# How to minimize the cost function

## Homework Statement

The cost function:

E(Cost)=E(F-LS)3

F is for Finished goods
L is for Lambda
S is for Sales

After expanding the function, what assumption minimized this function with respect to F?

## The Attempt at a Solution

F3-3F2LS+3F(LS)2-(LS)3

I know that I need to identify the terms that include both sales and inventories. The sales terms are the information needed for minimization.

Any ideas?
Thanks!

Sounds like a standard multivariate calculus minimization problem. I'm assuming that the cost is a function of both F and S. To find the min w.r.t F, take the partial derivative w.r.t F, set it equal to zero. Take the partial w.r.t S, set it equal to zero. Solve the simultaneous system to find the critical point(s). Then, classify these critical points as maxima, minima, or saddle points by taking the second derivative, plug in the critical points, note the sign of the second derivative w.r.t F both times. Then, calculate the determinant of the Hessian matrix. Using both the sign of the second derivative and determinant of the Hessian in order to classify each critical point. The details can be found in any calculus textbook.

Thank you!

Ray Vickson
Homework Helper
Dearly Missed

## Homework Statement

The cost function:

E(Cost)=E(F-LS)3

F is for Finished goods
L is for Lambda
S is for Sales

After expanding the function, what assumption minimized this function with respect to F?

## The Attempt at a Solution

F3-3F2LS+3F(LS)2-(LS)3

I know that I need to identify the terms that include both sales and inventories. The sales terms are the information needed for minimization.

Any ideas?
Thanks!

What does 'E' stand for? Typically in such problems, E is the expectation operator, and quantities like S are random (because we don't really control demand). Also, what is Lambda?

My guess is that S (and maybe Lambda) are either given data or given random variables with known probability distributions. That would leave F as the only 'variable' in the problem (and it is non-random), so you would just minimize the function ${\cal F}(F) = F^3 - 3 E(LS) F^2 + 3 E[(LS)^2] F - E[(LS)^3],$ as a univariate function of F. However, I could be wrong, depending on what the symbols in you problem actually mean. To minimize $\cal{F}$ you look where the derivative equals zero, and also worry about the possibility of an end-point optimum if there are lower and upper bounds on F; the derivative need not be zero at end-points.

RGV

OK, the author asks about the minimum information that includes both sales and inventories in order to minimize this function.
I took the second derivative of the function with respect to F (as asked in the problem), which is
6F-6LS

I know that the cost function is convex meaning that 6F-6LS is supposed to be positive.

So
6F-6LS>0 or
6(F-LS)>0 or
F-LS>0 or
F>LS, which means that the amount of finished goods should be greater than the amount of sales multiplied by Lambda.

I guess, this is the final answer.

Ray Vickson
Homework Helper
Dearly Missed
OK, the author asks about the minimum information that includes both sales and inventories in order to minimize this function.
I took the second derivative of the function with respect to F (as asked in the problem), which is
6F-6LS

I know that the cost function is convex meaning that 6F-6LS is supposed to be positive.

So
6F-6LS>0 or
6(F-LS)>0 or
F-LS>0 or
F>LS, which means that the amount of finished goods should be greater than the amount of sales multiplied by Lambda.

I guess, this is the final answer.

"I know that the cost function is convex meaning that 6F-6LS is supposed to be positive."

No: it is supposed to be >= 0, not necessarily positive.

"F>LS, which means that the amount of finished goods should be greater than the amount of sales multiplied by Lambda."

No: you should have F >= LS. It is convex in the region F >= LS and _strictly convex_ in the sub-region F > LS. It is important (sometimes) to be able to distinguish convexity from strict convexity. Certainly, in optimization it is crucial to distinguish between ">" and ">=".

RGV