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Inventory Management: Cycle Inventory

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第四單元： Inventory Management: Cycle Inventory

郭瑞祥教授

Understocking: Demand exceeds amount available

–Lost margin and future sales

Overstocking: Amount available exceeds demand

– Liquidation, Obsolescence, Holding

Role of Inventory in the Supply Chain

► Economies of scale

► Stochastic variability of supply and demand》 Batch size and cycle time》 Quantity discounts》 Short term discounts / Trade promotions》Service level given safety inventory》Evaluating Service level given safety inventory

Why hold inventory?

Improve Matching of Supplyand Demand

Improved Forecasting

Reduce Material Flow Time

Reduce Waiting Time

Reduce Buffer Inventory

Economies of ScaleSupply / Demand

Variability Seasonal Variability

Cycle Inventory Safety Inventory Seasonal Inventory

CostEfficiency

AvailabilityResponsiveness

Role of Inventory in the Supply Chain

Cycle Inventory

Improved Forecasting

Reduce Material Flow Time

Reduce Waiting Time

Reduce Buffer Inventory

Economies of ScaleSupply / Demand

Variability Seasonal Variability

Safety Inventory Seasonal Inventory

CostEfficiency

AvailabilityResponsiveness

Cycle inventory is the average inventory that built up in the supplychain because a stage of the supply chain either produces or

purchases in lots that are larger than those demanded by the customer.

Cycle Inventory

Inventory

Time t

Cycle inventory = lot size/2 = Q/2Cycle inventory = lot size/2 = Q/2

Little’s Law

► Average flow time = Average inventory / Average flow rate

Average flow time resulting from cycle inventory

Q: Lot size

D: Demand per unit time

► For any supply chain, average flow rate equals the demand,

= Cycle inventory / Demand = Q / 2D

Holding Cycle Inventory for Economies of Scale

► Fixed costs associated with lots

► Quantity discounts

► Trade Promotions

Economics of Scale to Exploit Fixed Costs

— Economic Order Quantity—

» D= Annual demand of the product

» S= Fixed cost incurred per order

» C= Cost per unit

» h=Holding cost per year as a fraction of product cost

» H=Holding cost per unit per year =hC

» Q=Lot size

» n=Order frequency

Lot Size

Lot Sizing for a Single Product (EOQ)

► Annual order cost =(D/Q)S=ns

Annual holding cost = (Q/2)H =(Q/2)hC

Annual material cost = CD

TC =CD + (D/Q)S + (Q/2)hC

Holding Cost

Material Cost

Order Cost

Total Cost

Lot Size

► Annual order cost =(D/Q)S

Annual holding cost = (Q/2)H =(Q/2)hc

Annual material cost = CD‧

TC =CD + (D/Q)S + (Q/2)hc

Holding Cost

Material Cost

Order Cost

Total Cost

Lot Sizing for a Single Product (EOQ)

Total annual cost, TC =CD + (D/Q)S + (Q/2)hcOptimal lot size, Q is obtained by taking the first derivative

Average flow time = Q*/2D S*

Example

Demand ， D =1,000 units/month = 12,000 units/year

Fixed cost, S = $4,000/order Unit cost, C = $500 Holding cost, h = 20% = 0.2

》 Optimal order size

》 Cycle inventory

》 Numbers of orders per year

》 Average flow time Q / 2D = 490 / 12000 =0.041 (year) =0.49(mounth)

Q = = 9800.2X500

2X12000X4000

Q/2 =490

D / Q = 12000 / 980 =12.24

Example - Continued

Demand ， D =1,000 units/month = 12,000 units/year

Fixed cost, S = $4,000/order Unit cost, C = $500 Holding cost, h = 20% = 0.2

》 Optimal order size

》 Cycle inventory

》 Numbers of orders per year

》 Average flow time Q / 2D = 490 / 12000 =0.041 (year) =0.49(mounth)

Q = = 9800.2X500

2X12000X4000

Q/2 =490

D / Q = 12000 / 980 =12.24

► If we want to reduce the optimal lot size from 980 to 200,

then how much the order cost per lot should be.

hC(Q*)2

S = 0.2X500X2002

2X12000= = $166.7

If we increase the lot size by 10% (from 980 to 1100), what the total cost would be.

Annual cost = $ 98,636 (from $ 97,980)(an increase by only 0.6%) (Note: material cost is not included)

Microsoft 。Microsoft 。

CoolCLIPS

Microsoft 。

Key Points from EOQ

Total order and holding costs are relatively stable around the economic order quantity. A firm is often better served by ordering a convenient lot size close to the EOQ rather than the precise EOQ.

To reduce the optimal lot size by a factor of k, the fixed order cost

S must be reduced by a factor of k2 .

If demand increases by a factor of k, the optimal lot size increases

by a factor of . The number of orders placed per year should

also increase by a factor of . Flow time attributed to cycle

inventory should decrease by a factor of .

Aggregating Multiple Products in a Single Order

► One of major fixed costs is transportation

► Ways to lower the fixed ordering and transportation costs:

► Ways to lower receiving or loading costs:

》ASN (Advanced Shipping Notice) with EDI

》Aggregating across the products from the same supplier

》Single delivery from multiple suppliers

》Single delivery to multiple retailers

Microsoft 。

Example: Lot Sizing with Multiple Products

Three computer models (L, M, H) are sold and the demand per year:

–DL = 12,000; DM = 1,200; DH = 120

► Common fixed (transportation) cost, S = $4,000

► Holding cost, h = 0.2

► Unit cost 》 CL = $500; CM = $500; CH = $500

► Additional product specific order cost

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

Microsoft 。Microsoft 。Microsoft 。

Delivery Options

► No aggregation

► Complete aggregation

► Tailored aggregation

》Each product is ordered separately

》All products are delivered on each truck

》Selected subsets of products on each truck

► No aggregation

Option 1: No Aggregation Result

Litepro Medpro Heavypro

Demand per year 12000 1200 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 holding cost $109,544 $34,642 $10,954

Annual total cost = $155,140 (no material cost)

► No aggregation

Option 2: Complete Aggregation

》Combined fixed cost per order is given by

》Let n be the number of orders placed per year. We have Total annual cost = Annual order cost + Annual holding cost

HML sssSS *

)]2/()2/()2/[()( * nhCDnhCDnhCDnS HHMMLL

hCDhCDhCDn HHMMLL

Option 2: Complete Aggregation

► No aggregation

► Tailored aggregation》Combined fixed cost per order is given by

》Let n be the number of orders placed per year. We have Total annual cost = Annual order cost + Annual holding cost

HML sssSS *

)]2/()2/()2/[()( * nhCDnhCDnhCDnS HHMMLL

hCDhCDhCDn HHMMLL

► No aggregation

Option 2: Complete Aggregation Result

Optimal order size 1,230 123 12.3

Annual holding cost $61,512 $6,151 $615

Annual order cost = 9.75×$7,000 = $68,250Annual total cost = $68,250+$61,512+$6,151+$615=$136,528

► No aggregation

Option 3: Tailored aggregation

A heuristic that yields an ordering policy whose cost is close to optimal.

► Step 2: Identify frequency of other products as a multiple of the order frequency of the most frequently ordered product.

► Step 3: Recalculate order frequency of most frequently ordered product.

► Step 4: Identify ordering frequency of all products.

► Step 1: Identify most frequently ordered product.

► No aggregation

►A heuristic that yields an ordering policy whose cost is close to optimal.

0.11 1.1 3.5, ,0.11)(2

nHnMnLsSLDLhC

}2(S+si)

hCiDiniMaxn

► No aggregation

Step 2: Identify frequency of other products as a multiple of the order frequency of the most frequently ordered product.

0.11 1.1 3.5, ,0.11)(2

nHnMnLsSLDLhC

2.4 7.72

55.4 Hm 24.1 7.7/0.11/

MM nnm

n= hCiDi

Option 3: Tailored aggregation► No aggregation

Step 4: Identify ordering frequency of all products.

0.11 1.1 3.5, ,0.11)(2

nHnMnLsSLDLhC

2.4 7.72

7.7/0.11/ MM nnm

)]/([2

55.4 Hm 24.1

TC= order cost + holding cost

Derivation of n

Step 3: Recalculate order frequency of most frequently ordered product.

0.11 1.1 3.5, ,0.11)(2

nHnMnLsSLDLhC

2.4 7.72

7.7/0.11/ MM nnm

)]/([2

Derivation of n

+= 2ni

hCiDiMi

TC= order cost + holding cost

(hCi)i

nisii 2ni

DinSTC=( )++

55.4 Hm 24.1 7.7/0.11/

MM nnm

Option 3: Tailored aggregation No aggregation

Complete aggregation

Tailored aggregation

0.11 1.1 3.5, ,0.11)(2

nHnMnLsSLDLhC

2.4 7.72

7.7/0.11/ MM nnm

)]/([2

Derivation of n

+= 2ni

hCiDiMi

TC= order cost +

(hCi)i

nisii 2ni

DinSTC=( )++

=0S+i mi

hCiDiMi

holding cost

)]/([2

0.11 1.1 3.5, ,0.11)(2

nHnMnLsSLDLhC

2.4 7.72

7.7/0.11/ MM nnm

)]/([2

mDhCn =11.47

55.4 Hm 24.1

Derivation of n

+= 2ni

hCiDiMi

TC= order cost +

(hCi)i

nisii 2ni

DinSTC=( )++

=0S+i mi

hCiDiMi

holding cost

)]/([2

Microsoft 。

Option 3: Tailored aggregation No aggregation

Complete aggregation

Tailored aggregation

0.11 1.1 3.5, ,0.11)(2

nHnMnLsSLDLhC

2.4 7.72

7.7/0.11/ MM nnm

)]/([2

mDhCn =11.47

► Step 4: nL=11.47/year, nM=11.47/2=5.74/year,

nH=11.47/5=2.29/year .

55.4 Hm 24.1

► No aggregation

Option 3: Tailored Aggregation Result

Optimal order size 1,046 209 52

Annual holding cost $52,310 $10,453 $2620

Annual order cost = nS + nLsL+ sMsM + nHsH =$65,380

Annual total cost = $130,763Complete aggregation (Annual total cost) =$136,528

► No aggregation

─ A fixed cost of (S+si) is allocated to each product i, and

)(2 frequency order frequently most The

i sSDhC

► No aggregation

► Step 4: Identify ordering frequency of all products.2si

hCi Dini

mi n / ni

► No aggregation

)]/([2

► No aggregation

ni=n/mi

Impact of Product Specific Order Cost

Product specificorder cost =$1,000

Product specificorder cost =$3,000

No aggregation $155,140 $183,564

Complete aggregation $136,528 $186,097

Tailored aggregation $130,763 $165,233 Ss ??

Lessons From Aggregation

Aggregation allows firm to lower lot size without increasing cost

Tailored aggregation is effective if product specific fixed cost is large

fraction of joint fixed cost

Complete aggregation is effective if product specific fixed cost is a

small fraction of joint fixed cost

► Economies of scale

► Stochastic variability of supply and demand

Why hold inventory?

》 Batch size and cycle time》 Quantity discounts》 Short term discounts / Trade promotions

Quantity Discounts

► Lot size based

► Volume based

How should buyer react? How does this decision affect the supply chain

in terms of lot sizes, cycle inventory, and flow time?

What are appropriate discounting schemes that suppliers should offer?

》Based on total quantity purchased over a given period

> All units> Marginal unit

》Based on the quantity ordered in a single lot

All Unit Quantity Discounts

q1 q2 q3

Average Cost per Unit

Quantity Purchased

Total Material Cost

Order Quantityq1 q2 q3

If an order that is at least as large as qi but smaller than qi+1 is placed,

then each unit is obtained at the cost of Ci. 38

► Evaluate EOQ for price in range qi to qi+1 ,

》Case 1:If qi Qi < qi+1 , evaluate cost of ordering Qi

》Case 2:If Qi < qi, evaluate cost of ordering qi

》Case 3:If Qi qi+1 , evaluate cost of ordering qi+1

► Choose the lot size that minimizes the total cost over all price ranges.

Evaluate EOQ for All Unit Quantity Discounts

DCihCiQiSD

DCihCiqiSD

DCi+1hCiSD

Order Quantity

Total CostLowest cost in the range

qi qi+1

Total Cost Lowest cost in the range

qi qi+1

Order Quantity

Total Cost Lowest cost in the range

qi qi+1

Assume the all unit quantity discounts

Based on the all unit quantity discounts, we have

If i = 0, evaluate Q0 as

Since Q0 > q1, we set the lost size at q1=5,000 and the total cost

Example

Order Quantity Unit Price

0-5,000 $ 3.00

5,000-10,000 $ 2.96

10,000 or more $ 2.92

D = 120,000/ yearS = $100/loth = 0.2

q0=0, q1=5,000, q2=10,000

C0=$3.00, C1=$2,96, C2=$2.92

2 = 6,324

q1 q2 q3

Quantity Purchased

Evaluate EOQ for All Unit Quantity Discounts

2 Evaluate EOQ for price in range qi to qi+1 ,

》Case 1:If qi Qi < qi+1 , evaluate cost of ordering Qi

》Case 2:If Qi < qi, evaluate cost of ordering qi

》Case 3:If Qi qi+1 , evaluate cost of ordering qi+1

Choose the lot size that minimizes the total cost over all price ranges.

DCi+1hCiSD

DCihCiqiSD

DCihCiQiSD

DC1hC1q1SD

Example

0-5,000 $ 3.00

5,000-10,000 $ 2.96

10,000 or more $ 2.92

D = 120,000/ yearS = $100/loth = 0.2

q0=0, q1=5,000, q2=10,000

C0=$3.00, C1=$2,96, C2=$2.92

2 = 6,324

= $359,080

Example

0-5,000 $ 3.00

5,000-10,000 $ 2.96

10,000 or more $ 2.92

D = 120,000/ yearS = $100/loth = 0.2

q0=0, q1=5,000, q2=10,000

C0=$3.00, C1=$2,96, C2=$2.92

2 = 6,324

= $359,080DC1hC1q1SD

All Unit Quantity Discounts

Total Material Cost

q1 q2 q3

Quantity Purchased Order Quantityq1 q2 q3

If an order is placed that is at least as large as qi but smaller than qi+1, then each unit is obtained at a cost of Ci.

Example

0-5,000 $ 3.00

5,000-10,000 $ 2.96

10,000 or more $ 2.92

D = 120,000/ yearS = $100/loth = 0.2

q0=0, q1=5,000, q2=10,000

C0=$3.00, C1=$2,96, C2=$2.92

2 = 6,324

= $359,080

DC1hC1q1SD

Example - Continued

For i = 1, we obtain Q1 = 6,367

Since 5,000 < Q1 <10,000 , we set the lot size at Q1 = 6,367.

Observe that the lowest total cost is for i = 2.

The optimal lot size = 10,000 (at the discount price of $2.92)

Since Q2 < q2 , we set the lot size at q2=10,000.

= $358,969DC1hC1Q1SD

DC2hC2q2SD

= $358,969DC1hC1Q1SD

Example - Continued

► Observe that the lowest total cost is for i = 2.

= $354,520

Example - Continued

= $354,520

= $358,969

DC2hC2q2SD

DC1hC1Q1SD

Example - Continued

DC1hC1Q1SD

= $358,969

= $354,520DC2hC2q2SD

The Impact of All Unit Discounts on Supply Chain

► In the above example

► If the fixed ordering cost S = $4,

► All unit quantity discounts encourage retailers to increase the size of their lots.

► This also increases cycle inventory and average flow time.

》The optimal order size = 6,324 when there is no discount.》The quantity discounts result in a higher order size = 10,000.

》The optimal order size without discount = 1,265》The optimal order size with all unit discounts = 10,000

Marginal Unit Quantity Discounts

q1 q2 q3

Marginal Cost per Unit

Quantity Purchased Order Quantity

q1 q2 q3

Total Material Cost

If an order of size q is placed, the first q1-q0 units are priced at C0, the next q2-q1 are priced at C1, and so on.

Optimal lot size

2D(S+Vi-qiCi)

► Evaluate EOQ for each marginal price Ci (or lot size between qi and qi+1)

Evaluate EOQ for Marginal Unit Discounts

》 Let Vi be the cost of order qi units. Define V0 = 0 and

Vi=C0(q1-q0)+C1(q2-q1)+ +C‧‧‧ i-1(qi-qi-1)

》 Consider an order size Q in the range qi to qi+1

Total annual cost = ( D/Q )S (Annual order cost)

+ (Q/2) h‧ ‧[ Vi+(Q-qi)Ci ] / Q (Annual holding cost)

+ D [ ‧ Vi+(Q-qi)Ci ] / Q(Annual material cost)

Optimal lot size

2D(S+Vi-qiCi)

► Evaluate EOQ for each marginal price Ci (or lot size between qi and qi+1)

Evaluate EOQ for Marginal Unit Discounts

》 Let Vi be the cost of order qi units. Define V0 = 0 and

Vi=C0(q1-q0)+C1(q2-q1)+ +C‧‧‧ i-1(qi-qi-1)

》 Consider an order size Q in the range qi to qi+1

Total annual cost = ( D/Q )S (Annual order cost)

+ (Q/2) h‧ ‧[ Vi+(Q-qi)Ci ] / Q (Annual holding cost)

+ D [ ‧ Vi+(Q-qi)Ci ] / Q(Annual material cost)

q0=0, q1=5,000, q2=10,000

C0=$3.00, C1=$2,96, C2=$2.92

V0=0 ; V1=3(5,000-0)=$15,000

V2=3(5,000-0)+2.96(10,000-5,000)=$29,800

2D(S+V0-q0C0)

Example

0-5,000 $ 3.00

5,000-10,000 $ 2.96

10,000 or more $ 2.92

D = 120,000/ yearS = $100/loth = 0.2

= 6,324

$363,900D

+[ V1+(q1-q1)C1] + [ V1+(q1-q1)C1]= q1q1

For i = 1, evaluate

Since Q1 > q2, we evaluate the cost of ordering q2=10,000

For i = 2, evaluate

Optimal order size = 16,961

Example - Continued

2D(S+V1-q1C1)

Q1= = 11,028

$361,780

2D(S+V2-q2C2)

Q2= = 16,961

$360,365DSDTC2

+[ V2+(Q2-q2)C2] + [ V2+(Q2-q2)C2 ]=Q2Q2

+[ V2+(q2-q2)C2] + [ V2+(q2-q2)C2]= q2

頁碼作品授權條件作者 /來源

12本作品轉載自Microsoft Office 2007多媒體藝廊，依據Microsoft 服務合約及著作權法第 46 、 52 、 65條合理使用。

12, 14本作品轉載自Microsoft Office 2007多媒體藝廊，依據Microsoft 服務合約及著作權法第 46 、 52 、 65條合理使用。

12, 14 本作品轉載自Microsoft Office 2007多媒體藝廊，依據Microsoft 服務合約及著作權法第 46 、 52 、 65條合理使用。

12 本作品轉載自Microsoft Office 2007多媒體藝廊，依據Microsoft 服務合約及著作權法第 46 、 52 、 65條合理使用。

12CoolCLIPS 。本作品轉載自 CoolCLIPS 網站（ http://dir.coolclips.com/Popular/World_of_Industry/Food/Shopping_cart_full_of_groceries_vc012266.html ），瀏覽日期 2011/12/28 。依據著作權法第 46 、 52 、 65 條合理使用。

頁碼作品授權條件作者 /來源

15, 27 本作品轉載自Microsoft Office 2007多媒體藝廊，依據Microsoft 服務合約及著作權法第 46 、 52 、 65條合理使用。

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