Inventory Management
Kinds of Inventory
•
•
•
•
•
•
Raw materials and purchased parts
Work-in-process (WIP)
Finished goods inventories or merchandise
Tools and supplies
Maintenance and repairs (MRO) inventory
Goods-in-transit to warehouses or customers
(pipeline inventory)
Functions of Inventory
– Anticipation stock
– Seasonal inventories
– Decoupling inventories
– Buffer inventories
– Cycle stock
– Price hedge
Objectives of Inventory Control
• Inventory management has two main
concerns:
1. Level of customer service
• Having the right goods available in the right quantity in
the right place at the right time
2. Costs of ordering and carrying inventories
• The overall objective of inventory management is to
achieve satisfactory levels of customer service while
keeping inventory costs within reasonable bounds
Inventory Management
• Requirements
– Keeping track of inventory
– Demand forecast
– Lead times
– Costs
• Holding
• Ordering
• Shortage
• Classification (A-B-C)
Inventory Costs
• Purchase cost
– The amount paid to buy the inventory
• Holding (carrying) costs
– Cost to carry an item in inventory for a length of time, usually a
year
• Ordering costs
– Costs of ordering and receiving inventory
• Setup costs
– The costs involved in preparing equipment for a job
– Analogous to ordering costs
• Shortage costs
– Costs resulting when demand exceeds the supply of inventory;
often unrealized profit per unit
INVENTORY MANAGEMENT
EOQ MODELS
How Much to Order: EOQ Models
• Economic order quantity models identify the
optimal order quantity by minimizing the sum
of annual costs that vary with order size and
frequency
1. The basic economic order quantity model
2. The economic production quantity model
3. The quantity discount model
Basic EOQ Model
• The basic EOQ model is used to find a fixed order quantity
that will minimize total annual inventory costs
• Assumptions:
1.
2.
3.
4.
5.
6.
Only one product is involved
Annual demand requirements are known
Demand is even throughout the year
Lead time does not vary
Each order is received in a single delivery
There are no quantity discounts
The Inventory Cycle
Profile of Inventory Level Over Time
Q
Usage
rate
Quantity
on hand
Reorder
point
Receive
order
Place
order
Receive
order
Lead time
Place
order
Receive
order
Time
EOQ
• Costs
– Annual Holding Cost = Q/2 * H (H is avg annual
holding cost per unit)
– Annual Ordering cost = D/Q*S
Annual Cost
Goal: Total Cost Minimization
The Total-Cost Curve Is U-Shaped
Q
D
TC  H  S
2
Q
Holding Costs
Ordering Costs
Order Quantity (Q)
QO (optimal order quantity)
EOQ
• 𝑇𝐶 = 𝐴𝑛𝑛𝑢𝑎𝑙 𝑐𝑎𝑟𝑟𝑦𝑖𝑛𝑔 𝑐𝑜𝑠𝑡 + 𝐴𝑛𝑛𝑢𝑎𝑙 𝑜𝑟𝑑𝑒𝑟𝑖𝑛𝑔 𝑐𝑜𝑠𝑡
• 𝑇𝐶 =
𝑄
𝐷
𝑄𝑋 𝑆
• Using calculus, we take the derivative of the total cost
function and set the derivative (slope) equal to zero and solve
for Q.
• 𝑄𝑜 =
2𝑋 𝐻 +
2𝐷𝑆
𝐻
EOQ
• A local distributor for a national tire company
expects to sell approximately 9,600 steel belted
radial tires of a certain size and tread design next
year. Annual carrying cost is Rs.192 per tire, and
ordering cost is Rs.900. The distributor operates
288 days a year.
• What is EOQ?
• How many times does the store reorder?
• What is the length of order cycle?
• What is the total annual cost?
EOQ
• D = 9600, S = 900, H = 192, Q = ?
• 𝑄𝑜 =
•
•
•
•
•
•
•
•
•
2𝐷𝑆
𝐻
=
2∗9600∗900
192
300
How many times the store reorders
9600/300 = 32
Length of order cycle
Daily consumption = 9600/288 = 33.3
How long every order will last, 300/33.3 = 9 days
Total cost, Annual Holding Cost + Annual Ordering Cost
Q/2*H + D/Q*S,
28800+28800
Economic Production Quantity (EPQ)
• Setup cost
– Whenever production requirements change (a
new product, different specs, different quality) the
machine has to be setup accordingly.
– While the machine is being setup, production
stops
– Tradeoff is between greater setups versus greater
inventory (because of larger batch size)
Economic Production Quantity (EPQ)
• The batch mode is widely used in production. In certain
instances, the capacity to produce a part exceeds its usage
(demand rate).
– Assumptions
1. Only one item is involved
2. Annual demand requirements are known
3. Usage rate is constant
4. Usage occurs continually, but production occurs periodically
5. The production rate is constant
6. Lead time does not vary
7. There are no quantity discounts
Economic Production Quantity
Q
Qp
Production
and usage
Usage
only
Production
and usage
Usage
only
Production
and usage
Cumulative
production
Imax
Amount
on hand
Time
EPQ
•
•
•
•
•
Setup cost = S = 𝐷 𝑄 𝑋 𝑆
Carrying/holding cost
How did we calculate holding cost in EOQ model
Holding cost was Q/2 X H
Where Q/2 was the average inventory in the
system
• And Q, which was the order quantity or the
maximum point of inventory
𝐼𝑚𝑎𝑥
• Carrying cost = ? =
2 𝑋𝐻
• Maximum inventory = 𝐼𝑚𝑎𝑥
EPQ
• 𝐼𝑚𝑎𝑥 is dependent on the production quantity,
the production rate and the consumption rate
• 𝐼𝑚𝑎𝑥 =
• TC =
𝑝−𝑢
𝑄𝑜 ( )
𝑝
𝐼𝑚𝑎𝑥
𝐻
2
+
𝐷
𝑄𝑜
S
EPQ
• 𝑄𝑝 =
2𝐷𝑆
𝐻
𝑝
𝑝−𝑢
• Cycle time =
• Run time =
𝑄𝑝
𝑢
𝑄𝑝
𝑝
EPQ
• A toy manufacturer uses 48,000 rubber wheels per
year for its popular dump truck series. The firm makes
its own wheels, which it can produce at a rate of 800
per day. The toy trucks are assembled uniformly over
the entire year. Carrying cost s Rs.1 per wheel per year.
Setup cost for a production run of wheels is Rs.45. The
firm operates 240 days per year. Determine the
• Optimal run size
• Minimum total annual cost for carrying and setup
• Cycle time for optimal run size
• Run time
EPQ
•
•
•
•
•
•
•
•
D = 48,000 wheels per year
S = $45
H = $1 per wheel per year
p = 800 wheels per day
u=?
Usage of 48000 spread over 240 working days
u = 48000/240 = 200 wheels per day
Qp = 2400 wheels
•
TC =
•
𝐼𝑚𝑎𝑥 =
•
•
•
𝐼𝑚𝑎𝑥 = 1800
TC = ((1800/2) X 1) + ((48000/2400) X 45
TC = 900 + 900 = 1800
•
Cycle time =
•
Run time =
𝐼𝑚𝑎𝑥
𝐻
2
𝐷
S
𝑄𝑜
𝑝−𝑢
𝑄𝑜 ( )
𝑝
+
𝑄𝑝
𝑢
𝑄𝑝
𝑝
= 2400/12 = 12 days
= 2400/800 = 3 days
Quantity Discount Model
• Quantity discount
– Price reduction for larger orders offered to
customers to induce them to buy in large
quantities
Total Cost  Carrying Cost  Ordering Cost  Purchasing Cost

Q
D
H  S  PD
2
Q
where
P  Unit price
Quantity Discounts
Adding PD does not change EOQ
Quantity Discounts (cont.)
The total-cost curve with
quantity discounts is
composed of a portion of
the total-cost curve for
each price
Quantity Discounts
• Calculate EOQ
• Calculating EOQ here depends on how holding
cost is calculated
• Scenario 1 = fixed holding cost (Rs X/yr)
• Scenario 2 = Percentage of price
Quantity Discounts with fixed HC
• Calculate EOQ
• Identify the price range in which the EOQ falls
• If the min point (calculated EOQ)is in the
lowest price range then this is the Optimal
Order Quantity
• If the calculated EOQ is in a higher cost range,
then calculate the total cost at this point and
then compare it with total cost at minimum
points of all lower price ranges
Quantity Discounts with fixed HC
• The maintenance department of a large
hospital uses about 816 cases of liquid
cleanser annually. Ordering costs are $12,
carrying costs are $4 per case a year and the
new price schedule indicates that orders of
less than 50 cases will cost $20 per case, 50 to
79 cases will cost $18 per case, 80 to 99 cases
will cost $17 per case, and larger orders will
cost $16 per case. Determine the optimal
order quantity and the total cost.
Quantity Discounts with fixed HC
• EOQ comes out to be approx. 70 cases
• This falls in range of 50-79 cases with per unit price of
$18
Range
Price
• TC @$18/unit and 70 cases as EOQ
1-49
20
50-79
18
• (70/2)4 + (816/70)12 + 18(816) = $14,968
• Because lower price ranges also exist so we will 80-99 17
calculate total cost at their initial points as well 100+ 16
• TC @$17 for EOQ of 80 cases
• (80/2)4 + (816/80)12 + 17(816) = $14,154
• TC @$16 for EOQ of 100 cases
• (100/2)4 + (816/100)12 + 16(816) = $13,354
Quantity Discount with carrying cost
expressed as percentage of price
• Calculate EOQ with minimum price range, if the
quantity (calculated) falls in the minimum price
bracket, this is your EOQ
• If the calculated quantity doesn’t fall in the range,
then calculate EOQ for next higher price range.
Once it falls in the same price bracket, calculate
total cost at this EOQ and price.
• Next calculate total cost at starting order points
of all subsequent lower price ranges
• The lowest total cost would be your EOQ
Quantity Discount with carrying cost
expressed as percentage of price
• A company uses 4,000 switches a year.
Switches are priced as follows: 1 to 499, 90
cents each; 500 to 999, 85 cents each; and
1,000 or more, 80 cents each. It costs $30 to
prepare an order and receive it, and carrying
costs are 40 percent of purchase price per unit
on an annual basis. Determine the optimal
order quantity and the total annual cost.
Quantity Discounts
•
•
•
•
•
•
•
•
•
•
•
•
•
Range
Unit price
H
Calculate EOQ at min price bracket
EOQ @ $0.80 price
1-499
0.9
.4*.9=.36
EOQ = 866 switches
500-999
0.85
.4*.85=.34
But quantity falls in $0.85 price bracket
1000+
0.8
.4*.8=.32
Calculate EOQ @ $0.85 price
EOQ = 840 switches (match)
TC @ $0.85 and 840 EOQ
(840/2)0.34 + (4000/840)30 + .85(4000) = $3686
(H @ $0.85 = 40% of price = .4*.85 = .34)
Calculate Total cost @ $0.80 price and min quantity in that range (1000)
TC @ $0.80 and 1000 EOQ
(1000/2)0.32 + (4000/1000)30 + .80(4000) = $3840
SO EOQ = 840 @ 0$0.85
When to Reorder
• Reorder point
– When the quantity on hand of an item drops to
this amount, the item is reordered.
– Determinants of the reorder point
1. The rate of demand
2. The lead time
Reorder Point: Under Certainty
ROP  d  LT
where
d  Demand rate (units per period, per day, per week)
LT  Lead time (in same time units as d )
Reorder Point: Under Uncertainty
• Demand or lead time uncertainty creates the possibility that
demand will be greater than available supply
• To reduce the likelihood of a stockout, it becomes necessary
to carry safety stock
– Safety stock
• Stock that is held in excess of expected demand due to variable
demand and/or lead time
Expected demand
ROP 
 Safety Stock
during lead time
Safety Stock
How Much Safety Stock?
• The amount of safety stock that is appropriate
for a given situation depends upon:
1. The average demand rate and average lead time
2. Demand and lead time variability
3. The desired service level
Expected demand
ROP 
 z dLT
during lead time
where
z  Number of standard deviations
 dLT  The standard deviation of lead time demand
Reorder Point
The ROP based on a
normal
distribution of lead
time demand
Reorder Point: Demand Uncertainty
ROP  d  LT  z d LT
where
z  Number of standard deviations
d  Average demand per period (per day, per week)
 d  The stdev. of demand per period (same time units as d )
LT  Lead time (same time units as d )
Note: If only demand is variable, then
 dLT   d LT
Reorder Point: Demand Uncertainty
Lead time is 4 days, so we need to look at the behavior of
demand during these four days with every day as a new
independent event
𝜎𝑑2 + 𝜎𝑑2 + 𝜎𝑑2 + 𝜎𝑑2
= 4𝜎𝑑
= 𝐿𝑇𝜎𝑑
= 4𝜎𝑑2
Reorder Point: Lead Time Uncertainty
ROP  d  LT  zd LT
where
z  Number of standard deviations
d  Demand per period (per day, per week)
 LT  The stddev. of lead time (same time units as d )
LT  Average lead time (same time units as d )
 dLT  d LT
Note: If only lead time is variable, then
When both demand and Lead time are variable
2
• ROP = 𝑑 ∗ 𝐿𝑇 + 𝑧 𝐿𝑇𝜎𝑑2 + 𝑑 2 𝜎𝐿𝑇
When To Reorder
• When only demand in variable
• ROP = 𝑑 ∗ 𝐿𝑇 + 𝑧𝜎𝑑 𝐿𝑇
• When only Lead time is variable
• ROP = 𝑑 ∗ 𝐿𝑇 + 𝑧𝑑𝜎𝐿𝑇
• When both demand and Lead time are variable
2
• ROP = 𝑑 ∗ 𝐿𝑇 + 𝑧 𝐿𝑇𝜎𝑑2 + 𝑑 2 𝜎𝐿𝑇
Reorder Point
• A restaurant uses an average of 50 jars of a
special sauce each week. Weekly usage of sauce
has a standard deviation of three jars. The
manager is willing to accept no more than a 10
percent risk of stockout during lead time, which is
two weeks. Assume the distribution of usage is
normal.
• Which formula is appropriate for this situation
• Determine the value of z
• Determine the ROP
Solution
•
•
•
•
•
•
•
𝑑 = 50 jars per week
LT = 2 weeks
𝜎𝑑 = 3 jars per week
Acceptable risk = 10 percent
Since only demand in variable we will be using
ROP = 𝑑 ∗ 𝐿𝑇 + 𝑧𝜎𝑑 𝐿𝑇
From appendix B, Table B of the book, using a
service level of 0.90, we get z = 1.28
• ROP comes out to be approx. 106 units
Fixed Order Interval Model
• Fixed order model
• Fixed interval model
Amount to Order
• Amount to order =
• Expected demand during protection interval +
Safety stock –
Amount on hand at reorder point
= 𝑑 𝑂𝐼 + 𝐿𝑇 + 𝑧𝜎𝑑 𝑂𝐼 + 𝐿𝑇 − 𝐴
𝑂𝐼 = 𝑂𝑟𝑑𝑒𝑟 𝑖𝑛𝑡𝑒𝑟𝑣𝑎𝑙 (𝑙𝑒𝑛𝑔𝑡ℎ 𝑜𝑓 𝑡𝑖𝑚𝑒 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 𝑜𝑟𝑑𝑒𝑟𝑠)
Amount to Order
• Given the following information, determine
the amount to order:
• 𝑑 = 30 𝑢𝑛𝑖𝑡𝑠 𝑝𝑒𝑟 𝑑𝑎𝑦
• Desired service level = 99%
• 𝜎𝑑 = 3 𝑢𝑛𝑖𝑡𝑠 𝑝𝑒𝑟 𝑑𝑎𝑦
• Amount on hand at reorder time = 71 units
• LT = 2 days
• OI = 7 days
Solution
• Z = 2.33 for 99 percent service level
• Q = 𝑑 𝑂𝐼 + 𝐿𝑇 + 𝑧𝜎𝑑 𝑂𝐼 + 𝐿𝑇 − 𝐴
• Q = 30(7+2) + 2.33(3)(3) – 71
• = 220 units
Single-Period Model
• Single-period model
– Model for ordering of perishables and other items with limited useful
lives
– Shortage cost
• Generally, the unrealized profit per unit
• Cshortage = Cs = Revenue per unit – Cost per unit
– Excess cost
• Different between purchase cost and salvage value of items left
over at the end of the period
• Cexcess = Ce = Cost per unit – Salvage value per unit
The Single Period Model
• Service level =
𝐶𝑠
𝐶𝑠 + 𝐶𝑒
Continuous Stocking level
• XYZ product has a mean weekly demand of
200 liters (normally distributed) with a
standard deviation of 10 liters per week.
Shortage cost is 60 cents per liter while excess
cost is 20 cents per liter. Find the optimal
stocking level.
Solution
•
•
•
•
•
Cs = 0.6
Ce = 0.2
S.L = .6/.6+.2
= 75%
This indicates that 75 percent of the area under the
normal curve must be to the left of the stocking level
• Appendix B, Table B shows that z value between .67
and .68 corresponds to area under the curve of 75%
• S.L = 200 + 0.67(10)
• 206.7 liters
Discrete Stocking Levels
• Shortage cost = $2400
• Excess cost = $800
• Demand distribution:
Demand
Relative Frequency
Cumulative frequency
0
.20
.20
1
.40
.60
2
.30
.90
3
.10
1.00
4 or more
.00
1.00
Cs = 0.6
Ce = 0.9
Demand
Probability
Cumulative
0
0.04
.04
1
0.07
.11
2
0.09
.2
3
0.12
.32
4
0.13
.45
5
0.17
.62
6
0.13
.75
7
0.10
.85
8
0.07
.92
9
0.05
.97
10
0.03
1.00
1.00
Decision
3
Per unit rev
1.5
Per unit cost
0.9
Demand
Prob
Sales
Rev
Cost
cont
Prob wt cont
0
0.04
0
0
2.7
-2.7
-0.108
1
0.07
1
1.5
2.7
-1.2
-0.084
2
0.09
2
3
2.7
0.3
0.027
3
0.12
3
4.5
2.7
1.8
0.216
4
0.13
3
4.5
2.7
1.8
0.234
5
0.17
3
4.5
2.7
1.8
0.306
6
0.13
3
4.5
2.7
1.8
0.234
7
0.1
3
4.5
2.7
1.8
0.18
8
0.07
3
4.5
2.7
1.8
0.126
9
0.05
3
4.5
2.7
1.8
0.09
10
0.03
3
4.5
2.7
1.8
0.054
Expected contribution
1.275