Two-stage fuzzy mixed integer optimization model for three-level location allocation problems under uncertain environment

Abstract

The problem of the optimal three-level location allocation of transfer center, processing factory and distribution center for supply chain network under uncertain transportation cost and customer demand are studied. We establish a two-stage fuzzy 0-1 mixed integer optimization model, by considering the uncertainty of the supply chain. Given the complexity of the model, this paper proposes a modified hybrid second order particle swarm optimization algorithm (MHSO-PSO) to solve the resulting model, yielding the optimal location and maximal expected return of supply chain simultaneously. A case study of clothing supply chain in Shanghai of China is then presented to investigate the specific influence of uncertainties on the transfer center, clothing factory and distribution center three-level location. Moreover, we compare the MHSO-PSO with hybrid particle swarm optimization algorithm and hybrid genetic algorithm, to validate the proposed algorithm based on the computational time and the convergence rate.

Keywords

Two-stage fuzzy 0-1 mixed integer optimization three-level location allocation uncertainty algorithm

1 Introduction

The problem of location allocation is one of the classic problems in operations research, which has been widely used in production, logistics, emergency service system construction and telecommunication network construction ([1 –15]). The importance of location allocation lies in that once the location is selected and the distribution mode is determined, it will directly affect the service mode, service quality, service efficiency, service cost, etc., thus affecting the profit and market competitiveness, and even determining the fate of enterprises. Once the site is improperly located, the consequences it brings are difficult to make up through other management follow-up measures. Therefore, the study of location allocation is of great economic and social significance.

In today’s competitive market, the product life cycle is getting shorter and shorter, which makes historical statistics less useful. We cannot get accurate information about transportation costs, demand and other parameters. In short, past data on consumer demand and transportation costs may not always be reliable or even available. Therefore, standard probability modeling may not be the best choice under such circumstances. Rather, the use of fuzzy transportation costs and demand better suits the new reality, and the parameters in the fuzzy transportation costs and demand can be estimated accordingly through the changes in actual transportation costs and consumer demand. In this case, fuzzy set theory provides an acceptable alternative approach to dealing with this uncertainty. Fuzzy set theory is proposed by Zadeh [16], it is widely used in many real problems. It has proved to be a useful tool for solving uncertain problems. In the past decades, many researchers have introduced fuzzy theory into location allocation problem.

Bhattacharya et al. [17] proposed a fuzzy goal programming approach to deal with the problem of facility location under multiple fuzzy criteria. Wenab [18] considered the problem of facility location allocation with fuzzy demand, established a alpha-cost model under the hurwicz criterion, and proposed a hybrid intelligent algorithm to solve the model. Based on queuing theory and fuzzy conditions, Shavandi et al. [19] established a fuzzy queuing consistency hierarchical location allocation model for congested systems. Shen et al. [20] proposed a class of fuzzy two-stage location allocation problem and designed a hybrid algorithm combining approximation method and particle swarm optimization to solve the problem. Liu and Xu [21] established a mixed integer programming model of random fuzzy facility location allocation problem by considering the uncertainty of demand and various related costs in the supply chain. This model is simplified to a deterministic model by using expected value and opportunity constrained programming techniques, and a genetic algorithm based on priority is designed to solve this model. Table 1 lists the representative literature and research focus on location allocation in the supply chain field. All these provide new ideas and methods for the research of this paper.

Table 1
Research on location allocation supply chains

Reference Research focus

[11 , 31] Fuzzy demand

[24, 33] Fuzzy cost

[2 , 31] facility location

[3 , 30] warehouse location

[10 , 29] distribution center location

[32] transfer center location

[22–28 , 30] One-stage model

[10 , 31] Two-stage model

Reference	Research focus
[11 , 31]	Fuzzy demand
[24, 33]	Fuzzy cost
[2 , 31]	facility location
[3 , 30]	warehouse location
[10 , 29]	distribution center location
[32]	transfer center location
[22–28 , 30]	One-stage model
[10 , 31]	Two-stage model

A supply chain system with optimal performance should maximize the value of internal activities while developing stable partnerships to maximize the value of external activities. Supply chain management realizes the integration and reconfiguration of nodal enterprise resources. In this paper, we consider the supply chain system, which is under the coordination of core enterprise, cancelled the original exists between suppliers, manufacturers and customers of buffer inventories, and transfer center is established between the supplier and the manufacturer, between manufacturers and customers to establish a warehouse center as the distribution center, the dispersed custody of stock material together, improve the efficiency of the storage, loading and unloading, etc, through rational planning transfer center and distribution center of transportation planning and distribution plan, thereby reducing the transportation cost of the entire supply chain, speed up the logistics turnover. In addition, as far as we know, there are few researches on the three-level location allocation supply chain with multi-product, multi-supplier, multi-transfer center, multi-processing factory, multi-distribution center and multi-customer under the fuzzy transportation cost and demand.

Thus, to fill this gap, this paper considers the coordination mechanism of the three-level location allocation supply chain system composed of multi-supplier, multi-transfer center, multi-processing factory, multi-distribution center and multi-customer. Considering the location of transfer center, processing factory, distribution center, a two-stage fuzzy 0-1 mixed integer programming model is established. In the first stage, the location variable is the decision variable in the supply chain network. These decision variables are determined according to different realisation of the scenarios in the second stage, hence the decision process is called “here and now”. The quantity of transportation between each member in the supply chain network is the variable of the second stage, which is related to the scenario.

The classical methods for solving mixed integer programming problems are Benders’ decomposition method ([34 –36]), cutting-plane method ([37, 38]), branch and bound method ([39, 40]), branch and cut method ([41]), etc. The main idea of these algorithms is to use dual theory to derive and generate the family of cutting planes in the iterative process, and add the optimal cutting to the constraint, so as to reduce the feasible region of solving the problem. If the above algorithm is used to solve two-stage fuzzy 0-1 mixed integer programming model, the two-stage problem is first transformed into a single-stage problem, and then the single-stage problem is solved. However, in second stage, for any scenario, we only know that the corresponding membership degree, but do not known the corresponding probability. If the two-stage problem is transformed into a single-stage problem, probability must be calculated. We must first figure out the second stage value in each scenario. In solving the iterative process, the corresponding probability value is not unique. That is to say, two-stage fuzzy 0-1 mixed integer optimization model cannot be converted into a single stage for solving. That is, the above algorithm cannot be used to solve proposed model. Moreover, the traditional optimization algorithm cannot solve the two-stage fuzzy 0-1 mixed integer programming problem.

Medsker [42] introduced many ideas for designing hybrid intelligent systems, which integrate a variety of intelligent algorithms to produce more powerful and effective algorithms. It lays a foundation for solving mixed integer programming problem. Therefore, another contribution of this study comes from the development of algorithms. In particular, we propose a MHSO-PSO, which combines the synchronous dynamic learning factor and compression factor, and adds them to second order particle swarm optimization algorithm (SO-PSO) to improve the convergence speed of the algorithm and ensure the global convergence of the algorithm. This algorithm has the advantages of fast searching speed, high efficiency and simple algorithm. In the iterative process, the simplex algorithm is used to solve the second stage problem (see Section 4). The main contributions of this paper are summarized as follows,

•Considering the uncertainty of supply chain, a two-stage fuzzy 0-1 mixed integer programming model is established.

•This paper proposes a MHSO-PSO to solve the two-stage fuzzy 0-1 mixed integer programming model.

•We compare the MHSO-PSO with hybrid particle swarm optimization algorithm (Hybrid PSO) and hybrid genetic algorithm (Hybrid GA), to validate the proposed algorithm based on the computational time and the convergence rate.

The rest of this paper is set as follows. Section 2 describes the mathematical preliminaries on fuzzy variables. The two-stage fuzzy 0-1 mixed integer optimization model is formulated in Section 3. Section 4 then presents a solution algorithm for the model. The computational results of a numerical example are presented and discussed in Section 5. Section 6 concludes with some future research directions.

2 Preliminaries

In this section, we review some basic concepts of fuzzy variables, which help readers understand the proposed two-stage fuzzy mixed integer optimization model. Converting fuzzy variables into an equivalent deterministic form has an important role in solving the fuzzy optimization problem, which is often encountered in real-life situations. Motivated by the need for an axiomatic approach to perform this step, Liu [43] has suggested the credibility measure and proposed the credibility theory. It is proved to be a suitable basis for fuzzy system optimization and numerical simulation.

Let $(Θ, A (Θ), Pos)$ be a possibility space, where $A (Θ)$ is the power set of Θ, and ξ be a fuzzy variable defined on $(Θ, A (Θ), Pos)$ whose membership function is μ_ξ. The possibility and credibility of the event ξ ≤ r, r beings a real number, is expressed as follows:

$Pos {ξ \leq r} = sup_{x \leq r} μ_{ξ} (x),$ (1)

$Cr {ξ \leq r} = \frac{1}{2} (sup_{x \leq r} μ_{ξ} (x) + 1 - sup_{x > r} μ_{ξ} (x)) .$ (2)

With the above concept, the expected value of the fuzzy variable can be defined as follows.

Definition 1. Let ξ be a fuzzy variable defined on the possibility space $(Θ, A (Θ), Pos)$ . The expected value E [ξ] is defined as $E [ξ] = \int_{0}^{+ \infty} Cr {ξ \geq r} dr - \int_{- \infty}^{0} Cr {ξ \leq r} dr$ provided that at least one of the two integrals is finite.

Particularly, for the discrete fuzzy variable ξ, the membership function is given by

$μ (x) = {\begin{matrix} μ_{1}, if x = x_{1} \\ ⋮ ⋮ \\ μ_{2}, if x = x_{2} \\ ⋮ ⋮ \\ μ_{n}, if x = x_{n} \end{matrix},$ (3) here μ₁ ∨ μ₂ ∨ ⋯ ∨ μ_n = 1 . Based on [44], using (1), (2) and (3), we obtain

$E [ξ] = \sum_{i = 1}^{n} p_{i} x_{i},$ (4) the corresponding weights p_i are given in the form

$\begin{matrix} p_{i} & = \frac{1}{2} (max_{1 \leq t \leq n} {μ_{t} | x_{t} \leq x_{i}} \\ - max_{1 \leq t \leq n} {μ_{t} | x_{t} < x_{i}}) \\ + \frac{1}{2} (max_{1 \leq t \leq n} {μ_{t} | x_{t} \geq x_{i}} \\ - max_{1 \leq t \leq n} {μ_{t} | x_{t} > x_{i}}), \end{matrix}$ and satisfy the following constraint $p_{i} \geq 0, \sum_{i = 1}^{n} p_{i} = 1 .$

Since the uncertainty factors considered in this paper include the fuzzy transportation cost and the fuzzy demand of customers, the membership function corresponding to the fuzzy vector is not easy to express. To solve this problem, the following definition of ordered weighted aggregation (OWA) operator is introduced.

Definition 2. [45] A mapping $F$ : $R^{n} \to R$ is called an OWA operator of dimension n, if $F (a_{1}, \dots, a_{n^{'}}) = \sum_{j^{'} = 1}^{n^{'}} δ_{j^{'}} b_{j^{'}},$ where δ_j′ are a collection of weights satisfying δ_j′ ∈ [0, 1] , and $\sum_{j^{'}}^{n^{'}} δ_{j^{'}} = 1,$ b_j′ is the j′ - th largest element in the collection a₁, ⋯ , a_n′ .

3 Two-stage fuzzy 0-1 mixed integer optimization model

In this section, we establish a two-stage fuzzy 0-1 mixed integer optimization model under the uncertainty of customer demand and transportation cost. The first stage decision is related to transfer center location, processing factory location and distribution center location. The second stage decision is related to the transportation cost, inventory cost and demand, whose goal is to minimize the expected value of the cost function. The structure of location allocation supply chain network is shown in Fig. 1.

Fig. 1

Network structure of supply chain.

In particular, we considered S suppliers, M transfer centers, I processing factory, J distribution centers and K types of customers. A typical supplier, transfer center, processing factory, distribution center, and customer are represented by s, m, i, j and k respectively. The links in the supply chain network denote the transaction links. The supplier supplies V kinds of raw materials, the processing factory produces L kinds of products, and then transport them to the distribution centers, who meets the customer’s fuzzy demand. The objective of the company is to maximize the expected profit by choosing the optimal number of transfer centers, processing factories and distribution centers in the market area on the premise of meeting customer fuzzy demand.

For simplicity, we apply the following symbols for this model. All the vectors used in this paper are assumed to be column vectors.

3.1 Notation

Indices and set

Suppliers index

Transfer centers index

Processing factories index

Distribution centers index

Customers index

Raw materials index

Products index

Set of the suppliers

Set of the transfer centers

Set of the processing factories

Set of the distribution centers

Set of the customers

Set of the raw materials

Set of the products

Parameters

c _m

Fixed cost of operating transfer center m

f _i

Fixed cost of operating processing factory i

g _j

Fixed cost of operating distribution center j

a _vs

The ability of supplier s to provide raw material v

r _vs

The cost of raw materials v provided by the supplier s

η _vm

The ability of transfer center m to transport raw materials v

n _vl

The quantity of raw material v required for processing product l

b _il

The ability of processing factory i to produce product l

q _il

The cost of producing product l in the processing factory i

τ _lj

The ability of distribution center j to store product l

w _lj

The inventory cost of product l stored in distribution center j

h _l

Retail price of unit product l

{\tilde{t}}_{vsm}

Fuzzy transportation cost of supplier s transporting raw material v to transfer center m

{\tilde{n}}_{vmi}

Fuzzy transportation cost of transfer center m transporting raw material v to processing factory i

{\tilde{m}}_{lij}

Fuzzy transportation cost of processing factory i transporting product l to distribution center j

{\tilde{u}}_{ljk}

Fuzzy distribution cost of distribution center j transporting product l to consumer k

{\tilde{d}}_{lk}

Fuzzy demand of consumer k for product l

ξ (ω)

Fuzzy transportation cost and demand vector ξ (ω) =( ${\tilde{t}}_{vsm} (ω)$ , ${\tilde{n}}_{vmi} (ω)$ , ${\tilde{m}}_{lij} (ω)$ , ${\tilde{u}}_{ljk} (ω)$ , ${\tilde{d}}_{lk} (ω)$ )

E (·)

Expectation

Variables

α _m

Binary variables, open transfer center m, value 1, otherwise 0, a first-stage decision variable

β _i

Binary variables, open processing factory i, value 1, otherwise 0, a first-stage decision variable

γ _j

Binary variables, open distribution center j, value 1, otherwise 0, a first-stage decision variable

e _vsm

The quantity of raw materials v transported by the supplier s to the transfer center m, a second-stage decision variable

x _vmi

The quantity of raw materials v transported by the transfer center m to the processing factory i, a second-stage decision variable

y _lij

The quantity of product l transported by the processing factory i to the distribution center j, a second-stage decision variable

z _ljk

The quantity of product l transported by the distribution center j to the consumer k, a second-stage decision variable

3.2 The model

Assumptions

1. Customer’s demand cannot be overserved, but it is possible that they are not fully satisfied.

2. Fuzzy transportation cost and demand vector ξ (ω) =( ${\tilde{t}}_{vsm} (ω)$ , ${\tilde{n}}_{vmi} (ω)$ , ${\tilde{m}}_{lij} (ω)$ , ${\tilde{u}}_{ljk} (ω)$ , ${\tilde{d}}_{lk} (ω)$ ) is defined on possibility space $(Θ, A (Θ), Pos)$ .

Based on the above assumptions and description, we present a two-stage fuzzy 0-1 mixed integer optimization model as follows, $min_{α, β, γ} \sum_{M} α_{m} c_{m} + \sum_{I} β_{i} f_{i} + \sum_{J} γ_{j} g_{j}$ (5) $\begin{matrix} + Q (α, β, γ) \\ s . t . α_{m} \in {0, 1}, \forall m \in M, \\ β_{i} \in {0, 1}, \forall i \in I, \\ γ_{j} \in {0, 1}, \forall j \in J, \end{matrix}$ where α = (α₁, ⋯ , α_M) , β = (β₁, ⋯ , β_I) , γ = (γ₁, ⋯ , γ_J) , $Q (α, β, γ) = E [Q (α, β, γ, ξ (ω))]$ , and Q ( α , β , γ , ξ (ω)) is the optimal value of the second-stage problem

$Q (α, β, γ, ξ (ω)) = min_{e, x, y, z} \sum_{V} \sum_{S} (r_{vs} \sum_{M} e_{vsm})$ (6) $\begin{matrix} + \sum_{I} \sum_{L} (q_{il} \sum_{J} y_{lij}) + \sum_{V} \sum_{S} \sum_{M} {\tilde{t}}_{vsm} (ω) e_{vsm} \\ + \sum_{V} \sum_{M} \sum_{I} {\tilde{n}}_{vmi} (ω) x_{vmi} + \sum_{L} \sum_{I} \sum_{J} {\tilde{m}}_{lij} (ω) y_{lij} \\ + \sum_{L} \sum_{J} \sum_{K} {\tilde{u}}_{ljk} (ω) z_{ljk} + \sum_{L} \sum_{J} (w_{lj} (\sum_{I} y_{lij} \\ - \sum_{K} z_{ljk})) - \sum_{L} (h_{l} \sum_{J} \sum_{K} z_{ljk}) \end{matrix}$

$s . t . \sum_{M} e_{vsm} \leq a_{vs} \forall v, s,$ (7a) $\sum_{I} x_{vmi} \leq \sum_{S} e_{vsm} \forall v, m,$ (7b) $\sum_{I} x_{vmi} \leq α_{m} η_{vm} \forall v, m,$ (7c) $\sum_{L} (n_{vl} \sum_{J} y_{lij}) \leq \sum_{M} x_{vmi} \forall v, i,$ (7d) $\sum_{J} y_{lij} \leq β_{i} b_{il} \forall i, l,$ (7e) $\sum_{K} z_{ljk} \leq \sum_{I} y_{lij} \forall l, j,$ (7f) $\sum_{K} z_{ljk} \leq γ_{j} τ_{lj} \forall l, j,$ (7g) $\sum_{J} z_{ljk} \leq {\tilde{d}}_{lk} (ω) \forall l, k,$ (7h) $e_{vsm}, x_{vmi}, y_{lij}, z_{ljk} \geq 0,$ (7i) $\forall v, m, s, i, j, k, l .$ where, ${\tilde{t}}_{vsm} (ω), {\tilde{n}}_{vmi} (ω), {\tilde{m}}_{lij} (ω), {\tilde{u}}_{ljk} (ω)$ and ${\tilde{d}}_{lk} (ω)$ are the realizations of the fuzzy transportation cost ${\tilde{t}}_{vsm}, {\tilde{n}}_{vmi}, {\tilde{m}}_{lij},$ ${\tilde{u}}_{ljk},$ and fuzzy demand ${\tilde{d}}_{lk}$ , respectively, for any ω ∈ Θ .

In the two-stage fuzzy 0-1 mixed-integer optimization model, the objective function (5) represents the expected cost minimization of the supply chain. The second-stage problem (6) consists of eight parts. The first term is the cost of raw materials, the second term is the production cost of the processing factory, the third, fourth and fifth terms are the transportation costs, the sixth term is the distribution cost, the seventh term is the inventory cost, and the last term is the revenue obtained by selling products.

Constraints (7a), (7c), (7e) and (7g) represent the raw material constraints of suppliers, the transport capacity constraints of transfer centers, the production capacity constraints of processing factories and the distribution capacity constraints of distribution centers, respectively. Constraints (7b), (7d) and (7f) represent the balance conditions of raw materials and products, respectively. Constraint (7h) ensures that customer’s demand can not be overserved. The final constraint (7i) guarantees the non-negativity of decision variables.

In model (5)-(6), the two-stage process of three-level location allocation problem is shown in Fig. 2. The decision vector α , β and γ are the first-stage decision which must be considered before the realizations of fuzzy transportation cost and demand vector ξ (ω) coming out. In the second-stage, the fuzzy demand and transportation cost ξ (ω), α , β and γ are known, the decision variable are e, x, y, z. The goal is to minimize costs in the supply chain.

Fig. 2

Two-stage process of fuzzy supply chain problem.

For each ξ (ω) , (7a) contains |V||S| constraints, (7b) contains |V||M| constraints, (7c) contains |V||M| constraints, (7d) contains |V||I| constraints, (7e) contains |I||L| constraints, (7f) and (7g) contains |L||J| constraints, respectively, (50) contains |L||K| constraints, (51) contains |V||S||M| + |V||M||I| + |L||I||J| + |L||J||K| constraints. So problem (6) has |V||S|+2|V||M| + |V||I| + |I||L|+2|L||J| + |L||K| + |V||S||M| + |V||M||I| + |L||I||J| + |L||J||K| constraints for each ξ (ω). This makes model (5)-(6) very complicated and difficult to solve. Based on the complexity of model (5)-(6), in the following part, we propose a hybrid intelligent algorithm that can effectively solve it.

4 Solution algorithm

This section focuses on the computation of the two-stage fuzzy 0-1 mixed integer optimization model (5)-(6). Firstly, we study the calculation of recourse function $Q (α, β, γ)$ . To get $Q (α, β, γ)$ , we need to compute Q ( α , β , γ , ξ (ω)) by solving the second-stage problem (6) of each ξ (ω). However, in general, for each ω, if ξ (ω) is a continuous fuzzy vector, computing $Q (α, β, γ)$ becomes solving an infinite linear programming problem. When fuzzy vectors are continuous, Wang et al. [33] proposed a method to construct a discrete fuzzy vector for approximating continuous fuzzy vectors. Without loss of generality, in the following discussion, we denote by ξ_n (ω) the n - th scenario and the corresponding nominal possibility is μ_n, n ∈ N, where N is the set of scenarios.

Among many intelligent algorithms, PSO has the advantages of fast search speed, high efficiency and simple. And the second order particle swarm optimization algorithm ([46]) avoids falling into local optimum. So, based on the above discussion and the complexity of the problem considered in this paper, a modified hybrid second order particle swarm optimization algorithm (MHSO-PSO) is proposed to solve the two-stage fuzzy 0-1 mixed integer optimization model (5)-(6).

In order to use MHSO-PSO to solve two-stage fuzzy 0-1 mixed integer optimization model more effectively, we make the following modifications to MHSO-PSO:

(a) We combine the synchronous dynamic learning factor and compression factor, and added them to the SO-PSO, which improve the convergence speed of the algorithm and guaranteed the global convergence of the algorithm.

(b) We introduce boundary conditions in MHSO-PSO to improve the search efficiency of particles and avoid the expansion and dispersal of particle swarm.

Next, we introduce the detailed solving process of the algorithm.

Let X = (α₁, ⋯ , α_M, β₁, ⋯ , β_I, γ₁, ⋯ , γ_J), D = M + I + J . Suppose that in the D dimensional objective search space, there are I′ particles forming a community, in which the i′ - th particle is represented as a D-dimensional vector X_i′ = (χ_i′1, ⋯ , χ_i′d, ⋯ , χ_i′D) , i′ = 1, 2, ⋯ , I′ . The flight velocity of the i′ - th particle is also a D-dimensional vector, which is written as $V_{i^{'}} = (v_{i^{'} 1}, v_{i^{'} 2}, \dots, v_{i^{'} D}),$ i′ = 1, 2, ⋯ , I′ . The personal best position and global best position of particle i′ in the k′ - th iteration denoted as $P_{{bi}^{'}}^{k^{'}}$ and $P_{{gi}^{'}}^{k^{'}}$ , respectively.

Denote Fit (·) is the fitness function, and let the fitness of each particle be the minus of the first stage value, i.e.,

$- Fit (X) = \sum_{M} α_{m} c_{m} + \sum_{I} β_{i} f_{i} + \sum_{J} γ_{j} g_{j} + Q (X),$

where

$Q (X) = \sum_{n = 1}^{N} p_{n} Q (X, ξ_{n}),$ the corresponding weights p_n are given in the form $p_{n} = \frac{1}{2} (max_{1 \leq t \leq N} {μ_{t} | Q (X, ξ_{t}) \leq Q (X, ξ_{n})}$ (8) $\begin{matrix} - max_{1 \leq t \leq N} {μ_{t} | Q (X, ξ_{t}) < Q (X, ξ_{n})}) \\ + \frac{1}{2} (max_{1 \leq t \leq N} {μ_{t} | Q (X, ξ_{t}) \geq Q (X, ξ_{n})} \\ - max_{1 < t \leq N} {μ_{t} | Q (X, ξ_{t}) > Q (X, ξ_{n})}), \end{matrix}$ and satisfy the following constraint $p_{n} \geq 0, \sum_{n = 1}^{N} p_{n} = 1, n = 1, \dots, N .$

Therefore, the particles of smaller the target value in the first stage are evaluated with higher fitness. Among them, for each ξ (ω), the second stage function value Q (X, ξ (ω)) is calculated by simplex algorithm.

In the process of calculation, the updated formula of particle swarm velocity vector in the k′ - th iteration is as follows: $V_{i^{'}}^{k^{'} + 1} = ϱ V_{i^{'}}^{k^{'}} + c_{1} \cdot rand \cdot (P_{{bi}^{'}}^{k^{'}} - 2 X_{i^{'}}^{k^{'}}$ (9) $+ X_{i^{'}}^{k^{'} - 1}) + c_{2} \cdot rand \cdot (P_{{gi}^{'}}^{k^{'}} - 2 X_{i^{'}}^{k^{'}} + X_{i^{'}}^{k^{'} - 1}),$ where c₁, c₂ ∈ [c_min, c_max] are the dynamic learning factors, the learning factor at the k′ - th iteration is

$c_{1} = c_{2} = c_{\min} + \frac{(c_{\max} - c_{\min}) k^{'}}{T},$ and satisfies c₁, c₂ ≥ 2, T is the maximum number of iterations. ϱ is the compression factor which is set by the following expression Clerc and Kennedy ([47])

$ϱ = \frac{2}{| 2 - ς - \sqrt{ς^{2} - 4 ς} |},$ where ς = c₁ + c₂ .

In (9), the first term represents the velocity of the previous iteration of particles, which is used to ensure the global convergence of the algorithm. The second to fourth terms guarantee the local convergence of the algorithm. As the number of iterations increases, c₁ and c₂ linearly increase and ϱ decreases. In this way, it can ensure that at the beginning of the algorithm, each particle can detect a better region in the global range with a greater speed step. Later in the iteration, a smaller ϱ ensures that the particle can do fine search around the extreme point, so that the algorithm has a greater probability to converge to the global optimal solution.

In order to improve the search efficiency of particles, avoid the expansion and divergence of the population, and avoid the blind search of particles in a wide range, we introduce the following boundary conditions: $(V_{i^{'}}^{k^{'} + 1})_{d} = {\begin{matrix} rand \cdot (V_{\max} - V_{\min}) + V_{\min}, \\ if (V_{i^{'}}^{k^{'} + 1})_{d} > V_{\max} \\ rand \cdot (V_{\max} - V_{\min}) + V_{\min}, \\ if (V_{i^{'}}^{k^{'} + 1})_{d} < V_{\min} \\ (V_{i^{'}}^{k^{'} + 1})_{d}, others . \end{matrix}$ (10)

In (10), $V_{\max}$ is the maximum velocity limit and $V_{\min}$ is the minimum velocity limit. Since the value and change of the particle in the state space are limited to the two values of 0 and 1, and the $(V_{i^{'}}^{k^{'} + 1})_{d}$ of each dimension of velocity represents the possibility that the value of position $(X_{i^{'}}^{k^{'} + 1})_{d}$ is 1. The location update equation [48] is expressed as follows:

$S (V_{i^{'}}^{k^{'} + 1}) = \frac{1}{1 + \exp (- V_{i^{'}}^{k^{'} + 1})},$ (11)

$(X_{i^{'}}^{k^{'} + 1})_{d} = {\begin{matrix} 1, S (V_{i^{'}}^{k^{'} + 1})_{d} > rand \\ 0, others \end{matrix} .$ (12)

In (11), $S (V_{i^{'}}^{k^{'} + 1})$ is a sigmoid function. In (12), $X_{i^{'}}^{k^{'} + 1}$ denotes the position of the i′ particle after the k′ - th iteration.

Based on the above description, we give the detailed calculation process of the hybrid intelligent algorithm.

Algorithm 1: Modified Hybrid Second Order Particle Swarm Optimization Algorithm (MHSO-PSO)

Step 0. Initialize the particle swarm. The population size is I′, maximum number of iterations T, $V_{\max} > V_{\min},$ c_max > c_min > 0, $P_{d}^{0} = ones (I^{'}, 1),$ $P_{g}^{0} = eps,$ X_best = ones (1, M + I + J), k′ = 0, i′ = 1 .

Step 1. Initializes particle velocity and position. Randomly obtain the initial population of binary code X⁰ = round (rand (I′, M + I + J)) , X^-1 = round (rand (I′, M + I + J)) , P = X⁰, $V^{0} = rand (I^{'}, M + I + J) \cdot (V_{\max} - V_{\min}) + V_{\min} .$

Step 2. If k′ > T, Stop. Return the particle X_best as the optimal solution, and -Fit (X_best) as the optimal value.

Step 3. If i′ > I′, set k′ = k′ + 1, i′ = 1, go to Step 2.

Step 4. Calculate the fitness $Fit (X_{i^{'}}^{k^{'}})$ through the simplex algorithm.

Step 5. Update the individual’s optimal position and value. If $Fit (X_{i^{'}}^{k^{'}}) > P_{{di}^{'}}^{k^{'}},$ $P_{i^{'}} = X_{i^{'}}^{k^{'}}$ , $P_{{di}^{'}}^{k^{'}} = Fit (X_{i^{'}}^{k^{'}}),$ else $P_{{di}^{'}}^{k^{'}} = P_{{di}^{'}}^{k^{'}} .$

Step 6. Update the global optimal position and value. If $P_{{di}^{'}}^{k^{'}} > P_{g}^{k^{'}},$ X_best = P_i′, $P_{g}^{k^{'}} = P_{{di}^{'}}^{k^{'}},$ else $P_{g}^{k^{'}} = P_{g}^{k^{'}} .$

Step 7. Update velocity particles by (9).

Step 8. Boundary condition treatment by (10).

Step 9. Update location particle by (11)-(12).

Step 10. Set i′ = i′ + 1, go to Step 3.

5 Numerical results

In this section, we apply the MHSO-PSO to solve a practical case and to provide a discussion of the results. All the program codes are written on MATLAB R2014a using Lenovo computers running on Intel(R) Core(TM) i7-8565U CPU @ 1.80 GHz, 8.00-GB memory. Throughout the computational experiments, the parameters in MHSO-PSO are taken as: I′ = 100, $V_{\max} = 10, V_{\min} = - 10,$ c_min = 2, c_max = 2.1. The stopping criterion for the algorithm is T = 1000 . In the respective tables of the numerical results, “Pro” denotes the total supply chain’s profit, ( α _best, β _best, γ _best) denote the optimal solutions for problem (5). $e_{vsm}^{*} = \sum_{N} p_{n} e_{vsm}^{n}$ denotes the expected quantity of raw material v transported from the supplier s to the transfer center m, $x_{vmi}^{*} = \sum_{N} p_{n} x_{vmi}^{n}$ denotes the expected quantity of raw material v transported from the transfer center m to the processing factory i, $y_{lij}^{*} = \sum_{N} p_{n} y_{lij}^{n}$ denotes the expected quantity of product l transported from the processing factory i to the distribution center j, $z_{ljk}^{*} = \sum_{N} p_{n} z_{ljk}^{n}$ denotes the expected quantity of product l transported from the the distribution center j to the consumer k, where, $(e_{vsm}^{n}, x_{vmi}^{n}, y_{lij}^{n}, z_{ljk}^{n})$ represents the optimal solution of problem (6) in scenarios n. “TI” denotes the computing time in seconds. In what follows, we provide a description of the test problem.

Taking Shanghai clothing supply chain network as an example, this paper studies the specific influence of uncertainty on three-level location allocation in supply chain. In the supply chain network, two fabric factories provide raw materials, in order to facilitate transportation and save costs, choose raw materials transport in four transfer centers, choose T-shirts production in four clothing factories, and then choose distribution in eight distribution centers to meet the clothing needs of customers in four demand areas of Shanghai. In addition, we assume that this batch of garments will be processed in 10 days and sold and distributed in 30 days.

As shown in Fig. 3, the specific location of the fabric market is marked by two grey-green houses, namely Suzhou fabric market (supplier 1) and Jiaxing fabric market (supplier 2). The specific location of the transfer center is marked by four blue cars, namely Anting transfer center, Shanghai transfer center, Songjiang transfer center and Jinshan transfer center. The specific location of the clothing factory is marked by four green triangles, namely Shanghai clothing factory, Weimin clothing factory, Mingchen clothing factory and Niukou clothing factory. They are recorded as processing factories 1-4. The specific location of distribution center is marked with 8 yellow houses. They are distribution center 1-8. The consumer demand area is marked by four minions, which are defined as “Demand area 1 (green), 2 (gray), 3 (red) and 4 (blue)”. Demand area 1 consists of jiading district, baoshan district, putuo district, changning district, xuhui district, huangpu district, jingan district, hongkou district and yangpu district. Demand area 2 consists of qingpu district, songjiang district and minhang district. Demand area 3 consists of jinshan district and fengxian district. Demand area 4 consists of pudong new district. As chongming district is an independent island, the consumer demand in chongming district is not considered in this calculation example.

Fig. 3

Location of fabric market, transfer center, clothing factory, distribution center and clothing demand area in Shanghai.

For the convenience of calculation, this paper only considers a class of clothes with the same price, and the unit raw material production unit clothes, i.e., V = 1, L = 1, n_vl = 1. Also, assume that distribution center can successfully deliver clothes to the demand area. The data for this case-study are partly calculated from actual data, and some are generated manually based on our personal communications with several players in this sector. The parameters related to fabric market, transfer center, clothing factory and distribution center are shown in Table 2, where, q_i1 includes the cost of work, packaging, water and electricity, equipment depreciation. f_i and g_j are the daily rent.

Table 2

Parameters for suppliers, transfer centers, processing factories, and distribution centers

Index	Suppliers		Transfer	centers	Processing		plants	Distribution		centers
s, m, i, j	a _1s	r _1s	η _1m	c _m	b _i1	q _i1	f_i * 10	τ _1j	w _1j	g_j * 30
1	6500	40	5100	1120	4500	28.80	5000	2200	0.25	235 * 30
2	6000	39	5200	1180	4700	28.50	5600	2100	0.25	230 * 30
3	/	/	5400	1210	4600	28.20	5200	2000	0.25	220 * 30
4	/	/	5150	1150	4800	28.70	5750	2050	0.25	225 * 30
5	/	/	/	/	/	/	/	2100	0.25	240 * 30
6	/	/	/	/	/	/	/	2150	0.25	220 * 30
7	/	/	/	/	/	/	/	2300	0.25	245 * 30
8	/	/	/	/	/	/	/	1950	0.25	243 * 30

Table 3 shows the transportation cost from the fabric market to the transfer center. Transportation costs include fuel costs, vehicle costs and driver wages. The formula for calculating the unit transportation cost is given below,

Table 3

Fuzzy transportation cost from fabric market to the transfer center

Transfer center\Fabric market	Anting	Shanghai	Songjiang	Jinshan
Suzhou	(0.21, 0.23, 0.25)	(0.22, 0.24, 0.26)	(0.22, 0.24, 0.26)	(0.24, 0.26, 0.28)
Jiaxing	(0.24, 0.26, 0.28)	(0.24, 0.26, 0.28)	(0.22, 0.24, 0.26)	(0.22, 0.24, 0.26)

fuel costs = distance * fuel consumption * oil price,

Unit transportation cost = (fuel costs + vehicle costs + driver wages) /1000.

For example, the distance from Suzhou fabric Market to Anting transfer center is 53 km, the fuel consumption is 12 L/100 km, the oil price is 6.67 CNY / L, the vehicle cost is 80 CNY (Chinese Yuan), and the driver wage is 90 CNY. Each shipment of 1000 units of raw materials, hence, (53 * 12/100 * 6.67 + 80 + 90)/1000 = 0.21 CNY. Due to traffic congestion, traffic flow and other uncertain factors, we use triangular fuzzy number (0.21, 0.23, 0.25) to represent the fuzzy transportation cost.

Similarly, Table 4 shows the fuzzy transportation cost of transfer center transporting fabric to clothing factory. Table 5 shows the fuzzy transportation cost of clothing factory transporting product to distribution center. Table 6 shows the fuzzy distribution cost of distribution center transporting product to consumer. According to the market survey statistics, the fuzzy demand of consumers is given in Table 7, where, ϱ represents the price elasticity of demand, ϱ = 8, h₁ = 89 CNY.

Table 4

Fuzzy transportation cost of transfer center transporting fabric to clothing factory

Transfer center\Clothing factory	Shanghai	Weimin	Mingchen	Niukou
Anting	(0.19, 0.21, 0.23)	(0.21, 0.23, 0.25)	(0.20, 0.22, 0.24)	(0.21, 0.23, 0.25)
Shanghai	(0.19, 0.21, 0.23)	(0.20, 0.22, 0.24)	(0.19, 0.21, 0.23)	(0.20, 0.22, 0.24)
Songjiang	(0.20, 0.22, 0.24)	(0.21, 0.23, 0.25)	(0.19, 0.21, 0.23)	(0.20, 0.22, 0.24)
Jinshan	(0.22, 0.24, 0.26)	(0.22, 0.24, 0.26)	(0.20, 0.22, 0.24)	(0.19, 0.21, 0.23)

Table 5

Fuzzy transportation cost of clothing factory transporting product to distribution center

Distribution center\Clothing factory	Shanghai	Weimin	Mingchen	Niukou
Distribution center 1	(0.19, 0.21, 0.23)	(0.18, 0.20, 0.22)	(0.20, 0.22, 0.24)	(0.21, 0.23, 0.25)
Distribution center 2	(0.18, 0.20, 0.22)	(0.19, 0.21, 0.23)	(0.19, 0.21, 0.23)	(0.20, 0.22, 0.24)
Distribution center 3	(0.19, 0.21, 0.23)	(0.21, 0.23, 0.25)	(0.19, 0.21, 0.23)	(0.20, 0.22, 0.24)
Distribution center 4	(0.19, 0.21, 0.23)	(0.19, 0.21, 0.23)	(0.18, 0.20, 0.22)	(0.19, 0.21, 0.23)
Distribution center 5	(0.20, 0.22, 0.24)	(0.19, 0.21, 0.23)	(0.20, 0.22, 0.24)	(0.20, 0.22, 0.24)
Distribution center 6	(0.20, 0.22, 0.24)	(0.19, 0.21, 0.23)	(0.19, 0.21, 0.23)	(0.19, 0.21, 0.23)
Distribution center 7	(0.21, 0.23, 0.25)	(0.21, 0.23, 0.25)	(0.19, 0.21, 0.23)	(0.19, 0.21, 0.23)
Distribution center 8	(0.21, 0.23, 0.25)	(0.20, 0.22, 0.24)	(0.19, 0.21, 0.23)	(0.18, 0.20, 0.22)

Table 6

Fuzzy distribution cost of distribution center transporting product to consumer

Distribution center\Demand area	Demand area 1	Demand area 2	Demand area 3	Demand area 4
Distribution center 1	(1.50, 1.70, 1.90)	(1.70, 1.90, 2.10)	(2.30, 2.50, 2.70)	(1.00, 1.20, 1.40)
Distribution center 2	(1.00, 1.20, 1.40)	(1.50, 1.70, 1.90)	(2.00, 2.20, 2.40)	(1.50, 1.70, 1.90)
Distribution center 3	(1.50, 1.70, 1.90)	(1.00, 1.20, 1.40)	(1.70, 1.90, 2.10)	(1.70, 1.90, 2.10)
Distribution center 4	(1.20, 1.40, 1.60)	(1.20, 1.40, 1.60)	(1.50, 1.70, 1.90)	(1.20, 1.40, 1.60)
Distribution center 5	(1.70, 1.90, 2.10)	(1.70, 1.90, 2.10)	(1.70, 1.90, 2.10)	(1.00, 1.20, 1.40)
Distribution center 6	(1.90, 2.10, 2.30)	(1.80, 2.00, 2.20)	(1.30, 1.50, 1.70)	(1.30, 1.50, 1.70)
Distribution center 7	(1.80, 2.00, 2.20)	(1.00, 1.20, 1.40)	(1.40, 1.60, 1.80)	(1.80, 2.00, 2.20)
Distribution center 8	(2.00, 2.20, 2.40)	(1.80, 2.00, 2.20)	(1.00, 1.20, 1.40)	(1.60, 1.80, 2.00)

Table 7

fuzzy demand of consumer

Demand area	Demand
Demand area 1	(2000, 2020, 2040) - ϱh₁
Demand area 2	(2900, 2920, 2940) - ϱh₁
Demand area 3	(2450, 2470, 2490) - ϱh₁
Demand area 4	(2550, 2570, 2590) - ϱh₁

In order to solve the fuzzy three-level location allocation problem, for any feasible solution, we first random generate 1000 fuzzy sampling points ξ_n (ω), n = 1, 2, ⋯ , 1000, for random simulation (such sample size is sufficient for simulation of fuzzy expected value). We calculate the corresponding membership function according to the Definition 2. Specifically as follows: suppose the triangular fuzzy number is represented by $\tilde{a} = ({\tilde{a}}^{L^{'}}, {\tilde{a}}^{M^{'}}, {\tilde{a}}^{U^{'}})$ , then according to the definition of aggregation operator, we get $\tilde{ξ} (ω)$ = (ξ^L′ (ω) , ξ^M′ (ω) , ξ^U′ (ω)), where ξ^L′ (ω) = ∑_VSM ϖ_vsm ${\tilde{t}}_{vsm}^{L^{'}} (ω)$ +∑_VMI ϖ_vmi ${\tilde{n}}_{vmi}^{L^{'}} (ω) +$ ∑_LIJ ϖ_lij ${\tilde{m}}_{lij}^{L^{'}} (ω) +$ ∑_LJK ϖ_ljk ${\tilde{μ}}_{ljk}^{L^{'}} (ω) +$ ∑_LK ϖ_lk ${\tilde{d}}_{lk}^{L^{'}} (ω)$ , ξ^M′ (ω) = ∑_VSM ϖ_vsm ${\tilde{t}}_{vsm}^{M^{'}} (ω)$ +∑_VMI ϖ_vmi ${\tilde{n}}_{vmi}^{M^{'}} (ω)$ +∑_LIJ ϖ_lij ${\tilde{m}}_{lij}^{M^{'}} (ω)$ +∑_LJK ϖ_ljk ${\tilde{μ}}_{ljk}^{M^{'}} (ω)$ +∑_LK ϖ_lk ${\tilde{d}}_{lk}^{L^{'}} (ω),$ ξ^U′ (ω) = ∑_VSMϖ_vsm ${\tilde{t}}_{vsm}^{U^{'}} (ω)$ +∑_VMI ϖ_vmi ${\tilde{n}}_{vmi}^{U^{'}} (ω)$ +∑_LIJ ϖ_lij ${\tilde{m}}_{lij}^{U^{'}} (ω)$ +∑_LJK ϖ_ljk ${\tilde{μ}}_{ljk}^{U^{'}} (ω)$ +∑_LK ϖ_lk ${\tilde{d}}_{lk}^{U^{'}} (ω)$ , and ∑_VSMϖ_vsm+ ∑_VMIϖ_vmi +∑_LIJ ϖ_lij +∑_LJK ϖ_ljk+ ∑_LK ϖ_lk = 1. In this paper, for ∀s, m, i, j, k, v, l, we take ϖ_vsm = ϖ_vmi = ϖ_lij = ϖ_ljk = ϖ_lk = 1/(|V||S||M| + |V||M||I| + |L||I||J| + |L||J||K| + |L||K|) . Then the membership function corresponding to ξ_n (ω) is

$μ (ξ_{n} (ω)) = {\begin{matrix} \frac{a^{0} - ξ_{n}^{L^{'}} (ω)}{ξ_{n}^{M^{'}} (ω) - ξ_{n}^{L^{'}} (ω)}, ξ_{n}^{L^{'}} (ω) \leq a^{0} \leq ξ_{n}^{M^{'}} (ω) \\ \frac{ξ_{n}^{U^{'}} (ω) - a^{0}}{ξ_{n}^{U^{'}} (ω) - ξ_{n}^{M^{'}} (ω)}, ξ_{n}^{M^{'}} (ω) \leq a^{0} \leq ξ_{n}^{U^{'}} (ω) \\ 0, Others \end{matrix},$

where $a^{0} = \sum_{VSM} ϖ_{vsm} {\tilde{t}}_{nvsm} (ω) +$ ∑_VMI ϖ_vmi ${\tilde{n}}_{nvmi} (ω) +$ ∑_LIJ ϖ_lij ${\tilde{m}}_{nlij} (ω) +$ ∑_LJK ϖ_ljk ${\tilde{μ}}_{nljk} (ω) +$ ∑_LK ϖ_lk ${\tilde{d}}_{nlk} (ω) .$

We put the data in Tables 2 and the generated sample points into two-stage fuzzy 0-1 mixed integer programming model (5)-(6) and used MHSO-PSO to solve it. The numerical results are given in Tables 8-12, and Fig. 4, 5, respectively.

Table 8

Numerical optimal solution and value of the example

α _best = (1, 0, 1, 0)	β _best = (0, 1, 1, 0)	γ _best = (0, 1, 1, 1,	0, 1, 0, 0)
$e_{111}^{*} = 1.0177 e + 03$	$e_{113}^{*} = 0.1186 e + 03$	$e_{121}^{*} = 1.0847 e + 03$	$e_{123}^{*} = 4.9153 e + 03$
$x_{112}^{*} = 1.4699 e + 03$	$x_{113}^{*} = 0.6326 e + 03$	$x_{132}^{*} = 1.0664 e + 03$	$x_{133}^{*} = 3.9674 e + 03$
$y_{122}^{*} = 0.7565 e + 03$	$y_{123}^{*} = 0.0815 e + 03$	$y_{124}^{*} = 0.3959 e + 03$	$y_{126}^{*} = 1.3024 e + 03$
$y_{132}^{*} = 0.5817 e + 03$	$y_{133}^{*} = 1.9185 e + 03$	$y_{134}^{*} = 1.6266 e + 03$	$y_{136}^{*} = 0.4732 e + 03$
$z_{121}^{*} = 1.3096 e + 03$	$z_{122}^{*} = 0.0262 e + 03$	$z_{124}^{*} = 0.0262 e + 03$	$z_{132}^{*} = 2.0000 e + 03$
$z_{142}^{*} = 0.1783 e + 03$	$z_{143}^{*} = 0.1742 e + 03$	$z_{144}^{*} = 1.6700 e + 03$	$z_{163}^{*} = 1.5867 e + 03$
$z_{164}^{*} = 0.1889 e + 03$	Pro = 9.9711e + 04	Others=0

Table 9

Comparisons of Different Algorithms

Algorithm	α _best	β _best	γ _best	Pro	TI
MHSO-PSO	(1, 0, 1, 0)	(0, 1, 1, 0)	(0, 1, 1, 1, 0, 1, 0, 0)	9.9711e + 04	13988.2385
Hybrid PSO	(1, 1, 0, 0)	(0, 1, 1, 0)	(0, 1, 1, 1, 0, 1, 0, 0)	9.9361e + 04	14659.1370
Hybrid GA	(0, 1, 0, 1)	(0, 1, 1, 0)	(1, 0, 1, 1, 0, 1, 0, 0)	9.9392e + 04	14840.2878

Table 10

Results of MHSO-PSO with different parameters

System	Parameters					Results
T	I′	c _min	c _max	α _best	β _best	γ _best	Pro	Error(%)
50	10	2.0	2.1	(1, 0, 0, 1)	(0, 1, 1, 0)	(1, 0, 0, 1, 0, 1, 1, 0)	9.9048e + 04	0.76
100	10	2.0	2.1	(0, 1, 0, 1)	(0, 1, 1, 0)	(0, 1, 1, 1, 0, 1, 0, 0)	9.9302e + 04	0.50
1000	10	2.0	2.1	(1, 0, 1, 0)	(0, 1, 1, 0)	(0, 1, 1, 1, 0, 1, 0, 0)	9.9414e + 04	0.39
2000	10	2.0	2.1	(1, 0, 0, 1)	(1, 0, 1, 0)	(0, 1, 1, 1, 0, 1, 0, 0)	9.9651e + 04	0.15
1000	100	2.0	2.1	(1, 0, 1, 0)	(0, 1, 1, 0)	(0, 1, 1, 1, 0, 1, 0, 0)	9.9711e + 04	0.09
1000	1000	2.0	2.1	(1, 0, 0, 1)	(0, 1, 1, 0)	(1, 0, 1, 1, 0, 1, 0, 0)	9.9803e + 04	0.00
1000	10	2.0	2.5	(1, 0, 1, 0)	(0, 1, 1, 0)	(0, 1, 1, 1, 0, 1, 0, 0)	9.9468e + 04	0.34
1000	10	2.0	3.0	(1, 1, 0, 0)	(0, 1, 1, 0)	(0, 1, 1, 1, 0, 1, 0, 0)	9.9459e + 04	0.35
1000	10	2.5	3.0	(0, 1, 0, 1)	(0, 1, 1, 0)	(1, 0, 1, 1, 0, 1, 0, 0)	9.9338e + 04	0.47

Table 11

Supply chain profit with fuzzy and expected transportation cost and demand

ξ (ω)	α _best	β _best	γ _best	Pro
Fuzzy	(1, 0, 1, 0)	(0, 1, 1, 0)	(0, 1, 1, 1, 0, 1, 0, 0)	9.9711e + 04
Expected	(1, 0, 0, 1)	(0, 1, 1, 0)	(1, 1, 1, 0, 0, 1, 0, 0)	9.9380e + 04

Table 12

Effect of raw material cost and retail price on supply chain profit

(r₁₁, r₁₂)	h ₁	α _best	β _best	γ _best	Pro
(40.0, 39.0)	80.1	(1, 0, 1, 0)	(0, 1, 1, 0)	(0, 1, 1, 1, 0, 1, 0, 0)	3.8674e + 04
(40.0, 39.0)	85.0	(1, 0, 1, 0)	(0, 1, 1, 0)	(0, 1, 1, 1, 0, 1, 0, 0)	7.2956e + 04
(40.0, 39.0)	89.0	(1, 0, 1, 0)	(0, 1, 1, 0)	(0, 1, 1, 1, 0, 1, 0, 0)	9.9711e + 04
(40.0, 39.0)	99.0	(1, 0, 0, 1)	(0, 1, 1, 0)	(1, 0, 0, 0, 0, 1, 1, 0)	1.6245e + 05
(39.0, 39.0)	89.0	(1, 0, 1, 0)	(0, 1, 1, 0)	(1, 1, 1, 0, 0, 1, 0, 0)	1.0055e + 05
(41.0, 39.0)	89.0	(0, 0, 1, 1)	(0, 1, 1, 0)	(1, 1, 1, 0, 0, 1, 0, 0)	9.8189e + 04
(40.0, 38.0)	89.0	(1, 1, 0, 0)	(1, 0, 1, 0)	(0, 1, 1, 1, 0, 1, 0, 0)	1.0535e + 05
(40.0, 40.0)	89.0	(1, 1, 0, 0)	(0, 1, 1, 0)	(1, 0, 1, 1, 0, 1, 0, 0)	9.3385e + 04

Fig. 4 shows the structure diagram of the three-level location allocation supply chain network, and the specific results are given in Table 8. In order to meet customers’ needs to the greatest extent, enterprises choose Anting, Songjiang Transfer Center, and choose Weimin, Mingchen clothing factory to produce T-shirts, and choose distribution centre 2, 3, 4 and 6 to deliver T-shirt to customers. At this time, the maximum profit of the supply chain is 9.9711e + 04 CNY. The expect demand scope of Demand area 1 is 1308, which is provided by distribution centre 2 with 1.3096e + 03. The expect demand scope of Demand area 2 is 2208, which is provided by distribution centre 2 with 0.0262e + 03, distribution centre 3 with 2.0000e + 03 and distribution centre 4 with 0.1783e + 03, respectively. The expect demand scope of Demand area 3 is 1758, which is provided by distribution centre 4 with 0.1742e + 03 and distribution centre 6 with 1.5867e + 03, respectively. The expect demand scope of Demand area 4 is 1858, which is provided by distribution centre 2 with 0.0262e + 03, distribution centre 4 with 1.6700e + 03 and distribution centre 6 with 0.1889e + 03, respectively. Numerical results show that the supply chain system has the ability to meet the demand, which reflects the reliability of the supply chain system. In addition, Anting transfer center transferred 1.0177e + 03 units of fabrics from Suzhou fabric market and 1.0847e + 03 units from Jiaxing fabric market respectively. Then it will ship 1.4699e + 03 units of fabrics to Weimin clothing factory and 0.6326e + 03 units to Mingchen clothing factory. Songjiang transfer center transferred 0.1186e + 03 units of fabrics from Suzhou fabric market and 4.9153e + 03 units from Jiaxing fabric market respectively. Then it will ship 1.0664e + 03 units of fabrics to Weimin clothing factory and 3.9674e + 03 units to Mingchen clothing factory. Weimin clothing factory processes the cloth into T-shirts, and then transports them to distribution center 2, 3, 4 and 6 respectively. The volume of transportation is respectively 0.7565e + 03, 0.0815e + 03, 0.3959e + 03 and 1.3024e + 03. Mingchen clothing factor processes the cloth into T-shirts, and then transports them to distribution center 2, 3, 4 and 6 respectively. The volume of transportation is respectively 0.5817e + 03, 1.9185e + 03, 1.6266e + 03, 0.4732e + 03. Meanwhile, it can be seen from Fig. 4 that the selected clothing factories and distribution centers are close to each other, both of which are located in the relative centers of the whole demand market, which is conducive to distribution, reducing transportation costs and increasing profits of the supply chain.

Fig. 4

Three-level location allocation supply chain network.

In order to further evaluate the performance of our proposed MHSO-PSO, we compare it with other discrete hybrid algorithms, such as binary PSO ([48]) and GA ([49, 50]). The numerical results are shown in Table 8 and Fig. 5.

Fig. 5

Comparisons of Different Algorithms.

In Table 8, we adopted three intelligent algorithms to solve the supply chain three-level location allocation problem, and obtained different optimal solutions and optimal values. Among them, when MHSO-PSO is applied to solve the problem, the supply chain profit is the largest, and the algorithm has a shorter computing time. It can be seen from Fig. 5 that MHSO-PSO converges faster than Hybrid PSO and Hybrid GA, indicating that MHSO-PSO is more suitable for two-stage fuzzy mixed integer programming.

Remark 1. In the computational experiments, the parameters in Hybrid PSO are taken as: I′ = 100, $V_{\max} = 10, V_{\min} = - 10,$ c_min = 2, c_max = 2.1, inertia weight W_max = 0.8, W_min = 0.4, maximum Iterations T = 1000 . The parameters in Hybrid GA are taken as: population size is 100, crossover operator p_c = 0.8, mutation operator p_m = 0.05, maximum Iterations T = 1000 .

In order to better prove the performance of MHSO-PSO, we verified the stability of the MHSO-PSO by setting different iteration numbers, population size and different parameters. The numerical results are shown in Table 10. The last column represents the relative error, which is defined by $Error = \frac{Optimal Pro - Pro}{Optimal Pro} \times 100 % .$

In Table 10, when we set different parameters, iteration numbers and population size in the MHSO-PSO, the relative error is no more than 0.76%, which indicates that the MHSO-PSO has strong robustness to parameters and can effectively solve the two-stage fuzzy mixed integer programming.

Table 11 shows the supply chain profit under fuzzy and expected cost demand. Table 11 shows that in these two cases, the location decision of the supply chain is different, and in the case of fuzzy cost demand, the supply chain profit is greater than expected, the supply chain profit increases by 0.33%. This indicates that the cost of ignoring the fuzziness of cost requirements in selection decisions is 331 CNY. This value is the motivation for fuzzy programming, which assesses the value of knowing and using distributions on future outcomes. In some cases, more information might be available through more extensive forecasting, sampling, or exploration. In these cases, there are more fuzziness in the problem, so using fuzzy programming modeling becomes more meaningful and practical.

With the advent of disruptive technology, effective cost reduction is an effective means to reap more benefits and greater profitability. Table 12 shows the effect of the raw materials cost on the optimal location and profitability of supply chain. As can be seen from Table 12, by reducing the raw materials cost, the corresponding supply chain profit increases. This suggests that the decision makers can choose low-cost raw materials to obtain more benefits and profitability. Moreover, the retailers can influence consumer demand by adjusting their prices in the face of consumer demand uncertainty. In Table 12, as retail prices increase, the corresponding consumer demand will decrease, but the supply chain profit will increase. This shows that, enterprises can set reasonable retail prices to obtain corresponding profits.

6 Conclusion

In this paper, the optimal three-level location allocation of supply chain network under uncertain transportation cost and customer demand are studied. A two-stage fuzzy 0-1 mixed integer optimization model is established. Given the complexity of the model, this paper proposes a MHSO-PSO to solve the problem studied. Taking Shanghai clothing supply chain as an example, this paper studies the influence of uncertainty on the three-level location of transfer center, processing factory and distribution center. By solving the problem, the optimal location and distribution of transfer center, processing factory and distribution center are obtained, and then the maximum expected profit is obtained. Moreover, the effects of raw material cost, retail price on the expected return are also analyzed. The computational results suggest that our MHSO-PSO is better suited to a two-stage fuzzy 0-1 mixed integer optimization model.

Future research can consider the robustness and uncertainty of supplier supply and consumer demand in the supply chain network simultaneously, to investigate the interaction among the various forms of uncertainty, and their effects on the supply chain system.

Footnotes

Acknowledgment

The work is supported by a research grant from the National Social Science Foundation of China (No. 17BGL083). The authors thank the editor and the reviewers for their highly constructive comments on the manuscript.

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Two-stage fuzzy mixed integer optimization model for three-level location allocation problems under uncertain environment

Abstract

Keywords

1 Introduction

Table 1 Research on location allocation supply chains Reference Research focus [11, 31] Fuzzy demand [24, 33] Fuzzy cost [2, 31] facility location [3, 30] warehouse location [10, 29] distribution center location [32] transfer center location [22–28, 30] One-stage model [10, 31] Two-stage model

3.2 The model

Footnotes

Acknowledgment

References