Gilbert Moss And Angela Pasaic Spent Several Summers During

Gilbert Moss And Angela Pasaic Spent Several Summers During Their Coll

Gilbert Moss and Angela Pasaic spent several summers during their college years working at archaeological sites in the Southwest. They learned to make ceramic tiles from local artisans, which they later used as a foundation for their tile manufacturing firm, Mosaics Tiles, Ltd. Their operation is based in New Mexico to leverage local clay resources essential for their production. The manufacturing process involves several steps: molding the tiles, baking them, and then glazing with color or patterns. They plan to produce two types of tiles: a larger, single-colored tile and a smaller, patterned tile. The process details include batch sizes of 100 tiles, specific processing times, resource constraints, and profit margins. The goal is to optimize production to maximize profit while respecting resource limitations. This problem involves formulating a linear programming model, solving it, analyzing the impact of process modifications, and assessing the benefit of additional resources.

Paper For Above instruction

To formulate a linear programming model for Mosaics Tiles, Ltd., we first identify decision variables, objective function, and constraints based on the problem description.

Decision Variables

• Let \( x_1 \) = number of batches of larger tiles produced per week
• Let \( x_2 \) = number of batches of smaller patterned tiles produced per week

Objective Function

Maximize profit, which is $190 per batch of larger tiles and $240 per batch of smaller tiles:

\[ \text{Maximize } Z = 190x_1 + 240x_2 \]

Constraints

Resource limitations are key. The constraints are based on molding time, baking time, glazing time, and clay availability.

1. Molding Time Constraints

Each batch of larger tiles requires 18 minutes (0.3 hours), and each batch of smaller tiles requires 15 minutes (0.25 hours). The total available molding time per week is 60 hours:

\[ 0.3x_1 + 0.25x_2 \leq 60 \]

2. Baking Time Constraints

Per batch, larger tiles take 0.27 hours, smaller tiles 0.58 hours, with 105 hours available weekly:

\[ 0.27x_1 + 0.58x_2 \leq 105 \]

3. Glazing Time Constraints

For glazing, larger tiles take 0.16 hours, smaller tiles 0.20 hours; total time is 40 hours weekly:

\[ 0.16x_1 + 0.20x_2 \leq 40 \]

4. Clay Derivative Constraints

Each batch of larger tiles requires 32.8 pounds; smaller requires 20 pounds; total available is 6,000 pounds:

\[ 32.8x_1 + 20x_2 \leq 6000 \]

5. Non-negativity Constraints

\[ x_1 \geq 0,\quad x_2 \geq 0 \]

Solution via Computer Optimization

Using linear programming software (such as Excel Solver or LINDO), the optimal production quantities are determined based on the above model. The solution gives the number of batches of each tile type to maximize profit considering resource constraints.

Sensitivity Analysis

Sensitivity analysis reveals how the optimal solution responds to changes in parameters such as resource availability, process times, and profit margins. For instance, if the profit for smaller tiles increases from $240 to a higher amount, it might change the optimal batch quantities. Similarly, resource constraints such as available hours or clay could shift the production plan. The computer solution indicates the ranges within which the current optimal solution remains optimal, helping assess robustness and potential for adjustments.

Impact of Reducing Molding Times

Mosaics consider reducing molding times to 16 minutes for larger tiles (from 18 minutes) and 12 minutes for smaller tiles (from 15 minutes). This change effectively relaxes the molding constraints, potentially allowing increased production of both tile types if molding was previously a bottleneck. Re-solving the LP with these new times would likely show increased optimal batch quantities and possibly higher total profit, demonstrating how process improvements translate into financial gains and production flexibility.

Additional Clay Supply

Another 100 pounds of clay per week can be obtained. Assessing whether to accept this involves analyzing the shadow price of the clay constraint. If the marginal value (shadow price) of clay is greater than zero, accepting the extra clay will increase profits. Given the current constraints and optimal solution, if the shadow price of the clay constraint is significant, this additional resource could enable higher production capacity and increased profits. Calculations based on LP analysis suggest that accepting the extra 100 pounds would be economically advantageous if the incremental profit exceeds the value of the additional clay used, particularly when the clay constraint is tight.

Conclusion

Implementing the linear programming model allows Mosaics Tiles, Ltd. to optimize production, maximize profit, and make informed decisions regarding process improvements and resource allocations. The model's sensitivity analysis and scenario evaluations equip management with insights essential for strategic planning and operational efficiency, reinforcing competitive advantage in their niche market.

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