Problem 51 - Applied Calculus: For Business, Economics, and the Social and Life Sciences

Question

A manufacturing firm receives an order for \(q\) units of a certain commodity. Each of the firm's machines can produce \(n\) units per hour. The setup cost is \(s\) dollars per machine, and the operating cost is \(p\) dollars per hour. a. Derive a formula for the number of machines that should be used to keep total cost as low as possible. b. Prove that when the total cost is minimal, the cost of setting up the machines is equal to the cost of operating the machines.

Step-by-Step Solution

Answer

Use \( m = \sqrt{\frac{pq}{sn}} \) machines. When optimal, setup cost equals operating cost.

Step 1: Understanding the Variables

Identify the given variables in the problem: - The total number of units to be produced: denoted as \( q \)- The number of units a single machine can produce per hour: denoted as \( n \)- The setup cost per machine: denoted as \( s \)- The operating cost per hour per machine: denoted as \( p \)

Step 2: Determining the Production Time

To find out the total time required to produce \( q \) units with one machine, use the formula \[ T = \frac{q}{n} \]where \( T \) is the production time in hours.

Step 3: Calculating the Number of Machines

Let \( m \) represent the number of machines. To minimize cost, the time taken by each machine to produce \( q \) units decreases as more machines are used. Thus, the time becomes \[ t = \frac{q}{mn} \]

Step 4: Formulating the Total Cost

The total cost \( C \) includes both the setup cost and the operating cost. Given by: \[ C = ms + mpt \] Substituting the value of \( t \), we get \[ C = ms + mp \left( \frac{q}{mn} \right) \] Simplifying, we obtain \[ C = ms + \frac{pq}{n} \]

Step 5: Minimizing the Total Cost

To minimize \( C \) in terms of \( m \), take the derivative of the cost function with respect to \( m \), and set it equal to zero:\[ \frac{dC}{dm} = s - \frac{pq}{mn^2} = 0 \]Solving for \( m \) we get: \[ m = \sqrt{\frac{pq}{sn}} \]

Step 6: Proving Equal Setup and Operating Costs

At the optimal number of machines \( m \), the setup cost and the operating cost are equal.The setup cost is:\[ ms = s\sqrt{\frac{pq}{sn}}\]The operating cost is:\[ mpt = p \times \frac{q}{mn} = p \times \frac{q}{\sqrt{\frac{pq}{sn}} \times n} \]Simplifying the operating cost formula, it also simplifies to\[ s\sqrt{\frac{pq}{sn}}\]Thus, the setup cost is equal to the operating cost at optimal \( m \).

Key Concepts

Cost Minimization

In industrial scenarios, firms aim to reduce expenses while maximizing productivity.
An important aspect is minimizing the costs associated with manufacturing.
For a machine-based production system, total cost comprises setup and operating costs.
The goal is to determine the number of machines (denoted as m) that minimizes these total costs.

To start with, let's break down the costs involved:

• Setup cost: This has a direct linear relationship with the number of machines.
• Operating cost: This depends on both the number of machines and the time required to produce the units, given that more machines reduce production time.

Combining these, we get the total cost formula:
\[ C = ms + mp \left( \frac{q}{mn} \right) \] where:

• C is the total cost.
• s is the setup cost per machine.
• p is the hourly operating cost per machine.
• q is the total quantity to produce.
• n is the production quantity of one machine per hour.

Using calculus, specifically the first derivative test, we derive an optimal formula for m, ensuring that total cost is at its minimum.

Derivative of Cost Function

To find the minimum cost, we need to analyze how total cost changes with the number of machines. We do this by taking derivatives.
The cost function is:\[ C = ms + mp \left( \frac{q}{mn} \right) \]First, simplify it to:
\[ C = ms + \frac{pq}{n} \]Next, take the derivative of this function with respect to m (number of machines):

\[ \frac{dC}{dm} = s - \frac{pq}{mn^2} \]To find the value of m where cost is minimized, set this derivative equal to zero:

\[ s - \frac{pq}{mn^2} = 0 \]Solving for m, we get:
\[ m = \sqrt{ \frac{pq}{sn} } \]This tells us the optimal number of machines required to minimize total cost.

Setup and Operating Costs

When manufacturing optimizes the number of machines, an interesting property arises: the setup cost and operating cost balance out.
Let's explore why and how that works.

At the optimal number of machines \left( m = \sqrt{ \frac{pq}{sn} } \right):

• Setup cost becomes:
\[ ms = s \sqrt{ \frac{pq}{sn} } \]
• Operating cost becomes:
\[ mpt = p \left( \frac{q}{mn} \right) = p \left( \frac{q}{ \sqrt{ \frac{pq}{sn} } \times n} \right) \]Simplifying, we observe that:\[ mpt = s \sqrt{ \frac{pq}{sn} } \]Thus, the setup and operating costs are equal when using the optimal number of machines. This balance emphasizes an important point:
• Optimizing operation ensures neither cost is excessively high.
• It stabilizes the total cost structure efficiently.
Understanding this principle helps firms plan better, ensuring resource allocation that maximizes efficacy while minimizing waste.