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Integrating Economic, Environmental, and Social Responsibility to Balance "Profit Maximization" and "Sustainable Development": A Mathematical Model for Corporate Sustainability Transition

1. Multi-Objective Optimization Model

This model combines profit maximization with minimizing sustainability indicators, forming a multi-objective optimization problem.

Objective Function:

max ( Z = λ_1 ⋅ Profit(t) - λ_2 ⋅ EnvironmentalImpact(t) - λ_3 ⋅ SocialImpact(t) )

Explanation:

• Profit(t): Profit function at time $t$, related to pricing strategy, sales volume, and costs.
• EnvironmentalImpact(t): Environmental impact function (e.g., carbon emissions, resource consumption), which should be minimized.
• SocialImpact(t): Social impact function (e.g., employee welfare, supply chain ethics), which reflects corporate social responsibility.
• ${\lambda }_1,{\lambda }_2,{\lambda }_3:$ Weight coefficients adjusting the priorities of different objectives.

Constraints:

1. Sales Constraints:
   $\text{Sales}(t)\ge \text{MinimumMarketDemand}(t)$
   Ensures market demand is met.
2. Environmental Constraints:
   $\text{EnvironmentalImpact}(t)\le \text{SustainableThreshold}(t)$
   Keeps environmental impact within sustainable limits (e.g., carbon emission regulations).
3. Social Responsibility Constraints:
   $\text{SocialImpact}(t)\ge \text{MinimumSocialResponsibility}(t)$
   Ensures minimum social responsibility standards are met, such as fair employee treatment and ethical trade.

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2. Balancing Profit and Carbon Emissions

This model aims to maximize profit while limiting carbon emissions.

Objective Function: $\text{max}P(t)=R(t)-C(t)-\gamma \cdot E(t)$

Explanation:

• $P(t):$ Profit function, equal to revenue minus costs and environmental costs.
• $R(t):$ Revenue function, dependent on product price and sales volume:
  $R(t)=\text{Price}(t)\cdot \text{Sales}(t)$
• $C(t):$ Cost function, including raw materials, labor, and transportation.
• $E(t):$ Carbon emissions function, linked to production and logistics activities.
• $\gamma :$ Carbon cost coefficient, reflecting the economic cost of carbon emissions (e.g., carbon tax or carbon credits).

Constraints:

1. Market Capacity Constraint:
   $\text{Sales}(t)\le \text{MarketPotential}(t)$
2. Carbon Emission Constraint:
   $E(t)\le \text{CarbonBudget}(t)$
   Ensures emissions remain within allocated carbon limits.
3. Price Elasticity Constraint:
   $\frac{\mathrm{\partial }R(t)/}{\mathrm{\partial }\text{Price}(t)}\ge 0$
   Ensures pricing strategies maintain market demand.

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3. Comprehensive Product Lifecycle Model

This model integrates a product's lifecycle impact ($P(t)$) with environmental and social factors.

Objective Function: 

max ( U = int_0 to T [ R(t) - C(t) - δ ⋅ E(t) - ϕ ⋅ S(t) ] dt )

Explanation:

• $U:$ Total utility function, accounting for economic benefits and negative impacts.
• $S(t):$ Social impact index, reflecting the company's contribution or negative effects on society.
• $\delta , \varphi :$ Environmental and social cost coefficients, measuring the weights of environmental and social responsibility.

Model Features:

• Time integration reflects the impact of the entire product lifecycle (from design to disposal).
• Carbon emissions ($E(t)$) and social impact ($S(t)$) are endogenized into decision-making.

Constraints:

1. Lifecycle Carbon Footprint:
   

   int_0 to T E(t)  dt ≤ TotalCarbonBudget

2. Minimum Social Responsibility Standards:
   $S(t)\ge \text{MinimumThreshold}$

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4. Dynamic System Modeling

Differential or difference equations are used to dynamically simulate the economic and environmental system:

dP(t) / dt = ProfitRate - γ ⋅ dE(t) / dt - ϕ⋅ dS(t) / dt

This model can be used to:

1. Simulate profit growth ($P(t)$) trends over time.
2. Study the long-term impact of carbon emissions ($E(t)$) and social factors ($S(t)$) on profit.

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5. Practical Applications and Implementation Recommendations

Implementation Steps:

1. Data Collection:

   • Product carbon footprint (including production, transportation, and disposal stages).
   • Production and operational costs.
   • Market demand and price elasticity data.

2. Model Calibration:

   • Adjust model parameters ($\lambda ,\gamma ,\delta ,\varphi$) based on industry characteristics.

3. Solving Multi-Objective Problems:

   • Use numerical methods (e.g., linear programming, simulated annealing, or particle swarm optimization) to solve the model.

4. Result Analysis and Policy Formulation:

   • Balance profit, environmental, and social objectives to develop sustainable pricing and innovation strategies.

These equations provide a framework to help businesses maximize profits while considering environmental and social responsibilities. However, specific applications require tailoring weights and parameters based on industry characteristics, with continuous monitoring of external changes to dynamically optimize strategies.

Case Study: Balancing Profit Growth with Carbon Emissions and Social Impact for an Electronics Manufacturer

Model Background:

An electronics manufacturer aims to maximize profit ($P(t)$) while reducing carbon emissions ($E(t)$) and improving social responsibility ($S(t)$). To achieve this, the following differential equation is used:

dP(t)/dt=ProfitRate−γ⋅dE(t)/dt−ϕ⋅dS(t)/dt

Assumptions and Initial Conditions:

• Initial Profit: $P(0)=100$ (in million USD)
• Initial Carbon Emissions: $E(0)=50$ (in kilotons)
• Initial Social Contribution: $S(0)=20$ (social responsibility index)
• Profit Growth Rate ($\text{ProfitRate}$): 30 million USD per year
• Carbon Emission Reduction Rate: $\frac{dE(t)/}{dt}=-5$ kilotons per year, with impact on profit $\gamma =2$ million USD/kiloton
• Social Responsibility Improvement Rate: $\frac{dS(t)}{dt}=3$, with impact on profit $\varphi =1.5$ million USD/unit

Dynamic Equations:

1. Profit Growth Equation: dP(t)/dt=30−2⋅dE(t)/dt−1.5⋅dS(t)/dt
2. Carbon Emission Reduction: $$\frac{dE(t)/}{dt}=-5$$
3. Social Responsibility Growth: $$\frac{dS(t)/}{dt}=3$$

Simulation Results:

Using these equations, we conduct a numerical simulation to observe changes over 10 years. The results provide insights into the balance between profit growth, carbon reduction, and improvements in social responsibility.