An uncertain sustainable supply chain network

Applied Mathematics and Computation 378 (2020) 125213

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An uncertain sustainable supply chain network Jiayu Shen Department of Public Basic Courses, Nanjing Institute of Industry Technology, Nanjing 210023, Jiangsu, People’s Republic of China

a r t i c l e

i n f o

Article history: Received 25 October 2019 Revised 25 February 2020 Accepted 7 March 2020

Keywords: Sustainable Supply chain Chance-constrained Uncertainty theory Genetic algorithm

a b s t r a c t As the concept of sustainable development and environmental awareness has aroused huge attention from the public, environmental and social factors have gradually been critical to the development of upstream and downstream enterprises in the supply chain. An uncertain sustainable supply chain involving the factors of cost, environmental impact and social benefits is considered. Some factors (e.g., demand, cost and capacity) are recognized as uncertain variables. In the present study, a multi-objective chance-constrained model in the uncertain scenario is developed to delve into the impact of uncertainties on decision variables. In accordance with the uncertainty theory, this study elucidates the deterministic equivalence of the model. To solve this model effectively, a hybrid genetic algorithm is proposed based on variable length chromosome coding. Lastly, numerical experiments are performed to verify the feasibility of the model and algorithm. © 2020 Elsevier Inc. All rights reserved.

1. Introduction As fueled by the advancement of modern industry, the environmental pollution problem has given rise to numerous social problems and threatened the survival and development of human beings. In the 1990s, to enlarge the economic benefits of enterprises, the concept of sustainable supply chain management (SCM) was proposed, i.e., the economic, environmental and social goals of the organizations can be strategically and transparently integrated by systematically coordinating the core business processes of the organizations. Accordingly, it is of profound theoretical and practical implication for the performance assessment of sustainable supply chain. Over the past few years, numerous researchers have studied the subject. Pinto-Varela [34] modeled an economic and environmental supply chain problem with the use of Resource-Task-Network (RTN) methodology; fuzzy programming was adopted to coordinate conflicting objectives. Yu et al. [63] studied different environmental tax policies in a multi-layer supply chain network; their results revealed that the implementing processes of environmental tax policy and consumers’ preferences could motivate enterprises to manage their business sustainably. Mota et al. [28] developed a multi-objective sustainable supply chain model and presented the triple bottom line optimization modeling tool. Das [12] proposed a lean system integrated sustainable supply chain design model, proved to be capable of boosting sustainable economic, environmental and social development. Pishvaee and Razmi [35] built an environmental supply chain model; they also designed an interactive approach for the model. Baud-Lavigne [1] formed a supply chain network with carbon emissions. Besides, the analysis was conducted on the product commonality, the cost and the carbon emission allocation. Taleizadeh et al. [52] designed a multi-cycle, multi-level and sustainable closed-loop supply chain considering social and environmental aspects. As revealed from the conclusion, the proposed model enabled decision makers to use appropriate solutions considering legislative, social preferences, costs, as well as environmental objectives. Nidhi and Pillai [30] proposed a model for a sustainable closed loop supply chain. The proposed model was employed to a real-life blood E-mail address: [email protected] https://doi.org/10.1016/j.amc.2020.125213 0 096-30 03/© 2020 Elsevier Inc. All rights reserved.

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J. Shen / Applied Mathematics and Computation 378 (2020) 125213

bag supply chain; its performance was verified as well. Chalmardi and Camacho-Vallejo [3] conducted an analysis of a sustainable supply chain network design. Moreover, a novel bi-level programming model and an algorithm based on simulated annealing were established. Sustainable supply chain profit allotment was optimized by negotiating outsource price. Rabbani et al. [37] analyzed a sustainable bioenergy supply chain network based on switchgrass. Two-stage algorithms were adopted to process the trade-off between sustainable factors. Saberi [38] established a sustainable multi-period supply chain network model; with this network, the effects of various tax policies on network activities and pollution stock were explored. Interested readers may also refer to more relevant literature [2,4,11,19,20,27,39,47–49,62,64,65]. Most of the mentioned literature considered the sustainable supply chain network under stochastic conditions. However, historical data are not practically reliable or available, and the probability distributions of some uncertain factors are hard to obtain. For instance, as impacted by several unexpected events and upgrading of products, economic costs, market demand, and production capacity are hard to estimate with historical data. The probability theory is likely to yield counterintuitive results. In this scenario, we can invite some domain experts to assess the belief degree that each event will occur. It is inappropriate to address belief degrees based on probability theory since it may cause contradictions. Here is a counterexample. Assume a vehicle crosses 50 bridges. The weight of the vehicle is 90 tons, and the bearing capacity of the bridge complies with the iid uniform distribution [95,110]. Assume that a bridge will collapse when its bearing capacity reaches lower than the weight of a vehicle. Noticeably, the probability that the vehicle will cross 50 bridges is 1. However, when no samples of bridge bearing capacity are observed at present, some bridge engineers should be invited to assess their belief degrees. As mentioned above, for conservatism, people usually estimate a wider range of values than the actual load-bearing capacity of bridges. The belief degree function is assumed as



0,

(x ) = (x − 80 )/40, 1,

if x < 80 if 80 ≤ x ≤ 120 if x > 120

What will happen if the belief degree function is considered a probability distribution? First, the strength of 50 bridges should be considered as iid uniform random variable over [80,120] tons. If we set the vehicle cross 50 bridges one by one, it yields

Pr{“the truck can cross over the 50 bridges”

} = 0.7550 ≈ 0

Thus, it is almost impossible for the vehicle to cross 50 bridges successfully. However, the results are at opposite poles. Such example reveals that the improper use of probability theory can make the inevitable event impossible. To address the degree of belief, the uncertainty theory was established in 2007 and refined by Liu [32]. Thus far, the uncertainty theory has advanced as a vital branch of mathematics, e.g., uncertain differential equation [16,17,21,22,46,54,57,58], uncertain programming [6,7,18,23,36,55,56,66], uncertain supply chain [8,24–26], uncertain scheduling [40–45], uncertain control [9,10,13,14], uncertain process [59–61]. In the present study, a sustainable supply chain under uncertainty is investigated. The concerned problem is inspired by uncertainty of environmental and social factors. In recent years, floods, droughts, epidemics, snowstorms, earthquakes and other natural disasters have taken place frequently, and man-made fires, mining accidents, food and drug safety and other major accidents are also continuous. After we ushered into the era of globalization, as fueled by the rapid flow of resource elements in the global scope, the globalization of politics, economy, culture and life style has been triggered, as well as the globalization of risks. Human society is currently facing more risks than ever. The risks in modern society exhibit concealment, relevance and diversity, as manifested by online rumor, nuclear risk, global economic crisis, regional environmental risk and terrorist extremism, causing the global chain reaction, whereas they are difficult to perceive, predict and control effectively. The mentioned uncertain environmental and social factors have significantly impacted the supply chain. The economic, environmental and social issues are considered simultaneously. Since historical data are scarce or untrustworthy practically on the whole, the demand, cost, carbon emission, employment opportunity is recognized as uncertain variables. To cope with such problem, a multi-objective chance-constrained model is developed in this study. The equivalent form of the uncertain model is obtained in accordance with uncertainty theory. The aim of the present study is to delve into the effects of these uncertain variables on the sustainable supply chain and how to formulate the optimal strategies in an uncertain environment. The innovations of the present study are as follows. (1) To better deal with various emergencies in the supply chain network, the demand, cost and capacity act as uncertain variables. A considerable number of variables cannot be considered random variables since they lack historical data, or the data is unreliable. (2) For the complexity of the uncertain scenario, the chance-constrained model cannot be solved directly; thus, only the inverse distribution method could be adopted to transform this model into a crisp one. (3) To down-regulate the search dimension of coding space, a binary mapping mode variable length chromosome coding method is proposed. (4) Numerical experiments reveal that the method could solve the problem effectively. The following parts of the paper are organized as follows. In Section 2, some concepts of uncertainty theory are introduced. In Section 3, the notations and the problem are presented. In Section 4, a chance-constrained model is built, and the equivalent form is obtained. In Section 5, a hybrid integer genetic algorithm is proposed to solve the model. To verify the feasibility of the mentioned modeling method, numerical experiments are presented in Section 6.

J. Shen / Applied Mathematics and Computation 378 (2020) 125213

3

2. Preliminaries Fundamental concepts of the uncertainty theory are introduced in the present section. Interested readers may also refer to more details in [32]. Let  be a nonempty set, L is a σ -algebra over  , and each element  in L is called an event. A set function M from L to [0, 1] is called an uncertain measure if it satisfies normality axiom, duality axiom, subadditivity axiom and product axiom [31,33]. The uncertain distribution  of an uncertain variable ξ is defined by (x ) = M{ξ ≤ x} for any real number x. The uncertain variables ξ1 , ξ2 , . . . , ξm are said to be independence [31] if

 M

m 

 (ξi ∈ Bi ) = min M{ξi ∈ Bi }

i=1

1≤i≤m

for any Borel sets B1 , B2 , . . . , Bn of real numbers. An uncertain distribution (x) is said to be regular if its inverse function −1 (x ) exists and is unique for each α ∈ (0, 1). Definition 1 [31]. Let ξ be an uncertain variable, and α ∈ (0, 1]. Then

ξsup (α ) = sup{r|M{ξ ≥ r} ≥ α} is called the α -optimistic value to ξ , and

ξinf (α ) = inf{r|M{ξ ≤ r} ≥ α} is called the α -pessimistic value to ξ . Theorem 1 [31]. Assume the constraint function g(x, ξ1 , ξ2 , . . . , ξn ) is strictly increasing with respect to ξ1 , ξ2 , . . . , ξk and strictly decreasing with respect to ξk+1 , ξk+2 , . . . , ξn . If ξ1 , ξ2 , . . . , ξn are independent uncertain variables with uncertainty distributions 1 , 2 , . . . , n , respectively, then the chance constraint

M{g(x, ξ1 , ξ2 , . . . , ξn ) ≤ 0 )} ≥ α holds if and only if −1 −1 −1 g(x, −1 1 (α ), . . . , k (α ), k+1 (1 − α ), . . . , n (1 − α )) ≤ 0.

3. Problem description This problem refers to a multi-level network including multiple suppliers, production centers, distribution centers (DCs) and customer areas. The aim is to ascertain the number and location of production centers and DCs, as well as to make decisions regarding suppliers selection and product flow between facilities, as an attempt to minimize total cost and environmental impact and maximize social benefits. The quality level of materials provided by suppliers exerts a far-reaching impact on total cost, environmental and social. Individual products are manufactured by different production centers with appropriate production technologies and materials in a set of potential locations. Subsequently, the potential DCs transport the product to fixed customer areas. The proposed model complied with the following common assumptions in the literature ([15,19,29,53]): • •

The locations of the customers and the supply zones are fixed. Production centers and DCs could be configured at a range of capacity levels.

On the whole, the location of customers and supply areas would not change, and the capacity of production center and distribution center could be different. Thus, the above assumption is reasonable. The sustainable supply chain issues address critical uncertainties (e.g., demand, cost, capacity, carbon dioxide emissions, number of jobs, generation of harmful by-products, and average working days lost attributed to the exploitation of novel technologies). Three objective functions are considered, namely, minimizing total costs, maximizing environmental impact and social benefits. The aim is to make a reasonable trade-off between them. The parameters and variables are as follows:

4

J. Shen / Applied Mathematics and Computation 378 (2020) 125213 i j p dp fi gk ai j b jp ci hj e ki mi j o jp u l

α β γ

ri qj si tj

the candidate locations of manufactories, i = 1, 2, . . . , I the candidate locations of DCs, j = 1, 2, . . . , J the fixed locations of customer zones, k = 1, 2, . . . , P the demand of customer zone p the fixed cost of opening manufactory i the fixed cost of opening DC k unit transportation cost from manufactory i to DC j unit transportation cost from DC j to customer zone p unit manufacturing cost of product at manufactory i unit holding cost of DC j unit cost of carbon dioxide equivalent trading with external markets carbon dioxide emissions per unit product produced in manufactory i carbon dioxide emissions per unit product from manufactory i to DC j carbon dioxide emissions per unit product from DC j to customer zone p average quantity of potential dangerous goods in production average number of days of loss due to health problems per worker during production weight of employment opportunity weight of potential dangerous goods weight of total days due to work losses number of jobs created by manufactory i number of jobs created by DC j maximum capacity of manufactory i maximum capacity of DC j

Decision variable



1 0

xi =

if a manufactory is opened at location i otherwise



1 0

yj =

if a DC is opened at location j otherwise

wi j

quantity of goods transport from manufactory i to DC j

v jp

quantity of goods transport from DC j to customer zone p

4. Chance-constrained model In the present section, an uncertain chance-constrained model is built to deal with the sustainable supply chain problem. The total costs of the sustainable supply chain is expressed as F1 , covering opening cost, production cost, transportation cost and inventory cost.

F1 =

I 

f i xi +

i=1

J 

g jy j +

J I  

( ci + ai j + h j )w i j +

i=1 j=1

j=1

J P  

b jp v jp .

(1)

j=1 p=1

The first two items express the fixed costs of opening factories and distribution centers. The third item expresses the cost of production, inventory and transportation. The last item describes the cost of transportation. The environmental impact of the sustainable supply chain is expressed as F2 , covering carbon dioxide emissions.



F2 = e

J I  

( ki + mi j )wi j +

i=1 j=1

J P  



o jp v jp .

(2)

j=1 p=1

The social benefits of the sustainable supply chain is expressed as F3 , covering employment opportunities.



F3 = α

I  i=1

ri xi +

J  j=1



q jy j

−β

J I   i=1 j=1

uwi j − γ

I 

lxi .

(3)

i=1

The first, second and third items describe employment opportunities created in the supply chain network, the number of dangerous by-products in the supply chain network, and the number of working days lost due to workplace hazards, respectively. The implementation of cost control is an important guarantee for enterprises to complete cost management. Whether the enterprise can make profits, in the case of other conditions, the level of cost control should be critically considered. The control of product cost is the key to enterprise competition. The basic foothold for enterprises to gain market competitive advantage. In the present study, we use the cost function as the primary objective function. Since costs are uncertain variables, the value of objective function is difficult to determine. In this scenario, the concept of chance-constrained [5] can be adopted to address this problem. A goal F1 should be determined, so there exists a solution x∗ satisfying M{F1 ≤ F1 } ≥ η, where η is a predetermined confidence level.

J. Shen / Applied Mathematics and Computation 378 (2020) 125213

5

Let  represent the uncertainty distribution of objective function, we have

(F1 ) = M{F1 ≤ F1 } ≥ η.

(4)

For any η ∈ (0, 1), we have the inverse distribution of F1 I 

F−1 (η ) = 1

i=1

+

−1 ( η )xi + fi

J 

P 

j=1 p=1

J 

−1 g j ( η )y j +

j=1

J I  

−1 −1 [−1 ci (η ) + ai j (η ) + h (η )]wi j j

i=1 j=1

(5)

−1 (η )v jp ≤ F1 . b jp

−1

where denotes the inverse distribution of uncertain variable. Chance-constrained is also considered for objective functions F2 and F3 . It is noteworthy that F2 and F3 are pessimistic and optimistic, respectively.



M e

J I  

( ki + mi j )wi j +

i=1 j=1

  M

α

I 

ri xi +

J 

i=1

J P  

−β

J I  

J I  

I 

 ( η )xi + −1 ri

i=1

≥ η,

 lxi ≥ F3

J P  

i

J  j=1

  ( η )y j − β −1 qj

≥η

(6)

i=1

[−1 (η ) + −1 m i j ( η )] w i j + k

i=1 j=1

  (η ) = α

uwi j − γ

I 



F−1 (η ) = −1 e (η ) 2 −1 F3

≤ F2

i=1 j=1

j=1

Similarly, we have

o jp v jp



j=1 p=1

 q jy j

 −1 ≤ F2 , o j p (η )v jp

j=1 p=1 J I  

−1 u ( η )wi j − γ

i=1 j=1

I 

−1 (η )xi ≥ F3 . l

(7)

i=1

The constraints of this problem are introduced below. First, to ensure that the needs of each customer area are met, we have



M

J 



v jp ≥ d p ≥ δ, ∀ p

j=1

Since the demand is an uncertain variable, and according to Theorem 1, we have the equivalent form

−1 (δ ) ≤ dp

J 

v jp , ∀ p

j=1

The output of all production centers should be less than their capacities.



M

J 



≥ δ, ∀i

wi j ≤ xi si

j=1

Likewise, we have the equivalent form of the above formula J 

wi j ≤ xi −1 si ( 1 − δ ), ∀i

j=1

The capacity of the distribution center should be satisfied.



M

P 



v jp ≤ y j t j ≥ δ, ∀ j

p=1

The above formula is equated with P 

v jp ≤ y j t−1 ( 1 − δ ), ∀ j j

p=1

To ensure the balance of the output and input of each distribution center, we have I  i=1

wi j =

P  p=1

v jp , ∀ j

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J. Shen / Applied Mathematics and Computation 378 (2020) 125213

Environmental impact should be minimized, and social benefits should be maximized while minimizing costs. The uncertain sustainable supply chain chance-constrained model can be built in accordance with the above instructions.

min F1

(8)

min F2

(9)

max F3

(10)

 M

subject to

I 

 M e

f i xi +

i=1

J I  

α

M

J 

 M

 M

−β



o jp v jp − F2 ≤ 0

J I  

uwi j − γ

i=1 j=1



I 

 b jp v jp − F1 ≤ 0

≥ η,

≥ η,

(11)

(12)

 lxi − F3 ≥ 0

≥ η,

(13)

i=1

v jp ≥ d p ≥ δ1 , ∀ p

(14)

 ≥ δ2 , ∀i

wi j ≤ xi si

j=1

P 

 q jy j

J P   j=1 p=1

j=1 p=1

j=1

j=1

J 

J 

( ci + ai j + h j )w i j +

i=1 j=1

J P  

( ki + mi j )wi j +

ri xi +

i=1

 M

I 

g jy j +

J I  

j=1

i=1 j=1

 

J 

(15)

 v jp ≤ y j t j ≥ δ3 , ∀ j

(16)

p=1 I 

wi j =

i=1

P 

v jp , ∀ j

p=1

wi j , v jp ≥ 0,

∀i, j, p ∀i, j.

xi , y j ∈ {0, 1},

(17)

where η and δi , i = 1, 2, 3 are confidence levels. This model attempts to solve the pessimistic values under chance-constrained. Since the model is a multi-objective programming, it cannot be solved directly. A weighted method is employed to transform a multi-objective linear combination into a single-objective programming problem. The objective functions can be transformed into the following the new objective function:

min

3 

wi f i ,

(18)

i=1

where wi refer to the weighting factors of the objective functions. The weighting factors can be adjusted according to the  decision maker’s bias, if wi = 1 for i = 1, 2, 3. i

The equivalent form of the model is achieved in accordance with uncertainty theory. Theorem 2. The chance-constrained model is equated with the following model:



min w1

I  i=1

+

J P  

−1 ( η )xi + fi

J P   j=1 p=1

j=1

−1 g j ( η )y j +





−1 (η )v jp + w2 −1 e (η ) b jp

j=1 p=1

+



J 

 −1 o j p (η )v jp

− w3

J I   −1 −1 [−1 ci (η ) + ai j (η ) + h (η )]wi j j

i=1 j=1 J I   [−1 (η ) + −1 m i j ( η )] w i j k i=1 j=1

i

  I  J   −1 α −1 ( η ) x +  ( η ) y i j ri qj i=1

j=1

J. Shen / Applied Mathematics and Computation 378 (2020) 125213

−β

J I  

−1 u ( η )wi j − γ

i=1 j=1

I 

7

 −1 ( η )xi l

i=1

subject to

−1 (δ1 ) ≤ dp

J 

v jp , ∀ p

j=1 J 

wi j ≤ xi −1 si (1 − δ2 ), ∀i

j=1 P 

v jp ≤ y j t−1 (1 − δ3 ), ∀ j j

p=1 I 

wi j =

i=1

P 

v jp , ∀ j

p=1

wi j , v jp ≥ 0,

∀i, j, p ∀i, j.

xi , y j ∈ {0, 1},

Proof: According to Definition 1 and Theorem 1, the formulations (8) and (11) are equivalent to I 

min F−1 (η ) = 1

i=1

−1 ( η )xi + fi

J  P 

+

j=1 p=1

J  j=1

−1 g j ( η )y j +

J I  

−1 −1 [−1 ci (η ) + ai j (η ) + h (η )]wi j j

i=1 j=1

(19)

−1 (η )v jp . b jp

Similarly, the formulations (9) and (12) are equivalent to



min 

−1 F2

(η ) = 

−1 e



J J I  P    −1 (η ) [−1 ( η ) +  ( η ) ] w + −1 i j m o j p (η )v jp . k ij i=1 j=1

i

(20)

j=1 p=1

The formulations (10) and (13) are equivalent to



min F−1 (η ) = α 3

 M

 M

 M

I  i=1

J  j=1 J 

p=1

J 

 J I  I   −1 −1 ( η ) y − β  ( η ) w − γ −1 ( η )xi . j i j qj u l

j=1

 J  v j p ≥ d p ≥ δ1 is equivalent to −1 (δ ) ≤ v j p. dp wi j ≤ xi si

i=1 j=1

(21)

i=1

j=1



j=1 P 

−1 ri ( η )xi +

≥ δ2 is equivalent to

 v j p ≤ y j t j ≥ δ3 is equivalent to

J  j=1

wi j ≤ xi −1 si ( 1 − δ ).

P  p=1

v j p ≤ y j t−1 (1 − δ ), ∀ j. j

The theorem is proved. Property 1. The objective function value remains stable with the confidence level η. Proof: Let A represent the feasible solution set of the objective function. If η2 ≥ η1 , then Aη2 ⊆ Aη1 . Noticeably, the objective function value under the confidence level η2 is greater than or equal to that of η1 . As we all know, because of the uncertainty of a large amount of information, decision makers will face the problem of multi-dimensional decision variables. The mentioned uncertain information will cause multiple integral problems in random environment, thereby making the solution more difficult to achieve. Fortunately, the operation law of inverse uncertainty distribution in uncertainty theory avoids the problem of multiple integrals. Accordingly, in numerous types of sustainable supply chain problems, the proposed uncertainty chance-constrained model outperforms the stochastic model. 5. Solution method Noticeably, this is a mixed integer programming model. As the scale increases and the complexity increases, the computation time will increase. The conventional exact algorithms e.g., branch and bound lack flexibility and have great limitations. Genetic algorithm (GA) refers to a probabilistic algorithm for adaptive global optimization, simulating the genetic and evolutionary processes of organisms in natural environment. Based on the pattern theorem, GA generates the optimal

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J. Shen / Applied Mathematics and Computation 378 (2020) 125213

Fig. 1. Binary hybrid coding.

individual in the group, instead of iterating the solution based on the gradient information; thus, its requirements for the objective function and constraints are relatively low. For the pure integer programming problem, the optimal GA as impacted by the limited feasible solution space can be obtained according to the optimal scheduling strategy. Besides, since it is a probabilistic algorithm, it is also possible to obtain a series of optimal scheduling strategies. In terms of nonlinear integer programming problems, the global optimization characteristics can also avoid the solution to the local optimal solution, thereby yielding the global optimal approximate solution. Based on the conventional integer programming algorithm, we propose a hybrid integer GA based on binary mapping mode variable length chromosome coding. The coding method solves the problem of infeasible coding to a certain extent, reduces the search dimension of the coding space, and greatly accelerates the convergence speed of the algorithm. Assume that there are n integer decision variables and m floating-point decision variables in the model. As shown in Fig. 1: Where Bt denotes the number of encoding bits of the corresponding variable Kt , t = 1, 2, . . . , m. Mapping rules: (1) Bt , t = 1, 2, . . . , m: ensuring the corresponding decision variables to be integral; given the value of each variable, different values can be selected. (2) Bt , t = m + 1, m + 2, . . . , m + n: under the category of guaranteeing the precision of variable search, any number of digits can be selected. In the evolution of the algorithm, with the evolution process, the length of the chromosome and the number of bits of the integer variable code Bi are down-regulated, thereby narrowing the search space of the code and noticeably increasing the convergence speed of the algorithm. Steps of hybrid integer GA: Step 1: The initial population is randomly generated to determine the feasibility of the individual; if it is an infeasible individual, it will be generated continuously until it is feasible. Step 2: Assessment of the fitness of the population. Step 3: One-point crossover method is used. Determine the feasibility of the individual. If it is an infeasible individual, randomly generate a crossover position to continue the crossover operation until a feasible individual is generated. Step 4: The flip mutation operation is used. Determine the feasibility of the individual. If it is an infeasible individual, randomly generate a crossover position to continue the crossover operation until a feasible individual is generated. Step 5: The chromosome is selected using the roulette wheel method. Step 6: Assessment of the fitness of the population. Decrease the corresponding number of bits in the mapping mode of the chromosome based on the optimal individual in the population with a specified probability. If the digits of any mapping pattern vary, the length of chromosomes is altered, and they are recorded in the novel mapping pattern in accordance with to the decoding value of the existing chromosomes. Step 7: If the maximum number of iterations is reached, then stop; otherwise, go to step 2 to perform another iteration. 6. Numerical experiments In the present section, numerical experiments are used to verify the efficiency of the proposed model and algorithm. The size of the problem is listed in Table 1. Assume that the uncertain variables in the present study are expressed by uncertain linear distributions. w1 = w2 = 0.4, w3 = 0.2. The parameters of genetic algorithm cover popsize=50, crossover probability Pc =0.75, mutation probability Pm =0.25, maximum number of iterations 10 0 0. Unit transportation costs are listed in Tables 2 and 3. All the instances were solved by Matlab on an AMD FX(tm)-8300 Eight-core processor @3.3 GHz with RAM 8GB on Windows 7. In general, policymakers define the weight assigned to CSR-related factors, such as job creation, generation of hazardous substances, and the number of working days lost due to workplace hazards. However, these weights should take into account

Table 1 The size of the problem. Manufactory

Distribution center

Customer zone

5

7

14

J. Shen / Applied Mathematics and Computation 378 (2020) 125213

9

Table 2 Unit transportation cost from manufactories (Mfy) to DCs. Mfy

1 2 3 4 5

DCs 1

2

3

4

5

6

7

L (10, 12 ) N (12, 15 ) L (8, 10 ) L (9, 11 ) N (7, 10 )

L ( 7, 9 ) L (8, 10 ) L (13, 16 ) L (10, 13 ) L (9, 11 )

L ( 5, 6 ) L (10, 12 ) L (15, 17 ) L (11, 13 ) L ( 8, 9 )

L (9, 12 ) L (11, 13 ) L (12, 14 ) L (8, 11 ) L (10, 13 )

L (12, 15 ) L ( 6, 8 ) L ( 5, 7 ) L ( 7, 9 ) L ( 6, 7 )

L (15, 18 ) L (7, 10 ) L (10, 13 ) L (15, 18 ) L (11, 13 )

L ( 6, 9 ) L ( 5, 7 ) L (9, 12 ) L (12, 15 ) L (12, 15 )

Table 3 Unit transportation cost from DCs to customer zones DC

Customer zone 1

2

3

4

5

6

7

1 2 3 4 5 6 7 8

L ( 5, 6 ) L ( 4, 5 ) L ( 8, 9 ) L ( 4, 5 ) L ( 3, 4 ) L ( 8, 9 ) L ( 7, 8 ) L (9, 10 )

L ( 4, 5 ) L ( 8, 9 ) L ( 4, 5 ) L (9, 10 ) L ( 5, 6 ) L ( 7, 8 ) L (8, 10 ) L ( 6, 8 )

L ( 8, 9 ) L (9, 10 ) L ( 5, 6 ) L ( 7, 8 ) L ( 6, 8 ) L ( 3, 5 ) L ( 6, 8 ) L ( 4, 5 )

L ( 7, 8 ) L ( 4, 5 ) L ( 3, 4 ) L (12, 13 ) L ( 4, 6 ) L ( 6, 8 ) L ( 7, 8 ) L ( 5, 6 )

L (9, 10 ) L (12, 14 ) L (9, 10 ) L ( 8, 9 ) L (9, 10 ) L ( 5, 6 ) L (9, 10 ) L ( 7, 8 )

L (12, 14 ) L ( 8, 9 ) L (12, 14 ) L ( 3, 4 ) L (10, 12 ) L ( 7, 8 ) L (11, 12 ) L (8, 10 )

L (9, 10 ) L ( 5, 6 ) L ( 7, 8 ) L ( 8, 9 ) L ( 4, 5 ) L (10, 12 ) L (10, 12 ) L (11, 12 )

1 2 3 4 5 6 7 8

8 L (8, 10 ) L ( 7, 8 ) L ( 5, 6 ) L (10, 12 ) L (11, 12 ) L (12, 14 ) L (8, 10 ) L ( 7, 8 )

9 L ( 3, 4 ) L ( 4, 5 ) L (8, 10 ) L (11, 12 ) L ( 5, 6 ) L (8, 10 ) L ( 7, 8 ) L (9, 10 )

10 L (9, 10 ) L ( 7, 8 ) L (10, 12 ) L ( 7, 8 ) L ( 6, 8 ) L (9, 10 ) L ( 3, 4 ) L ( 4, 5 )

11 L ( 5, 6 ) L ( 7, 8 ) L ( 5, 6 ) L (9, 10 ) L (8, 10 ) L ( 5, 6 ) L ( 6, 8 ) L ( 5, 6 )

12 L (10, 12 ) L (9, 10 ) L ( 3, 5 ) L (8, 10 ) L ( 4, 6 ) L ( 7, 8 ) L (8, 10 ) L (8, 10 )

13 L ( 7, 8 ) L ( 3, 4 ) L ( 4, 5 ) L ( 3, 5 ) L (10, 12 ) L ( 6, 8 ) L ( 4, 5 ) L (10, 12 )

14 L (9, 10 ) L ( 4, 6 ) L ( 7, 8 ) L (8, 10 ) L ( 3, 5 ) L (10, 12 ) L (9, 10 ) L (11, 12 )

goals related to regional development. For example, in areas with high unemployment, the weight assigned to the number of job creations should be higher than the weight assigned to other factors. Parameters related to social factors are set based on this. The remaining parameters are set based on production, manufacturing, transportation and sales. d1 ∼ L(200, 250 ), d2 ∼ L(300, 350 ), d3 ∼ L(100, 150 ), d4 ∼ L(100, 120 ), d5 ∼ L(250, 300 ), d6 ∼ L(350, 400 ), d7 ∼ L(150, 180 ), d8 ∼ L(80, 90 ), d9 ∼ L(220, 270 ), d10 ∼ L(130, 160 ), d11 ∼ L(210, 260 ), d12 ∼ L(100, 130 ), d13 ∼ L(90, 110 ), d14 ∼ L(50, 60 ). f1 ∼ L(10 0 0 0, 1050 0 ), f2 ∼ L(150 0 0, 1550 0 ), f3 ∼ L(120 0 0, 1250 0 ), f4 ∼ L(140 0 0, 1450 0 ), f5 ∼ L(150 0 0, 160 0 0 ), g1 ∼ L(130 0 0, 140 0 0 ), g2 ∼ L(120 0 0, 130 0 0 ), g3 ∼ L(150 0 0, 160 0 0 ), g4 ∼ L(140 0 0, 150 0 0 ), g5 ∼ L(110 0 0, 120 0 0 ), g6 ∼ L(10 0 0 0, 1130 0 ), g7 ∼ L(160 0 0, 1650 0 ), c1 ∼ L(100, 120 ), c2 ∼ L(80, 100 ), c3 ∼ L(90, 100 ), c4 ∼ L(50, 60 ), c5 ∼ L(60, 80 ), h1 ∼ L(15, 20 ), h2 ∼ L(12, 15 ), h3 ∼ L(10, 12 ), h4 ∼ L(14, 20 ), h5 ∼ L(11, 15 ), h6 ∼ L(9, 12 ), h7 ∼ L(10, 12 ), e ∼ L(20, 24 ), ki ∼ L(2, 3 ), mi j ∼ L(3, 4 ), o j p ∼ L(3, 4 ), α = β = γ = 1/3, r1 ∼ L(15, 20 ), r2 ∼ L(10, 12 ), r3 ∼ L(12, 15 ), r4 ∼ L(9, 10 ), r5 ∼ L(14, 15 ), q1 ∼ L(15, 16 ), q2 ∼ L(16, 18 ), q3 ∼ L(14, 16 ), q4 ∼ L(12, 15 ), q5 ∼ L(11, 14 ), q6 ∼ L(10, 12 ), q7 ∼ L(18, 20 ), s1 ∼ L(180 0, 190 0 ), s2 ∼ L(190 0, 20 0 0 ), s3 ∼ L(20 0 0, 2150 ), s4 ∼ L(170 0, 180 0 ), s5 ∼ L(160 0, 170 0 ), t1 ∼ L(220 0, 230 0 ), t2 ∼ L(240 0, 250 0 ), t3 ∼ L(230 0, 250 0 ), t4 ∼ L(250 0, 260 0 ), t5 ∼ L(210 0, 220 0 ), t6 ∼ L(2450, 250 0 ), t7 ∼ L(2350, 2450 ), u = 10, l = 10. Test results of the chance-constrained model are listed in Table 4. For each case of η, run the algorithm five times to calculate the minimal value of the objective function. According to Table 4, the objective function value remains stable with the confidence level η. At a high confidence level, the minimal acceptable objective function becomes higher, indicating that the decision maker should open more factories and distribution centers to meet customers’ needs, suggesting that more resources are consumed to deal with uncertainty. The decision maker can determine the confidence level based on his preferences and actual situation. The number of jobs created depends on the number of facilities. Table 4 shows that the number of facilities will increase as the confidence level increases, resulting in a significant increase in employment opportunities. Furthermore, the computational times of GA are acceptable.

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J. Shen / Applied Mathematics and Computation 378 (2020) 125213 Table 4 Test results of the chance-constrained model.

η

Objective function

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

8.7377E+05 9.1504E+05 9.5177E+05 9.9496E+05 1.0388E+06 1.0901E+06 1.1363E+06 1.1830E+06 1.2241E+06

Number of open factories Mfy

DC

2 2 2 2 3 4 4 5 5

3 4 4 5 5 6 7 7 7

CPU time

23 23 22 23 22 22 21 22 22

Table 5 Setting of uncertain parameters. aij [5, 15]

bjp [5, 15]

ci [50, 120]

dp [50, 400]

e [20, 24]

fi [10, 0 0 0, 16, 0 0 0]

gk [10, 0 0 0, 16, 0 0 0]

hj [10, 20]

ki [2, 5]

mij [2, 5]

ojp [2, 5]

ri [10, 20]

qj [10, 20]

si [1500, 2500]

tj [2000, 2500]

Table 6 Test results of the large-scale situation.

η 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Objective function

Number of open factories

3.2343E+07 3.6423E+07 4.1255E+07 4.3624E+07 4.9204E+07 5.3441E+07 5.7216E+07 6.3099E+07 6.8227E+07

Mfy

DC

12 12 12 13 14 16 16 18 18

14 14 14 15 17 18 18 20 20

CPU time

323 331 332 319 322 326 323 326 323

Table 7 Results of chance-constrained model . N

500 1000 1500 2000 2500 3000

Small-scale

Large-scale

RPD(%)

MAD(%)

RPD(%)

MAD(%)

1.93 1.33 0.78 0.32 0.18 0.05

0.29 0.17 0.06 0.02 0.00 0.00

2.12 1.57 0.83 0.41 0.23 0.12

0.42 0.23 0.14 0.05 0.02 0.00

To verify the feasibility of the hybrid algorithm, large-scale numerical experiments are considered. The uncertain variables are randomly generated according to the following Table 5. The settings for other parameters are identical to the mentioned example. There are 20 manufactories, 30 distribution centers, as well as 80 customer zones. Test results of the large-scale numerical experiments are listed in Table 6. For respective case of η, run the algorithm five times to calculate the minimal value of the objective function. As can be seen from Table 6, when the total number of factories, distribution centers and customer areas increases, the performance of the objective function values is similar to the test results in Table 5. The results indicate that the algorithm can still solve the problem effectively, and that the computational time satisfies an acceptable range in large-scale scenarios. Moreover, relative percentage deviation (RPD) and mean absolute deviation (MAD) also used to verify the feasibility of the hybrid algorithm. They are defined as follows.

RP D =

objective value − the optimal objective value × 100%, the optimal objective value

J. Shen / Applied Mathematics and Computation 378 (2020) 125213

MAD =

11

objective value − the mean value of the objective function × 100%. the mean value of the objective function

N denotes the maximum number of iterations of the algorithm. Table 7 implies that the error will decrease as the number of iterations increases. Especially in the large-scale case, the RPD value is less than 1% in the 1500th generation; the RPD value is close to 0.1% in the 30 0 0th generation. The results suggest that the performance of the GA is satisfactory. 7. Conclusions In the present study, a sustainable supply chain problem considering both environmental impact and social benefits was analyzed. To achieve the more accurate right decision in the turbulent and complex modern society, the uncertain factors were considered in the study. The customer demand, cost, and capacity were considered uncertain variables due to numerous practical historical data are unavailable or untrustworthy. To measure the environmental impact of related logistics networks, carbon dioxide emissions were employed. Employment opportunities were used to assess social benefits. To study the effects of uncertain variables on supply chain, a novel multi-objective chance-constrained model was developed. In such model, pessimistic value was adopted to solve the imprecise objective function, and chance-constrained programming method was adopted to regulate the confidence level of imprecise constraint satisfaction. The equivalence of the chance-constrained model was verified using the uncertainty inverse distribution method. An effective hybrid integer genetic algorithm based on binary mapping mode variable length chromosome coding was proposed. Numerical experiments implied that the proposed method can efficiently find solutions in the conflicting objective functions and solve problems on a large scale. Furthermore, as revealed from the experimental results, at the high confidence level, the decision maker should consider consuming more resources to cope with the uncertainties in the system. 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