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91Ó°ÊÓ

A toy manufacturer estimates the demand for a doll to be 10,000 per year. Each doll costs \(\$ 5\) to manufacture, plus setup costs of \(\$ 800\) for each production run. If it costs \(\$ 4\) to store a doll for a year, how many should be manufactured at a time and how many production runs should there be to minimize costs?

Short Answer

Expert verified
Manufacture 2,000 dolls per run with 5 production runs per year to minimize costs.

Step by step solution

01

Understanding the Problem

To minimize costs, we need to determine the optimal batch size for manufacturing the dolls and the number of production runs per year. We have the annual demand, setup costs per run, manufacturing cost per doll, and storage cost per doll per year. The goal is to find the batch size that minimizes the total cost, which includes setup costs and storage costs.
02

Introducing Relevant Formula

We use the Economic Order Quantity (EOQ) model, which helps minimize the total cost. The EOQ is given by the formula: \[ Q = \sqrt{\frac{2DS}{H}} \] where:- \( D = 10,000 \) is the annual demand,- \( S = 800 \) is the setup cost per production run,- \( H = 4 \) is the holding cost per unit per year.
03

Calculating EOQ

Plug the values into the EOQ formula: \[ Q = \sqrt{\frac{2 \times 10,000 \times 800}{4}} \] Simplify the expression: \[ Q = \sqrt{\frac{16,000,000}{4}} = \sqrt{4,000,000} = 2,000 \] So, 2,000 dolls should be manufactured at a time to minimize costs.
04

Determining Number of Production Runs

Once we have the EOQ, the number of production runs per year is calculated by dividing the annual demand by the EOQ: \[ \text{Number of Runs} = \frac{D}{Q} = \frac{10,000}{2,000} = 5 \] Therefore, there should be 5 production runs per year.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Inventory Management
Inventory management plays a crucial role in ensuring that businesses have the right amount of stock available to meet demand while minimizing costs. In our toy manufacturer's example, the main goal is to determine how many dolls should be manufactured and stored efficiently, adhering to the demand of 10,000 dolls annually. Efficient inventory management helps in limiting overproduction or underproduction.

The Economic Order Quantity (EOQ) is particularly useful in inventory management as it determines the optimal order quantity by balancing different costs. When we calculate the EOQ, we aim to:
  • Minimize storage costs - the expense of keeping inventory on hand.
  • Optimize the production schedule - reducing setup costs by determining a sensible batch size.
  • Meet customer demand without delays.
Implementing effective inventory management techniques, like using the EOQ model, is essential for achieving financial efficiency and fulfilling the desired market demand.
Cost Minimization
Cost minimization focuses on reducing the overall expenses associated with manufacturing and storing products. In the given problem, minimizing costs involves two primary focuses: setup costs and holding costs.

By utilizing the EOQ formula, the toy manufacturer can find the perfect balance between these costs. This formula assumes demand is steady, and it helps calculate that making 2,000 dolls per batch minimizes total costs. Here's how:
  • Setup costs decrease as larger quantities result in fewer production runs.
  • Storage costs increase with larger batch sizes since more items are held for a longer period, but these costs are balanced by fewer setups.
  • The EOQ helps find the point where these conflicting costs are optimized for minimum expenditure.
Reducing unnecessary expenses through such strategic cost management offers a competitive advantage, allowing companies to invest savings back into the business, improve pricing strategies, or increase profits.
Production Management
For efficient production management, balancing the manufacturing and logistical needs with cost efficiency is crucial. From the example, effective production management is achieved by determining the EOQ and then organizing how often dolls should be produced.

Producing 2,000 dolls per run and conducting 5 runs a year aligns with the computed EOQ. This setup ensures that:
  • The production schedule is consistent with demand, enabling streamlined operations.
  • 91Ó°ÊÓ are allocated efficiently, reducing downtime and maximizing equipment use.
  • Production runs are optimized, balancing labor, material planning, and facility utilization.
Production management is an ongoing process of planning and control. By using tools like EOQ, manufacturers can maintain production stability and flexibility, ensuring they meet customer demand in a cost-effective manner. Proper production planning facilitates business growth and enhances overall operational efficiency.

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