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First page of scanned document
This presentation includes selected slides from
Operations Management (2011) Jay Heizer and Barry Render 10e Global edition, Pearson Publication, ISBN: 13 978-0-13-511143-7
RMIT University 2012
The Basic Economic Order Quantity (EOQ) Model
Minimizing Costs
Reorder Points
Other Probabilistic Models
Amazon.com
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*
Amazon.com
Each order is assigned by computer to the closest distribution center that has the product(s)
A “flow meister” at each distribution center assigns work crews
Lights indicate products that are to be picked and the light is reset
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Amazon.com
Crates arrive at central point where items are boxed and labeled with new bar code
Gift wrapping is done by hand at 30 packages per hour
Completed boxes are packed, taped, weighed and labeled before leaving warehouse in a truck
Order arrives at customer within 2 - 3 days
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Inventory Management
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Importance of Inventory
One of the most expensive assets of many companies representing as much as 50% of total invested capital
Operations managers must balance inventory investment and customer service
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To decouple or separate various parts of the production process
To decouple the firm from fluctuations in demand and provide a stock of goods that will provide a selection for customers
To take advantage of quantity discounts
To hedge against inflation
A function of cycle time for a product
Maintenance/repair/operating (MRO)
Finished goods
The Material Flow Cycle
Input Wait for Wait to Move Wait in queue Setup Run Output
inspection be moved time for operator time time
Cycle time
95% 5%
How inventory items can be classified
ABC divides inventory into three classes based on annual dollar volume
Class A - high annual dollar volume
Class B - medium annual dollar volume
Class C - low annual dollar volume
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*
Silicon Chips, Inc. maker of Dram chips, wants to categorise
its 10 major inventory items using ABC analysis.
See OMGT2228 Lecture 7 ABC analysis example.xlsx on the Blackboard
Dollar Volume Rank
Category
1
10286
90
1000
$ 90,000.00
38.78%
38.78%
A
2
11526
154
500
$ 77,000.00
33.18%
71.97%
A
3
12760
17
1550
$ 26,350.00
11.35%
83.32%
B
4
10867
42.86
350
$ 15,001.00
06.46%
89.78%
B
5
10500
12.5
1000
$ 12,500.00
05.39%
95.17%
B
6
12572
14.17
600
$ 8,502.00
03.66%
98.83%
C
7
14075
0.6
2000
$ 1,200.00
00.52%
99.35%
C
8
01036
8.5
100
$ 850.00
00.37%
99.72%
C
9
01307
0.42
1200
$ 504.00
00.22%
99.94%
C
10
10572
0.6
250
$ 150.00
00.06%
100.00%
C
Total
$ 232,057.00
72%
23%
5%
80 –
70 –
60 –
50 –
40 –
30 –
20 –
10 –
0 –
10 20 30 40 50 60 70 80 90 100
Percent of inventory items
Tighter physical inventory control for A items
More care in forecasting A items
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Independent Versus
Dependent Demand
Independent demand - the demand for item is independent of the demand for any other item in inventory
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Holding, Ordering, and Setup Costs
Holding costs - the costs of holding or “carrying” inventory over time
Ordering costs - the costs of placing an order and receiving goods
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Housing costs (building rent or depreciation, operating costs, taxes, insurance)
6% (3 - 10%)
Material handling costs (equipment lease or depreciation, power, operating cost)
3% (1 - 3.5%)
11% (6 - 24%)
Housing costs (building rent or depreciation, operating costs, taxes, insurance)
6% (3 - 10%)
Material handling costs (equipment lease or depreciation, power, operating cost)
3% (1 - 3.5%)
11% (6 - 24%)
26%
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Basic economic order quantity
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Receipt of inventory is instantaneous and complete
Quantity discounts are not possible
Only variable costs are setup and holding
Stockouts can be completely avoided
Important assumptions
Usage rate
Annual cost
Order quantity
Holding cost
Q* = Optimal number of pieces per order (EOQ)
D = Annual demand in units for the inventory item
S = Setup or ordering cost for each order
H = Holding or carrying cost per unit per year
Annual setup cost = (Number of orders placed per year)
x (Setup or order cost per order)
Annual demand
=
Q* = Optimal number of pieces per order (EOQ)
D = Annual demand in units for the inventory item
S = Setup or ordering cost for each order
H = Holding or carrying cost per unit per year
Annual holding cost = (Average inventory level)
x (Holding cost per unit per year)
Order quantity
Q
2
= (H)
Q* = Optimal number of pieces per order (EOQ)
D = Annual demand in units for the inventory item
S = Setup or ordering cost for each order
H = Holding or carrying cost per unit per year
Optimal order quantity is found when annual setup cost equals annual holding cost
Solving for Q*
EOQ Example 1
Sharp Inc. a company that markets painless hypodermic needles to hospitals, would like to reduce its inventory cost by determining the optimal number of hypodermic needles to obtain per order. The annual demand is 1000 units, ordering cost is $10 per order, and inventory holding cost is $0.50 per unit per year. Determine optimal number of needles to order.
D = 1,000 units
2DS
H
Q* =
Q* =
2(1,000)(10)
0.50
D = 1,000 units Q* = 200 units
S = $10 per order
Demand
*
*
D = 1,000 units Q* = 200 units
S = $10 per order N = 5 orders per year
H = $.50 per unit per year, 1 year =250 days
Number of working
days per year
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*
S = $10 per order N = 5 orders per year
H = $.50 per unit per year T = 50 days
1 year =250 days
TC = (5)($10) + (100)($.50) = $50 + $50 = $100
D
Q
Q
2
The EOQ model is robust
It works even if all parameters and assumptions are not met
*
S = $10 per order N = 5 orders per year
H = $.50 per unit per year T = 50 days
Total annual cost increases by only 25%
Sharp Inc (in example 1) underestimates the total annual
demand (the actual demand is 1500 rather than 1000).
How the annual inventory cost can be affected?
(note: compare old EOQ with new one)
D
Q
Q
2
1,500 units
D = 1,000 units Q* = 244.9 units
S = $10 per order N = 5 orders per year
H = $.50 per unit per year T = 50 days
TC = $61.24 + $61.24 = $122.48
Only 2% less than the total cost of $125 when the order quantity was 200
D
Q
Q
2
The reorder point (ROP) tells “when” to order
= d x L
Demand per day
d =
Q*
Reorder Point Example
An Apple distributor has a demand of 8000 ipads per year. With the following information calculate ROP.
Demand = 8,000 iPads per year
250 working day year
ROP = d x L
D
d =
Production Order Quantity Model
Used when inventory builds up over a period of time after an order is placed
Used when units are produced and sold simultaneously
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*
Demand part of cycle with no production
Part of inventory cycle during which production (and usage) is taking place
t
Q = Number of pieces per order p = Daily production rate
H = Holding cost per unit per year d = Daily demand/usage rate
t = Length of the production run in days
Annual inventory holding cost
= –
Q = Number of pieces per order p = Daily production rate
H = Holding cost per unit per year d = Daily demand/usage rate
t = Length of the production run in days
However, Q = total produced = pt ; thus t = Q/p
Maximum inventory level
= –
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Q = Number of pieces per order p = Daily production rate
H = Holding cost per unit per year d = Daily demand/usage rate
D = Annual demand
1
2
S = $10 d = 4 units per day
H = $0.50 per unit per year
Nathan Manufacturing makes and sells hubcaps for the retail automobile aftermarket. Nathan’s forecast for hubcaps is 1000 units next year, with an average daily demand of 4 units. The production process is most efficient at 8 units per day. So the company produces 8 units but uses 4 units per day. The company wants to solve for the optimum number of units per order. (note: the plant schedules production of the hubcaps only as needed, during 250 days per year the shop operates). S= $10, H =$0.50 per unit per year
2DS
Note:
1,000
250
Reduced prices are often available when larger quantities are purchased
Trade-off is between reduced product cost and increased holding cost
Total cost = Setup cost + Holding cost + Product cost
D
Q
For each discount, calculate Q*
If Q* is too low to qualify for discount, choose the smallest possible order size to get the discount
Compute the total cost for each Q* or adjusted value from Step 2
Select the Q* that gives the lowest total cost
Steps in analyzing a quantity discount
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Wohls Discount Store stocks toy race cars. Recently, the store
has been given a quantity discount schedule for these cars. The
quantity schedule is shown in the following table. Ordering cost
is $49 per order and annual demand is 5000 race cars, and
inventory carrying cost, as percentage of cost, is 20%. What
is EOQ to minimize the total inventory cost?
Discount Number
Discount Quantity
2DS
HP
Q* =
2DS
HP
Q* =
Quantity Discount Example
Choose the price and quantity that gives the lowest total cost
Buy 1,000 units at $4.80 per unit
Discount Number
Unit Price
Order Quantity
Used when demand is not constant or certain
Use safety stock to achieve a desired service level and avoid stockouts
ROP = d x L + ss
Annual stockout costs = the sum of the units short x the probability x the stockout cost/unit
x the number of orders per year
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Safety Stock Example
David Rivera Optical has determined that its reorder point (lead time )
for eyeglass frames is 50 (d * L) units. Its carrying cost per frame per
year is $5 and stockout (or lost sale) cost is $40 per frame. The store has
experienced the following probability distribution for inventory demand
during the lead time (reorder point). The optimum number of orders is 6.
How much safety stock should David keep on hand?
ROP = 50 units Stockout cost = $40 per frame
Orders per year = 6 Carrying cost = $5 per frame per year
Number of Units
ROP = 50 units Stockout cost = $40 per frame
Orders per year = 6 Carrying cost = $5 per frame per year
A safety stock of 20 frames gives the lowest total cost
ROP = 50 + 20 = 70 frames
Safety Stock
Probabilistic Demand
Use prescribed service levels to set safety stock when the cost of stockouts cannot be determined
ROP = demand during lead time + ZsdLT
where Z = number of standard deviations
sdLT = standard deviation of demand during lead time
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*
kit that has a normally distributed demand during the reorder
Point (lead time). The mean demand during the reorder point is 350 kits, with a standard deviation of 10 kits. The hospital
wants to follow a policy that results in stockout only 5 % of the
time (or service level of 95%)
*
*
*
Standard deviation of demand during lead time = sdLT = 10 kits
5% stockout policy (service level = 95%)
Using Appendix I, for an area under the curve of 95%, the Z = 1.65
Safety stock = ZsdLT = 1.65(10) = 16.5 kits
Reorder point = expected demand during lead time + safety stock
= 350 kits + 16.5 kits of safety stock
= 366.5 or 367 kits
Normal distribution probability of demand during lead time
Expected demand during lead time (350 kits)
ROP = 350 + safety stock of 16.5 = 366.5
Receive order
Lead time
When both demand and lead time are variable
*
*
ROP = (average daily demand
where sd = standard deviation of demand per day
sdLT = sd lead time
Probabilistic Example
The average daily demand for Apple ipads at a Circuit Town store is 15, with standard deviation of 5 units. The lead time is constant at 2 days. Find the reorder point if management wants a 90% service level ( risk of stockout is 10%). What is safety stock?
Average daily demand (normally distributed) = 15
Standard deviation = 5
90% service level desired
Z for 90% = 1.28
ROP = (15 units x 2 days) + ZsdLT
= 30 + 1.28(5)( 2)
2. Lead time is variable and demand is constant
ROP = (daily demand x average lead time in days)+ Z (daily demand) x sLT
where sLT = standard deviation of lead time in days
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*
Probabilistic Example
Circuit Town store sells about 10 cameras a day (almost constant quantity). Lead time for camera delivery is normally distributed with a mean time of 6 days and standard deviation of 3 days. With a 98% service level what is ROP?
Daily demand (constant) = 10
Standard deviation of lead time = sLT = 3
98% service level desired
Z for 98% = 2.055
= 60 + 61.65 = 121.65
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*
ROP = (average daily demand
where sd = standard deviation of demand per day
sLT = standard deviation of lead time in days
sdLT = (average lead time x sd2)
+ (average daily demand)2 x sLT2
*
*
Standard deviation = sd = 16
Standard deviation = sLT = 1 day
95% service level desired
Z for 95% = 1.65
From Appendix I
Circuit Town store’s most popular item is six-packs of 9-volt batteries.
About 150 packs are sold every day, following a normal distribution
with standard deviation of 16 packs. Batteries are ordered from an out -
of -state distributor, lead time has normal distribution with
mean = 5 days and standard deviation of 1 days. To maintain 95%
service level what ROP is appropriate?
ROP = (150 packs x 5 days) + 1.65sdLT
= (150 x 5) + 1.65 (5 days x 162) + (1502 x 12)
= 750 + 1.65(154) = 1,004 packs
Only one order is placed for a product
Units have little or no value at the end of the sales period
To calculate the service level (the probability of not stocking out):
Cs = Cost of shortage = Sales price/unit – Cost/unit
Co = Cost of overage = Cost/unit – Salvage value
Cs
Single Period Example
Chris runs a newsstand who usually sells 120 copies of a specific newspaper. The sales of the newspaper is normally distributed with standard deviation of 15 papers. Chris pays 70 cents for
*
Service level =
The optimal stocking level
= 1 – .578 = .422 = 42.2%