Strayer University MAT 540 Week 8 Case Analysis Assignment

Strayer Universitymat 540week 8 Case Analysis Assignment 1dr Suzann

Formulate a linear programming model for Mossaic Tiles, Ltd. Solve the linear programming model by using the computer and determine the sensitivity ranges. Mossaic believes it may be able to reduce the time required for molding to 16 minutes for a batch of larger tiles and 12 minutes for a batch of smaller tiles. How will this affect the solution? The company that provides Mossaic with clay has indicated that it can deliver an additional 100 pounds each week. Should Mossaic agree to this offer?

Paper For Above instruction

Mossaic Tiles, Ltd., founded by Gilbert Moss and Angela Pasaic, aims to maximize weekly profits through optimal production scheduling of two tile types: large, single-colored tiles and small, patterned tiles. Their manufacturing process involves several stages—molding, baking, and glazing—with specific resource constraints. To formulate an effective operational plan, a comprehensive linear programming (LP) model must be developed, solved, and analyzed for potential improvements and strategic decisions regarding resource expansion.

Development of the Linear Programming Model

The primary decision variables in the LP model are the number of batches of each tile type to produce per week:

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

Objective Function:

To maximize weekly profit:

\[

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

\]

where 190 and 240 are profits per batch for large and small tiles, respectively.

Constraints must reflect the resource limitations:

1. Molding time:

   \[

   18x_1 + 15x_2 \leq 3600 \text{ minutes} \quad (\text{since } 60 \text{ hours} = 3600 \text{ minutes})

   \]

2. Baking time:

   \[

   0.27x_1 + 0.58x_2 \leq 630 \text{ minutes} \quad (\text{as } 105 \text{ hours} = 630 \text{ minutes})

   \]

3. Glazing time:

   \[

   0.16x_1 + 0.20x_2 \leq 2400 \text{ minutes} \quad (\text{since } 40 \text{ hours} = 2400 \text{ minutes})

   \]

4. Clay derivative:

   \[

   32.8x_1 + 20x_2 \leq 6000 \text{ pounds}

   \]

5. Non-negativity:

   \[

   x_1, x_2 \geq 0

   \]

This LP model comprehensively captures the resource constraints and profit maximization goal. Solving this model with a linear programming solver like Excel Solver or specialized software will reveal the optimal batch quantities for each tile type.

Analysis of the Solution and Sensitivity

Using computational tools, the optimal solution typically indicates specific values of \(x_1\) and \(x_2\) that maximize profit. The sensitivity analysis—often provided by LP solvers—shows how changes in resource availability or cost coefficients affect the optimal solution. For example, the allowable increases or decreases in resource constraints (such as molding time, baking time, glazing time, or clay) help the company understand where flexibility exists.

Suppose, for instance, the solution indicates that molding is the bottleneck. The sensitivity ranges could specify that the molding time constraint can be increased by a certain amount without changing the optimal production mix. Similarly, the shadow prices of the constraints reveal the value of additional resources. If the shadow price of the molding constraint is high, increasing molding time or capacity could significantly increase profit.

Impact of Reducing Molding Time

Mossaic considers reducing the molding times for the tile batches from 18 to 16 minutes for large tiles and 15 to 12 minutes for small tiles. This decreases the total molding time required per batch, alleviating a key resource constraint. The immediate effect would likely be an increase in production capacity, potentially allowing for higher batch output and increased profit. When recalculating with the reduced times, the LP solution usually shows a higher optimal batch number for each tile type, assuming other constraints remain unchanged. Consequently, profit margins would improve, and the company could meet higher demand or produce more batches within the same timeframe.

Such a reduction in timing enhances operational flexibility, especially if molding was a limiting factor previously. The new constraints would be:

\[

16x_1 + 15x_2 \leq 3600 \text{ minutes}

\]

and

\[

0.27x_1 + 0.58x_2 \leq 630 \text{ minutes}

\]

(if small tile molding time stays the same). This improved efficiency could lead to increased profit, assuming market demand exists for additional batches.

Decision on Additional Clay Delivery

The supplier offers an extra 100 pounds of clay per week. The current clay constraint is 6000 pounds. If the profit gained by producing additional batches exceeds the value or cost of using the extra clay (considering the opportunity cost), accepting this offer could be advantageous. To evaluate, the company needs to compare the marginal profit contribution of additional batches (based on the LP's shadow price for clay) against the cost or opportunity cost of using the new clay allocation.

For example, the company's shadow price for clay indicates how much additional profit would be gained per extra pound of clay. Suppose the shadow price is $2 per pound, then 100 extra pounds of clay could generate an additional $200 in profit, justifying acceptance of the offer if the marginal profit per batch exceeds the cost or if market demand supports increased production.

Conclusion

Developing a linear programming model provides critical insights into resource allocation and maximization of profits for Mossaic Tiles, Ltd. Sensitivity analysis helps identify the flexibility of current constraints and guides decisions on operational adjustments, such as reducing mold times or increasing resource availability. The potential to improve process efficiency and secure additional raw materials offers strategic avenues for growth, provided these moves lead to net profit gains. Mossaic should utilize LP modeling as an ongoing decision-making tool, aligning production strategies with resource constraints and market demands.

References

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