Transportation Research Part E 45 (2009) 572–582
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Transportation Research Part E
journal homepage: www.elsevier.com/locate/tre
A technical note on ‘‘Optimizing inventory decisions in a multi-stage
multi-customer supply chain”
Kit Nam Francis Leung *
Department of Management Sciences, City University of Hong Kong, 83 Tat Chee Avenue, Kowloon Tong, Hong Kong
a r t i c l e
i n f o
Article history:
Received 9 September 2008
Received in revised form 4 January 2009
Accepted 24 January 2009
Keywords:
Inventory
Production
Without derivatives
The perfect squares method
a b s t r a c t
We first generalize Khouja [Khouja, M., 2003. Optimizing inventory decisions in a multistage multi-customer supply chain. Transportation Research Part E: Logistics and Transportation Review 39 (3), 193–208] integrated model considering the integer multipliers
mechanism and next individually derive the optimal solution to the three- and four-stage
model using the perfect squares method, which is a simple algebraic approach so that
ordinary readers unfamiliar with differential calculus can understand the optimal solution
procedure with ease. We subsequently deduce the optimal expressions for Khouja (2003)
and Cárdenas-Barrón [Cárdenas-Barrón, L.E., 2007. Optimal inventory decisions in a multistage multi-customer supply chain: a note. Transportation Research Part E: Logistics and
Transportation Review 43 (5), 647–654] model, and identify the associated errors in Khouja
(2003). We present two numerical examples for illustrative purposes. We finally shed light
on some future research by extending or modifying the generalized model.
Ó 2009 Elsevier Ltd. All rights reserved.
1. Introduction
Increasing attention has been given to the management of a multi-stage multi-firm (or multi-customer) supply chain in
recent years. This is due to increasing competitiveness, short life cycles of modern electronic products and the quick global
changes in today’s businesses. The integration of the supply chain provides a key to successful international business operations. This is because the integrated approach improves the global system performance and cost effectiveness. Besides integrating all members in a supply chain, to improve the traditional method of solving inventory problems is also necessary.
Without using derivatives, Grubbström (1995) first derived the optimal expressions for the classical economic order quantity
(EOQ) model using the unity decomposition method, which is an algebraic approach. Adopting this method, Grubbström and
Erdem (1999) and Cárdenas-Barrón (2001), respectively, derived the optimal expressions for an EOQ and economic production quantity (EPQ) model with complete backorders. In this note, a generalized model for a three- or four-stage multi-firm
production-inventory integrated system is solved using the revised version of the perfect squares method, which is also an
algebraic approach; whereby optimal expressions of decision variables and the objective function are derived.
In addition to the papers with regard to solving some inventory models without derivatives surveyed by and classified in
Table 1 of Cárdenas-Barrón (2007), we review some recently relevant papers as follows: using the unity decomposition
method, Chiu et al. (2006) derived the optimal expressions for an EPQ model with complete backorders, a random proportion
of defectives, and an immediate imperfect rework process while Cárdenas-Barrón (2008) derived those for an EPQ model
with no shortages, a fixed proportion of defectives, and an immediate or a N-cycle perfect rework process. Using the complete squares method and perfect squares method proposed by Chang et al. (2005), Wee and Chung (2007) and Chung and
* Tel.: +852 27888589; fax: +852 27888560.
E-mail address: [email protected]
1366-5545/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved.
doi:10.1016/j.tre.2009.01.007
K.N.F. Leung / Transportation Research Part E 45 (2009) 572–582
573
Wee (2007), respectively, derived the optimal expressions for a two- and three-stage single-firm supply chain inventory
model with complete backorders, and lot streaming (which means that any shipments can be made from a production batch
before the whole batch is finished). Leung (2008a) proposed revised versions of the complete and perfect squares methods to
derive the optimal expressions for an EOQ model with partial backorders and Leung (2008b) also adopted them to derive
those for an EOQ model when the quantity backordered and the quantity received are both uncertain. Teng (2008) proposed
the arithmetic–geometric-mean-inequality method to derive the optimal expressions for the classical EOQ model. Wee et al.
(2009) proposed a modified version of the cost-difference comparisons method originated from Minner (2007) to individually derive the optimal expressions for an EOQ and EPQ model with complete backorders.
2. Assumptions and notation
Our multi-stage multi-firm supply chain inventory-production model is based on the assumptions stated by Khouja
(2003), with the following five main exceptions:
(1)
(2)
(3)
(4)
(5)
The setup or ordering costs are different for all firms in the chain.
The holding costs of raw materials are different from those of finished products.
The holding costs of raw materials are different for all firms in the chain.
The holding costs of finished goods are different for all firms in the chain.
There are three or more stages.
We thus generalize Khouja’s (2003) model by incorporating these five realistic conditions. The following notation (almost
all as defined in Khouja, 2003) is used in the expression of the joint total relevant cost per year.
Dij = demand rate of firm jð¼ 1; . . . ; J i Þ in stage ið¼ 1; . . . ; nÞ [units per year]
Pij = production rate of firm jð¼ 1; . . . ; J i Þ in stage ið¼ 1; . . . ; n 1Þ [units per year]
Sij = setup or ordering cost of firm jð¼ 1; . . . ; J i Þ in stage ið¼ 1; . . . ; nÞ [$ per cycle]
ðrmÞ
g ij hi1;j = holding cost of incoming raw material of firm jð¼ 1; . . . ; J i Þ in stage ið¼ 1; . . . ; n 1Þ [$ per unit per year]
hij = holding cost of finished goods of firm jð¼ 1; . . . ; J i Þ in stage ið¼ 1; . . . ; nÞ [$ per unit per year]
T nj ¼ T = basic cycle time of firm jð¼ 1; . . . ; J n Þ in stage n (T is a decision variable with non-negative real values) [a fraction
of a year]
Q
T ij ¼ T n1
k¼i K k = integer–multiplier cycle time of firm jð¼ 1; . . . ; J i Þ in stage ið¼ 1; . . . ; n 1Þ (K 1 ; . . . ; K n1 are decision variables, each with positive integral values) [a fraction of a year]
TC ij = total relevant cost of firm jð¼ 1; . . . ; J i Þ in stage ið¼ 1; . . . ; nÞ [$ per year]
P PJi
TC ij = joint total relevant cost as a function of K 1 ; . . . ; K n1 and T (the objective function) [$
JTCðK 1 ; . . . ; K n1 ; TÞ ¼ ni¼1 j¼1
per year]
To simplify the presentation of the subsequent mathematical expressions, we designate
Dij
Pij
Ji
X
uij ¼
SiJ ¼
for i ¼ 1; . . . ; n 1; j ¼ 1; . . . ; J i ;
Sij
ð1Þ
for i ¼ 1; . . . ; n;
ð2Þ
D1j ½u1j ðg 1j þ h1j Þ þ h1j ;
ð3Þ
j¼1
H1 ¼
J1
X
j¼1
Hi ¼
Ji
X
Dij ½uij ðg ij þ hij Þ þ hij J i1
X
j¼1
Di1;j hi1;j
for i ¼ 2; . . . ; n 1;
ð4Þ
j¼1
and
Hn ¼
Jn
X
Dnj hnj j¼1
J n1
X
Dn1;j hn1;j :
ð5Þ
j¼1
The total relevant cost per year of firm jð¼ 1; . . . ; J i Þ in stage ið¼ 1; . . . ; n 1Þ is given by
Qn1
TC ij ¼
k¼i
K k TD2ij
ð
ðg ij þ hij Þ þ
2Pij
Qn1
k¼i
Kk Qn1
k¼iþ1 K k ÞTDij
2
Sij
;
hij þ Qn1
k¼i K k T
ð6Þ
where term 1 represents the sum of the holding cost of raw material as it is being converted into finished goods and that of
finished goods during the production portion of a cycle, term 2 represents the holding cost of finished goods during the nonproduction portion of a cycle, and term 3 represents the setup cost.
574
K.N.F. Leung / Transportation Research Part E 45 (2009) 572–582
To simplify the notation and facilitate the computation, we are better off adopting designation (1) to Eq. (6) rather than
Q
employing the symbols P ¼ i;j P ij and Pij ¼ P Pij (not P Pij Þ as in Khouja (2003) or Cárdenas-Barrón (2007). Hence, Eq. (6)
after some manipulations becomes
TC ij ¼
Q
Qn1
TDij ½uij ðg ij þ hij Þ þ hij n1
TDij hij k¼iþ1 K k
Sij
k¼i K k
þ Qn1
2
2
T k¼i K k
for i ¼ 1; . . . ; n 1; j ¼ 1; . . . ; J i ;
and
n1
Y
K k 1:
ð7Þ
k¼n
The total relevant cost per year of firm jð¼ 1; . . . ; J n Þ in stage n is given by
TC nj ¼
TDnj hnj Snj
þ
2
T
for j ¼ 1; . . . ; J n ;
ð8Þ
where term 1 represents the holding cost of finished goods and term 2 represents the ordering cost.
3. An algebraic solution to an integrated model of a three-stage multi-firm supply chain with an integer multiplier at
each stage without lot streaming (which means that any shipments cannot be made from a production batch until the
whole batch is finished)
The joint total relevant cost per year for the supply chain integrating multiple suppliers ði ¼ 1; j ¼ 1; . . . ; J 1 Þ, multiple manufacturers ði ¼ 2; j ¼ 1; . . . ; J 2 Þ and multiple retailers ði ¼ 3; j ¼ 1; . . . ; J 3 Þ is given by
JTCðK 1 ; K 2 ; TÞ ¼
J1
X
j¼1
TC 1j þ
J2
X
j¼1
TC 2j þ
J3
X
TC 3j :
ð9Þ
j¼1
Substituting Eqs. (7) and (8) with n ¼ 3 into Eq. (9) and using designations (2)–(5) with n ¼ 3 yield
JTCðK 1 ; K 2 ; TÞ ¼
J1
J1
h
i TK X
TK 1 K 2 X
S1J
2
Dij u1j ðg 1j þ h1j Þ þ h1j D1j h1j þ
2
2
TK
1K2
j¼1
j¼1
J2
J2
J3
h
i T X
TK 2 X
S2J
T X
S3J 1 S1J
S2J
D2j u2j ðg 2j þ h2j Þ þ h2j D2j h2j þ
þ
D3j h3j þ
þ
þ S3J
¼
2 j¼1
T K1K2 K2
2 j¼1
TK 2 2 j¼1
T
(
( J
)
J
J
h
i
h
i
1
2
1
X
X
X
T
þ
D1j u1j ðg 1j þ h1j Þ þ h1j þ K 2
D2j u2j ðg 2j þ h2j Þ þ h2j D1j h1j
K1K2
2
j¼1
j¼1
j¼1
!)
J3
J2
X
X
1 S1J
S2J
H1 K 1 K 2 þ H 2 K 2 þ H3
¼
þ
D3j h3j D2j h2j
þ
þ S3J þ T
:
ð10Þ
T K1K2 K2
2
j¼1
j¼1
þ
Using the perfect squares method advocated in Leung (2008a) on p. 279, we have
"sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
#2
ffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 S1J
S2J
H1 K 1 K 2 þ H2 K 2 þ H 3
JTCðK 1 ; K 2 ; TÞ ¼
þ
þ S3J T
T K1K2 K2
2
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
S1J
S2J
þ 2
þ
þ S3J ðH1 K 1 K 2 þ H2 K 2 þ H3 Þ:
K1K2 K2
ð11Þ
For two fixed positive integral values of the decision variables K 1 and K 2 , Eq. (11) has a unique minimum value when the
quadratic non-negative term, depending on T, is made equal to zero. Therefore, the optimal value of the decision variable
and the resulting minimum cost are denoted and determined by
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ffi
S1J
S2J
1
T ðK 1 ; K 2 Þ ¼ 2
þ
þ S3J
;
H1 K 1 K 2 þ H2 K 2 þ H3
K1K2 K2
and
JTC ðK 1 ; K 2 Þ ¼
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
S1J
S2J
2
þ
þ S3J ðH1 K 1 K 2 þ H2 K 2 þ H3 Þ:
K1K2 K2
Multiplying out the two factors inside the square root in Eq. (13) yields
pffiffiffi
JTC ðK 1 ; K 2 Þ ¼ 2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
S1J H2
S2J H3
S1J H3
þ S2J H1 K 1 þ
þ S3J H2 K 2 þ
þ S3J H1 K 1 K 2 þ S1J H1 þ S2J H2 þ S3J H3 :
K1
K2
K1K2
ð12Þ
ð13Þ
575
K.N.F. Leung / Transportation Research Part E 45 (2009) 572–582
Clearly, to minimize JTC ðK 1 ; K 2 Þ is equivalent to minimize
fðK 1 ; K 2 Þ ¼
S1J H2
S2J H3
S1J H3
þ S2J H1 K 1 þ
þ S3J H2 K 2 þ
þ S3J H1 K 1 K 2 :
K1
K2
K1K2
ð14Þ
The two ð¼ 2 1Þ options to determine the optimal integral values of K 1 and K 2 according to Eq. (14) are shown below.
Option (1): Eq. (14) can be written as
H3
S1J H2
f ðK 1 ; K 2 Þ ¼
þ S2J H1 K 1 þ
K1
ð1Þ
S1J
K1
þ S2J
K2
þ S3J ðH1 K 1 þ H2 ÞK 2 :
To minimize fð1Þ ðK 1 ; K 2 Þ is equivalent to separately minimize
ð1Þ
/2 ðK 1 ; K 2 Þ
H3
S1J
K1
þ S2J
K2
þ S3J ðH1 K 1 þ H2 ÞK 2 ;
ð15Þ
and
ð1Þ
/1 ðK 1 Þ S1J H2
þ S2J H1 K 1 :
K1
ð16Þ
Hence, the joint total relevant cost per year can be minimized by first choosing K 1 ¼
that
ð1Þ
ð1Þ
ð1Þ
ð1Þ
K1
and next K 2 ¼
ð1Þ
K2
ð1Þ
K 2 ðK 1 Þ
ð1Þ
/1 ðK 1 Þ < /1 ðK 1 1Þ and /1 ðK 1 Þ 6 /1 ðK 1 þ 1Þ;
such
ð17Þ
and
ð1Þ
ð1Þ
ð1Þ
ð1Þ
ð1Þ
ð1Þ
ð1Þ
ð1Þ
/2 ðK 1 ; K 2 Þ < /2 ðK 1 ; K 2 1Þ and /2 ðK 1 ; K 2 Þ 6 /2 ðK 1 ; K 2 þ 1Þ:
ð18Þ
Two closed-form expressions, derived in Appendix A, for determining the optimal integral values of K 1 and K 2 are denoted
and given by
ð1Þ
K1
and
ð1Þ
K2
¼
$sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
%
S1J H2
þ 0:25 þ 0:5 ;
S2J H1
ð19Þ
ffi
7
6vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
7
6u
7
6u H S1J þ S
2J
7
6u 3 K ð1Þ
7
6t
1
þ 0:25 þ 0:57;
¼6
5
4 S3J ðH1 K ð1Þ þ H2 Þ
1
ð20Þ
where bxc is the largest integer 6 x.
Option (2): Eq. (14) can also be written as
S1J H2 þ HK 23
S2J H3
f ðK 1 ; K 2 Þ ¼
þ S3J H2 K 2 þ
þ H1 ðS2J þ S3J K 2 ÞK 1 :
K1
K2
ð2Þ
To minimize fð2Þ ðK 1 ; K 2 Þ is equivalent to separately minimize
ð2Þ
/2 ðK 1 ; K 2 Þ
S1J H2 þ HK 23
K1
þ H1 ðS2J þ S3J K 2 ÞK 1 ;
and
ð2Þ
/1 ðK 2 Þ S2J H3
þ S3J H2 K 2 :
K2
ð2Þ
Similarly, the joint total relevant cost per year can be minimized by first choosing K 2 ¼ K 2
determined by
ð2Þ
K2
and
ð2Þ
K1
¼
$sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
%
S2J H3
þ 0:25 þ 0:5 ;
S3J H2
ffi
7
6vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
7
6u
7
6u S H þ H3
2
ð2Þ
7
6u 1J
K2
7
6t
¼6
þ
0:25
þ
0:5
7:
ð2Þ
5
4 H1 ðS2J þ S3J K Þ
2
ð2Þ
and next K 1 ¼ K 1
ð2Þ
K 1 ðK 2 Þ
ð21Þ
ð22Þ
576
K.N.F. Leung / Transportation Research Part E 45 (2009) 572–582
4. Deduction of two special models of a three-stage multi-firm supply chain without lot streaming
4.1. Khouja’s (2003) model based on integer–multiplier coordination mechanism
Suppose that for all j; S1j ¼ S1 ; S2j ¼ S2 ; S3j ¼ S3 ; g 1j ¼ h0 ; g 2j ¼ h1j ¼ h1 ; h2j ¼ h2 and h3j ¼ h3 . Then designations (2)–(5)
become
S1J ¼ J 1 S1 ;
S2J ¼ J 2 S2 ;
H1 ¼ ðh0 þ h1 Þ
J1
X
S3J ¼ J 3 S3 ;
ð23Þ
D1j u1j þ Dh1 ;
ð24Þ
D2j u2j þ Dðh2 h1 Þ;
ð25Þ
j¼1
H2 ¼ ðh1 þ h2 Þ
J2
X
j¼1
and
H3 ¼ Dðh3 h2 Þ;
ð26Þ
where the total demand at each stage is given by
D¼
J1
X
D1j ¼
j¼1
J2
X
D2j ¼
j¼1
J3
X
D3j :
ð27Þ
j¼1
Substituting designations (23)–(27) into Eqs. (19) and (20) [or (21) and (22)], (12) and (13), we have the optimal solution
ðK 1 ; K 2 ; T ; JTC Þ to Khouja’s (2003) model with an integer multiplier at each stage. Notice that the optimal expressions of
ðK 1 ; K 2 ; T Þ given by Eqs. (8)–(10) of Khouja (2003) are wrong.
4.2. Khouja’s (2003) model based on equal-cycle-time coordination mechanism
Substituting designations (24)–(27) into Eqs. (12) and (13) with K 1 ¼ K 2 ¼ 1, we have
T ð1; 1Þ ¼
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2ðS1J þ S2J þ S3J Þ
;
PJ1
PJ 2
D1j u1j þ ðh1 þ h2 Þ j¼1
D2j u2j þ Dh3
ðh0 þ h1 Þ j¼1
ð28Þ
and
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
"
#ffi
u
J1
J2
X
X
u JTC ð1; 1Þ ¼ t2 S1J þ S2J þ S3J ðh0 þ h1 Þ
D1j u1j þ ðh1 þ h2 Þ
D2j u2j þ Dh3 ;
j¼1
ð29Þ
j¼1
which are Eqs. (7) and (8) with n ¼ 3 of Cárdenas-Barrón (2007).
5. An algebraic solution to an integrated model of a four-stage multi-firm supply chain with an integer multiplier at
each stage without lot streaming
The joint total relevant cost per year for the supply chain integrating multiple suppliers ði ¼ 1; j ¼ 1; . . . ; J 1 Þ, multiple
manufacturers ði ¼ 2; j ¼ 1; . . . ; J 2 Þ, multiple assemblers ði ¼ 3; j ¼ 1; . . . ; J 3 Þ and multiple retailers ði ¼ 4; j ¼ 1; . . . ; J 4 Þ is given
by
JTCðK 1 ; K 2 ; K 3 ; TÞ ¼
J1
X
j¼1
TC 1j þ
J2
X
TC 2j þ
j¼1
J3
X
j¼1
TC 3j þ
J4
X
TC 4j :
ð30Þ
j¼1
Substituting Eqs. (7) and (8) with n ¼ 4 into Eq. (30), using designations (2)–(5) with n ¼ 4 and manipulating yield
TCðK 1 ; K 2 ; K 3 ; TÞ ¼
1
S1J
S2J
S3J
H 1 K 1 K 2 K 3 þ H2 K 2 K 3 þ H 3 K 3 þ H4
:
þ
þ
þ S4J þ T
T K1K2K3 K2K3 K3
2
ð31Þ
Using the perfect squares method obtains
"sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ffi#2
ffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1
S1J
S2J
S3J
H 1 K 1 K 2 K 3 þ H2 K 2 K 3 þ H3 K 3 þ H4
TCðK 1 ; K 2 ; K 3 ; TÞ ¼
þ
þ
þ S4J T
T K1K2K3 K2K3 K3
2
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
S1J
S2J
S3J
þ 2
þ
þ
þ S4J ðH1 K 1 K 2 K 3 þ H2 K 2 K 3 þ H3 K 3 þ H4 Þ:
K1K2K3 K2K3 K3
ð32Þ
577
K.N.F. Leung / Transportation Research Part E 45 (2009) 572–582
For three fixed positive integral values of K 1 ; K 2 and K 3 , the optimal value of the decision variable T and the resulting minimum cost are denoted and determined by
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
S1J
S2J
S3J
1
T ðK 1 ; K 2 ; K 3 Þ ¼ 2
þ
þ
þ S4J
;
H 1 K 1 K 2 K 3 þ H2 K 2 K 3 þ H 3 K 3 þ H4
K1K2K3 K2K3 K3
ð33Þ
and
JTC ðK 1 ; K 2 ; K 3 Þ ¼
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
S1J
S2J
S3J
2
þ
þ
þ S4J ðH1 K 1 K 2 K 3 þ H2 K 2 K 3 þ H3 K 3 þ H4 Þ:
K1K2K3 K2K3 K3
ð34Þ
Multiplying out the two factors inside the square root in Eq. (34) yields
JTC ðK 1 ; K 2 ; K 3 Þ
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
pffiffiffi
S1J H2
S2J H3
S3J H4
S1J H3
S2J H4
S1J H4
þ S2J H1 K 1 þ
þ S3J H2 K 2 þ
þ S4J H3 K 3 þ
þ S3J H1 K 1 K 2 þ
þ S4J H2 K 2 K 3 þ
þ S4J H1 K 1 K 2 K 3 þ S1J H1 þ S2J H2 þ S3J H3 þ S4J H4 :
¼ 2
K1
K2
K3
K1K2
K2K3
K1K2K3
Clearly, to minimize JTC ðK 1 K 2 K 3 Þ is equivalent to minimize
fðK 1 ; K 2 ; K 3 Þ ¼
S1J H2
S2J H3
S3J H4
S1J H3
S2J H4
þ S2J H1 K 1 þ
þ S3J H2 K 2 þ
þ S4J H3 K 3 þ
þ S3J H1 K 1 K 2 þ
þ S4J H2 K 2 K 3
K1
K2
K3
K1K2
K2K3
S1J H4
þ
þ S4J H1 K 1 K 2 K 3 :
K1K2K3
ð35Þ
There are six ð¼ 3 2 1Þ options to determine the optimal integral values of K 1 ; K 2 and K 3 ; however, we show below that
only five options are distinct according to Eq. (35).
Option (1):
H3
S1J H2
f ðK 1 ; K 2 ; K 3 Þ ¼
þ S2J H1 K 1 þ
K1
ð1Þ
S1J
K1
þ S2J
þ S3J ðH1 K 1 þ H2 ÞK 2 þ
K2
H4
S1J
K1 K2
S
þ K2J2 þ S3J
þ S4J ðH1 K 1 K 2 þ H2 K 2 þ H3 ÞK 3 :
K3
Following the derivation of Eq. (19), we have
ð1Þ
K1
ð1Þ
K3
$sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
%
S1J H2
¼
þ 0:25 þ 0:5 ;
S2J H1
ð1Þ
K2
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ffi
7
6v
7
6u
S1J
7
6u
H3 ð1Þ þ S2J
7
6u
K
6u
þ 0:25 þ 0:57
¼ 6t 1
7;
ð1Þ
5
4 S H K
þH
3J
1
2
1
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ffi
7
6v
7
6u
S1J
S2J
7
6u
H4 ð1Þ ð1Þ þ ð1Þ þ S3J
7
6u
K1 K2
K2
7
6u
t
¼6
þ 0:25 þ 0:57:
5
4 S H K ð1Þ K ð1Þ þ H K ð1Þ þ H
4J
1
1
2
2
2
ð36Þ
3
Option (2):
S1J H2
S3J H4
fð2Þ ðK 1 ; K 2 ; K 3 Þ ¼
þ S2J H1 K 1 þ
þ S4J H3 K 3 þ
K1
K3
S1J
K1
þ S2J
H3 þ HK 34
K2
þ ðS3J þ S4J K 3 ÞðH1 K 1 þ H2 ÞK 2 ;
and
ð2Þ
K1
ð1Þ
ð2Þ
¼ K1 ; K3
$sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
%
S3J H4
¼
þ 0:25 þ 0:5 ;
S4J H3
ð2Þ
K2
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
7
6v
7
6u
S1J
H4
7
6u
H3 þ ð2Þ
ð2Þ þ S2J
7
6u
K1
K3
6u
þ 0:25 þ 0:57
¼ 6t
7:
ð2Þ
ð2Þ
5
4
þH
H K
S þS K
3J
4J
1
3
ð37Þ
2
1
Option (3):
fð3Þ ðK 1 ; K 2 ; K 3 Þ ¼
S2J H3
þ S3J H2 K 2 þ
K2
S1J H2 þ HK 23
K1
þ H1 ðS2J þ S3J K 2 ÞK 1 þ
H4
S1J
K1K2
S
þ K2J2 þ S3J
K3
þ S4J ðH1 K 1 K 2 þ H2 K 2 þ H3 ÞK 3 ;
578
K.N.F. Leung / Transportation Research Part E 45 (2009) 572–582
and
ð3Þ
K2
ð3Þ
K3
$sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
%
S2J H3
¼
þ 0:25 þ 0:5 ;
S3J H2
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ffi
7
6v
7
6u
H3
7
6u
S
H
þ
1J
2
ð3Þ
7
6u
K
6u
2
þ 0:25 þ 0:57
¼ 6t 7;
5
4 H S þ S K ð3Þ
ð3Þ
K1
1
2J
3J
2
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ffi
7
6v
7
6u
S1J
S2J
7
6u
H
þ
þ
S
u
4
3J
ð3Þ ð3Þ
ð3Þ
7
6u
K
K
K
6t 1
2
2
þ 0:25 þ 0:57
¼6
7:
5
4 S H K ð3Þ K ð3Þ þ H K ð3Þ þ H
4J
1
1
2
2
ð38Þ
3
2
Option (4):
H4
S2J H3
f ðK 1 ; K 2 ; K 3 Þ ¼
þ S3J H2 K 2 þ
K2
ð4Þ
S2J
K2
þ S3J
K3
þ S4J ðH2 K 2 þ H3 ÞK 3 þ
S1J H2 þ HK 23 þ KH2 K4 3
K1
þ H1 ðS2J þ S3J K 2 þ S4J K 2 K 3 ÞK 1 ;
and
ð4Þ
K2
ð3Þ
ð4Þ
¼ K2 ;
K3
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ffi
7
6v
7
6u
S2J
7
6u
H
þ
S
u
3J
7
6u 4 K ð4Þ
6t 2
þ 0:25 þ 0:57
¼6
7;
5
4 S H K ð4Þ þ H
4J
ð4Þ
K1
2
3
2
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ffi
7
6v
7
6u
7
6u
3
S1J H2 þ Hð4Þ
þ ð4ÞH4 ð4Þ
7
6u
K
K
K
u
6t 2
2
3
þ 0:25 þ 0:57
¼6
7:
5
4 H S þ S K ð4Þ þ S K ð4Þ K ð4Þ
1
2J
3J
2
4J
2
ð39Þ
3
Option (5):
S3J H4
S1J H2
f ðK 1 ; K 2 ; K 3 Þ ¼
þ S4J H3 K 3 þ
þ S2J H1 K 1 þ
K3
K1
ð5Þ
S1J
K1
þ S2J
H3 þ HK 34
K2
þ ðS3J þ S4J K 3 ÞðH1 K 1 þ H2 ÞK 2 ;
and
ð5Þ
K3
ð2Þ
ð5Þ
¼ K3 ;
K1
ð2Þ
¼ K1
ð1Þ
¼ K1 ;
ð5Þ
K2
ð2Þ
¼ K2 ;
which is the same as Eq. (37). In other words, Option (5) is the same as Option (2).
Option (6):
S2J H3 þ HK 34
S1J H2 þ HK 23 þ KH2 K4 3
S3J H4
f ðK 1 ; K 2 ; K 3 Þ ¼
þ S4J H3 K 3 þ
þ H2 ðS3J þ S4J K 3 ÞK 2 þ
þ H1 ðS2J þ S3J K 2 þ S4J K 2 K 3 ÞK 1 ;
K3
K2
K1
ð6Þ
and
ð6Þ
K3
ð6Þ
K1
ð5Þ
¼ K3
ð2Þ
¼ K3 ;
ð6Þ
K2
ffi
7
6vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
7
6u
7
6u S H þ H4
3
ð6Þ
7
6u 2J
K
t
7
6
3
¼6
þ 0:25 þ 0:57;
ð6Þ
5
4 H2 ðS3J þ S4J K Þ
3
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ffi
7
6v
7
6u
H3
H4
7
6u
S1J H2 þ ð6Þ þ ð6Þ ð6Þ
7
6u
K2
K2 K3
6u
þ 0:25 þ 0:57
¼ 6t 7:
5
4 H S þ S K ð6Þ þ S K ð6Þ K ð6Þ
1
2J
3J
2
4J
2
ð40Þ
3
PJ 3
In particular, substituting designations (24) and (25), H3 ¼ ðh2 þ h3 Þ j¼1
D3j u3j þ Dðh3 h2 Þ; H4 ¼ Dðh4 h3 Þ and
PJ2
PJ 3
PJ 4
PJ1
D ¼ j¼1 D1j ¼ j¼1 D2j ¼ j¼1 D3j ¼ j¼1 D4j into Eqs. (33) and (34) with K 1 ¼ K 2 ¼ K 3 ¼ 1, we have
T ð1; 1; 1Þ ¼
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2ðS1J þ S2J þ S3J þ S4J Þ
;
PJ1
PJ 2
PJ3
D1j u1j þ ðh1 þ h2 Þ j¼1
D2j u2j þ ðh2 þ h3 Þ j¼1
D3j u3j þ Dh4
ðh0 þ h1 Þ j¼1
ð41Þ
579
K.N.F. Leung / Transportation Research Part E 45 (2009) 572–582
and
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
"
#ffi
u
J1
J2
J3
X
X
X
u
JTC ð1; 1; 1Þ ¼ t2ðS1J þ S2J þ S3J Þ ðh0 þ h1 Þ
D1j u1j þ ðh1 þ h2 Þ
D2j u2j þ ðh2 þ h3 Þ
D3j u3j þ Dh4 ;
j¼1
j¼1
j¼1
which are Eqs. (7) and (8) with n ¼ 4 of Cárdenas-Barrón (2007).
6. Numerical examples
Example 1. A three-stage multi-firm supply chain
Suppose that an item has the same characteristics as those on p. 203 of Khouja (2003) as follows:
Holding costs: h0 ¼ 0:08; h1 ¼ 0:8; h2 ¼ 2; h3 ¼ 5 ($ per unit per year)
One supplier ði ¼ 1; j ¼ 1Þ:
D11 ¼ 133; 000 units per year; P11 ¼ 399; 000 units per year; S1 ¼ $800 per setup:
Three manufacturers ði ¼ 2; j ¼ 1; 2; 3Þ
D21 ¼ 70; 000; P 21 ¼ 140; 000 ðunits per yearÞ;
D22 ¼ 36; 000; P22 ¼ 108; 000 ðunits per yearÞ;
D23 ¼ 27; 000; P23 ¼ 108; 000 ðunits per yearÞ;
S2 ¼ $200 per setup:
Seven retailers ði ¼ 3; j ¼ 1; . . . ; 7Þ
D31 ¼ 10; 000;
D35 ¼ 24; 000;
D32 ¼ 20; 000; D33 ¼ 40; 000; D34 ¼ 12; 000;
D36 ¼ 9000; D37 ¼ 18; 000 ðunits per yearÞ;
S3 ¼ $50 per order:
Designations (1) and (23)–(27) give
1
3
S1J ¼ 800;
1
3
S2J ¼ 600; S3J ¼ 350;
1
þ 133; 000ð0:8Þ ¼ 145; 413:33;
H1 ¼ ð0:08 þ 0:8Þ 133; 000 3
1
H2 ¼ ð0:8 þ 2Þ 70; 000 0:5 þ 36; 000 þ 27; 000 0:25 þ 133; 000ð2 0:8Þ ¼ 310; 100;
3
u11 ¼ ; u21 ¼ 0:5; u22 ¼ ; u23 ¼ 0:25;
H3 ¼ 133; 000ð5 2Þ ¼ 399; 000;
D ¼ 133; 000:
Eqs. (19), (20) and (13) give
$sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
%
800ð310; 100Þ
¼
þ 0:25 þ 0:5 ¼ b2:26c ¼ 2;
600ð145; 413:33Þ
7
6sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
800
7
6
7
6
399;
000
þ
600
ð1Þ
2
K2 ¼ 4
þ 0:25 þ 0:55 ¼ b1:97c ¼ 1;
350ð145; 413:33 2 þ 310; 100Þ
ffi
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
800 600
JTC ð2; 1Þ ¼ 2
þ
þ 350 ð145; 413:33 2 þ 310; 100 1 þ 399; 000Þ
2
1
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
¼ 2ð1350Þð999; 926:26Þ ¼ $51; 959:62 per year:
ð1Þ
K1
Eqs. (21), (22) and (13) give
$sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
%
600ð399; 000Þ
¼
þ 0:25 þ 0:5 ¼ b2:07c ¼ 2;
350ð310; 100Þ
7
6sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
7
6
399;000
7
6
800ð310;
100
þ
Þ
ð2Þ
2
K1 ¼ 4
þ 0:25 þ 0:55 ¼ b2:05c ¼ 2;
145; 413:33ð600 þ 350 2Þ
ffi
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
800 600
JTC ð2; 2Þ ¼ 2
þ
þ 350 ð145; 413:33 4 þ 310; 100 2 þ 399; 000Þ
4
2
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
¼ 2ð850Þð1; 600; 853:32Þ ¼ $52; 157:52 per year:
ð2Þ
K2
ð42Þ
580
K.N.F. Leung / Transportation Research Part E 45 (2009) 572–582
Because JTC ð2; 1Þ < JTC ð2; 2Þ, the optimal integral values of K 1 and K 2 are 2 and 1. Hence, Eq. (12) gives the optimal basic
cycle time
T ð2; 1Þ ¼
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2ð1350Þ
¼ 0:05196 year ffi 19 days:
999; 926:26
Thus, the supplier fixes a setup every 38 days, each of the three manufacturers fixes a setup every 19 days and each of the
seven retailers places an order every 19 days. Notice that Table 2 of Khouja (2003) shows the non-optimal basic cycle time
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2ð850Þ
¼ 0:03259 year ffi 12 days.
T ð2; 2Þ ¼ 1;600;853:32
In particular, the optimal solution to the model based on the equal-cycle-time coordination mechanism is as follows:
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2ð800 þ 600 þ 350Þð145; 413:33 þ 310; 100 þ 399; 000Þ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
¼ 2ð1750Þð854; 513:33Þ ¼ $54; 688:18 per year ½5:25% higher than JTC ð2; 1Þ;
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2ð1750Þ
¼ 0:063999 year ffi 24 days:
T ð1; 1Þ ¼
854; 513:33
JTC ð1; 1Þ ¼
Example 2. A four-stage multi-firm supply chain
We extend Example 1 by incorporating two assemblers between the manufacturers and retailers, and assume all the
parameter values as follows:
One supplier ði ¼ 1; j ¼ 1Þ
g 11 ¼ $0:08 per unit per year; h11 ¼ $0:8 per unit per year;
D11 ¼ 133; 000 units per year; P 11 ¼ 399; 000 units per year; S11 ¼ $800 per setup:
Three manufacturers ði ¼ 2; j ¼ 1; 2; 3Þ
g 21 ¼ 0:83;
h21 ¼ 2;
D21 ¼ 70; 000;
P21 ¼ 140; 000;
S21 ¼ 200;
g 22 ¼ 0:81;
h22 ¼ 2:1;
D22 ¼ 36; 000;
P22 ¼ 108; 000;
S22 ¼ 210;
g 23 ¼ 0:79;
h23 ¼ 1:8;
D23 ¼ 27; 000;
P23 ¼ 108; 000;
S23 ¼ 205:
Two assemblers ði ¼ 3; j ¼ 1; 2Þ
g 31 ¼ 2:2;
h31 ¼ 3;
g 32 ¼ 1:9;
h32 ¼ 3:1;
D31 ¼ 75; 000;
D32 ¼ 58; 000;
P31 ¼ 150; 000;
P32 ¼ 116; 000;
S31 ¼ 125;
S32 ¼ 130:
Seven retailers ði ¼ 4; j ¼ 1; . . . ; 7Þ
h41 ¼ 5;
D41 ¼ 10; 000;
S41 ¼ $50 per order;
h42 ¼ 5:1;
D42 ¼ 20; 000;
S42 ¼ 48;
h43 ¼ 4:8;
D43 ¼ 40; 000;
S43 ¼ 51;
h44 ¼ 4:7;
D44 ¼ 12; 000;
S44 ¼ 53;
h45 ¼ 5:2;
D45 ¼ 24; 000;
S45 ¼ 50;
h46 ¼ 5;
D46 ¼ 9000;
S46 ¼ 49;
h47 ¼ 4:9;
D47 ¼ 18; 000;
S47 ¼ 52:
Designations (1)–(5) give
1
3
S1J ¼ 800;
1
3
S2J ¼ 615; S3J ¼ 255; S4J ¼ 353;
1
H1 ¼ 133; 000
0:88 þ 133; 000ð0:8Þ ¼ 39; 013:33 þ 106; 400 ¼ 145; 413:33;
3
1
H2 ¼ ½70; 000ð0:5 2:83Þ þ 36; 000
2:91 þ 27; 000ð0:25 2:59Þ
3
u11 ¼ ; u21 ¼ 0:5; u22 ¼ ; u23 ¼ 0:25; u31 ¼ 0:5; u32 ¼ 0:5;
þ ½70; 000ð2Þ þ 36; 000ð2:1Þ þ 27; 000ð1:8Þ 106; 400 ¼ 151; 452:5 þ 264; 200 106; 400 ¼ 309; 252:5;
H3 ¼ ½75; 000ð0:5 5:2Þ þ 58; 000ð0:5 5Þ þ ½75; 000ð3Þ þ 58; 000ð3:1Þ 264; 200
¼ 340; 000 þ 404; 800 264; 200 ¼ 480; 600;
H4 ¼ ½10; 000ð5Þ þ 20; 000ð5:1Þ þ 40; 000ð4:8Þ þ 12; 000ð4:7Þ þ 24; 000ð5:2Þ
þ 9000ð5Þ þ 18; 000ð4:9Þ 404; 800 ¼ 253; 600:
K.N.F. Leung / Transportation Research Part E 45 (2009) 572–582
581
Eqs. (36) and (34) give
$sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
%
800ð309; 252:5Þ
¼
þ 0:25 þ 0:5 ¼ b2:24c ¼ 2;
615ð145; 413:33Þ
6sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
7
6
7
6
7
480; 600 800
þ
615
ð1Þ
2
K2 ¼ 4
þ 0:25 þ 0:55 ¼ b2:35c ¼ 2;
255ð145; 413:33 2 þ 309; 252:5Þ
7
6sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ffi
800 615
7
6
7
6
253;
600
þ
þ
255
ð1Þ
4
2
K3 ¼ 4
þ 0:25 þ 0:55 ¼ b1:26c ¼ 1;
353ð145; 413:33 4 þ 309; 252:5 2 þ 480; 600Þ
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ffi
800 615 255
JTC ð2; 2; 1Þ ¼ 2
þ
þ
þ 353 ð145; 413:33 4 þ 309; 252:5 2 þ 480; 600 1 þ 253; 600Þ
4
2
1
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
¼ 2ð1115:5Þð1; 934; 358:32Þ ¼ $65; 692:87 per year:
ð1Þ
K1
Similarly, Eqs. (37)–(40) and (34) give
ð2Þ
K1
ð3Þ
K2
ð4Þ
K2
ð6Þ
K3
ð2Þ
¼ 2;
K3
¼ 2;
ð3Þ
K1
ð3Þ
¼ K2
¼ 2;
¼ 1;
K2
ð6Þ
¼ 1;
K2
ð2Þ
¼ 2;
JTC ð2; 2; 1Þ ¼ $65; 692:87 per year;
¼ 2;
ð3Þ
K3
¼ 1;
JTC ð2; 2; 1Þ ¼ $65; 692:87 per year;
ð4Þ
K3
¼ 2;
ð4Þ
¼ 1;
ð6Þ
K1
K1
¼ 2;
¼ 2;
JTC ð2; 2; 1Þ ¼ $65; 692:87 per year;
JTC ð2; 2; 1Þ ¼ $65; 692:87 per year:
Hence, the optimal integral values of K 1 ; K 2 and K 3 are 2, 2 and 1, and Eq. (33) gives the optimal basic cycle time
T ð2; 2; 1Þ ¼
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2ð1115:5Þ
¼ 0:03396 year ffi 13 days:
1; 934; 358:32
Thus, the supplier fixes a setup every 26 days, each of the three manufacturers fixes a setup every 26 days, each of the two
assemblers fixes a setup every 13 days, and each of the seven retailers places an order every 13 days.
In particular, the optimal solution to the model based on the equal-cycle-time coordination mechanism is as follows:
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2ð800 þ 615 þ 255 þ 353Þð145; 413:33 þ 309; 252:5 þ 480; 600 þ 253; 600Þ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
¼ 2ð2023Þð1; 188; 865:83Þ ¼ $69; 355:25 per year ½5:58% higher than JTC ð2; 2; 1Þ;
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2ð2023Þ
¼ 0:05834 year ffi 22 days:
T ð1; 1; 1Þ ¼
1; 188; 865:83
JTC ð1; 1; 1Þ ¼
7. Conclusions and future research
We present a review of some work relating to algebraic solutions to inventory systems from 2006 to 2009 in Section 1.
The main contribution of the note to the literature is twofold: first, we establish a more pragmatic model than that of Khouja
(2003) by including the five realistic conditions listed in Section 2. Secondly, we provide a much more simplified optimal
solution procedure which is developed using the perfect squares method so that users may easily understand and apply than
that provided in Khouja (2003) or Cárdenas-Barrón (2007), and illustrate the procedure through the two numerical
examples.
The limitation of our model and its optimal solution procedure may be that both cannot be readily modified to adapt the
use of the integer powers of two multipliers mechanism in order that the joint total relevant cost can be further reduced.
Khouja (2003) shows numerically that saving may be significant when going from equal-cycle-time to integer multipliers
mechanism, while the saving is less pronounced when going from integer multipliers to integer powers of two multipliers
mechanism. We generally believe that it is not worthwhile to employ the integer powers of two multipliers mechanism with
which inherently complicated computations are accompanied.
Three ready extensions of our model that constitutes future research endeavors in this field are as follows: first, following
the evolutions of a three- and four- stage multi-firm supply chain shown in Sections 3 and 5, we can readily formulate and
algebraically analyze the integrated model of a five- or higher-stage multi-firm supply chain. Secondly, modifying designations
(3)–(5) and following the evolutions shown in Sections 3 and 5, we can solve the integrated model of a n-stage ðn ¼ 2; 3; . . .Þ
multi-firm supply chain for an equal-cycle-time or an integer multiplier at each stage with lot streaming, and deduce the optimal solution to Ben-Daya and Al-Nassar’s (2008) model. Thirdly, using not only the perfect squares method but also the complete squares method advocated in Leung (2008a,b), we can solve the integrated model of a n-stage multi-firm supply chain for
an equal-cycle-time or an integer multiplier at each stage with complete backorders allowed for some/all downstream firms
(or retailers), and without or with lot streaming, and individually deduce the optimal solution to Wee and Chung’s (2007) model (where the three inspection costs are assigned to be zero) and Wee and Chung’s (2007) model.
582
K.N.F. Leung / Transportation Research Part E 45 (2009) 572–582
Acknowledgements
The author is grateful to the two anonymous reviewers and to the Editor-in-Chief, Dr. Wayne K. Talley, for helpful comments and suggestions on the original version of this note.
Appendix A
Substituting Eq. (16) into the two conditions of (17) yields the following inequality:
K 1 ðK 1 1Þ <
S1J H2
6 K 1 ðK 1 þ 1Þ:
S2J H1
ð1Þ
We can derive a closed-form expression concerning the optimal integer K 1
as follows:
S1J H2
þ 0:25 6 K 1 ðK 1 þ 1Þ þ 0:25
S2J H1
S1J H2
þ 0:25 6 ðK 1 þ 0:5Þ2
() ðK 1 0:5Þ2 <
S2J H1
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
S1J H2
() K 1 0:5 <
þ 0:25 6 K 1 þ 0:5
S2J H1
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
S1J H2
S1J H2
()
þ 0:25 0:5 6 K 1 <
þ 0:25 þ 0:5:
S2J H1
S2J H1
K 1 ðK 1 1Þ þ 0:25 <
ð1Þ
From the last inequality, we can deduce that the optimal integer K 1 is represented by expression (19). In an analogous
ð1Þ
manner, the optimal integer K 2 represented by expression (20) is derived.
References
Ben-Daya, M., Al-Nassar, A., 2008. An integrated inventory production system in a three-layer supply chain. Production Planning and Control 19 (2), 97–104.
Cárdenas-Barrón, L.E., 2001. The economic production quantity (EPQ) with shortage derived algebraically. International Journal of Production Economics 70
(3), 289–292.
Cárdenas-Barrón, L.E., 2007. Optimal inventory decisions in a multi-stage multi-customer supply chain: a note. Transportation Research Part E: Logistics
and Transportation Review 43 (5), 647–654.
Cárdenas-Barrón, L.E., 2008. Optimal manufacturing batch size with rework in a single-stage production system: a simple derivation. Computers and
Industrial Engineering 55 (4), 758–765.
Chang, S.K.J., Chuang, P.C.J., Chen, H.J., 2005. Short comments on technical note – the EOQ and EPQ models with shortages derived without derivatives.
International Journal of Production Economics 97 (2), 241–243.
Chiu, Y.S.P., Lin, H.D., Cheng, F.T., 2006. Technical note: optimal production lot sizing with backlogging, random defective rate, and rework derived without
derivatives. Proceedings of the Institution of Mechanical Engineers Part B: Journal of Engineering Manufacture 220 (9), 1559–1563.
Chung, C.J., Wee, H.M., 2007. Optimizing the economic lot size of a three-stage supply chain with backordered derived without derivatives. European
Journal of Operational Research 183 (2), 933–943.
Grubbström, R.W., 1995. Modeling production opportunities: an historical overview. International Journal of Production Economics 41 (1–3), 1–14.
Grubbström, R.W., Erdem, A., 1999. The EOQ with backlogging derived without derivatives. International Journal of Production Economics 59 (1–3), 529–
530.
Khouja, M., 2003. Optimizing inventory decisions in a multi-stage multi-customer supply chain. Transportation Research Part E: Logistics and
Transportation Review 39 (3), 193–208.
Leung, K.N.F., 2008a. Technical note: A use of the complete squares method to solve and analyze a quadratic objective function with two decision variables
exemplified via a deterministic inventory model with a mixture of backorders and lost sales. International Journal of Production Economics 113 (1),
275–281.
Leung, K.N.F., 2008b. Using the complete squares method to analyze a lot size model when the quantity backordered and the quantity received are both
uncertain. European Journal of Operational Research 187 (1), 19–30.
Minner, S., 2007. A note on how to compute order quantities without derivatives by cost comparisons. International Journal of Production Economics 105
(1), 293–296.
Teng, J.T., 2008. A simple method to compute economic order quantities. European Journal of Operational Research. doi:10.1016/j.ejor.2008.05.019.
Wee, H.M., Chung, C.J., 2007. A note on the economic lot size of the integrated vendor–buyer inventory system derived without derivatives. European
Journal of Operational Research 177 (2), 1289–1293.
Wee, H.M., Wang, W.T., Chung, C.J., 2009. A modified method to compute economic order quantities without derivatives by cost-difference comparisons.
European Journal of Operational Research 194 (1), 336–338.