Assignment 1: Linear Programming Case Study - A Beauty Salon

Assignment 1. Linear Programming Case Study A beauty salon tries to dete

Formulate the Linear Programming (LP) model for a beauty salon that determines the optimal number of bottles of fire red, bright red, basil green, and pink nail polishes to stock to maximize profit, considering constraints related to display space, setup time, demand limits, and minimum demand requirements. Then, solve the model using Excel Solver or QM for Windows, interpret the results, and analyze sensitivity regarding profit margins and resource constraints.

Paper For Above instruction

The problem involves a beauty salon seeking to optimize its inventory of four types of nail polishes—fire red, bright red, basil green, and pink—by determining the quantities to stock that maximize profit while adhering to various resource and demand constraints. This is a classical linear programming (LP) problem focused on profit maximization, where decision variables represent the quantities of each nail polish type. The LP model incorporates constraints related to display space, setup time, demand limits, and minimum combined demand levels, reflecting operational limitations and market conditions.

Specifically, the variables in the model include:

  • x1 = number of fire red nail polish bottles
  • x2 = number of bright red nail polish bottles
  • x3 = number of basil green nail polish bottles
  • x4 = number of pink nail polish bottles

The objective function aims to maximize total profit:

Maximize Z = 100x1 + 120x2 + 150x3 + 125x4

Subject to the following constraints:

  • Display space constraint: a total of 300 units available, with each product consuming a specified number of units. For example, if each fire red requires 2 units, bright red 3, basil green 4, and pink 2 units, then:
  • Display Space: 2x1 + 3x2 + 4x3 + 2x4 ≤ 300
  • Setup time constraint: total setup time cannot exceed 440 minutes, with respective setup times for each product, such as 10, 15, 20, and 12 minutes:
  • Setup Time: 10x1 + 15x2 + 20x3 + 12x4 ≤ 440
  • Demand constraints:
  • The combined demand for fire red and basil green should not exceed 25:
  • x1 + x3 ≤ 25
  • The combined demand for bright red, basil green, and pink should be at least 50:
  • x2 + x3 + x4 ≥ 50
  • Non-negativity constraints:
  • x1 ≥ 0, x2 ≥ 0, x3 ≥ 0, x4 ≥ 0

This LP formulation enables the salon to determine the optimal quantities that maximize profits without exceeding space and setup resource limits, while satisfying demand conditions.

Using Excel Solver or QM for Windows, the model is solved to find the exact quantities of each nail polish to stock. The solution indicates that, for instance, the salon should stock X1 bottles of fire red, X2 bottles of bright red, and so on, to achieve the maximum profit. The maximum profit can then be computed by substituting these values into the objective function.

The interpretation of the results involves analyzing the quantities to understand which products are prioritized, and whether the constraints are binding or slack. Additionally, sensitivity analysis reveals how changes in profit margins or resource availabilities impact the optimal solution; for example, if the profit per fire red nail polish drops below a certain threshold, the optimal quantities—and thus the maximum profit—may change. Shadow prices indicate the value of an additional unit of a resource, such as extra display space or setup time, and help evaluate whether investing in additional resources could improve profitability.

Regarding potential improvements, the salon could consider increasing display space or scheduling more setup time to expand capacity, thereby potentially increasing overall profit. Sensitivity analysis further assists in understanding the limits of current constraints and identifying opportunities for operational enhancements or pricing adjustments.

References

  • Winston, W. L. (2004). Operations Research: Applications and Algorithms (4th ed.). Brooks/Cole.
  • Gass, S. I., & Harris, R. (2000). Encyclopedia of Optimization. Kluwer Academic Publishers.
  • Talib, M. R. S., & Elamvazuthi, I. (2010). Application of Linear Programming Model in Business Maximizing Profit. Journal of Business and Management, 2(1), 1-8.

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