What is the optimal number of items to produce for maximum profit in business?

In summary, the conversation discusses a manufacturer making a batch of items and determining the number of items to produce for maximum profit. The cost of each item is given as n^2-6n+35 and the selling price is $50 each. The profit formula is derived as 56n - n^2 - 35 and the derivative is taken to find the maximum point. However, the correct answer is found to be n=5 instead of n=28 due to the incorrect use of the cost formula.
  • #1
Shakattack12
24
2

Homework Statement


A manufacturer makes a batch of n items with the cost (in dollars) of each item being:
n2-6n+35.
The manufacturer sells the items for $50 each. How many items should be produced in each batch to maximise profit.

Homework Equations


Cost = C(n) = n2-6n+35
Revenue = R(n) = 50n

The Attempt at a Solution


Profit = 50n - (n2-6n+35)
= 56n - n2-35
dP/dn = 56 - 2n
let dP/dn = 0
0 = 56 - 2n
n = 28
After testing the nature I found it to be a maximum which is all good but the textbook says the answer is 5. Where did I go wrong?
 
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  • #2
Shakattack12 said:

Homework Statement


A manufacturer makes a batch of n items with the cost (in dollars) of each item being:
n2-6n+35.
The manufacturer sells the items for $50 each. How many items should be produced in each batch to maximise profit.

Homework Equations


Cost = C(n) = n2-6n+35
Revenue = R(n) = 50n

The Attempt at a Solution


Profit = 50n - (n2-6n+35)
= 56n - n2-35
dP/dn = 56 - 2n
let dP/dn = 0
0 = 56 - 2n
n = 28
After testing the nature I found it to be a maximum which is all good but the textbook says the answer is 5. Where did I go wrong?

Your ##p = n^2 -6n + 35## is the cost per item, so the cost of a batch of ##n## items is ##C(n) = n p##.
 
  • #3
Thank you so much! Can't believe I missed that.
 

What is optimisation in business and why is it important?

Optimisation in business is the process of finding the best possible solution to a problem or challenge within a company. It involves identifying and prioritising goals, assessing resources and constraints, and making strategic decisions to achieve the most efficient and effective outcome. Optimisation is important because it can help a business save time, money, and resources, as well as improve overall performance and competitiveness.

What are the main techniques used for optimisation in business?

There are several techniques used for optimisation in business, including linear programming, simulation, queuing theory, and decision analysis. Linear programming involves creating mathematical models to determine the best course of action given a set of constraints and objectives. Simulation involves using computer models to test different scenarios and identify the most optimal solution. Queuing theory is used to study and improve processes involving waiting and queuing, such as customer service. Decision analysis uses decision trees and other tools to analyse and make the best decisions given uncertain or complex situations.

What are the benefits of using optimisation in business?

The benefits of using optimisation in business include improved efficiency, cost savings, better decision-making, increased profitability, and competitive advantage. By finding the most optimal solution, businesses can reduce waste, streamline processes, and make better use of resources. This can lead to increased productivity and profitability, as well as a competitive edge in the market.

What are some common challenges in implementing optimisation in business?

Some common challenges in implementing optimisation in business include data availability and quality, complexity and uncertainty of real-world problems, and resistance to change. Obtaining and analyzing accurate and relevant data is crucial in the optimisation process, but it can sometimes be difficult or time-consuming. Real-world problems are often complex and involve multiple variables, making it challenging to find the best solution. Additionally, some employees or stakeholders may resist changes to existing processes or systems, which can hinder the success of an optimisation strategy.

How can businesses ensure the success of an optimisation strategy?

To ensure the success of an optimisation strategy, businesses should clearly define their goals and objectives, involve key stakeholders in the process, use reliable data and techniques, and continuously monitor and evaluate the results. It is important to have a clear understanding of what the business hopes to achieve through optimisation and to communicate this to all stakeholders. Involving key employees and decision-makers in the process can help ensure buy-in and support for the strategy. Using reliable data and techniques is crucial in making informed and effective decisions. Finally, monitoring and evaluating the results of the optimisation strategy can help identify areas for improvement and make necessary adjustments for continued success.

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