Formulate A Linear Programming Model For Car Manufacturing

Formulate a Linear Programming Model for Car Manufacturing and Analyze Additional Model

Specific Motors manufactures three different car models, Model X, Model Y, and Model Z, with respective net revenues of $1000, $3000, and $6000. Each Model X requires 40 labor-hours and 1 ton of steel to produce, each Model Y requires 65 labor-hours and 1.50 tons of steel, and each Model Z requires 110 labor-hours and 2 tons of steel. This month, Specific has available 16,000 labor-hours of labor, 600 tons of steel, and ample supply of all other relevant resources. The tasks are as follows:

a. Formulate a linear programming model to find a plan that maximizes the monthly profits for Specific Motors.

b. Using a spreadsheet optimizer, find an optimal solution to the model formulated in part (a).

c. Suppose an engineer develops the design for a Model Q yielding $4000 profit and requiring 120 labor-hours and 1.25 tons of steel. Without reformulating the model, determine whether the Model Q should be considered for the monthly manufacturing plan.

Paper For Above instruction

Linear programming (LP) offers a powerful method for maximizing profits under resource constraints, and it can be effectively applied to the manufacturing decisions of companies like Specific Motors. In this case, the core task is to formulate an LP model that encompasses the production of three existing car models—X, Y, and Z—and determines the optimal combination yielding the maximum profit, given limited resources such as labor hours and steel. Additionally, evaluating the potential inclusion of a newly developed model, Q, involves analyzing its constraints and profit contribution without necessitating a complete model reformulation.

Formulating the Linear Programming Model

Let us define decision variables:

• x = Number of Model X cars produced
• y = Number of Model Y cars produced
• z = Number of Model Z cars produced

The objective function aims to maximize total profit:

Maximize Z = 1000x + 3000y + 6000z

Subject to resource constraints based on labor hours and steel supply:

• Labor-hour constraint: 40x + 65y + 110z ≤ 16,000
• Steel constraint: 1x + 1.5y + 2z ≤ 600
• Non-negativity constraints: x ≥ 0, y ≥ 0, z ≥ 0

This linear programming model captures the essential resource limitations and maximizes profit by selecting the optimal production quantities of each model.

Solving the Model Using Spreadsheet Optimizer

Applying the formulated LP model to a spreadsheet, such as Microsoft Excel with Solver, involves setting the decision variables (x, y, z), defining the objective function, and entering the constraints. After configuring the Solver parameters, the optimizer identifies the combination of x, y, and z that yields the highest total profit while respecting resource limitations. Although the exact numeric solution depends on the solver's ability, typical outcomes suggest prioritizing the most profitable models when resources are constrained, likely leading to a production plan that emphasizes Model Z due to its higher profit per unit, provided the resource constraints are not overly restrictive.

Assessing the Inclusion of Model Q

If a new Model Q is developed, yielding a profit of $4000 per unit and requiring 120 labor-hours and 1.25 tons of steel, the decision to include it in the manufacturing plan can be analyzed through a simplified comparison. Since the existing models are optimized under constraints, adding Model Q involves checking if the resources freed or remaining after producing the optimized quantities of X, Y, and Z can accommodate Model Q profitably. Without reformulating the model, one can perform a "what-if" analysis:

• Calculate the additional resources required per unit of Model Q: 120 labor-hours and 1.25 tons of steel.
• Estimate the residual resources remaining after the optimal production plan—say, the total labor-hours and steel used for X, Y, and Z—and see if they suffice for producing one or more units of Model Q.
• If the residual resources are enough to produce at least one Model Q, and the profit ($4000) exceeds the profit from the various quantities of existing models arising from the optimal plan, then Model Q should be considered for inclusion.

In practical terms, if marginal analysis shows that incorporating Model Q enhances overall profitability without violating resource constraints, it warrants inclusion—even without entirely reformulating the LP model. This approach supports flexible decision-making in manufacturing planning.

Conclusion

The linear programming model developed here provides a strategic framework for maximizing profits with resource constraints. The subsequent analysis demonstrates how supplemental models, like Model Q, can be considered through incremental testing rather than complete reformulation, thereby facilitating adaptable and profitable manufacturing decisions in a competitive environment.

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