The Economic Production Quantity EPQ Model

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The Economic Production Quantity (EPQ) Model
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• Similar assumptions to the EOQ model, except
  that production/delivery is not instantaneous
• Units are produced and delivered one unit at a
  time
• Production capacity is finite with a finite
  production rate P

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Notation

• P production rate (number of units/time period)
• TP production cycle (time facility is producing
  per order cycle)
• TD withdrawal cycle (time facility is idle per
  order cycle)
• T total inventory cycle (time between setups)
• Qmax maximum inventory level (units)

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Inventory versus Time
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• TP Q/P
• TD Qmax/D
• Qmax TP(P - D) Q(1 - D/P)
• Average inventory Qmax/2
• Number of orders per unit time D/Q

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Costs

• Total holding cost hQmax/2hQ(1-D/P)/2
• Total ordering/setup cost AD/Q
• Total production/purchasing cost cD
• Total cost AD/Q hQ(1 - D/P)/2 cD
• Unit cost A/Q hQ(1 - D/P)/2D c

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The Economic Production Quantity
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• The EPQ is equivalent to an EOQ model with
  holding cost hh(1-D/P).
• Consequently, the optimal cost under the EPQ
  model is lower than the optimal cost under the
  EOQ model with holding cost h.

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Production Facility Utilization

• U D/P (capacity utilization)
• We must not operate above capacity (i.e., always
  keep U ? 1)
• What happens when D gt P?
• What happens when D P?

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EOQ vs. EPQ

• Q(EPQ) ? Q(EOQ) when U ? 1
• Q(EPQ) Q(EOQ) when U ? 0
• Q(EPQ) ? infinity when U ? 1 (continuous
  production)
• Y(Q(EPQ)) ? cD when U ? 1

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Systems with Multiple Products
N number of products Di demand rate for
product i Pi production rate for product i hi
holding cost per unit per unit time for product
i Ai Ordering/setup cost for product i ci
production cost for product i
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Objective
Minimize total cost while guaranteeing that no
stockouts occur for any product.
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• In order to ensure feasibility, we must have
• Choosing can
  lead to stockouts

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A Cyclic Policy

• A strictly cyclic policy is used (in each cycle,
  there is exactly one setup per product)
• Cycle time, T, is the time between two
  consecutive setups for any given product
• During T, a quantity Qi of each product i is
  produced and consumed therefore, Qi DiT

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Inventory versus Time
P1 P2 P3
Inventory
Time
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Order Quantities and Order Interval

• Cost for Product i

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Order Quantities and Order Interval

• Cost for Product i
• Total Cost

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Order Quantities and Order Interval

• Cost for Product i
• Total Cost
• Since Qi/Di T,

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Optimal Order Interval and Order Quantities