# EOQ & EPQ Calculators

Steve Myles
January 2016

### Background

• Economic Order Quantity (EOQ)
• a classic model in production scheduling
• results in the optimal order quantity that should be purchased with each order to minimize the total cost
• total cost = the cost of holding excess inventory plus the cost of placing orders
• Economic Production Quantity (EPQ)
• an extension of the EOQ model that assumes that the company will produce its own inventory
• There is a new online EOQ & EPQ Calculator (GitHub repo)
• based on user inputs, this tool calculates the EOQ and EPQ as well as related costs
• Extensions to both models exist that allow for inclusion of shortage (stockout) and back-ordering costs, as well as minimum order quantities (MOQ)
• these extensions are outside the scope of this tool

### Formulas

• EOQ and EPQ are both functions of annualized demand $$R$$, cost per order $$C$$ (not dependent on the quantity ordered), unit cost $$P$$ (price for EOQ, production cost for EPQ), and holding cost $$H$$ (defined as a percentage $$F$$ of unit cost, so $$H = PF$$)
• Additionally, EPQ is a function of production rate (units produced per day) $$p$$ and demand rate (daily demand) $$r$$
• Derivations: EOQ | EPQ

• EOQ: $Q^* = \sqrt{\frac{2CR}{H}} = \sqrt{\frac{2CR}{PF}}$

• EPQ: $Q^* = \sqrt{\frac{2CRp}{H(p - r)}}$

### Economic Order Quantity Example

eoq <- function(demand, order_cost, holding_cost_percent, unit_cost) {
sqrt(2 * order_cost * demand /
## avoid division by 0 by setting 0% holding cost to 0.001
(ifelse(holding_cost_percent == 0, 0.001, holding_cost_percent) * unit_cost)) }
demand <- 1000; order_cost <- 10; holding_cost_percent <- 0.1; unit_cost <- 5
eoqty <- eoq(demand, order_cost, holding_cost_percent, unit_cost)
eoqty

[1] 200

• For the above scenario, the EOQ is 200
• If this reflected the need to order a partial unit, one would need to round up to meet demand

### Economic Production Quantity Example

demand_rate <- function(demand) {demand / 250}
epq <- function(demand, order_cost, holding_cost_percent, unit_cost, production_rate) {
sqrt(2 * order_cost * demand * production_rate /
((ifelse(holding_cost_percent == 0, 0.001, holding_cost_percent)
* unit_cost) * (production_rate - demand_rate(demand)))) }
production_rate <- 5 ## the other parameters are the same as those in the EOQ example
epqty <- epq(demand, order_cost, holding_cost_percent, unit_cost, production_rate)
epqty

[1] 447.2136

• For the above scenario, the EPQ is 447.2
• Each production run should be 448 in this case (assuming partial units cannot be produced)