# Robust Optimization of Fourth Party Logistics Network Design under Disruptions

Authors: Jia Li; Yanqiu Liu; Ying Zhang; Zhongjun Hu
Journal: Discrete Dynamics in Nature and Society
 (2015)

---

## Abstract

The Fourth Party Logistics (4PL) network faces disruptions of various sorts under the dynamic and complex environment. In order to explore the robustness of the network, the 4PL network design with consideration of random disruptions is studied. The purpose of the research is to construct a 4PL network that can provide satisfactory service to customers at a lower cost when disruptions strike. Based on the definition ofβ-robustness, a robust optimization model of 4PL network design under disruptions is established. Based on the NP-hard characteristic of the problem, the artificial fish swarm algorithm (AFSA) and the genetic algorithm (GA) are developed. The effectiveness of the algorithms is tested and compared by simulation examples. By comparing the optimal solutions of the 4PL network for different robustness level, it is indicated that the robust optimization model can evade the market risks effectively and save the cost in the maximum limit when it is applied to 4PL network design.


---

## Body

## 1. Introduction

With the development of outsourcing and the rising expectations on service level of customers, unique and comprehensive supply chain solutions can no longer be achieved by a single third party logistics (3PL) provider. Thus, the fourth party logistics (4PL) need to be developed eagerly for the modern logistics challenges [1]. Depending on its strong capability of supply chain integration, 4PL greatly improves the efficiency and quality of logistics operations and reduces logistics costs [2].Many researchers have studied the 4PL and it has received increasing attention. A tabu search algorithm for model of integration of 4PL was developed in [3]. A conceptual model based on 4PL implementation was presented in [4]. Cui et al. [5] described the 4PL routing problem with fuzzy duration time and established a chance-constrained programming model. Many scholars research on 4PL from different angles, but how to construct a 4PL network that not only performs efficiently under normal but also performs its intended function relatively well in the presence of disruptions is the essential core issue in 4PL mode. The robust optimization can be considered as an approach for dealing with the failures of components or subsystems. Three of the most recent robust optimization approaches are introduced by Mulvey et al. [6], Ben-Tal and Nemirovski [7–9], and Bertsimas and Sim [10, 11]. Many robustness criteria have been proposed in the literature, and most have been applied to facility location or network design problems [12].The 4PL network is a complex engineered system due to its size, span, and the network flow. Many potential threats can lead disruptions, such as industrial accidents, natural disasters, and terrorist strikes [13]. Bunschuh et al. [14] add supplier sourcing constraints to a supply chain network to mitigate disruptions. Various disruptions may make network cannot work normally and pose a threat to the security of 4PL network. Also, they could cause serious operational consequences, such as higher transportation costs, inventory shortages, order delays, and loss of market shares. So, it is very significant to explore the robustness of the network and improve its anti-interference ability for 4PL.To our best knowledge, there is no research work about robust optimization of 4PL network design under uncertain conditions. However, the related problem about logistics and supply chain network optimization considering security and robustness has been studied by researchers. A review on robust supply chain network design is given by Klibi et al. [15]. A robust model for an integrated logistics network design which takes facility disruptions into consideration was proposed by Hatefi and Jolai [16]. Peng et al. [17] presented a strategic supply chain management problem to design reliable logistics networks considering robustness level. Baghalian et al. [18] developed a mathematical model for supply chain network design comprising distribution centers, several capacitated production facilities, and retailers in markets under uncertainty. Chen [19] presented an efficient algorithm to solve the problem of the optimal capacity planning in robust supply chain management. Hu et al. [20] researched the robust scheduling optimization problem of resource constrained emergency rescue. Pishvaee et al. [21] and Hasani et al. [22] considered the robust optimization approach developed by Ben-Tal and Nemirovski [7–9] to design robust closed-loop supply chain networks.The existing studies just can cope with the robust optimization of the 3PL network problems, while this paper studies robust optimization of 4PL network design under disruptions. According to the mentioned literature, we are the first to use theβ-robustness measure to design the 4PL network to ensure high-level performance during disruptions. We construct 4PL networks which can provide satisfactory service to customers at a lower cost when disruptions strike. We formulate the robust optimization model of 4PL network design which incorporates the β-robustness measure in the constraints. For the NP-hard characteristic of the problem, we will use artificial fish swarm algorithm (AFSA) and genetic algorithm (GA) to optimize and simulate the model. The effectiveness of the algorithms is tested and compared by simulation examples. The results indicate that the AFSA outperforms the classic GA. Meanwhile, they validate that the robust optimization model can evade the market risks effectively and provide optimized strategy support for the 4PL providers. Section 2 describes the formulation of the 4PL network design under disruptions. Section 3 discusses the process of AFSA to solve this problem. The experimental results and comparisons are analyzed in Section 4. Finally, Section 5 provides the conclusions and prospects to this work.


## 2. Model Formulation

### 2.1. Description of Problem

We consider a multisource to multidestination 4PL network consisting of 3PL providers, suppliers, intermediate facilities, and demanders.Figure1 is a directed graph G
(
V
,
E
) for the potential 4PL network with nine nodes. The arcs denote 3PL providers. Since there may exist several 3PL providers for the transportation between any two nodes, there may have been multiple arcs between two nodes. 4PL network design decisions include determining 3PL providers and the locations of facilities. The objective is to construct a 4PL network under disruptions, which minimize the sum of the fixed cost and the expected transportation cost subjective to the robustness level constraint.Figure 1
Potential 4PL network.For the convenience of model establishment, we make the following assumptions:(1)
The nodes are all secure; they can not be disrupted.
(2)
Each 3PL provider may be disrupted randomly and independent to each other.The variables and parameters that will be used in the model are defined as follows.Sets
V: set of nodes including suppliers, transit facilities, and demanders, V
=
O
∪
M
∪
D.
O: sets of suppliers.
M: set of potential transit facilities.
D: set of demanders.
E: set of potential arcs.
K
i
j: set of potential 3PL providers between nodes i and j.
S: set of scenarios.Parameters
a
i
j
k: the kth 3PL provider between nodes i and j, ∀
i
,
j
∈
V, ∀
k
∈
K
i
j.
F
i
j
k: the fixed cost of a
i
j
k, ∀
i
,
j
∈
V, ∀
k
∈
K
i
j.
C
i
j
k: the unit transportation cost of a
i
j
k, ∀
i
,
j
∈
V, ∀
k
∈
K
i
j.
L
i
j
k: the transportation capacity of a
i
j
k, ∀
i
,
j
∈
V, ∀
k
∈
K
i
j.
P
i
j
k: the failure rate of a
i
j
k, ∀
i
,
j
∈
V, ∀
k
∈
K
i
j.
W
d: the quantity of demand of demander d, d
∈
D.
U
o: the maximum supply quantity of supplier o, o
∈
O.
F
m
′: the fixed cost of transit facility m, m
∈
M.
C
m
′: the unit processing cost at facility m, m
∈
M.
L
m
′: the processing capacity of facility m,  m
∈
M.
λ: the unit penalty cost for not satisfying demand due to disruption.Decision Variables
x
i
j
k
=
1
if
a
i
j
k
is
selected
0
otherwise
.
y
u
=
1
if
facility
u
is
selected
0
otherwise
.
X: the vector including all decision variables x
i
j
k.
Y: the vector including all decision variables y
u.
f
i
j
k
s: the flow amount of a
i
j
k when scenario s occurs, ∀
i
,
j
∈
V, ∀
k
∈
K
i
j, ∀
s
∈
S.Assistant Variables
R
i
j
k
s
=
1
if
a
i
j
k
disrupts
under
scenario
s
0
otherwise
.


### 2.2. Robustness of the 4PL Network

The robustness of the 4PL network is the ability of maintaining operation function at a lower cost when the system faces to the external and internal management in emergencies and other uncertainties of disturbance.Definition ofβ-Robustness. For a given set S of scenarios, let β
s be a deterministic (i.e., single-scenario) optimization problem for scenario s. Let C
s
* be the optimal objective value for β
s. Let (
X
,
Y
) be a solution to β
s for all s
∈
S and let C
s
(
X
,
Y
) be the objective value of problem β
s under solution (
X
,
Y
). Then (
X
,
Y
) is called β-robust if, for all s
∈
S,
(1)
C
s
(
X
,
Y
)
-
C
s
*
C
s
*
≤
β
or equivalently,
(2)
C
s
X
,
Y
≤
1
+
β
C
s
*
,
where β
≥
0 is a given constant which denotes the robustness level. (
C
s
(
X
,
Y
)
-
C
s
*
)
/
C
s
* is the regret value for scenario s. The β-robust measure sets upper bounds on the maximum allowable regret value for each scenario.Theβ-robust measure, in which the relative regret in each scenario is required to be no more than a constant β, was first applied in a facility layout problem [23] and later in an uncapacitated network design problem [24]. Snyder and Daskin [25] adopt the term “β-robust” to describe this measure. They formulate the stochastic β-robust k-median problem and the β-robust uncapacitated fixed charge location problem.


### 2.3. Robust Optimization Model of 4PL Network Design under Disruptions

The robust model for 4PL network design under disruptions can be described as follows:(3)
min
⁡
∑
i
∈
V
∑
j
∈
V
∑
k
∈
K
i
j
x
i
j
k
F
i
j
k
+
∑
m
∈
M
y
m
F
m
′
+
∑
s
∈
S
q
s
C
s
(
X
,
Y
)
subject to
(4)
C
s
X
,
Y
≤
1
+
β
C
s
*
,
∀
s
∈
S
,
(5)
x
m
j
k
≤
y
m
∀
m
∈
M
,
∀
j
∈
V
,
∀
k
∈
K
m
j
,
(6)
x
j
m
k
≤
y
m
∀
m
∈
M
,
∀
j
∈
V
,
∀
k
∈
K
j
m
,
(7)
∑
j
∈
V
∑
k
∈
K
m
j
x
m
j
k
≤
y
m
∀
m
∈
M
,
(8)
∑
j
∈
V
∑
k
∈
K
j
m
x
j
m
k
≤
y
m
∀
m
∈
M
,
(9)
x
i
j
k
,
y
m
∈
0,1
∀
i
,
j
∈
V
,
∀
m
∈
M
,
∀
k
∈
K
i
j
.The objective function (3) minimizes the sum of the fixed cost and the expected operation cost. Constraint (4) requires the cost in each scenario cannot be more than 100
(
1
+
β
)
% of the optimal operation cost C
s
*. Constraints (5) and (6) ensure that no 3PL provider is selected when the connected facility is not selected. Constraints (7) and (8) ensure that when a facility is selected, a 3PL provider will be selected at least, which transports products into or out of the facility. Constraint (9) denotes the binary variables.The optimal operation costsC
s
* for each of the scenarios s can be calculated as follows:
(10)
C
s
*
=
min
⁡
∑
i
∈
V
∑
j
∈
V
∑
k
∈
K
i
j
f
i
j
k
s
C
i
j
k
+
∑
i
∈
V
∑
m
∈
M
∑
k
∈
K
i
m
f
i
m
k
s
C
i
j
k
′
+
λ
∑
d
∈
D
W
d
-
∑
d
∈
D
∑
i
∈
V
∑
k
∈
K
i
d
f
i
d
k
s
subject to
(11)
∑
j
∈
V
∑
k
∈
K
o
j
f
o
j
k
s
≤
U
o
∀
o
∈
O
,
s
∈
S
(12)
f
i
j
k
s
≤
(
1
-
R
i
j
k
s
)
x
i
j
k
L
i
j
k
∀
i
,
j
∈
V
,
∀
k
∈
K
i
j
(13)
∑
j
∈
V
∑
k
∈
K
m
j
f
m
j
k
s
≤
y
m
L
m
′
∀
m
∈
M
(14)
∑
j
∈
V
∑
k
∈
K
m
j
f
m
j
k
s
-
∑
j
∈
V
∑
k
∈
K
j
m
f
j
m
k
s
=
0
∀
m
∈
M
,
s
∈
S
(15)
f
i
j
k
s
≥
0
∀
i
,
j
∈
V
,
∀
k
∈
K
i
j
.The objective function (10) is to minimize the operation cost in scenario s. It is the sum of the costs of transportation and penalty costs for not satisfying the demand. Constraint (11) requires the flow out to be less than or equal to maximum supply quantity of suppliers. Constraint (12) represents capacity of the selected 3PL provider must be not less than the transportation capacityrequired by customer. Constraint (13) ensures that the total flow through a facility does not exceed its processing capacity. Equation (14) means to keep a balance of the network flow. Constraint (15) is the nonnegative and integer condition constraints.


## 2.1. Description of Problem

We consider a multisource to multidestination 4PL network consisting of 3PL providers, suppliers, intermediate facilities, and demanders.Figure1 is a directed graph G
(
V
,
E
) for the potential 4PL network with nine nodes. The arcs denote 3PL providers. Since there may exist several 3PL providers for the transportation between any two nodes, there may have been multiple arcs between two nodes. 4PL network design decisions include determining 3PL providers and the locations of facilities. The objective is to construct a 4PL network under disruptions, which minimize the sum of the fixed cost and the expected transportation cost subjective to the robustness level constraint.Figure 1
Potential 4PL network.For the convenience of model establishment, we make the following assumptions:(1)
The nodes are all secure; they can not be disrupted.
(2)
Each 3PL provider may be disrupted randomly and independent to each other.The variables and parameters that will be used in the model are defined as follows.Sets
V: set of nodes including suppliers, transit facilities, and demanders, V
=
O
∪
M
∪
D.
O: sets of suppliers.
M: set of potential transit facilities.
D: set of demanders.
E: set of potential arcs.
K
i
j: set of potential 3PL providers between nodes i and j.
S: set of scenarios.Parameters
a
i
j
k: the kth 3PL provider between nodes i and j, ∀
i
,
j
∈
V, ∀
k
∈
K
i
j.
F
i
j
k: the fixed cost of a
i
j
k, ∀
i
,
j
∈
V, ∀
k
∈
K
i
j.
C
i
j
k: the unit transportation cost of a
i
j
k, ∀
i
,
j
∈
V, ∀
k
∈
K
i
j.
L
i
j
k: the transportation capacity of a
i
j
k, ∀
i
,
j
∈
V, ∀
k
∈
K
i
j.
P
i
j
k: the failure rate of a
i
j
k, ∀
i
,
j
∈
V, ∀
k
∈
K
i
j.
W
d: the quantity of demand of demander d, d
∈
D.
U
o: the maximum supply quantity of supplier o, o
∈
O.
F
m
′: the fixed cost of transit facility m, m
∈
M.
C
m
′: the unit processing cost at facility m, m
∈
M.
L
m
′: the processing capacity of facility m,  m
∈
M.
λ: the unit penalty cost for not satisfying demand due to disruption.Decision Variables
x
i
j
k
=
1
if
a
i
j
k
is
selected
0
otherwise
.
y
u
=
1
if
facility
u
is
selected
0
otherwise
.
X: the vector including all decision variables x
i
j
k.
Y: the vector including all decision variables y
u.
f
i
j
k
s: the flow amount of a
i
j
k when scenario s occurs, ∀
i
,
j
∈
V, ∀
k
∈
K
i
j, ∀
s
∈
S.Assistant Variables
R
i
j
k
s
=
1
if
a
i
j
k
disrupts
under
scenario
s
0
otherwise
.


## 2.2. Robustness of the 4PL Network

The robustness of the 4PL network is the ability of maintaining operation function at a lower cost when the system faces to the external and internal management in emergencies and other uncertainties of disturbance.Definition ofβ-Robustness. For a given set S of scenarios, let β
s be a deterministic (i.e., single-scenario) optimization problem for scenario s. Let C
s
* be the optimal objective value for β
s. Let (
X
,
Y
) be a solution to β
s for all s
∈
S and let C
s
(
X
,
Y
) be the objective value of problem β
s under solution (
X
,
Y
). Then (
X
,
Y
) is called β-robust if, for all s
∈
S,
(1)
C
s
(
X
,
Y
)
-
C
s
*
C
s
*
≤
β
or equivalently,
(2)
C
s
X
,
Y
≤
1
+
β
C
s
*
,
where β
≥
0 is a given constant which denotes the robustness level. (
C
s
(
X
,
Y
)
-
C
s
*
)
/
C
s
* is the regret value for scenario s. The β-robust measure sets upper bounds on the maximum allowable regret value for each scenario.Theβ-robust measure, in which the relative regret in each scenario is required to be no more than a constant β, was first applied in a facility layout problem [23] and later in an uncapacitated network design problem [24]. Snyder and Daskin [25] adopt the term “β-robust” to describe this measure. They formulate the stochastic β-robust k-median problem and the β-robust uncapacitated fixed charge location problem.


## 2.3. Robust Optimization Model of 4PL Network Design under Disruptions

The robust model for 4PL network design under disruptions can be described as follows:(3)
min
⁡
∑
i
∈
V
∑
j
∈
V
∑
k
∈
K
i
j
x
i
j
k
F
i
j
k
+
∑
m
∈
M
y
m
F
m
′
+
∑
s
∈
S
q
s
C
s
(
X
,
Y
)
subject to
(4)
C
s
X
,
Y
≤
1
+
β
C
s
*
,
∀
s
∈
S
,
(5)
x
m
j
k
≤
y
m
∀
m
∈
M
,
∀
j
∈
V
,
∀
k
∈
K
m
j
,
(6)
x
j
m
k
≤
y
m
∀
m
∈
M
,
∀
j
∈
V
,
∀
k
∈
K
j
m
,
(7)
∑
j
∈
V
∑
k
∈
K
m
j
x
m
j
k
≤
y
m
∀
m
∈
M
,
(8)
∑
j
∈
V
∑
k
∈
K
j
m
x
j
m
k
≤
y
m
∀
m
∈
M
,
(9)
x
i
j
k
,
y
m
∈
0,1
∀
i
,
j
∈
V
,
∀
m
∈
M
,
∀
k
∈
K
i
j
.The objective function (3) minimizes the sum of the fixed cost and the expected operation cost. Constraint (4) requires the cost in each scenario cannot be more than 100
(
1
+
β
)
% of the optimal operation cost C
s
*. Constraints (5) and (6) ensure that no 3PL provider is selected when the connected facility is not selected. Constraints (7) and (8) ensure that when a facility is selected, a 3PL provider will be selected at least, which transports products into or out of the facility. Constraint (9) denotes the binary variables.The optimal operation costsC
s
* for each of the scenarios s can be calculated as follows:
(10)
C
s
*
=
min
⁡
∑
i
∈
V
∑
j
∈
V
∑
k
∈
K
i
j
f
i
j
k
s
C
i
j
k
+
∑
i
∈
V
∑
m
∈
M
∑
k
∈
K
i
m
f
i
m
k
s
C
i
j
k
′
+
λ
∑
d
∈
D
W
d
-
∑
d
∈
D
∑
i
∈
V
∑
k
∈
K
i
d
f
i
d
k
s
subject to
(11)
∑
j
∈
V
∑
k
∈
K
o
j
f
o
j
k
s
≤
U
o
∀
o
∈
O
,
s
∈
S
(12)
f
i
j
k
s
≤
(
1
-
R
i
j
k
s
)
x
i
j
k
L
i
j
k
∀
i
,
j
∈
V
,
∀
k
∈
K
i
j
(13)
∑
j
∈
V
∑
k
∈
K
m
j
f
m
j
k
s
≤
y
m
L
m
′
∀
m
∈
M
(14)
∑
j
∈
V
∑
k
∈
K
m
j
f
m
j
k
s
-
∑
j
∈
V
∑
k
∈
K
j
m
f
j
m
k
s
=
0
∀
m
∈
M
,
s
∈
S
(15)
f
i
j
k
s
≥
0
∀
i
,
j
∈
V
,
∀
k
∈
K
i
j
.The objective function (10) is to minimize the operation cost in scenario s. It is the sum of the costs of transportation and penalty costs for not satisfying the demand. Constraint (11) requires the flow out to be less than or equal to maximum supply quantity of suppliers. Constraint (12) represents capacity of the selected 3PL provider must be not less than the transportation capacityrequired by customer. Constraint (13) ensures that the total flow through a facility does not exceed its processing capacity. Equation (14) means to keep a balance of the network flow. Constraint (15) is the nonnegative and integer condition constraints.


## 3. Intelligent Optimization Algorithms

The problem of 4PL network design under disruptions is NP-hard. The AFSA is developed to solve the problem. It is a stochastic search algorithm which simulates the food-seeking behaviors of the fish swarm [26]. It is a swarm intelligence algorithm and has been used successfully in various optimization problems [27]. W. Tian and Y. Tian [28] used the AFSA to solve task scheduling problems in autonomic multirobot group. Liu et al. [29] presented an AFSA to solve the multicast routing problem, which was abstracted as a Steiner tree problem in graphs. Yang [30] presented an AFSA by employing some improvements for the Multiple Sequence Alignment. The AFSA is capable of solving the problems by inspiration from the en masse movement of fishes. Fishes show different behaviors including seeking for food, following other fishes, protecting the group against threats, and stochastic search. This algorithm has many advantages including high convergence speed, flexibility, and high accuracy. The AFSA shows more intelligent behavior and obtains more optimized results compared with other algorithms. The classic GA is also used to compare the quality of the solutions obtained by AFSA.


### 3.1. Relevant Definitions

LetQ
=
(
Q
1
,
Q
2
,
…
,
Q
N
P
) be the state of the artificial fish individual, and let N
P be the population size of the artificial fish swarm.Q
i
=
{
q
i
1
,
q
i
2
,
…
,
q
i
r
,
…
,
q
i
n
}
(
i
=
1,2
,
…
,
N
P
)
(
r
=
1,2
,
…
,
n
) are the decision variables, which denote the 4PL network structure (
X
,
Y
). Fitness function Fitness
(
Q
) presents the food concentration of the artificial fish. distance
(
Q
i
,
Q
j
)
=
Q
i
-
Q
j is the distance between artificial fishes Q
i and Q
j.Visual is the artificial fish sight range.step is the step length. δ is a congestion factor.try_number is the maximum test number.Definition 1.
The distance betweenQ
i
=
{
q
i
1
,
q
i
2
,
…
,
q
i
r
,
…
,
q
i
n
} and Q
j
=
{
q
j
1
,
q
j
2
,
…
,
q
j
r
,
…
,
q
j
n
} is presented as follows:
(16)
distance
(
Q
i
,
Q
j
)
=
∑
i
=
1
N
P
∑
j
=
1
N
P
∑
r
=
1
n
sign
⁡
q
i
r
-
q
j
r
,
where
(17)
sign
⁡
x
=
0
,
x
=
0
1
,
x
>
0
-
1
,
x
<
0
is the number of elements which belong to neither Q
i nor Q
j.Definition 2.
Consider(18)
center
Q
1
,
Q
2
,
…
,
Q
N
P
=
most
i
=
1
,
…
,
N
P
q
i
1
,
q
i
2
,
…
,
q
i
n
is the center of variables Q
1
,
Q
2
,
…
,
Q
N
P.


### 3.2. Encoding

In order to use AFSA, we design a method of encoding the solution. As shown in Figure2, the roads between two corresponding nodes of the 4PL network can be coded by Roads 1 to 5. The arcs (3PL providers) between facilities can be coded by integer number. The element encoded as 1 indicates that no 3PL provider is selected between two corresponding nodes. The element encoded as 2 means that the first 3PL provider is selected. The element encoded as 3 means that the second 3PL provider is selected. And the element encoded as 4 means that all the 3PL providers between two corresponding nodes are selected. For example, string [
2
3
1
3
2
] represents the 4PL network structure in Figure 3.Figure 2
Encoding.Figure 3
Decoding.


### 3.3. Calculation of Food Concentration

The fitness functionFitness
(
Q
) denotes the food concentration of the artificial fish.In this paper, the robustness level constraint (4) is used as a penalty added to the objective function. The fitness function is represented by
(19)
Fitness
Q
i
=
min
⁡
∑
i
∈
V
∑
j
∈
V
∑
k
∈
K
i
j
x
i
j
k
F
i
j
k
+
∑
m
∈
M
y
m
F
m
′
+
∑
s
∈
S
q
s
C
s
(
X
,
Y
)
+
ω
·
max
⁡
0
,
∑
s
∈
S
C
s
X
,
Y
-
1
+
β
C
s
*
.
ω is the penalty factor. A smaller fitness value Fitness
(
Q
i
) indicates better fitness.


### 3.4. Artificial Fish Behavior

#### 3.4.1. Prey Behavior

Let the current state of artificial fish beQ
i, and select a state Q
j which is selected randomly in its visual distance (distance
(
Q
i
,
Q
j
)
<
Visual). If Fitness
(
Q
j
)
<
Fitness
(
Q
i
), it goes forward a step in this direction; that is, let Q
i
=
Q
i
+
rand
(
)
·
step
·
(
Q
j
-
Q
i
)
/
Q
i
-
Q
j, where rand
(
) produces random numbers between 0 and 1. Otherwise, select a state Q
j randomly; that is, Q
i
=
Q
i
+
rand
(
)
·
Visual. Repeat the proceduretry_number times. If it cannot satisfy the forward condition, it moves a step randomly.


#### 3.4.2. Swarm Behavior

LetQ
center be the center position. n
f is the number of its companions in the current neighborhood (distance
(
Q
i
,
Q
j
)
<
Visual), and n is total fish number. If Fitness
(
Q
center
)
<
Fitness
(
Q
i
) and n
f
/
n
<
δ, it means that the center position has more food and is not very crowded. Then it goes forward a step to the center position; that is, let Q
i
=
Q
i
+
rand
(
)
·
step
·
(
Q
center
-
Q
i
)
/
Q
center
-
Q
i. Otherwise, execute the prey behavior.


#### 3.4.3. Follow Behavior

LetQ
i explore the companion Q
best in the neighborhood (distance
(
Q
i
,
Q
j
)
<
Visual), which has the best Fitness
(
Q
best
). If Fitness
(
Q
best
)
<
Fitness
(
Q
i
) and n
f
/
n
<
δ, it goes forward a step to the companion Q
best; that is, let Q
i
=
Q
i
+
rand
(
)
·
step
·
(
Q
best
-
Q
i
)
/
Q
best
-
Q
i. Otherwise, execute the prey behavior.


#### 3.4.4. Random Move Behavior

Choose a stateQ
new randomly in the visual distance (distance
(
Q
i
,
Q
j
)
<
Visual); then move towards this state; that is, let Q
i
=
Q
i
+
rand
(
)
·
step
·
(
Q
new
-
Q
i
)
/
Q
new
-
Q
i.


#### 3.4.5. Bulletin Board

The bulletin board keeps a record of the optimal artificial fish state and fitness value.


### 3.5. The Procedure of AFSA

Step 1.
Generate the initial artificial fish population randomly. Set population size of the artificial fishN
P, the maximum number of iterations N
G, the maximum test numbertry_number, the artificial fish sight rangeVisual, the step lengthstep, and the congestion factor δ.Step 2.
Calculate the food concentration, that is, fitness functionFitness
(
Q
).Step 3.
Simulate fish preying, following, swarming, and random moving behavior respectively. Update the bulletin board if a smaller fitness value arises.Step 4.
Check whether iteration times are overN
G. If it is true, stop the procedure and output the optimum solution; else turn to Step 2.


### 3.6. Calculation of the Expected Operation Cost

The number of the disruption scenarios is very large. The expected operation cost∑
s
∈
S
q
s
C
s
(
X
,
Y
) can be approximated using Monte Carlo simulation.The following are steps of emulation computation.Step 1.
GenerateN samples of scenarios, according to the disruption probability of the 3PL providers P
i
j
k.Step 2.
Calculate the value ofC
s
(
X
,
Y
) for each sample scenario s.Step 3.
Estimate the expected operation cost as follows:∑
s
∈
S
q
s
C
s
(
X
,
Y
)
=
(
1
/
N
)
∑
s
=
1
N
C
s
(
X
,
Y
).


## 3.1. Relevant Definitions

LetQ
=
(
Q
1
,
Q
2
,
…
,
Q
N
P
) be the state of the artificial fish individual, and let N
P be the population size of the artificial fish swarm.Q
i
=
{
q
i
1
,
q
i
2
,
…
,
q
i
r
,
…
,
q
i
n
}
(
i
=
1,2
,
…
,
N
P
)
(
r
=
1,2
,
…
,
n
) are the decision variables, which denote the 4PL network structure (
X
,
Y
). Fitness function Fitness
(
Q
) presents the food concentration of the artificial fish. distance
(
Q
i
,
Q
j
)
=
Q
i
-
Q
j is the distance between artificial fishes Q
i and Q
j.Visual is the artificial fish sight range.step is the step length. δ is a congestion factor.try_number is the maximum test number.Definition 1.
The distance betweenQ
i
=
{
q
i
1
,
q
i
2
,
…
,
q
i
r
,
…
,
q
i
n
} and Q
j
=
{
q
j
1
,
q
j
2
,
…
,
q
j
r
,
…
,
q
j
n
} is presented as follows:
(16)
distance
(
Q
i
,
Q
j
)
=
∑
i
=
1
N
P
∑
j
=
1
N
P
∑
r
=
1
n
sign
⁡
q
i
r
-
q
j
r
,
where
(17)
sign
⁡
x
=
0
,
x
=
0
1
,
x
>
0
-
1
,
x
<
0
is the number of elements which belong to neither Q
i nor Q
j.Definition 2.
Consider(18)
center
Q
1
,
Q
2
,
…
,
Q
N
P
=
most
i
=
1
,
…
,
N
P
q
i
1
,
q
i
2
,
…
,
q
i
n
is the center of variables Q
1
,
Q
2
,
…
,
Q
N
P.


## 3.2. Encoding

In order to use AFSA, we design a method of encoding the solution. As shown in Figure2, the roads between two corresponding nodes of the 4PL network can be coded by Roads 1 to 5. The arcs (3PL providers) between facilities can be coded by integer number. The element encoded as 1 indicates that no 3PL provider is selected between two corresponding nodes. The element encoded as 2 means that the first 3PL provider is selected. The element encoded as 3 means that the second 3PL provider is selected. And the element encoded as 4 means that all the 3PL providers between two corresponding nodes are selected. For example, string [
2
3
1
3
2
] represents the 4PL network structure in Figure 3.Figure 2
Encoding.Figure 3
Decoding.


## 3.3. Calculation of Food Concentration

The fitness functionFitness
(
Q
) denotes the food concentration of the artificial fish.In this paper, the robustness level constraint (4) is used as a penalty added to the objective function. The fitness function is represented by
(19)
Fitness
Q
i
=
min
⁡
∑
i
∈
V
∑
j
∈
V
∑
k
∈
K
i
j
x
i
j
k
F
i
j
k
+
∑
m
∈
M
y
m
F
m
′
+
∑
s
∈
S
q
s
C
s
(
X
,
Y
)
+
ω
·
max
⁡
0
,
∑
s
∈
S
C
s
X
,
Y
-
1
+
β
C
s
*
.
ω is the penalty factor. A smaller fitness value Fitness
(
Q
i
) indicates better fitness.


## 3.4. Artificial Fish Behavior

### 3.4.1. Prey Behavior

Let the current state of artificial fish beQ
i, and select a state Q
j which is selected randomly in its visual distance (distance
(
Q
i
,
Q
j
)
<
Visual). If Fitness
(
Q
j
)
<
Fitness
(
Q
i
), it goes forward a step in this direction; that is, let Q
i
=
Q
i
+
rand
(
)
·
step
·
(
Q
j
-
Q
i
)
/
Q
i
-
Q
j, where rand
(
) produces random numbers between 0 and 1. Otherwise, select a state Q
j randomly; that is, Q
i
=
Q
i
+
rand
(
)
·
Visual. Repeat the proceduretry_number times. If it cannot satisfy the forward condition, it moves a step randomly.


### 3.4.2. Swarm Behavior

LetQ
center be the center position. n
f is the number of its companions in the current neighborhood (distance
(
Q
i
,
Q
j
)
<
Visual), and n is total fish number. If Fitness
(
Q
center
)
<
Fitness
(
Q
i
) and n
f
/
n
<
δ, it means that the center position has more food and is not very crowded. Then it goes forward a step to the center position; that is, let Q
i
=
Q
i
+
rand
(
)
·
step
·
(
Q
center
-
Q
i
)
/
Q
center
-
Q
i. Otherwise, execute the prey behavior.


### 3.4.3. Follow Behavior

LetQ
i explore the companion Q
best in the neighborhood (distance
(
Q
i
,
Q
j
)
<
Visual), which has the best Fitness
(
Q
best
). If Fitness
(
Q
best
)
<
Fitness
(
Q
i
) and n
f
/
n
<
δ, it goes forward a step to the companion Q
best; that is, let Q
i
=
Q
i
+
rand
(
)
·
step
·
(
Q
best
-
Q
i
)
/
Q
best
-
Q
i. Otherwise, execute the prey behavior.


### 3.4.4. Random Move Behavior

Choose a stateQ
new randomly in the visual distance (distance
(
Q
i
,
Q
j
)
<
Visual); then move towards this state; that is, let Q
i
=
Q
i
+
rand
(
)
·
step
·
(
Q
new
-
Q
i
)
/
Q
new
-
Q
i.


### 3.4.5. Bulletin Board

The bulletin board keeps a record of the optimal artificial fish state and fitness value.


## 3.4.1. Prey Behavior

Let the current state of artificial fish beQ
i, and select a state Q
j which is selected randomly in its visual distance (distance
(
Q
i
,
Q
j
)
<
Visual). If Fitness
(
Q
j
)
<
Fitness
(
Q
i
), it goes forward a step in this direction; that is, let Q
i
=
Q
i
+
rand
(
)
·
step
·
(
Q
j
-
Q
i
)
/
Q
i
-
Q
j, where rand
(
) produces random numbers between 0 and 1. Otherwise, select a state Q
j randomly; that is, Q
i
=
Q
i
+
rand
(
)
·
Visual. Repeat the proceduretry_number times. If it cannot satisfy the forward condition, it moves a step randomly.


## 3.4.2. Swarm Behavior

LetQ
center be the center position. n
f is the number of its companions in the current neighborhood (distance
(
Q
i
,
Q
j
)
<
Visual), and n is total fish number. If Fitness
(
Q
center
)
<
Fitness
(
Q
i
) and n
f
/
n
<
δ, it means that the center position has more food and is not very crowded. Then it goes forward a step to the center position; that is, let Q
i
=
Q
i
+
rand
(
)
·
step
·
(
Q
center
-
Q
i
)
/
Q
center
-
Q
i. Otherwise, execute the prey behavior.


## 3.4.3. Follow Behavior

LetQ
i explore the companion Q
best in the neighborhood (distance
(
Q
i
,
Q
j
)
<
Visual), which has the best Fitness
(
Q
best
). If Fitness
(
Q
best
)
<
Fitness
(
Q
i
) and n
f
/
n
<
δ, it goes forward a step to the companion Q
best; that is, let Q
i
=
Q
i
+
rand
(
)
·
step
·
(
Q
best
-
Q
i
)
/
Q
best
-
Q
i. Otherwise, execute the prey behavior.


## 3.4.4. Random Move Behavior

Choose a stateQ
new randomly in the visual distance (distance
(
Q
i
,
Q
j
)
<
Visual); then move towards this state; that is, let Q
i
=
Q
i
+
rand
(
)
·
step
·
(
Q
new
-
Q
i
)
/
Q
new
-
Q
i.


## 3.4.5. Bulletin Board

The bulletin board keeps a record of the optimal artificial fish state and fitness value.


## 3.5. The Procedure of AFSA

Step 1.
Generate the initial artificial fish population randomly. Set population size of the artificial fishN
P, the maximum number of iterations N
G, the maximum test numbertry_number, the artificial fish sight rangeVisual, the step lengthstep, and the congestion factor δ.Step 2.
Calculate the food concentration, that is, fitness functionFitness
(
Q
).Step 3.
Simulate fish preying, following, swarming, and random moving behavior respectively. Update the bulletin board if a smaller fitness value arises.Step 4.
Check whether iteration times are overN
G. If it is true, stop the procedure and output the optimum solution; else turn to Step 2.


## 3.6. Calculation of the Expected Operation Cost

The number of the disruption scenarios is very large. The expected operation cost∑
s
∈
S
q
s
C
s
(
X
,
Y
) can be approximated using Monte Carlo simulation.The following are steps of emulation computation.Step 1.
GenerateN samples of scenarios, according to the disruption probability of the 3PL providers P
i
j
k.Step 2.
Calculate the value ofC
s
(
X
,
Y
) for each sample scenario s.Step 3.
Estimate the expected operation cost as follows:∑
s
∈
S
q
s
C
s
(
X
,
Y
)
=
(
1
/
N
)
∑
s
=
1
N
C
s
(
X
,
Y
).


## 4. Experimental Results and Comparisons

In this section, an instance is developed to evaluate the performance of the AFSA and the GA. The algorithms are coded in Matlab 7.0 and run on an Intel Core 2 Quad 2.66 GHz computer.Assume that a 4PL provider plans to transport a consignment of goods from the supply nodes to the demand nodes. Decision makers need to design the 4PL network which can provide satisfactory service to customers at a lower cost when disruptions strike. The topological structure of the potential 4PL network is shown in Figure1. The failure rate of each 3PL provider is drawn uniformly from [
0
,
1
]. The unit penalty cost for not satisfying demand due to disruption λ is set to be 500. By running a lot of experiments firstly, the parameters in AFSA are set below:
(20)
ω
=
2
,
N
P
=
50
,
N
G
=
200
,
Visual
=
20
,
step
=
1
,
δ
=
0.3
,
try
_
number
=
20
.The costs of the facilities and the 3PL providers are recorded by dollar. The capacity is recorded by ton, and the CPU time is recorded by second. The node and arc data are given in Tables1 and 2.Table 1
The data of nodes for the 9-node problem.
V
i
U
o
W
d
F
m
′
C
m
′
L
m
′
1
157
∖
∖
∖
∖
2
142
∖
∖
∖
∖
3
∖
∖
636
34
120
4
∖
∖
763
44
142
5
∖
∖
651
37
119
6
∖
∖
645
24
166
7
∖
∖
761
48
179
8
∖
62
∖
∖
∖
9
∖
44
∖
∖
∖Table 2
The data of arcs for the 9-node problem.
V
i
V
j
a
i
j
k
F
i
j
k
C
i
j
k
L
i
j
k
1
3
1
743
34
123
2
728
40
189
5
1
722
43
102
2
961
42
162
2
3
1
968
26
118
2
947
26
192
4
1
907
24
106
2
601
16
101
3
6
1
873
11
120
2
733
47
145
4
6
1
919
37
120
2
690
37
102
7
1
855
30
183
2
595
22
143
5
6
1
849
16
154
2
927
44
138
8
1
950
30
159
2
909
36
182
6
8
1
864
31
134
2
784
44
131
9
1
773
38
137
2
811
38
144
7
9
1
990
17
188
2
938
20
127The AFSA and GA are run 200 times. The experimental results and comparisons are shown in Table3. It can be seen that the AFSA outperforms the classic GA under different regret value β. The AFSA has better stabilization and it can obtain better solutions.Table 3
Comparison of two algorithms with different regret valueβ constraint.
Algorithm
Regret valueβ constraint
Objective value ($) best
Objective value ($) worst
Objective value ($) ave.
Ave. CPU time (second)
AFSA
0.4
26160
28530
26856
576
0.2
32970
33190
33070
0.1
33150
34880
33278
GA
0.4
27180
29980
28742
868
0.2
33010
33260
33192
0.1
34860
35980
35276In order to analyze the influence of regret valueβ constraint to the 4PL network construction, we perform numerical experiments with different regret value β.Figure4 shows the results of the different 4PL network structures. V
2, V
3, V
6, V
8, and V
9 are selected with β
=
0.4. The minimum total cost is $26,160. V
1, V
5, V
6, V
8, and V
9 are selected with β
=
0.2. The minimum total cost is $32,970. V
1, V
2, V
3, V
5, V
6, V
8, and V
9 are selected with β
=
0.1. The minimum total cost is $33,150. It is indicated that the 4PL network becomes more complex when the higher robustness level is desired. Besides, the total cost increases when the regret value β decreases.Figure 4
4PL network structures under different regret valueβ.Hence, the 4PL provider is allowed to adjust the robustness level and has options to design 4PL network.


## 5. Conclusions and Prospects

This paper studies the robust optimization of 4PL network design under disruptions. In order to make the 4PL network operate safely and efficiently, the concept ofβ-robustness is introduced into the network design problem. A robust optimization model with β-robust constraints is established to construct a 4PL network at a lower cost among disruption scenarios. The objective function of the mathematical model is minimizing the sum of the fixed cost and the expected operation cost. Based on the NP-hard characteristic of the problem, the AFSA and GA are presented and the performance of the algorithms is tested by simulation examples. According to the simulation of Matlab, the AFSA performs better in effectiveness and stability than the classic GA. After solving the model, we obtain different 4PL network structures which satisfy different robustness requirements with the lowest cost. We also demonstrate that the robust optimization model can evade the market risks effectively and save the cost in the maximum limit when it is applied to 4PL network design.For the future research, more realistic problem such as a multiperiod and multiproduct integrated 4PL network design with disruptions may be taken into consideration. The scenario approach can also be adopted to model other sources of uncertainties, such as customer demands and transportation costs. It is also possible to consider network design together with other supply chain problems, such as inventory management, capacity expansion, and vehicle routing.



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*Source: 720628-2015-02-23.xml*