# Inventory

Inventory is the raw material, component parks, or finished goods that are held at some location in the supply chain.

## Types of Inventory

• Raw materials (wood, steel, oil, ect)
• Component parts (bought from somewhere else)
• Finished goods (what you made)

## Inventory Management

• Balance and matching of supply and demand in supply chain.
• It’s important because inventories are a big company asset (big investment)

## Reason to Keep Inventory

• Protect against spikes in demand
• Protects against supply disruptions (can’t get stuff)
• Discount on high quantity order
• Good for uncertain conditions

## Reason to Not Have Inventory

• Inventory can become obsolete
• Things can spoil
• Opportunity costs (money not liquid)
• Holding costs
• insurance, security, spacing, warehouse costs

## Important Decisions

• HOW MUCH should we order?
• WHEN should we order more?

### How much should I order

Simple models start by assuming:

• D: Demand is known and constant
• S: The ordering cost is known and fixed.
• Instant replenishment
• No limit on order quality
• H: Annual holding cost per unit
• Q: The limit to which we build back up
• P: Price per unit

This would mean Q units would be bought. They would be depleted linearly till 0, then a price S would be paid to order more back up to Q level. Hence the average inventory would be Q/2

EOQ - Economic Order Quantity - The Size

$TotalCost=OrderingCost+InventoryCost$

$OrderingCost=NumberOrdersPerYear \cdot CostPerOrder$

In this case, $OrderingCost=\frac{D}{Q}\cdot S$

$InventoryCost=AverageInventory \cdot HoldingCostPerYear$

In this case $HoldingCost=\frac{Q}{2}\cdot H$

To find the minimum and optimal total cost, it is when holding cost equals ordering costs. Could also set the derivative to 0.

$Q^{*} \sqrt{\frac{2SD}{H}}=EOQ$

Expected orders: $N=\frac{D}{Q^*}$

Expected time between orders: $T=\frac{NumberOfDaysPerYear}{N}$

## How Can We Account for a Quantity Discount?

$Expand Equation for Total Cost (ETC) = \frac{D}{Q}\cdot S + \frac{Q}{2}\cdot H + P\cdot D$

This will allow for discount in price/unit on large quantity orders. Discounts encourage to hold more inventory.

## At What Point do you reorder?

In real life, a instant replenishment time is unrealstic and impossible. Hence, there are lead times.

• L: Lead time in days
• d: average daily demand

ROP = Reorder Point.

$ROP = d\cdot L$

But what if demand is not constant? How does ROP change?

Well we can model it with a probability distribution. Will add a buffer of extra inventory and call it our safety pile. This size will depend on demand uncertainty, the penalty of running out, lead time.

Use a normal curve for the probability of having enough stock.

The safety stock can be lead to be represented as

$Safety Stock = Safety Factor \cdot STD(Lead Time Demand)$

The safety factor is just the Z from the % you don’t want to be out of stock.

Instead of standard deviation of the lead time, is set to be

$STD(LeadTime) = Z\cdot \sigma_D \cdot \sqrt{LT}$

Hence we end up with $ROP = d\cdot L + SS$