## 2010年2月13日 星期六

### Optimum Service Levels

Very odd formulas in the notes. Let me try to have some proof.

Find a service level such that profit is maximized:
"Revenue" at service level S for a year
= P(S)
= (Price - "Cost") * Annual Sales * Sales Response(S)
= Trading Margin * Annual Sales * Sales Response(S)
(Why this is "Revenue" ? ...)

Logistics Cost at service level S for a year
= C(S)
= Annual Holding Cost(S)
(I assume those ordering cost, material cost or whatever cost except holding cost are counted in that "Cost".)

Objective function: Y(S) = P(S) - C(S)
Take the derivative, Y'(S) = P'(S) - C'(S)
Function maximized when Y'(S) = 0 <=> P'(S) = C'(S)

Marginal "Revenue" = Trading Margin * Annual Sales * Sales Response Rate
(Sales Response Rate = Change in Sales Response per Unit change in Service Level. Assume it is a constant.)

Marginal Logistics Cost = Change in Annual Holding Cost per Unit change in Service Level
= Annual Holding Cost * Change in number of Inventory per Unit change in Service Level
= (Percentage Annual Holding Cost * Standard Product Cost) * SD of demand during lead time * Change in Z per Unit change in Service Level

Inventory during lead time = Expected Demand + Safety Stock
Increasing Service Level needs to increase Safety Stock (for the extra demand during lead time)

P(Demand < Inventory) = Service Level
P((Demand - Expected Demand)/SD < Safety Stock/SD) = Service Level
P(Z < Safety Stock/SD) = Service Level
Safety Stock = Z_Service Level * SD

OK. The proof is not fun.
Calculator program will be provided later. (Still thinking.)