## 3 Mathematical Model  
Mathematical methods have been used to calculate the environmental impact of energy resources. In the past, managers were only focusing on the basic objectives i.e., cost, time and quality. Now, the current and future situations of the global warming compelled researchers and experts to set an eco-friendly objectives. This is the reason that managers are also manufacturing and dealing with sustainable elements including energy and carbon emissions in the supply chain. This paper deals with the supply chain management, which integrates multiobjective goal programming associated with the manufacturing phase of a Taiwan-based automobile part with integrated suppliers. The importance of proposed mathematical model is not only limited to the production system but also extended to the supply chain management by integrating sustainable suppliers to undertake the limitations of carbon footprint and renewable energy. Taiwan is known for manufacturers and exporters of a variety of auto parts among developing countries. The production flow diagram of automobile parts manufacturing is illustrated in two steps as in Figures 2 and 3, respectively. Figure 2: Production flow of automobile parts. The three parts \(A\), \(B\), and \(C\) are produced in-house by a sequence of manufacturing processes, i.e., sealing, bending, stuffing, stamping 1 and 2, cutting, cleaning, laser, coating, welding, fastening, and packaging. The sources of energy used in the manufacturing processes are electricity, solar, diesel, and gasoline. Solar energy as a renewable energy is considered to represent \(h\%\) of the electricity consumed. By this consideration, the requirement of electricity is reduced to \(1-h\) in each process category. The economic feasibility of solar energy as a renewable is analyzed by using the levelized cost of electricity (LCOE) generation to compare with other electricity technologies. LCOE calculates the cost of solar energy annually by considering the lifetime of the solar system using discounted cash flow techniques i.e., by calculating the present value of cash flows by mean of discount rate, \(r\). Therefore the sum of present value of LCOE multiplied by the energy generated should be equal to the present valued net costs in Equation (1) [17,50,51]:  
\[\sum_{t=1}^{T}\left(\frac{LCOE_{t}}{(1+r)^{t}}\times E_{t}\right) = \sum_{t=1}^{T}\frac{C_{t}}{(a+r)^{t}}, \tag{1}\]  
where \(C_{t}\) is the net cost of the solar system, consisting cash outflows in the form of initial investment \((I_{t})\), operations costs \((O_{t})\), maintenance cost \((M_{t})\), and fuel expenses \((F_{t})\) as given in Equation (2):  
\[LCOE = \frac{\sum_{t=1}^{T}\left(I_{t}+O_{t}+M_{t}+F_{t}\right)/(1+r)^{t }}{\sum_{t=1}^{T}E_{t}/(1+r)^{t}}. \tag{2}\]  
The limited capacity of resources and the aim of low operational cost compel manufacturing firms to outsource coating and laser operations to suppliers i.e., \(U\), \(V\), and \(W\). There is an inspection workstation after each laser and coating operation to sort pass, rework and rejection parts. Pass parts move to the next station, rework parts backtracks to repeat the same operation and rejections move to scrap section, which are further salvaged at a low price. Let \(\beta\) be the proportion of defective items of total production, and \(\alpha\) is the percentage of scrap among defective items, then \(\alpha\beta\) is the scrap quantity of total production, and \(\beta-\alpha\beta\) represents the rework parts backtracked on the same workstation. The manufacturing industry evaluates the suppliers for outsourcing on the basis of quality, cost and carbon footprint. This model is proposed to replace MQL with cutting fluid in the cutting operations of the process flow to enhance environmental, societal, and economic fronts. Different techniques are used to find the optimal solution of a multiobjective mathematical model in the form of a system of equations. Goal programming (GP), developed by [PERSON] et al.