International Journal of the Information Systems for Logistics and Management (IJISLM), Vol. 1, No. 1, pp. 1-7 (2005)
1
Optimal Issuing Policy for Fish-Breeding Supply Center
with Items Weibull Ameliorating
Heung-Suk Hwang1*, Ho-Gyun Kim2 and Chun-Hyun Paik2
1*Department
of Business Administration, Kainan University,
No. 1 Kainan Rd., Lu-jhu, Taoyuan, 338, Taiwan
2Department of Information and Industrial Engineering, Dongeui University
San 24, Gaya-dong, Pusanjin-gu, Pusan, 614-714, Korea
ABSTRACT
In conventional inventory models, the items are based on the assumptions that the values (or
utility) of items in inventory remain constant over time, but in the most practical cases, it is not constant
and changes increasing or decreasing over time. In this study, a special case of inventory model is
studied for the items of which utilities are increasing over time. We develop ameliorating inventory
models to find optimal issuing policy in fish-breeding supply center. The two issuing policies are
considered, FIFO (first-in-first-out) and LIFO (last-in last-out), to find optimal issuing policy. We
derived the equations of inventory levels for both issuing policies, FIFO and LIFO, and we developed
two ameliorating inventory models to find the economic order quantity (EOQ) and the economic selling
quantity (ESQ). We developed a computer program for this research and applied it in a fish-breeding
supply center problem to show the effectiveness of the proposed models. From the sensitivity analysis,
we can see that the ameliorating inventory effect affects the issuing policy, EOQ and ESQ of the fishbreeding supply center significantly.
Keywords: ameliorating inventory, issuing polilcy, fish-breeding supply center management.
1. INTRODUCTION
Most of the previous works on the supply chain inventory models have been based on the ideal assumption
that the value of inventory items remains constant over
time (Gupta, 1982; Lars and Tayfur, 1998; Nakamura et
al., 1998). Only few studies dealt with deteriorating or
perishable inventory models (Hwang and Hwang, 1982;
Lin et al., 2000; Raafat and Eldin, 1991). In this paper,
the term ameliorating inventory (or item) is introduced
to represent the one whose utility (value or quantity) is
increased over time. This amelioration phenomenon can
often be observed in the areas such as cattle breeding, fish
culture, and fund managing under the inflation, etc. The
supply centers of these areas usually have a combined
hierarchy structure consisting of a supply center and several retailers with ameliorating items. Our study is concerned on an ameliorating inventory model and optimal
issuing policy for the fish-breeding supply center. There
are several researches on supply chain inventory models
(Garg, 1996; He, 1996; Leung et al., 2000; Ross, 1996),
but most of these studies assumed unchanged utility during the storage period. However, due to the breeding ac-
tivities the items in the fish-breeding supply center will
ameliorate. Recently, these special inventory problems
with ameliorating items are considered as an important
area for inventory management in supply centers.
There are several studies on ameliorating inventory
models for supply centers in the literature. Hwang (1982,
1997, 1999) considered an inventory model for items with
Weibul ameliorating and deteriorating. Sparker et al.
(2000) developed a supply chain model to determine an
optimal ordering policy for deteriorating items under
inflation. Chung and Lin (2001) used a discounted cash
flow approach to investigate deteriorating inventory
problems. For the application of ameliorating inventory,
Hwang (2004) studied a stochastic set-covering location
problem for both ameliorating and deteriorating items. He
determined the minimum number of supply center facilities via 0-1 programming.
In ameliorating inventory system, the amount of
items increase by ameliorating activities within a time
period depends on the ameliorating rate of each items
which are varied by the ages of items in storage. Thus,
the inventory issuing policy is an important factor in ameliorating inventory modeling, but most of previous works
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International Journal of the Information Systems for Logistics and Management (IJISLM), Vol. 1, No. 1 (2005)
did not take into account of issuing polishes to consider
the varying rate of ameliorating effect. In this study, we
considered two issuing policies, LIFO and FIFO, to consider the varying rate of ameliorating items in storage of
fish-breeding supply center. Fig. 1 shows an explanatory
example for two-echelon hierarchy structure in which the
items are in ameliorating state at the upper echelon, while
the items are in deteriorating state at the lower echelon.
This study is focused on the issuing policies in the upperechelon, supply center.
Under the assumption that the time to amelioration
of an item is distributed according to the Weibull
distribution, we consider two different inventory models:
the well-known economic order quantity-type (EOQ-type)
model and the economic selling quantity (ESQ) model.
While the former case is developed for the case of the
inventory increment ratio (ameliorating rate) is less than
the demand rate, while the latter is for the case that the
ameliorating rate is greater than the demand rate. With
the time varying rate of ameliorating, the amount of ameliorated items during a given time interval depends on
the amount of on-hand inventory of which the ages of
items in inventory are not same. Thus, it is reasonable to
consider an appropriate issuing policy minimizing the
inventory-related costs. In this study we consider the two
issuing policies: FIFO and LIFO. We derive the equations of inventory levels for ameliorating items to find the
minimizing inventory-related cost under each issuing
policy, and also provide an efficient numerical method to
solve the equations.
The extensive computational experiments are performed to show the effectiveness of the developed models and the two issuing policies are compared in terms of
inventory level.
Notations:
R: demand rate given in number of units/time,
α, β: two parameters of Weibull amelioration,
Co: ordering cost ($/unit),
Ca: ameliorating cost ($/unit),
Cp: purchasing cost ($/unit),
Ps: selling price ($/unit),
Ch: carrying cost ($/unit/time),
Q: partial ordering size,
S: partial selling amount,
A(t): instantaneous ameliorating rate,
TC: total cost/unit time.
Assumptions:
(1) Demand rate R is known and constant,
(2) Units are available for satisfying demand
after amelioration,
(3) The ameliorated units are immediately added
to inventory level,
(4) Shortages are not allowed,
(5) The number of units is treated as a continuous variable,
(6) The time for an item to ameliorate follows
a Weibull distribution of which probability
density function, f(t), is given by
f(t) = αβ t β - 1 e - α t
β
and the instantaneous amelioration rate is
given by
f (t)
A(t) =
∞
= αβt β – 1, where α, β > 0.
f (x)dx
t
2.1 EOQ-type Model
2. MODEL DEVELOPMENT
Throughout this paper, the time to amelioration of
an item is assumed to be distributed according to the
Weibull distribution which is one of the most commonly
used distributions in the literature (Hwang and Hwang,
1982; Raafat and Eldin, 1991) because it can represent a
variety of real-life situations simply by varying the values
of two parameters of the distribution.
The following notations and assumptions are used
for model development:
When the inventory increment rate by ameliorating
activity is less than the demand rate, the inventory level
decreases monotonically until the inventory level decreases
to 0. Hence, the inventory behavior of this case is similar
to that of the classical EOQ model except the amelioration phenomenon. This is the reason why we call our
model EOQ-type one. We propose two EOQ-type models under different issuing policies LIFO and FIFO as
followings.
2.1.1 Inventory level under LIFO issuing policy
Fish-breeding
Supply Center
Retailer
Retailer
Upper-echelon
Retailer
Lower-echelon
Fig. 1. Example of two-echelon hierarchy structure of supply chain
We define the cycle time T by the duration from the
replenishment epoch of items to the depletion epoch of
the inventory consisting of replenished items, Io and the
ameliorated items.
We consider that during the time interval (t, t + ∆t)
t < T, the instantaneous amelioration occurs according to
the rate of A(t) and the demand occurs at a constant rate
of R units.
3
H. S. Hwang et al.: Optimal Issuing Policy for Fish-Breeding Supply Center with Items Weibull Ameliorating
Invt,
Level
Inv.
Level
R•T
R•T
I0
I0
Time
0
t
T
t+∆t
Fig. 2. Inventory level of EOQ-type model under LIFO
where 0 ≤ t ≤ T,
and after some algebra,
β
I t = e αt – R
t
β
e – α x dx + k ,
(1)
From the boundary conditions at 0 and T, the value of
constant k can be given by
T
k = I0 = R
T
of demand, R . ∆t1, during the time interval (t1, t1 + ∆t1)
where t1 ≤ T, is satisfied from the amount ameliorated
during (t(t1), t(t1) + ∆t1), as shown in Fig. 3. Note that
the time t(t 1) indicates the epoch at which the items
ameliorated up to t 1 are exhausted by the demands.
Under FIFO Issuing policy, the amount of inventory
ameliorated during (0, t(t1)) is exhausted by the demands
occurred during (0, t1).
This gives, R . ∆t1 = I0 . exp(−α (t1 − t) β) . ∆t
and ∆t = R ⋅ exp ( – α(t 1 – t )β )
∆t 1 I 0
(4)
Let α (t1 − t) β = x
(5)
β
e – α x dx .
dt = h(x) . dx
(6)
where h (x ) = 1 / (α 1 / β ⋅ βx β – 1(
I0
⋅ exp ( – x ) – 1))
R
Then the solution of equation (6) can be given by
0
x
Thus, the inventory level at time t is given by
β
t1
then, differentiation of equation (5) and substitution of
equation (4) into the result yields
0
where k is a constant value
Time
t(t1) t(t1) + = t1
Fig. 3. Inventory level of EOQ model under FIFO policy
During this time interval the entire amount of ameliorated items is supplied for demands because the amount
of ameliorated per unit time is less than the demand rate
R. LIFO policy implies the ameliorated items do not enter the inventory as shown in Fig. 2.
The inventory level at time t It can be derived as:
dI = I t ⋅ A (t ) dt – Edt
dI – I (αβ t β – 1) = – R ,
t
dt
0
I t = e αt – R
t
0
β
e – α x dx + I 0
t (t 1) =
(2)
As a special case, if β = 1then the ameliorating rate becomes constant and the following is satisfied.
R (1 – e α T ) and I = R (1 – e α (t – T ) )
I0 = α
t
α
(3)
2.1.2 Inventory level under FIFO issuing policy
The similar manner of the LIFO case can be applied
to the FIFO one besides the fact that in this case, the amount
h (y ) dy + K ,
(7)
0
where x = α(t1 − t) β , and
h (x ) = 1 / (α 1 / β ⋅ βx β – 1(
I0
⋅ exp ( – x ) – 1)).
R
It is not easy to derive as closed-form solution of equation
(7). An approximation method based on perturbation technique (Kevorkian and Cole, 1981) is used as follows:
t(t1) ≅ f0 +αf1(t1) + α 2f2(t1)
where, f = R t 1
0
I0
(8)
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International Journal of the Information Systems for Logistics and Management (IJISLM), Vol. 1, No. 1 (2005)
f 1(t 1) =
R (t 1 – f 0)β + 1
(I 0 – R )(β + 1)
f 2(t 1) =
(t 1 – f 0)β R (I 0 + R )(t 1 – f 0)β + 1 R f 1(t 1) R βf 1(t 1)
–
–
(I 0 – R )
2(I 0 – R )(2β + 1)
β +1
β +1
The boundary condition, at time t1 = 0, t = 0, and x = 0
gives k = 0. Since the inventory level at time t1 is the
amount ameliorated during (t(t1), t1), the inventory level
at time t1 is given by:
t1
It1 =
t (t 1)
Z(T)
T = Z(T)
(0, β)
I 0 exp ( – α(t 1 – y )β ) dy ,
where, t1 ≤ T
(9)
T+
T
2.1.3 Optimal cycle time
Fig. 4. Graphical representation of T and Z(T)
To illustrate the effect of amelioration on the inventory level, we have to consider the total cost function. To
obtain an optimal cycle time T* minimizing the total cost
per cycle time, we first have to find the total cost (TC)
function which consists of ameliorating cost, holding cost,
and ordering cost. With the inventory equations provided
above, the total cost function can be obtained by
TC = I 0(
Letting T = Z(T), where,
∞
Z (T ) = 1 R Σ
C0 k =0
Cp –Ca Ch
C
+
) + RC a + 0
T
2
T
( – α)k T k β + 3
k βT – 1 C h
+
⋅ (C p – C a )
K!
kβ + 1
2
The solution search procedure is then summarized
as:
( – α)k T k β
k = 0 (k β + 1)k !
∞
= R (C p – C a ) Σ
+
C h ∞ ( – α)k T k β + 1
C
R Σ
+ R C a + 0 (10)
2 k = 0 (k β + 1)k !
T
An optimal cycle time T* can be obtained by letting
dTC = 0
dT
Fig. 4 shows a schematic diagram of the computer
search method. Once T* is given, the optimal inventory
level and total cost can be obtained by the equations (2),
(9) and (10) respectively by this computer search method.
k
kβ –1
∞
dTC = (C – C )R Σ ( – α) ⋅ k β ⋅ T
p
a
dT
(k β + 1)k !
k =0
+
C h ∞ ( – α)k T k β C 0
R Σ
– 2 =0
2 k =0
k!
T
(11)
To solve the above equation, we adopt a numerical
solution search method based on a graphical procedure
after transforming the equation (11) into
∞
T = 1 R Σ
C0 k =0
Step 1. Give a few values to T and find corresponding values of Z(T) from equation (10) for a
particular value of α and β.
Step 2. Plot these points for different values of α
and β.
Step 3. The points, where the line T = Z(T) cuts these
curves, will give optimal values of T*.
( – α)k T k β + 3
k βT – 1 C h
+
⋅ (C p – C a )
k!
kβ + 1
2
(12)
2.2 ESQ Model
When the inventory increment by amelioration is
greater than the demand rate, the ameliorated items will
be accumulated in the inventory system as time elapses.
In this case the carrying cost per unit time would be increased as the surplus amount of inventory increases.
Hence, at a proper point of time, it will be better to sell
out this surplus amount of inventory So. Immediately after selling out this surplus, the inventory level will be
dropped to I0 (base line of inventory level). An example
5
H. S. Hwang et al.: Optimal Issuing Policy for Fish-Breeding Supply Center with Items Weibull Ameliorating
The amount of economic selling
Inventory Level
Z (T ) =
It + S0
S0
It
β
(P s – C a )
I 0(e α T – 1)
C0
β
C
– 1 (P s – C a ) – k T I 0αβT β + 3e α T
2
C0
I0
I0
+ αβe α T
t dt
∞ ( – α)k T (k + 1)β – 1
– C a)
⋅R ⋅T3 Σ
C0
(k β + 1)k !
k =0
Time
T
k (k + 1)β – 1
β ∞ ( – α) T
C
– 1 α ⋅ β h ⋅ R ⋅ T 3e α T Σ
2
C0
(k β + 1)k !
k =0
T
Fig. 5. Inventory cycle of ESQ model
for the inventory cycle of ESQ model when the inventory
increment by amelioration is decreasing is shown in
Fig. 5.
The initial inventory level (base-line inventory) I0
and the economic selling quantity So can be found by the
equation (1). From the boundary condition IT = I0 + So,
equation (1) yields
T
β
I 0 + S 0 = e αT – R
0
1
β
e αT – 1
e
αx β
t
R
e
– αx β
0
dx + S 0
If the value of I0 is given, So is represented a function of cycle time T as follows:
T
β
S 0 = e αT – R
0
β
β
e α x dx + I 0(e α T – 1)
Let us now find the total cost per cycle time. Using
ESQ model in which the selling cost is additionally included to the total cost, we can obtain the total cost TC
by equation (13).
TC = (P s – C a )R +
I 0(e
αT β
Ps – C a C h
–
T
2
– 1) – Re
C
– C hI0 – 0
T
αT β
+
k (k + 1)β – 1
β ∞ ( – α) T
Ps – C a
⋅ R ⋅ e αT Σ
C0
(k β + 1)k !
k =0
∞ ( – α)k T k β
C
–1 h R Σ
2 C 0 k = 0 (k β + 1)k !
(14)
3. NUMERICAL EXAMPLES
To illustrate the developed EOQ-type and ESQ
models for ameliorating inventory, three examples are introduced for a fish-breeding supply center inventory
problems.
β
e – α x dx + I 0 ,
and alternatively
I0 =
β (P s
Example 1: Inventory level
The inventory is depleted according to an issuing
policy. Inventory levels at given times according to issuing policies are summarized in Table 1 under the identical
cycle times, T = 4, and considered the Weibull ameliorating inventory with parameters, α = 0.3 and β = 1.4, and
the results are depicted in Fig. 6. Note that the inventory
level in case 1 decreases with a constant demand rate, R =
1000 units/unit time.
In Table 1, we can see that the inventory levels
needed for a cycle time, T are different by issuing policies.
In case 1 (non-ameliorating) required amount of inventory level in fish-breeding supply centre is 4,000, while,
in case 2 (FIFO issuing policy) it is 2,773 and in case
Table 1. Comparison of inventory levels under LIFO
and FIFO:
( – α)k T k β + 1
Σ
k = 0 (k β + 1)k !
α = 0.3, β = 1.4, R = 1000
∞
(13)
Using dTC and the equation (14) given as following,
dT
we can find the optimal cycle time T* by the same method
used for the case of EOQ-type model,
Inventory level at time
Issuing policy
Case 1:
Non-ameliorating
Case 2:
FIFO
Case 3:
LIFO
0
1
2
3
4
4,000
3,000
2,000
1,000
0
2,773
2,199
1,438
737
0
1,999
1,502
1,158
738
0
6
International Journal of the Information Systems for Logistics and Management (IJISLM), Vol. 1, No. 1 (2005)
Inv. Level
In Supply
Center
Z(T)
α = 0.30
Z(T) = T
α = 0.20
α = 0.15
Case 1. Non-Ameliorating
Case 2. Amelisorating (FIFO)
4,000
2,773
α = 0.10
α = 0.05
Case 3. Ameliorating (LIFO)
1,999
4
9
14
20
27 Cycle Time, T*
Fig. 7. Graphical representation for the EOQ-type model: β = 0.30
0
1
2
3
4
Time
Fig. 6. Comparison of issuing policies (LIFO and FIFO)
3 (LIFO policy) it is only 1,999. In this case we can conclude that LIFO issuing policy better than FIFO policy.
Example 2: EOQ-type model (T *, I *0, TC *)
We consider the same fish culture company to sell
raw fishes in a small sea village. The company periodically orders fishes when the tank is almost empty. The
optimal values of cycle time, order quantity and corresponding total cost are obtained under various parameters
of Weibull amelioration. The data needed are given as:
R: 1,000 kg/day,
Co: 300,000 /unit,
Ca: 4,000 /unit,
Ch: 400 /kg/day,
Cp: 10,000 /kg
Numerical results for the EOQ-type model under
LIFO are summarized in Table 2 and the graphical representation of T and Z(T) with β = 0.30 is given as Fig. 7.
Example 3: ESQ model ((T*, S*0, TC*)
Table 3 and Fig. 8 show the results for the ESQ
model under LIFO. Given β = 0.6 and other parameter
Table 2. Sample results of EOQ-type model
- Optimal Cycle Time, T *
- Optimal Order Level, I *0
- Minimum Cost, TC *
Description
β = 0.10
β = 0.15
β = 0.20
β = 0.30
α = 0.05
α = 0.1
α = 0.2
α = 0.3
α = 0.4
3
2579.84
9661438
3
2579.84
9388672
3
2453.32
9129316
4
2945.28
8640190
4
3420.48
964625
4
3420.48
9376753
4
3246.75
9107472
9
4331.91
8598514
4
3584.23
9630564
6
5019.88
9320881
7
5478.88
9012989
20
7363.61
8411970
6
5262.66
9575801
9
7211.60
9201653.
14
10003.92
8809016
27
14689.99
8010052
8
6868.91
9532634
13
9926.00
9100634
21
13758.32
8633150
30
17655.00
7243463
values as following. The optimal values of cycle time,
selling quantity, and total cost per cycle time are summarized in Table 3. The input data for ESQ model
(example 3) are given as follows.
R: 1,000 unit/day
Co: 8,000 /unit
Ca: 1,000 /kg,
Ch: 100 /unit/time
Cp: 8,000 /unit
Ps: 10,000 /unit,
In case of α = 0.6 and β = 0.6, the optimal economic
selling quantity and the total cost per cycle time are given
by; S*0 = 35,944, TC* = 7.61 × 106, T * = 12, I *0 = 30,000.
From Fig. 8, we can find that the optimal cycle time
in the ESQ model under LIFO does not become sensitive
to the value of α as it increases. This example is the case
that the amount of unit ameliorated is greater than the
Table 3. Results for the ESQ model: β =0.6
T*
26
12
4
2
2
1
1
Description
α = 0.5
α = 0.6
α = 0.7
α = 0.8
α = 0.9
α = 1.0
α = 1.2
S *0
845832
359444
80794
67551
83818
50043
67956
TC *
6.40 × 106
7.61 × 106
2.01 × 106
5.65 × 106
1.28 × 106
2.03 × 106
3.64 × 106
Z(T)
α = 0.5
α = 0.6
α = 0.7
α = 0.8, 0.9
α = 1.0, 1.2
1 2
4
12
26
T, Cycle Time
Fig. 8. Graphical representation of the ESQ model, β = 0.6
H. S. Hwang et al.: Optimal Issuing Policy for Fish-Breeding Supply Center with Items Weibull Ameliorating
demand rate and the surplus amount of inventory is accumulated in fish-breeding supply center.
4. CONCLUSIONS
In this paper, we have considered a special case of
problem, the inventory issuing problem of a fish-breeding supply center for items with Weibull ameliorating that
the utility in inventory items increase over time. We developed two supply chain inventory models, EOQ-type
and ESQ model for items with Weibull amelioration. Since
the inventory depletion policies of fish-breeding supply
center are closely related to the ameliorating inventory,
we have considered two issuing policies, LIFO and FIFO.
The formulae for the inventory level and the total cost per
cycle time have been obtained. Due to the non-closed
forms of the solutions of the proposed models, we have
solved by a computerized search technique and graphical
method. Three numerical examples have been performed
to illustrate and demonstrate the proposed models as: (1)
computing the inventory levels by issuing policies and
compared, (2) using the EOQ-type mode to find the optimal cycle time, T*, order point, I *0, and optimal cost per
cycle, TC*), and (3) using the ESQ model to find optimal
cycle time, T*, optimal selling quantity S*0, and related
cycle time cost, TC*. For the further research we are
to develop a more practical method to solve the optimal
values for, T*, I *0, S*0, and TC*, related with ameliorating
inventory management and develop GUI-type computer
program for users.
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