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