An Effective Particle Swarm Optimization for Lot-Sizing Problem with Particulate Matter Emission 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 Lot-Sizing 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 lot-sizing 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 two-stage 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 multi-criteria 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 multi-product, multi-period 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 two-level 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 NP-hard [16].

Table 1. Literature review

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 SKIKDA-Algeria.

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Figure 1. Particle matter emissions in the cement industry in SKIKDA-Algeria

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].

Now we adopt this algorithm to solve capacitated lot-sizing 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 multi-level capacitated lot-sizing 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 lot-sizing 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.

4.png

Figure 4. Way for calculation

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.

Everything is ready to calculate fitness. The next step is to use PSO to solve the capacitated lot-sizing 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

(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.

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Figure 5. Framework of the proposed BPSO

Most of the research’s scholars are interested in integrating carbon emission constraint within the lot-sizing 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 lot-sizing 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 mixed-integer programming model has been constructed the problem is NP-hard. We have developed a binary particle swarm optimization one for solving it. The BPSO approach that was proposed is powerful and delivers high-quality 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 lot-sizing 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 real-world constraints must be considered if we want to solve lot-sizing problem in the various industrial environments. Through considering the real-life 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 lot-sizing problem.

• Efficient hybrid swarm intelligence optimization with the local search is vital for solving the green lot-sizing problem.

• The models and planning strategies for multi-product and multi-objective optimization remain a challenging issue, which needs to be further studied for the lot-sizing problem.

• Comparing the results of the study with metaheuristic such as ant colonies