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CDAE 266 - Class 23
Nov. 15
Last class:
Result of Problem set 3
4. Queuing analysis and applications
Group project 3
5. Inventory decisions
Quiz 5 (sections 4.1 – 4.4)
Today:
Result of Quiz 5
5. Inventory decisions
Quiz 6 (take-home)
Next class:
5. Inventory decisions
CDAE 266 - Class 23
Nov. 15
Important dates:
Problem set 4: due today
Group project 3: due today
Final exam, 8:00-11:00am, Monday, Dec. 10
Result of Quiz 5
N = 54
Range = 4 -- 10
Average = 8.1
5. Inventory analysis and applications
5.1. Basic concepts
5.2. Inventory cost components
5.3. Economic order quantity (EOQ) model
5.4. Inventory policy with backordering
5.5. Inventory policy and service level
5.6. Production and inventory model
5.1. Basic concepts
-- Inventories:
-- Inventory policy (decision problems):
Q = Inventory order quantity?
R = Inventory reorder point (R = level of
inventory when you make the order)?
-- Optimal inventory policy: Determine the
order quantity (Q) and reorder point (R)
that minimize the inventory cost.
-- SKU: Stock-keeping units
5.1. Basic concepts
-- Lead time (L): the time between placing an
order and receiving delivery
-- Inventories models:
(1) Economic order quantity (EOQ) model
(2) Backordering model
(3) Production and inventory model
5.2. Inventory cost components:
-- Inventory ordering and item costs
-- Ordering costs (telephone, checking
the order, labor, transportation, etc.)
-- Item cost (price x quantity)
-- Inventory holding costs (interest, insurance,
storage, etc.)
-- Inventory shortage costs (customer
goodwill and satisfaction costs)
5.3. The economic order quantity (EOQ) model
5.3.1. Assumptions:
-- One item with constant demand (A)
-- Lead time = 0
-- No backordering
-- All the cost parameters are known
5.3. The economic order quantity (EOQ) model
5.3.2. A graphical presentation
5.3.3. Mathematical model
-- Variable definitions:
k = fixed cost per order
A = annual demand (units per yr.)
c = price
h = annual holding cost per $ value
T = time between two orders
A graphical presentation of the EOQ model:
– The constant environment described by the EOQ
assumptions leads to the following observation
THE OPTIMAL EOQ POLICY ORDERS
THE SAME AMOUNT EACH TIME.
This observation results in the inventory profile below:
Q
Q
Q
5.3. The economic order quantity (EOQ) model
5.3.3. Mathematical model
-- Objective: choose order quantity
(Q) to minimize the total annual
inventory cost
What is the reorder point (R)?
5.3. The economic order quantity (EOQ) model
5.3.3. Mathematical model
-- Total annual inventory cost
= annual ordering costs + annual
holding cost + annual item costs
(a) Annual order costs
Annual demand = A
Quantity of each order = Q
Number of orders per yr. = A/Q
Fixed cost per order = k
Annual ordering costs = k (A/Q)
5.3. The economic order quantity (EOQ) model
5.3.3. Mathematical model
-- Total annual inventory cost
= annual ordering costs + annual
holding cost + annual item costs
(b) Annual holding costs
Average inventory = Q/2
Annual holding cost per unit = hc
Annual holding cost = (Q/2)*hc
Annual unit holding cost
• Holding Costs (Carrying
costs)
– Cost of capital
– Storage space cost
– Costs of utilities
– Labor
– Insurance
– Security
– Theft and breakage
h*c
h = Annual holding cost rate
(cost per dollar value)
c = Unit value (price)
5.3. The economic order quantity (EOQ) model
5.3.3. Mathematical model
-- Total annual inventory cost
= annual ordering costs + annual
holding cost + annual item costs
(c) Annual item cost
Average demand = A
Cost per unit (price) = c
Annual item cost = Ac
5.3. The economic order quantity (EOQ) model
5.3.3. Mathematical model
-- Total annual inventory cost
= annual order costs + annual
holding cost + annual item costs
 A
Q
 k  hc   Ac
2
Q
-- Total annual relevant (variable) cost:
 A
Q
TC   k  hc 
2
Q
-- Examples
5.3. The economic order quantity (EOQ) model
5.3.3. Mathematical model
-- Derive the optimal solution:
(1) A graphical analysis: the sum of
the two costs is at the minimum
level when the annual holding
cost is equal to the annual
ordering cost:
 A
Q
 k  hc  => => =>
2
Q
2 Ak
Q 
hc
*
5.3. The economic order quantity (EOQ) model
5.3.3. Mathematical model
-- Derive the optimal solution:
(2) A mathematical analysis:
At the minimum point of the
curve, the slope (derivative) is
equal to zero:
=> =>
2 Ak
Q 
hc
*
5.3. The economic order quantity (EOQ) model
5.3.4. Examples
(1) Liquor store (pp. 209-211)
Available information:
A = 5200 cases/yrk = $10/order
c = $2 per case
h = $0.20 per $ per yr.
(a) Current policy: Q = 100 cases/order
R = (5200/365) * 1 = 15 cases
T = Q/A = 100/5200 (year) = 7 days
TC = $540 per year (see page 210)
(b) Optimal policy: Q* = 510 cases/order
R = (5200/365) * 1 = 15 cases
T = Q*/A = 510/5200 (year) =36 days
TC = $204 per year
If the retail price is $3 per case,
Gross profit = 5200*3 – 5200*2 – 204 = $4996
Take-home class exercise
1. Draw a graph to show the following inventory policy for
a business with no backordering: the annual demand is
3650 units and the business opens 365 days a year, the
order quantity is 305 units and the lead time is 4 days.
2. If some customers of the above business are willing to
take backorders and the maximum backorders are 50
units, draw another graph to show the inventory policy
(there is no change in order quantity and lead time)
3. Take-home exercise: Example on pp. 215-216 with the
annual demand (A) increased to 1200 units and the lead
time to be 3 days.