Question

Prove that the EOQ cost function can be rewritten as

g(Q) = h/2 * Q(Q - underroor 2K/h) + underroot 2Kh
Use this to prove without using calculus.

Answer 1

the EOQ cost function
`\mathbf{g(Q)= (h)/(2 \lambda Q)( Q- \sqrt{(2 \lambda K )/(2)})^2 + \sqrt{(2 h K)/(\lambda)}}`

Explanation:

Given that:

`g(Q) = (h)/(2 \lambda Q) \begin {pmatrix} Q- \sqrt{(2 K \lambda)/(h)} \end {pmatrix}^2 + \sqrt{(2 Kh)/(\lambda)}`

Total cost = Purchase Cost + Ordering Cost + Holding Cost

i.e

T =
`P \lambda + h (Q)/(2)+(\lambda K)/(Q)`

By differentiating with respect to Q and equating to zero, we have:

`0 = 0 +(h)/(2)- (\lambda K)/(Q^2)`

`Q* = \sqrt{(2 k \lambda)/(h)}`

Now;

`T = P \lambda + (h)/(2Q)(Q-Q^*)^2 + hQ^*`

`T = (h)/(2Q)(Q- \sqrt{(2 \lambda K)/(h)})^2 + √(2 hK \lambda ) + P \lambda`

`((T - P \lambda)/(\lambda )) = (h)/(2 \lambda Q)( Q- \sqrt{(2 \lambda K )/(2)})^2 + \sqrt{(2 h K)/(\lambda)}`

Therefore:

the EOQ cost function
`\mathbf{g(Q)= (h)/(2 \lambda Q)( Q- \sqrt{(2 \lambda K )/(2)})^2 + \sqrt{(2 h K)/(\lambda)}}`