Inventory Control and EOQ Models

Inventory models decide how much to order and when. The classic answer is the economic order quantity (EOQ), which balances ordering cost against holding cost.

Costs

• Ordering (setup) cost S per order.
• Holding (carrying) cost H per unit per year (often a percentage of unit price).
• Shortage cost if demand cannot be met.

Basic EOQ model

Assumptions: constant known demand D per year, instant replenishment, no shortages.

• Annual ordering cost = (D/Q) S. Annual holding cost = (Q/2) H.
• EOQ: Q* = √(2DS/H).
• At Q*, ordering cost = holding cost, and total cost (excluding purchase) = √(2DSH).
• Number of orders = D/Q; time between orders = Q/D.

The total cost curve is flat near Q*: ordering 20% more or less than EOQ raises cost only a little.

Production (EPQ) model

When stock is produced at rate P (> demand rate d), not received at once:

Q* = √(2DS/(H(1 − d/P))), maximum inventory = Q(1 − d/P).

Model with planned shortages (backorders)

With backorder cost B per unit per year: Q* = √(2DS/H) × √((H + B)/B).

Quantity discounts

Compute EOQ at each price; if it is below the price break, raise it to the break quantity. Then compare total cost (purchase + ordering + holding) at each candidate and choose the lowest.

Reorder point and safety stock

• Reorder point = demand during lead time + safety stock = d L + SS.
• With uncertain demand: SS = z σ_L (z from the normal table for the desired service level).

ABC analysis

Classifies items by annual value: A items (about 10–20% of items, 70–80% of value) get tight control; C items (many items, little value) get simple control. VED analysis classifies by criticality (vital, essential, desirable).

Worked example

D = 12,000 units/year, S = ₹100 per order, H = ₹6 per unit per year.

• Q* = √(2 × 12,000 × 100/6) = √400,000 = 632.5 units.
• Total cost = √(2 × 12,000 × 100 × 6) = ₹3794.7.

Practice