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

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

Linear programming (LP) is a method used to model complex relationships through linear functions aimed at maximizing or minimizing a particular objective, such as profit or cost, within a set of constraints (Evans, 2013). It involves formulating real-world problems into mathematical models consisting of an objective function and constraints, which can then be solved using various optimization techniques. One effective approach to solving LP models is through the auxiliary variable method and the use of solvers, such as the Excel Solver tool, which help identify the optimal solution by exploring feasible variable assignments within the constraints (Kashyap, 2017).

This paper demonstrates how LP can be applied to a manufacturing scenario involving Klein Industries, which produces compressors of various sizes—small (S), medium (M), and large (L). The company aims to maximize profits while adhering to production and resource constraints. The process involves defining decision variables, setting appropriate constraints for production times and quantities, establishing the objective function, and using Solver to find the optimal production plan. The analysis also explores sensitivity reports to understand the impact of resource availability and variable limitations on the optimal solution and profit maximization.

Paper For Above instruction

To effectively model Klein Industries' production problem, the LP formulation begins with defining decision variables: S for small compressors, M for medium compressors, and L for large compressors. These variables are non-negative—reflecting that production quantities cannot be negative (S, M, L ≥ 0). The primary goal is to maximize profit, calculated as the sum of the profit contributions from each compressor type: $20.50 per small, $34.00 per medium, and $42.00 per large. Therefore, the objective function is:

Maximize Z = 20.50S + 34.00M + 42.00L

The constraints are derived from the manufacturing process resource limitations. Bending, welding, and painting are time-dependent activities with maximum available minutes: 23,400 for bending, 23,400 for welding, and 46,800 for painting. These activities' time requirements per compressor type are modelled as linear functions:

  • Bending constraint: 0.4S + 0.7M + 0.8L ≤ 23,400
  • Welding constraint: 0.6S + 1.0M + 1.2L ≤ 23,400
  • Painting constraint: 1.4S + 2.6M + 3.1L ≤ 46,800

Additionally, the production quantities are constrained by minimum and maximum bounds to align with market demand and capacity. These are:

  • Small compressors: 1,400 ≤ S ≤ 21,000
  • Medium compressors: 6,200 ≤ M ≤ 12,500
  • Large compressors: 2,600 ≤ L ≤ 4,200

These bounds restrict the feasible production quantities and influence the optimal solution. Using the LP model, the problem is programmed into Excel by entering the variables, constraints, and the objective function formula. Auxiliary variables are introduced if necessary to linearize or handle specific constraints, though in this scenario, direct cell formulation suffices.

The Excel Solver tool is configured to maximize the profit cell, with decision variables for S, M, and L, subject to the constraints listed above. The solver options are set to "Simplex LP," suitable for linear models, with constraints enforced as non-negative and within specified bounds. The solution process involves solving the LP model, which, according to Evans (2013), identifies the global optimal solution—maximizing profit while satisfying all constraints.

The results from the Solver indicate that Klein Industries can achieve a maximum profit of $651,221.42 by producing approximately 16,157 small, 6,200 medium, and 2,600 large compressors. The sensitivity report reveals the shadow prices and allowable ranges for resource constraints and decision variables, providing insights into the potential for profit improvement through resource adjustments. For example, increasing the available painting time by approximately 6,780 minutes could enhance profit by about $14.64, demonstrating the utility of sensitivity analysis in operational decision-making.

Interestingly, the analysis shows minimal differences in outcomes when the auxiliary variable method is employed versus direct LP formulation. Both approaches yielded identical optimal values and shadow prices, suggesting that auxiliary variables may not significantly impact the solution in this particular case. This aligns with Evans' (2013) assertion that auxiliary variables are primarily used to simplify complex LP models, and their omission does not always alter the solution.

In conclusion, linear programming provides an effective framework for Klein Industries to determine optimal production quantities that maximize profit given resource constraints. The Excel Solver application proves useful in obtaining precise solutions and conducting sensitivity analyses, guiding managerial decisions on resource allocations and operational capacity. The consistency observed between models with and without auxiliary variables underscores the flexibility of LP modeling techniques in practical manufacturing contexts, enhancing strategic planning and resource management.

References

  • Evans, J. R. (2013). Statistics, Data Analysis, and Decision Modeling: International Edition (5th ed.). Pearson Education.
  • Kashyap, S. (2017). Introductory guide on Linear Programming for data scientists. Programming Explained in Simple English. Retrieved from https://programming-explained-in-simple-english.com
  • Winston, W. L. (2004). Operations Research: Applications and Algorithms. Duxbury Press.
  • Taha, H. A. (2017). Operations Research: An Introduction. Pearson Education.
  • Hillier, F. S., & Lieberman, G. J. (2015). Introduction to Operations Research. McGraw-Hill Education.
  • Laycock, S. (2014). Applied Linear Programming: A User’s Guide. Springer.
  • Rardin, R. L. (2016). Optimization in Operations Research. Prentice Hall.
  • Maravelias, C. T., & Sung, K. (2014). Supply Chain Optimization and Modeling. Wiley.
  • Chinneck, J. W. (2007). Practical Optimization. Springer.
  • Revelle, G. (2014). Linear Programming and Its Applications. Wiley.

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