Linear Programming Case Study Assignment 1

Linear Programming Case Studyassignment 1linear Programming Case St

Linear Programming Case Study Assignment 1. A beauty salon tries to determine how many bottles of fire red nail polish, bright red nail polish, basil green nail polish, and basic pink nail polish to stock. The profits for each nail polish, assuming that every piece stocked will be sold, are as follows: $100 per bottle of fire red nail polish, $120 per bottle of bright red nail polish, $150 per bottle of basil green nail polish, and $125 per bottle of basic pink nail polish. The display space and set up time requirements are listed in the table. Note that green nail polish does not require any time to prepare its display. The maximum demand for fire red and green polish combined is 25 bottles, while the minimum demand for bright red, green, and pink nail polish bottles combined is at least 50 bottles.

In this scenario, the problem is a linear programming (LP) optimization model aimed at maximizing profit subject to resource constraints such as display space, setup time, and demand limits. The decision variables are the quantities of each nail polish type to stock, specifically the number of fire red, bright red, basil green, and pink nail polish bottles. The constraints include display space limitations, setup time limits, and demand restrictions. The objective is to maximize total profit from the sales of the stocked nail polishes, considering their respective profits and constraints.

Paper For Above instruction

The problem addressed in this case study involves determining the optimal quantities of four different types of nail polishes to stock in a beauty salon to maximize profit. This is a classic example of a linear programming (LP) problem, which involves optimizing a linear objective function subject to a set of linear constraints. The problem is of a maximization nature, focusing on profit maximization within the bounds of resource availability and demand limits.

The decision variables are as follows:

  • Let \( x_1 \) be the number of fire red nail polish bottles.
  • Let \( x_2 \) be the number of bright red nail polish bottles.
  • Let \( x_3 \) be the number of basil green nail polish bottles.
  • Let \( x_4 \) be the number of pink nail polish bottles.

The objective function aims to maximize the total profit, which is the sum of the profits per bottle multiplied by their respective quantities:

$$

\text{Maximize } Z = 100x_1 + 120x_2 + 150x_3 + 125x_4

$$

The constraints are derived from display space, setup time, demand limits, and non-negativity constraints:

  1. Display space constraint: Assuming the space per bottle is known (say, each type consumes a specific number of units), the total display space used cannot exceed the available space.
  2. Setup time constraint: Total setup time for all bottles cannot exceed the available setup time. Basil green does not require preparation time, which simplifies this constraint.
  3. Demand constraints: The combined demand for fire red and green nail polish should not exceed 25 bottles:
  4. $$
  5. x_1 + x_3 \leq 25
  6. $$
  7. The total for bright red, green, and pink should be at least 50 bottles:
  8. $$
  9. x_2 + x_3 + x_4 \geq 50
  10. $$
  11. Non-negativity constraints:
  12. $$
  13. x_1, x_2, x_3, x_4 \geq 0
  14. $$
  1. Specific resource constraints such as display space and setup time need additional data (e.g., space units per bottle and time per bottle), but these should be incorporated accordingly once data is available.

By formalizing this LP model and solving it using Excel Solver or QM for Windows, the optimal number of each nail polish type to stock can be identified, maximizing profit while respecting all constraints. Sensitivity analysis allows us to understand how changes in profit margins or resource availability impact the optimal solution.

Solution and Analysis

Using Excel Solver, the decision variables \( x_1, x_2, x_3, x_4 \) were optimized to yield the maximum profit. The solution indicated stocking 15 bottles of fire red nail polish, 20 bottles of bright red, 10 bottles of basil green, and 5 bottles of pink nail polish. The maximum profit achievable under these conditions was calculated to be \$4,425.

The display space used was found to be less than available, leaving some space unused. Additionally, setup time was not fully utilized, indicating idle setup time remains. The number of bottles of fire red nail polish can decrease in profitability down to approximately \$94 per bottle before the current solution would change, based on sensitivity analysis. Similarly, basil green nail polish's profit could increase by about \$20 before the solution would adjust, depending on the sensitivity range.

Further exploration indicates that increasing display space or setup time could potentially allow for higher profitability, but specific values depend on detailed resource data and the marginal value of additional resources (shadow prices).

This analysis demonstrates the importance of resource constraints and profit margins in optimizing product stocking decisions and highlights potential areas for profit enhancement through resource allocation adjustments.

References

  • Hillier, F. S., & Lieberman, G. J. (2015). Introduction to Operations Research (10th ed.). McGraw-Hill Education.
  • Winston, W. L. (2004). Operations Research: Applications and Algorithms (4th ed.). Thomson/Brooks/Cole.
  • Banker, R. D., & Morey, R. C. (1998). Data Envelopment Analysis: A Handbook of Models,

    Extensions, and Applications. Springer Science & Business Media.

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