The best quantity to order One of the formula...

Question

The best quantity to order One of the formulas for inventory management says that the average weekly cost of ordering, paying for, and holding merchandise is

A(q) = (km / q) + cm + (hq / 2),

where q is the quantity you order when things run low (shoes, TVs, brooms, or whatever the item might be); k is the cost of placing an order (the same, no matter how often you order); c is the cost of one item (a constant); m is the number of items sold each week (a constant); and h is the weekly holding cost per item (a constant that takes into account things such as space, utilities, insurance, and security).

Find dA/dq and d²A/dq².

Accepted Answer

Welcome back, everyone. A formula for the cost of maintaining a fleet of vehicles is given by C V equals Km divided by V plus CV + H2 divided by 2, where V is the number of vehicles in the fleet, K is the cost of operating a vehicle, C is the cost of maintaining each vehicle, M is the number of vehicles used per month. And these the cost of storing each vehicle per month. Find DC divided by DV and the second derivative of C with respect to B. So let's begin by finding the first derivative of DC. Divided by DV. What we can do is simply focus on. The expression of CV and what we can do is simply apply the power rule, but we have to rewrite our function as KM multiplied by V to the power of -1, so we're using the reciprocal rule. We're adding CV and then we're adding H divided by 2 multiplied by V2. We have to differentiate this expression and as we can see, We have factored out our constants KM. C and H divided by 2. So here we can begin applying the power rule which gives us KM multiplied by -1. We're multiplying by the exponent. Followed by multiplication by V with an exponent of -1 minus 1, which is -2. Where addingc multiplied by the derivative of B which is 1. And then the derivative of B squared is going to be 2V. So 2V multiplied by H divided by 2 is going to give us the final derivative. What we want to do from here is simplify. This gives us negative KM. Divided by B squad, so we can also write the negative sign. In the middle of the fraction. Where then adding C. Followed by H divided by 2 multiplied by 2 V. So this is simply HV, right? Well, then we have our first derivative negative km divided by V2 + C + HV and we can start finding the second derivative. So the second derivative of C with respect to V. Is going to be the derivative of negative KMB to the power of -2. Once again we're applying the reciprocal rule. Plus the derivative of C plus the derivative of HV. So let's differentiate each term. First of all, factoring out the constant, we have negative KM. Multiplied by the exponent of -2, multiplied by V to the power of -3, so -2 minus 1. The derivative of C is 0 because it's a constant. The derivative of V is one so we're simply adding H. And now simplifying we get 2 km divided by the cubed. Once again applying the reciprocal rule. Plus H. And that's how our second derivative of. That's it. Thank you for watching.

Key Concepts

Derivative

Second Derivative

Optimization

Watch next