Finance:Fixed time period model

The fixed time period model is a deterministic lot size model in inventory theory, a variation of the classic economic order quantity (EOQ) model. In this model a product order is placed with a fixed interval of time.

Overview

Consider an inventory system in which an order is placed every T units of time, all demands are met from inventory, the demand rate λ is know with certainty and does not vary.

We want to determine the optimal quantity to order in each period.

Assumptions

1. Products can be analyzed individually
2. Unfilled demand is back-ordered (no lost sales)
3. Replenishments are ordered one at a time

Variables

• [math]\displaystyle{ T }[/math] = time period
• [math]\displaystyle{ T^* }[/math] = optimal time period
• [math]\displaystyle{ \lambda }[/math] = demand rate
• [math]\displaystyle{ Q }[/math] = order quantity
• [math]\displaystyle{ Q^* }[/math] = optimal order quantity
• [math]\displaystyle{ A }[/math] = annual demand quantity
• [math]\displaystyle{ C }[/math] = unit cost
• [math]\displaystyle{ I }[/math] = carrying charge

The total cost function and the derivation of fixed time period formula

The average annual variable cost is given by:

[math]\displaystyle{ K = {\frac{A}{T}} + IC {\frac{\lambda T}{2}} }[/math]

We can minimize K by setting the first derivative equal to zero and finding the optimal value of T:

[math]\displaystyle{ T^* = \sqrt{{\frac{2A}{IC\lambda}}} }[/math]

Which multiplied by λ gives the optimal order quantity:

Fixed Time Period formula

[math]\displaystyle{ Q^*=\lambda T^*= \sqrt{{\frac{2\lambda A}{IC}}} }[/math]

See also

• Reorder point
• Safety stock
• Infinite fill rate for the part being produced: Economic order quantity
• Constant fill rate for the part being produced: Economic production quantity
• Demand is random: classical Newsvendor model
• Demand varies deterministically over time: Dynamic lot size model
• Several products produced on the same machine: Economic lot scheduling problem