© 2024 The author. This article is published by IIETA and is licensed under the CC BY 4.0 license (http://creativecommons.org/licenses/by/4.0/).
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The green supply chain is the reduction of the atmospheric release emissions including gases, vapour, smoke, solid or liquid particles. This atmospheric reduction will concern each stage of the chain: supply, production, distribution, warehousing, transport and delivery. The design of this loop is based on industrial ecological perspectives, particularly in the production, and the transport stage. In this work, we present a lotsizing problem with capacitated one warehouse multi retailers (OWMR) under the minimization of particles matter (PM) emission from production and delivery, knowing that the problem is an NPhard. We have developed a logistics structure containing a production unit connected to a distribution network characterized by (size, number and location) retailers specializing in a single type of product. Then, we will introduce our mathematical problem modelling using mixedinteger programming and develop an approach based on the metaheuristic called binary particle swarm optimization (BPSO) in this approach; we will study new strategies and techniques concerning the particle swarm parameters. The improved BPSO will be tested on a series of benchmark data sets and compared with CPLEX. According to the experimental results, this approach is effective in minimizing the total cost of the supply chain and promoting green technology by reducing the number of the particles emitted into the air. It also provides a decision support system to answer key questions about when and how much produce and distribute in a sustainable environment.
planning problem, one warehouse multi retailers, particle swarm optimization, environmental constraint
Supply chain management (SCM) can be defined as the interconnection of three basic functions: planning, design and control (activities and flows). It starts through the supply that ends with customer satisfaction [1, 2].
Effective management of the logistics chain in a competitive environment requires effective governance in production planning. The problem of a single product, multiple periods, and inventory size are among the basic problems that affect trading and have been addressed by a group of researchers [3, 4].
Green LotSizing Problem (GLSP) is considered as a tradeoff between setup and inventory holding costs to determine the minimum cost of a production plan for one or several machines, in order to meet the demand for each item with respecting the environmental constraints.
The atmospheric release inside the supply chain management is carbon emission constraint and particles matter emission constraint.
Concerning the first environmental constraint (carbon emission constraint), in Table 1, we’ve compared the different studies about lotsizing problem with different carbon emission constraints thanks to the literature review. The main research gaps here are: (1) research authors, (2) model studies, (3) carbon emission policy and (4) resolution method.
Their study was to shed light on the integration of two important dimensions [5]: production planning and the principle of sustainability. The study aimed to maximize the expected gross profit of the twostage newsstand model with environmental constraints: using the cost of licenses, emission limit values, and fines imposed in case of exceeding their permissible limits: Many authors have also focused on this topic. El Saadany et al. [6] focused on two basic approaches, one of which displays the relationship between price and demand with constant quality, and the others present the supply chain precisely affects the criteria of quality, demand, price, in addition the relationships between these criteria under the environmental constraint. In fact, the authors developed a multicriteria decision support system based on the Pareto method under environmental (carbon emission with MRL) constraints [7] has been used by Bouchery et al. [8] in order to perform operational optimization. Benjaafar et al. [9] mentioned four different types of carbon emission constraints, which are: strict carbon caps, carbon tax, carbon emission trading and carbon. Absi et al. [10] have proposed a new classification of carbon emission constraint, unlike Benjaafar. The four types of carbon emissions are: periodic carbon emission, cumulative carbon emission, global carbon emission and rolling carbon emission [10], other Emission of pollutants such as waste and dust. Although carbon emission limits have been addressed in the majority of articles, the penalty resulting from exceeding these emission limit values has only been addressed by three authors [11, 12]. Four criteria were addressed in this study by focusing on the economic model of quantity scaling with multiple replenishment modes. Suppliers of means of transportation from an economic and environmental perspective (cost and emission level) In a study [13], the authors focused on the consumption of environmental products and how they affect carbon emissions in a complex supply chain [14]. This paper addressed a novel multiproduct, multiperiod replenishment problem, and proposed the nonlinear model solved by GA and PSO. The researchers in this work [15] based it on attaching the quantity of economic demand in a twolevel supply chain model with a carbon tax and emission penalties [9]. Where the researcher and his colleagues were interested in developing improvement models to reduce the carbon footprint, where the relationship was found between the discrepancy in the quantity produced and the quantity of carbon emitted.
The second environmental constraint (particulate matter emissions) is a global concern for environmental monitoring and regulating particulate matter emissions of industrial systems. The Environmental Protection Agency (EPA) impose, therefore, legal penalties for those whose emissions exceed the reference limit values. The EPA defines particulate matter as “particulate pollutants,” which consist of acid and chemical particles, soil particles, and dust. In this study, we are interested in Particulate Matter (PM). In the production of plants, the processed PM is discharged via stacks or pipe. This present paper proposes a solution to the planning problem with OWMR under particle matters emission constraints. In this work, we have expanded the research [10] in different directions to make it more realizable. At the beginning, we describe a logistics structure under an environmental constraint then, we consider that the main source of PM emission at the level of production and transport functions. We’ve developed an approach based on a metaheuristic algorithm called the binary particle swarm optimization (BPSO). This approach can be used as a resolution method to assist company managers in determining how and when to trigger production in order to satisfy a customer service rate with a minimum total cost while respecting PM emission constraints knowing that this problem is NPhard [16].
Table 1. Literature review
Authors 
Carbon Emission Constraint 
Description Model 
Approach 

CAP 
CAP & TRADE 
TAX 
PENALTY 

[17] 
 
 
 
 
Inventory model transportation 
Dynamic programming 
[7] 
* 
* 
 
 
Single echelon inventory 
EOQ 
[18] 
 
 
 
 
Stochastic model 
Tabu search 
[19] 
* 
* 
 
 
Classical singleperiod model 
NEWSVENDOR 
[15] 
 
 
* 
* 
Two echelon supply chain 
EOQ 
[10] 
* 
 
 
 
Multi sourcing deterministic lotsizing problems 
Dynamic programming 
[20] 
* 
* 
* 
 
Single echelon inventory model 
EOQ 
[2124] 
* 
* 
* 
 
Multiitem production extended 
NEWSVENDOR 
[25] 
 
* 
* 
 
Dual sourcing 
NEWSVENDOR 
[26] 
* 
* 
 
 
Inventory model with truck capacities 
Heuristic local search algorithm 
[12] 
* 
* 
* 
* 
Replenishment and supplier/transportation 
CPLEX 
[18, 27] 
* 
* 
 
 
Multi product singleperiod production model stochastic demand 
Classical Newsboy model 
[11] 
* 
* 
 
* 
Single period, single product inventory problem stochastic D 
Classical newsboy model 
[18, 27] 
* 
 
 
 
One plant, multiple distribution centers (DCs) and multiple retailers 
Genetic algorithm 
[28] 
* 
* 
 
 
Multi (Manufacturing plant, warehouse, product) with transport mode 
Crossentropy 
[29] 
* 
 
* 
 
Multiechelon productioninventory model with lead time 
CPLEX 
[30] 
* 
* 
* 
 
Single echelon inventory 
EOQ 
[18, 27] 
 
 
* 
 
Two echelon inventory (Distributors and retailers) model 

[31] 
* 
* 
* 
 
Supply chain network design model (inventory, production and transport with product, network and facility parameters 
CPLEX 
[32] 
 
 
 
 
Stochastic capacitated lot sizing problem 
CPLEX 
[18, 27] 
* 
 
 
 
Thirdparty logistics providers (3PLs). multi warehouse 
CPLEX 
[33] 
 
* 
 
 
Multistage dynamic optimization problem 
Dynamic programming 
[34] 
 
 
 
 
Nonstationary stockastic demand 
Mixed integer linear programming 
[33] 
 
* 
 
 
Multistage dynamic optimization problem 
AMPL/CPLEX 
[35] 
 
* 
 
 
Two echelon multiproduct supply chain 
EOQ and EPQ 
After a brief introduction, we have described the planning problem with OWMR under cumulative emission of particulate matter constraint developed in Section 2. Then, the appropriate BPSO is provided in Section 3. The numerical experiment results are reported in Section 4. Section 5 concludes the work and suggests research opportunities and directions for further work.
Figure 1 presents an example of PMcement production process SKIKDAAlgeria.
Figure 1. Particle matter emissions in the cement industry in SKIKDAAlgeria
2.1 Problem definition
Environment
We can define the Justintime logistics structure by a production unit and a distribution network (size, number and location) of the different retailers specified by a single product, as show in Figure 2.
Figure 2. Structure studies
Assumptions
The main assumptions are as follows:
Objective
Minimizing the total cost of structure logistic.
We will introduce the mathematical formulation of Mixed Integer Linear Programming (MILP) in this next part.
2.2 Problem formulation
Objectif function
$\begin{gathered}\operatorname{Min} Z=\sum_{t=1}^T(Production\ cos+Distribution\ Center\ cost)+\sum_{t=1}^T \sum_{i=1}^{N R}(Retailers\ cost)\end{gathered}$ (1a)
$\begin{aligned} & \operatorname{Min} Z=\sum_{t=1}^T\left(f p_i \boldsymbol{y}_t+p_i \boldsymbol{x}_{\boldsymbol{t}}\right)+\left(f d_t \boldsymbol{y} \boldsymbol{d}_{\boldsymbol{t}}+s d_t \boldsymbol{I} \boldsymbol{d}_{\boldsymbol{t}}\right)+\sum_{t=1}^T \sum_{i=1}^{N R}\left(f r_{i t} \boldsymbol{y} r_{i t}+s r_{i t} I \boldsymbol{r}_{i t}\right) \\ & \operatorname{Min} Z 2=\sum_{t=1}^T\left(f p_i \boldsymbol{y}_{\boldsymbol{t}}+p_i \boldsymbol{x}_{\boldsymbol{t}}\right)+\left(f d_t \boldsymbol{y} \boldsymbol{d}_{\boldsymbol{t}}+s d_t \boldsymbol{I} \boldsymbol{d}_{\boldsymbol{t}}\right)+\sum_{t=1}^T \sum_{i=1}^{N R}\left(f r_{i t} \boldsymbol{y} r_{i t}+s r_{i t} \boldsymbol{I} \boldsymbol{r}_{i t}\right)+\sum_{t=1}^T \sum_{i=1}^{N R}\left(u t_{i t} \boldsymbol{q} \boldsymbol{l}_{i t}\right)(1') \\ & \operatorname{Min} Z=\sum_{t=1}^T\left(f p_i \boldsymbol{y}_{\boldsymbol{t}}+p_i \boldsymbol{x}_{\boldsymbol{t}}\right)+\left(f d_t \boldsymbol{y} \boldsymbol{d}_{\boldsymbol{t}}+s d_t \boldsymbol{I} \boldsymbol{d}_{\boldsymbol{t}}\right)+\sum_{t=1}^T \sum_{i=1}^{N R}\left(f r_{i t} \boldsymbol{y} \boldsymbol{r}_{i t}+s r_{i t} \boldsymbol{I} \boldsymbol{r}_{i t}\right)+\sum_{t=1}^T \sum_{i=1}^{N R}\left(u t_{i t} \boldsymbol{q} \boldsymbol{l}_{\boldsymbol{i t}}\right) \\ & \end{aligned}$ (1b)
Subject to
The production capacity constraint
${{x}_{t}}\le {{y}_{t}}ca{{p}_{t}}$ (2)
The inventory level of DC and retailers’ constraints
$I d_t=I d_{t1}+x_t\sum_{i=1}^{N R} d_{i t}$ (3)
$I{{r}_{it}}=I{{r}_{it1}}+q{{l}_{it}}{{d}_{it}}$ (4)
The PM emission constraint
$\sum_{k=1}^t\left(p e_k^mP E_k^m\right) x_k \leq 0$ (5)
$h_{t}^{m}=h_{t1}^{m}\left( pe_{k}^{m}PE_{k}^{m} \right){{x}_{t}}$ (6)
${{h}_{t}}\ge 0~;~{{h}_{0}}=0$ (7)
The domain of definition of decision variables
$x_t,\ I d_t,\ I r_{i t} \geq 0$;$\ intergers$ $\forall\ \mathrm{i},\ \mathrm{t},\ \mathrm{k}$ (8)
The definition of decision variables
$y_t=\left\{\begin{array}{c}0 \text { if } x_t=0 \\ 1 \text { Otherwise }\end{array}\right.$ (9)
$y d_t=\left\{\begin{array}{c}0 \text { if } I d_t=0 \\ 1 \text { Otherwise }\end{array}\right.$ (10)
$y r_{i t}=\left\{\begin{array}{l}0 \text { if } I d_t=0 \\ 1 \text { Otherwise }\end{array}\right.$ (11)
Table 2 presents the measurements of these particles over a 20182019 horizon in this company.
Table 2. Presentation of dust measurements in 20182019 at the cement industry in SKIKDA, Algeria

2018 
2019 

Location 
Production Tons 
Particle Matter (Mg/Nm³) 
Production Tons 
Particle Matter (Mg/Nm³) 
GP120 Filter 
1008783.84 
25.67 
1086120.3 
30.92 
Handle filter outlet L01 
0.81 
6.34 

Handle filter outlet L02 
0.78 
3.72 

L01 Chiller Bag Filter Outlet 
0.68 
2.16 

L02 Chiller Bag Filter Outlet 
1.21 
1.65 
Table 3 explains the principle of inventory emission variable.
Table 3. Example for cumulative emission
t 
x 
pe 
Emission 
PE 
pePE 
x(pePE) 
H 
1 
100 
10 
1000 
15 
5 
500 
500 
2 
120 
5 
600 
10 
5 
600 
1100 
3 
50 
15 
750 
17 
2 
100 
1200 
4 
0 
17 
0 
20 
3 
0 
1200 
5 
250 
10 
2500 
15 
5 
1250 
2450 
6 
25 
9 
225 
10 
1 
25 
2475 
7 
40 
12 
480 
10 
2 
80 
2395 
8 
90 
8 
720 
4 
4 
360 
2035 
9 
0 
14 
0 
8 
6 
0 
2035 
10 
200 
5 
1000 
5 
0 
0 
2035 
In Table 3, example for cumulative emission ${{\text{h}}_{\text{t}}}$ values increase until t=6, because $\text{P}{{\text{E}}_{\text{t}}}$ is greater then $\text{p}{{\text{e}}_{\text{t}}}$ ($\text{P}{{\text{E}}_{\text{t}}}>\text{p}{{\text{e}}_{\text{t}}}$); it means permission is sufficient for production. Till $\text{t}$=7 h decrease $\text{P}{{\text{E}}_{\text{t}}}$ is less then $\text{p}{{\text{e}}_{\text{t}}}$ ($\text{P}{{\text{E}}_{\text{t}}}<\text{p}{{\text{e}}_{\text{t}}}$), it means we need permissions from inventory emission to produce x quantity. This is what explain clearly Figure 3. In another manner:
For t=1 to T do
If $\text{P}{{\text{E}}_{\text{t}}}\text{*}{{\text{x}}_{\text{t}}}>\text{p}{{\text{e}}_{\text{t}}}{{\text{x}}_{\text{t}}}$ then
Do not need ${{\text{h}}_{\text{t}}}$
Else
Need ${{\text{h}}_{\text{t}}}$
End.
Figure 3. Cumulative emission
Another scenario presents itself, when h takes negative values; that mean in this period plant cannot produce all quantity desired because it has not permission for particle emission. It has consumed all its reserve during the previous periods. Therefore, in this case plant must reduce production to get at least zero inventory of emission and satisfy environmental constraint.
After we have defined the mathematical model in MIPL with different constraints, we will go through proposing a way to solve this problem.
3.1 The basis of PSO
Particle swarm optimization (PSO) is developed by Kennedy and Eberhart, and it the main product is through consistency and competition by conveying specific information to guide the improvement process [36].
Algorithm General PSO 
Begin 
Initialize randomly swarm, velocity/*a set of particles 
Calculate Pbest, Gbest 
For i=1 to NI/*number of iterations 
Calculate new velocity 
Calculate new swarm 
Calculate Pbest and Gbest Seek fitness 
End for 
Write fitness/*the best value from all. 
End 
Now we adopt this algorithm to solve capacitated lotsizing problem.
3.2 Structure of the binary PSO algorithm
Better efficiency of PSO based search could be achieved by modifying the particle representation and its related operators to generate feasible solutions [37].
Sazvar et al. [14] proposed an integer presentation of the particle; each particle refers to the number of batch sizes ordered for each product for each period to solve a supply chain with perishable items. Boonmee and Sethanan [38] developed a new decoding representation where each particle in the swarm is separated into two parts, to solve multilevel capacitated lotsizing and scheduling problems. The first part is the number of chicks purchased and delivered to the poultry industry in each period, and the second part is the allocation of chicks and pullets to farms.
However, Chen and Lin [39] developed a representation of complex particles encoding type is integers, while Izakian et al. [40] only encoded with binary values, which speeds up the algorithm and deals with large solution spaces.
We have designed an effective particle representation with an accelerated algorithm on the basis of the analysis of the approach adopted from the above literature. PSO simulates the movement of a group of volatile particles. It can search very large spaces of candidate solutions.
Now, we adopt this algorithm to solve capacitated lotsizing problem.
3.2.1 Binary particle encoding
In this case, we use two particles:
(1) ${{L}_{it}}$: binary matrix $L_{i t}=\left[\begin{array}{lllllll}1 & 0 & 0 & 1 & 0 & 1 & 0 \\ 1 & 1 & 0 & 0 & 0 & 1 & 0 \\ 1 & 0 & 0 & 0 & 1 & 1 & 1\end{array}\right]$
(2) $y_t=\left[\begin{array}{lllllll}1 & 0 & 0 & 1 & 0 & 0 & 1\end{array}\right]$
3.2.2 Calculating fitness
To calculate $z$, we need all the values of decision variables ${{x}_{t}},\text{ }\!\!~\!\!\text{ }I{{d}_{t}},\text{ }\!\!~\!\!\text{ }I{{r}_{it}},\text{ }\!\!~\!\!\text{ }y{{d}_{t}},y{{r}_{it}}$, from particles one; such parameters are mentioned in Figure 4, which illustrates the way of calculation.
Figure 4. Way for calculation
Algorithm quantity delivered 
Begin 
ql=0 
For i=1 to NR do 
For t=1 to T 
If L(i,t)=1 
ql(i;t)=d(i;t) 
k=t 
Else 
While L(i,t)=0 
ql(i;k)= ql(i;k)+d(i;t) 
t=t+1 
End for 
Ir(I, t)=Ir(I,t1)+ql(i,t)d(i,t) 
If Ir(i, t)= 0 then 
yr(i, t)=0 Else 
yr(i, t)=1 
End if 
The next step is to find the produced quantity. We should use the delivered quantity instead of the demand written in the produced quantity algorithm.
Algorithm quantity produced 
Begin 
x=0; h=0 
For t=1 to T do 
If y(t)=1 
x(t)=sum ql(i; t) if $\mathbf{PE}\left( \mathbf{t} \right)>\mathbf{pe}\left( \mathbf{t} \right)$ then h=h+(PE(t)pe(t)); end 
k=t 
Else 
While y(t)=0 and x(k)$\le $cap(t) and $\mathbf{p}\le \mathbf{h}$ 
x(k)=x(k)+sum ql(i,t) h=h+(PE(t)pe(t)); 
t=t+1 
End while 
If (x(k)>cap(t) or pe(t)>PE(t) then/*test constraint capacity and CAP emission 
x(k)=x(k)sum ql(i, t) 
Y(t)=1 
End if 
t=t1 
Id(t)=Id(t1)+x(t)sum ql(i, t) 
End for 
If Id(t)=0 then 
yd(t)=0 Else 
yd(t)=1 
End if 
End 
Everything is ready to calculate fitness. The next step is to use PSO to solve the capacitated lotsizing problems.
3.2.3 Binary PSO algorithm
(1) Generate randomly X such as
$X\left( i,t,p \right)=\left[ \begin{matrix} y \\ L \\\end{matrix} \right]=\left[ \begin{matrix} \left[ \begin{matrix} 1 & 0 & \begin{matrix} 0 & 1 & \begin{matrix} 0 & 0 & 1 \\\end{matrix} \\\end{matrix} \\\end{matrix} \right] \\ \left[ \begin{matrix} \begin{matrix} 1 & 0 & 0 \\ 1 & 1 & 0 \\ 1 & 0 & 0 \\\end{matrix} & \begin{matrix} 1 & 0 & 1 \\ 0 & 0 & 1 \\ 0 & 1 & 1 \\\end{matrix} & \begin{matrix} 0 \\ 0 \\ 1 \\\end{matrix} \\\end{matrix} \right] \\\end{matrix} \right]$
where, i=1, i=1, t=7, p=1
(2) Generate randomly$~V$ in $\left[ v;\text{ }\!\!~\!\!\text{ }+v \right]$ when Dim [X]=Dim [V]
(3) Calculate fitness $gbest$ and $pbest$ as shown in Table 4.
Table 4. Fitness values
P 
Iter1 
Iter2 
Iter3 
Iter4 
1 
25 
20 
18 
16 
2 
40 
35 
25 
15 
3 
20 
19 
30 
25 
4 
60 
45 
20 
22 
(4) Next iterations to calculate the new velocity V for$~V_{\left( i,t,p \right)}^{iter+1}$
Using the following equation:
$\begin{array}{r}V_{(i, t, p)}^{iter+1}=\omega V_{(i, t, p)}^{iter}+C 1 r 1\left(pbest^{iter}X_{(i, t, p)}^{i t e r}\right) +C 2 r 2\left(gbest^{iter}X_{(i, t, p)}^{i t e r}\right)\end{array}$ (12)
(5) Calculate new $X, X_{(i, t, p)}^{\text {iter }+1}=\left\{\begin{array}{c}1 \text { if } \operatorname{Sig}\left(V_{(i, t, p)}^{i t e r+1}>r\right) \\ 0 \text { otherwise }\end{array}\right.$
Such as $Sig=\frac{1}{1+e^{V_{(i, t, p)}^{i t e r+1}}}$
In Figure 5, the detailed procedure of the Binary Particle Swarm Optimization for the planning problem with OWMR.
Figure 5. Framework of the proposed BPSO
In order to investigate the performance of the proposed algorithm (BPSO), a concrete analysis of the proposed algorithm is made. The BPSO is coded on Lenovo PC with 8G RAM and 2GHz. The software is MATLAB 2013. I have run my proposed algorithm for ten times on the same instance with the same best values of the selected parameters as it is shown in Table 5 [4143].
Table 5. The best parameters values for BPSO
Parameters 
Values 
Best Parameters 
$\omega $ 
[1,10] 
1 
$C1$ 
[1,4] 
3 
$C2$ 
[1,4] 
3 
$P$ 
[50,200] 
100 
$V$ 
[10,90] 
50 
The instance problems case for the planning problem with OWMR under environment (PM emission) constraint are presented as following (see Table 6):
Table 6. Instances variation for the proposed BPSO
T 
NR 
T 
NR 
T 
NR 
Small size 
Medium size 
Big size 

3,6,9,12 
[2,10] 
15,18,21,27 
[5,25] 
40,50,60,70 
[30,70] 
• Case 1: little size/30 instance problems/(T$\in $[3;12], NR$\in $[2; 10])
• Case 2: middle size/30 instance problems (T$\in $[15; 27], NR$\in $[5; 25])
• Case 3: great size/30 instance problems/(T$\in $[40; 70], NR$\in $[30; 70])
Table 7 displays the computational results of our proposed BPSO. In this table, the problem instances are listed in the first column, the second PM emissions and the third columns represent the number of period and retailers respectively, the fourth and five column that refer to our algorithms are still compared with CPLEX lower bound.
Table 7. Performance of BPSO
Little size 
Instances 
PM Emission 
T 
NR 
T*NR 
PSO 
CPLEX 
Err 

PE 
Pe 
Z_{PSO} 
Time(s) 
Z_{CPLEX} 
Time (s) 

1 
1 
U[3,6] 
U[2,5] 
3 
2 
6 
1030 
27.931 
1030 
0.614 
0 

2 
2 
U[3,6] 
U[2,5] 
3 
4 
12 
574 
61.1 
575 
1.343 
0 

3 
3 
U[3,6] 
U[2,5] 
6 
2 
12 
1311 
54.977 
1295 
1.208 
0.01 

4 
4 
U[3,6] 
U[2,5] 
3 
6 
18 
2953 
65.26 
2954 
1.434 
0 

5 
5 
U[3,6] 
U[2,5] 
6 
3 
18 
587 
60.45 
589 
1.302 
0 

6 
6 
U[3,6] 
U[2,5] 
9 
2 
18 
3732 
60.918 
3589 
1.339 
0.04 

7 
7 
U[3,6] 
U[2,5] 
3 
8 
24 
3831 
33.709 
3831 
0.741 
0 

8 
8 
U[3,6] 
U[2,5] 
6 
4 
24 
3831 
33.709 
3794 
0.741 
0.01 

9 
9 
U[3,6] 
U[2,5] 
12 
2 
24 
4982 
62.244 
4152 
1.368 
0.2 

10 
10 
U[3,6] 
U[2,5] 
3 
9 
27 
5841 
67.205 
5841 
1.469 
0 

11 
11 
U[3,6] 
U[2,5] 
9 
3 
27 
9653 
33.835 
9553 
0.819 
0.01 

12 
12 
U[3,6] 
U[2,5] 
3 
10 
30 
7325 
36.504 
7261 
0.802 
0.01 

13 
13 
U[3,6] 
U[2,5] 
6 
5 
30 
8652 
33.705 
8635 
0.823 
0 

14 
14 
U[3,6] 
U[2,5] 
6 
6 
36 
5906 
130.52 
5791 
2.869 
0.02 

15 
15 
U[3,6] 
U[2,5] 
9 
4 
36 
5870 
66.989 
5755 
1.472 
0.02 

16 
16 
U[3,6] 
U[2,5] 
9 
5 
45 
9565 
67.301 
9326 
1.467 
0.02 

17 
17 
U[3,6] 
U[2,5] 
6 
8 
48 
7662 
67.405 
7439 
1.482 
0.03 

18 
18 
U[3,6] 
U[2,5] 
12 
4 
48 
9197 
70.07 
7075 
1.54 
0.3 

19 
19 
U[3,6] 
U[2,5] 
6 
9 
54 
9383 
73.098 
9122 
1.6 
0.03 

20 
20 
U[3,6] 
U[2,5] 
9 
6 
54 
17012 
73.086 
16421 
1.606 
0.04 

21 
21 
U[3,6] 
U[2,5] 
6 
10 
60 
14651 
73.008 
14406 
1.605 
0.02 

22 
22 
U[3,6] 
U[2,5] 
12 
5 
60 
14651 
73.326 
14406 
1.626 
0.02 

23 
23 
U[3,6] 
U[2,5] 
9 
8 
72 
11919 
79.014 
11572 
1.737 
0.03 

25 
25 
U[3,6] 
U[2,5] 
12 
6 
72 
11700 
79.313 
8299 
1.743 
0.41 

26 
26 
U[3,6] 
U[2,5] 
15 
5 
75 
5952 
82.654 
3608 
1.816 
0.65 

27 
27 
U[3,6] 
U[2,5] 
9 
9 
81 
9565 
82.126 
9326 
1.923 
0.02 

28 
28 
U[3,6] 
U[2,5] 
9 
10 
90 
21803 
86.489 
14536 
1.901 
0.5 

29 
29 
U[3,6] 
U[2,5] 
12 
8 
96 
16971 
89.284 
11624 
1.962 
0.4 

30 
30 
U[3,6] 
U[2,5] 
12 
9 
108 
15245 
91.326 
10136 
1.852 
0.501 

Middle size 
31 
31 
U[3,4] 
U[2,3] 
18 
5 
90 
6892 
85,501 
1887 
33.879 
2.652 
32 
32 
U[3,4] 
U[2,3] 
21 
5 
105 
11771 
91.286 
2243 
35.875 
2.006 

33 
33 
U[3,4] 
U[2,3] 
12 
10 
120 
15140 
97.955 
9961 
33.321 
2.153 

34 
34 
U[3,4] 
U[2,3] 
27 
5 
135 
9661 
103.35 
1281 
60.732 
2.271 

35 
35 
U[3,4] 
U[2,3] 
15 
10 
150 
16887 
108.212 
8529 
60.378 
2.378 

36 
36 
U[3,4] 
U[2,3] 
18 
10 
180 
20284 
117.195 
8992 
65.576 
2.576 

37 
37 
U[3,4] 
U[2,3] 
21 
10 
210 
36845 
129.675 
7066 
71.089 
2.85 

38 
38 
U[3,4] 
U[2,3] 
15 
15 
225 
40385 
136.695 
17949 
70.004 
3.004 

39 
39 
U[3,4] 
U[2,3] 
18 
13 
234 
30254 
137.025 
7542 
70.52 
3.011 

40 
40 
U[3,4] 
U[2,3] 
18 
15 
270 
73815 
152.321 
18641 
73.348 
3.348 

41 
41 
U[3,4] 
U[2,3] 
27 
10 
270 
30805 
152.75 
4247 
73.023 
3.357 

43 
43 
U[3,4] 
U[2,3] 
21 
13 
273 
73262 
152.98 
15478 
73.828 
3.735 

44 
44 
U[3,4] 
U[2,3] 
15 
19 
285 
33585 
165.25 
8754 
74.110 
2.83 

45 
45 
U[3,4] 
U[2,3] 
15 
20 
300 
76534 
165.386 
33955 
74.080 
3.635 

46 
46 
U[3,4] 
U[2,3] 
21 
15 
315 
40363 
160.417 
6397 
73.584 
3.745 

47 
47 
U[3,4] 
U[2,3] 
18 
19 
342 
76982 
171.123 
11548 
70.365 
5.66 

48 
48 
U[3,4] 
U[2,3] 
15 
23 
345 
54576 
170.502 
11587 
73.58 
3.710 

49 
49 
U[3,4] 
U[2,3] 
18 
20 
360 
50575 
189.371 
15120 
44.623 
4.162 

50 
50 
U[3,4] 
U[2,3] 
15 
25 
375 
30120 
196.586 
8991 
44.320 
4.32 

51 
51 
U[3,4] 
U[2,3] 
21 
19 
399 
77895 
200.93 
13254 
44.523 
4.887 

52 
52 
U[3,4] 
U[2,3] 
27 
15 
405 
87628 
201.76 
10622 
44.623 
4.434 

53 
53 
U[3,4] 
U[2,3] 
18 
23 
414 
82585 
205.36 
15236 
49.326 
4.423 

55 
55 
U[3,4] 
U[2,3] 
21 
20 
420 
38178 
213.98 
5796 
50.356 
4.703 

56 
56 
U[3,4] 
U[2,3] 
18 
25 
450 
56901 
225.81 
13376 
52.963 
3.25 

57 
57 
U[3,4] 
U[2,3] 
21 
23 
483 
79852 
223.52 
13845 
52.627 
4.985 

58 
58 
U[3,4] 
U[2,3] 
27 
19 
513 
88956 
225.66 
15852 
52.071 
4.611 

59 
59 
U[3,4] 
U[2,3] 
21 
25 
525 
46904 
227.27 
9003 
52.962 
5.654 

60 
60 
U[3,4] 
U[2,3] 
27 
20 
540 
43615 
256.62 
4714 
52.987 
5.64 

Great size 
61 
61 
U[2,4] 
U[1,3] 
40 
30 
1200 
180659 
538.2 
18530 
448.5 
8.75 
62 
62 
U[2,4] 
U[1,3] 
40 
35 
1400 
338803 
528.19 
31459 
440.158 
9.77 

63 
63 
U[2,4] 
U[1,3] 
50 
30 
1500 
435249 
661.31 
41217 
551.092 
9.56 

64 
64 
U[2,4] 
U[1,3] 
40 
40 
1600 
227569 
733.98 
23015 
611.65 
8.89 

65 
65 
U[2,4] 
U[1,3] 
50 
35 
1750 
515273 
777.53 
51766 
647.942 
8.95 

66 
66 
U[2,4] 
U[1,3] 
40 
45 
1800 
178438 
897.52 
17402 
747.933 
9.25 

67 
67 
U[2,4] 
U[1,3] 
60 
30 
1800 
334717 
789.1 
28152 
970.593 
10.89 

68 
68 
U[2,4] 
U[1,3] 
40 
50 
2000 
186797 
1006.33 
22626 
838.608 
7.26 

69 
69 
U[2,4] 
U[1,3] 
50 
40 
2000 
548875 
950.56 
55125 
792.133 
8.96 

70 
70 
U[2,4] 
U[1,3] 
40 
52 
2080 
183852 
973.83 
19856 
797.362 
8.259 

71 
71 
U[2,4] 
U[1,3] 
60 
35 
2100 
305751 
953.94 
29830 
1173.346 
9.25 

72 
72 
U[2,4] 
U[1,3] 
70 
30 
2100 
379088 
901.94 
30231 
2899.737 
11.54 

73 
73 
U[2,4] 
U[1,3] 
40 
55 
2200 
182973 
1109.73 
17895 
983.408 
9.224 

75 
75 
U[2,4] 
U[1,3] 
50 
45 
2250 
664356 
1109.81 
57371 
924.842 
10.58 

76 
76 
U[2,4] 
U[1,3] 
60 
40 
2400 
365551 
1119.3 
36464 
1376.739 
9.03 

77 
77 
U[2,4] 
U[1,3] 
70 
35 
2450 
703450 
1115.66 
60176 
3586.847 
10.69 

78 
78 
U[2,4] 
U[1,3] 
50 
50 
2500 
776186 
1300 
61897 
1599 
11.54 

79 
79 
U[2,4] 
U[1,3] 
40 
65 
2600 
285658 
1300.86 
22025 
1602.258 
11.656 

80 
80 
U[2,4] 
U[1,3] 
60 
45 
2700 
399334 
1301.56 
36254 
1600.919 
10.02 

81 
81 
U[2,4] 
U[1,3] 
50 
55 
2750 
778985 
1112.63 
66584 
964.057 
10.699 

82 
82 
U[2,4] 
U[1,3] 
70 
40 
2800 
271554 
1560 
N/A 
N/A 
N/A 

83 
83 
U[2,4] 
U[1,3] 
60 
50 
3000 
446246 
3258.97 
N/A 
N/A 
N/A 

84 
84 
U[2,4] 
U[1,3] 
50 
65 
3250 
487589 
1658.25 
N/A 
N/A 
N/A 

85 
85 
U[2,4] 
U[1,3] 
60 
55 
3300 
332557 
4165.35 
N/A 
N/A 
N/A 

86 
86 
U[2,4] 
U[1,3] 
70 
55 
3850 
798258 
1325.36 
N/A 
N/A 
N/A 

87 
87 
U[2,4] 
U[1,3] 
60 
65 
3900 
273854 
2265.23 
N/A 
N/A 
N/A 

88 
88 
U[2,4] 
U[1,3] 
70 
60 
4200 
458272 
2210.02 
N/A 
N/A 
N/A 

89 
89 
U[2,4] 
U[1,3] 
70 
65 
4550 
985025 
3873.58 
N/A 
N/A 
N/A 

90 
90 
U[2,4] 
U[1,3] 
70 
70 
4900 
575092 
4160 
N/A 
N/A 
N/A 
For each instance, results are summarized in Table 7, in which we compare our proposed BPSO algorithms and CPLEX in terms of total cost ${{Z}_{BPSO}}$ time and running Time(s).
• Little size
From Table 7 in the little size, the difference between ${{Z}_{BPSO}}\text{ }\!\!~\!\!\text{ and }\!\!~\!\!\text{ }{{Z}_{CPLEX}}$ is $\text{Err}\in $[0; 0.65] s is small, when BPSO speed reaches 0.65s, it gives us solution optimal. We have also noticed a proportionality between the running time and (T, NR).
• Middle size
From Table 7 in the Middle size, the difference between ${{Z}_{BPSO}}\text{ }\!\!~\!\!\text{ and }\!\!~\!\!\text{ }{{Z}_{CPLEX}}$ is $\text{Err}\in $[0.98; 3.29] s for an increased instance (T, NR), and the Err increases too. We have also noticed a proportionality between the running time and (T, NR).
• Great size
From Table 7 in Great size, the difference between ${{Z}_{BPSO}}\text{ }\!\!~\!\!\text{ and }\!\!~\!\!\text{ }{{Z}_{CPLEX}}$ is $\text{Err}\in$[7.26; 11.65] s for an increased instance (T, NR) and the Err increases more as well.
$Err=\frac{{{Z}_{PSO}}{{Z}_{CPLEX}}}{{{Z}_{CPLEX}}}$
The running time BPSO is better than CPLEX.
Figure 6 depicts the convergence behavior of BPSO for the (T=50 and NR=40) instance. This figure shows the improvement of average solution quality of this instance over the number of Iterations.
Figure 6. Decrease of cost function
Figure 7 illustrates the running time (s) (BPSO, CPLEX). The running time is exponential linear.
Figure 7. The running time (BPSOCPLEX)
Most of the research’s scholars are interested in integrating carbon emission constraint within the lotsizing problem. In this article, we have focused on integrating hard particles different in their nature. These emissions contribute riskily to increasing air pollution, which negatively affects the environment, especially by major industrial companies, in general, and companies of construction materials and cement, in particular. In this article, we have solved a lotsizing problem. The description model of this later is production unit and a distribution network (size, number and location) in different retailers in time under cumulative particulate matters emission constraint.
A mixedinteger programming model has been constructed the problem is NPhard. We have developed a binary particle swarm optimization one for solving it. The BPSO approach that was proposed is powerful and delivers highquality solutions within a short running time.
Based on the discussions and analysis of this study trend of Binary swarm intelligence optimization for solving the green lotsizing problem,
Through this study, we have developed a specialized software that is also a decision support system Creates a decision support system to help business managers make decisions in time and space, quantity to trigger production and distribute in a sustainable environment while satisfying the customer at the lowest total cost and on the other hand respecting cumulative emissions.
We give some perspectives from the problems aspects, approaches and constraints for future works.
• The realworld constraints must be considered if we want to solve lotsizing problem in the various industrial environments. Through considering the reallife constraints, we can put the planning results included within the theoretical research into a specific field or a specific transport and products storage as a helping tool for making decision.
• Minimizing energy consumption in transports and production are two new objectives to seek in integrating the problem of turning the vehicle and production scheduling problems with the green lotsizing problem.
• Efficient hybrid swarm intelligence optimization with the local search is vital for solving the green lotsizing problem.
• The models and planning strategies for multiproduct and multiobjective optimization remain a challenging issue, which needs to be further studied for the lotsizing problem.
• Comparing the results of the study with metaheuristic such as ant colonies
National Higher School of Technology and Engineering has generously supported the research. The author would like to express their sincere appreciation for all support provided. We would like to thank Pr. A. Driss and R. Zeggada for checking our English phrasing.
Indices 

T 
Number of periods 
NR 
Number of retailers 
t 
Index of periods, $\text{t}$=1, 2, ..., T 
i 
Index of retailers,$\text{ }\!\!~\!\!\text{ i}$=1, 2, ..., NR 
M 
Number of particulate matters 
m 
Index of particulate matters m=1, 2, …M 
Parameters 

d_{it} 
Amount of demand of retailer i at the end period t 
$pe_{k}^{m}$ 
PM emission quota per unitary product in 
$PE_{k}^{m}$ 
Maximum unitary PM emission per period 
Production 

Cap_{t} 
Total production capacity of plant during period t 
p_{t} 
Unitary cost of production 
fp_{t} 
Setup cost of production 
Delivery 

ut_{t} 
delivery cost for unit of product 
Holding 

sd_{t} 
Unitary holding cost for distribution center in period $\text{i}$ 
fd_{t} 
Setup cost for distribution center in period i 
sr_{it} 
Unitary holding cost for retailer $\text{i}$ in period t 
fr_{it} 
Setup cost for retailer $\text{i}$ in period t 
Variable decision 

x_{t} 
Quantity produced in period t 
y_{t} 
Binary variable there is or no production in period t 
id_{t} 
Inventory level in distribution center at period t 
yd_{t} 
Binary variable there is or no stock in DC in period t 
ir_{it} 
Inventory level for retailer i in period t 
yr_{it} 
Binary variable there is or no stock in retailer i in period t 
qlr_{it} 
Quantity delivered to retailer I at period t 
X 
Swarm of particles 
p 
Index of swarm 
P 
Maximum number of particles 
iter 
Index of iterations 
ω 
Inertia weight 
C1, C2 
Positive acceleration which control the influence of gbest and pbest in search process 
r1, r2, r 
Random variable uniform distribution 
[1] Sarker, B.R. (2014). Consignment stocking policy models for supply chain systems: A critical review and comparative perspectives. International Journal of Production Economics, 155: 5267. https://doi.org/10.1016/j.ijpe.2013.11.005
[2] Tempelmeier, H., Hilger, T. (2015). Linear programming models for a stochastic dynamic capacitated lot sizing problem. Computers & Operations Research, 59: 119125.https://doi.org/10.1016/j.cor.2015.01.007
[3] Lee, A.H., Kang, H.Y., Lai, C.M., Hong, W.Y. (2013). An integrated model for lot sizing with supplier selection and quantity discounts. Applied Mathematical Modelling, 37(7): 47334746. https://doi.org/10.1016/j.apm.2012.09.056
[4] Rapine, C., Penz, B., Gicquel, C., Akbalik, A. (2018). Capacity acquisition for the singleitem lot sizing problem under energy constraints. Omega, 81: 112122. https://doi.org/10.1016/j.omega.2017.10.004
[5] Manikas, A., Godfrey, M. (2010). Inducing green behavior in a manufacturer. Global Journal of Business Research, 4(2): 2738.
[6] El Saadany, A.M.A., Jaber, M.Y., Bonney, M. (2011). Environmental performance measures for supply chains. Management Research Review, 34(11): 12021221. https://doi.org/10.1108/01409171111178756
[7] Hua, G., Cheng, T.C.E., Wang, S. (2011). Managing carbon footprints in inventory management. International Journal of Production Economics, 132(2): 178185. https://doi.org/10.1016/j.ijpe.2011.03.024
[8] Bouchery, Y., Ghaffari, A., Jemai, Z., Dallery, Y. (2012). Including sustainability criteria into inventory models. European Journal of Operational Research, 222(2): 229240.https://doi.org/10.1016/j.ejor.2012.05.004
[9] Benjaafar, S., Li, Y., Daskin, M. (2012). Carbon footprint and the management of supply chains: Insights from simple models. IEEE Transactions on Automation Science and Engineering, 10(1): 99116.https://doi.org/10.1109/TASE.2012.2203304
[10] Absi, N., DauzèrePérès, S., KedadSidhoum, S., Penz, B., Rapine, C. (2013). Lot sizing with carbon emission constraints. European Journal of Operational Research, 227(1): 5561. https://doi.org/10.1016/j.ejor.2012.11.044
[11] Dye, C.Y., Yang, C.T. (2015). Sustainable trade credit and replenishment decisions with creditlinked demand under carbon emission constraints. European Journal of Operational Research, 244(1): 187200. https://doi.org/10.1016/j.ejor.2015.01.026
[12] Palak, G., Ekşioğlu, S.D., Geunes, J. (2014). Analyzing the impacts of carbon regulatory mechanisms on supplier and mode selection decisions: An application to a biofuel supply chain. International Journal of Production Economics, 154: 198216. https://doi.org/10.1016/j.ijpe.2014.04.019
[13] Nouira, I., Hammami, R., Frein, Y., Temponi, C. (2016). Design of forward supply chains: Impact of a carbon emissionssensitive demand. International Journal of Production Economics, 173: 8098. https://doi.org/10.1016/j.ijpe.2015.11.002
[14] Sazvar, Z., Mirzapour Alehashem, S.M.J., Govindan, K., Bahli, B. (2016). A novel mathematical model for a multiperiod, multiproduct optimal ordering problem considering expiry dates in a FEFO system. Transportation Research Part E: Logistics and Transportation Review, 93: 232261. https://doi.org/10.1016/j.tre.2016.04.011
[15] Jaber, M.Y., Glock, C.H., El Saadany, A.M. (2013). Supply chain coordination with emissions reduction incentives. International Journal of Production Research, 51(1): 6982. https://doi.org/10.1080/00207543.2011.651656
[16] Florian, M., Klein, M. (1971). Deterministic production planning with concave costs and capacity constraints. Management Science, 18(1): 1220. https://doi.org/10.1287/mnsc.18.1.12
[17] Bonney, M., Jaber, M.Y. (2011). Environmentally responsible inventory models: non classical models for a nonclassical era. International Journal of Production Economics, 133(1): 4353. https://doi.org/10.1016/j.ijpe.2009.10.033
[18] Mirabelli, G., Solina, V. (2022). Optimization strategies for the integrated management of perishable supply chains: A literature review. Journal of Industrial Engineering and Management, 15(1): 5891. https://doi.org/10.3926/jiem.3603
[19] Song, J.P., Leng, M.M. (2012). Analysis of the singleperiod problem under carbon emissions policies. In: Choi, TM. (eds) Handbook of Newsvendor Problems. International Series in Operations Research & Management Science, vol. 176. Springer, New York, NY. https://doi.org/10.1007/9781461436003_13
[20] Chen, X., Benjaafar, S., Elomri, A. (2013). The carbonconstrained EOQ. Operations Research Letters, 41(2): 172179. https://doi.org/10.1016/j.orl.2012.12.003
[21] Zhang, G.Q., Ma, L.P. (2009). Optimal acquisition policy with quantity discounts and uncertain demands. International Journal of Production Research, 47(9): 24092425. https://doi.org/10.1080/00207540701678944
[22] Zhang, M.J., Kucukyavuz, S., Yaman, H. (2012). A polyhedral study of multiechelon lot sizing with intermediate demands. Operations Research, 60(4): 918935. https://doi.org/10.1287/opre.1120.1058
[23] Zhang, Z.H., Jiang, H., Pan, X.Z. (2012). A Lagrangian relaxation based approach for the capacitated lot sizing problem in closedloop supply chain. International Journal of Production Economics, 140(1): 249255. https://doi.org/10.1016/j.ijpe.2012.01.018
[24] Zhang, B., Xu, L., (2013). Multiitem production planning with carbon cap and trade mechanism. International Journal of Production Economics, 144(1): 118127. https://doi.org/10.1016/j.ijpe.2013.01.024
[25] Rosic, H., Jammernegg, W. (2013). The economic and environmental performance of dual sourcing: A newsvendor approach, International Journal of Production Economics, 143(1): 109119. https://doi.org/10.1016/j.ijpe.2012.12.007
[26] Konur, D., Schaefer, B. (2014). Integrated inventory control and transportation decisions under carbon emissions regulations: LTL vs. TL carriers. Transportation Research Part E: Logistics and Transportation Review, 68: 1438. https://doi.org/10.1016/j.tre.2014.04.012
[27] Ghosh, A., Jha, J.K., Sarmah, S.P. (2017). Optimal LoTsizing under strict carbon cap policy considering stochastic demand. Applied Mathematical Modelling, 44: 688704. https://doi.org/10.1016/j.apm.2017.02.037
[28] Fahimnia, B., Sarkis, J., Choudhary, A., Eshragh, A. (2015). Tactical supply chain planning under a carbon tax policy scheme: A case study. International Journal of Production Economics, 164: 206215. https://doi.org/10.1016/j.ijpe.2014.12.015
[29] Hammami, R., Nouira, I., Frein, Y. (2015). Carbon emissions in a multiechelon productioninventory model with lead time constraints. International Journal of Production Economics, 164: 292307. https://doi.org/10.1016/j.ijpe.2014.12.017
[30] He, Y., Li, Y., Wu, T., Sutherland, J. (2015). An energyresponsive optimization method for machine tool selection and operation sequence in flexible machining job shops. Journal of Cleaner Production, 87(1): 245254. https://doi.org/10.1016/j.jclepro.2014.10.006
[31] Martí, J.M.C., Tancrez, J.S., Seifert, R.W. (2015). Carbon footprint and responsiveness tradeoffs in supply chain network design. International Journal of Production Economics, 166: 129142. https://doi.org/10.1016/j.ijpe.2015.04.016
[32] Koca, E., Yaman, H., Aktürk., M.S. (2015). Stochastic lot sizing problem with controllable processing times. Omega, 53: 110. https://doi.org/10.1016/j.omega.2014.11.003
[33] Zhou., S.H., Zhou, Y.L., Zuo, X.R., Xiao, Y.Y., Cheng Y. (2018). Modeling and solving the constrained multiitems lotsizing problem with timevarying setup cost. Chaos, Solitons and Fractals, 116: 202207. https://doi.org/10.1016/j.chaos.2018.09.012
[34] Purohit, A.K., Shankar, R., Dey, K.P., Choudhary, A. (2016). Nonstationary stochastic inventory lotsizing with emission and service level constraints in a carbon capandtrade system. Journal of Cleaner Production, 113: 654661. https://doi.org/10.1016/j.jclepro.2015.11.004
[35] Claassen G.D.H., Kirst, P., Thai Thi Van, A., Snels, J.C.M.A., Guo, X., van Beek, P. (2024). Integrating timetemperature dependent deterioration in the economic order quantity model for perishable products in multiechelon supply chains. Omega, 125: 103041. https://doi.org/10.1016/j.omega.2024.103041
[36] Kennedy, J., Eberhart, R. (1995). Particle swarm optimization. In Proceedings of ICNN'95International Conference on Neural Networks, Perth, WA, Australia, pp. 19421948. https://doi.org/10.1109/ICNN.1995.488968
[37] Driss, I. (2021). Binary particle swarm optimization for one warehouse multi retailer problem with cumulative particulate matter constraint. In 2021 International Conference on Control, Automation and Diagnosis (ICCAD), Grenoble, France, pp. 16. https://doi.org/10.1109/ICCAD52417.2021.9638769
[38] Boonmee, A., Sethanan, K. (2016). A GLNPSO for multilevel capacitated lotsizing and scheduling problem in the poultry industry. European Journal of Operational Research, 250(2): 652665. https://doi.org/10.1016/j.ejor.2015.09.020
[39] Chen, Y.Y., Lin, J.T. (2009). A modified particle swarm optimization for production planning problems in the TFT Array process. Expert Systems with Applications, 36(10): 1226412271. https://doi.org/10.1016/j.eswa.2009.04.072
[40] Izakian, H., Ladani, B.T., Abraham, A., Snasel, V. (2010). A discrete particle swarm optimization approach for grid job scheduling. International Journal of Innovative Computing, Information and Control, 6(9): 115.
[41] Han, Y., Tang, J., Kaku, I., Mu, L. (2009). Solving uncapacitated multilevel lotsizing problems using a particle swarm optimization with flexible inertial weight. Computers & Mathematics with Applications, 57(1112): 17481755. https://doi.org/10.1016/j.camwa.2008.10.024
[42] Pant, M., Thangaraj, R., Abraham, A. (2009). Particle swarm optimization: Performance tuning and empirical analysis. In Foundations of Computational Intelligence. Berlin, Heidelberg: Springer, pp. 101128. https://doi.org/10.1007/9783642010859_5
[43] Parapar, J., Vidal, M.M., Santos, J. (2012). Finding the best parameter setting: Particle swarm optimisation. In 2nd Spanish Conference on Information Retrieval, pp. 4960.