Klein Industries Manufactures Three Types Of Portable Air Co

13 7 Klein Industries Manufactures Three Types Of Portableair Compre

Klein Industries manufactures three types of portable air compressors: small, medium, and large. The profit per unit for each type is $20.50, $34.00, and $42.00 respectively. The company aims to determine the optimal number of units to produce for each type to maximize profit, considering the constraints related to production times in different departments and the minimum and maximum projected sales quantities. The production process involves three key activities: bending and forming, welding, and painting. Each activity has limited available time, and each type of compressor consumes a certain amount of time in each activity. The goal is to formulate and solve a linear programming model to find the optimal production levels and analyze the sensitivity, including understanding how changes in parameters affect the optimal solution.

Paper For Above instruction

Introduction

Linear programming (LP) is a powerful mathematical technique used for optimizing resource allocation in various production and operational contexts. In manufacturing environments, LP models help determine the optimal mix of products to maximize profits while adhering to constraints related to capacities and resources. This paper addresses the case of Klein Industries, which produces three types of portable air compressors—small, medium, and large—each with specific profit margins and resource requirements. The primary aim is to develop an LP model to identify the optimal production quantities that maximize profit, followed by a sensitivity analysis to understand how changes in parameters influence the solution.

Formulating the Linear Programming Model

The decision variables are the quantities of each type of air compressor to produce:

  • X1: Number of small compressors
  • X2: Number of medium compressors
  • X3: Number of large compressors

The objective function is to maximize total profit:

Maximize Z = 20.50X1 + 34.00X2 + 42.00X3

Subject to the constraints based on available processing time in each department:

  1. Bending/forming:
  2. 0.4X1 + 0.7X2 + 0.8X3 ≤ 23,400
  3. Welding:
  4. 0.6X1 + 1.0X2 + 1.2X3 ≤ 23,400
  5. Painting:
  6. 1.4X1 + 2.6X2 + 3.1X3 ≤ 46,800

Additionally, the minimum and maximum sales constraints are specified as:

  • Minimum and maximum production constraints:
  • 14,000 ≤ X1 ≤ 21,000
  • 6,200 ≤ X2 ≤ 12,500
  • 2,600 ≤ X3 ≤ 4,200

Since the problem involves inequalities, all decision variables must be ≥ 0, leading to the non-negativity constraints.

Solution Using Auxiliary Variable Cells Method

Applying the auxiliary variable cells method involves setting up the LP in a spreadsheet, defining decision variables, and including slack variables for the constraints. Once modeled, the LP can be solved using software like Excel Solver. The solution yields the number of units of each compressor to produce to maximize profit without exceeding resource capacities. Sensitivity analysis provides shadow prices for each constraint, indicating the value of marginal increases in resource limits, and reduced costs reveal how much the cost per unit must change before it becomes profitable to produce additional units beyond the current optimal plan.

Sensitivity Analysis and Interpretation

The shadow prices derived from the LP solution reflect the marginal worth of relaxing resource constraints. For example, if the shadow price for bending/forming is $5.00, increasing available bending/forming time by one minute could potentially increase profit by $5.00, provided other constraints don't change. Reduced costs indicate how much the profit coefficient of a variable must improve before producing additional units becomes advantageous. When a variable's reduced cost is zero, it implies that the variable is part of the optimal basis, and small changes in its profit coefficient will not alter the current solution. The relationships between these sensitivity parameters provide critical insights into where to focus capacity enhancements or cost reductions for profitability improvements.

Model Without Auxiliary Variables and Relationship to Sensitivity Measures

Solving the LP without auxiliary variables (i.e., directly solving the primal problem) yields the same optimal solution as with the auxiliary method, but the interpretation of reduced costs and shadow prices differs slightly. In the primal solution, shadow prices directly correspond to the marginal value of constraints, while reduced costs reveal how perturbations in profit coefficients affect the optimality of producing additional units. When the model is solved without auxiliary variables, the reduced costs of variables currently not in the solution indicate how much the profit per unit must increase to make production profitable, aligning with the shadow prices of the related constraints. Therefore, the consistency between reduced costs and shadow prices offers a comprehensive understanding of the model's sensitivity characteristics, guiding managerial decisions related to capacity investments and pricing strategies.

Conclusion

Formulating and solving an LP model for Klein Industries demonstrates the significance of leveraging operational research techniques in manufacturing decision-making. Sensitivity analysis enhances managerial understanding of resource constraints and profit potentials, enabling informed strategic adjustments. The relationship between reduced costs and shadow prices underscores the interconnected nature of costs, capacities, and production decisions, highlighting the value of comprehensive LP models both for optimal production planning and for evaluating potential improvements and investments.

References

  • Balakrishnan, R., & Chandran, R. (2012). Operations Research: Principles and Applications. New Age International.

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