# An Inventory Model under Trapezoidal Type Demand, Weibull-Distributed Deterioration, and Partial Backlogging

Authors: Lianxia Zhao
Journal: Journal of Applied Mathematics
 (2014)

---

## Abstract

This paper studies an inventory model for Weibull-distributed deterioration items with trapezoidal type demand rate, in which shortages are allowed and partially backlogging depends on the waiting time for the next replenishment. The inventory models starting with no shortage is are to be discussed, and an optimal inventory replenishment policy of the model is proposed. Finally, numerical examples are provided to illustrate the theoretical results, and a sensitivity analysis of the major parameters with respect to the optimal solution is also carried out.


---

## Body

## 1. Introduction

The effect of deteriorating for items cannot be disregarded in many inventory systems and it is a general phenomenon in real life. Deterioration is defined as any process that decreases the usefulness or the value of the original item, such as decay or physical depletion. For example, fruits, vegetables, or foodstuffs are subject to spoilage directly while being kept in store, and electronic products, radioactive substances, and photographic film deteriorate through a gradual loss of potential or utility with the passage of time.Due to the variability in economic circumstances, the basic assumptions of the EOQ model should be constantly modified according to the studied inventory model. In recent years, many researchers have studied kinds of EOQ models for deteriorating items. Ghare and Schrader [1] established the classical no-shortage inventory model with a constant rate of decay. Wu et al. [2] studied an inventory model with a Weibull-distributed deteriorating rate for items and assumed the demand rate with a continuous function of time. Wee [3] developed an inventory model with quantity discount, pricing, and partial backordering when the product in stock deteriorates with time. Related literature also includes Skouri and Papachristos [4], Wee [5], and Dye et al. [6].practically, the demand rate of deterioration items is impossible to increase continuously all the time. Hill [7] proposed an inventory model with ramp type demand rate. Mandal and Pal [8] extended the inventory model with ramp type demand for deterioration items and allowed shortage. Wu [9] considered an inventory model with Weibull distribution deterioration and ramp type demand rate in which shortages are allowed and the backlogging rate is dependent on waiting time. Giri et al. [10] extended the ramp type demand inventory model with a more generalized Weibull deterioration distribution. Manna and Chaudhuri [11] developed an inventory model for time-dependent deteriorating items with ramp type demand rate. Skouri et al. [12] considered an inventory model with general ramp type demand rate, partial backlogging, and Weibull deterioration rate. Hung [13] extended their inventory model from ramp type demand rate and Weibull deterioration rate to arbitrary demand rate and arbitrary deterioration rate. Kumar et al. [14] studied fuzzy EOQ models with ramp type demand rate, partial backlogging, and time-dependent deterioration rat. Cheng et al. [15] considered an inventory model for time-dependent deteriorating items with trapezoidal type demand rate and partial backlogging. Uthayakumar and Rameswari [16] studied an inventory model for defective items with trapezoidal type demand rate to determine the optimal product reliability. Tan and Weng [17] considered a discrete-in-time inventory model for deteriorating items with partially backlogged. Ahmed et al. [18] proposed a method for finding the economic order quantity for an inventory model with ramp type demand rate, partial backlogging, and general deterioration rate. Lin [19] explored the inventory model with a general demand rate in which both the Weibull-distributed deterioration and partial backlogging are considered.In the above mentioned research, one of assumptions was considered: the ramp type demand rate, partial backlogging, and Weibull-distributed deterioration rate. However, for fashionable commodities, high-tech products, and other short life cycle products, the demand rate should increase with the time up to certain point at first stage then reach a stabilized period and finally the demand rate decrease to zero and the products retreat from market in their product life cycle, that is, the demand rate with continuous trapezoidal function of time. On the other hand, in many real situations, customers encountering shortages will respond differently. Some customers are willing to wait until the next replenishment, while others may be impatient and go elsewhere as waiting time increases; that is, the willingness for a customer to wait for backlogging is diminishing with the length of the waiting time. In this paper, we consider an inventory model with Weibull-distributed deterioration items, trapezoidal type demand rate, and time-dependent partial backlogging. By analyzing the inventory model, a useful inventory replenishment policy is proposed. Finally, numerical examples are provided to illustrate the theoretical results, and a sensitivity analysis of the optimal solution with respect to major parameters is also carried out.The rest of the paper is organized as follows. Section2 describes the notation and assumptions used throughout this paper. Section 3 analyzes the inventory model, and some numerical examples to illustrate the solution procedure are provided. Sensitivity analysis of the major parameters is also carried out in Section 4, and the final Section concludes this paper.


## 2. Notations and Assumptions

The fundamental notations and assumptions used in inventory model and considered in this paper are given as below.(i)
I
(
t
) the level of inventory at time t, 0
≤
t
≤
T.
(ii)
T the fixed length of each ordering cycle.
(iii)
t
1 the time when the inventory level reaches zero for the inventory model.
(iv)
t
1
* the optimal point.
(v)
S the maximum inventory level for each ordering cycle.
(vi)
Q
* the optimal ordering quantity.
(vii)
A
0 the fixed cost per order.
(viii)
c
1 the cost of each deteriorated item.
(ix)
c
2 the inventory holding cost per unit per unit of time.
(x)
c
3 the shortage cost per unit per unit of time.
(xi)
c
4 the lost sales cost per unit.
(xii)
C
i
(
t
1
)
i
=
1,2
,
3, the average total cost per unit time under different conditions, respectively.
(xiii)
T
C
(
t
1
) the average total cost per unit time.
(xiv)
The demand rate,D
(
t
), which is positive and consecutive, is assumed to be a trapezoidal type function of time; that is,
(1)
D
(
t
)
=
{
f
(
t
)
,
t
≤
μ
1
;
D
0
,
μ
1
<
t
<
μ
2
;
g
(
t
)
,
μ
2
≤
t
<
T
,
where μ
1 is time point changing from the increasing demand function f
(
t
) to constant demand D
0, and μ
2 is time point changing from the constant demand D
0 to the decreasing demand function g
(
t
).
(xv)
The replenishment rate is infinite; that is, replenishment is instantaneous.
(xvi)
The deterioration rate of the item is defined as Weibull(
α
,
β
); that is the deterioration rate is θ
(
t
)
=
α
β
t
β
-
1
(
α
>
0
,
β
>
0
,
t
>
0
).
(xvii)
Shortages are allowed and they adopt the notation used in Abad [20], where the unsatisfied demand is backlogged and the fraction of shortages backordered is e
-
δ
t, where t is the waiting time up to the next replenishment. We also assume that t
e
-
δ
t is an increasing function, which had appeared in Skouri et al. [12].
(xviii)
The time horizon of the inventory model is finite.


## 3. Model Formulation

In this section, we consider an inventory model starting with no shortage. The behavior of the model during a given cycle is depicted in Figure1. Replenishment occurs at time t
=
0 and the inventory level attains its maximum. From t
=
0 to t
1, the inventory level reduces due to demand and deterioration. At t
1, the inventory level achieves zero, then shortage is allowed to occur during the time interval (
t
1
,
T
), and all of the demand during the shortage period (
t
1
,
T
) is partially backlogged. According to the notations and assumptions mentioned above, the behavior of the model at any time can be described by the following differential equations:
(2)
d
I
(
t
)
d
t
=
{
-
θ
(
t
)
I
(
t
)
-
D
(
t
)
,
0
<
t
<
t
1
;
-
e
-
δ
(
T
-
t
)
D
(
t
)
,
t
1
<
t
<
T
,
with boundary conditions I
(
0
)
=
S, I
(
t
1
)
=
0.Figure 1
Graphical representation of inventory level over the cycle.In the following, we consider three possible cases based on the values oft
1, μ
1, and μ
2. These three cases are shown.Case 1 (0
<
t
1
≤
μ
1).
Due to the deteriorating and trapezoidal type demand rate, the inventory level gradually diminishes during the time interval[
0
,
t
1
] and ultimately falls to zero at time t
1. Thus, from (2), we have
(3)
d
I
(
t
)
d
t
=
{
-
α
β
t
β
-
1
I
(
t
)
-
f
(
t
)
,
0
<
t
<
t
1
;
-
e
-
δ
(
T
-
t
)
f
(
t
)
,
t
1
<
t
<
μ
1
;
-
e
-
δ
(
T
-
t
)
D
0
,
μ
1
<
t
<
μ
2
;
-
e
-
δ
(
T
-
t
)
g
(
t
)
,
μ
2
<
t
<
T
.
By using the boundary condition I
(
t
1
)
=
0, the solutions of (3) are given by
(4)
I
(
t
)
=
{
∫
t
t
1
‍
f
(
x
)
e
α
(
x
β
-
t
β
)
d
x
,
0
<
t
<
t
1
;
-
∫
t
1
t
‍
e
δ
(
x
-
T
)
f
(
x
)
d
x
,
t
1
<
t
<
μ
1
;
D
0
δ
(
e
δ
(
μ
1
-
T
)
-
e
δ
(
t
-
T
)
)
-
∫
t
1
μ
1
‍
e
δ
(
x
-
T
)
f
(
x
)
d
x
,
μ
1
<
t
<
μ
2
;
-
∫
μ
2
t
‍
g
(
x
)
e
δ
(
x
-
T
)
d
x
+
D
0
δ
(
e
δ
(
μ
1
-
T
)
-
e
δ
(
μ
2
-
T
)
)
-
∫
t
1
μ
1
‍
e
δ
(
x
-
T
)
f
(
x
)
d
x
,
μ
2
<
t
<
T
.
The maximum inventory level per cycle is
(5)
S
=
I
(
0
)
=
∫
0
t
1
‍
f
(
x
)
e
α
x
β
d
x
.
Then, the total number of deteriorated items D
T in the interval [
0
,
t
1
] is
(6)
D
T
=
S
-
∫
0
t
1
‍
D
(
t
)
d
t
=
∫
0
t
1
‍
f
(
x
)
(
e
α
x
β
-
1
)
d
x
.
The total number of inventory H
T carried during the interval [
0
,
t
1
] is
(7)
H
T
=
∫
0
t
1
‍
I
(
t
)
d
t
=
∫
0
t
1
‍
∫
t
t
1
‍
f
(
x
)
e
α
(
x
β
-
t
β
)
d
x
d
t
.
The total shortage quantity B
T during the interval [
t
1
,
T
] is
(8)
B
T
=
-
∫
t
1
T
‍
I
(
t
)
d
t
=
∫
t
1
μ
1
‍
[
∫
t
1
t
‍
e
δ
(
x
-
T
)
f
(
x
)
d
x
]
d
t
-
∫
μ
1
μ
2
‍
[
∫
t
1
μ
1
‍
D
0
δ
(
e
δ
(
μ
1
-
T
)
-
e
δ
(
t
-
T
)
)
-
∫
t
1
μ
1
‍
e
δ
(
x
-
T
)
f
(
x
)
d
x
]
d
t
+
∫
μ
2
T
‍
[
∫
μ
2
t
‍
e
δ
(
x
-
T
)
g
(
x
)
d
x
-
D
0
δ
(
e
δ
(
μ
1
-
T
)
-
e
δ
(
μ
2
-
T
)
)
+
∫
t
1
μ
1
‍
e
δ
(
x
-
T
)
f
(
x
)
d
x
∫
μ
2
t
‍
]
d
t
=
∫
t
1
μ
1
‍
(
T
-
t
)
e
δ
(
t
-
T
)
f
(
t
)
d
t
+
∫
μ
2
T
‍
(
T
-
t
)
e
δ
(
t
-
T
)
g
(
t
)
d
t
+
D
0
δ
2
(
e
δ
(
μ
2
-
T
)
-
e
δ
(
μ
1
-
T
)
)
+
D
0
δ
[
(
T
-
μ
2
)
e
δ
(
μ
2
-
T
)
-
(
T
-
μ
1
)
e
δ
(
μ
1
-
T
)
]
.
The total of lost sales L
T during the interval [
t
1
,
T
] is
(9)
L
T
=
∫
t
1
μ
1
‍
(
1
-
e
δ
(
t
-
T
)
)
f
(
t
)
d
t
+
∫
μ
1
μ
2
‍
(
1
-
e
δ
(
t
-
T
)
)
D
0
d
t
+
∫
μ
2
T
‍
(
1
-
e
δ
(
t
-
T
)
)
g
(
t
)
d
t
.
Therefore, the average total cost per unit time under the condition t
1
≤
μ
1 can be given by
(10)
C
1
(
t
1
)
=
1
T
[
A
0
+
c
1
D
T
+
c
2
H
T
+
c
3
B
T
+
c
4
L
T
]
=
1
T
{
∫
μ
2
T
‍
A
0
+
c
1
∫
0
t
1
‍
f
(
x
)
(
e
α
x
β
-
1
)
d
x
+
c
2
∫
0
t
1
‍
∫
t
t
1
‍
f
(
x
)
e
α
(
x
β
-
t
β
)
d
x
d
t
+
c
4
[
∫
μ
2
T
‍
∫
t
1
μ
1
‍
(
1
-
e
δ
(
t
-
T
)
)
f
(
t
)
d
t
+
∫
μ
1
μ
2
‍
(
1
-
e
δ
(
t
-
T
)
)
D
0
d
t
+
∫
μ
2
T
‍
(
1
-
e
δ
(
t
-
T
)
)
g
(
t
)
d
t
]
+
c
3
[
∫
t
1
μ
1
‍
e
δ
(
t
-
T
)
(
T
-
t
)
f
(
t
)
d
t
+
∫
μ
2
T
‍
e
δ
(
t
-
T
)
(
T
-
t
)
g
(
t
)
d
t
+
D
0
δ
2
(
e
δ
(
μ
2
-
T
)
-
e
δ
(
μ
1
-
T
)
)
+
D
0
δ
(
(
T
-
μ
2
)
e
δ
(
μ
2
-
T
)
-
(
T
-
μ
1
)
e
δ
(
μ
1
-
T
)
)
∫
t
1
μ
1
‍
]
∫
μ
2
T
‍
}
.Case 2 (μ
1
≤
t
1
≤
μ
2).
The differential equations governing the inventory model can be expressed as follows:(11)
d
I
(
t
)
d
t
=
{
-
α
β
t
β
-
1
I
(
t
)
-
f
(
t
)
,
0
<
t
<
μ
1
;
-
α
β
t
β
-
1
I
(
t
)
-
D
0
,
μ
1
<
t
<
t
1
;
-
e
-
δ
(
T
-
t
)
D
0
,
t
1
<
t
<
μ
2
;
-
e
-
δ
(
T
-
t
)
g
(
t
)
,
μ
2
<
t
<
T
.
Solving the differential equation (11) with I
(
t
1
)
=
0, we have
(12)
I
(
t
)
=
{
∫
t
μ
1
‍
f
(
x
)
e
α
(
x
β
-
t
β
)
d
x
+
D
0
∫
μ
1
t
1
‍
e
α
(
x
β
-
t
β
)
d
x
,
0
<
t
<
μ
1
;
D
0
∫
t
t
1
‍
e
α
(
x
β
-
t
β
)
d
x
,
μ
1
<
t
<
t
1
;
D
0
δ
(
e
δ
(
t
1
-
T
)
-
e
δ
(
t
-
T
)
)
,
t
1
<
t
<
μ
2
;
-
∫
μ
2
t
‍
e
δ
(
x
-
T
)
g
(
x
)
d
x
+
D
0
δ
(
e
δ
(
t
1
-
T
)
-
e
δ
(
μ
2
-
T
)
)
,
μ
2
<
t
<
T
.
The beginning inventory level can be computed as
(13)
S
=
I
(
0
)
=
∫
0
μ
1
‍
f
(
x
)
e
α
x
β
d
x
+
D
0
∫
μ
1
t
1
‍
e
α
x
β
d
x
.
The total number of items which perish in the interval [
0
,
t
1
] is
(14)
D
T
=
∫
0
μ
1
‍
f
(
x
)
(
e
α
x
β
-
1
)
d
x
+
D
0
∫
μ
1
t
1
‍
(
e
α
x
β
-
1
)
d
x
.
The total number of inventory carried during the interval [
0
,
t
1
] is
(15)
H
T
=
∫
0
μ
1
‍
∫
t
μ
1
‍
e
α
(
x
β
-
t
β
)
f
(
x
)
d
x
d
t
+
D
0
∫
μ
1
t
1
‍
∫
t
t
1
‍
e
α
(
x
β
-
t
β
)
d
x
d
t
.
The total shortage quantity during the interval [
t
1
,
T
] is
(16)
B
T
=
∫
μ
2
T
‍
(
T
-
t
)
e
δ
(
t
-
T
)
g
(
t
)
d
t
+
D
0
δ
[
e
δ
(
t
1
-
T
)
(
T
-
t
1
+
1
δ
)
-
e
-
δ
(
μ
2
-
T
)
(
T
-
μ
2
+
1
δ
)
]
.
The total of lost sales during the interval [
t
1
,
T
] is
(17)
L
T
=
D
0
∫
t
1
μ
2
‍
(
1
-
e
δ
(
t
-
T
)
)
d
t
+
∫
μ
2
T
‍
(
1
-
e
δ
(
t
-
T
)
)
g
(
t
)
d
t
.
Therefore, the average total cost per unit time under the condition μ
1
≤
t
1
≤
μ
2 can be given by
(18)
C
2
(
t
1
)
=
1
T
[
A
0
+
c
1
D
T
+
c
2
H
T
+
c
3
B
T
+
c
4
L
T
]
=
1
T
{
+
∫
μ
2
T
‍
A
0
+
c
1
[
∫
μ
1
t
1
‍
∫
0
μ
1
f
(
x
)
(
e
α
x
β
-
1
)
d
x
+
D
0
∫
μ
1
t
1
‍
(
e
α
x
β
-
1
)
d
x
]
+
c
2
[
∫
0
t
1
‍
∫
t
t
1
‍
∫
0
μ
1
‍
∫
t
μ
1
‍
e
α
(
x
β
-
t
β
)
f
(
x
)
d
x
d
t
+
D
0
∫
0
t
1
‍
∫
t
t
1
‍
e
α
(
x
β
-
t
β
)
d
x
d
t
]
+
c
3
[
∫
μ
2
T
‍
(
T
-
t
)
e
δ
(
t
-
T
)
g
(
t
)
d
t
+
D
0
δ
(
e
δ
(
t
1
-
T
)
(
T
-
t
1
+
1
δ
)
-
e
δ
(
μ
2
-
T
)
(
T
-
μ
2
+
1
δ
)
)
∫
μ
2
T
‍
]
+
c
4
[
∫
μ
2
T
‍
D
0
∫
t
1
μ
2
‍
(
1
-
e
δ
(
t
-
T
)
)
d
t
+
∫
μ
2
T
‍
(
1
-
e
δ
(
t
-
T
)
)
g
(
t
)
d
t
]
}
.Case 3 (μ
2
≤
t
1
<
T).
The differential equations governing the inventory model can be expressed as follows:(19)
d
I
(
t
)
d
t
=
{
-
α
β
t
β
-
1
I
(
t
)
-
f
(
t
)
,
0
<
t
<
μ
1
;
-
α
β
t
β
-
1
I
(
t
)
-
D
0
,
μ
1
<
t
<
μ
2
;
-
α
β
t
β
-
1
I
(
t
)
-
g
(
t
)
,
μ
2
<
t
<
t
1
;
-
e
-
δ
(
T
-
t
)
g
(
t
)
,
t
1
<
t
<
T
.
Solving the differential equation (19) with I
(
t
1
)
=
0, we have(20)
I
(
t
)
=
{
∫
t
μ
1
‍
e
α
(
x
β
-
t
β
)
f
(
x
)
d
x
+
D
0
∫
μ
1
μ
2
‍
e
α
(
x
β
-
t
β
)
d
x
+
∫
μ
2
t
1
‍
e
α
(
x
β
-
t
β
)
g
(
x
)
d
x
,
0
<
t
<
μ
1
;
D
0
∫
t
μ
2
‍
e
α
(
x
β
-
t
β
)
d
x
+
∫
μ
2
t
1
‍
e
α
(
x
β
-
t
β
)
g
(
x
)
d
x
,
μ
1
<
t
<
μ
2
;
∫
t
t
1
‍
e
α
(
x
β
-
t
β
)
g
(
x
)
d
x
,
μ
2
<
t
<
t
1
;
-
∫
t
1
t
‍
e
δ
(
x
-
T
)
g
(
x
)
d
x
,
t
1
<
t
<
T
.The beginning inventory level can be computed as
(21)
S
=
I
(
0
)
=
∫
0
μ
1
‍
e
α
x
β
f
(
x
)
d
x
+
D
0
∫
μ
1
μ
2
‍
e
α
x
β
d
x
+
∫
μ
2
t
1
‍
e
α
x
β
g
(
x
)
d
x
.
The total number of items which perish in the interval [
0
,
t
1
] is
(22)
D
T
=
∫
0
μ
1
‍
(
e
α
x
β
-
1
)
f
(
x
)
d
x
+
D
0
∫
μ
1
μ
2
‍
(
e
α
x
β
-
1
)
d
x
+
∫
μ
2
t
1
‍
(
e
α
x
β
-
1
)
g
(
x
)
d
x
.
The total number of inventory carried during the interval [
0
,
t
1
] is
(23)
H
T
=
∫
0
μ
1
‍
∫
t
μ
1
‍
e
α
(
x
β
-
t
β
)
f
(
x
)
d
x
d
t
+
D
0
∫
0
μ
1
‍
∫
μ
1
μ
2
‍
e
α
(
x
β
-
t
β
)
d
x
d
t
+
D
0
∫
μ
1
μ
2
‍
∫
t
μ
2
‍
e
α
(
x
β
-
t
β
)
d
x
d
t
+
∫
0
μ
2
‍
∫
μ
2
t
1
‍
e
α
(
x
β
-
t
β
)
g
(
x
)
d
x
d
t
+
∫
μ
2
t
1
‍
∫
t
t
1
‍
e
α
(
x
β
-
t
β
)
g
(
x
)
d
x
d
t
.
The total shortage quantity during the interval [
t
1
,
T
] is
(24)
B
T
=
∫
t
1
T
‍
(
T
-
t
)
e
-
δ
(
T
-
t
)
g
(
t
)
d
t
.
The total of lost sales during the interval [
t
1
,
T
] is
(25)
L
T
=
∫
t
1
T
‍
(
1
-
e
-
δ
(
T
-
t
)
)
g
(
t
)
d
t
.
Therefore, the average total cost per unit time under the condition μ
2
≤
t
1
≤
T can be given by
(26)
C
3
(
t
1
)
=
1
T
[
A
0
+
c
1
D
T
+
c
2
H
T
+
c
3
B
T
+
c
4
L
T
]
=
1
T
{
+
c
4
[
∫
t
1
T
‍
(
1
-
e
δ
(
t
-
T
)
)
g
(
t
)
d
t
]
A
0
+
c
1
[
+
∫
μ
2
t
1
‍
(
e
α
x
β
-
1
)
g
(
x
)
d
x
∫
0
μ
1
‍
(
e
α
x
β
-
1
)
f
(
x
)
d
x
+
D
0
∫
μ
1
μ
2
‍
(
e
α
x
β
-
1
)
d
x
+
∫
μ
2
t
1
‍
(
e
α
x
β
-
1
)
g
(
x
)
d
x
]
+
c
2
[
+
∫
μ
2
t
1
‍
∫
t
t
1
‍
e
α
(
x
β
-
t
β
)
∫
0
μ
1
‍
∫
t
μ
1
‍
e
α
(
x
β
-
t
β
)
f
(
x
)
d
x
d
t
+
D
0
∫
0
μ
1
‍
∫
μ
1
μ
2
‍
e
α
(
x
β
-
t
β
)
d
x
d
t
+
D
0
∫
μ
1
μ
2
‍
∫
t
μ
2
‍
e
α
(
x
β
-
t
β
)
d
x
d
t
+
∫
0
μ
2
‍
∫
μ
2
t
1
‍
e
α
(
x
β
-
t
β
)
g
(
x
)
d
x
d
t
+
∫
μ
2
t
1
‍
∫
t
t
1
‍
e
α
(
x
β
-
t
β
)
g
(
x
)
d
x
d
t
]
+
c
3
[
∫
t
1
T
‍
(
T
-
t
)
e
δ
(
t
-
T
)
g
(
t
)
d
t
]
+
c
4
[
∫
t
1
T
‍
(
1
-
e
δ
(
t
-
T
)
)
g
(
t
)
d
t
]
}
.
From the above analysis, we obtain that the total average cost of the model over the time interval [
0
,
T
] is
(27)
T
C
(
t
1
)
=
{
C
1
(
t
1
)
,
0
<
t
1
≤
μ
1
;
C
2
(
t
1
)
,
μ
1
<
t
1
≤
μ
2
;
C
3
(
t
1
)
,
μ
2
<
t
1
<
T
,
where C
1
(
t
1
), C
2
(
t
1
), and C
3
(
t
1
) are obtained from (10), (18), and (26), respectively.
In the following, we will provide the results which ensure the existence of a uniquet
1, say t
1
*, so as to minimize the total average cost for the model system starting with no shortages.
If0
<
t
1
≤
μ
1, taking the first-order derivative of C
1
(
t
1
) with respect to t
1, we obtain
(28)
d
C
1
(
t
1
)
d
t
1
=
f
(
t
1
)
T
h
(
t
1
)
,
where
(29)
h
(
t
1
)
=
c
1
(
e
α
t
1
β
-
1
)
+
c
2
∫
0
t
1
‍
e
α
(
t
1
β
-
t
β
)
d
t
+
c
3
(
t
1
-
T
)
e
δ
(
t
1
-
T
)
+
c
4
(
e
δ
(
t
1
-
T
)
-
1
)
,
then we can obtain h
(
0
)
<
0 and h
(
T
)
>
0. By using the assumption (t
e
-
δ
t is an increasing function, where t is the waiting time up to the next replenishment), we have
(30)
d
h
(
t
1
)
d
t
1
=
α
β
t
1
β
-
1
(
c
1
e
α
t
1
β
+
c
2
∫
0
t
1
‍
e
α
(
t
1
β
-
t
β
)
d
t
)
+
[
c
3
(
δ
(
t
1
-
T
)
+
1
)
+
c
4
δ
]
e
δ
(
t
1
-
T
)
+
c
2
>
0
,
which implies that h
(
t
1
) is a strictly monotone increasing function. Therefore, the equation
(31)
h
(
t
1
)
=
c
1
(
e
α
t
1
β
-
1
)
+
c
2
∫
0
t
1
‍
e
α
(
t
1
β
-
t
β
)
d
t
+
c
3
(
t
1
-
T
)
e
δ
(
t
1
-
T
)
+
c
4
(
e
δ
(
t
1
-
T
)
-
1
)
=
0
has a unique root t
1
*
∈
(
0
,
T
) obtained by using Mathematica 9.0. Further, t
1
* is the only zero-point of d
C
1
(
t
1
)
/
d
t
1
=
0 since f
(
t
1
)
>
0.
If0
<
t
1
*
≤
μ
1, for this t
1
*, we have
(32)
d
2
C
1
(
t
1
)
d
t
1
2
|
t
1
=
t
1
*
=
f
(
t
1
*
)
d
h
(
t
1
*
)
T
d
t
1
*
>
0
,
which means that the total average cost C
1
(
t
1
) can obtain its minimum value at t
1
*.
The optimal value of the order level,S
=
I
(
0
), is
(33)
S
*
=
∫
0
t
1
*
‍
f
(
x
)
e
α
x
β
d
x
,
and the optimal order quantity Q
* is
(34)
Q
*
=
S
*
+
∫
t
1
*
μ
1
‍
e
δ
(
t
-
T
)
f
(
t
)
d
t
+
D
0
∫
μ
1
μ
2
‍
e
δ
(
t
-
T
)
d
t
+
∫
μ
2
T
‍
e
δ
(
t
-
T
)
g
(
t
)
d
t
.
If t
1
*
≥
μ
1, then the optimal value of C
1
(
t
1
) is obtained at t
1
=
μ
1.
Ifμ
1
<
t
1
≤
μ
2, taking the first-order and second-order derivative of C
2
(
t
1
) with respect to t
1, respectively, we obtain
(35)
d
C
2
(
t
1
)
d
t
1
=
D
0
T
h
(
t
1
)
.
If μ
1
<
t
1
*
≤
μ
2, for this t
1
*, we have
(36)
d
2
C
2
(
t
1
)
d
t
1
2
|
t
1
=
t
1
*
=
D
0
d
h
(
t
1
*
)
T
d
t
1
*
>
0
,
where the function h
(
t
1
) is given by (31), and (36) implies that C
2
(
t
1
) is a strictly convex function in t
1 and obtained its minimum value at t
1
*. Therefore, the equation h
(
t
1
)
=
0 has a unique root t
1
* in (
0
,
T
).
The optimal value of the order level,S
=
I
(
0
), is
(37)
S
*
=
∫
0
μ
1
‍
f
(
x
)
e
α
x
β
d
x
+
D
0
∫
μ
1
t
1
*
‍
e
α
x
β
d
x
,
and the optimal order quantity Q
* is
(38)
Q
*
=
S
*
+
∫
t
1
*
μ
2
‍
e
δ
(
t
-
T
)
D
0
d
t
+
∫
μ
2
T
‍
e
δ
(
t
-
T
)
g
(
t
)
d
t
.
Ift
1
*
≤
μ
1, then the optimal value of C
2
(
t
1
) is obtained at t
1
*
=
μ
1, and if t
1
*
≥
μ
2, then the optimal value of C
2
(
t
1
) is obtained at t
1
*
=
μ
2.
Ifμ
2
<
t
1
≤
T, taking the first-order and second-order derivative of C
3
(
t
1
) with respect to t
1, respectively, we obtain
(39)
d
C
3
(
t
1
)
d
t
1
=
g
(
t
1
)
T
h
(
t
1
)
.
If μ
2
<
t
1
*
≤
T, for this t
1
*, we have
(40)
d
2
C
3
(
t
1
)
d
t
1
2
|
t
1
=
t
1
*
=
g
(
t
1
*
)
d
h
(
t
1
*
)
T
d
t
1
*
>
0
.
The function h
(
t
1
) is given by (31), and (40) implies that C
3
(
t
1
) can obtain its minimum value at t
1
*.
The optimal value of the order level,S
=
I
(
0
), is
(41)
S
*
=
∫
0
μ
1
‍
e
α
x
β
f
(
x
)
d
x
+
D
0
∫
μ
1
μ
2
‍
e
α
x
β
d
x
+
∫
μ
2
t
1
*
‍
e
α
x
β
g
(
x
)
d
x
,
and the optimal order quantity Q
* is
(42)
Q
*
=
S
*
+
∫
t
1
*
T
‍
e
δ
(
t
-
T
)
g
(
t
)
d
t
.
If t
1
*
≤
μ
2, then the optimal value of C
3
(
t
1
) is obtained at t
1
*
=
μ
2.
The above analysis shows that the three average cost functionsC
1
(
t
1
), C
2
(
t
2
), and C
3
(
t
1
) can obtain their minimum value at t
1
*
∈
(
0
,
T
) which is determined by (31). Therefore, based on the results analyzed above, this paper derives a procedure to locate the optimal replenishment policy starting with no shortage for the three cases. The procedure is as follows.
Step
1. Solve t
1
* from (31).
Step
2. Compare t
1
* to μ
1 and μ
2, respectively.
Step
2.1. If t
1
*
∈
(
0
,
μ
1
], then the optimal total average cost and the optimal order quantity can be obtained by (10) and (34), respectively.
Step
2.2. If t
1
*
∈
(
μ
1
,
μ
2
], then the optimal total average cost and the optimal order quantity can be obtained by (18) and (38), respectively.Step
2.3. If t
1
*
∈
(
μ
2
,
T
], then the optimal total average cost and the optimal order quantity can be obtained by (26) and (42), respectively.Remark 1.
In such considered inventory model starting with no shortage, ift
1 satisfies μ
1
<
t
1
≤
T
<
μ
2, the considered inventory model reduces to that of Skouri et al. [12].


## 4. Numerical Example

In order to demonstrate the above procedure which can be applied to obtain the optimal solution of the model, this paper presents several examples for the model, respectively. Examples are based on piecewise demand rate, such asf
(
t
)
=
a
1
+
b
1
t and g
(
t
)
=
a
2
e
-
b
2
t.Example 1.
The parameter values are given as follows:T
=
12 weeks, μ
1
=
4 weeks, μ
2
=
8 weeks, α
=
0.005, β
=
2, δ
=
0.04, a
1
=
30 unit, b
1
=
5 unit, a
2
=
100 unit, A
0
=
$
500, c
1
=
$
2, c
2
=
$
3, c
3
=
$
12, and c
4
=
$
8.
The model starting with no shortage; by solving the equationh
(
t
1
)
=
0, we have t
1
*
=
8.7622. From (42) and (26), we obtain Q
*
=
582.5217 and T
C
(
t
1
*
)
=
793.9986, respectively.Example 2.
The parameter values are given as follows:T
=
12 weeks, μ
1
=
4 weeks, μ
2
=
8 weeks, α
=
0.005, β
=
2, δ
=
0.02, a
1
=
30 unit, b
1
=
5 unit, a
2
=
100 unit, A
0
=
$
500, c
1
=
$
5, c
2
=
$
10, c
3
=
$
12, and c
4
=
$
8.
The model starting with no shortage, by solving the equationh
(
t
1
)
=
0, we have t
1
*
=
5.8330. From (38) and (18), we obtain Q
*
=
528.4725 and T
C
(
t
1
*
)
=
1601.4013, respectively.Example 3.
The parameter values are given as follows:T
=
12 weeks, μ
1
=
4 weeks, μ
2
=
6 weeks, α
=
0.005, β
=
1.6, δ
=
0.2, a
1
=
30 unit, b
1
=
5 unit, a
2
=
100 unit, A
0
=
$
500, c
1
=
$
5, c
2
=
$
10, c
3
=
$
12, and c
4
=
$
8.
The model starting with no shortage, solving the equationh
(
t
1
)
=
0, the optimal value of t
1 is t
1
*
=
2.3235. The optimal ordering quantity is Q
*
=
272.6678, and the minimum cost T
C
(
t
1
*
)
=
1258.82.
In order to clearly indicate the effects of parameters such asδ, α, β, c
1, c
2, c
3, and c
4 on the optimal on-hand inventory S
*, the optimal ordering quantity Q
*, and the optimal total cost T
C
(
t
1
*
), respectively, the paper will study the sensitivity of the optimal solution to changes in the value of different parameter associated with the studied inventory model. The sensitivity analysis is performed on the base of Example 1, and the results are shown in Table 1–7.
By studying the results of Table1, it is found that the shortage time t
1
*, inventory level S
*, order quantity Q
*, and the total average cost T
C
(
t
1
*
) gradually decrease as the shortage parameter δ increases for the model, respectively. We also find that the percentage increase of δ from 14.3% to 100% causes T
C
(
t
1
*
) to decrease from 0.45% to 0.34%, Q
* decrease from 0.75% to 0.52%, t
1
* decrease from 0.78% to 0.53%, and S
* decrease from 1.02% to 0.69%. It is also observed that the value of t
1
*, S
*, Q
*, and T
C
(
t
1
*
) all are lowly sensitive to the changes of δ for the considered inventory model.
By studying the results of Table2, it is found that S
*, Q
*, and T
C
(
t
1
*
) coordinates to the deterioration parameter α; the shortage time t
1
* decreases as α increases for the model. It is also found that the percentage increase of α from 16.7% to 100% causes T
C
(
t
1
*
) to decrease by 2.066%–2.595%, Q
* to increase by 1.597%–2.44%, the shortage time t
1
* to decrease by 1.513%–1.455%, and S
* to increase by 0.917%–1.651%. It also observes that the value of t
1
*, S
*, Q
*, and T
C
(
t
1
*
) all are moderately sensitive to the changes of α for the considered inventory model.
By studying the results of Table3, it is found that S
*, Q
*, and T
C
(
t
1
*
) coordinate to the deterioration parameter β, while the shortage time t
1
* decreases as β increases for the model. It is also found that the increase of β from 1.4 to 2.2 causes S
* to increase, while the increase of β from 2.4 to 2.8 causes S
* to decrease, Q
* to increase by 2.44%–1.597%, T
C
(
t
1
*
) to increase by 2.595%–2.066%, and the shortage time t
1
* to decrease by 1.513%–1.455%. It is also observed that the value of t
1
*, S
*, Q
*, and T
C
(
t
1
*
) all are moderately sensitive to the changes of β for the considered inventory model.
By studying the results of Table4, it is found that T
C
(
t
1
*
) coordinate to c
1, while the shortage time t
1
*, S
*, and Q
* decrease as c
1 increases for the model. It is also found that c
1 increases from 8.3% to 150%, T
C
(
t
1
*
) decreases by 0.269%–0.383%, Q
* decreases by 0.083%–0.058%, t
1
* decreases by 0.264%–0.181%, and S
* decreases by 0.203%–0.138%, respectively. It is also observed that the values of t
1
*, S
*, Q
*, and T
C
(
t
1
*
) all are lowly sensitive to the changes of c
1 for the considered inventory model.
By studying the results of Table5, it is found that T
C
(
t
1
*
) coordinates to c
2, while S
*, Q
*,and t
1
* decrease as c
1 increases for the model. It is also found that c
2 increases by 100%, T
C
(
t
1
*
) decreases by 0.269%–0.383%, Q
* decreases by 0.083%–0.058%, t
1
* decreases by 0.264%–0.181%, and S
* decreases by 0.203%–0.138%.
By studying the results of Table6, it is found that t
1
*, S
*, Q
*, and T
C
(
t
1
*
) coordinate to c
3. It is also observed that the value of t
1
*, S
*, Q
*, and T
C
(
t
1
*
) all are lowly sensitive to the changes of c
3 for the inventory models; that is, c
3 increases from 1.9% to 3.2%, the change of all the parameters is no more than 1%.
By studying the results of Table7, it is found that t
1
*, S
*, Q
*, and T
C
(
t
1
*
) coordinate to c
4. It is also observed that the value of t
1
*, S
*, Q
*, and T
C
(
t
1
*
) all are lowly sensitive to the changes of c
4 for the inventory models; that is, c
4 increases from 14.3% to 100%, the change of all the parameters is no more than 1%.Table 1
The sensitivity ofδ for the models in Example 1.
δ
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
t
1
*
8.9664
8.9187
8.8689
8.8167
8.7622
8.7049
8.6449
8.5817
8.5152
S
*
472.9889
469.7083
466.2823
462.7001
458.9498
455.0167
450.8910
446.5519
441.9829
Q
*
595.6569
592.5872
589.3822
586.0308
582.5217
578.8412
574.9771
570.9112
566.6265
TC
(
t
1
*
)
805.6323
802.8699
800.0139
797.0588
793.9986
790.8268
787.5365
784.1200
780.5690Table 2
The sensitivity ofα for the models in Example 1.
α
0
0.001
0.002
0.003
0.004
0.005
0.006
0.007
t
1
*
9.4545
9.3115
9.1702
9.0312
8.8950
8.7622
8.6328
8.5072
S
*
428.3286
435.4002
442.0048
448.1317
453.7785
458.9498
463.6555
467.9088
Q
*
523.8866
536.6685
548.9558
560.7091
571.9025
582.5217
592.5618
602.0259
TC
(
t
1
*
)
704.6045
722.8867
741.0107
758.9297
776.6035
793.9986
811.0872
827.8474Table 3
The sensitivity ofβ for the models in Example 1.
β
1.4
1.6
1.8
2.0
2.2
2.4
2.6
2.8
t
1
*
9.2876
9.1810
9.0142
8.7622
8.4047
7.9422
7.4073
6.8496
S
*
442.9058
447.5894
453.1802
458.9076
462.8345
461.6422
453.4686
438.8904
Q
*
545.1340
554.1055
566.4485
582.4772
601.2274
774.6099
766.4364
751.8582
TC
(
t
1
*
)
745.7703
764.3043
791.7624
790.7114
829.3653
588.7417
473.1282
391.1266Table 4
The sensitivity ofc
1 for the models in Example 1.
c
1
0
0.4
1
1.6
2
2.4
2.6
3
3.6
t
1
*
8.8225
8.8103
8.7921
8.7741
8.7622
8.7503
8.7443
8.7326
8.7150
S
*
463.0980
462.2595
461.0101
459.7707
458.9498
458.1333
457.7266
456.9164
455.7090
Q
*
584.1909
583.8529
583.3499
582.8514
582.5217
582.1940
582.5217
581.7061
581.2226
TC
(
t
1
*
)
783.5429
785.6518
788.7984
791.9251
793.9986
796.0633
797.0925
799.1443
802.2061Table 5
The sensitivity ofc
2 for the model in Example 1.
c
2
0
0.4
0.8
1.2
1.8
2.4
3
3.4
3.8
t
1
*
11.8343
11.267
10.7698
10.326
9.7377
9.2216
8.7622
8.4819
8.2196
S
*
673.7217
633.0852
597.9589
566.9228
526.1187
490.5382
458.9498
439.6922
421.6606
Q
*
679.6039
659.4483
642.7261
628.4787
610.4744
595.4175
582.5217
574.8648
567.8284
TC
(
t
1
*
)
67.2679
193.9216
308.8975
413.9636
555.8150
681.6692
793.9986
862.3253
925.9516Table 6
The sensitivity ofc
3 for the models in Example 1.
c
3
10.4
10.6
10.8
11
11.2
11.6
12
12.4
12.8
t
1
*
8.4424
8.4859
8.5284
8.5698
8.6102
8.6879
8.7622
8.8329
8.9005
S
*
436.9771
439.9713
442.889
445.7332
448.5071
453.854
458.9498
463.8114
468.4587
Q
*
573.7973
574.9747
576.1255
577.2506
578.3509
580.4809
582.5217
584.4787
586.3587
TC
(
t
1
*
)
763.8057
767.9203
771.9299
775.8387
779.6505
786.9976
793.9986
800.6779
807.0579Table 7
The sensitivity ofc
4 for the models in Example 1.
c
4
0
2
4
6
8
10
12
14
16
t
1
*
8.6989
8.7150
8.7309
8.7466
8.7622
8.7775
8.7928
8.8078
8.82272
S
*
454.6069
455.7103
456.8017
457.8815
458.9498
460.0048
461.0527
462.0877
463.1120
Q
*
580.7818
581.2231
581.6601
582.0929
582.5217
582.9455
583.367
583.7837
584.1966
TC
(
t
1
*
)
788.1626
789.6445
791.1109
792.5621
793.9986
795.4203
796.8276
798.2207
799.5999


## 5. Conclusion

An inventory model starting without shortage for Weibull-distributed deterioration with trapezoidal type demand rate and partial backlogging is considered in this paper. The optimal replenishment policy for the inventory model is proposed, and numerical examples are provided to illustrate the theoretical results. A sensitivity analysis of the optimal solution with respect to major parameters is also carried out. From Table1–7, it can be found that the shortage time point t
1
*, order quantity Q
*, and the total average cost T
C
(
t
1
*
) are moderately sensitive to the changes of α and β and lowly sensitive to the changes of δ, c
i
(
i
=
1,2
,
3,4
), respectively. The paper provides an interesting topic for further study, such that the joint influence from some of these parameters may be investigated to show the effects; the model starting with shortage will be studied and other types of models for deteriorating items in supply chain situation are also to be studied in the future.



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*Source: 747419-2014-03-06.xml*