# Applying Fuzzy Multiobjective Integrated Logistics Model to Green Supply Chain Problems

Authors: Chui-Yu Chiu; Yi Lin; Ming-Feng Yang
Journal: Journal of Applied Mathematics
 (2014)

---

## Abstract

The aim of this paper is attempting to explore the optimal way of supply chain management within the domain of environmental responsibility and concerns. The background of this research involves the issue of green supply chain management (GSCM) and the concept of the multiobjective integrated logistics model. More specifically, in this paper, we suggest the fuzzy multiobjective integrated logistics model with the transportation cost and demand fuzziness to solve green supply chain problems in the uncertain environment which is illustrated via the detailed numerical example. Results and the sensitivity analysis of the numerical example indicate that when the governmental subsidy value increased the profits of the reverse chain also increased. The finding shows that the governmental subsidy policy could remain of significant influence for used-product reverse logistics chain.


---

## Body

## 1. Introduction

In today’s highly competitive global market, preferable environmental concerns and reserved considerations are acknowledged as having high potential to impact the benefits of the supply chain management. Both academicians and practitioners tend to find an optimal way to keep business competition with more environmental concerns. Hence, the manufacturers have to consider carefully many aspects that govern their performance in order to ensure end user satisfaction [1]. In recent years considerable concern has been arising over the issues of environmental pollution accompanying industrial development in the operational process of the supply chain management research. Environment protection is becoming a vital issue for enterprises, government, and nonprofit organizations because of stronger public awareness. Indeed, the green supply chain management has emerged as an approach to balance these competitive requirements [2].This paper is an attempt to fill the gap in the existing literature of GSCM, much of which focuses on the manufacturing chain, reverse logistics chain, chain members, cash flow, and so forth. However, major issues of uncertainty and real-world applications have not been many researches attentions. In view of this gap, we consider the fuzzy demand and transportation cost more closely in the real-world situation.


## 2. Literature Review

In reality, the environment of global market is complicated and uncertain; therefore, some researchers have been devoting themselves to research the supply chain management with the aspect of uncertainty. Petrovic et al. [3] applied fuzzy set to handle uncertain demands and external raw material problems and to deal with linguistic expressions and uncertain issues. Giannoccaro et al. [4] also proposed fuzzy sets theory to the uncertainties associated with both market demand and inventory costs. In this paper, center of gravity is proposed to convert the fuzzy number into a crisp number.As a result of increasing challenges in highly competitive global supply chain environment, both academicians and practitioners have increasing interests of finding the optimal way to remain competitive, and manufacturers have also been seeking to deliver to their customers high-quality products in the right time at the right price [1]. Drzymalski and Odrey [5] proposed that each entity within a supply chain is also different in its constraints, operations, and objectives, resulting in varying performance measures within each organization. Hence, supply chain modeling has been becoming crucial as companies face more complicated and global interactions and increased customer expectations. When an item moves through more than one step before reaching the final customer, the terms “multilevel” or “multiechelon” production and distribution supply chain are also synonymous with such supply chains [6–8]. Initial multiechelon inventory theory was introduced by Clark and Scarf [9] as a way of proving the optimality of a base stock policy for the pure serial inventory system and developed an efficient decomposing method to compute the optimal base stock ordering policy. In addition, some researchers considered a strategic supply chain design problem with three echelons, multiple commodities, and technology selection [10]. And then, Disney et al. [11] proposed the research on a coordination scheme in a two-echelon supply chain which involves sharing details of lead times, replenishment rules, and demand patterns and tuning the replenishment rules to exploit the supply chain’s cost structure. Yazlali and Erhun [12] noted a single-product dual-supply problem under a periodically reviewed, finite planning horizon. As we mentioned in the earlier section, with the promotion of environmental consciousness in recent years, the logistics and supply chain managers have to balance efforts to reduce costs and innovate while maintaining good environmental (ecological) performance [13]. The enterprises also have to take more applicable measures to improve reverse logistics. Indeed, the green supply chain management (GSCM) has appeared as an approach to balance these competitive requirements [2].Recently, greening the supply chain has become the vital issue. Many organizations, universities, and governments held conferences of supply chain management with environmental thinking. The purpose of these conferences was to demonstrate and exchange the green technology, green knowledge, and green research. It means that the field of green supply chain management has been becoming more and more important in both academic research and practice in the real market [14]. Also, there have been a number of studies that have developed and implemented how to green the supply chain. For example, Hsu et al. [15] applied the Decision-Making Trial and Evaluation Laboratory (DEMATEL) approach to select the green suppliers in green supply chain management. Pishvaee et al. [16] created the model to minimize the environmental impacts and costs in the green supply chain. And then, Trappey et al. [17] designed the green production planning and discussed how to minimize a product’s carbon footprint.However, due to the imprecision of the information related to parameters, the deterministic models are inappropriate for obtaining an effective solution for supply chain implications. In uncertainty, the method to carry out this study is applying the fuzzy set theory, which provides a way to obtain feasible answers to overcome the natural difficulties. There have been a number of studies that have investigated the implications of the fuzzy set theory. The fuzzy decision-making concept, introduced by Bellman and Zadeh [18] as a way of handling imprecise data, proposed the most suitable fuzzy decision analysis procedure in the fuzzy environment. Afterwards, Zimmermann [19] combined the fuzzy sets theory, the traditional linear programming, and the fuzzy decision-making and linear membership function as the modified method and it became a new academic direction.In fuzzy multiobjective programming aspect, the paper by Zimmermann [20] provided the fuzzy goals of the decision-maker with the fuzzy decisions. Bellman and Zadeh [18] noted in their research that the linear membership functions and the fuzzy decisions can be adopted to convert the problem to be solved into an ordinary single-goal mathematical programming (Max L) problem. Then, Li et al. [21] and Kumar et al. [22] developed a fuzzy mixed integer goal programming approach for supplier evaluation and selection model. Furthermore, Kumar et al. [23] and Amid et al. [24] extended their research and developed a fuzzy multiobjective integer programming model to deal with the different weights in various criteria. Numerous studies noted that the fuzzy multiobjective programming decision models remained consistently under different situations [25–28].


## 3. Model Formulation

The purpose of this section is to establish a mathematical model. In this work, we modified the integrated logistics model which is indicated by Sheu et al. [29]. We use the linear multiobjective programming model into the integrated logistics operational problems of green supply chain management. The objective of our proposed model is to maximize the total profit function with respect to general manufacturing chain and the reverse chain. Also, we consider the fuzzy demand and transportation cost and adopt the triangular fuzzy number to represent these variables [29].


### 3.1. Assumptions

The assumptions of this research are as follows.(1)
Only single product is considered in the proposed model.
(2)
The time-varying quantity of product demands from end customers in any given time interval is given.
(3)
Shortages are not allowed.
(4)
There is a given return ratio, referring to the proportion of the quantity of used products returned from end customers and through the reverse logistics chain.
(5)
The time horizon is infinite.
(6)
Facility capacities associated with chain members of the proposed integrated logistics system are known.
(7)
The lead time is known in both the general and the reverse supply chains.
(8)
Pollution such as heavy metal pollution, toxic pollution, and water pollution is not considered in this green supply chain.A mathematical model was developed to seek equilibrium solutions with the target of maximizing the systematic net profit which aggregated from the chain-based net profits associated with the respect of the manufacturing supply chain and the reverse logistics chain. Figure1 illustrates the conceptual framework for integrated logistics control across a green supply chain.Figure 1
Conceptual framework for integrated logistics control across a green supply chain.As we can see in Figure1, there are two groups in the green supply chain system: (
1
) manufacturing supply chain and (
2
) used-product reverse supply chain. In addition, the typical 5-layer manufacturing supply chain is proposed; these layers are given as follows: raw material suppliers (m
1 for short), manufacturers (m
2 for short), wholesalers (m
3 for short), retailers (m
4 for short), and end customers (m
5 for short). Also, the typical 5-layer used-product reverse supply chain is proposed as well with its layers given as follows: collecting points (r
1 for short), recycling plants (r
2 for short), disassembly plants (r
3 for short), secondary material market (r
4 for short), and final disposal (r
5 for short). The mathematical formulation of our proposed model is indicated below. All notations of the proposed model are listed in the appendix.


### 3.2. Manufacturing Chain

According to the aforementioned assumptions, objective optimization models are formulated to seek the maximum net profit of the general supply chain and the reverse supply chain.MP and RP are measured by subtracting the corresponding aggregate costs from the respective aggregate revenues and costs, as expressed, respectively, in (1) and (2) as follows:
(1)
Max
⁡
MP
=
MTR
-
MC
=
MTR
-
(
MPC
+
MMC
+
MIC
+
MTC
+
MRC
+
MLC
)
.As in (1), the total cost of manufacturing chain consists of six parts: raw material procurement, manufacturing, inventory, transportation, recycling fees paid to the EPA, and labor cost.


### 3.3. Reverse Chain

Consider the following(2)
Max
⁡
RP
=
RTR
-
RC
=
(
RR
+
RS
)
-
(
RTCC
+
RTTC
+
RIC
+
RTC
+
RFC
+
RLC
)
.Similarly, in (2), the total cost of the reverse chain is composed of six parts: the cost of collecting used products, transitional treatment, inventory, transportation, final disposal, and labor cost. Besides, there are two sources of revenue in the reverse chain: the general revenue and the subsidies from EPA.From the proposed integrating logistics system architecture as shown in Figure1, the objective functions are composed of (
1
) manufacturing chain-based net profit (MP) maximization and (
2
) reverse chain-based net profit (RP) maximization. And our target is to maximize the total revenue in the supply chain system as shown below:
(3)
Max
⁡
Total
Revenue
=
MP
+
RP
=
(
MTR
-
MC
)
+
(
RTR
-
RC
)
.


### 3.4. Revenues

MTR = Profit oriented from the raw material flows + Profit oriented from the physical flows of the manufactured product in any given distribution channel of the manufacturing chain:(4)
∑
∀
t
{
[
∑
∀
m
1
∑
∀
m
2
R
m
1
,
m
2
(
t
)
×
Q
m
1
,
m
2
(
t
)
]
[
∑
i
=
2
4
∑
i
+
1
5
∑
∀
m
i
∑
∀
m
j
R
m
i
,
m
+
1
(
t
)
×
Q
m
i
,
m
+
1
(
t
)
]
h
h
h
+
[
∑
i
=
2
4
∑
i
+
1
5
∑
∀
m
i
∑
∀
m
j
R
m
i
,
m
+
1
(
t
)
×
Q
m
i
,
m
+
1
(
t
)
]
}
.RR = Refund obtained by end customer for returning the used product + Revenue associated with the used-product collection for selling the collected unprocessed used product to other reverse chains’ members + Revenue from selling the processed used product to another member in the same reverse chain + Revenue of the secondary material market for selling the processed used raw materials to the layer of manufacturing of the given manufacturing chain:(5)
∑
∀
t
{
[
∑
j
=
1
3
∑
∀
m
5
∑
∀
r
j
R
m
5
,
r
j
(
t
)
×
Q
m
5
,
r
j
(
t
)
]
h
h
h
h
+
[
∑
j
=
2
3
∑
∀
r
1
∑
∀
r
j
R
r
1
,
r
j
(
t
)
×
Q
r
1
,
r
j
(
t
)
]
h
h
h
h
+
[
∑
j
=
2
3
∑
∀
r
j
∑
∀
r
j
+
1
R
r
j
,
r
j
+
1
(
t
)
×
Q
r
j
,
r
j
+
1
(
t
)
]
h
h
h
h
[
∑
j
=
1
3
∑
∀
m
5
∑
∀
r
j
R
m
5
,
r
j
(
t
)
×
Q
m
5
,
r
j
(
t
)
]
+
[
∑
∀
r
4
∑
∀
m
2
R
r
4
,
m
2
(
t
)
×
Q
r
4
,
m
2
(
t
)
]
}
.RS = Subsidies associated with the reverse chains that are oriented from the reverse flows of the returned used products transported to the layer of disassembly plants for government subsidies:(6)
∑
∀
t
{
∑
i
=
1
2
∑
∀
r
i
∑
∀
r
3
EPO
×
Q
r
i
,
r
3
(
t
)
}
.As we mentioned in the earlier section, the revenues from manufacturing chain were composed of raw material and physical flows. And the revenues from the reverse chain were constituted of the regular returned used-product flows and subsidies.


### 3.5. Costs of Manufacturing Chain

The costs of manufacturing chain are composed of procurement, manufacturing, inventory, transportation, recycle, and labor cost. In this part, the detailed discussions of manufacturing chain are presented as follows.MPC = Initialized cost of raw materials + Procurement cost oriented from raw material suppliers and secondary material market + Manufactured-product procurement costs in any distribution channels of manufacturing chain:(7)
∑
∀
t
{
[
∑
∀
m
1
C
R
(
t
)
×
Q
R
(
t
)
]
[
∑
i
=
2
3
∑
i
+
1
4
∑
∀
m
i
∑
∀
m
j
C
m
i
,
m
i
+
1
P
(
t
)
×
Q
m
i
,
m
i
+
1
(
t
)
]
h
h
h
+
[
∑
∀
m
1
∑
∀
m
2
C
m
1
,
m
2
P
(
t
)
×
Q
m
1
,
m
2
(
t
)
h
h
h
h
h
h
+
∑
∀
r
4
∑
∀
m
2
C
r
4
,
m
2
P
(
t
)
×
Q
r
4
,
m
2
(
t
)
]
h
h
h
+
[
∑
i
=
2
3
∑
i
+
1
4
∑
∀
m
i
∑
∀
m
j
C
m
i
,
m
i
+
1
P
(
t
)
×
Q
m
i
,
m
i
+
1
(
t
)
]
}
.MMC = Manufacturing cost of all manufactured products:(8)
∑
∀
t
{
∑
∀
m
2
C
M
(
t
)
×
Q
M
(
t
)
}
.MIC = Inventory cost of raw materials oriented from raw material suppliers and manufactures + Inventory cost of products in any chain member of the manufacturing chain:(9)
∑
∀
t
{
[
∑
i
=
1
2
∑
∀
m
i
C
m
i
IR
(
t
)
×
Q
m
i
IR
(
t
)
]
+
[
∑
i
=
2
4
∑
∀
m
i
C
m
i
I
(
t
)
×
Q
m
i
I
(
t
)
]
}
.MTC
≅ Transportation cost of raw materials transported from suppliers to manufactures + Transportation cost of products transported in any chain member of the manufacturing chain:
(10)
∑
∀
t
{
[
∑
∀
m
1
∑
∀
m
2
C
T
~
m
1
,
m
2
(
t
)
×
Q
m
1
,
m
2
(
t
)
]
+
[
∑
i
=
2
4
∑
i
+
1
5
∑
∀
m
i
∑
∀
m
j
C
T
~
m
i
,
m
i
+
1
(
t
)
×
Q
m
i
,
m
i
+
1
(
t
)
]
h
h
h
+
[
∑
i
=
2
4
∑
i
+
1
5
∑
∀
m
i
∑
∀
m
j
C
T
~
m
i
,
m
i
+
1
(
t
)
×
Q
m
i
,
m
i
+
1
(
t
)
]
}
.MRC = Recycling fees of the products:(11)
∑
∀
t
∑
∀
m
2
F
×
Q
M
(
t
)
.MLC = Labor cost of any member in the manufacturing chain:(12)
∑
∀
t
{
∑
i
=
1
4
∑
∀
m
i
C
m
i
L
×
Q
m
i
L
(
t
)
}
.


### 3.6. Costs of Reverse Chain

The costs of the reverse chain are composed of collected returned used products, transitional treatment procedures, inventory, transportation, final disposal, and labor cost. And we will discuss the reverse chain in this section.RTCC = The amount of used products collected from the end customer to the members of the reverse chain:
(13)
∑
∀
t
{
∑
j
=
1
3
∑
∀
m
5
∑
∀
r
j
C
m
5
,
r
j
C
(
t
)
×
Q
m
5
,
r
j
(
t
)
}
.RTTC = The transitional treatment procedures executed potentially in all the reverse chain layers, except the final disposal:
(14)
∑
∀
t
{
∑
j
=
1
4
∑
∀
r
j
C
r
j
TT
(
t
)
×
Q
r
j
TT
(
t
)
}
.RIC = The inventory cost of unprocessed used product + Inventory cost of processed used product + Inventory cost of processed used product that may be stored in disassembly plant which is for final disposal and selling to the secondary market:
(15)
∑
∀
t
{
[
∑
j
=
1
4
∑
∀
r
j
C
r
j
IT
-
(
t
)
×
Q
r
j
IT
-
(
t
)
]
h
h
h
+
[
∑
j
=
1
,
i
≠
3
4
∑
∀
r
j
C
r
j
IT
(
t
)
×
C
r
j
IT
(
t
)
]
h
h
h
[
∑
j
=
1
4
∑
∀
r
j
C
r
j
I
T
-
(
t
)
×
Q
r
j
I
T
-
(
t
)
]
+
∑
r
3
[
C
r
3
,
r
4
I
(
t
)
×
Q
r
3
,
r
4
I
(
t
)
+
C
r
3
,
r
5
I
(
t
)
×
Q
r
3
,
r
5
I
(
t
)
]
}
.RTC
≅ The summation of transporting costs of any distribution channels of the reverse chain, excluding the collection costs that are oriented from the reverse flows associated with end customers:
(16)
∑
∀
t
{
[
∑
j
=
2
3
∑
r
1
∑
∀
r
j
C
T
~
r
1
,
r
j
(
t
)
×
Q
r
1
,
r
j
(
t
)
]
h
h
h
+
[
∑
i
=
2
3
∑
r
i
∑
∀
r
i
+
1
C
T
~
r
i
,
r
i
+
1
(
t
)
×
Q
r
i
,
r
i
+
1
(
t
)
]
h
h
h
+
[
∑
∀
r
3
∑
∀
r
5
C
T
~
r
3
,
r
5
(
t
)
×
Q
r
3
,
r
5
(
t
)
]
h
h
h
[
∑
j
=
2
3
∑
r
1
∑
∀
r
j
C
T
~
r
1
,
r
j
(
t
)
×
Q
r
1
,
r
j
(
t
)
]
+
[
∑
∀
r
4
∑
∀
m
2
C
T
~
r
4
,
m
2
(
t
)
×
Q
r
4
,
m
2
(
t
)
]
}
.RFC = Total amount of used products disposed in the layer of final disposal:
(17)
∑
∀
t
{
∑
∀
r
5
C
F
(
t
)
×
Q
F
(
t
)
}
.RLC = Labor cost of any member in the reverse chain:
(18)
∑
∀
t
{
∑
r
=
1
5
∑
∀
r
j
C
r
j
L
×
Q
r
j
L
(
t
)
}
.


### 3.7. Constraints

In this part, the detailed discussions of three constraints such as inventory constraints, demand constraints, and return resource constraints are presented as follows.(i) Inventory Constraints. In order to reach the reality, we set safety stock for every member in the two chains. (a)
Forraw material suppliers,
(19)
S
S
m
1
IR
≤
Q
m
1
IR
(
t
)
=
Q
m
1
IR
(
t
-
1
)
+
Q
R
(
t
)
-
∑
∀
m
2
Q
m
1
,
m
2
(
t
)
≤
S
m
1
IR
∀
(
m
1
,
t
)
.
(b)
Forproduct manufacturers,
(1)
for raw materials,(20)
S
S
m
2
IR
≤
Q
m
2
IR
(
t
)
=
Q
m
2
IR
(
t
-
1
)
+
[
∑
∀
m
1
Q
m
1
,
m
2
(
t
)
+
∑
∀
r
4
Q
r
4
,
m
2
(
t
)
]
-
γ
m
2
r
/
m
×
Q
M
(
t
)
≤
S
m
2
IR
∀
(
m
2
,
t
)
,
(2)
for products,(21)
S
S
m
2
I
≤
Q
m
2
I
(
t
)
=
Q
m
2
I
(
t
-
1
)
+
Q
M
(
t
)
-
∑
j
=
3
5
∑
∀
m
j
Q
m
2
,
m
j
(
t
)
≤
S
m
2
I
∀
(
m
2
,
t
)
.
(c)
Forwholesalers and retailers,
(22)
S
S
m
j
I
≤
Q
m
j
I
(
t
)
=
Q
m
j
I
(
t
-
1
)
+
[
∑
i
=
2
j
-
1
∑
∀
m
i
Q
m
i
,
m
j
(
t
)
]
-
[
∑
n
=
j
+
1
5
∑
∀
m
n
Q
m
j
,
m
n
(
t
)
]
≤
S
m
j
I
∀
(
m
j
,
t
)
,
j
=
3
or
4
.
(d)
Forcollecting points,
(23)
S
S
r
1
IT
¯
≤
Q
r
1
IT
¯
(
t
)
=
Q
r
1
IT
¯
(
t
-
1
)
+
[
∑
∀
m
5
Q
m
5
,
r
1
(
t
)
]
-
Q
r
1
TT
(
t
)
≤
S
r
1
IT
¯
∀
(
r
1
,
t
)
,
S
S
r
1
IT
≤
Q
r
1
IT
(
t
)
=
Q
r
1
IT
(
t
-
1
)
+
Q
r
1
TT
(
t
)
-
∑
j
=
2
3
∑
∀
r
j
Q
r
1
,
r
j
(
t
)
≤
S
r
1
IT
∀
(
r
1
,
t
)
.
(e)
Forrecycle plants,
(24)
S
S
r
2
IT
¯
≤
Q
r
2
IT
¯
(
t
)
=
Q
r
2
IT
¯
(
t
-
1
)
+
[
∑
∀
m
5
Q
m
5
,
r
2
(
t
)
+
∑
∀
r
1
Q
r
1
,
r
2
(
t
)
]
-
Q
r
2
TT
(
t
)
≤
S
r
2
IT
¯
∀
(
r
2
,
t
)
,
S
S
r
2
IT
≤
Q
r
2
IT
(
t
)
=
Q
r
2
IT
(
t
-
1
)
+
Q
r
2
TT
(
t
)
-
∑
∀
r
3
Q
r
2
,
r
3
(
t
)
≤
S
r
2
IT
∀
(
r
2
,
t
)
.
(f)
Fordisassembly plants,
(25)
S
S
r
3
IT
¯
≤
Q
r
3
IT
¯
(
t
)
=
Q
r
3
IT
¯
(
t
-
1
)
+
[
∑
∀
m
5
Q
m
5
,
r
3
(
t
)
+
∑
i
=
1
2
∑
∀
r
i
Q
r
i
,
r
3
(
t
)
]
-
Q
r
3
TT
(
t
)
≤
S
r
3
IT
¯
∀
(
r
3
,
t
)
,
S
S
r
3
,
r
4
IT
≤
Q
r
3
,
r
4
I
(
t
)
=
Q
r
3
,
r
4
I
(
t
-
1
)
+
γ
r
3
,
r
4
×
Q
r
3
TT
(
t
)
-
∑
∀
r
4
Q
r
3
,
r
4
(
t
)
≤
S
r
3
,
r
4
I
∀
(
r
3
,
t
)
,
S
S
r
3
,
r
5
I
≤
Q
r
3
,
r
5
I
(
t
)
=
Q
r
3
,
r
5
I
(
t
-
1
)
+
γ
r
3
,
r
5
×
Q
r
3
TT
(
t
)
-
∑
∀
r
5
Q
r
3
,
r
5
(
t
)
≤
S
r
3
,
r
5
I
∀
(
r
3
,
t
)
.
(g)
Forsecondary material markets,
(26)
S
S
r
4
IT
¯
≤
Q
r
4
IT
¯
(
t
)
=
Q
r
4
IT
¯
(
t
-
1
)
+
[
∑
∀
r
3
Q
r
3
,
r
4
(
t
)
]
-
Q
r
4
TT
(
t
)
≤
S
r
4
IT
¯
∀
(
r
4
,
t
)
,
S
S
r
4
IT
≤
Q
r
4
IT
(
t
)
=
Q
r
4
IT
(
t
-
1
)
+
γ
t
4
t
r
e
×
Q
t
r
r
4
(
t
)
-
∑
∀
m
2
Q
r
4
,
m
2
(
t
)
≤
S
r
4
IT
∀
(
r
4
,
t
)
.
(h)
Forfinal disposal locations,
(27)
S
S
r
5
IT
¯
≤
Q
r
5
IT
¯
(
t
)
=
Q
r
5
IT
¯
(
t
-
1
)
+
[
∑
∀
r
3
Q
r
3
,
r
5
(
t
)
]
-
Q
r
5
F
(
t
)
≤
S
r
5
IT
¯
∀
(
r
5
,
t
)
.(ii) Demand Constraints. Consider the following
(28)
D
(
t
)
~
≥
∑
i
=
2
4
∑
∀
m
i
∑
∀
m
5
Q
m
i
,
m
5
(
t
)
≥
0
∀
t
.(iii) Return Resource Constraints. Consider the following
(29)
Q
RE
(
t
)
=
∑
j
=
1
3
∑
∀
m
5
∑
∀
r
j
Q
m
5
,
r
j
(
t
)
=
β
×
D
(
t
)
~
≥
0
∀
t
.


## 3.1. Assumptions

The assumptions of this research are as follows.(1)
Only single product is considered in the proposed model.
(2)
The time-varying quantity of product demands from end customers in any given time interval is given.
(3)
Shortages are not allowed.
(4)
There is a given return ratio, referring to the proportion of the quantity of used products returned from end customers and through the reverse logistics chain.
(5)
The time horizon is infinite.
(6)
Facility capacities associated with chain members of the proposed integrated logistics system are known.
(7)
The lead time is known in both the general and the reverse supply chains.
(8)
Pollution such as heavy metal pollution, toxic pollution, and water pollution is not considered in this green supply chain.A mathematical model was developed to seek equilibrium solutions with the target of maximizing the systematic net profit which aggregated from the chain-based net profits associated with the respect of the manufacturing supply chain and the reverse logistics chain. Figure1 illustrates the conceptual framework for integrated logistics control across a green supply chain.Figure 1
Conceptual framework for integrated logistics control across a green supply chain.As we can see in Figure1, there are two groups in the green supply chain system: (
1
) manufacturing supply chain and (
2
) used-product reverse supply chain. In addition, the typical 5-layer manufacturing supply chain is proposed; these layers are given as follows: raw material suppliers (m
1 for short), manufacturers (m
2 for short), wholesalers (m
3 for short), retailers (m
4 for short), and end customers (m
5 for short). Also, the typical 5-layer used-product reverse supply chain is proposed as well with its layers given as follows: collecting points (r
1 for short), recycling plants (r
2 for short), disassembly plants (r
3 for short), secondary material market (r
4 for short), and final disposal (r
5 for short). The mathematical formulation of our proposed model is indicated below. All notations of the proposed model are listed in the appendix.


## 3.2. Manufacturing Chain

According to the aforementioned assumptions, objective optimization models are formulated to seek the maximum net profit of the general supply chain and the reverse supply chain.MP and RP are measured by subtracting the corresponding aggregate costs from the respective aggregate revenues and costs, as expressed, respectively, in (1) and (2) as follows:
(1)
Max
⁡
MP
=
MTR
-
MC
=
MTR
-
(
MPC
+
MMC
+
MIC
+
MTC
+
MRC
+
MLC
)
.As in (1), the total cost of manufacturing chain consists of six parts: raw material procurement, manufacturing, inventory, transportation, recycling fees paid to the EPA, and labor cost.


## 3.3. Reverse Chain

Consider the following(2)
Max
⁡
RP
=
RTR
-
RC
=
(
RR
+
RS
)
-
(
RTCC
+
RTTC
+
RIC
+
RTC
+
RFC
+
RLC
)
.Similarly, in (2), the total cost of the reverse chain is composed of six parts: the cost of collecting used products, transitional treatment, inventory, transportation, final disposal, and labor cost. Besides, there are two sources of revenue in the reverse chain: the general revenue and the subsidies from EPA.From the proposed integrating logistics system architecture as shown in Figure1, the objective functions are composed of (
1
) manufacturing chain-based net profit (MP) maximization and (
2
) reverse chain-based net profit (RP) maximization. And our target is to maximize the total revenue in the supply chain system as shown below:
(3)
Max
⁡
Total
Revenue
=
MP
+
RP
=
(
MTR
-
MC
)
+
(
RTR
-
RC
)
.


## 3.4. Revenues

MTR = Profit oriented from the raw material flows + Profit oriented from the physical flows of the manufactured product in any given distribution channel of the manufacturing chain:(4)
∑
∀
t
{
[
∑
∀
m
1
∑
∀
m
2
R
m
1
,
m
2
(
t
)
×
Q
m
1
,
m
2
(
t
)
]
[
∑
i
=
2
4
∑
i
+
1
5
∑
∀
m
i
∑
∀
m
j
R
m
i
,
m
+
1
(
t
)
×
Q
m
i
,
m
+
1
(
t
)
]
h
h
h
+
[
∑
i
=
2
4
∑
i
+
1
5
∑
∀
m
i
∑
∀
m
j
R
m
i
,
m
+
1
(
t
)
×
Q
m
i
,
m
+
1
(
t
)
]
}
.RR = Refund obtained by end customer for returning the used product + Revenue associated with the used-product collection for selling the collected unprocessed used product to other reverse chains’ members + Revenue from selling the processed used product to another member in the same reverse chain + Revenue of the secondary material market for selling the processed used raw materials to the layer of manufacturing of the given manufacturing chain:(5)
∑
∀
t
{
[
∑
j
=
1
3
∑
∀
m
5
∑
∀
r
j
R
m
5
,
r
j
(
t
)
×
Q
m
5
,
r
j
(
t
)
]
h
h
h
h
+
[
∑
j
=
2
3
∑
∀
r
1
∑
∀
r
j
R
r
1
,
r
j
(
t
)
×
Q
r
1
,
r
j
(
t
)
]
h
h
h
h
+
[
∑
j
=
2
3
∑
∀
r
j
∑
∀
r
j
+
1
R
r
j
,
r
j
+
1
(
t
)
×
Q
r
j
,
r
j
+
1
(
t
)
]
h
h
h
h
[
∑
j
=
1
3
∑
∀
m
5
∑
∀
r
j
R
m
5
,
r
j
(
t
)
×
Q
m
5
,
r
j
(
t
)
]
+
[
∑
∀
r
4
∑
∀
m
2
R
r
4
,
m
2
(
t
)
×
Q
r
4
,
m
2
(
t
)
]
}
.RS = Subsidies associated with the reverse chains that are oriented from the reverse flows of the returned used products transported to the layer of disassembly plants for government subsidies:(6)
∑
∀
t
{
∑
i
=
1
2
∑
∀
r
i
∑
∀
r
3
EPO
×
Q
r
i
,
r
3
(
t
)
}
.As we mentioned in the earlier section, the revenues from manufacturing chain were composed of raw material and physical flows. And the revenues from the reverse chain were constituted of the regular returned used-product flows and subsidies.


## 3.5. Costs of Manufacturing Chain

The costs of manufacturing chain are composed of procurement, manufacturing, inventory, transportation, recycle, and labor cost. In this part, the detailed discussions of manufacturing chain are presented as follows.MPC = Initialized cost of raw materials + Procurement cost oriented from raw material suppliers and secondary material market + Manufactured-product procurement costs in any distribution channels of manufacturing chain:(7)
∑
∀
t
{
[
∑
∀
m
1
C
R
(
t
)
×
Q
R
(
t
)
]
[
∑
i
=
2
3
∑
i
+
1
4
∑
∀
m
i
∑
∀
m
j
C
m
i
,
m
i
+
1
P
(
t
)
×
Q
m
i
,
m
i
+
1
(
t
)
]
h
h
h
+
[
∑
∀
m
1
∑
∀
m
2
C
m
1
,
m
2
P
(
t
)
×
Q
m
1
,
m
2
(
t
)
h
h
h
h
h
h
+
∑
∀
r
4
∑
∀
m
2
C
r
4
,
m
2
P
(
t
)
×
Q
r
4
,
m
2
(
t
)
]
h
h
h
+
[
∑
i
=
2
3
∑
i
+
1
4
∑
∀
m
i
∑
∀
m
j
C
m
i
,
m
i
+
1
P
(
t
)
×
Q
m
i
,
m
i
+
1
(
t
)
]
}
.MMC = Manufacturing cost of all manufactured products:(8)
∑
∀
t
{
∑
∀
m
2
C
M
(
t
)
×
Q
M
(
t
)
}
.MIC = Inventory cost of raw materials oriented from raw material suppliers and manufactures + Inventory cost of products in any chain member of the manufacturing chain:(9)
∑
∀
t
{
[
∑
i
=
1
2
∑
∀
m
i
C
m
i
IR
(
t
)
×
Q
m
i
IR
(
t
)
]
+
[
∑
i
=
2
4
∑
∀
m
i
C
m
i
I
(
t
)
×
Q
m
i
I
(
t
)
]
}
.MTC
≅ Transportation cost of raw materials transported from suppliers to manufactures + Transportation cost of products transported in any chain member of the manufacturing chain:
(10)
∑
∀
t
{
[
∑
∀
m
1
∑
∀
m
2
C
T
~
m
1
,
m
2
(
t
)
×
Q
m
1
,
m
2
(
t
)
]
+
[
∑
i
=
2
4
∑
i
+
1
5
∑
∀
m
i
∑
∀
m
j
C
T
~
m
i
,
m
i
+
1
(
t
)
×
Q
m
i
,
m
i
+
1
(
t
)
]
h
h
h
+
[
∑
i
=
2
4
∑
i
+
1
5
∑
∀
m
i
∑
∀
m
j
C
T
~
m
i
,
m
i
+
1
(
t
)
×
Q
m
i
,
m
i
+
1
(
t
)
]
}
.MRC = Recycling fees of the products:(11)
∑
∀
t
∑
∀
m
2
F
×
Q
M
(
t
)
.MLC = Labor cost of any member in the manufacturing chain:(12)
∑
∀
t
{
∑
i
=
1
4
∑
∀
m
i
C
m
i
L
×
Q
m
i
L
(
t
)
}
.


## 3.6. Costs of Reverse Chain

The costs of the reverse chain are composed of collected returned used products, transitional treatment procedures, inventory, transportation, final disposal, and labor cost. And we will discuss the reverse chain in this section.RTCC = The amount of used products collected from the end customer to the members of the reverse chain:
(13)
∑
∀
t
{
∑
j
=
1
3
∑
∀
m
5
∑
∀
r
j
C
m
5
,
r
j
C
(
t
)
×
Q
m
5
,
r
j
(
t
)
}
.RTTC = The transitional treatment procedures executed potentially in all the reverse chain layers, except the final disposal:
(14)
∑
∀
t
{
∑
j
=
1
4
∑
∀
r
j
C
r
j
TT
(
t
)
×
Q
r
j
TT
(
t
)
}
.RIC = The inventory cost of unprocessed used product + Inventory cost of processed used product + Inventory cost of processed used product that may be stored in disassembly plant which is for final disposal and selling to the secondary market:
(15)
∑
∀
t
{
[
∑
j
=
1
4
∑
∀
r
j
C
r
j
IT
-
(
t
)
×
Q
r
j
IT
-
(
t
)
]
h
h
h
+
[
∑
j
=
1
,
i
≠
3
4
∑
∀
r
j
C
r
j
IT
(
t
)
×
C
r
j
IT
(
t
)
]
h
h
h
[
∑
j
=
1
4
∑
∀
r
j
C
r
j
I
T
-
(
t
)
×
Q
r
j
I
T
-
(
t
)
]
+
∑
r
3
[
C
r
3
,
r
4
I
(
t
)
×
Q
r
3
,
r
4
I
(
t
)
+
C
r
3
,
r
5
I
(
t
)
×
Q
r
3
,
r
5
I
(
t
)
]
}
.RTC
≅ The summation of transporting costs of any distribution channels of the reverse chain, excluding the collection costs that are oriented from the reverse flows associated with end customers:
(16)
∑
∀
t
{
[
∑
j
=
2
3
∑
r
1
∑
∀
r
j
C
T
~
r
1
,
r
j
(
t
)
×
Q
r
1
,
r
j
(
t
)
]
h
h
h
+
[
∑
i
=
2
3
∑
r
i
∑
∀
r
i
+
1
C
T
~
r
i
,
r
i
+
1
(
t
)
×
Q
r
i
,
r
i
+
1
(
t
)
]
h
h
h
+
[
∑
∀
r
3
∑
∀
r
5
C
T
~
r
3
,
r
5
(
t
)
×
Q
r
3
,
r
5
(
t
)
]
h
h
h
[
∑
j
=
2
3
∑
r
1
∑
∀
r
j
C
T
~
r
1
,
r
j
(
t
)
×
Q
r
1
,
r
j
(
t
)
]
+
[
∑
∀
r
4
∑
∀
m
2
C
T
~
r
4
,
m
2
(
t
)
×
Q
r
4
,
m
2
(
t
)
]
}
.RFC = Total amount of used products disposed in the layer of final disposal:
(17)
∑
∀
t
{
∑
∀
r
5
C
F
(
t
)
×
Q
F
(
t
)
}
.RLC = Labor cost of any member in the reverse chain:
(18)
∑
∀
t
{
∑
r
=
1
5
∑
∀
r
j
C
r
j
L
×
Q
r
j
L
(
t
)
}
.


## 3.7. Constraints

In this part, the detailed discussions of three constraints such as inventory constraints, demand constraints, and return resource constraints are presented as follows.(i) Inventory Constraints. In order to reach the reality, we set safety stock for every member in the two chains. (a)
Forraw material suppliers,
(19)
S
S
m
1
IR
≤
Q
m
1
IR
(
t
)
=
Q
m
1
IR
(
t
-
1
)
+
Q
R
(
t
)
-
∑
∀
m
2
Q
m
1
,
m
2
(
t
)
≤
S
m
1
IR
∀
(
m
1
,
t
)
.
(b)
Forproduct manufacturers,
(1)
for raw materials,(20)
S
S
m
2
IR
≤
Q
m
2
IR
(
t
)
=
Q
m
2
IR
(
t
-
1
)
+
[
∑
∀
m
1
Q
m
1
,
m
2
(
t
)
+
∑
∀
r
4
Q
r
4
,
m
2
(
t
)
]
-
γ
m
2
r
/
m
×
Q
M
(
t
)
≤
S
m
2
IR
∀
(
m
2
,
t
)
,
(2)
for products,(21)
S
S
m
2
I
≤
Q
m
2
I
(
t
)
=
Q
m
2
I
(
t
-
1
)
+
Q
M
(
t
)
-
∑
j
=
3
5
∑
∀
m
j
Q
m
2
,
m
j
(
t
)
≤
S
m
2
I
∀
(
m
2
,
t
)
.
(c)
Forwholesalers and retailers,
(22)
S
S
m
j
I
≤
Q
m
j
I
(
t
)
=
Q
m
j
I
(
t
-
1
)
+
[
∑
i
=
2
j
-
1
∑
∀
m
i
Q
m
i
,
m
j
(
t
)
]
-
[
∑
n
=
j
+
1
5
∑
∀
m
n
Q
m
j
,
m
n
(
t
)
]
≤
S
m
j
I
∀
(
m
j
,
t
)
,
j
=
3
or
4
.
(d)
Forcollecting points,
(23)
S
S
r
1
IT
¯
≤
Q
r
1
IT
¯
(
t
)
=
Q
r
1
IT
¯
(
t
-
1
)
+
[
∑
∀
m
5
Q
m
5
,
r
1
(
t
)
]
-
Q
r
1
TT
(
t
)
≤
S
r
1
IT
¯
∀
(
r
1
,
t
)
,
S
S
r
1
IT
≤
Q
r
1
IT
(
t
)
=
Q
r
1
IT
(
t
-
1
)
+
Q
r
1
TT
(
t
)
-
∑
j
=
2
3
∑
∀
r
j
Q
r
1
,
r
j
(
t
)
≤
S
r
1
IT
∀
(
r
1
,
t
)
.
(e)
Forrecycle plants,
(24)
S
S
r
2
IT
¯
≤
Q
r
2
IT
¯
(
t
)
=
Q
r
2
IT
¯
(
t
-
1
)
+
[
∑
∀
m
5
Q
m
5
,
r
2
(
t
)
+
∑
∀
r
1
Q
r
1
,
r
2
(
t
)
]
-
Q
r
2
TT
(
t
)
≤
S
r
2
IT
¯
∀
(
r
2
,
t
)
,
S
S
r
2
IT
≤
Q
r
2
IT
(
t
)
=
Q
r
2
IT
(
t
-
1
)
+
Q
r
2
TT
(
t
)
-
∑
∀
r
3
Q
r
2
,
r
3
(
t
)
≤
S
r
2
IT
∀
(
r
2
,
t
)
.
(f)
Fordisassembly plants,
(25)
S
S
r
3
IT
¯
≤
Q
r
3
IT
¯
(
t
)
=
Q
r
3
IT
¯
(
t
-
1
)
+
[
∑
∀
m
5
Q
m
5
,
r
3
(
t
)
+
∑
i
=
1
2
∑
∀
r
i
Q
r
i
,
r
3
(
t
)
]
-
Q
r
3
TT
(
t
)
≤
S
r
3
IT
¯
∀
(
r
3
,
t
)
,
S
S
r
3
,
r
4
IT
≤
Q
r
3
,
r
4
I
(
t
)
=
Q
r
3
,
r
4
I
(
t
-
1
)
+
γ
r
3
,
r
4
×
Q
r
3
TT
(
t
)
-
∑
∀
r
4
Q
r
3
,
r
4
(
t
)
≤
S
r
3
,
r
4
I
∀
(
r
3
,
t
)
,
S
S
r
3
,
r
5
I
≤
Q
r
3
,
r
5
I
(
t
)
=
Q
r
3
,
r
5
I
(
t
-
1
)
+
γ
r
3
,
r
5
×
Q
r
3
TT
(
t
)
-
∑
∀
r
5
Q
r
3
,
r
5
(
t
)
≤
S
r
3
,
r
5
I
∀
(
r
3
,
t
)
.
(g)
Forsecondary material markets,
(26)
S
S
r
4
IT
¯
≤
Q
r
4
IT
¯
(
t
)
=
Q
r
4
IT
¯
(
t
-
1
)
+
[
∑
∀
r
3
Q
r
3
,
r
4
(
t
)
]
-
Q
r
4
TT
(
t
)
≤
S
r
4
IT
¯
∀
(
r
4
,
t
)
,
S
S
r
4
IT
≤
Q
r
4
IT
(
t
)
=
Q
r
4
IT
(
t
-
1
)
+
γ
t
4
t
r
e
×
Q
t
r
r
4
(
t
)
-
∑
∀
m
2
Q
r
4
,
m
2
(
t
)
≤
S
r
4
IT
∀
(
r
4
,
t
)
.
(h)
Forfinal disposal locations,
(27)
S
S
r
5
IT
¯
≤
Q
r
5
IT
¯
(
t
)
=
Q
r
5
IT
¯
(
t
-
1
)
+
[
∑
∀
r
3
Q
r
3
,
r
5
(
t
)
]
-
Q
r
5
F
(
t
)
≤
S
r
5
IT
¯
∀
(
r
5
,
t
)
.(ii) Demand Constraints. Consider the following
(28)
D
(
t
)
~
≥
∑
i
=
2
4
∑
∀
m
i
∑
∀
m
5
Q
m
i
,
m
5
(
t
)
≥
0
∀
t
.(iii) Return Resource Constraints. Consider the following
(29)
Q
RE
(
t
)
=
∑
j
=
1
3
∑
∀
m
5
∑
∀
r
j
Q
m
5
,
r
j
(
t
)
=
β
×
D
(
t
)
~
≥
0
∀
t
.


## 4. Treatment of the Fuzzy Formulation

In practice, Liang [30] noted that decision-makers are more familiar with estimating optimistic, pessimistic, and most likely parameters. The pattern of triangular distribution is commonly adopted due to ease in defining the maximum and minimum limit of deviation of the fuzzy number of its central value. Furthermore, according to the definition of Rommelfanger [31], when knowledge of the distribution is limited, triangular distribution is appropriate for representing a fuzzy number. And then, other studies also indicated that the primary advantages of the triangular fuzzy number are the simplicity and flexibility of the fuzzy arithmetic operations [31–35].In order to discuss the fuzziness of the integrated logistics model for the demand and transportation cost, there are some definitions stated as follows [34].Definition 1.
The fuzzy setB
~
=
(
a
,
b
,
c
), where a
<
b
<
c is defined on R, is called thetriangular fuzzy number, if the membership function of B
~ is given by
(30)
μ
B
~
(
x
)
=
{
(
x
-
a
)
(
b
-
a
)
,
a
≤
x
≤
b
,
(
c
-
x
)
(
c
-
b
)
,
b
≤
x
≤
c
,
0
,
otherwise
.Definition 2.
The fuzzy set[
a
α
,
b
α
] defined on R, 0
≤
α
≤
1, is called an α-level fuzzy interval if the membership function of [
a
α
,
b
α
] is given by
(31)
μ
[
a
α
,
b
α
]
(
x
)
=
{
α
,
a
≤
x
≤
b
,
0
,
otherwise
.Definition 3.
LetB
~ be a fuzzy set on R, and 0
≤
α
≤
1; then the α-cut, B
(
α
), of B
~ consists of pointsx such that μ
B
~
(
x
)
≥
α; that is, B
(
α
)
=
{
x
∣
μ
B
~
(
x
)
≥
α
}.With (10) and (15), if the minimum acceptable membership level α is given, the corresponding auxiliary crisp inequality expression of these two equations can be presented as follows:
(32)
MTC
≅
∑
∀
t
{
[
w
1
∑
∀
m
1
∑
∀
m
2
C
t
~
m
1
,
m
2
p
,
α
(
t
)
×
Q
m
1
,
m
2
(
t
)
w
3
∑
i
=
2
4
∑
j
=
i
+
1
5
∑
∀
m
i
∑
∀
m
j
C
t
~
m
i
,
m
j
o
,
α
(
t
)
×
Q
m
i
,
m
j
(
t
)
h
h
h
h
h
+
w
2
∑
∀
m
1
∑
∀
m
2
C
t
~
m
1
,
m
2
m
,
α
(
t
)
×
Q
m
1
,
m
2
(
t
)
h
h
h
h
h
+
w
3
∑
∀
m
1
∑
∀
m
2
C
t
~
m
1
,
m
2
o
,
α
(
t
)
×
Q
m
1
,
m
2
(
t
)
]
h
h
h
+
[
w
1
∑
i
=
2
4
∑
j
=
i
+
1
5
∑
∀
m
i
∑
∀
m
j
C
t
~
m
i
,
m
j
p
,
α
(
t
)
×
Q
m
i
,
m
j
(
t
)
h
h
h
h
h
h
+
w
2
∑
i
=
2
4
∑
j
=
i
+
1
5
∑
∀
m
i
∑
∀
m
j
C
t
~
m
i
,
m
j
m
,
α
(
t
)
×
Q
m
i
,
m
j
(
t
)
h
h
h
h
h
h
+
w
3
∑
i
=
2
4
∑
j
=
i
+
1
5
∑
∀
m
i
∑
∀
m
j
C
t
~
m
i
,
m
j
o
,
α
(
t
)
Q
m
i
,
m
j
(
t
)
]
}
,
RTC
≅
∑
∀
t
{
[
w
1
∑
j
=
2
3
∑
r
1
∑
∀
r
j
C
t
~
m
i
,
m
j
p
,
α
(
t
)
×
Q
r
1
,
r
j
(
t
)
h
h
h
h
h
+
w
2
∑
j
=
2
3
∑
r
1
∑
∀
r
j
C
t
~
m
i
,
m
j
m
,
α
(
t
)
×
Q
r
1
,
r
j
(
t
)
h
h
h
h
h
+
w
3
∑
j
=
2
3
∑
r
1
∑
∀
r
j
C
t
~
m
i
,
m
j
o
,
α
(
t
)
×
Q
r
1
,
r
j
(
t
)
]
h
h
h
h
+
[
w
1
∑
i
=
2
3
∑
r
i
∑
∀
r
i
+
1
C
t
~
r
i
,
r
i
+
1
p
,
α
(
t
)
×
Q
r
i
,
r
i
+
1
(
t
)
h
h
h
h
h
h
h
+
w
2
∑
i
=
2
3
∑
r
i
∑
∀
r
i
+
1
C
t
~
r
i
,
r
i
+
1
m
,
α
(
t
)
×
Q
r
i
,
r
i
+
1
(
t
)
h
h
h
h
h
h
h
+
w
3
∑
i
=
2
3
∑
r
i
∑
∀
r
i
+
1
C
t
~
r
i
,
r
i
+
1
o
,
α
(
t
)
×
Q
r
i
,
r
i
+
1
(
t
)
]
h
h
h
h
+
[
w
1
∑
∀
r
3
∑
∀
r
5
C
t
~
r
3
,
r
5
p
,
α
(
t
)
×
Q
r
3
,
r
5
(
t
)
h
h
h
h
h
h
h
+
w
2
∑
∀
r
3
∑
∀
r
5
C
t
~
r
3
,
r
5
m
,
α
(
t
)
×
Q
r
3
,
r
5
(
t
)
h
h
h
h
h
h
h
+
w
3
∑
∀
r
3
∑
∀
r
5
C
t
~
r
3
,
r
5
o
,
α
(
t
)
×
Q
r
3
,
r
5
(
t
)
]
h
h
h
h
+
[
w
1
∑
∀
r
4
∑
∀
m
2
C
t
~
r
4
,
m
2
p
,
α
(
t
)
×
Q
r
4
,
m
2
(
t
)
h
h
h
h
h
h
h
+
w
2
∑
∀
r
4
∑
∀
m
2
C
t
~
r
4
,
m
2
m
,
α
(
t
)
×
Q
r
4
,
m
2
(
t
)
h
h
h
h
h
h
h
w
1
∑
j
=
2
3
∑
r
1
∑
∀
r
j
C
t
~
m
i
,
m
j
p
,
α
(
t
)
×
Q
r
1
,
r
j
(
t
)
+
w
3
∑
∀
r
4
∑
∀
m
2
C
t
~
r
4
,
m
2
o
,
α
(
t
)
×
Q
r
4
,
m
2
(
t
)
]
}
.Similarly, if the minimum acceptable membership levelα is given, the constraint of demand is presented as follows:
(33)
w
1
D
(
t
)
~
p
,
α
+
w
2
D
(
t
)
~
m
,
α
+
w
3
D
(
t
)
~
o
,
α
≥
∑
i
=
2
4
∑
∀
m
i
∑
∀
m
5
Q
m
i
,
m
5
(
t
)
≥
0
∀
t
.The weights ofw
1, w
2, and w
3
(
w
1
+
w
2
+
w
3
=
1
) must represent the weight of the most pessimistic, most likely, and most optimistic values of the fuzzy variables. We set w
2
=
4
/
6, w
1
=
w
3
=
1
/
6, and α
=
0.5 for all formulations. However, in reality, the value α and the relative weightings among three critical points can be adjusted subjectively based on the experience and knowledge of decision-makers and/or experts. The most likely values are used here because they are generally the most important ones and, thus, should be assigned greater weights [30]. However, the most pessimistic and most optimistic values which provided the boundary solutions of fuzzy market demand for each destination are too pessimistic and optimistic, respectively, and, thus, should be assigned smaller weights [35].For convenience, we will use fuzzy summation calculation (FSC) to integrate all the experts’ optimistic, most likely and pessimistic estimation values into one value, respectively. IfA is a parameter and A
p, A
m, and A
o represent the fuzzy number of A’s optimistic, most likely and pessimistic estimation values. The formulation of integration is given as follows [36, 37]:
(34)
A
p
=
min
⁡
{
A
k
p
}
,
A
m
=
min
⁡
{
A
k
m
}
,
A
o
=
max
⁡
{
A
k
o
}
.


### 4.1. Solving Procedure

Step 1.
Based on (1)~(27), we formulate the original fuzzy multiobjective linear programming model for the integrated logistic problems.Step 2.
Based on (32) and (33), we provide the acceptable minimum membership level α and then convert the fuzzy inequality constraints into crisp ones using the weighted average method.Step 3.
This step is to integrate all optimistic, most likely and pessimistic estimation values into one value, respectively, according to (34).Step 4.
Final step is to solve the ordinary LP problem. If the DM is dissatisfied with the initial solutions, the model should be adjusted until a preferred satisfactory solution is obtained.


## 4.1. Solving Procedure

Step 1.
Based on (1)~(27), we formulate the original fuzzy multiobjective linear programming model for the integrated logistic problems.Step 2.
Based on (32) and (33), we provide the acceptable minimum membership level α and then convert the fuzzy inequality constraints into crisp ones using the weighted average method.Step 3.
This step is to integrate all optimistic, most likely and pessimistic estimation values into one value, respectively, according to (34).Step 4.
Final step is to solve the ordinary LP problem. If the DM is dissatisfied with the initial solutions, the model should be adjusted until a preferred satisfactory solution is obtained.


## 5. Numerical Result

To illustrate the effectiveness of the models presented above, we examine a simplified numerical study. Sheu et al. [29] have conducted a local operational case of a well-known Taiwanese notebook computer manufacturer which is one of the top three domestic brands in Taiwan. This case study is based on the logistics distribution channels and built a simplified integrated logistics network of the given notebook manufacturers in the northern region of Taiwan. The collected historical and interview survey data were used to estimate both the input data such as annual sales and used-product returns and primary parameters like logistics-induced operational costs for use in formulating the integrated logistics management problem. With fuzzy formulations, the decision-maker can consider the environment under variable demand and transportation cost which are more close to the real. Details of the primary procedures in the numerical study and corresponding results are presented below.


### 5.1. Parameter Settings

Based on Sheu et al. [29], the parameters will be used as the average of Sheu’s data as shown in Tables 1, 2, 3, and 4.Table 1
Estimates of unit revenues.
Layer
Unit revenue (US$)
Manufacturing chain
(1) Raw material supplier
42
(2) Manufacturer
485
(3) Wholesaler
673
(4) Retailer
873
(5) End customer
2.5
Reverse chain
(1) Collecting point
5.55
(2) Recycle plant
7.8
(3) Disassembly plant
10.85
(4) Secondary material market
22.5
Subsidy
8.7Table 2
Estimates of manufacturing chain unit costs.
Layer
Parameter
Unit cost (US$)
(1) Raw material supplier
C
R
19.5
C
m
1
IR
1.75
C
m
1
L
5
(2) Manufacturer
C
m
1
,
m
2
P
24
C
r
4
,
m
2
P
22.5
C
M
115
C
m
2
I
69.5
C
m
2
IR
50
C
m
2
L
5
(3) Wholesaler
C
m
2
,
m
3
P
531.5
C
m
3
I
86.5
C
m
3
L
5
(4) Retailer
C
m
3
,
m
4
P
621.5
C
m
4
I
101.5
C
m
4
L
5Table 3
Estimates of used-product reverse chain unit costs.
Layer
Parameter
Unit cost (US$)
(1) Collecting point
C
m
5
,
r
1
C
1
C
TT
6.5
C
IT
-
0.45
C
IT
0.45
C
r
1
L
3.5
(2) Recycle plant
C
m
5
,
r
2
C
1.85
C
TT
2.85
C
IT
-
0.45
C
IT
0.45
C
r
2
L
3.5
(3) Disassembly plant
C
m
5
,
r
3
C
2.15
C
TT
1.75
C
IT
-
0.45
C
r
3
,
r
4
I
1.8
C
r
3
,
r
5
I
0.23
C
r
3
L
3.5
(4) Secondary material market
C
r
3
,
r
4
C
4.4
C
TT
2.05
C
IT
-
1.05
C
IT
1.6
C
r
4
L
3.5Table 4
Inventory capacity.
(a)
Manufacturing chain
S
m
1
IR
S
m
2
IR
S
m
2
I
S
m
3
I
S
m
4
I
7000
5000
5000
500
100
S
S
m
1
IR
S
S
m
2
IR
S
S
m
2
I
S
S
m
3
I
S
S
m
4
I
3500
2500
2500
250
50
(b)
Reverse chain
S
r
1
IT
-
S
r
1
IT
S
r
2
IT
-
S
r
2
IT
S
r
3
IT
-
S
r
3
,
r
4
I
S
r
3
,
r
5
I
S
r
4
IT
-
S
r
4
IT
S
r
5
IT
-
500
500
1000
1000
2500
1500
500
1000
800
500
S
S
r
1
IT
-
S
S
r
1
IT
S
S
r
2
IT
-
S
S
r
2
IT
S
S
r
3
IT
-
S
S
r
3
,
r
4
I
S
S
r
3
,
r
5
I
S
S
r
4
IT
-
S
S
r
4
IT
S
S
r
5
IT
-
250
250
500
500
1200
750
250
500
400
250Because demand and transportation cost were fuzzy numbers, these were conducted with high-level decision-makers of the targeted notebook manufacturing enterprise and its potential channel members, including the chain members in both the manufacturing and the reverse chains. These parameters will be estimated as optimistic, pessimistic, and most likely numbers and will be integrated by using the center of gravity method.We invited five experts to estimate the demand and transportation cost, while the last column is the data after integrating into one number through fuzzy summation calculation [36, 37]. In addition, Table 5 shows the other parameters of the model.Table 5
Other parameters.
Return ratio (β)
0.25
Unit recycle fee (F)
1.1
Unit subsidy (S)
10
Used-product return
1538In convenience, the initial inventory condition of each member in the manufacturing and the reverse chains was generated here using a respective uniform distribution bounded by the range between 0 and the corresponding inventory capacity estimated.


### 5.2. Results

According to the corresponding regulations of the Taiwan EPA for 2005, the unit subsidy is set to be 10 (US$). The numerical results are summarized in Table6.Table 6
Numerical result.
Net profit (US$)
Manufacturing chain
5,399,846
Reverse chain
96,308
Aggregate
5,454,023Table6 indicates the maximum profit of the manufacturing chain and the reverse chain. When we aggregate manufacturing and reverse chains together, there will be some factors influencing each other, so the maximum aggregate profit is not the combination of the single profit of each chain.Accordingly, manufacturers can be convinced more easily to coordinate all the chain members for the promotion of GSCM. In addition, subsidy strategies also have a strong impact on reverse logistics chain system. Due to the above mentions, we take the unit subsidy into further investigation. In practice, the unit subsidy may rely on the corresponding environmental protection regulations imposed by the EPA [29].Figure2 shows the sensitivity analysis of the unit subsidy parameter to investigate the potential effects of these parameters on the performance of the reverse logistics chain.Figure 2
Numerical result of sensitivity analysis.In Figure2, when the governmental subsidy value increased, the profit of the reverse chain also increased. This indicates that the governmental subsidy policy remains as a critical determinant in influencing the performance of used-product reverse chain.


## 5.1. Parameter Settings

Based on Sheu et al. [29], the parameters will be used as the average of Sheu’s data as shown in Tables 1, 2, 3, and 4.Table 1
Estimates of unit revenues.
Layer
Unit revenue (US$)
Manufacturing chain
(1) Raw material supplier
42
(2) Manufacturer
485
(3) Wholesaler
673
(4) Retailer
873
(5) End customer
2.5
Reverse chain
(1) Collecting point
5.55
(2) Recycle plant
7.8
(3) Disassembly plant
10.85
(4) Secondary material market
22.5
Subsidy
8.7Table 2
Estimates of manufacturing chain unit costs.
Layer
Parameter
Unit cost (US$)
(1) Raw material supplier
C
R
19.5
C
m
1
IR
1.75
C
m
1
L
5
(2) Manufacturer
C
m
1
,
m
2
P
24
C
r
4
,
m
2
P
22.5
C
M
115
C
m
2
I
69.5
C
m
2
IR
50
C
m
2
L
5
(3) Wholesaler
C
m
2
,
m
3
P
531.5
C
m
3
I
86.5
C
m
3
L
5
(4) Retailer
C
m
3
,
m
4
P
621.5
C
m
4
I
101.5
C
m
4
L
5Table 3
Estimates of used-product reverse chain unit costs.
Layer
Parameter
Unit cost (US$)
(1) Collecting point
C
m
5
,
r
1
C
1
C
TT
6.5
C
IT
-
0.45
C
IT
0.45
C
r
1
L
3.5
(2) Recycle plant
C
m
5
,
r
2
C
1.85
C
TT
2.85
C
IT
-
0.45
C
IT
0.45
C
r
2
L
3.5
(3) Disassembly plant
C
m
5
,
r
3
C
2.15
C
TT
1.75
C
IT
-
0.45
C
r
3
,
r
4
I
1.8
C
r
3
,
r
5
I
0.23
C
r
3
L
3.5
(4) Secondary material market
C
r
3
,
r
4
C
4.4
C
TT
2.05
C
IT
-
1.05
C
IT
1.6
C
r
4
L
3.5Table 4
Inventory capacity.
(a)
Manufacturing chain
S
m
1
IR
S
m
2
IR
S
m
2
I
S
m
3
I
S
m
4
I
7000
5000
5000
500
100
S
S
m
1
IR
S
S
m
2
IR
S
S
m
2
I
S
S
m
3
I
S
S
m
4
I
3500
2500
2500
250
50
(b)
Reverse chain
S
r
1
IT
-
S
r
1
IT
S
r
2
IT
-
S
r
2
IT
S
r
3
IT
-
S
r
3
,
r
4
I
S
r
3
,
r
5
I
S
r
4
IT
-
S
r
4
IT
S
r
5
IT
-
500
500
1000
1000
2500
1500
500
1000
800
500
S
S
r
1
IT
-
S
S
r
1
IT
S
S
r
2
IT
-
S
S
r
2
IT
S
S
r
3
IT
-
S
S
r
3
,
r
4
I
S
S
r
3
,
r
5
I
S
S
r
4
IT
-
S
S
r
4
IT
S
S
r
5
IT
-
250
250
500
500
1200
750
250
500
400
250Because demand and transportation cost were fuzzy numbers, these were conducted with high-level decision-makers of the targeted notebook manufacturing enterprise and its potential channel members, including the chain members in both the manufacturing and the reverse chains. These parameters will be estimated as optimistic, pessimistic, and most likely numbers and will be integrated by using the center of gravity method.We invited five experts to estimate the demand and transportation cost, while the last column is the data after integrating into one number through fuzzy summation calculation [36, 37]. In addition, Table 5 shows the other parameters of the model.Table 5
Other parameters.
Return ratio (β)
0.25
Unit recycle fee (F)
1.1
Unit subsidy (S)
10
Used-product return
1538In convenience, the initial inventory condition of each member in the manufacturing and the reverse chains was generated here using a respective uniform distribution bounded by the range between 0 and the corresponding inventory capacity estimated.


## 5.2. Results

According to the corresponding regulations of the Taiwan EPA for 2005, the unit subsidy is set to be 10 (US$). The numerical results are summarized in Table6.Table 6
Numerical result.
Net profit (US$)
Manufacturing chain
5,399,846
Reverse chain
96,308
Aggregate
5,454,023Table6 indicates the maximum profit of the manufacturing chain and the reverse chain. When we aggregate manufacturing and reverse chains together, there will be some factors influencing each other, so the maximum aggregate profit is not the combination of the single profit of each chain.Accordingly, manufacturers can be convinced more easily to coordinate all the chain members for the promotion of GSCM. In addition, subsidy strategies also have a strong impact on reverse logistics chain system. Due to the above mentions, we take the unit subsidy into further investigation. In practice, the unit subsidy may rely on the corresponding environmental protection regulations imposed by the EPA [29].Figure2 shows the sensitivity analysis of the unit subsidy parameter to investigate the potential effects of these parameters on the performance of the reverse logistics chain.Figure 2
Numerical result of sensitivity analysis.In Figure2, when the governmental subsidy value increased, the profit of the reverse chain also increased. This indicates that the governmental subsidy policy remains as a critical determinant in influencing the performance of used-product reverse chain.


## 6. Conclusion

With the encouragement of environmental consciousness recently, the issue of environmental protection has been becoming a trend of supply chain management. In addition, the enterprises also have to take more applicable measures to improve the reverse logistics. Therefore, we formulated a fuzzy multiobjective integrated logistics model which coordinated the cross-functional product logistics flows and used-product reverse logistics flows with green supply chain consideration.The findings of the numerical result indicated that the maximum profit is $5,399,846 in the manufacturing chain and $96,308 in the reverse chain. And the results showed that the maximum profit was up to $5,454,023 when we aggregated these two chains together. In addition, we found that when the governmental subsidy value increased, the profit of the reverse chain also increased. The findings of this study identified that the governmental subsidy policy could remain of significant influence for used-product reverse logistics chain.Nevertheless, this paper is not totally without merit. We are hopeful that our experimental results are of great interest for both application and scientific research. Most importantly, we will make every endeavor to cooperate with more real-world cases to acquire more perusable and actual data as our future works. Finally, the future research might usefully extend the present use of the proposed models to examine more green supply chain management aspects and solve more GSCM-related problems.



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*Source: 767095-2014-07-07.xml*