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EOQ Model
Economic Order Quantity


     Ken Homa
EOQ Assumptions

• Known & constant demand
• Known & constant lead time
• Instantaneous receipt of material
• No quantity discounts
• Only order (setup) cost & holding cost
• No stockouts
Inventory Holding Costs
        Reasonably Typical Profile

                                 % of
   Category                Inventory Value
Housing (building) cost             6%
Material handling costs             3%
Labor cost                          3%
Inventory investment costs         11%
Pilferage, scrap, & obsolescence    3%
Total holding cost                 26%
EOQ Model
Annual Cost




                      Order Quantity
EOQ Model
Annual Cost




                          Holding Cost



                      Order Quantity
Why Order Cost Decreases

  • Cost is spread over more units
  Example: You need 1000 microwave ovens
1 Order (Postage $ 0.35)    1000 Orders (Postage $350)

    Purchase Order                 Purchase Order
                                  Purchase Order
                                 Purchase Order
                                 Description Qty.
                                Purchase OrderQty.
  Description Qty.              Description Qty.
   Microwave 1000              Description
                              Description Qty.11
                                  Microwave
                                 Microwave 1
                                Microwave 1
                               Microwave
                  Order
                 quantity
EOQ Model
Annual Cost




                          Holding Cost
                    Order (Setup) Cost


                      Order Quantity
EOQ Model
Annual Cost

         Total Cost Curve


                              Holding Cost
                        Order (Setup) Cost


                            Order Quantity
EOQ Model
Annual Cost

         Total Cost Curve


                                 Holding Cost
                            Order (Setup) Cost


           Optimal            Order Quantity
      Order Quantity (Q*)
EOQ Formula Derivation
D=      Annual demand (units)
C=      Cost per unit ($)             Total cost =    (Q/2) x I x C + S x (D/Q)
Q=      Order quantity (units)                        inv carry cost    order cost
S=      Cost per order ($)
I =     Holding cost (%)              Take the 1st derivative:
H=      Holding cost ($) = I x C
                                      d(TC)/d(Q) =     (I x C) / 2     - (D x S) / Q²

Number of Orders = D / Q              To optimize: set d(TC)/d(Q) = 0
Ordering costs   = S x (D / Q)
                                                         DS/ Q² = IC / 2
Average inventory
             units = Q / 2                               Q²/DS = 2 / IC
               $ = (Q / 2) x C
                                                         Q²= (DS x 2 )/ IC
Cost to carry
average inventory = (Q / 2) x I x C                      Q = sqrt (2DS / IC)
                  = (Q /2) x H
Economic Order Quantity

         2× D× S
   EOQ =
            H
   D=    Annual demand (units)
   S=    Cost per order ($)
   C=    Cost per unit ($)
   I =   Holding cost (%)
   H=    Holding cost ($) = I x C
EOQ Model Equations
                               2⋅ D ⋅S
Optimal Order Quantity = Q * =
                                  H
                               D
Expected Number Orders = N =
                               Q*
                                     Working Days / Year
Expected Time Between Orders = T =
                                               N
              D
d=                         D = Demand per year
     Working Days / Year   S = Setup (order) cost per order
                           H = Holding (carrying) cost
ROP = d ⋅ L
                           d = Demand per day
                           L = Lead time in days
EOQ
             Example
You’re a buyer for SaveMart.


SaveMart needs 1000 coffee makers per
year. The cost of each coffee maker is
$78. Ordering cost is $100 per order.
Carrying cost is 40% of per unit cost. Lead
time is 5 days. SaveMart is open 365
days/yr.


What is the optimal order quantity & ROP?
SaveMart EOQ

                    2× D× S
              EOQ =
                       H
D=    1000
                         2 ×1000 × $100
S=   $100          EOQ =
C=   $ 78                    $31.20
I=   40%
H=   CxI
H=   $31.20       EOQ = 80 coffeemakers
SaveMart ROP
ROP = demand over lead time
    = daily demand x lead time (days)
    =dxl


D = annual demand = 1000
Days / year = 365
Daily demand = 1000 / 365 = 2.74
Lead time = 5 days


ROP = 2.74 x 5 = 13.7 => 14
SaveMart
       Average (Cycle Stock) Inventory


Avg. CS = OQ / 2
        = 80 / 2 = 40 coffeemakers
        = 40 x $78 = $3,120

Inv. CC = $3,120 x 40% = $1,248


Note: unrelated to reorder point
Economic Order Quantity

         2× D× S
   EOQ =
            H
   D=    Annual demand (units)
   S=    Cost per order ($)
   C=    Cost per unit ($)
   I =   Holding cost (%)
   H=    Holding cost ($) = I x C
2× D× S
      EOQ =
               H
     What if …
2.   Interest rates go up ?
3.   Order processing is automated ?
4.   Warehouse costs drop ?
5.   Competitive product is introduced ?
6.   Product is cost-reduced ?
7.   Lead time gets longer ?
8.   Minimum order quantity imposed ?

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EOQ Ken Homa

  • 1. EOQ Model Economic Order Quantity Ken Homa
  • 2. EOQ Assumptions • Known & constant demand • Known & constant lead time • Instantaneous receipt of material • No quantity discounts • Only order (setup) cost & holding cost • No stockouts
  • 3. Inventory Holding Costs Reasonably Typical Profile % of Category Inventory Value Housing (building) cost 6% Material handling costs 3% Labor cost 3% Inventory investment costs 11% Pilferage, scrap, & obsolescence 3% Total holding cost 26%
  • 4. EOQ Model Annual Cost Order Quantity
  • 5. EOQ Model Annual Cost Holding Cost Order Quantity
  • 6. Why Order Cost Decreases • Cost is spread over more units Example: You need 1000 microwave ovens 1 Order (Postage $ 0.35) 1000 Orders (Postage $350) Purchase Order Purchase Order Purchase Order Purchase Order Description Qty. Purchase OrderQty. Description Qty. Description Qty. Microwave 1000 Description Description Qty.11 Microwave Microwave 1 Microwave 1 Microwave Order quantity
  • 7. EOQ Model Annual Cost Holding Cost Order (Setup) Cost Order Quantity
  • 8. EOQ Model Annual Cost Total Cost Curve Holding Cost Order (Setup) Cost Order Quantity
  • 9. EOQ Model Annual Cost Total Cost Curve Holding Cost Order (Setup) Cost Optimal Order Quantity Order Quantity (Q*)
  • 10. EOQ Formula Derivation D= Annual demand (units) C= Cost per unit ($) Total cost = (Q/2) x I x C + S x (D/Q) Q= Order quantity (units) inv carry cost order cost S= Cost per order ($) I = Holding cost (%) Take the 1st derivative: H= Holding cost ($) = I x C d(TC)/d(Q) = (I x C) / 2 - (D x S) / Q² Number of Orders = D / Q To optimize: set d(TC)/d(Q) = 0 Ordering costs = S x (D / Q) DS/ Q² = IC / 2 Average inventory units = Q / 2 Q²/DS = 2 / IC $ = (Q / 2) x C Q²= (DS x 2 )/ IC Cost to carry average inventory = (Q / 2) x I x C Q = sqrt (2DS / IC) = (Q /2) x H
  • 11. Economic Order Quantity 2× D× S EOQ = H D= Annual demand (units) S= Cost per order ($) C= Cost per unit ($) I = Holding cost (%) H= Holding cost ($) = I x C
  • 12. EOQ Model Equations 2⋅ D ⋅S Optimal Order Quantity = Q * = H D Expected Number Orders = N = Q* Working Days / Year Expected Time Between Orders = T = N D d= D = Demand per year Working Days / Year S = Setup (order) cost per order H = Holding (carrying) cost ROP = d ⋅ L d = Demand per day L = Lead time in days
  • 13. EOQ Example You’re a buyer for SaveMart. SaveMart needs 1000 coffee makers per year. The cost of each coffee maker is $78. Ordering cost is $100 per order. Carrying cost is 40% of per unit cost. Lead time is 5 days. SaveMart is open 365 days/yr. What is the optimal order quantity & ROP?
  • 14. SaveMart EOQ 2× D× S EOQ = H D= 1000 2 ×1000 × $100 S= $100 EOQ = C= $ 78 $31.20 I= 40% H= CxI H= $31.20 EOQ = 80 coffeemakers
  • 15. SaveMart ROP ROP = demand over lead time = daily demand x lead time (days) =dxl D = annual demand = 1000 Days / year = 365 Daily demand = 1000 / 365 = 2.74 Lead time = 5 days ROP = 2.74 x 5 = 13.7 => 14
  • 16. SaveMart Average (Cycle Stock) Inventory Avg. CS = OQ / 2 = 80 / 2 = 40 coffeemakers = 40 x $78 = $3,120 Inv. CC = $3,120 x 40% = $1,248 Note: unrelated to reorder point
  • 17. Economic Order Quantity 2× D× S EOQ = H D= Annual demand (units) S= Cost per order ($) C= Cost per unit ($) I = Holding cost (%) H= Holding cost ($) = I x C
  • 18. 2× D× S EOQ = H What if … 2. Interest rates go up ? 3. Order processing is automated ? 4. Warehouse costs drop ? 5. Competitive product is introduced ? 6. Product is cost-reduced ? 7. Lead time gets longer ? 8. Minimum order quantity imposed ?