cover some realistic situations which relax one or more of the eoq assumptions
DESCRIPTION
TRANSCRIPT
04/09/23 Special Inventory Models CJA10.14.1
• Cover some realistic situations which relax one or more of the EOQ assumptions– Non-Instantaneous Replenishment– Quantity Discounts– One-Period Decisions
Operations Management
Special Inventory Models
04/09/23 Special Inventory Models CJA10.14.2
• Item used or sold as they are completed, without waiting for a full lot to be completed
• Usual case is where production rate, p, exceeds the demand rate, d, so there is a buildup rate of (p – d) units during time when both production and demand occur.
• Both p and d expressed in same time interval.
Special Inventory Models
Non-Instantaneous Replenishment
04/09/23 Special Inventory Models CJA10.14.3
• Recall…TC = annual holding costs plus annual setup costs
=
Ave. Inv. Level =
• Now...
Avg. Inv. Level =
Non-Instantaneous Replenishment
Economic Lot Size (ELS)
2
)0(min)(max levelQlevel
2
)0(min)(max max levelIlevel
04/09/23 Special Inventory Models CJA10.14.4
• Buildup at a rate of ( p-d) (units/day) during production phase until a lot size of Q is produced
• Buildup continues for Q / p days (units/units/day)
• Imax = units per day, ( p-d),
x number of days (Q / p )
where: p = production rated = demand rateQ = lot size
Non-Instantaneous Replenishment
Economic Lot Size (ELS)
04/09/23 Special Inventory Models CJA10.14.5
Total Cost = Annual holding costs + annual ordering costs
Setting up the total cost equation, where D is the annual demand:
Non-Instantaneous Replenishment
Economic Lot Size (ELS)
2
max SQ
DH
ITC
04/09/23 Special Inventory Models CJA10.14.6
• Differentiation of this equation with respect to Q,setting the result equal to zero, and solving for Q results in the
Economic Production Lot Size, ELS:
• Since p > d, the second term is greater than 1,
– so the ELS is __________ than the EOQ
Non-Instantaneous Replenishment
Economic Lot Size (ELS)
H
DS2 ELS
04/09/23 Special Inventory Models CJA10.14.7
Demand = 30 barrels/day Setup cost = $200Production rate = 190 barrels/day Annual holding cost = $0.21/barrelAnnual demand = 10,500 barrels Plant operates 350 days/year
ELS = pp - d
2DSH
Non-Instantaneous Replenishment
Example
04/09/23 Special Inventory Models CJA10.14.8
Demand = 30 barrels/day Setup cost = $200Production rate = 190 barrels/day Annual holding cost = $0.21/barrelAnnual demand = 10,500 barrels Plant operates 350 days/year
Non-Instantaneous Replenishment
Example
SELS
DH
p
dpELSTC
2
04/09/23 Special Inventory Models CJA10.14.9
TBOELS = (350 days/year)ELSD
Demand = 30 barrels/day Setup cost = $200Production rate = 190 barrels/day Annual holding cost = $0.21/barrelAnnual demand = 10,500 barrels Plant operates 350 days/year
Non-Instantaneous Replenishment
Example
04/09/23 Special Inventory Models CJA10.14.10
Production time = ELSp
Demand = 30 barrels/day Setup cost = $200Production rate = 190 barrels/day Annual holding cost = $0.21/barrelAnnual demand = 10,500 barrels Plant operates 350 days/year
Non-Instantaneous Replenishment
Example
04/09/23 Special Inventory Models CJA10.14.11
• Quantity discounts are price incentives to purchase large quantities
• Price break is the minimum purchase quantity to get a certain discount price
• The item’s price is no longer fixed so there arethree relevant cost components
– annual purchase costs in addition to annual holding costs and annual ordering (setup) costs
Special Inventory Models
Quantity Discounts
PDSQ
DH
QTC
2
04/09/23 Special Inventory Models CJA10.14.12
• There are cost curves for each price level
• The feasible total cost begins at the top curve,
then drops down, curve by curve,
at the price breaks.
• The EOQs do not necessarily produce the
best (“minimum total annual cost”) lot size.
Quantity Discounts
Feasible Price-Quantity Combinations
04/09/23 Special Inventory Models CJA10.14.13
C for P = $4.00
C for P = $3.50
C for P = $3.00
Total annualcost, $
Purchase quantity, Q
0 100 200
(a) Total cost curves with purchased materials added
Purchase Discounts
Total Cost Curves
Firstprice break
Second price break
04/09/23 Special Inventory Models CJA10.14.14
Step 1:
Beginning with the lowest price, calculate the EOQ for each price level until a feasible EOQ is found.
– it is feasible if the quantity lies in the range corresponding to its price.
As subsequent prices are larger than the previous one,the holding cost, H, (H = i·P ) gets larger.
Since H is in the denominator of the EOQ formula,
the EOQ gets smaller.
Purchase Discounts
Solution Procedure
04/09/23 Special Inventory Models CJA10.14.15
Annual demand = 936 unitsOrdering cost = $100.00Holding cost = 25% of unit price
Order Quantity Price per Unit
0 - 249 $60.00250 - 499 $59.00500 or more $58.00
Purchase Discounts
Example
Pi
SDEOQ
2
00.58$
04/09/23 Special Inventory Models CJA10.14.16
$55,000
$56,000
$57,000
$58,000
$59,000
$60,000
$61,000
$62,000
0 250 500 750
Order Quantity, Q
Tota
l Ann
ual C
ost,
$
Purchase Discounts
Example
Price = $60.00
Price = $59.00
Price = $58.00
04/09/23 Special Inventory Models CJA10.14.17
Step 2:
If the first feasible EOQ found is for the lowest price level, this quantity is the best lot size.
Otherwise, calculate the total cost for the first feasible EOQ and for the most economical, feasible order quantity at each lower price level.
The quantity with the lowest total cost is optimal.
Purchase Discounts
Solution Procedure
04/09/23 Special Inventory Models CJA10.14.18
$55,000
$56,000
$57,000
$58,000
$59,000
$60,000
$61,000
$62,000
0 250 500 750
Order Quantity, Q
Tota
l Ann
ual C
ost,
$
Purchase Discounts
Example
Price = $60.00
Price = $59.00
Price = $58.00
04/09/23 Special Inventory Models CJA10.14.19
Annual demand = 936 unitsOrdering cost = $100.00Holding cost = 25% of unit price
Order Quantity Price per Unit
0 - 249 $60.00250 - 499 $59.00500 or more $58.00
Purchase Discounts
Example
DPSQ
DPi
QTCQ
2
04/09/23 Special Inventory Models CJA10.14.20
• The best purchase quantity is 250 units, which does not correspond to the deepest discount price.
• This is not always true - EOQ is affected by:
– small discounts, quantity break points,
– large holding cost, and
– small demand.
• Small lot sizes may be better even though the price is not the lowest
Quantity Discounts
Example
04/09/23 Special Inventory Models
• 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)
Special Inventory Models
Inventory Models
04/09/23 Special Inventory Models CJA10.14.22
• Problem for seasonal and high fashion goods.
• Only allowed to order one time.
• Short selling seasons and long lead times prohibit the possibility of placing a second order.
• A balance between ordering enough to meet demand and not having any left over at the end of the season.
• Sometimes referred to as the ”Newsvendor” problem
Special Inventory Models
One-Period Decisions
04/09/23 Special Inventory Models CJA10.14.23
• List different demand levels and probabilities
• Develop a payoff table, where each new row represents a different order quantity and each column represents a different demand.
One-Period Decisions
Selecting the Purchase Quantity
04/09/23 Special Inventory Models CJA10.14.24
• The payoff is:
where: p = profit per unit sold during the season l = loss per unit disposed of after the seasonQ = purchase quantityD = demand level
One-Period Decisions
Selecting the Purchase Quantity
demand) exceedsquantity (purchase if
markup) fullat sold units (all if Payoff
DQ
DQ
DQlpD
pQ
04/09/23 Special Inventory Models CJA10.14.25
• Calculate the expected payoff of each Q. For a specific Q, first multiply each payoff by its demand probability, and then add the products.
• Choose the order quantity Q with the highest expected payoff.
One-Period Decisions
Selecting the Purchase Quantity
04/09/23 Special Inventory Models CJA10.14.26
For one item, p = $10 and l= $5. The probability distribution for the season’s demand is:
Demand Demand(D) Probability10 0.220 0.330 0.340 0.150 0.1
One-Period Decisions
Example
04/09/23 Special Inventory Models CJA10.14.27
Complete the following payoff matrix, as well as the column on the right showing expected payoff.
D ExpectedQ 10 20 30 40 50 Payoff--- (.2) (.3) (.3) (.1) (.1) ---
10 $100 $100 $100 $100 $100 $10020 50 200 200 200 200 17030 0 ____ 300 ____ 300 ____40 –50 100 250 400 400 17550 –100 50 200 350 500 140
One-Period Decisions
Example
04/09/23 Special Inventory Models CJA10.14.28
Payoff if Q = 30 and D = 20:
Payoff if Q = 30 and D = 40:
Expected payoff if Q = 30:
What is the best choice for Q?
One-Period Decisions
Example