Impact of Finite Life Cycle to a Consignment Stocking Supply Chain with Uncertain Demand
Impact of Finite Life Cycle to a Consignment Stocking Supply Chain
with Uncertain Demand
Hui Zhi Yi1, Bhaba R. Sarker1*, and Cun Rong Li2
* Corresponding author: Tel.: (225) 578-5370; Fax: (225) 578-5924; E-mail: bsarker@lsu.edu
1Department of Mechanical and Industrial Engineering
Louisiana State University
Baton Rouge, LA, 70803, USA
2School of Mechanical & Electronics Engineering
Wuhan University of Technology
Wuhan, Hubei, 430070, P.R. China
ABSTRACT
There are usually two reasons that force a supply chain to adopt the consignment stocking policy: the high demand
variance and the high risk of obsolescence. This research aims to study the influences of these two issues to a
consignment stocking integrated supply chain. To better reflect the real situation in the engineering practice, the
buyer’s warehouse limitation and a reducible lead time is also considered. A five-variable mixed-integer
optimization model that aimed to minimize the annual joint total expected cost (JTEC) is developed under the
general assumption that the production rate, the number of shipments and the shipping quantities are all fixed. To
distinguish the Consignment Stocking policy from traditional ones, the holding cost is considered to consist of a
storage component and a financial component. Due to the complexity of the objective function, traditional
mathematical methods are cumbersome. As a result, closed-form solutions for the model are not provided. Instead,
a novel doubly-hybrid meta-heuristic algorithm is employed to solve the problem. A numerical example is used to
illustrate the solution procedure and the efficiency of the doubly-hybrid meta-heuristic algorithm.
1. INTRODUCTION
Consignment Stocking (CS) is a novel approach of cooperation between suppliers and customers and has become
popular in variety of industries (Chen and Liu, 2008). The risk of obsolescence is important to the success of a
buyer-vendor integrated system taking the CS policy. Failure to consider the impact of obsolescence, resulting in
unsold products, will decrease the profit of the supply chain. However, it is rare in the literature that that factor is
considered, especially in the context of an integrated CS system. Traditionally, the risk of obsolescence is borne solely
by the buyer of a two-echelon supply chain. On the contrary, under a CS scenario, this risk is shared by both the vendor
and the buyer, since the unsold products remain owned by the vendor. As a result, the replenishment policies of both the
vendor and the buyer have to be modified to accommodate this risk. In order to provide the decision makers with such
a system the optimal decisions against the obsolescence risk, a four-factor CS model that considers obsolescence,
variability in demand rate, buyer’s space limitation, and controllable lead time, is developed in this paper. The
objective is to jointly decide the optimal ordering size, number of shipments within each production cycle, the number
of delay shipments within each cycle, the lead time, and the safety stock, that minimize the annual joint total expected
cost (JTEC) of the system.
Under a traditional CS policy, either the vendor keeps most of the inventory (when the buyer’s unit inventory cost is
much higher than that of the vendors) while maintaining a minimum amount of inventory in the buyer’s warehouse, or
the buyer stores the majority of the products (when the buyer’s unit inventory cost is lower to the vendor), keeping the
lower amount at the vendor’s site. This is defined by Yi and Sarker (2013a, 2013b) as the CS (k = 0) and CS (k = n – 1)
policy. The four-factor CS model to be developed here need to consider several system constraints.
Firstly, the buyer may want to place a space limitation to each of his/her supplier for each of the product. When there
is an upper limit capacity in the space, the vendor cannot put as much inventory as s/he wants to the buyer. In the
beginning, when the buyer’s space limitation is not reached, the vendor is obliged, based on a CS agreement, to keep the
buyer’s inventory above a certain safety level. Toward this end, the products are shipped to the buyer in small quantities
without having to wait until the up-time of each production cycle is ended. As a result, the buyer’s inventory level
Flexible Automation and Intelligent Manufacturing, FAIM2014
gradually increases in the beginning of each cycle. In the meanwhile, the vendor maintains a minimum level of average
inventory (equal to half the shipping size) during this period. However, when the inventory level in buyer’s warehouse
reaches its maximum level, Imax, which is close or equal to the buyer’s space limitation, all later shipments from the
vendor are delayed for a certain period so that the late arrival of a new shipment brings the buyer’s inventory level back
to Imax. As a result, the vendor’s inventory level is forced to be increased to a certain level.
Secondly, the shipments are taking a certain period to reach the buyer, which incurs a shipping cost and a holding
cost of the products in transit. However, the lead time can be reduced, often required by the buyer, with an extra charge,
which is also shared by both the vendor and buyer under a long-term agreement. The lead time crushing cost is a
function of both the time to be reduced and the quantity of the products to be shipped. The trade-off is that, while the
shipping cost might be increased, the holding cost in transit is reduced, and the safety stock level is also reduced
accordingly because of a shorter lead time.
The third constraint is that the demand rate may be uncertain. Based upon historical data; however, it may follow a
stochastic distribution with known mean and variance. Because of the variation in demand, there might be back orders
and/or some extra space may be needed in the buyer’s warehouse. Both of the two cases cost the system an extra.
Finally, the product may be obsolescent sometime, which may occur at any time within the last production cycle.
Thus, the last cycle may be incomplete. After that point, the unsold products and the material are considered to be lost.
Hence, it incurs an obsolescence cost to the system.
The remainder of this paper is organized as follows. Section 2 defines parameters and assumptions used throughout
the paper. The mathematical model is formulated in Section 3. Computational results are illustrated in Section 4.
Conclusions are summarised in Section 5.
2. NOTATION AND ASSUMPTIONS
In order to develop the integrated models, the notations and assumptions used throughout this paper are given
below:
2.1. NOTATIONS
Av: vendor’s batch setup cost ($/setup),
Ab: Buyer’s ordering cost ($/order),
ai: Maximum duration of the ith segment of lead time Si (year),
bi: Minimum duration of the ith segment of lead time Si (year),
cb: Unit backorder cost ($/unit),
Cb: Expected annual backorder cost ($/year),
cil: Unit crashing cost for reducing one time unit of the ith segment of lead time Si when the ordering
quantity q is between ql-1and ql ($/year),
co: Unit outsourcing cost ($/unit),
Co: Expected annual extra space cost ($/year),
D: Yearly demand rate at the buyers’ level (units/year), D ~ N(μ, σ ) .
E(▪): Mathematical expectation of ▪,
h: Annual unit holding cost ($/unit/year),
Hv: Vendor’s average annual inventory holding cost ($/year),
hv: Unit annual holding cost for inventory at the vendor ($/unit/year), f
v
s
hv = hv + h ,
f
hv : Financial components of hv ($/unit/year), v
f
hv = rp ,
s
hv : Storage component of hv ($/unit/year),
Hb: Buyer’s average annual inventory holding cost ($/year),
hb: Unit annual holding cost for inventory at the buyer ($/unit/year), f
v
s
hb = hb + h ,
s
hb : Storage component of hb ($/unit/year),
Hd: Average annual holding cost for inventory in transit ($/year),
hd: Unit annual holding cost in transit ($/unit/year), f
v
s
hd = hd + h
s
hd : Storage component of the holding cost of hd ($/unit/year),
r: Opportunity cost of capital (%
/year),
I: Average inventory (units),
Ib: Buyer’s average inventory (units),
Imax: Buyer’s maximum inventory level (units),
Impact of Finite Life Cycle to a Consignment Stocking Supply Chain with Uncertain Demand
Is : Average system inventory (units),
Iv: Vendor’s average inventory (units),
k: Number of delayed deliveries due to buyer’s stock capacity,
L: Length of lead time (year), = Σ L Si ,
Li: Lead time (year) where the ith component Si was crashed to its minimum duration ai,
n: Number of delivery operations per full production batch,
n * : Number of full production cycles during the entire planning horizon, n* = μT / nq ,
P: Vendor’s production rate (units/year),
p v : Vendor’s unit production cost ($/unit),
pb : Buyer’s unit purchasing cost ($/unit),
q: Size of each delivery or shipped lot (batch size Q = nq ),
R(q, L): Lead time crashing cost per replenishment cycle ($/shipment),
s: Safety factor,
Si: ith segment of lead time (year), ai ≤ Si ≤ bi ,
ss: Safety stock level,
t: Length of the last incomplete production cycle (year), t = T − n *×nq /μ ,
I
tli : Lower bound of the ith scenario associated with Model I (year), where i is a positive integer less or
equal to four.
I
tui : Upper bound of the ith scenario associated with Model I (year), where i is a positive integer less or
equal to four.
II
tli : Lower bound of the ith scenario associated with Model II (year), where i is a positive integer less or
equal to five.
II
tui : Upper bound of the ith scenario associated with Model II (year), where i is a positive integer less or
equal to five.
T : Item life period (year),
U: Space limitation placed by the buyer to the vendor (units),
μ : Expectation value of annual demand rate D (units/year), ∫ +∞
−∞ μ = Df (D)dD ,
σ : Standard deviation of annual demand rate D (units),
v: Number of price segments associated with ordering quantity,
x+: Maximum value of x and 0, i.e. x+ = max{x, 0},
X1: The random demand during the period (q/P + L), having a mean μ(q / P + L) and standard deviation
σ q / P + L ,
X2: The random demand during the period q/P, having a mean qμ / P and standard deviation σ q / P ,
X3: The random demand during the period q /μ , having a mean q and standard deviation σ q /μ ,
X4: A random variable associated with the four-factor CS model, i.e., the during the period
(n − k)q /μ − (n − k)q / P − L , having a mean (n − k)q −(n − k)qμ / P − μL and standard deviation
σ (n − k)q /μ − (n − k)q / P − L ,
X5: The random demand during the last incomplete production cycle t, having a mean μt and standard
deviation σ t ,
X6: The random demand during the period t − L − q / P − (t − L − q / P)P / qq / P , having a standard deviation
σ t − L − q / P − (t − L − q / P)P / qq / P and a mean μ(t − L − q / P) −μ (t − L − q / P)P / q q / P ,
X7: A random variable associated with scenarios 3 and 4 of the four-factor CS model I, i.e., the demand
during the period t − L − (n − k)q / P − [t − L − (n − k)q / P]μ / qq /μ , having a mean
μ[t − L − (n − k)q / P] − [t − L − (n − k)q / P]μ / qq and a standard deviation
σ t − L − (n − k)q / P − [t − L − (n − k)q / P]μ / qq /μ ,
X8: The random demand during the period (n − k +1)q /μ − (n − k −1)q / P −U /μ − sσ L + q / P μ , having a
mean (n − k +1)q − μ(n − k −1)q / P −U − sσ L + q / P and a standard deviation
σ (n − k +1)q /μ − (n − k −1)q / P −U /μ − sσ L + q / P μ ,
X9: The random demand during the period U /μ − q / P − L , which is the duration of the first delayed
shipment in the modified four-factor CS model. X9 has a mean U − μq / P − μL and a standard deviation
σ U /μ − q / P − L ,
X10: A random variable associated the modified four-factor CS model, i.e., the demand during the period
t − (n − k)q / P − L , having a mean μ[t − (n − k)q / P − L] and a standard deviation σ t − (n − k)q / P − L ,
X11: A random variable associated the modified four-factor CS model, i.e., the demand during the period
( )μ / /μ 4 t t t II q q
l
− − , having a mean t t t II qq
l ( ) / 4
μ − − μ and a standard deviation σ ( )μ / /μ 4 t t t II q q
l
− − .
Flexible Automation and Intelligent Manufacturing, FAIM2014
2.2. ASSUMPTIONS
The following assumptions are necessary to the model developed in this paper:
(1) The yearly production rate P is finite.
(2) The demand of the buyer D follows a normal distribution with a mean μ and a standard deviation σ, i.e.,
D ~ N(μ, σ ) .
(3) The demand during lead-time L also follows normal distribution with a mean μL and a standard deviation
σ L .
(4) The inventory is continuously reviewed.
(5) The products are shipped to the buyer in n batches.
(6) Due to the demand uncertainty, shortage is allowed in the system and is backordered with a shortage cost.
(7) Also due to the demand uncertainty, extra inventories beyond the buyer’s capacity are allowed. The extra
products may be still stored in the buyer’s warehouse but in a space reserved for other products/suppliers of
the buyer or they may be stored by a third party. In both cases the vendor will be charged an extra penalty cost.
(8) Unlike most other researchers who consider the shipping cost to be a function of the guaranteed lead time,
this study assume that it is an incremental function of both the guaranteed lead time and the shipment quantity.
(9) The predicted time between the beginning of the first production cycle and the date of obsolescence is
deterministic (Persona et al, 2005, Battini et al, 2010a, Battini et al, 2010b).
3. MODEL FORMULATION
In this Section, the CS-k model considering four practical factors: (a) obsolescence (Obs), (b) controllable lead time
(CLT), (c) buyer’s capacity limitation (BCL), and (d) stochastic demand (SD) is developed according to the pattern of
the system inventory, the vendor inventory, the inventory in transit, and the buyer inventory in a full production cycle of
this model is shown in Figure 1. Specifically, the inventories in a full production cycle is shown in with dashed lines,
whereas the inventory patterns during the last incomplete production cycle is shown with solid lines.
Unlike normal full production cycles, the last cycle may be incomplete due to the product obsolescence. To avoid
further lost caused by the obsolescence, the vendor is assumed to cease its production at the moment when the
obsolescence occurs. The remaining stocks are considered to be lost and will be cleared from the warehouses
immediately. The in-transit inventory, if any, will be cleared when it reaches the buyer. Because the obsolescence can
occur anytime within the last cycle, the inventory patterns can be categorized into four scenarios according to the time
that the obsolescence may occur (Cases 1, 2, 3, and 4 in Figure 1). Specifically, Case 1 is the situation when the
obsolescence occurs before the arrival of the first shipment. Case 2 reflects when the obsolescence occurs after the
arrival of the first shipment, but before the maximum inventory level Imax is reached. Case 3 describes when
obsolescence occurs after the beginning of the first delayed shipment, but before the end of the up-time Tup = nq / P .
Case 4 illustrates when obsolescence occurs during the down-time. To save space, only case 3 are used to develop the
four-factor CS model.
According to Valentini and Zavanella (2003), the buyer’s ordering cost is zero under a CS policy. As the result, the
joint total expected cost of the system, JTEC, can be written as the sum of the vendor’s expected setup cost, the buyer’s
expected backorder cost, the buyer’s expected extra space cost, the extra expected lead time crushing cost, the expected
system holding cost, and the obsolescence cost. The calculations of all these costs are discussed below.
The vendor’s expected annual setup cost v
Cs can be written as
=
nq
T
T
Cv Av
s
μ
. (1)
The expected annual lead time crushing cost can be written as
− + −
=
=
Σ −
− =
q
T
T
qc L L q c b a
q
T
T
C R q L
i
il i j jl j j
L
μ μ 1
( , ) ( 1 ) 1 ( ) , (2)
where, i = 1, 2, ..., m , l = 1, 2, ..., r for q (ql 1, ql ) − ∈ and ( , −1 ) ∈ L Li Li .
Impact of Finite Life Cycle to a Consignment Stocking Supply Chain with Uncertain Demand
Figure 1. Composition of the inventory in the four-factor CS Model.
The expected annual holding cost incurred in the buyer’s inventory Hb of case 3 can be written as:
( f )
v
s
Hb = hb + h
μ
σ
μ μ μ nq k k s L q P n
nP
q P
P
nq P
P
q + / *
2
( 1)
2 2
+
− +
−
−
+
2
2 2
2
( )(2 ) ( )( 1)( )
2
1
P
n k P q n k n k P q
P
L q
P
L s L q
P
q μ μ
μ σ
μ − − + − − − −
+
+
+ + + +
P
s L q
P
n k q n k q
P
s L q
P
i q n k q n k q + +
− −
+ − −
+ +
− −
−
+ − − σ
μ
σ
μ
μ
( 1) ( ) ( 1)
2
' 1 i q i q T
−
− −
μ
τ
μ
τ
μ ' ' ' '
2
, (3)
where, n* = μT / nq , τ '= t − L − (n − k)q / P , and i'= τ 'μ q .
The expected holding cost incurred in the vendor’s warehouse Hv of case 3 is written as
( f )
v
s
Hv = hv + h
2
2
2
2 2
2
'( ' 1)( )
2
'
2
* ( )
2
( 1)
2 μ
μ
μ μ
μ μ n Pq n n P q
P
k k n nq n k q
nP
q P
P
q + −
+ +
−
+
− +
+
n q T
P
n q P t n k q
P
n q P t n k q
−
−
− +
−
−
−
−
+
2 ( ) '
2
( ) '
2
' ( )
μ μ μ
μ , (4)
, e r e h w q n T n / * μ = d n a 1 ) ( ' −
−
= −
P
n k
q
n t
μ μ .
The expected annual holding cost in transit Hd for case 3 is given by:
qL T
P
n k
q
H h h f nqLn n n k t
v
s
d d
−
= + + − + − μ
μ
( ) * min , , (5)
U
Systems and
Buyer
Vendor In Transit
μ
μ nq
nq
t T T
= −
P
q
q/P+L
q
q/P
T cycle =nq/μ
(n-k-1)q/P
q/μ
kq/μ
q/P
T down
L T up T down
P-μ
Case 1 Case 2 Case 3 Case 4
Flexible Automation and Intelligent Manufacturing, FAIM2014
The expected annual backorder cost Cb of case 3 can be written as:
Σ
− −
=
+ +
+ + − + + − +
= + − − − +
2
0
( * 1) 1 ( * 1) 2 ( 1)
n k
i
b b P
s L q
P
n E X i q iq
P
L s L q
P
C c n E X q σ
μ
μ σ
μ
+
− − + − − − +
+ − +
P
s L q
P
E X n k q n k q
q
n k σ
τ μ μ
*( 1) ' 3 ( ) ( 1)
+
+ − − + − − − +
P
s L q
P
E X n k q n k q σ
μ
4 ( ) ( 1)
T
P
s L q
P
E X n k q n k q
+ − − + − − − +
+
σ
μ
7 ( ) ( 1) , (6)
where, n* = μT / nq and τ '= t − L − (n − k)q / P .
The expected annual extra space cost Co of case 3 can be given as:
+
− + + − − − − + −
= *+ ' ( 1) ( 1) / X3
P
E n k q U n k q P s L q
q
kn
T
c
C o
o μ σ
τ μ , (7)
The expected annual obsolescence cost Cob of case 3 can be calculated as:
−
− −
−
− +
−
−
−
=
μ
μ μ μ μ
μ
μ q
P
n k
q
q t
P
P t n k
P
n k
q
C p P q t ob v
( ) ( ) ( )
+ − − − − + + − −
μ
τ μ
σ μ τ
μ q
P q
s L q
P
p n k q n k q b
( ) ( 1) ' ' [ ( ) ]
− − − − +
+
L
Max t t t t q q q L
p q
I
l
I
l
b
0, 3 3 μ / /μ /μ , (8)
Finally, the annual joint total expected cost JTEC(q,n,k, s,L) for given L∈(Li , Li−1) and q∈(ql −1, ql ) can be written as
JTEC(q, n, k, s, L) = v +
Cs CL + Hv + Hd + Hb + Cb + Co + Cob . (9)
The Four-factor CS model can, therefore, be written as:
Min JTEC(q, n, k, s, L) (10)
Subject to: (n − k)q − (n − k −1)qμ / P + sσ L + q / P ≤U , (11)
k ≤ n , (12)
(n − k)q / P + L + kq /μ ≤ nq /μ , (13)
q, n, k, L are positive and integers. (14)
The problem is to jointly decide the optimal ordering quantity q, safety factor s, number of shipments within a
production cycle n, number of delayed shipments k, and lead time L that minimize the JTEC as expressed by Equation
(10), under the constraints in (11) – (14). In which, the constraint in Equation (11) ensures that the buyer’s space
limitation is greater than the buyer’s potential maximum inventory level, and the constraint in formula (13) ensures the
arrival of all shipments to the buyer within each full production cycle.
The model developed here is a constrained nonlinear mixed-integer optimization problem. Due to its complexity,
the traditional optimization methods are cumbersome [Persona, et al. (2005), Battini, et al. (2010a, 2010b)]. A newly
developed doubly-hybrid meta-heuristic algorithm (DHMHA) (Yi, et al., 2013), which has been shown by Yi and
Sarker (2013b, and 2013c) to be a satisfactory tool to solve complicated inventory models with multiple variables, is
adopted in the next Section to locate the solutions. Also, to perform a numerical study, the normal distribution is
adopted for the demand rate so that the expected backorder cost and the expected extra space cost can be evaluated.
4. COMPUTATIONAL RESULTS
In this Section, a numerical example is used to demonstrate the solution process. The outcomes of the numerical
example, found by the two doubly-hybrid algorithms, are compared to that of an exhaustive search algorithm (ESA) to
verify that the solutions found by the doubly-hybrid are global optimum. All the algorithms were coded in Matlab and
were executed on a HP Pavilion Dv8 notebook PC with an Intel® QuadCore i7 CPU and Q 720@ 1.6 GHz processor.
For each setting of the test parameters, 10 runs were made by both doubly-hybrid algorithms (PSO+IHS+HJ and
MDE’+IHS+HJ). A run is declared as ‘successful’ when the global optimum (or the best-known) was found within 10-6
error.
A numerical example is framed here to illustrate the optimal solutions (ESA) along with PSO+IHS+HJ and
MDE’+IHS+HJ. Most of the values of the parameters used in this example are adopted from Braglia and Zavanella
Impact of Finite Life Cycle to a Consignment Stocking Supply Chain with Uncertain Demand
(2003), and Huang and Chen (2009). The lead time consists of three segment the composition of which is shown on
Table 1 [the original time data are in days].
Table 1. The composition of the lead time components.
Lead time component Normal duration Minimum duration
i bi (years) ai (years)
1 20/365=0.05479 6/365=0.01644
2 20/365=0.05479 6/365=0.01644
3 16/365=0.04384 9/365=0.02466
Similar as in Yi and Sarker (2013c), the unit lead time crushing cost in this example is a function of both the reduced
period and the shipping size. A unified quantity discount is used to represent the relationship between the lead time
crushing cost and the quantity. Table 2 illustrates the detailed composition of this cost.
Table 2. The composition of the unit lead time crushing cost cij.
q ≥ 100 100 > q ≥ 20 q < 20
Lead time component i ci1($/unit/year) ci2($/unit/year) ci3($/unit/year)
1 (0.8)(0.1)(365)=29.2 (0.9)(0.1)(365)=32.85 (1)(0.1)(365)=36.5
2 (0.8)(1.2)(365)=350.4 (0.9)(1.2)(365)=394.2 (1)(1.2)(365)=438
3 (0.8)(5.0)(365)=1,460 (0.9)(5.0)(365)=1,642.5 (1)(5.0)(365)=1,825
The values of other parameters are: Av = $400 /setup, Ab = $25 /order, μ = 1000 units/year, σ = 100 units, r = 10% ,
P = 3200 units/year, pv = $20/unit, s = $3
hv /unit/year, = $1.50 s
hb /unit/year, = $4 s
hd /unit/year, co = $10/unit, cb = $50/unit,
and U = 150 units.
In the model developed in this study, all decision variables are integers except for the safety factor s, which is
continuous. Two decimal values are allowed for s so that the ESA can be used to determine the optimal solutions and
verify whether the solutions found by the two doubly-hybrid meta-heuristic methods are global optimal. Table 3 shows
the optimal solutions of the decision variables and expected system cost found by the three algorithms.
It is observed that all the three algorithms lead to the same solutions, which is guaranteed to be the global optimum
since the ESA is used as a comparison. The ESA requires more than 30 days to find the optimal solutions
(exhaustively), whereas the two doubly-hybrid approaches take a few minutes. This result shows that the accuracy of
both the hybrid algorithms is as good as that of the ESA, but the efficiency of the doubly-hybrid algorithms are superior.
Table 3. Optimal solutions of the three algorithms.
Algorithm JTEC($/yr) n q k L
(day) s CPU/M_CPU (s) NFE/M_NFE Suc_
Rate
ESA 3,297.64 123 3 87 56 1.90 3,199,302.72 1,023,814,404 -
PSO+IHS+HJ 3,297.64 123 3 87 56 1.90 432.89 100,131 0.50
MDE’+IHS+HJ 3,297.64 123 3 87 56 1.90 258.80 100,252 0.60
5. CONCLUSIONS
This paper studies the effects of introducing the risk of obsolescence into an integrated lead time controllable
consignment stock inventory system with buyer’s space limitation and stochastic demand. A four-factor inventory
model is developed to jointly determine the optimal value of five decision variables that minimize the annual JTEC of
the system. Due to the complexity of the problem structure, analytical solutions are not presented. Instead, two novel
doubly-hybrid meta-heuristic algorithms are utilized to find the global optimum of the models. Numerical examples
Flexible Automation and Intelligent Manufacturing, FAIM2014
showed that both the doubly-hybrid algorithms perform well both in the sense of the CPU time and the success rate. The
results obtained in this study help understand the role of obsolescence, stochastic demand rate, the buyer’s space
limitation, controllable lead time, and the CS mechanism. Moreover, the successful use of doubly-hybrid
meta-heuristic algorithms to inventory problems provides a possible way of solving more difficult and complicated
models in the future.
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