Chapter 2: Constant and Time Varying
Demand
 Costs in inventory models
 Economic order quantity (EOQ)
 Optimal Order Quantity, Reorder point.
 Safety stock
 Discount
 Planned shortage models
1. Type of Costs in
Inventory Models
 Inventory analyses can be thought of as cost-control
techniques.
 Categories of costs in inventory models:
 Holding (carrying costs)
 Order/ Setup costs
 Customer satisfaction costs
 Procurement/Manufacturing costs
Type of Costs in
Inventory Models
Holding Costs (Carrying costs): depend on the order size
 Cost of capital
 Storage space rental cost
 Costs of utilities
 Labor
 Insurance
 Security
 Theft and breakage
 Deterioration or Obsolescence
Type of Costs in
Inventory Models
Order/Setup Cost: is independent of the order size.
 Order costs are incurred when purchasing a good
from a supplier. They include costs such as
 Telephone
 Order checking
 Labor
 Transportation
 Setup costs are incurred when producing goods for
sale to others. They can include costs of
 Cleaning machines
 Calibrating equipment
 Training staff
Type of Costs in
Inventory Models
Customer Satisfaction Costs
 Measure the degree to which a customer is satisfied.
 Unsatisfied customers may:
 Switch to the competitors (lost sales).
 Wait until an order is supplied.
 When customers are willing to wait there are two types of
costs incurred:
Type of Costs in
Inventory Models
Procurement/Manufacturing Cost
 Represents the unit purchase cost (including
transportation) in case of a purchase.
 Unit production cost in case of in-house manufacturing
2. Economic Order Quantity
(EOQ) Model - Assumptions
 Demand occurs at a known and
reasonably constant rate.
 The item has a sufficiently long
shelf life.
 The item is monitored using a
continuous review system.
 All the cost parameters remain
constant forever (over an infinite
time horizon).
 A complete order is received in one
The EOQ Model –
Inventory Profile
The constant environment described by the EOQ assumptions
leads to the following observation:
 The optimal EOQ policy consists of same-size orders.
 This observation results in the following inventory profile :
2.1. Cost Equation for the
EOQ Model
 Let is
the order quantity or lot size
total annual
inventory cost
total annual
holding cost
total annual total annual
ordering cost procurement cost
Costs in the EOQ Model
cost
total cost
at the optimal order size
total holding costs and ordering costs
are equal
total ordering cost
Order quantity
Sensitivity Analysis in
EOQ Models
cost
The curve is reasonably flat around
Deviations from the optimal order size
cause only small increase in the total cost.
Order quantity
Cycle Time
 The cycle time, T, represents the time that elapses
between the placement of orders.
 Note, if the cycle time is greater than the shelf life, items
will go bad, and the model must be modified.
Number of Orders per Year
 To find the number of orders per years, take the reciprocal of the
cycle time
Example: The demand for a product is 1000 units per year.
The order size is 250 units under an EOQ policy.
 How many orders are placed per year? N = 1000/250 = 4 orders.
 How often orders need to be placed (what is the cycle time)?
T = 250/1000 = ¼ years. {Note: the four orders are equally spaced}.
Lead Time and Reorder Point
 In reality, lead time L always exists, and must be
accounted for when deciding when to place an order.
 The reorder point, R, is the inventory position when an
order is placed.
 R is calculated by:
 L and D must be expressed in the same time unit.
Lead Time and Reorder Point –
Graphical demonstration: Short Lead Time
R = Inventory at hand at the beginning of lead time
reorder point
place the order now
Lead Time and Reorder Point –
Graphical demonstration: Long Lead Time
outstanding
order
place the order now
2.2. Safety Stock
 Safety stocks act as buffers to handle:
 Higher than average lead time demand.
 Longer than expected lead time.
 With the inclusion of safety stock (SS), R is calculated by
 The size of the safety stock is based on having a desired service level.
Safety Stock
reorder point
place the order now
Safety Stock
reorder point
place the order now
The safety stock
prevents excessive
shortages.
Inventory Costs
Including Safety Stock
total annual
total annual
inventory cost holding cost
total annual total annual safety stock
ordering cost procurement holding cost
cost
ALLEN APPLIANCE
COMPANY (AAC)
 AAC wholesales small appliances.
 AAC currently orders 600 units of the Citron brand juicer
each time inventory drops to 205 units.
 Management wishes to determine an optimal ordering
policy for the Citron brand juicer
ALLEN APPLIANCE
COMPANY (AAC)
Data
 Co = $12 ($8 for placing an order) + (20 min. to check)($12 per hr)
 C =
$10.
 H =
14% (10% ann. interest rate) + (4% miscellaneous)
 Ch = $1.40 [HC = (14%)($10)]
 D =
demand information of the last 10 weeks was collected:
Sales of Juicers over the last 10 weeks
Week
1
2
3
4
5
Sales
105
115
125
120
125
Week
6
7
8
9
10
Sales
120
135
115
110
130
ALLEN APPLIANCE
COMPANY (AAC)
Data
 The constant demand rate seems to be a good
assumption.
 Annual demand = (120/week)(52weeks) = 6240 juicers.
AAC – Solution:
EOQ and Total Variable Cost
 Current ordering policy calls for Q = 600 juicers.
TV( 600) = (600/2)($1.40) + (6240 / 600)($12) =$544.8
TV is total variable cost
 The EOQ policy calls for orders of size:
TV(327) = (327 / 2)($1.40) + (6240 / 327) ( $12) = $457.89
Savings of 16% is achieved by applying the EOQ solution.
AAC – Solution:
Reorder Point and Total Cost
 Under the current ordering policy AAC holds 13 units safety stock (how
come? Observe):
 AAC is open 5 day a week.
 The average daily demand = (120/week)/5 = 24 juicers.
 Lead time is 8 days. Lead time demand is (8)(24) = 192 juicers.
 Reorder point without Safety stock = LD = 192.
 Current policy: R = 205.
 Safety stock = 205 – 192 = 13.
 For safety stock of 13 juicers the total cost is
TC(327) = 457.89 + 6240($10) + (13)($1.40) = $62,876.09
TV(327) + procurement cost + safety stock holding cost
AAC – Solution:
Sensitivity of the EOQ Results
Changing the order size
 Suppose juicers must be ordered in increments of 100 (order 300 or 400)
 AAC will order Q = 300 juicers in each order.
 There will be a total variable cost increase of $1.71.
 This is less than 0.5% increase in variable costs.
Changes in input parameters
 Suppose there is a 20% increase in demand. D=7500 juicers.
 The new optimal order quantity is Q* = 359.
 The new variable total cost = TV(359) = $502
 If AAC still orders Q = 327, its total variable costs becomes
TV(327) = (327/2)($1.40) + (7500/327)($12) = $504.13 → only increase 0.4%
AAC – Solution: Cycle Time
 For an order size of 327 juicers we have:
T = (327/ 6240) = 0.0524 year.
= 0.0524(52)(5) = 14 days.
working days per week
 This is useful information because:

Shelf life may be a problem.

Coordinating orders with other items might be desirable.
AAC – Excel Spreadsheet
2.3. EOQ Models with
Quantity Discounts
Quantity Discounts are Common Practice in Business
 By offering discounts buyers are encouraged to increase
their order sizes, thus reducing the seller’s holding
costs.
 Quantity discounts reflect the savings inherent in large
orders.
 With quantity discounts sellers can reward their biggest
customers.
EOQ Models with
Quantity Discounts
Quantity Discount Schedule
 This is a list of per unit discounts and their corresponding purchase
volumes.
 Normally, the price per unit declines as the order quantity increases.
 The order quantity at which the unit price changes is called a break point.
 There are two main discount plans:
 All unit schedules - the price paid for all the units purchased is based
on the total purchase.
 Incremental schedules - The price discount is based only on the
additional units ordered beyond each break point.
All Units Discount Schedule
 To determine the optimal order quantity, the total
purchase cost must be included
 Ci represents the unit cost at the ith pricing level.
AAC - All Units
Quantity Discounts
 AAC is offering all units quantity discounts to its customers.
 Data
Quantity Discount Schedule
1-299
$10.00
300-599
$9.75
600-999
$9.40
1000-4999
$9.50
5000
$9.00
Should AAC increase its regular order of
327 juicers, to take advantage of the discount?
AAC – All units
discount procedure
 Step 1: Find the optimal order Qi* for each discount level “i” by using the formula
 Step 2: For each discount level “i” modify Qi* as follows

If Q* < qi, then Qi* = qi.

If qi  Q* < qi+1, then Qi* = Q*

If qi+1  Q*, eliminate this level from further consideration.
 Step 3: Substitute the modified Qi* value in the total cost formula TC(Qi*).
 Step 4: Select the Qi* that minimizes TC(Qi*)
AAC – All units
discount procedure
 Step 1: Find the optimal order Qi* for each discount level
“i” by using the formula
Lowest cost order size per discount level
Discount
Qualifying
Price
level
order
per unit
Q*
0
1-299
10.00
327
1
300-599
9.75
331
2
600-999
9.50
337
3
1000-4999
9.40
336
4
5000
9.00
345
AAC – All units
discount procedure
 Step 2:
Lowest cost order size per discount level
Discount Qualifying Price
level
order
per unit
Q*
Qi*
0
1-299
10.00
327
****
1
300-599
9.75
331
331
2
600-999
10004999
5000
9.50
337
600
9.40
336
1000
9.00
345
5000
3
4
AAC – All units
discount procedure
 Step 3: Substitute the modified Qi* value in the total cost
formula TC(Qi*).
Modified Q* and total Cost
Qualified
Urder
Price
per Unit
Q*
Modified
Qi*
Total
Cost
1-299
10.00
300
****
***
300-599
9.75
331
331
$61,309.88
600-999
9.50
336
600
$59,192.71
1000-4999
9.40
337
1000
$60,037.17
5000
9.00
345
5000
$59,341.36
AAC – All units
discount procedure
 Step 4: AAC should order 600 juicers as it results in the
minimum total annual cost
Modified Q* and total Cost
Qualified
Urder
Price
per Unit
Q*
Modified
Q*
Total
Cost
1-299
10.00
300
****
****
300-599
9.75
331
331
$61,309.88
600-999
9.50
336
600
$59,192.71
1000-4999
9.40
337
1000
$60,037.17
5000
9.00
345
5000
$59,341.36
AAC – All Units Discount Excel
Worksheet
3. Planned Shortage Model
 When an item is out of stock, customers may:
 Go somewhere else (lost sales).
 Place their order and wait (backordering).
 In this model we consider the backordering case.
 All the other EOQ assumptions are in place.
Planned Shortage Model –
the Total Variable Cost Equation
 The parameters of the total variable costs function are similar to
those used in the EOQ model.
 In addition, we need to incorporate the shortage costs in the
model.
 Backorder cost per unit per year (loss of good will cost) - Cs.
 Reflects future reduction in profitability.
 Can be estimated from market surveys and focus groups.
 Backorder administrative cost per unit - Cb.
 Reflects additional work needed to take care of the backorder.
Planned Shortage Model –
the Total Variable Cost Equation
 The Annual holding cost =
Ch[T1/T](Average inventory) = Ch[T1/T] (Q-S)/2
 The Annual shortage cost =
Cb(number of backorders per year) +
CS(T2/T)(Average number of backorders).
 To calculate the annual holding cost and
shortage cost we need to find
 The proportion of time inventory is carried, (T1/T)
 The proportion of time demand is backordered, (T2/T).
Finding T1/ T and T2/ T
average inventory (Q-S)/2
Proportion of time
inventory exists
= T1/T
= (Q - S) / Q
Proportion of time
shortage exists
= T2/T
=S/Q
Average shortage = S / 2
Planned Shortage Model –
The Total Variable Cost Equation
 Annual holding cost:
Ch[T1/T](Q-S)/2 = Ch[(Q-S) /Q](Q-S)/2
= Ch(Q-S)2/2Q
 Annual shortage cost:
Cb(Units in short per year) +
Cs[T2/T](Average number of backorders) =
Cb(S)(D/Q) + CsS2/(2Q)
Planned Shortage Model –
The Total Variable Cost Equation
 The total annual variable cost equation
Time independent
backorder costs
Time independent
backorder costs
 The optimal solution to this problem is obtained under the
following conditions

Cs > 0 ;

Cb < (2CoCh / D)1/2
Planned Shortage Model –
The Optimal Inventory Policy
 The Optimal Order Size
 The Optimal Backorder level
 Reorder Point
SCANLON PLUMBING
CORPORATION
 Scanlon distributes a portable sauna from Sweden.
 Data
 A sauna costs Scanlon $2400.
 Annual holding cost per unit $525.
 Fixed ordering cost $1250 (fairly high, due to costly transportation).
 Lead time is 4 weeks.
 Demand is 15 saunas per week on the average.
SCANLON PLUMBING
CORPORATION
 Backorder costs
 Scanlon estimates a $20 goodwill cost for each week
a customer who orders a sauna has to wait for
delivery.
 Administrative backordrer cost is $10.
 Management wishes to know:
 The optimal order quantity.
 The optimal number of backorders.
SCANLON PLUMBING –
Solution
 Input for the total variable cost function
 D = 780 saunas
[(15)(52)]
 Co = $1,250
 Ch = $525
 Cs = $1,040
 Cb = $10
SCANLON PLUMBING –
Spreadsheet Solution