Mathematical Modeling for Sustainable Production Line

This paper examines how to design the sustainable production line in the factories with minimum environmental impacts and maximum social benefits. For modeling, the literature review is carried out first and according to researches and papers, environmental and social factors are extracted and then with using Sustainable Supply Chain Management structure, Integer Linear programming with using binary variables and Multi Goal Programming, to design an overall sustainable and environmentally friendly production line model with the most positive social effect.


Introduction
In this paper, in order to review the impacts of factories and production lines on the environment and society, we first review the literature survey which are presented by different researches for using mathematical models. By searching for papers, the papers closest to this issue under investigation were Sustainability papers and the most relevant studies are on supply chain and production line. Then we checked the environmental and social parameters and variables which are used in the different mathematical models. Finally, by using the Multi Goal Programming, Integer linear program sustainable supply chain management structure, we make a model for sustainable production line.
The first time, sustainable supply chain was introduced in 2008. (Seuring, S. & Muller, M., 2008) found that the concept of integrating environmental, social and economic factors to improve information flow, workflow and environmental protection.
According to searches in papers, most of the environmental and product impact discussions have been found in the Sustainable Supply Chain Management field. Hence the structure used in the production line is based on Sustainability and using SSCM structures.
In this paper, we first review the thematic literature on the three categories of SSCM and the mathematical models solved in SSCM, environmental factors and social factors, then apply the modeling used for the production line based on the literature review and modeling experiences in other fields. Finally, it is summarized and recommendations for further research are given.
In this case, we present a new mathematical model for introducing Sustainable Production line like that some researchers applied responsibilities for environmental protections to the production processes through Cleaner Production. For making our model, we considered the three dimensions of sustainability or triple line model which are social, environmental and economic dimensions.
Research problem is specifically defined by example where the production line undertakes minimize consumed energy and environmental impact, while a factory management wanted minimum costs and also social rules defined social benefits for factories. In this case, that is necessary both parties (production and management) may consider these sustainable dimensions in their efforts. For SPL modeling and implementation, the below steps are done:  Modeling which is used the SSCM structure and Integer programming.
The results show that for implementation SPL, we have different alternative and this model can be as a dynamic program model, because of we should check different alternatives continually and in programming time need to review the production process for re-engineering SPL model.

Sustainable Supply Chain Management
Sustainability is not a new approach, and the first time (Costanza, R., Patten, B., , 1995) was applied sustainability when he wanted to defining and predicting sustainability. After that (Seuring, S. & Muller, M., 2008) used sustainability in definition of SCM for introducing an integration between environments, social and economic goals in the industries. Sustainability also known as a triple bottom line (TPL) for designing model. (Devika, K., Jafarian, A. & Nourbakhsh, V., 2014). The triple line bottom refers to three dimensions of environmental, economic and social. Majority of researches used triple line bottom for making their models either conceptual or mathematical in the field of supply chain management.

Sustainable Supply Chain Management Mathematical Models
For several years, researchers try to define the conceptual designs for this concept, however after 2010, authors designed different mathematical models for solving problems in the scope of SSCM. Mathematical Modeling approaches are applied into SSCM in different ways for making solutions like Multi-Objective programming, integer Linear Programming, fuzzy logic-based Multi Criteria Decision Making (MCDM), Data Envelopment Analysis (DEA), Game Theory, Analysis of Variance (ANOVA), and Goal Programming and so on which is shown in the below table. Given the question of whether designing and modeling the optimal production line is the least harmful to the environment and the most beneficial to the social, a review of the subject literature has been conducted. By looking at previous research and papers, we have come to the category of Sustainability and SSCM, and the researchers have identified environmental factors as variables or parameters in these two cases.

Sustainable Production Line
Supply chain management has different modules which are including Marketing Designing, Supplier Selection, Purchasing and Procurement, Manufacturing and Production, Logistic and transportation, etc. There are limited researches for Sustainable production line which are designed mathematical model. Sustainable production module in SCM refers to different process of manufacturing including designing, supplying raw material, manufacturing, packing and marking and so on. (Alinso,L., Rubino,E.M., Agustina,B., & Domingo,R., , 2017). The cleaner production program is a methodology and useful model for designing the environmental dimension of sustainable production line (Tran, T.V., Schnitzer, H., Braunegg, G., & Lee, H.T., 2017). The sustainable production line focus on the process of production and manufacturing which should meet the environmental and social requirement.

Corporate Social Responsibility
Nowadays social responsibility is a common requirement for all factories. There are a huge scholar researches for social responsibility including corporate social responsibility model (Tsalis, T.A. & Evangelinos, K.L., 2019) (Govindan, K., Shankar, M. & Devika, K. , 2018). The corporate social responsibility (CSR) is defined as a concept of management which is used by companies and factories for integration of environmental and social concerns in the process and system design and their interactions (Motevali, H.S., Torabi, S.A., Ghasemi, R., 2016). The purpose of CSR is that the companies and factories involve is social responsibility and philanthropic causes which increase the social benefits. CSR highlight the social benefit enablers, and we use this approach for designing our model.
Our aim in this research is to make a mathematical model with using cleaner production and social benefits in the supply chain. The Goal of social benefits are through the lens of two kind of parameters that we considered: negative social impact and positive social impact. (We emphasis on the environmental and social parameters which are related to Sustainable Supply Chain Management). Finally, in this paper we explore how theory from the field of Cleaner Production Program and SSCM can apply for building a mathematical model and, find a solution for implementation the Sustainable Production Line.

Modeling
As presented in the Introduction, this paper presents a new method for implementation Sustainable Production Line. We used the Sustainable Supply Chain Management structure as the main structure for making model and used Goal Programming through applying the integer linear program for developing an optimal mathematical model. It is clearly that every production line has different process and if we want to make a model, first we need to define process map of the production line.
In the initial literature review, Goal Programming is an effective decision support tool for alternative selection. (Jones, D.F. & Tamiz, M., 2002) said that Goal Programming has a widely field in the different researches such as engineering, academic management, agricultural management, energy planning, production planning, and green environmental management.
As per our searching, below papers were found that present a goal-programming methodology related to SSCM, CP and CSR. Table 2 presents the papers which are used mathematical models for solving SCM problems and achieving the best solution for environment protection and social benefit.

Problem Definition
The research question is how to design a model for production processes that has the least environmental detrimental effect, benefits the community most, and costs the least.
To cope with this challenge, describe in the introduction, this paper suggest an optimization mathematical model based on goal programming into "cleaner production methodology", "Social Benefits" and "management of costs" as follows: First, we define Assumptions, Decision variables and parameters. Then, model developing including objective functions and constraints.

Definition of the Assumptions, Decision Variables
This model is based on production process and for improving the production processes and making a sustainable production line, first we define all the process of production line, then we consider different alternatives of every processes.

Production Processes
-Assuming production line has different processes (Pi, i=1,2,…n) and every processes have different alternatives (xij, j=1,2,...mi) for performing, we propose a model to find an optimum selections of alternatives which bring the best results for our model. Figure 1. The diagram of processes and alternatives for production line -We consider n process for production in SPL modeling and each process has m i alternatives for doing process. It is clearly that if one alternative of process "i" is chosen, the other alternatives will be cancelled and it means that we define the variables for alternatives as x ij = 1 (if selected) or 0 (if not selected). Table 3 presents the variables matrix for processes alternatives. On the other hand, we define "X io " as baseline production line processes of Pi (without improvement); -Every alternatives of process have own costs, environmental impact and social impact. We calculate costs, environmental impact and social impact of every alternatives of processes by using the models that propose by some researchers like Cleaner Production for environmental impact, SCR for social impact and using Accounting models for costs. Then, we make a model for choosing the optimum solution for combination of processes and make a SPL (Sustainable Production Line.) Table 3 shows the decision variable matrix for different Alternatives of production processes. Every column shows the alternatives for every process.

Environmental Impact
Every alternative of the process has own environmental impact which can be seen in the environment impact matrix as per below table. For calculating the environmental impact, we have different index and parameters. We use the sustainability development index (2019) and the summary of parameters which are suggested by different researchers through cleaner production planning researches (Table 5). For every alternatives of process, we have above parameters. It means that for Calculating Eij of alternative j of process i, referring to table (4), we have different amount of environmental impact like E(P) ij :

Equation 1: E(P) ij = {E (energy)ij , E (water)ij , E (co2)ij , E (CH4)ij , E (N2O)ij , E (HFCs)ij , E (PFCs)ij , E (NP3)ij , E (sw)ij ,E (Hw)ij , E (bio)ij }.
For Calculating E ij , we have to normalize different environmental impact variables and then we can calculate the total environmental impact of alternative j process i. We use Normal Distribution for normalizing the environmental parameters. The formula that we use is: The result of Equations 2, 3 and 4 is :

Economic Parameters
Assuming the development SPL economic variables as per below: o Cc: Cost for purchasing carbon credit for production line according to rules and regulation of government for environmental protection.
Equation 6: We have three matrixes for Costs which are Economic Costs (C pij ) Matrix, Environmental reduction costs Matrix (C Eij ) and Social impact costs (C Sij ). We define C Xij as per Equation 6.
Equation 7: We have a cost matrix for processes alternatives as per below table.

Social Parameters
Nowadays one of the important responsibilities of factories is social responsibility. For designing social responsibilities systems in the factories and implementation and measurement, researchers proposed different models. We study these models and most important is the parameters and criteria for measurements and modeling the social responsibilities. Different researchers offer various parameters such as decent work conditions, Human rights and abuse human rights, personal training, health and safety, rewards and punishments criteria and so on. (Table 7). According to the research results and literature review, we define two kind of social impacts: positive impacts and negative impacts.  With using Equation (8) , Equation (9) and Equation (10): The result of the Equations 7-11 are a matrix as per below table which is shown the social impacts in every alternatives:

Model Development
The main issues for the production line is which alternatives for every process should be selected in order to minimum environmental impacts and maximum social benefits with production constraints and budget constraints. The Goals are defined according to research question as per below:  Objective function of Goal 2: The goal is the minimization of total improvement cost including environmental and social costs. Investment cost is the total cost for improvement of all processes and selected correspondence alternatives. Thus, the objective function of case 1 is given by Equation (19): By using equation (11) and equation (15), we have equation (16): ))*X ij

Constraints
Always resources are limited and main constraints are related to limitation of resources. The main resources for every production lines are Budget and Time. Some other constraints are related to rules and regulation like as: Government limitation for Carbon Cap, Environmental Protection rules for using water and electricity, Social rules, Stockholders and management rules and so on.

 Time Limitation:
As the commitment of factory to customers for their Demand and also time limitation for production line, we define time constraint as in Equation (17). We consider the budget for decreasing environmental impact as C o . Thus, total investment cost for environmental impact must be less than C o . Constraint of investment cost can be formulated as Equations (18,19,20): The budget of payment to Personal Staff for new job opportunities and training workers is CS o , thus, the total costs of staff payment should be less than budget : In addition, total costs including investment for environmental impact and staff payment must be less than S o + C 0 . Constraint of investment cost can be formulated as Equation (17): We consider TB for total budget of production line and we have below constraint:  Environmental Limitation: If we consider the objective of environmental impact is ER and the existing situation is E 0 , Thus, total decreasing environmental impact is not less than ER. It means that E 0 *X ij -E ij * X ij should be less than ER. The constraint of environmental impact reduction potential can be formulated as Equation (22): Social Limitation: (Stindt, 2017) defined that the constraints in social dimension are related to human right, labor practice and decent work and Social Life Cycle Assessment (SLCA) of Products. The human rights in different countries are defined in the labor rules and conditions. For decent work, normally is 160 hours/month and 250 days in the year. We formulate constraint as per below for working time, we consider NYW parameter as normal yearly work (Man/Day for one year): Equation 23: ∑ ∑

 Definition of Decision Variables limitations:
x ij is the binary variable, x ij = 0, 1, if X ij The following constraints are related to the X ij which are decision variables in the model.

Conclusion
The research of this paper is in the context of designing production processes appropriate to environmental and social impacts. The question of this research is how we can design sustainable production line which has the minimum bad effects on environmental and social and fulfill the minimum economic requirements. For this design, the literature on environmental and social impacts has been reviewed at first, and the closest research framework to this topic has been SSCM. By studying the structures and models common in SSCM and modeling, we came up with the Sustainable Production Line model for designing production line processes. This model uses the structure introduced in the SSCM and based on the minimum environmental impact and the maximum social benefits.
Using the literature review, environmental and social impact parameters are identified in tables. Normal distribution is used to apply the parameters in the mathematical model and to match them. Based on the mathematical models of integer programming and multi goal programming the basic mathematical model of SPL was developed. The variables of decision making are the alternatives of processes for production lines which are impacts on costs of production lines, also have social and environmental impacts.
It is recommended to develop this model in different modules of supply chain management like Transportation and Logistics processes, Purchasing and procurement, Design, and so on. And fit the relevant environmental and social parameters into the model for each module. In light of the conclusions above, the study recommends that making mathematical model for the whole SSCM process with considering Environmental and social parameters.