Statistic Help Solve Problem And Applications Klein Industri

Statistic Helpsolveproblem And Applicationsklein Industries Manufact

Klein Industries manufactures three types of portable air compressors: small, medium, and large, with unit profits of $20.50, $34.00, and $42.00, respectively. The projected monthly sales are: Small (14,000 to 21,000 units), Medium (6,200 to 12,500 units), and Large (2,600 to 4,200 units). The production process involves bending and forming, welding, and painting, with specific time requirements for each product in each activity. The available time for bending/forming and welding is 23,400 minutes each, and for painting, 46,800 minutes. The goal is to determine how many of each type of air compressor to produce to maximize profit, given these constraints. The problem requires formulating and solving a linear optimization model using the auxiliary variable cells method, and then analyzing sensitivity information via a memo to the production manager. Additionally, the problem asks to solve the model without auxiliary variables, interpret the relationship between reduced costs and shadow prices, and ensure correct setup of the Solver tool in Excel, including constraints and proper cell assignments.

Paper For Above instruction

Optimizing Production for Profit Maximization at Klein Industries

Introduction

Klein Industries, a manufacturer of portable air compressors, faces the challenge of determining the optimal production quantities of small, medium, and large compressors to maximize profit while adhering to production constraints. This problem encapsulates core principles of linear programming, including modeling constraints, objective functions, and analyzing sensitivity information to inform managerial decision-making.

Formulating the Linear Programming Model

The decision variables are the number of units produced for each compressor type: \(x_1\) (small), \(x_2\) (medium), and \(x_3\) (large). The objective function aims to maximize total profit:

Maximize Z = 20.50x_1 + 34.00x_2 + 42.00x_3

Subject to constraints based on sales projections, manufacturing times, and capacities:

• Sales constraints (minimums):
• \(x_1 \geq 14,000\)
• \(x_2 \geq 6,200\)
• \(x_3 \geq 2,600\)
• Sales constraints (maximums):
• \(x_1 \leq 21,000\)
• \(x_2 \leq 12,500\)
• \(x_3 \leq 4,200\)
• Time constraints:
• Bending/Forming: \( 0.4x_1 + 0.7x_2 + 0.8x_3 \leq 23,400\)
• Welding: \( 0.6x_1 + 1.0x_2 + 1.2x_3 \leq 23,400\)
• Painting: \( 1.4x_1 + 2.6x_2 + 3.1x_3 \leq 46,800\)

Supply, demand, and process constraints collectively define the feasible region of production quantities. Solving this LP model involves setting up the decision variables, objective function, and constraints, then applying Excel Solver utilizing the auxiliary variable cells method, which helps in conducting sensitivity analysis effectively.

Solving Using Auxiliary Variables

Auxiliary variables are introduced to model the slack or surplus of constraints, which makes sensitivity analysis more transparent. For example, slack variables \(s_1, s_2, s_3\) could be added to the capacity constraints to measure unused time in each process. The LP model then becomes:

• \( 0.4x_1 + 0.7x_2 + 0.8x_3 + s_1 = 23,400\)
• \( 0.6x_1 + 1.0x_2 + 1.2x_3 + s_2 = 23,400\)
• \( 1.4x_1 + 2.6x_2 + 3.1x_3 + s_3 = 46,800\)

In Excel, decision variables (\(x_1\), \(x_2\), \(x_3\)), slack variables (\(s_1, s_2, s_3\)), and their respective coefficients are inputted into cells. Solver then optimizes the profit objective while adjusting variables within their constraints.

Sensitivity Analysis and Memo

After obtaining the optimal solution, Solver provides shadow prices for constraints, indicating how much the objective value would improve with an additional unit of available capacity. Reduced costs indicate the cost of increasing the decision variables from zero, reflecting the economic value or opportunity cost of adjusting production levels.

The sensitivity report informs management about the robustness of the solution: if shadow prices are high, small increases in capacity can significantly boost profit; if reduced costs are high, producing certain products may not be economically justifiable unless their costs decrease.

Without Auxiliary Variables: Relationship with Shadow Prices and Reduced Costs

Solving the LP model without auxiliary variables involves directly analyzing primal variables and their associated coefficients. Shadow prices in the sensitivity report correspond to the dual variables linked to constraints—these represent the marginal worth of additional capacity. Reduced costs relate to the discrepancy between the current solution and potential candidates for improvement; non-zero reduced costs imply that the variable's inclusion or increase doesn't currently improve the objective function unless conditions change.

This relationship underscores the economic interpretation in LP: shadow prices measure the value of relaxing a constraint, while reduced costs indicate how much the objective function would need to improve before producing one additional unit of a product becomes profitable.

Conclusion

Optimizing the production of portable air compressors at Klein Industries through careful LP modeling enables maximization of profit within operational constraints. The auxiliary variable method supports sensitivity analysis, providing managerial insights into capacity utilization and product profitability. Understanding the relationship between reduced costs and shadow prices further enhances decision-making, guiding resource allocation and production planning strategies to ensure sustained profitability.

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