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 DISCOUNTSReconsider 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*QL QU

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.

CDQCQDCQTC 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(2365.1

331)6240)(12()331( TC

CDQCQDCQTC 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(233.1

600)6240)(12()600( TC

CDQCQDCQTC 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(2316.1

1000)6240)(12()1000( TC

CDQCQDCQTC 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(226.1

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 $61,292600-1000 $ 9.50 $1.33 336 600 $59,8041000-5000 $ 9.40 $1.316 337 1000 $59,3895000 $ 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