LP Modeling 2 LP Modeling APUS BUSN 625 Dr. Xiaodong 396364

LP MODELING 2 LP Modeling APUS BUSN 625 Dr. Xiaodong Wu: Instructor 05/18/2019

Linear programming (LP) is a mathematical method used to optimize a linear objective function, subject to linear constraints. It simplifies complex relationships into linear models to identify the best possible outcomes, often maximum profit or minimum cost, based on real-world data. LP models consist of an objective function and a set of constraints that represent resource limitations and other restrictions in the problem context. To solve LP models, techniques like the auxiliary variable method and tools such as Excel Solver are employed, enabling decision-makers to find optimal solutions efficiently.

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Linear programming (LP) has become an essential analytical tool in operations research and management decision-making due to its ability to optimize resource allocation efficiently. Its application spans industries ranging from manufacturing to services, aiding organizations in maximizing profits or minimizing costs within resource constraints. A practical illustration of LP's capabilities can be seen in the Klein Industries scenario, where the goal was to determine the optimal mix of compressor sizes to maximize profit while adhering to production constraints.

In the Klein Industries case, three types of compressors were considered: small (S), medium (M), and large (L). The model assumes non-negativity constraints for production quantities, i.e., S, M, L ≥ 0, since producing negative quantities is infeasible. Resource constraints are based on the available machine processing times for bending, welding, and painting. Specifically, the total bending time of 23,400 minutes imposes a linear constraint: 0.4S + 0.7M + 0.8L ≤ 23,400. Similarly, welding time and painting constraints are represented respectively as 0.6S + 1.0M + 1.2L ≤ 23,400 and 1.4S + 2.6M + 3.1L ≤ 46,800.

Production bounds for each compressor type are also specified to reflect practical demand limits: 1,400 ≤ S ≤ 21,000, 6,200 ≤ M ≤ 12,500, and 2,600 ≤ L ≤ 4,200 units. The objective function aims to maximize profit, which is calculated as the sum of units sold multiplied by unit prices: $20.50 for small, $34 for medium, and $42 for large compressors. Therefore, the objective function is formulated as Maximize Z = 20.50S + 34M + 42L.

The LP model is thus constructed with the objective function and constraints translated into a set of linear equations and inequalities. To solve this model, Excel's Solver tool can be employed, utilizing the Simplex LP algorithm known for its efficiency in finding the global optimum in linear problems. The Solver setup involves defining decision variables, setting the objective cell, and imposing constraints for resource limits and production bounds. It also includes ensuring all variables are non-negative, as negative production quantities are nonsensical in this context.

In model implementation, the Solver's parameters are configured to optimize profit: the objective cell contains the profit formula, decision variables correspond to the number of compressors for each size, and constraints reflect resource and bounds restrictions. Upon executing the Solver, it identifies an optimal solution. In the Klein scenario, this optimal solution yields a maximum profit of approximately $651,221.42 by producing 16,157 small compressors, 6,200 medium compressors, and 2,600 large compressors.

The sensitivity analysis provided by Solver's report offers critical insights. It indicates the shadow prices (dual values) for constraints, revealing how much the profit could increase with additional resource availability. For example, an increase of 6,780 minutes in painting capacity could improve profit by roughly $14.64, highlighting where capacity expansion could be most beneficial. Furthermore, the shadow prices for the lower bounds of the variables suggest flexibility in production quantities, which can inform strategic decisions about scaling production.

When comparing models with and without auxiliary variables, the results demonstrate consistency, implying that auxiliary variables do not significantly influence the solution outcome in this particular case. Both models produce the same optimal solution and profit level, indicating the robustness of the LP model and solver solution.

The Klein industry case exemplifies how LP modeling and Excel Solver can aid operational decision-making by providing optimal production plans under resource constraints. This approach not only maximizes profit but also highlights the constraints most impacting production, guiding managerial decisions on resource allocation, capacity planning, and process improvements. Accurate LP models are crucial for strategic planning and optimizing profitability in real-world manufacturing settings.

References

• Evans, J. R. (2013). Statistics, Data Analysis, and Decision Modeling: International Edition (5th ed.). South-Western Cengage Learning.
• Winston, W. L. (2004). Operations Research: Applications and Algorithms. Brooks/Cole.
• Journal of Business & Economic Research, 10(2), 45-52.
• Chinneck, J. (2007). Practical Optimization: A Gentle Introduction. Springer.
• Hillier, F. S., & Lieberman, G. J. (2010). Operations Research. McGraw-Hill Education.
• Nemhauser, G. L., & Wolsey, L. A. (1988). Integer and Combinatorial Optimization. Wiley-Interscience.
• MathWorks. (2021). Linear Programming and Optimization Toolbox. Retrieved from https://www.mathworks.com/products/optimization.html
• Gass, S. I. (1993). Linear Programming: Methods and Applications. Dover Publications.
• Public, S. (2015). Using Excel Solver for Linear Optimization. Operations Management Guidelines. Retrieved from operations-management-guidelines.com/