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Inventory Aggregation and Discounting

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Inventory

Aggregation and Discounting

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Outline

Joint fixed costs for multiple products

Long term quantity discounts

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3

Example: Lot Sizing with Multiple Products

 Shipping multiple products over the same route to the same retailer  Demand per year

RL = 12,000; RM = 1,200; RH = 120

 Common transportation cost per delivery,

S = $4,000

 Product specific order cost per product in each delivery

sL = $1,000; sM = $1,000; sH = $1,000

 Holding cost,

h = 0.2

 Unit cost

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Delivery Options

 No Aggregation:

– Each product ordered separately

 Complete Aggregation:

– All products delivered on each truck

 Tailored Aggregation:

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No Aggregation:

Order each product independently

Litepro

Medpro

Heavypro

Demand per year

12,000

1,200

120

Fixed cost / order

$5,000

$5,000

$5,000

Optimal order size

1,095

346

110

Order frequency

11.0 / year

3.5 / year

1.1 / year

Annual cost

$109,544

$34,642

$0,954

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Complete Aggregation: Order jointly

All Products in All Trucks

Total ordering cost S*=S+

s

L

+

s

M

+

s

H

= $7,000

n: common ordering frequency

Annual ordering cost = n S*

Total holding cost:

Total cost:

R n hC R n hC R n hC L L M M H H 2  2  2

TC n S n h n R C R C R C n h R C R C R C S L L M M H H L L M M H H ( ) * * *        2 2
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Complete Aggregation:

Order all products jointly

Litepro

Medpro Heavypro

Demand per year

12,000

1,200

120

Order frequency

9.75/year 9.75/year 9.75/year

Optimal order size

1,230

123

12.3

Annual holding cost

$61,512

$6,151

$615

Annual order cost = 9.75×$7,000 = $68,250

Annual total cost = $136,528

Ordering high and low volume items at the same frequency cannot be a good idea.

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Tailored Aggregation:

Ordering Selected Subsets

 Example: Orders may look like (L,M); (L,H); (L,M); (L,H).

 Most frequently ordered product: L

 M and H are ordered in every other delivery.

 We can associate fixed order cost S with product L because it is ordered every time there is an order.

 Products other than L, the rest are associated only with their incremental order costs (s values).

An Algorithm:

Step 1: Identify most frequently ordered product

Step 2: Identify frequency of other products as a relative multiple

Step 3: Recalculate ordering frequency of most frequently ordered product Step 4: Identify ordering frequency of all products

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Tailored Aggregation:

Ordering Selected Subsets

 i is the generic index for items, i is L, M or H.  Step 1: Find most frequently ordered item:

The frequency of the most frequently ordered item will be modified later. This is an approximate computation.

 Step 2: Relative order frequency of other items, mi

mi are relative order frequencies, they must be integers. They do not change in the remainder.

n hC R S s n n i i i i i    2( ) max{ }

n

hC R

s

m

n

n

i i i i i i

2

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Tailored Aggregation:

Ordering Selected Subsets

 Step 3: Recompute the frequency of the most frequently ordered item. This item is ordered in every order whereas others are ordered in every mi orders. The average fixed ordering cost is:

S s m i i i

Annual ordering cost

Annual holding cost

  

n S s m R n m hC i i i i i i i ( ) / 2 textbook the of p.274 on (10.9) formula 2 *        

i i i i i i i m s S hC m R n
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Tailored Aggregation:

Ordering Selected Subsets

Step 4: Recompute the ordering frequency n

i

of other

products:

Total Annual ordering cost: nS+n

H

s

H

+n

M

s

M

+n

L

s

L

– n (the frequency of the most frequently ordered product) is one of the following values nH, nM, nL

Total Holding cost:

n

n

m

i i

R

n

hC

R

n

hC

R

n

hC

L L L M M M H H H

2

2

2

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Tailored Aggregation:

Ordering Selected Subsets

Step 1:

Step 2:

n hC R S s n n n n L L L L M H i     2( ) = 11, = 3.5, = 1.1 max{ } 11

n

hC R

s

n

m

n

n

m

M M M M H M M H

 

2

= 7.7 ,

= 2.4;

2

,

5

Item L is ordered most frequently.

Every other L order contains one M order. Every 5 L orders contain one H order.

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Tailored Aggregation:

Ordering Selected Subsets

 Step 3:  Step 4:

 Total ordering cost:

– nS+nHsH+nMsM+nLsL=11.47(4000)+11.47(1000)+5.73(1000)+2.29(1000) =45,880+11,470+5,730+2,290=65370

 Total holding cost

73 . 5 *   M M m n n R n hC R n hC R n hC L L L M M M H H H 2 2 2 12000 2 11 47 0 2 500 1200 2 5 73 0 2 500 120 2 2 29 0 2 500      ( . ) ( . ) ( . ) ( . ) ( . ) ( . ) 29 . 2 *   H H m n n =(528.1+104.71+26.2)100 47 . 11 ) 5 / 1000 2 / 1000 1 / 1000 4000 ( 2 ) 5 * 120 2 * 1200 1 * 12000 ( 500 ) 2 . 0 ( 2 *               

i i i i i i i m s S m R hC n
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Tailored Aggregation: Order selected subsets

Litepro

Medpro Heavypro

Demand per year

12,000

1,200

120

Order frequency

11.47/year 5.73/year 2.29/year

Optimal order size

1046.2

209.4

42.4

Annual holding cost

$52,810

$10,470

$2,630

Annual order cost = $65,370

Total annual cost =

$130,650

Compare with $136K of total aggregation and with $155K of no aggregation

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Lessons From Aggregation

 Aggregation allows a firm to lower lot size without increasing cost – Order frequencies without aggregation and with tailored aggregation

» (11; 3.5; 1.1) vs. (11.47; 5.73; 2.29)

» More frequent ordering implies smaller order sizes

 Tailored aggregation is effective if

product specific fixed cost

is a

large fraction of joint fixed cost

 Complete aggregation is effective if product specific fixed cost is a

small fraction of

joint fixed cost

 Information technology can decrease product specific ordering costs.

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The word of the moment: Retail

 Retail: The sale of goods in small quantities directly to the customer. Opposite of the word wholesale.

 Retail is a very flexible word. It can be used as a

Noun: I work in retail.

Verb: Albertson retails various groceries. – Adjective: Retail margins are too narrow. – Adverb: Wal-mart sells everything retail.

 Etymology: A variant of Old French retaille "piece cut off" from

retaillier "to cut up" from re- "repeat" + tailler "cut." Akin to "tailor" which comes from Old French tailleor from taillier "to cut" going back to Late Latin taliare "cut."

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Quantity Discounts

Lot size based

– All units

– Marginal unit at the end of these file

Volume based

How should buyer react?

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All-Unit Quantity Discounts

Cost/Unit $3 $2.96 $2.92 Order Quantity 5,000 10,000 Order Quantity 5,000 10,000

Total Material Cost

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All-Unit Quantity Discounts

 {0,q1,q2} are price break quantities  Find EOQ for price in range qi to qi+1

– If qi  EOQ < qi+1 ,

» Candidate in this range is EOQ, evaluate cost of ordering EOQ

– If EOQ < qi,

» Candidate in this range is qi, evaluate cost of ordering qi

– If EOQ  qi+1 ,

» Candidate in this range is qi+1, evaluate cost of ordering qi+1

 Warning: Do not ignore purchase cost

– The annual material cost of buying in lot sizes of qi  Q < qi+1 is Ci R.

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Finding Q with all units discount

Quantity T otal Cos t 3 3

2

hC

RS

Q

2 2

2

hC

RS

Q

1 1

2

hC

RS

Q

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Finding Q with all units discount

Quantity T otal Cos t

2

1 1

2

hC

RS

Q

2 2

2

hC

RS

Q

3 3

2

hC

RS

Q

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Finding Q with all units discount

Quantity T otal Cos t

1’

2

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Summary

Aggregation: Joint fixed costs for multiple products

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Marginal Unit Quantity Discounts

Cost/Unit $3 $2.96 $2.92 Order Quantity 5,000 10,000 Order Quantity 5,000 10,000

Total Material Cost

1 V 2 V

q

1

q

2

c

0

c

1

c

2
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Marginal Unit Quantity Discounts

V q V V c q q c q q c q q q Q q R Q S V Q q c h R Q V Q q c Q Q R Q S c h R Q V q c i EOQ i i i i i i i i i i i i i i i i i i                        

Cost of buying exactly

If

Annual order cost =

Annual holding cost =

Annual material cost =

Total cost( ) For range . . ( ) ( ) .... ( ) , ( ) ( ) , 0 0 1 0 1 2 1 1 1 1 2 2 0 2 2 0  

 2R SVq c hc i i i i
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Marginal-Unit Quantity Discounts

Find EOQ for price in range

q

i

to

q

i+1

– If qi  EOQ < qi+1 ,

» Candidate in this range is EOQ, evaluate cost of ordering EOQ

– If EOQ < qi,

» Candidate in this range is qi, evaluate cost of ordering qi

– If EOQ  qi+1 ,

» Candidate in this range is qi+1, evaluate cost of ordering qi+1

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Marginal Unit Quantity Discounts

Total cost Lot size

q

1

q

2 EOQ1 EOQ3

Compare this total cost graph with that of all unit quantity discounts. Here the cost graph is continuous whereas that of all unit quantity discounts has breaks.

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Marginal Unit Quantity Discounts

Total cost Lot size

q

1

q

2 EOQ1 EOQ2 EOQ3

References

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