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Inventory ModelsInventory Models
Quantity Discount Models
TYPES OF QUANTITY TYPES OF QUANTITY DISCOUNT MODELSDISCOUNT MODELS
• Quantity Discount Models are of two types– All Units Discounts
• Above a certain order level, Q1, all units in the order are discounted (Usual Case)
– Incremental Discounts (Not Discussed Here)• Above a certain order level, Q1, all items above Q1
are discounted – the first Q1 items are sold at a non-discounted price
ALL UNIT QUANTITY DISCOUNTSALL UNIT QUANTITY DISCOUNTS
Reconsider the juicer example where:
H = .14, CO = 12, D = 6240/yr
Suppose the following all-unit discount pricing plan applies:
Quantity Unit Ordered Cost
<300 $10.00 300- 600 $ 9.75 600-1000 $ 9.50 1000-5000 $ 9.40 ≥5000 $ 9.00
PIECEWISE APPROACHPIECEWISE APPROACH• Consider each quantity discount for Ci as if it were
valid everywhere from 0 -
• STEP 1: For each value of Ci, calculate the corresponding value of Qi* -- it will change slightly since Ch = HCi and Ci changes slightly.
• STEP 2: For each quantity discount for Ci, consider the interval (from a lower limit QL to an upper limit QU) over which the price Ci is valid and determine the value of Q that gives the lowest cost for the interval. (See next slide.)
• STEP 3: Compare the lowest costs for each interval, and choose the lowest of these interval lowest costs.
DETERMINIG THE BEST VALUE OF Q DETERMINIG THE BEST VALUE OF Q FOR EACH INTERVAL AND OVERALLFOR EACH INTERVAL AND OVERALLBEST VALUE OF Q FOR AN INTERVAL
BETWEEN QL AND QU
–If Q* >QU, ignore this interval
–if Qi* is in interval, Qopt = Qi*
–If Q* < QL, Qopt = QL
QQOPTOPT for the interval in which Q for the interval in which Qii* > Q* > QUU
TC
QQi*QL QU
Lowest point occurs at QU.Qopt = QU
But we can see that Q*(which will have an evendeeper discount) gives a
lower total cost, TC.
QQOPTOPT When Q When Qii* Is In the Interval * Is In the Interval
TC
QQ*QLQU
Q* is the lowest point in the interval.Qopt = Q*
QQOPTOPT for the interval in which Q for the interval in which Qii* < Q* < QLL
TC
QQ* QL QU
Lowest point occurs at QL.Qopt. = QL
Calculations for C = $10Calculations for C = $10Interval (0,300)Interval (0,300)
327)10(14.
)6240)(12(2* Q
Since Q* > 300, the optimal solution for the model cannot come from this interval.
CDQC
Q
DCQTC opt
H
opt
Oopt
2)(
Calculations for C = $9.75Calculations for C = $9.75Interval (300,600)Interval (300,600)
331)75.9(14.
)6240)(12(2* Q
Since Q* = 331 is in the interval (300,600), for this interval: Qopt = Q* = 331
292,61$)6240)(75.9()331(2
365.1
331
)6240)(12()331( TC
CDQC
Q
DCQTC opt
H
opt
Oopt
2)(
Calculations for C = $9.50Calculations for C = $9.50Interval (600,1000)Interval (600,1000)
336)50.9(14.
)6240)(12(2* Q
Since Q* = 336 < 600,for this interval: Qopt = 600
804,59$)6240)(50.9()600(2
33.1
600
)6240)(12()600( TC
CDQC
Q
DCQTC opt
H
opt
Oopt
2)(
Calculations for C = $9.40Calculations for C = $9.40Interval (1000,5000)Interval (1000,5000)
337)40.9(14.
)6240)(12(2* Q
Since Q* = 337 < 1000,for this interval: Qopt = 1000
389,59$)6240)(40.9()1000(2
316.1
1000
)6240)(12()1000( TC
CDQC
Q
DCQTC opt
H
opt
Oopt
2)(
Calculations for C = $9.00Calculations for C = $9.00Interval (5000,Interval (5000,))
345)00.9(14.
)6240)(12(2* Q
Since Q* = 345 < 5000,for this interval: Qopt = 5000
325,59$)6240)(00.9()5000(2
26.1
5000
)6240)(12()5000( TC
QUANTITY DISCOUNT QUANTITY DISCOUNT APPROACH FOR ALLENAPPROACH FOR ALLEN
Quantity Unit Cost Ch Q* Qopt TC
< 300 $10.00 $1.40 327 ---- ----
300-600 $ 9.75 $1.365 331 331 $61,292
600-1000 $ 9.50 $1.33 336 600 $59,804
1000-5000 $ 9.40 $1.316 337 1000 $59,389
5000 $ 9.00 $1.26 345 5000 $59,325• ORDER 5000ORDER 5000
Note: This is over a 9 month supply -- is this OK?Note: This is over a 9 month supply -- is this OK?
REORDER POINT ANALYSISREORDER POINT ANALYSIS
• Reorder point and safety stock determination are not affected by quantity discounts.
• They are found in the same way as before:– r* = LD + SS
if demand is constant over lead time – r* is found using service levels
if demand varies during lead time
Using the TemplateUsing the Template
All-Units Worksheet
Enter Values
Enter Discount Breaksand Discount Prices
Optimal Values
ReviewReview• Types of Discount Models:
– All-unit and Incremental
• All-Units Quantity Discount Model– Qi* is found for each interval from QL to QU
– Best point for interval is Qi* if Qi* is in interval
– If Q* < QL, QL is best point for interval
• Best value of Q is found by finding the lowest cost for each interval and taking the Q with the lowest total cost.
• Reorder point analysis is not affected.
• Use of template