2)(2.96)+ (120000)(2.96) =
$358,969TC2 < TC1 The optimal order quantity Q* is q2 = 10001
41
All-Unit Quantity Discounts
What is the effect of such a discount schedule? Retailers are encouraged to increase the size of
their orders Average inventory (cycle inventory) in the supply
chain is increased Average flow time is increased Is an all-unit quantity discount an advantage in the
supply chain?
42
5. Model with “re-order points”
• The reorder point is the inventory level at which a new order is placed.
• Order must be made while there is enough stock in place to cover demand during lead time.
• Formulation: R = dL, where d = demand rate per time period, L = lead time
Then R = dL = (10,000/311)(10) = 321.54
Working days/yr
43
Reorder Point• Inventory level might be depleted at slower or faster rate during lead time.
• When demand is uncertain, safety stock is added as a hedge against stockout.
Two possible scenarios
Safety stock!
No Safetystocks!
We should then ensureSafety stock is secured!
44
Determining Safety Stocks Using Service Levels
• We apply the Z test to secure its safety level,
)( LZLdR d
Reorder point
Safety stock
Average sample demand
How these values are represented in the diagram of normal distribution?
45
Reorder Point with Variable Demand
stocksafety
yprobabilit level service toingcorrespond deviations standard ofnumber demanddaily ofdeviation standard the
timeleaddemanddaily average
pointreorder where
LZ
Z
Ld
R
LZLdR
d
d
d
46
Reorder Point with Variable DemandExample
Example: determine reorder point and safety stock for service level of 95%.
26.1. : formulapoint reorder in termsecond isstock Safety
yd 1.3261.26300)10)(5)(65.1()10(30
1.65 Zlevel, service 95%For
dayper yd 5 days, 10 L day,per yd 30 d
LZLdR
d
d