Industrial learning curve. A company is producing a new...

Industrial learning curve. A company is producing a new product. Due to the nature of the product, the time required to produce each unit decreases as workers become more \intamiliar with the production procedure. It is determined that the function for the learning process is $$T(x)=2+0.3\left(\frac{1}{x}\right)$$ where $T(x)$ is the time, in hours, required to produce the $x$ th unit. Use this information. Find the total time required for a new worker to produce units 1 through $20 ;$ units 20 through 40

Industrial learning curve. A company is producing a new product. Due to the nature of the product, the time required to produce each unit decreases as workers become more \intamiliar
with the production procedure. It is determined that the function for the learning process is
$$T(x)=2+0.3\left(\frac{1}{x}\right)$$
where $T(x)$ is the time, in hours, required to produce the $x$ th unit. Use this information.
Find the total time required for a new worker to produce units 1 through $20 ;$ units 20 through 40

Key Concepts

Industrial Learning Curve

The industrial learning curve refers to the observed phenomenon where the time, cost, or effort required to produce a unit decreases as the total number of units produced increases. This decrease is due to improved worker proficiency, optimized processes, and increased production efficiency as experience is gained over time.

Production Time Function

A production time function models the time required to produce each individual unit, often incorporating factors related to learning and experience. It mathematically represents how production efficiency evolves, allowing for the calculation of time estimates for different production quantities based on the work completed so far.

Cumulative Production Time

Cumulative production time is the total time needed to complete a series of tasks or produce a batch of units, computed by summing the individual time requirements for each unit. This concept is crucial in operational planning and cost estimation, especially when production times change due to increased proficiency or learning effects.

Transcript

00:01 Okay, so we're given this equation which tells us the amount of time in hours to reduce the xth unit.

00:07 And what we want to do is we want to use this equation to find how long it takes a worker to produce units 1 through 20 and then 20 through 40.

00:16 So for 1 through 20, the way that we're going to do this is we're just going to find the integral of our equation from 1 to 20.

00:22 And again, because this tells us the amount of time for the x unit, if we take the integral of that, then we'll find the amount of time for the total.

00:31 Or the total amount of time to produce those units.

00:35 So that's why you're taking the integral.

00:37 So we have the integral of 2 plus 0 .3 times 1 over x, dx.

00:44 And this is equal to just you doing anti -differentiation.

00:49 The anti -derivative of 2 is 2x, and then you're plusing 0 .3 times the natural log of x, and this is from 1 to 20.

01:00 And the reason that the natural log of x is the anti -derivative of 1 over x is because the derivative of the natural log of x is just 1 over x.

01:09 Okay, so now let's plug in our points 1 and 20.

01:13 So for 1, we would have 2 plus 0 .3 times the natural log of 1, and the natural log of 1 is just 0.

01:20 So this would just be 2.

01:22 We're minusing.

01:23 And if we plug in 20, we have 20 times 2, which is actually 40, and then we're adding 0 .3.

01:30 Three times the natural log of 20.

01:34 And so if we put this into a calculator, the natural log of 20 is about 2 .99, and if we times that by 0 .3, and if we add that to 40, and we minus 2, should get an answer that is equal to about 38 .9 hours.

02:00 So the total amount of time that it would take to produce units 1 through 20 would be 38 .9 hours...

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