Minimization problem: Economics: quantity to order

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1. The problem statement, all variables and given/known data
Capture1.JPG

Capture.JPG

2. Relevant equations
Minimum/Maximum occurs when the first derivative=0

3. The attempt at a solution
$$Q=\sqrt{\frac{2(K+pQ)}{h}}~\rightarrow~Q=\frac{2}{h}(KM+pM)$$
##Q'=0~## gives no sense result
 

Attachments

Ray Vickson

Science Advisor
Homework Helper
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1. The problem statement, all variables and given/known data
View attachment 230767
View attachment 230768
2. Relevant equations
Minimum/Maximum occurs when the first derivative=0

3. The attempt at a solution
$$Q=\sqrt{\frac{2(K+pQ)}{h}}~\rightarrow~Q=\frac{2}{h}(KM+pM)$$
##Q'=0~## gives no sense result
You cannot just write ##Q = \sqrt{2(K+pQ)/h}##. You must start the whole modelling procedure over again.

In the old model, the average cost per week was given as
$$C(Q) = \frac{KM}{Q} + \frac{1}{2} h Q \; \hspace{2cm} (1) $$
Solving ##C'(Q)=0## gave the economic order quantity as ##Q = \sqrt{2KM/h}.##

What is the equation that replaces (1) in the new model?
 
Last edited:
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22
In the old model:
$$A(Q)=\frac{KM}{Q}+cM+\frac{hQ}{2}$$
Where c is the purchase cost of one item. in the new model:
$$A(Q)=\frac{K+pQ}{Q/M}+cM+\frac{hQ}{2}=\frac{KM}{Q}+(c+p)M+\frac{hQ}{2}$$
And differentiating gives the same result
 

Ray Vickson

Science Advisor
Homework Helper
10,639
1,667
In the old model:
$$A(Q)=\frac{KM}{Q}+cM+\frac{hQ}{2}$$
Where c is the purchase cost of one item. in the new model:
$$A(Q)=\frac{K+pQ}{Q/M}+cM+\frac{hQ}{2}=\frac{KM}{Q}+(c+p)M+\frac{hQ}{2}$$
And differentiating gives the same result
Correct.

BTW: why do you call this thread a "min max" problem? It is a "minimization" problem, but there is no "max" involved. In mathematics, operations research, economics and other such fields the term "min-max" has a definite meaning, and it is nothing like how you use it.
 
Last edited:
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22
Thank you Ray, you are correct, this is only a minimum problem.
I just automatically wrote the heading.
Thanks
 

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