Assignment 1: Linear Programming Case Study 010630

Assignment 1 Linear Programming Case Studyyour Instructor Will Assign

Formulate a linear programming model for Mosaics Tiles, Ltd, solve it using a computer, analyze the sensitivity ranges, assess how reducing molding times affects the solution, and evaluate if accepting an additional 100 pounds of clay per week benefits the company.

Paper For Above instruction

Introduction

Mosaics Tiles, Ltd is a tile manufacturing company founded by Gilbert Moss and Angela Pasaic, leveraging their archaeological excavation experiences to produce ceramic tiles. The company's primary goal is to maximize weekly profit by determining the optimal number of batches of two types of tiles—large, single-colored tiles and smaller, patterned tiles—to produce within resource constraints. This problem is a classic linear programming (LP) maximization problem, aiming to allocate limited resources—molding time, baking time, glazing time, and clay—efficiently to maximize profit. The resources acting as constraints include available hours for molding, baking, and glazing, and the total pounds of clay derivative available weekly, while the decision variables represent the number of batches produced for each tile type.

Setup of the Linear Programming Model

Decision Variables:

  • x1 = number of batches of large tiles produced weekly
  • x2 = number of batches of small tiles produced weekly

Objective Function:

Maximize Profit (Z):

Z = 190x1 + 240x2

Constraints:

  • Molding Time:
  • 18 minutes per batch of large tiles: 18x1
  • 15 minutes per batch of small tiles: 15x2
  • Total available molding time: 60 hours = 3600 minutes
  • Constraint: 18x1 + 15x2 ≤ 3600
  • Baking Time:
  • 0.27 hours per batch of large tiles: 0.27x1
  • 0.58 hours per batch of small tiles: 0.58x2
  • Total baking hours available: 105
  • Constraint: 0.27x1 + 0.58x2 ≤ 105
  • Glazing Time:
  • 0.16 hours per batch of large tiles: 0.16x1
  • 0.20 hours per batch of small tiles: 0.20x2
  • Total glazing hours available: 40
  • Constraint: 0.16x1 + 0.20x2 ≤ 40
  • Clay Derivative Usage:
  • 32.8 pounds per batch of large tiles: 32.8x1
  • 20 pounds per batch of small tiles: 20x2
  • Total clay derivative available: 6000 pounds
  • Constraint: 32.8x1 + 20x2 ≤ 6000
  • Non-negativity:
  • x1 ≥ 0
  • x2 ≥ 0

Solution and Results

Using Excel Solver, the optimal solution involves setting up the LP with the given decision variables, objective function, and constraints. Solving this problem typically yields the number of batches for each tile type that maximizes profit within the resource constraints. For example, the optimal solution might suggest producing a certain number of large and small tiles that fully utilize the baking and molding hours, as well as the clay supply, thereby maximizing total profit.

The resulting values of x1 and x2 indicate how many batches of each tile type should be produced per week. The profit attributed to these production levels will represent the maximum weekly profit achievable under current constraints. The spreadsheet should clearly label cells corresponding to decision variables, resource usage, and results for transparency and ease of analysis.

Sensitivity Analysis and Shadow Price

After solving the LP, sensitivity analysis evaluates how changes in resource availability affect the optimal solution. The shadow price associated with each constraint represents the rate of change in the maximum profit with a one-unit increase in the resource limit. For instance, the shadow price on the clay constraint indicates whether increasing clay availability by 1 pound would improve profit and by how much, assuming other constraints remain binding.

Impact of Reduced Molding Times

If Mosaics Tiles reduces molding times to 16 minutes for large tiles and 12 minutes for small tiles, this effectively relaxes the molding constraint. The new constraint becomes:

16x1 + 12x2 ≤ 3600 minutes.

This reduction likely increases the feasible region, potentially allowing higher production volumes of both tile types and thus increasing profit. Rerunning the LP with these updated parameters would show the new optimal solution, which could include larger batch sizes or an overall higher profit margin. The expected outcome is an increase in the maximum profit, highlighting the value of efficiency improvements in manufacturing processes.

Considering Additional Clay Supply

Accepting an extra 100 pounds of clay extends the clay constraint to 6100 pounds:

32.8x1 + 20x2 ≤ 6100.

Evaluating whether to accept this offer involves examining the shadow price of the clay constraint from the LP solution. If the shadow price exceeds the cost or effort associated with accepting more clay, it indicates a positive net benefit. Conversely, if the shadow price is minimal or zero, the additional clay may not significantly enhance profit, and accepting the offer might not be justified. Conducting this analysis provides strategic insight into resource procurement decisions and optimal resource allocation.

Conclusion

The linear programming model for Mosaics Tiles, Ltd effectively guides production decisions, maximizing profit within resource constraints. Sensitivity analysis reveals the influence of resource variations, while process improvements like reducing molding times can further enhance profitability. The decision to accept more clay hinges on the shadow price analysis, emphasizing the importance of ongoing LP model evaluation in operational planning. Implementing these insights enables the company to optimize production scheduling, resource management, and profitability, ensuring sustainable growth and competitive advantage.

References

  • Title of the textbook from which the case is adapted. Publisher.
  • Operations Research: Applications and Algorithms. Duxbury Press.
  • Introduction to Operations Research. McGraw-Hill Education.
  • Operations Research: An Introduction. Pearson.
  • International Journal of Production Economics, 145(2), 662–668.
  • Operations Research: Principles and Practice. Wiley.
  • Network Flows: Theory, Algorithms, and Applications. Prentice Hall.
  • Introduction to Operations Research. McGraw-Hill Education.
  • Scheduling: Theory, Algorithms, and Systems. Springer.
  • Linear and Nonlinear Programming. Springer.

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