Firm Manufactures Bicycle Wheels Given The Following Costs

Firm Manufactures Bicycle Wheels Given The Following Costs And Qu

1) A firm manufactures bicycle wheels. Given the following costs and quarterly sales forecasts, use the transportation method to design a production plan that will economically meet demand. What is the cost of the plan? Quarter 1: Sales Forecast 50,000 Quarter 2: Sales Forecast 150,000 Quarter 3: Sales Forecast 200,000 Quarter 4, Sales Forecast 52,000 Inventory carrying cost: $3 per pair of wheels per quarter Production per employee: 1,000 pairs of wheels per quarter Regular workforce: 50 workers Overtime capacity: 50,000 pairs of wheels Subcontracting capacity: 40,000 pairs of wheels Cost of regular production: $50 per pair of wheels Cost of overtime production: $75 per pair of wheels Cost of subcontracting: $85 per pair of wheels

2) A firm makes five different integrated chip products and has five production lines each of which is dedicated to a particular integrated chip product. The production lines differ by sophistication of machines, sites, and the experience of the production personnel. The following estimate of processing times (in hours) is given; assign the integrated chip products so that the order can be completed as soon as possible. Integrated Chips Production Lines A B C D E

Paper For Above instruction

The assignment requires developing an optimal production plan for manufacturing bicycle wheels over the four quarters to meet forecasted demands while minimizing total costs, using the transportation method. Additionally, an optimal assignment of five different integrated chip products to five dedicated production lines should be determined to minimize completion time.

Part One: Bicycle Wheel Production Planning Using Transportation Method

The problem involves designing a production scheduling strategy to satisfy the quarterly demand of bicycle wheels at minimal total cost, considering capacity constraints and production costs. The transportation method is suitable here because the demand (sales forecast) acts as the demand nodes over four quarters, and production capacities with associated costs serve as supply nodes.

Firstly, the demand for each quarter is: Quarter 1: 50,000, Quarter 2: 150,000, Quarter 3: 200,000, Quarter 4: 52,000. The total annual demand sums to 452,000 wheels. Given that production per employee per quarter is 1,000 wheels, with a workforce of 50 workers, regular production capacity is 50,000 wheels per quarter, matching the demand for Q1, but falling short for subsequent quarters, necessitating overtime and subcontracting options.

Production costs are: $50 per wheel for regular, $75 for overtime, and $85 for subcontracting. The inventory cost is $3 per wheel per quarter, which applies to either held inventory from one quarter to the next or excess stock produced. The objective is to assign production quantities across the available capacity, choosing between regular, overtime, or subcontracting, to fulfill demand at minimal cost, while considering inventory holding costs.

The transportation model can be constructed with supply nodes representing the factory's capacity segments (regular capacity, overtime, subcontracting), each with their associated costs per wheel, and demand nodes representing each quarter's needs. The supply can be set with the maximum feasible production capacities, considering the maximum regular production (50,000 per quarter), overtime capacity (50,000 per quarter), and subcontracting capacity (40,000 per quarter). The costs associated with each route guide the optimal allocation.

Applying linear programming or a specialized transportation algorithm (like the Vogel’s approximation or stepping-stone method), the optimal solution will allocate regular capacity where cheapest, then overtime, and finally subcontracting, to either fulfill demand or carry excess inventory, factoring in inventory costs. The total cost includes production and inventory costs across all quarters.

Part Two: Assigning Integrated Chips to Production Lines to Minimize Completion Time

The problem involves assigning five distinct integrated chip products to five dedicated production lines, each with different processing times (not specified in detail here, but typically provided as a matrix). The goal is to assign tasks so that the overall completion time (makespan) is minimized.

This constitutes a classic assignment problem, solvable through the Hungarian Algorithm, which finds the assignment with the minimum total processing time across all line-product pairs. Each product must be assigned to one line, and each line handles exactly one product, ensuring an optimal distribution that reduces total completion time.

To solve, construct a cost matrix where each cell corresponds to the processing time of a product on a particular line. Applying the Hungarian Algorithm will yield the optimal assignment, balancing the load and minimizing the total processing time. This approach ensures the earliest possible completion of all products, thus improving efficiency and throughput in the manufacturing process.

Conclusion

In conclusion, the transportation method provides an effective framework for planning bicycle wheel production costs over multiple quarters, considering capacities and inventory costs. Simultaneously, the Hungarian Algorithm addresses the assignment problem efficiently, optimizing the allocation of product to production line to minimize total processing time. Both methods demonstrate the importance of operational research techniques in manufacturing optimization, contributing significantly to cost reduction and process efficiency.

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