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MATH 415
Oct 17, 2023
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Yanqian Pan Scenario : Profit Optimization for a Furniture Manufacturing Business In this scenario, I am the owner of a furniture manfacture business. I produce two types of furniture: chairs and tables. The production process involves two main activities: carfting the furniture and painting the furniture. I want to maximize monthly profit given the constraints related to labor hours and material costs. Decision Variables : x = number of chairs produced per month y = number of tables produced per month Px = profit per chair, which is $8 Py = profit per tables, which is $10 Cx: craft time per chair, which is 2hrs Cy: craft time per table, which is 3hrs Tc: total craft time, which is 20 hrs Objective Function : To maximize monthly profit: Z = Px*x+Py*y Constraints : 1. Craft time: limited amount of labor hours available per month. Cx*x + Cy*y<=Tc, which is 2. Painting time: a limited amount of labor hours available per month. 3. Material Usage: a limited amount of material, such as woods, that can be used per month. 4. Demand: The number of chairs and tables produced must be less than or equal to the demand. 5. Non-negativity: The production quantity cannot be negative. Mathematically, the constraints can be represented as follows: 1. **Carpentry Time:** \[ C_{\text{chair}} \cdot x + C_{\text{table}} \cdot y \leq T_{\text{carpentry}} \] where: - \( C_{\text{chair}} \) = carpentry time per chair - \( C_{\text{table}} \) = carpentry time per table - \( T_{\text{carpentry}} \) = total available carpentry time
2. **Finishing Time:** \[ F_{\text{chair}} \cdot x + F_{\text{table}} \cdot y \leq T_{\text{finishing}} \] where: - \( F_{\text{chair}} \) = finishing time per chair - \( F_{\text{table}} \) = finishing time per table - \( T_{\text{finishing}} \) = total available finishing time 3. **Material Usage:** \[ M_{\text{chair}} \cdot x + M_{\text{table}} \cdot y \leq M_{\text{total}} \] where: - \( M_{\text{chair}} \) = material used per chair - \( M_{\text{table}} \) = material used per table - \( M_{\text{total}} \) = total available material 4. **Demand:** \[ x \leq D_{\text{chair}} \] \[ y \leq D_{\text{table}} \] where: - \( D_{\text{chair}} \) = demand for chairs - \( D_{\text{table}} \) = demand for tables 5. **Non-negativity:** \[ x, y \geq 0 \] ### Example Parameter Values: - Profit per chair (\( P_{\text{chair}} \)): $50 - Profit per table (\( P_{\text{table}} \)): $120 - Carpentry time per chair (\( C_{\text{chair}} \)): 5 hours - Carpentry time per table (\( C_{\text{table}} \)): 15 hours - Total available carpentry time (\( T_{\text{carpentry}} \)): 450 hours - Finishing time per chair (\( F_{\text{chair}} \)): 2 hours - Finishing time per table (\( F_{\text{table}} \)): 4 hours - Total available finishing time (\( T_{\text{finishing}} \)): 180 hours
- Material used per chair (\( M_{\text{chair}} \)): 10 units - Material used per table (\( M_{\text{table}} \)): 25 units - Total available material (\( M_{\text{total}} \)): 600 units - Demand for chairs (\( D_{\text{chair}} \)): 40 units - Demand for tables (\( D_{\text{table}} \)): 20 units Now, let's solve the LP problem with the provided parameter values and interpret the results. ### Solution and Interpretation: - **Chairs to Produce:** \(40\) units - **Tables to Produce:** \(8\) units - **Maximum Possible Profit:** $\(2960\) per month **Interpretation:** The optimal production plan to maximize profit, given the constraints, involves producing 40 chairs and 8 tables per month. By adhering to this plan, the maximum attainable profit is $2960 per month. This implies that the demand constraints for chairs are fully utilized (i.e., producing exactly as much as the demand), while the production of tables is restricted by the available carpentry time, finishing time, and materials, not reaching the demand limit. ### Further Analysis: 1. **Sensitivity Analysis:** It could be valuable to analyze how sensitive the profit is to changes in the constraints and coefficients, providing insights into which parameters are critical for enhancing profitability. 2. **Scalability:** Assessing whether the production plan is scalable and understanding how changes in demand or resource availability (such as increased carpentry time or materials) might impact the profit and production quantities. 3. **Alternative Scenarios:** Exploring alternative scenarios, such as introducing new product lines or adjusting the production process, to understand how these changes might influence the optimal production plan and profitability. ### Conclusion: Linear programming provides a powerful tool for making strategic production decisions that maximize profitability while considering various constraints like labor hours, material availability, and demand. This methodology enables business owners to make informed, data-driven decisions, enhancing the efficiency and sustainability of their operations.
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