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Chapter 3: Problem 7

A toy manufacturer estimates the demand for a game to be 2000 per year. Each game costs \(\$ 3\) to manufacture, plus setup costs of \(\$ 500\) for each production run. If a game can be stored for a year for a cost of \(\$ 2,\) how many should be manufactured at a time and how many production runs should there be to minimize costs?

Short Answer

Expert verified
Produce 1000 games per run, with 2 production runs per year.

Step by step solution

01

Understanding the Problem

The goal is to minimize the total cost by determining the optimal number of games to produce at a time, which will in turn determine the number of production runs per year. We have various costs: manufacturing costs, setup costs for each production run, and storage costs.
02

Set up Variables and Formula

Let \( q \) be the number of games produced in each production run. The fixed setup cost for each run is \( \\(500 \) and the production cost is \( \\)3 \) per game. The inventory holding cost is \( \\(2 \) per game per year. We need to calculate the Economic Production Quantity (EPQ) using the formula: \[ q = \sqrt{\frac{2DS}{H}} \] where \( D \) is the demand (2000 games/year), \( S \) is the setup cost (\\)500), and \( H \) is the holding cost per unit (\$2).
03

Calculate Optimal Production Quantity

Substitute the values into the EPQ formula:\[ q = \sqrt{\frac{2 \times 2000 \times 500}{2}} = \sqrt{1000000} = 1000 \] Thus, the optimal number of games to produce in each run is 1000.
04

Determine Number of Production Runs

The number of production runs per year is determined by dividing the total demand by the number of games produced per run: \( \frac{2000}{1000} = 2 \). Thus, there should be 2 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 involves planning how much stock to produce and store. It ensures that a company has enough products to meet demand without overproducing. Proper inventory management enhances efficiency by keeping costs low while satisfying customer needs.
In our toy manufacturer example, it's essential to balance the number of games produced with the storage space and costs available. Efficient inventory management helps identify the optimal Economic Production Quantity (EPQ).
  • EPQ helps determine how much stock to produce in each production batch.
  • By calculating EPQ, companies manage inventory levels effectively to meet annual demands.
Maintaining the correct inventory level avoids unnecessary storage costs and ensures that products are available when needed.
Production Optimization
Production optimization is all about maximizing production efficiency. It's a strategy used by manufacturers to carefully plan and control the production process to minimize waste. In our example, calculating the EPQ is a key part of production optimization.
For the toy manufacturer, the goal is to find the perfect balance between setup costs and production costs while meeting demand.
  • The EPQ formula calculates the ideal production quantity to reduce costs.
  • Finding this quantity helps streamline manufacturing processes.
  • It ensures the production process is economically efficient.
Optimizing production means fewer production runs, leading to cost savings and potentially higher profits.
Cost Minimization
Cost minimization involves lowering the total expenses associated with production and inventory management. The aim is to find the most economical way to produce and store items. In the toy manufacturing scenario, several cost factors must be considered, including setup costs, production costs, and holding costs.
  • Setup costs: These are incurred every time production is initiated, like the $500 in the example.
  • Production costs: The expense of manufacturing each unit; here, it's $3 per game.
  • Holding costs: Costs for storing inventory, such as the $2 per game annually.
By calculating EPQ, the manufacturer can determine the optimal production run size that minimizes these costs. Selecting an appropriate batch size decreases total costs, ensuring more resources are available for other strategic purposes.

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Most popular questions from this chapter

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