Eoq
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Chapter 3 Economic Order Quantity
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Background to the model The approach is to build a model of an idealized inventory system and calculate the fixed order quantity that minimizes total costs. This optimal order size is called the economic order quantity (EOQ).
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Assumption for a basic model • The demand is known exactly, is continuous and is constant over time. • All costs are known exactly and do not vary. • No shortages are allowed. • Lead time is zero – so a delivery is made as soon as the order is placed.
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Other assumptions implicitly in the model • We can consider a single item in isolation, so we cannot save money by substituting other items or grouping several items into a single order. • Purchase price and reorder costs do not vary with the quantity ordered. • A single delivery is made for each order. • Replenishment is instantaneous, so that all of an order arrives in stock at the same time and can be used immediately.
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• The most important assumption here is that demand is known exactly, is continuous and constant over time (Fig. 3.2) • The assumptions give an idealized pattern for a stock level. (Fig. 3.3)
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Variables used in the analysis • 1. 2. 3. 4.
Four costs of inventory Unit cost (UC) Reorder cost (RC) Holding cost (HC) Shortage cost (SC)
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Three other variables: • Order quantity (Q) • Cycle time (T) • Demand (D)
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Derivation of the economic order quantity • Three steps: 1. Find the total cost of one stock cycle. 2. Divided this total cost by the cycle length to get a cost per unit time. 3. Minimize this cost per unit time.
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• Amount entering stock in cycle = amount leaving stock in cycle So Q=DxT Total cost per cycle = unit cost + reorder cost + holding cost (component) Page 13 of 61
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The optimal time between orders is: To = Qo/D =500/6,000 = 0.083 years = 1 month The associate variable cost is: VCo = HC × Qo = 6 × 500 = $3,000 a year This gives a total cost of: TCo = UC × D + VCo = 30 × 60000 + 3000 = $183,000 a year
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Summary • We have built a model of an idealized inventory system that relates order size to costs and demand. This shows that large, infrequent orders have a high holding cost component, so the total cost is high: small, frequent orders have a high reorder cost component, so the total cost is also high. A compromise finds the optimal order size – or economic order quantity – that minimizes inventory costs. Page 24 of 61
Adjusting the economic order quantity •
• • •
Moving away from the EOQ The EOQ suggests fractional value for things which come in discrete units (e.g. an order for 2.7 lorries) Suppliers are unwilling to split standard package sizes. Deliveries are made by vehicles with fixed capacities. It is simply more convenient to round order sizes to a convenient number.
Page 25 of 61
How much costs would rise if we do not use the EOQ Example: D = 6000 units a year Unit cost, UC = £30 a unit Reorder cost, RC = £125 an order Holding cost, HC = £7 a unit a year
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Order for discrete items This suggests a procedure for checking whether it is better to round up or round down discrete order quantities: 1. Calculate the EOQ, Qo. 2. Find the integers Q’ and Q’-1 that surround Qo. 3. If Q’×(Q’-1) is less than or equal to Qo2, order Q’. 4. If Q’×(Q’-1) is greater than Qo2, order Q’-1. Page 34 of 61
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Uncertainty in demand and costs Error in parameters • Few organizations know exactly what demand they have to meet in the future. • The variable cost is stable around the EOQ and small errors and approximations generally make little difference.
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Adjusting the order quantity
Page 41 of 61
Qo =
2 × RC × D HC
or
Qo × HC RC = 2× D 2
Page 42 of 61
2 × RC × D Q= K× HC or
Q=
2 × RC × D HC × K
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Page 45 of 61
Adding a finite lead time • Time for order preparation • Time to get the order to the right place in suppliers • Time at the supplier • Time to get materials delivered from suppliers • Time to process the delivery
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Reorder level The amount of stock needed to cover the lead time is also constant at: Lead time × demand per unit time Reorder level = lead time demand = lead time × demand per unit time ROL = LT × D
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(b) Substituting value LT = 2 and D = 100, gives ROL = LT × D = 2 × 100 = 200 units As soon as the stock level declines to 200 units, Carl should place an order for 250 units.
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Longer lead times The cycle length is: T = Q/D = 250/100 = 2.5 weeks What happens when the lead time is increased to 3 weeks ? ROL = LT × D = 3 × 100 = 300 Page 52 of 61
Stock on hand + stock on order = LT × D =300 units Reorder level = lead time demand – stock on order n × T < LT < (n+1) × T Reorder level = lead time demand – stock on order ROL = LT × D – n × Qo
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Some practical points • This chapter has shown when to place an order – set by the reorder level. • How much to order – set by the economic order quantity. • A two-bin system gives a simple procedure for controlling stock without computers or continuous monitoring of stock levels. Page 56 of 61
Three-bin system • The two-bin system can be extended to a three-bin system which allows for some uncertainty. • The third bin holds a reserve that is only used in an emergency. • The normal stock is used from bin A.
Page 57 of 61
• Our calculations assume that the lead time is known exactly and constant. • In practice, there can be quite wide variation, allowing for availability, supplier reliability, checks on deliveries, transport conditions, customs clearance, delays in administration, and so on.
Page 58 of 61
• Some stock levels are not recorded continuously but are checked periodically, perhaps at the end of the week. • Then an unexpectedly large demand might reduce stocks well below the reorder level before they are checked.
Page 59 of 61
• Another problem appears with large stocks, such as chemical tanks, coal tips or raw materials, where the stock level is only known approximately. Then the reorder level might be passed without anyone noticing.
Page 60 of 61
• We have assumed that the order size is independent of the lead time and the reorder level. • In practice, people often prefer larger orders if there are longer lead times, and they raise the reorder level to add an element of safety. • The reorder level can be also influence the order size, with people typically placing smaller orders with higher reorder levels.
Page 61 of 61
Page 1 of 61
Background to the model The approach is to build a model of an idealized inventory system and calculate the fixed order quantity that minimizes total costs. This optimal order size is called the economic order quantity (EOQ).
Page 3 of 61
Page 4 of 61
Assumption for a basic model • The demand is known exactly, is continuous and is constant over time. • All costs are known exactly and do not vary. • No shortages are allowed. • Lead time is zero – so a delivery is made as soon as the order is placed.
Page 5 of 61
Other assumptions implicitly in the model • We can consider a single item in isolation, so we cannot save money by substituting other items or grouping several items into a single order. • Purchase price and reorder costs do not vary with the quantity ordered. • A single delivery is made for each order. • Replenishment is instantaneous, so that all of an order arrives in stock at the same time and can be used immediately.
Page 6 of 61
• The most important assumption here is that demand is known exactly, is continuous and constant over time (Fig. 3.2) • The assumptions give an idealized pattern for a stock level. (Fig. 3.3)
Page 7 of 61
Page 8 of 61
Variables used in the analysis • 1. 2. 3. 4.
Four costs of inventory Unit cost (UC) Reorder cost (RC) Holding cost (HC) Shortage cost (SC)
Page 9 of 61
Three other variables: • Order quantity (Q) • Cycle time (T) • Demand (D)
Page 10 of 61
Derivation of the economic order quantity • Three steps: 1. Find the total cost of one stock cycle. 2. Divided this total cost by the cycle length to get a cost per unit time. 3. Minimize this cost per unit time.
Page 11 of 61
Page 12 of 61
• Amount entering stock in cycle = amount leaving stock in cycle So Q=DxT Total cost per cycle = unit cost + reorder cost + holding cost (component) Page 13 of 61
Page 14 of 61
Page 15 of 61
Page 16 of 61
Page 17 of 61
Page 18 of 61
Page 19 of 61
The optimal time between orders is: To = Qo/D =500/6,000 = 0.083 years = 1 month The associate variable cost is: VCo = HC × Qo = 6 × 500 = $3,000 a year This gives a total cost of: TCo = UC × D + VCo = 30 × 60000 + 3000 = $183,000 a year
Page 20 of 61
Page 21 of 61
Page 22 of 61
Page 23 of 61
Summary • We have built a model of an idealized inventory system that relates order size to costs and demand. This shows that large, infrequent orders have a high holding cost component, so the total cost is high: small, frequent orders have a high reorder cost component, so the total cost is also high. A compromise finds the optimal order size – or economic order quantity – that minimizes inventory costs. Page 24 of 61
Adjusting the economic order quantity •
• • •
Moving away from the EOQ The EOQ suggests fractional value for things which come in discrete units (e.g. an order for 2.7 lorries) Suppliers are unwilling to split standard package sizes. Deliveries are made by vehicles with fixed capacities. It is simply more convenient to round order sizes to a convenient number.
Page 25 of 61
How much costs would rise if we do not use the EOQ Example: D = 6000 units a year Unit cost, UC = £30 a unit Reorder cost, RC = £125 an order Holding cost, HC = £7 a unit a year
Page 26 of 61
Page 27 of 61
Page 28 of 61
Page 29 of 61
Page 30 of 61
Page 31 of 61
Page 32 of 61
Page 33 of 61
Order for discrete items This suggests a procedure for checking whether it is better to round up or round down discrete order quantities: 1. Calculate the EOQ, Qo. 2. Find the integers Q’ and Q’-1 that surround Qo. 3. If Q’×(Q’-1) is less than or equal to Qo2, order Q’. 4. If Q’×(Q’-1) is greater than Qo2, order Q’-1. Page 34 of 61
Page 35 of 61
Page 36 of 61
Uncertainty in demand and costs Error in parameters • Few organizations know exactly what demand they have to meet in the future. • The variable cost is stable around the EOQ and small errors and approximations generally make little difference.
Page 37 of 61
Page 38 of 61
Page 39 of 61
Page 40 of 61
Adjusting the order quantity
Page 41 of 61
Qo =
2 × RC × D HC
or
Qo × HC RC = 2× D 2
Page 42 of 61
2 × RC × D Q= K× HC or
Q=
2 × RC × D HC × K
Page 43 of 61
Page 44 of 61
Page 45 of 61
Adding a finite lead time • Time for order preparation • Time to get the order to the right place in suppliers • Time at the supplier • Time to get materials delivered from suppliers • Time to process the delivery
Page 46 of 61
Reorder level The amount of stock needed to cover the lead time is also constant at: Lead time × demand per unit time Reorder level = lead time demand = lead time × demand per unit time ROL = LT × D
Page 47 of 61
Page 48 of 61
Page 49 of 61
(b) Substituting value LT = 2 and D = 100, gives ROL = LT × D = 2 × 100 = 200 units As soon as the stock level declines to 200 units, Carl should place an order for 250 units.
Page 50 of 61
Page 51 of 61
Longer lead times The cycle length is: T = Q/D = 250/100 = 2.5 weeks What happens when the lead time is increased to 3 weeks ? ROL = LT × D = 3 × 100 = 300 Page 52 of 61
Stock on hand + stock on order = LT × D =300 units Reorder level = lead time demand – stock on order n × T < LT < (n+1) × T Reorder level = lead time demand – stock on order ROL = LT × D – n × Qo
Page 53 of 61
Page 54 of 61
Page 55 of 61
Some practical points • This chapter has shown when to place an order – set by the reorder level. • How much to order – set by the economic order quantity. • A two-bin system gives a simple procedure for controlling stock without computers or continuous monitoring of stock levels. Page 56 of 61
Three-bin system • The two-bin system can be extended to a three-bin system which allows for some uncertainty. • The third bin holds a reserve that is only used in an emergency. • The normal stock is used from bin A.
Page 57 of 61
• Our calculations assume that the lead time is known exactly and constant. • In practice, there can be quite wide variation, allowing for availability, supplier reliability, checks on deliveries, transport conditions, customs clearance, delays in administration, and so on.
Page 58 of 61
• Some stock levels are not recorded continuously but are checked periodically, perhaps at the end of the week. • Then an unexpectedly large demand might reduce stocks well below the reorder level before they are checked.
Page 59 of 61
• Another problem appears with large stocks, such as chemical tanks, coal tips or raw materials, where the stock level is only known approximately. Then the reorder level might be passed without anyone noticing.
Page 60 of 61
• We have assumed that the order size is independent of the lead time and the reorder level. • In practice, people often prefer larger orders if there are longer lead times, and they raise the reorder level to add an element of safety. • The reorder level can be also influence the order size, with people typically placing smaller orders with higher reorder levels.
Page 61 of 61
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