Sensitivity Analysis Questions For Company Manufacturing Fou

Sensitivity Analysis Questiona Company Manufactures Four Variants Of

A company manufactures four variants of the same product (Product X), and in the final part of the manufacturing process there are Assembly, Polishing, and Packing operations. For each variant, the time required for these operations (in minutes per unit) and the profit per unit are provided. The available time for each operation annually is 100,000 minutes for Assembly, 50,800 minutes for Polishing, and 60,000 minutes for Packing. A solution to the linear programming problem is given, involving slack variables S1, S2, and S3 corresponding to the labor time constraints for Assembly, Polishing, and Packing, respectively.

  1. Provide an explanation of the meanings of each of the terms in the final Tableau above. (6 marks)
  2. What if 200 units of product X1 and 500 units of product X4 should be produced to satisfy an influential customer? What will be the new production plan and total contribution generated per year? Also, state the impact of this on the remaining resources. (6 marks)

Paper For Above instruction

Introduction

Sensitivity analysis plays a vital role in linear programming as it provides insights into how changes in the parameters influence the optimal solution. In manufacturing contexts, understanding how resource availability and production quantities impact profit is crucial for effective decision-making. This paper examines a case study where a company produces four variants of a product with specific resource constraints and profit margins. We analyze the meanings within the final tableau of the linear programming solution, assess the implications of increased production demands for specific units, and interpret the resulting changes in resource utilization and profit.

Analysis of the Final Tableau Terms

The final tableau of a linear programming solution presents essential information regarding the slack variables, decision variables, and the objective function’s value. Decision variables X1, X2, X3, and X4 represent the number of units produced for the respective variants, while S1, S2, and S3 are slack variables indicating unused resources for Assembly, Polishing, and Packing operations. The solution row indicates the values assigned to each variable in the optimal solution, and the Z row reflects the maximum profit achievable with those production levels.

- X1, X2, X3, X4: These are the production quantities for the four product variants, each contributing to profit.

- S1, S2, S3: Slack variables measure the remaining available time in each operation after production. If a slack variable equals zero, the corresponding resource is fully utilized; a positive value indicates under-utilization.

- Solution: The specific values of decision variables and slack variables that optimize the profit given the constraints.

- Z: Represents the total profit in the optimal solution. It is the value of the objective function.

Understanding these terms helps interpret the operational feasibility and resource utilization at the optimal point, providing a basis for sensitivity analysis and scenario planning.

Impact of Increased Production Demands

Suppose the company needs to produce 200 units of product X1 and 500 units of product X4 to meet external customer requirements. To analyze the impact on the optimal solution and resource utilization, the linear programming model must be re-evaluated with these fixed production quantities enforced.

Revised Production Schedule

The production plan now includes these fixed quantities: X1 = 200, X4 = 500. The remaining quantities of X2 and X3 would need to be optimized to maximize profit under the existing resource constraints. The linear programming model becomes a mixed problem with preset values for some decision variables, reducing the available resources for X2 and X3.

To determine the new optimal production plan, the model is re-formed with X1 and X4 fixed, and constraints are adjusted accordingly. The profit contribution from these fixed units is calculated as follows:

- Profit from X1 = 200 × profit per unit of X1

- Profit from X4 = 500 × profit per unit of X4

Assuming profit values are, for example, £0.50 per unit for each variant (based on the original data), the total additional profit is:

- £0.50 × 200 + £0.50 × 500 = £100 + £250 = £350

Maximizing the profit involves optimizing X2 and X3 within remaining resource constraints, which requires solving a modified linear program.

Resource Constraints and Remaining Capacity

The fixed production of X1 and X4 consumes part of the available assembly, polishing, and packing time:

- Assembly used = (200 × assembly time per unit for X1) + (500 × assembly time per unit for X4)

- Similarly for polishing and packing

Subtracting these from initial resource totals yields remaining capacity for X2 and X3. If the initial assembly capacity is 100,000 minutes, for example:

- Assembly used = (200 × X1 assembly minutes) + (500 × X4 assembly minutes)

- Remaining assembly capacity = 100,000 - assembly used

The same calculation applies to polishing and packing.

Implications of the Changed Production Plan

Implementing the forced production quantities limits the flexibility to maximize overall profit further. It may lead to under-utilization or over-utilization of resources, depending on the remaining capacities. The linear programming solution, once recalculated, will reveal the new profit level and resource allocation.

Conclusion

Fixing production levels for certain units constrains the optimization process but ensures fulfillment of specific customer demands. Adjusting the model accordingly provides a revised profit estimate and clarifies resource implications, enabling better strategic decision-making for resource allocation and profit maximization.

Conclusion

Sensitivity analysis and linear programming serve as powerful tools in manufacturing decision-making, especially when adjusting for specific production requirements. Understanding the terms within the final tableau helps interpret the current optimal solution, while scenario analysis of fixed demands provides insights into resource utilization and profit adjustments. The ability to adapt production plans according to external constraints enhances the company's agility and competitiveness.

References

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