Inventory Management: Cycle Inventory

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Inventory Management Cycle Inventory
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Role of Inventory in the Supply Chain

• Understocking Demand exceeds amount available
• Lost margin and future sales

• Overstocking Amount available exceeds
  demand
• Liquidation, Obsolescence, Holding

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Why hold inventory?

• 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

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Role of Inventory in the Supply Chain
Improve Matching of Supply and Demand
Improve Matching of Supply and Demand
Improved Forecasting
Reduce Material Flow Time
Cost Efficiency
Availability Responsiveness
Reduce Waiting Time
Reduce Buffer Inventory
Supply / Demand Variability
Seasonal Variability
Economies of Scale
Cycle Inventory
Safety Inventory
Seasonal Inventory
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Cycle Inventory
Cycle inventory is the average inventory that
built up in the supply chain because a stage of
the supply chain either produces or purchases in
lots that are larger than those demanded by the
customer.
Improve Matching of Supply and Demand
Improved Forecasting
Reduce Material Flow Time
Cost Efficiency
Availability Responsiveness
Q
Reduce Waiting Time
Reduce Buffer Inventory
Supply / Demand Variability
Cycle inventory lot size/2 Q/2
Seasonal Variability
Economies of Scale
Safety Inventory
Cycle Inventory
Seasonal Inventory
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Littles Law

• Average flow time Average inventory / Average
  flow rate

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

Cycle inventory / Demand Q / 2D
Q Lot size D Demand per unit time
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Holding Cycle Inventory for Economies of Scale

• Fixed costs associated with lots
• Quantity discounts
• Trade Promotions

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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
• hHolding cost per year as a fraction of product
  cost
• HHolding cost per unit per year hC
• QLot size
• nOrder frequency

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Lot Sizing for a Single Product (EOQ)

• Annual order cost (D/Q)Sns
• Annual holding cost (Q/2)H (Q/2)hC
• Annual material cost CD

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Lot Sizing for a Single Product (EOQ)
Total annual cost, TC CD (D/Q)S
(Q/2)hc Optimal lot size, Q is obtained by taking
the first derivative
Average flow time Q/2D
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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
Q/2 490
Cycle inventory
Numbers of orders per year
D / Q 12000 / 980 12.24
Average flow time
Q / 2D 490 / 12000 0.041 (year)
0.49(mounth)
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Example - Continued

• If we want to reduce the optimal lot size from
  980 to 200,
• then how much the order cost per lot should
  be.

• 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)
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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 .

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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

• Aggregating across the products from the same
  supplier
• Single delivery from multiple suppliers
• Single delivery to multiple retailers

• ASN (Advanced Shipping Notice) with EDI

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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

• Additional product specific order cost
• sL 1,000 sM 1,000 sH 1,000

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• Holding cost, h 0.2

• Unit cost
• CL 500 CM 500 CH 500

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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

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Option 1 No Aggregation Result

• No aggregation
• Complete aggregation
• Tailored aggregation

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)
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Option 2 Complete Aggregation

• No aggregation
• Complete 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

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Option 2 Complete Aggregation
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Option 2 Complete Aggregation Result

• No aggregation
• Complete aggregation
• Tailored aggregation

Litepro Medpro Heavypro
Demand per year 12000 1200 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 9.757,000 68,250 Annual
total cost 68,25061,5126,151615136,528
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Option 3 Tailored aggregation

• No aggregation
• Complete aggregation
• Tailored aggregation

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

• Step 1 Identify most frequently ordered product.

• 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.

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Option 3 Tailored aggregation

• Step 1 Identify most frequently ordered product.

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Option 3 Tailored aggregation

• Step 1 Identify most frequently ordered product.

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

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Option 3 Tailored aggregation
Derivation of n

• Step 1 Identify most frequently ordered product.

TC order cost holding cost

• 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.

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Option 3 Tailored aggregation
Derivation of n

• Step 1 Identify most frequently ordered product.

TC order cost holding cost
å(hCi)
ånisi

nS
TC( )
i
i
å
hCiDiMi
å
n

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

si
i



nS
mi
2n
i

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

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Option 3 Tailored aggregation
Derivation of n
holding cost

• Step 1 Identify most frequently ordered product.

TC order cost
å(hCi)
ånisi
Di

nS
TC( )
2ni
i
i
å
hCiDiMi
å
n

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

si
i



nS
mi
2n
i

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

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Option 3 Tailored aggregation

• Step 1 Identify most frequently ordered product.

• 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.

11.47
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Option 3 Tailored aggregation

• Step 1 Identify most frequently ordered product.

• 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.

11.47
nL11.47/year, nM11.47/25.74/year,
nH11.47/52.29/year .

• Step 4

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Option 3 Tailored Aggregation Result

• No aggregation
• Complete aggregation
• Tailored aggregation

Litepro Medpro Heavypro
Demand per year 12000 1200 120
Order frequency 11.47/year 5.74/year 2.29/year
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 130,763
Complete aggregation (Annual total cost)
136,528
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Option 3 Tailored aggregation

• No aggregation
• Complete aggregation
• Tailored aggregation

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

• Step 1 Identify most frequently ordered product.

• 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.

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Option 3 Tailored aggregation

• No aggregation
• Complete aggregation
• Tailored aggregation

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

• Step 1 Identify most frequently ordered product.

• 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.

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Option 3 Tailored aggregation

• No aggregation
• Complete aggregation
• Tailored aggregation

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

• Step 1 Identify most frequently ordered product.

• 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.

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Option 3 Tailored aggregation

• No aggregation
• Complete aggregation
• Tailored aggregation

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

• Step 1 Identify most frequently ordered product.

• 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.

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Impact of Product Specific Order Cost
Product specific order cost 1,000 Product specific order cost 3,000
No aggregation 155,140 183,564
Complete aggregation 136,528 186,097
Tailored aggregation 130,763 165,233
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Lessons From Aggregation

• Aggregation allows firm to lower lot size without
  increasing cost

• Complete aggregation is effective if product
  specific fixed cost is a
• small fraction of joint fixed cost

• Tailored aggregation is effective if product
  specific fixed cost is large
• fraction of joint fixed cost

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Why hold inventory?

• Economies of scale
• Stochastic variability of supply and demand

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

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Quantity Discounts

• Lot size based
• Volume based

• Based on the quantity ordered in a single lot

• All units
• Marginal unit

• Based on total quantity purchased over a given
  period

• 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?
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All Unit Quantity Discounts
Total Material Cost
Average Cost per Unit
C0
C1
C2
Quantity Purchased
Order Quantity
q1
q2
q3
q1
q2
q3
If an order that is at least as large as qi but
smaller than qi1 is placed, then each unit is
obtained at the cost of Ci.
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Evaluate EOQ for All Unit Quantity Discounts

• Evaluate EOQ for price in range qi to qi1 ,
• Case 1If qi ? Qi lt qi1 , evaluate cost of
  ordering Qi
• Case 2If Qi lt qi, evaluate cost of ordering qi
• Case 3If Qi ? qi1 , evaluate cost of ordering
  qi1
• Choose the lot size that minimizes the total cost
  over all price ranges.

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Example

• Assume the all unit quantity discounts
• Based on the all unit quantity discounts, we have
• If i 0, evaluate Q0 as
• Since Q0 gt q1, we set the lost size at
  q15,000 and the total cost

D 120,000/ year S 100/lot h 0.2
Order Quantity Unit Price
0-5,000 3.00
5,000-10,000 2.96
10,000 or more 2.92
q00, q15,000, q210,000
C03.00, C12,96, C22.92
6,324
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Evaluate EOQ for All Unit Quantity Discounts

• Evaluate EOQ for price in range qi to qi1 ,
• Case 1If qi ? Qi lt qi1 , evaluate cost of
  ordering Qi
• Case 2If Qi lt qi, evaluate cost of ordering qi
• Case 3If Qi ? qi1 , evaluate cost of ordering
  qi1
• Choose the lot size that minimizes the total cost
  over all price ranges.

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Example

• Assume the all unit quantity discounts
• Based on the all unit quantity discounts, we have
• If i 0, evaluate Q0 as
• Since Q0 gt q1, we set the lost size at
  q15,000 and the total cost

D 120,000/ year S 100/lot h 0.2
Order Quantity Unit Price
0-5,000 3.00
5,000-10,000 2.96
10,000 or more 2.92
q00, q15,000, q210,000
C03.00, C12,96, C22.92
6,324
359,080
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Example

• Assume the all unit quantity discounts
• Based on the all unit quantity discounts, we have
• If i 0, evaluate Q0 as
• Since Q0 gt q1, we set the lost size at
  q15,000 and the total cost

D 120,000/ year S 100/lot h 0.2
Order Quantity Unit Price
0-5,000 3.00
5,000-10,000 2.96
10,000 or more 2.92
q00, q15,000, q210,000
C03.00, C12,96, C22.92
6,324
359,080
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All Unit Quantity Discounts
Total Material Cost
Average Cost per Unit
C0
C1
C2
Quantity Purchased
Order Quantity
q1
q2
q3
q1
q2
q3
If an order is placed that is at least as large
as qi but smaller than qi1, then each unit is
obtained at a cost of Ci.
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Example

• Assume the all unit quantity discounts
• Based on the all unit quantity discounts, we have
• If i 0, evaluate Q0 as
• Since Q0 gt q1, we set the lost size at
  q15,000 and the total cost

D 120,000/ year S 100/lot h 0.2
Order Quantity Unit Price
0-5,000 3.00
5,000-10,000 2.96
10,000 or more 2.92
q00, q15,000, q210,000
C03.00, C12,96, C22.92
6,324
359,080
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Example - Continued

• For i 1, we obtain Q1 6,367
• Since 5,000 lt Q1 lt10,000 , we set the lot
  size at Q1 6,367.

358,969

• For i 2, we obtain Q2 6,410
• Since Q2 lt q2 , we set the lot size at
  q210,000.

• Observe that the lowest total cost is for i 2.
• The optimal lot size 10,000 (at the
  discount price of 2.92)

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Example - Continued

• For i 1, we obtain Q1 6,367
• Since 5,000 lt Q1 lt10,000 , we set the lot
  size at Q1 6,367.

• For i 2, we obtain Q2 6,410
• Since Q2 lt q2 , we set the lot size at
  q210,000.

354,520

• Observe that the lowest total cost is for i 2.
• The optimal lot size 10,000 (at the
  discount price of 2.92)

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Example - Continued

• For i 1, we obtain Q1 6,367
• Since 5,000 lt Q1 lt10,000 , we set the lot
  size at Q1 6,367.

Q1
D
358,969



DC1
hC1
S
TC1
2
Q1

• For i 2, we obtain Q2 6,410
• Since Q2 lt q2 , we set the lot size at
  q210,000.

q2
D



354,520
DC2
hC2
S
TC2
2
q2

• Observe that the lowest total cost is for i 2.
• The optimal lot size 10,000 (at the
  discount price of 2.92)

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Example - Continued

• For i 1, we obtain Q1 6,367
• Since 5,000 lt Q1 lt10,000 , we set the lot
  size at Q1 6,367.

Q1
D
358,969



DC1
hC1
S
TC1
2
Q1

• For i 2, we obtain Q2 6,410
• Since Q2 lt q2 , we set the lot size at
  q210,000.

354,520

• Observe that the lowest total cost is for i 2.
• The optimal lot size 10,000 (at the
  discount price of 2.92)

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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

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Marginal Unit Quantity Discounts
Marginal Cost per Unit
Total Material Cost
C0
C1
C2
Quantity Purchased
Order Quantity
q1
q2
q3
q1
q2
q3
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.
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Evaluate EOQ for Marginal Unit Discounts

• Evaluate EOQ for each marginal price Ci (or lot
  size between qi and qi1)

• Let Vi be the cost of order qi units. Define V0
  0 and
• ViC0(q1-q0)C1(q2-q1)???Ci-1(qi-
  qi-1)

• Consider an order size Q in the range qi to qi1
• 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)

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Evaluate EOQ for Marginal Unit Discounts

• Evaluate EOQ for each marginal price Ci (or lot
  size between qi and qi1)

• Let Vi be the cost of order qi units. Define V0
  0 and
• ViC0(q1-q0)C1(q2-q1)???Ci-1(qi-
  qi-1)

• Consider an order size Q in the range qi to qi1
• 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)

2D(SVi-qiCi)
Qi
hCi
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Example

• Assume the all unit quantity discounts
• q00, q15,000, q210,000
• C03.00, C12,96, C22.92
• V00 V13(5,000-0)15,000
• V23(5,000-0)2.96(10,000-5,000)29,800
• If i 0, evaluate Q0 as
• Since Q0 gt q1, we set the lost size at
  q15,000 and the total cost

D 120,000/ year S 100/lot h 0.2
Order Quantity Unit Price
0-5,000 3.00
5,000-10,000 2.96
10,000 or more 2.92
6,324
363,900
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Example - Continued

• For i 1, evaluate
• Since Q1 gt q2, we evaluate the cost of
  ordering q210,000
• For i 2, evaluate
• Optimal order size 16,961

11,028
361,780
16,961
360,365
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