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KING FAHD UNIVERSITY OF PETROLEUM & MINERALSINDUSTRIAL & SYSTEMS ENGINEERING DEPARTMENT
Report on:
CONSIGNMENT AND VENDOR MANAGED INVENTORY IN SINGLE-VENDOR MULTIPLE BUYERS SUPPLY CHAINS
Mohamed Ben-Daya, Elkafi Hassini, Moncer Hariga, and Mohammad M. AlDurgam
International Journal of Production Research 51(2013), 1347-1365.
By
Mohammed Al-Marhoun
Term 131
1. Objective
1
Study the values of using a hybrid Vendor Managed Inventory and Consignment
(VMI&CS) partnership inventory program. Through finding the number of shipments
and replenishment cycle time of each batch such that the joint manufacturer and retailer
cost is minimized
2. Introduction
Vendor-buyer partnerships:
i. The vendor and the buyers act independently.
ii. The vendor enters in a VMI&CS partnership with the buyers.
iii. The vendor and the buyer belong to a vertically integrated firm where a single
decision maker decides about the ordering policy.
Consignment (CS):
The process of a supplier placing goods at a customer location without receiving payment
until after the goods are used or sold.
Vendor Managed Inventory (VMI):
The vendor is responsible for managing the inventory for the buyer, including initiating
orders on behalf of the buyer. The vendor in return gets more visibility about the
product’s demand.
Share of cost and decisions in a supply chain under VMI, CS and VMI&CS inventory
management programs
2
Decision Cost
Order quantity Number of shipments Ordering Holding
VMI Vendor Vendor Shared Buyer
CS Buyer Buyer Buyer Shared
VMI&CS Vendor Vendor Shared Shared
3. Notation
A
bpi
: the cost of placing an order by the i
th buyer ($/order)
A
bri
: the cost of receiving a shipment by the i
th buyer ($/order)
Abi
: i
th buyer’s total ordering cost composed of the cost of placing an order and
the cost of receiving a shipment (Abi
= Abpi
+ Abri
) ($/order)
h
boi
: i
th buyer’s opportunity cost of holding one unit in stock for one unit of time
($/unit/unit time)
H
bsi
: i
th buyer’s physical storage cost for one unit of stock held for one unit of
time ($/unit/unit time)
hbi
: i
th buyer’s total holding cost per unit of stock per unit of time (h
bi= h
boi +
hbsi
)
Avs
:
Vendor’s setup cost ($/order)
A :Vendor’s shipment release cost to the i
th buyer ($/order)
3
vri
hv :
Vendor’s total cost of holding one unit in stock for one unit of time ($/unit/unit time)
c :
Unit purchase price paid by the buyers ($/unit).
n :
equal number of shipment that is sent to buyers during a cycle
N :
Number of buyers
di :
demand from buyer i (units)
D :total demand of buyers = ∑
i=1
N
d i (units)
P :
vendor’s production rate (units/unit time)
Decision and consequence variables
qi :
shipment size for buyer i
Q : Total shipments sizes to all buyers = ∑i=1
N
qi
T :
replenishment cycle length
T
Cks
:
Total cost for supply chain party k, where k = v (vendor) and k = bi (buyer
i) under system s=1 (no partnership), s = 2 (Vendor managed inventory and
consignment) and s = 3 (centralized)
4
TCs
:
Total cost for system s=1 (no partnership), s = 2 (Vendor-managed
inventory and consignment) and s = 3 (centralized).
4. Assumptions
1. Share of ordering and holding costs in the different supply chain scenarios
Supply Chain Structure
Supply
Chain Partner
Independent parties
VMI&CS Centralized
Costs Costs Costs
Ordering
Holding Ordering
Holding
Ordering
Holding
Vendor Avs
Avri hv
Abpi
Avs
Avri
hboi
hv Abi
Avs
Avri
hbi
hv
Buyer Abi hbi Abri hbsi
2. The shipments to the buyers are time-phased and their sizes are not proportional to
the buyer’s demand (equal shipments).
3. A cyclic delivery policy where the shipment is sent to each buyer and then repeat this
cycle until all shipments are delivered.
5
Figure 1. Cyclic delivery policy
5. Models
1.1 No partnership
No coordination between the vendor and buyers and all parties act independently and
attempt to optimize their own cost without taking into consideration the decision of the
other parties.
Buyer:
TCbi1 =
Abi
T bi
+hbi
qi1
2=
Abi d i
q i1 +hbi
qi1
2
6
Figure 2. Buyer inventory cycle in no partnership policy
∂
∂ qi1
TC bi1 =0 q i
1=√ 2 Abi d i
hbi
, i=1 ,2 , …, N
∂
∂ Tbi1
TC bi1 =0 T bi
1 =√ 2 Abi
d i hbi
, i=1 , 2 ,…, N
Substituting by the value of qi1 and Tbi
1 into TCbi1
TCbi1 = √2 Abihbid i ,i=1,2 , …, N
Vendor:
TC v1 =
Avs
T v1 +hv
DT v1
2 (1− DP )+∑
i=1
N Avri
T bi1 +hv∑
i=1
N
q i1
Figure 3. Vendor inventory cycle in no partnership policy
D=∑i=1
N
d i
7
∂
∂ Qv1
TC v1=0
Qv1=√ 2 Avs D
hv(1−DP )
∂
∂ T v1
TC v1=0
T v1=√ 2 Avs
hv D(1− DP )
Substituting by the value of Qv1, Tv
1and Tbi1 into TCv
1
TC v1 = √2 Avs D hv (1−D
P )+hv∑i=1
N
qi1+∑
i=1
N (Avri√ hbid i
2 Abi)
TC 1 = TC v1+∑
i=1
N
TCbi1
TC 1 = √2 Avs D hv (1−DP )+hv∑
i=1
N
qi1+∑
i=1
N (Avri√ hbid i
2 Abi
+√2 Abid i hi)Example 1:
P = 3200 items/year D = 1500 items/year
d1 = 500 items/year d2 = 1000 items/year
Avs = $400 per setup Avri = 0 per shipment
Ab1 = $25 per order Ab2 = $75 per order
hb1 = $5 per item per year hb2 = $5 per item per year
hv = $4 per item per year
Tv= 0.501
T1= 0.141 T2= 0.173
q1= 70.71 q2= 173.20
TCv= 2572.53
TC1= 353.55 TC2= 866.03
TC= 3792.11
1.2 VMI&CS Partnership
8
Figure 4. Inventory cycle in VMI&CS partnership policy
Buyer average inventory:
Figure 5. Triangle of type I and rectangle for average inventory
R1 = (QP )(qi−
QP
d i)=QP
q i+Q2
P2 di
T 11 =12 (Q
P )(QP
d i)=12
Q2
P2 d i
9
Figure 6. Triangle type II for average inventory
T 2 =12 (qi+(n−1 )(q i−
QP
d i))( 1d i
(q i+ (n−1 )(qi−QP
d i)))=
12 (qi+(n−1 ) qi−( n−1 ) Q
Pd i)( qi
d i
+ (n−1 )q i
d i
−(n−1 ) QP
qi
d i)
=12 (nqi− (n−1 ) Q
Pd i)( n q i
d i
−(n−1 ) QP )
=12 ( n2 q i
2
di
−n (n−1 )qiQP
−n (n−1 )qi QP
+(n−1 )2 Q2
P2d i)
=n2 qi
2
2 d i
−n (n−1 )q iQ
P+
(n−1 )2
2Q2
P2 d i
T = (n−1 ) QP
+(n q i
d i
−(n−1 ) QP )
= nq i
d i
10
I =1T ((n−1 )T 11+T 12+
n (n−1 )2
R1)= 1
T ( n−12
Q2
P2di+
n2 qi2
2 d i
−n (n−1 ) QP
qi+(n−1 )2
2Q2
P2d i+
n (n−1 )2
QP
qi−n (n−1 )
2Q 2
P2d i)
=1T ( n2 qi
2
2 d i
−n (n−1 )
2QP
qi)Substitute by T=
n qi
d i
=n qi
2−nQ
2 Pd i+
QP
d i
=T d i
2−nTD
2Pnd i+
TDnP
d i
=T2
d i(1− DP
+ DnP )
Vendor :
TC v2 =
Avs+n∑i=1
N
Avri
T+
hv
2nT∑
i=1
N d i2
P
+n∑i=1
N
Abpi
T+ T
2∑i=1
N
hboi di(1−DP
+ DnP )
∂∂ T
TC = 0
11
−Avs+n∑i=1
N
Avri
T 2 +hv
2 n∑i=1
N d i2
P−
n∑i=1
N
Abpi
T 2 + 12∑i=1
N
hboi d i(1− DP
+ DnP )=0
1T2 (Avs+n∑
i=1
N
Avri+n∑i=1
N
Abpi)= hv
2n∑i=1
N d i2
P+ 1
2∑i=1
N
hboi d i(1− DP
+ DnP )
T ¿ = √ 2(Avs+n∑i=1
N
( Avri+ Abpi))∑i=1
N
( hv
nPdi
2+hboi d i(1− DP
+ DnP ))
TC v2(n) = √2(Avs+n∑
i=1
N
( Avri+ Abpi ))∑i=1
N
( hv
nPd i
2+hboi di(1−DP
+ DnP ))
Minimizing TC v2(n)
Means minimizing
(Avs+n∑i=1
N
( Avri+Abpi ))∑i=1
N
( hv
nPd i
2+hboi d i(1− DP
+ DnP ))
Which equivalent to minimizing
Avs
nP∑i=1
N
(hv d i2+hboi d i D )+n(1−D
P )∑i=1
N
( Avri+ Abpi )∑i=1
N
hboi d i
Applying the first difference approach
TC v2 (n )<TC v
2 (n+1 )
Avs
nP∑i=1
N
(hv d i2+hboi d i D )+n(1−D
P )∑i=1
N
( Avri+ Abpi )∑i=1
N
hboi d i
12
¿Avs
(n+1)P∑i=1
N
(hv d i2+hboi d i D )+(n+1)(1− D
P )∑i=1
N
( A vri+ Abpi)∑i=1
N
hboi d i
n (n+1 )>Avs(hv∑
i=1
N
di2+D∑
i=1
N
hboi d i)( P−D )∑
i=1
N
( Avri+ Abpi )∑i=1
N
hboi d i
Let β=Avs (hv∑
i=1
N
d i2+D∑
i=1
N
hboi d i)( P−D )∑
i=1
N
( Avri+ Abpi )∑i=1
N
hboi d i
n2+n−β>0
n=−1 ±√1+4 β2
Since n can’t be negative n=−1+√1+4 β2
TC v2 (n )<TC v
2 (n−1 )
Avs
nP∑i=1
N
(hv d i2+hboi d i D )+n(1−D
P )∑i=1
N
( Avri+ Abpi )∑i=1
N
hboi d i
¿Avs
(n−1) P∑i=1
N
(hv d i2+hboi d i D )+(n−1)(1−D
P )∑i=1
N
( Avri+ Abpi)∑i=1
N
hboi d i
n (n−1 )<Avs(hv∑
i=1
N
d i2+ D∑
i=1
N
hboi d i)(P−D )∑
i=1
N
( Avri+ Abpi)∑i=1
N
hboi d i
Let β=Avs (hv∑
i=1
N
d i2+D∑
i=1
N
hboi d i)( P−D )∑
i=1
N
( Avri+ Abpi )∑i=1
N
hboi d i
13
n2−n−β<0
n=1±√1+4 β2
Since n can’t be negative n=1+√1+4 β2
−1+√1+4 β2
<n< 1+√1+4 β2
n2¿=⌈−1+√1+4 β
2⌉
TC bi2 =
n Abri
T+ T
2d i(1−D
P+ D
nP )
Example 2: Consider the same data as in example 1
Abp1= 15 Abp2= 50 hbo1= 2.5 hb02= 2
n2* = 3 T2* = 0.657
TCv2= 1810.72 < TCv
1 TCb12= 328.04 < TCb1
1 TCb22= 791.86 < TCb2
1
TC2= 2930.61 < TC1
The partnership is efficient since all members realized cost savings
“Efficient partnership”
Example 3: Consider the same data as in example 1
Abp1= 20 Abp2= 65 hbo1= 4.5 hbo2=4.5
14
n2* = 3 T2
* = 0.504
TCv2= 2600.29 > TCv
1 TCb12= 73.07 < TCb1
1 TCb22= 146.14 <TCb2
1
TC2= 2819.50 < TC1
The supply chain cost is smaller than the one with no partnership, but the vendor is
now worse off.
“potentially efficient partnership”
Partnership coordination through side payments
• If vendor is worse off, some of the buyers’ savings can be transferred to the vendor
through a unit price increase.
• The maximum price increase is the one that makes at least one buyer indifferent to
go for the partnership.
• The minimum price increase is the one that makes the vendor no worse off without
partnership.
• Similarly, when the partnership achieves system-wide savings and some buyers (or
all of them) are worse off, the vendor can offer a price discount to these buyers as an
incentive to accept the partnership.
Proposition 1.
To achieve coordination the vendor can vary the unit price c in the range [Cmin, Cmax]
where
cmin =c+
TCv2−TC v
1
∑i=1
N
d i
15
cmax = c+min1≤i ≤ N [TCb2−TC b
1
di]
Proof.
The minimum price increase is the one that makes the vendor no worse off without
partnership.
cmin−c =
TCv2−TCv
1
∑i=1
N
d i
By definition the maximum price increase is obtained by finding the maximum price that
satisfies all the following inequalities:
c d i+TCbi1 ≥ cmax d i+TC bi
2 For i= 1, 2, … , N
Example 4:
Abp2= 40
TCv2= 1759.99 TCb1
2= 342.31 TCb22= 883.99
TC2= 2986.29
Only Buyer 1 is better off
1.3 Centralized supply chain
The vendor and buyers are part of a vertically integrated supply chain under a common
control.
16
TC 3 = TC v3+∑
i=1
N
TCbi3
TC 3 =Avs+n∑
i=1
N
Avri
T+ T
2 ( hv
nP∑i=1
N
d i2)
+ ∑i=1
N
( n Abi
T+ T
2 (1−DP
+ DnP )hbi d i)
Let Ai = Avri + Abi be the total ordering cost
TC 3 =Avs+n∑
i=1
N
A i
T+ T
2 ( hv
nP∑i=1
N
d i2+(1− D
P+ D
nP )∑i=1
N
hbi d i)
∂∂ T
TC3
= 0
−Avs+n∑i=1
N
A i
T 2 + 1T2 ( hv
nP∑i=1
N
di2+(1− D
P+ D
nP )∑i=1
N
hbid i)=0
T ¿ = √ 2(Avs+n∑i=1
N
A i)hv
nP ∑i=1
N
d i2+(1−D
P+ D
nP )∑i=1
N
hbi d i
TC 3 = √2(Avs+n∑i=1
N
Ai)( hv
nP∑i=1
N
di2+(1− D
P+ D
nP )∑i=1
N
hbi di)
Minimizing TC 3(n)
17
Means minimizing
(Avs+n∑i=1
N
Ai)( hv
nP∑i=1
N
d i2+(1− D
P+ D
nP )∑i=1
N
hbi d i)Which equivalent to minimizing
Avs
hv
nP∑i=1
N
d i2+Avs
DnP
∑i=1
N
hbid i+n∑i=1
N
A i(1− DP )∑
i=1
N
hbi d i
Avs
nP (hv∑i=1
N
d i2+D∑
i=1
N
hbid i)+n(1−DP )∑
i=1
N
A i∑i=1
N
hbid i
Applying the first difference approach
TC 3 (n )<TC3 (n+1 )
Avs
nP (hv∑i=1
N
d i2+D∑
i=1
N
hbid i)+n(1−DP )∑
i=1
N
A i∑i=1
N
hbid i
¿Avs
(n+1)P (hv∑i=1
N
d i2+D∑
i=1
N
hbi di)+(n+1)(1− DP )∑
i=1
N
A i∑i=1
N
hbi di
n (n+1 )>
Avs
P (hv∑i=1
N
d i2+D∑
i=1
N
hbid i)(1− D
P )∑i=1
N
A i∑i=1
N
hbi d i
Let β=Avs (hv∑
i=1
N
d i2+D∑
i=1
N
hbi d i)( P−D )∑
i=1
N
A i∑i=1
N
hbid i
n2+n−β>0
18
n=−1 ±√1+4 β2
Since n can’t be negative n=−1+√1+4 β2
TC 3 (n )<TC3 (n−1 )
Avs
nP (hv∑i=1
N
d i2+D∑
i=1
N
hbid i)+n(1−DP )∑
i=1
N
A i∑i=1
N
hbid i
¿Avs
(n−1) P (hv∑i=1
N
d i2+D∑
i=1
N
hbi d i)+(n−1)(1−DP )∑
i=1
N
A i∑i=1
N
hbid i
n (n−1 )<
Avs
P (hv∑i=1
N
d i2+D∑
i=1
N
hbi d i)(1− D
P )∑i=1
N
Ai∑i=1
N
hbid i
Let β=Avs (hv∑
i=1
N
d i2+D∑
i=1
N
hbi d i)( P−D )∑
i=1
N
A i∑i=1
N
hbid i
n2+n−β>0
n=−1 ±√1+4 β2
Since n can’t be negative n=−1+√1+4 β2
−1+√1+4 β2
<n< 1+√1+4 β2
n2¿=⌈ −1+√1+4 β
2⌉
19
n3¿
= ⌈ 0.5(−1+√1+4 Avs
hv∑i=1
N
d i2+ D∑
i=1
N
hbid i
( P−D )∑i=1
N
Ai∑i=1
N
hbi d i)⌉
Proposition 2. TC3(n3*) < TCc(nc*)
Where TCc the total cost for the consecutive delivery policy proposed by Zavanella and
Zanoni (2009)
Consecutive delivery policy: At least one shipment will be sent to each buyer within one
cycle and if there are two or more shipments they will be sent consecutively
TC c∗¿¿ = √2(Avs+n∑i=1
N
Ai)( hv∑i=1
N
d i2+∑
i=1
N
hbi d i2
nP+∑
i=1
N
hbid i−∑i=1
N
hbid i2
P )Proof. Let nc
* be the optimal number of orders for the consecutive delivery policy. Then
TC 3 ( nc¿ )=√2(Avs+n∑
i=1
N
Ai)( hv
nc¿ P
∑i=1
N
d i2+∑
i=1
N
hbi d i−∑i=1
N
d i∑i=1
N
hbi d i
P (1− 1nc
¿ ))
TC c (nc¿)=√2(Avs+n∑
i=1
N
Ai)( hv
nc¿ P
∑i=1
N
d i2+∑
i=1
N
hbi d i−∑i=1
N
hbi d i2
P (1− 1nc
¿ ))
20
Given that ∑i=1
N
d i∑i=1
N
hbid i>∑i=1
N
hbid i2
Then TC 3 ( nc¿ )<TCc (nc
¿ )
and therefore TC 3 ( n3¿ )<TC c (nc
¿ )
21