Chapter 3. Managing economies of scale in a supply
chain: cycle inventory
Learning objectives:
1. Balance the appropriate costs to choose the optimal amount of
cycle inventory in a supply chain
2. Understand the impact of quantity discount on lot size and
cycle inventory
3. Devise appropriate discounting schemes for a supply chain
4. Understand the impact of trade promotions on lot size and
cycle inventory
5. Identify managerial levers that reduce lot size and cycle
inventory in a supply chain without increasing cost
1
Role of cycle inventory
2
Why do companies hold inventory?
Why might they avoid doing so?
• WHY?
– To take advantage of economic purchase order size :
economy of scale (cycle inventory)
– To meet anticipated customer demand
– To account for differences in production timing
(smoothing)
– To protect against uncertainty (demand surge, price
increase, lead time slippage)
– To maintain independence of operations (buffering)
• WHY NOT?
– Requires additional space
– Opportunity cost of capital
– Spoilage / obsolescence
3
The role of cycle inventory in a
supply chain
• A lot or batch size is the quantity that a stage of a SC
either produces or purchases at a time.
• The lot size is usually larger than the quantities
demanded by the customer.
• Cycle inventory is the average inventory in a SC due to
this difference.
Key point : Cycle inventory exists in a SC bcs different stages
exploit the economies of scale to lower total cost. The costs
considered include: material cost, fixed ordering cost, and
holding cost.
4
The role of cycle inventory in a
supply chain
On-hand
Inventory
• Example: Consider a computer store selling an average of D = 4
printers a day but ordering Q = 80 printers from the
manufacturer each time.
• Cycle inventory = lot size/2 = Q/2 = 40
• Average flow time = cycle inventory/demand rate = 40/4 = 10
days (inventory holding time)
• Inventory turnover (taux de rotation), inventory coverage (taux
de couverture)
Q
Q/2
Demand
Rate, D
Average Cycle
Inventory, Q/2
Time
5
Two Decisions in
Inventory Management
• When is it time to reorder?
• If it is time to reorder, how much?
6
Economies of scale to exploit fixed costs:
Economic Order Quantity Model
7
Economies of scale to exploit fixed costs:
Economic Order Quantity Model
Time Between Orders
On-hand Inventory
(Cycle Time) T = Q/D
Q
Demand
Rate, D
Average Cycle
Inventory, Q/2
Q/2
Reorder
Point, R
Place
Order
Receive
order
Time
Lead
Time,
L
8
Basic EOQ Assumptions
•
•
•
•
Constant Demand Rate
Constant Lead Time
Orders received in full after lead-time.
Constant Unit Price (no discounts)
9
Economic Order Quantity Cost Model:
Constant Demand, No Shortages
TC
D
Q
K
c
I
=
=
=
=
=
=
total annual inventory cost
annual demand (units / year)
order quantity (units)
cost of placing an order or setup cost ($)
cost per unit
annual interest rate
Total Annual
Inventory =
Cost
TC
=
Annual
Ordering
Cost
Annual
+ Holding
Cost
(D / Q) K + (Q / 2) Ic
10
Cost Relationships for Basic EOQ
(Constant Demand, No Shortages)
Total
Cost
Carrying
Cost
Ordering
Cost
Q*
Order Quantity (how much)
EOQ balances carrying
costs and ordering
costs in this model.
11
Many orders,
low inventory
level
On-hand Inventory
Trade-off in EOQ Model:
Inventory Level vs. Number of Orders
Q
Time
Few orders,
high inventory
level
On-hand Inventory
Q
Time
12
EOQ Results (How Much to Order)
(Constant Demand, No Shortages)
Economic Order Quantity = Q* =
2DK
Ic
Number of Orders per year = D / Q*
Length of order cycle T = Q* / D
Total cost = TC = (D / Q*) K + (Q* / 2) Ic
13
Determining When to Reorder
• Quantity to order (how much…) determined by EOQ
• Reorder point (when…)determined by finding the
inventory level that is adequate to protect the
company from running out during delivery lead time
• With constant demand and constant lead time,
(EOQ assumptions), the reorder point is exactly the
amount that will be sold during the lead time.
Example:
14
EOQ Example
D = 1,000 units per year
BE CAREFUL!
S = $20 per order
IC = $8.33 per unit per month
HOW MUCH TO ORDER?
WHEN TO ORDER?
Number of orders per year =
Length of order cycle = T =
Total cost =
15
Exercise
Question: What if the company can only order in multiples
of 12? (That is, order either 0 or 12 or 24 or 36, etc…)?
16
Robustness of EOQ model
Very Flat Curve - Good!!
Total
Cost
DTC
Q*-DQ
Q*
Q*+DQ
Order Quantity
Would have to mis-specify Q* by quite a bit
before total annual inventory costs would
change significantly.
17
Example: EOQ Robustness
• Suppose that in the last problem, you have mis-specified the
order costs by 100% and the holding costs by 50%. That is,
– S used in the computation = $40/order (actual cost = $20 / order)
– IC used in computation = $150 / unit / year (actual = $ 100)
– Then, using these wrong costs, you would have gotten
Q' 
2(1,000)40
 23.1
150
Your actual TC (computed substituting Q’ into TC, using correct costs of S = $20, and h = $100:
TC 
1,000
23
20  100  $2,019
23
2
Only 1% above minimum TC!
18
Key points
KP1 : Total ordering 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
(robustness).
KP2 : If the demand increases by a factor of k, the optimal lot size
increases by a factor of k0,5 . The number of orders placed per year
increases by a factor of k0,5. Flow time due to cycle inventory decreases
by a factor of k0,5.
KP3 : To reduce the optimal lot size by a factor of k, the fixed cost K must
be reduced by a factor of k2.
KP4 : Aggregating replenishment across products, retailers, or suppliers in
a single order allow for a reduction of lot size of individual products bcs
the fixed costs are now spread across differents aggregated entities. 19
Lot sizing with multiple products or
customers
20
• In general, the ordering, transportation, and
receiving cost of an order grows with the
variety of products or pickup points.
• A portion of the fixed cost of an order can be related to
transportation (this depends only on the load but not
on the product variety)
• A portion of the fixed cost is related to loarding and
receiving (this cost increases with variety on the truck)
21
Assumptions :
• similar to EOQ model except the followings.
• Di : annual demand for product i
• S: order cost incurred each time an order is placed,
independent of the variety of products included
• si: additional order cost incurred if product i is
included in the order.
22
Three approaches :
1. Each product manager orders his model
independently (highest cost)
2. The product managers jointly order every product in
each lot (easy to administer and implement, but not
selective enough and expensive joint ordering if
product specific order cost high)
3. Product managers order jointly but not every order
contains every product, i.e. each lot contains a
selected subset of products.
23
Example:
• Best Buy sells 3 models of computers, the Litepro, the
Medpro, the Heavypro.
• The annual demands are DL = 12000, DM = 1200, DH =
120.
• Each model costs Best Buy 500$.
• A fixed transportation cost of 4000$ is incurred each
time an order is delivered. For each model ordered and
delivered on the same truck, an additional fixed cost of
1000$ is incurred for receiving and storage.
• Best Buy has an annual holding cost of 20%.
24
Approach 1 : Independent ordering
• QL = 1095, QM = 346, QH = 110.
• Oder frequencies : 11/year, 3,5/year, 1.1/year.
• Total inventory cost = 155140 $
• Other measures of interest : cycle inventory, annual
holding cost/prod, annual ordering cost, flow time.
25
Approach 2 : Lots ordered and delivered for all
• Combined fixed order cost/order : K = S +  si
• The optimal order frequency is (to explain, express
total cost in T):
k
n* 
D I c
i 1
i i i
2K
• Example : n* = 9.75, annual inventory cost = 136528$,
i.e. a reduction of 13% over approach 1.
26
Approach 3 : Lots ordered and delivered jointly for a
selected subset of products
Step 1. Identify most frequently ordered product
assuming each being ordered independently.
n  max ni
i
Di I i ci
ni 
2  S  si 
The most frequently order products i* is included each
time an order is placed
27
Step 2. Identify the frequency with which other products
are included.
• Calculate the order frequency as a multiple of n
• As the most frequently ordered product is in each
order, the inclusion of a product i incurs an additional
product specific fixed order cost of si.
• Product i is included once every mi orders
mi   n ni 
ni 
Di I i ci
2 si
28
Step 3. Recalculate the order frequency of the most
frequently order product n.
n
D I c m
2S   s m 
i i i
i
i
i
Why? (order cycle T for n, order cycle miT for i)
29
Step 4. For each product, evaluate the order frequency ni
= n/mi and the total cost of such an ordering policy.
Example :
n = 11.47, mL = 1, mM = 2, mH = 5,
annual total inventory cost = 130767$, a reduction of
4% over approach 2.
30
Key point:
• A key to reducing cycle inventory is the reduction of
lot size.
• A key to reducing lot size without increasing costs is
to reduce the fixed cost associated with each lot.
• This may be achieved by reducing the fixed cost itself
or by aggregating lots across products, customers,
suppliers.
• When aggregating, tailored aggregation is best,
especially if product specific costs are large.
31
Economies of scale to exploit quantity
discounts
32
Introduction
• Pricing schedule often displays economies of scale, with prices
decreasing as lot size increases.
• A discount is lot size based if the pricing schedule offers
discounts based on the quantity ordered in a single lot.
• A discount is volume based if the discount is based on the total
quantity purchased over a given period.
• Two commonly used lot size based discount schemes : all unit
quantity discounts, marginal unit quantity discount or
multiblock tarriffs
33
Two basic questions
• Given a pricing schedule with quantity discount, what is the
optimal purchasing decision for a buyer? How does this affect
the SC in terms of lot size, cycle inventories, flow times?
• Under what conditions should a supplier offer quantity
discounts? What are appropriate pricing schedules that a
supplier should offer?
34
EOQ with all quantity discount
• Pricing schedule :
The unit purchase cost is Ci if the order quantity is at least qi
with q0 = 0 < q1 < q2 < … < qr = ∞ and c0 > c1 > c2 > …
• The retailer’s objective is to maximise its profit, i.e. minimise
the sum of material, order, and holding costs.
35
Example
• Drug Online (DO) is an online retailer of prescription drugs.
Demand for vitamins is 10000 bottles per month. DO incurs a
fixed order cost of 100$ each time an occurs is placed with the
manufacturer. DO has an annual holding cost of 20%.
• The pricing schedule of the manufacturer is the all unit discount
schedule:
Order quantity
Unit Price ($)
0-5000
3
5000-10000
2,96
10000 or more
2,92
36
Solution
Step 1. Determine the EOQ Qi for each price Ci
Qi 
2 DK
Ici
Step 2. Determine the total annual cost TCi for each price range
Case 1: Qi >= qi+1, ignored as it is considered for Qi+1
Case 2: Qi < qi,
D
q
TCi    K  i Ici  Dci
2
 qi 
Case 3: qi <= Qi < qi+1,
D
Q
TCi    K  i Ici  Dci
2
 Qi 
Step 3. Determine the optimal order quantity.
37
Example (draw TC(Q))
Step 1: Q0 = 6324, Q1 = 6367, Q2 = 6410
Step 2:
Order quantity
Q0 >= 5000, TC0 ignore
0-5000
5000-10000
5000 < Q1 < 10000, TC1 = 358969 $
10000 or more
Q2 < 10000, TC2 = 354520$
Step 3: Optimal order size = q2 = 10000, TC = TC2.
Unit Price ($)
3
2,96
2,92
Remarks :
• Presence of quantity discount leads to Larger order size of
10000 units than the normal EOQ = 6324
• If S = 4$, order size under all unit discount schedule is still
10000 units and is 8 times the normal EOQ = 1265.
38
EOQ with marginal quantity discount
• Pricing schedule :
The pricing schedule contains specified break points q0 = 0 <
q1 < q2 < … < qr = ∞. The marginal cost of a unit decreases at
the break points to ci if the order quantity is at least qi with c0
> c1 > c2 > …
• The purchasing cost Vi of qi units is determined as follows:
V0 = 0, Vi+1 = Vi + ci (q i+1 – qi), for i = 0, 1, …
• Purchasing cost of an order of Q such that qi <= Q < qi+1 is:
C(q) = Vi + ci(Q-qi)
39
Solution
Step 1. Determine the EOQ Qi for each price range Ci (why?)
Qi 
2 D  K  Vi  qi ci 
Ici
Step 2. Determine the total annual cost TCi for each price range
Case 1: Qi < qi, Qi* = qi
Case 2: Qi > qi+1, Qi* = qi+1
Case 3: qi <= Qi < qi+1, Qi* = Qi
TCi  Q 
Vi   Q  qi  ci
K
1 Vi   Q  qi  ci 

 I
Q 
Q D 2 
Q
Q D


V   Q  qi  ci
K
1
 I Vi   Q  qi  ci   i
Q D 2
Q D
Step 3. Determine the optimal order quantity.
40
Example (draw TC(Q))
V0 = 0, V1 = 15000, V2 = 29800
Step 1: Q0 = 6324, Q1 = 11028, Q2 = 16961
Step 2:
Order quantity
0-5000
Q0 >= 5000, TC0 = 363900$
5000-10000
Q1 > 10000, TC1 = 361780 $
10000 or more
10000 < Q2, TC2 = 360365$
Step 3: Optimal order size = Q2 = 16961, TC = TC3.
Unit Price ($)
3
2,96
2,92
Remarks :
• Much larger order size of 16961 units than the normal EOQ =
6324
• If S = 4$, order size 15755 is 12,5 times the normal EOQ =
1265.
41
Key point
• There can be significant increase of order size and
cycle inventory in the absence of fixed order costs as
long as quantity discounts are offered.
• Quantity discounts lead to a significant buildup of
cycle inventory in a supply chain.
• In many SC, quantity discounts contribute more to
cycle inventory than fixed ordering cost.
• Value of quantity discount in a supply chain?
42
Why quantity discount?
43
Coordination to increase total SC profits
Quantity discount for commodity products
• For commodity products, a competitive market exists,
the market sets the price, the firm’s objective is the
lower costs.
44
A two-stage supply chain example
Retailer
• Drug Online (DO) is an online retailer of prescription drugs.
Demand for vitamins is 10000 bottles per month. DO incurs a
fixed order cost of 100$ each time an occurs is placed with the
manufacturer. DO has an annual holding cost of 20%.
• Normal wholesale price of the manufacturer cr = 3$ per unit.
Manufacturer : processing, packing & shipping DO orders
• A line packing bottles at a steady rate matching the demand.
• Fixed setup cost Km = 250$ / order
• Production cost cm = 2$/bottle
• Holding cost = 20%
DO makes its lot sizing decision based on costs it faces.
45
Manufacturer
Inventory
production
for next DO
delivery
DO Inventory
46
Coordination to increase total SC profits
Quantity discount for commodity products
DO Local optimum face wholesale unit price cr = 3$
DO : D = 120000/year, Kr = 100$, I = 20%, cr = 3$
EOQ = 6324,
TC-DO = 3795 $ (DO cost excluding purchasing cost crD)
Manufacturer :
• Setup cost Km = 250$ / order, unit production cost cm = 2$
• Annual setup cost = KmD/EOQ = 4744$
• Annual holding cost = (Icm)EOQ/2 = 1265$
• TC-M = 6009$ (Manuf cost excluding sales revenue crD)
Total SC cost = TC-DO + TC-M = 6009 + 3795 = 9804$
47
Coordination to increase total SC profits
Quantity discount for commodity products
Supply Chain Global Optimum
DO cost : TC-DO = KrD/Q + (Icr)Q/2
Manufacturer cost: TC-M = KmD/Q + (Icm)Q/2
SC cost: TC-SC = TC-DO + TC-M = (Km+Kr)D/Q + I(cr+cm)Q/2
SC lot size Q = [2D(Km+Kr)/ (Icr+Icm)]0,5=9165
Opt SC cost = 9165 $
Gain of SC optimum = 9804 – 9165 = 638$
48
Coordination to increase total SC profits
Quantity discount for commodity products
Pricing scheme for achieving opt SC profit:
• C = 3$/bottle if Q < 9165, C = 2.9978$ otherwise.
DO :
• has an incentive to order EOQ = 9165,
• material cost reduction just enough to offset the
increase of ordering & holding cost
Total SC cost = opt SC cost = 9165 $
In practice, the manufacturer may have to share the
increase of SC profit of 638$.
49
Coordination to increase total SC profits
Quantity discount for commodity products
Key point
• For commodity products for which price is set by the
market, manufacturer with large fixed costs per lot can
use lot-size quantity discounts to maximise total SC
profits.
• Lot size-based discounts, however, increase cycle
inventory in the SC.
• The benefit of quantity discount decreases as the setup
cost of the manufacturer decreases. (Importance of
coordination between marketing & production)
50
Coordination to increase total SC profits
Quantity discount for products for which the firm has market
power
• Consider the scenario in which the manufacturer has
invented a new vitamin pill, vitaherb, for which few
competitors exist.
• The price p at which DO sells vitaherb influence
demand.
• Assume that: D = 360000 – 60000p.
• Production cost Cs = 2$/bottle
• The manufacturer decides the price Cr to charge DO
51
Coordination to increase total SC profits
Quantity discount for products for which the firm has market
power
When decisions are coordinated:
• SC profil :
Prof_SC = (p – Cs)(360000 – 60000p)
Results:
• p = 3 + Cs/2 = 4
• D = 120000,
• Prof_SC = 240000$
52
Coordination to increase total SC profits
Quantity discount for products for which the firm has market
power
When decisions are made independently:
• DO sets p : MAXp Prof_r = (p – Cr)(360000 – 60000p)
p = 3+0.5Cr
• Manufacturer sets Cr: MAXCr Prof_m
• Prof_m = (Cr – Cs)(360000 – 60000p)
= (Cr – Cs)(180000 – 30000Cr)
• Cr = 3 + 0.5Cs = 4  p = 5, D = 60000
Results:
• Prof_m = 120000$, Prof_r = 60000$, SC profil = 180000
• Loss of 60000$ due to independent price setting, phenomenon
known as double marginalization
53
Coordination to increase total SC profits
Quantity discount for products for which the firm has market
power
Key point
• The supply chain profit is lower if each stage of the
supply chain makes its pricing decisions
independently, with the objective of maximizing its
own profit.
• A coordinated solution results in higher profit.
54
Coordination to increase total SC profits
Quantity discount for products for which the firm has market
power
Pricing schemes to achieve the coordinated solution
Two-part tariff :
• The manufacturer charges its entire profit as an up-front
franchise fee and then sells to the retailer at cost.
• It is then optimal for the retailer to price as though the two
stages are coordinated.
Example :
• Opt Prof_SC = 240000 $, Prof_DO = 60000$ (when no
coordination)
• Pricing scheme: charge the DO of the franchise fee of 180000$
and material cost of Cr = 2$/bottle.
• DO maximises its profit if it sets p = 4$ (Why?).
55
Coordination to increase total SC profits
Quantity discount for products for which the firm has market
power
Pricing schemes to achieve the coordinated solution
Volume-based quantity discount:
• The two-tariff is a volume-based quantity discount as the
average material cost of DO declines as the purchase increases.
• Design discount scheme to encourage DO the purchase the opt
quantity 120000.
• Pricing scheme : Cr = 4$ if the purchase < 120000, and Cr =
3.5$ otherwise.
• DO optimal solution: p = 4, Prof_DO = 60000$, D = 120000,
Prof_SC = 240000$ (Why?).
56
Coordination to increase total SC profits
Quantity discount for products for which the firm has market
power
Key point
• For products for which the firm has market power, two-part tariffs or
volume-based discounts can be used to achieve SC coordination and
maximize SC profits.
• Lot size-based discounts are not optimal even in the presence of
inventory costs. In such as setting, either two-part tariff or a volumebased discount, with the supplier passing on some of its fixed cost to
the retailer, is needed for the SC to be coordinated.
• Lot size based discount tends to raise the cycle inventory. In contrast,
volume based discounts are compatible with small lots. Use lot size
based discount only when the supplier has high fixed cost.
• Volume-based discounts suffer from orders peak toward the end of
financial horizon. Volume discount based on a rolling horizon could
57
help.
Short-term discounting: trade
promotions
58
Introduction
• Manufacturers use trade promotions to offer a discounted price
and a time period over which the discount is effective.
• Ex: 10% off for any purchase from 12/15 to 01/25.
• The goal is to influence retailers to act in a way that helps the
manufacturer achieve its objectives.
59
Introduction
Key goals (from the manufacturer perspective)
• Induce retailers to use price discount, displays or advertising to
spur sales
• Shift inventory from the manufacturer to the retailer and the
customer
• Defend a brand against competition
Need to understand the impact of trade promotion on the
behaviour of a retailer and SC performances.
60
Introduction
Retailer’s options facing a trade promotion
1. Pass through some or all of the promotion to customers to spur
sales (increase the sales of the whole SC)
2. Pass through very little of the promotion to customer but
purchase in greater quantity during promotion period to exploit
the temporary reduction in price (forward buy and no increase
of sales)
61
Forward buy
• d$: discount per product offered
• Q* : EOQ at normal price
• Qd : lot size ordered at discounted price
Assumptions:
• Discount is offered once
• Retailer takes no action to influence demand
• Qd is an integer multiple of Q*.
62
Qd
Q*
Time
63
Forward buy
dD
cQ *
Q 

c  d  I c  d
d
Forward buy = Qd – Q*
• Why? Profit maximization (gain in fix cost, gain in purchase,
loss of inventory cost).
64
Forward buy
• Let T = Q/D be the period covered by short-term promotion
• Cost during T without promotion
Q*
 Q Q * K  Qc  Ic T
2
• Cost during T with promotion
Q
K  Q c  d   I c  d  T
2
• Cost Gain during T
D  Q    Q Q *  1 K  Qd  0.5  I  c  d  Q  IcQ * T
Q
  Q Q *  1 K  Qd  0.5  I  c  d  Q  IcQ *
D
65
Forward buy
• The optimal Q is obtained from
IcQ * I  c  d  Q

D
D
I c  d  Q
1
  KD Q *  0.5IcQ *  d 
D
D
I c  d  Q
1
 IcQ *  d 
(From property of EOQ, KD Q *  0.5IcQ*)
D
D
D  Q  Q  K Q *  d  0.5
I c  d  Q
1
D  Q  Q  0  IcQ *  d 
0
D
D
IcQ *  dD
Q
I c  d 
66
Forward buy
Example:
• DO is a retailer selling vitaherb. Demand is 120000 bottles/year.
The manufacturer currently charges 3$/bottle and DO has an
annual holding cost of 20%. Fixed order cost K = 1000 $. What
is the current lot size Q* of DO, cycle time, average flow time?
• The manufacturer has offered a discount of 0.15$ for all bottles
purchased by the retailer over the coming month. How many
bottles should DO order given the promotion?
Answer:
• Q* = 6324, Qd = 38236, Forward buy = 31912
Remark:
• 5% discount causes the lot size to increase by 500+%.
67
Forward buy
Key point :
• Trade promotions lead to a significant increase in lot size and
cycle inventory because of forward buying by retailer.
• This generally results in reduced SC profits unless the trade
promotion reduces demand fluctuations.
68
Impact on the demand
Example:
• Assume DO selling at price p faces a demand of D = 300000 –
60000p. The normal price charged by the manufacturer is Cr =
3$/bottle. Ignoring the inventory related costs, evaluate the
optimal response of DO to a discount of 0.15$ per bottle.
Answer:
• Without discount and Cr = 3$, p = 4$, D = 60000
• With discount of 0.15$ and Cr = 2.85$, p = 3.925$, D = 64500.
• 7.5% increase in demand, DO pass only half of the trade
promotion discount to Customers.
Why:
• Maxp Profit-DO = (p – Cr)*(300000 – 60000p)  p = 2.5+0.5Cr
69
Impact on the demand
Key point
• Faced with a short term discount, it is optimal for retailers to
pass through only a fraction of the discount to the customer,
keeping the rest for themselves.
• Simultaneously, it is optimal for retailers to increase the
purchase lot size and forward buy for future periods.
• Thus, trade promotions often lead to an increase of cycle
inventory in a SC without a significant increase in customer
demand.
• Trade promotion should be designed so that retailers limit their
forward buying and pass along more of the discount to end
customers.
70
Managing multiechelon cycle inventory
71
A mutliechelon distribution supply chain
stage
1
stage
2
stage
3
stage
4
A multiechelon supply chain has multiple stages and possibly many
players et each stage.
Goal: decrease the total costs by coordinating orders across the SC
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One manufacturer supplying one retailer
(Instantaneuous production, lotsize Q)
No synchronization : production right after delivery, average INV = 3Q/2
mfg
inventory
retailer
inventory
shipping
production
Synchronization : production after before delivery, average INV = Q/2
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Simple multiechelon with one player at each stage
Integer replenishment policy:
• lot size at each stage = integer multiple of the lot size of its
immediate customer
• Coordination of ordering across stages allows for a portion of
the delivery to a stage to be cross-docked on to the next stage
• Extent of cross-docking depends on the ratio of fixed ordering
cost S and holding cost H at each stage. The closer the ratio, the
higher the optimal percentage of cross-docked product.
Shown to be quite close to optimal.
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distributor replenshment order arrives
Distributer replenishes every two weeks
retailer shipment is cross-docked
Retailer replenishes every week
retailer shipment is from inventory
Retailer replenishes every two weeks
retailer shipment is cross-docked
retailer shipment is cross-docked
Retailer replenishes every four weeks
One distributor supplies multiple retailers
Integer replenishment policy:
• Distinguish retailers with high demand from those with low
demand
• Group retailers such that all retailers in one group order together
• fr = n*fd or fd = n*fr, for each retailer r where n is an integer
and fr and fd are retailer and distributor order frequencies
• Each player orders periodically with reorder interval equal to an
integer multiple of some base period
Shown to be near optimal by Roundy.
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Integer replenishment policies
• Divide all parties within a stage into groups such that all parties of a
group order from the same supplier and have the same reorder interval
• Set reorder intervals across stages such that the receipt of a
replenishment order at any stage is synchronized with the shipment of a
replenishment order to at least one of its customers. The synchronized
portion can be cross-docked.
• For customers with a longer reorder interval than the supplier, make the
customer reorder interval an integer multiple of the suppliers' interval
and synchronize their replenishment to facilitate cross-docking
• For customers with a shorter reorder interval, make the supplier's
reorder interval an integer multiple of the customer's interval and
synchronize the replenishment
• The relative frequency of reordering depends on the setup cost, holding
cost and demand at different parties.
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Key points
• Integer replenishment policies can be synchronized in multiechelon
supply chains to keep cycle inventory and order costs low.
• Under such policies, the reorder interval at any stage is an integer
multiple of a base reorder interval.
• Synchronized integer replenishment policies facilitate a high level of
cross-docking.
• Whereas the integer policies synchronize replenishment and decrease
cycle inventories, they increase safety inventories because of the lack of
flexibility with the timing of a reorder
• These policies make the most sense for supply chains in which cycle
inventories are large and demand is relatively predictable.
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Integer replenishment policies
• Divide all parties within a stage into groups such that all parties of a
group order from the same supplier and have the same reorder interval
• Set reorder intervals across stages such that the receipt of a
replenishment order at any stage is synchronized with the shipment of a
replenishment order to at least one of its customers. The synchronized
portion can be cross-docked.
• For customers with a longer reorder interval than the supplier, make the
customer reorder interval an integer multiple of the suppliers' interval
and synchronize their replenishment to facilitate cross-docking
• For customers with a shorter reorder interval, make the supplier's
reorder interval an integer multiple of the customer's interval and
synchronize the replenishment
• The relative frequency of reordering depends on the setup cost, holding
cost and demand at different parties.
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Echelon inventory
• Ordering policies based on echelon inventory (s, S), (r, Q)
• Problems: where to locate the inventory, how to allocate the inventory
warehouse echelon inventory
supplier
warehouse
warehouse echelon lead time
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Echelon inventory
Local installation based-stock policy:
• Order-up-to local base stock Sj if the local inventory position < Sj
• Local inventory position of j = inventory on hand at stage j + on order
of stage j + in transit to stage j – backorder (on order) of stage j-1
S j  L j E  D   z L j  D
On
order
L4
L3
On
hand
3
4
in
transit
L2
2
L1
1
Demand D
Echelon inventory
Echelon based-stock policy:
• Order-up-to echelon base stock ESj if the echelon inventory
position < ESj
• Echelon inventory position of j = on order of stage j + inventories
at stages (j, j-1, …, 1) + in transit to stages (j, j-1, …, 1) –
backorder of stage 1
ES j   L j  L j 1  ...  L1  E  D   z L j  L j 1  ...  L1 D
L4
4
L3
3
L2
2
L1
Echelon 3
1
Demand D