What Is Inventory?
Inventory Management
Three measures of process performance p erformance
Homework (page 602): 13.5, 13.6, 13.19, 13.22, 13.34, 13.41
Little’s Law:
Types of Inventories
Raw materials & purchased parts Partially completed goods called work in process (WIP) Finished-goods inventories (manufa ( manufacturi cturing ng firms ) or merchandise (retail ( retail stores )
Replacement parts, tools, & supplies
Goods-in-transit to warehouses or
Flow Time
Periodic System
Perpetual Inventory System (Continual System) System that keeps track of removals from inventory continuously, thus monitoring current levels of each item
Inventory Flow rate
Inventory turnover
=
1 Flow time
Inventory Counting Systems (Cont’d)
Physical count of items made at periodic intervals
=
Inventory turnover is the ratio o f annual cost of goods sold to average inventory investment. The turnover ratio indicates how many times a year the inventory is sold. The higher the ratio, the better.
Inventory Counting Systems
Average Inventory=average flow rate * average flow time
Inventory Turnover
customers (Pipeline inventory)
Flow rate (throughput rate (throughput rate): the rate at which the process is delivering output Flow Time : the time it takes a flow unit to get through the process Inventory : The number of units contained within the process.
Two-Bin System - Two - Two containers of inventory; reorder when the first is empty (ROP: the amount contained in the 2 nd bin) Universal Bar Code - Bar - Bar code printed on a label that has information about the item 0 to which it is attached 214800 232087768
ABC Classification System
Inventory Costs
Lead time : time interval between ordering and receiving the order 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
Interest, insurance, depreciation, warehousing cost, deterioration, etc.
Preparing invoices, shipping cost, inspecting goods upon arrival, etc.
Shortage costs : costs when demand exceeds supply
Opportunity cost of not making a sale, loss of customer goodwill, late charges.
Form of Pareto analysis (80/20 rule) Classifying inventory according to some measure of importance, usually annual dollar usage, and allocating control efforts accordingly.
A - very important B - mod. important C - least important
High
A
Annual $ volume of items
B C
Low Few
Many
Number of Items
ABC Classification System-Example Item 1 2 3 4 5 6 7 8 9 10 11 12
Demand Unit Cost 1,000 $4,300 5,000 $720 1,900 $500 1,000 $710 2,500 $250 2,500 $192 400 $200 500 $100 200 $210 1,000 $35 3,000 $10 9,000 $3 Total
Dollar Value $4,300,000 $3,600,000 $950,000 $710,000 $625,000 $480,000 $80,000 $50,000 $42,000 $35,000 $30,000 $27,000 $10,929,000
Inventory Models
Fixed order quantity models Economic order quantity Production order quantity Quantity discount
Probabilistic models
Fixed order interval models
Dollar Usuage 39.34% 32.94% 72.28% 8.69% 6.50% 5.72% 4.39% 25.30% 0.73% 0.46% 0.38% 0.32% 0.27% 0.25% 6.81%
Help Helpanswer answer the the inventory inventory planning planning questions! questions!
Basic Inventory Planning Questions
How much to order When to order? Purchase Order Description Qty. Microw ave
Economic Order Quantity (EOQ) Model
1000
Assumptions of EOQ Model
Inventory Cost Components
Purchase cost, P ($/unit) Holding cost, H ($/unit/period) Ordering cost, S ($/order)
An EOQ System Inventory Level
Order size=350 Usage rate=50/day Lead time=2 days
Average Inventory (Q/2)
Order Quantity (Q) (Q=350 )
When the inventory of microwaves gets down to 15 units (reorder point), order 35 units (EOQ).
Purchase Order Description Qty. Microwave 35
left 5
Time
12 14
7
Lead Time
EOQ Model: total cost (Carrying Cost + Ordering Cost)
1 year Low Q
EOQ Model Output Example
15
Reorder Point (ROP) ROP=100 (2 days’ supply)
Known & constant demand Known & constant lead time Demand is even throughout the year Each order is received in a single d elivery There are no quantity discounts No stockouts
Average Inventory (Q/2)
Annual Cost Total Cost Curve
Time
Holding (Carrying) Cost
Many orders produce a low average inventory High Q
Average Inventory (Q/2)
Order (Setup) Cost Optimal Order Quantity (Q*)
Few orders produce a high average inventory
Time
Order Quantity
Carrying costs are linearly related to order size Ordering costs are inversely and nonlinearly related to order size
EOQ Model Equations
Total Cost (Carrying Cost + Ordering Cost) Annual Annual Total cost = carrying + ordering cost cost Q H 2
TC =
H Q*
Q
2
=
=
D Q
+
D S Q
S
=Q
*=
Number of Orders = m =
2 ⋅ D ⋅ S H
D Q* Q*
Expected Time Between Orders = T =
D
Total Cost (Carrying and Ordering Cost) = Total Annual Cost = pD +
Q
*
2
H +
D Q
2 DS
*
Q
*
2
H +
D Q
*
∗
S = Q H
S
D = Demand rate (e.g., per year) S = Setup (ordering) cost per order H = Holding (carrying) cost
H
EOQ Model: Example You’re a buyer for Wal-Mart. WalMart needs 1000 coffee makers per year. The cost of each coffee maker is $78. Ordering cost is $100 per order. Carrying cost is 40% of per unit cost. Lead time is 5 days. Wal-Mart is open 365 days/yr. What is the optimal order quantity & ROP?
D=1000/year
Optimal Order Quantity
S=$100/order
H=0.4*78=$31.2/unit/year
EOQ Solution Optimal Order Quantity:
Q* =
2 ⋅ D ⋅ S H
Number of orders per year: m =
D Q
Length of order cycle: T = Daily usage:
d =
* Q D
=
(2)(1000)(100) = 80 units (.40)(78)
=
∗
=
1000 = 12. 5 ≈ 13 (Orders) 80
80 = 0.08 (Year) = 29.2 (Days) 1000
1000 = 2.74 units/day 365
Reorder Point: ROP = d LT . = (2.74 )(5) = 13.7 units Total Cost: =(Q*/2)*H+(D/Q*)*S =80/2*31.2+13*100=$2548 Inventory Turnover: =1000/40=25
Sensitivity Analysis (Order Quantity) A. As demand increases, EOQ increases in proportion of square root of D rather than in di rect proportion to demand. B. What if the optimal policy is followed, but values of D, S, or h are incorrectly specified? Specifically, suppose we don’t know S, but rather only have ˆ . Given this estimate, we’d compute an order an estimate S quantity which is optimal with respect to the estimate: Qˆ = optimal order quantity based on estimates
Qˆ =
ˆ D 2 S h
Sensitivity Analysis (Total Cost)
Total Cost is not particularly sensitive to the optimal order quantity
T (Q) ∗
T (Q )
=
1 Q ( * 2 Q
+
Q* Q
)
Order Quantit 50% 80% 90% 100% 110% 120% 150% 200% Cost Increase 125% 103% 101% 100% 101% 102% 108 % 125%
EOQ “Zone”
When to Reorder with EOQ Ordering
Annual Cost
Total Cost Curve
Q*
Order Quantity
Reorder Point (ROP) - When the quantity on hand of an item drops to this amount, the item is reordered Safety Stock(SS) - Stock that is held in excess of expected demand due to variable demand rate and/or lead time. Service Level (SL)- Probability that demand will not exceed supply during lead time.
EOQ “Zone”
The total cost curve is relatively flat near the EOQ
Reorder Point Under Uncertainty
Safety Stock Inventory Level Maximum probable demand during lead time Expected demand during lead time
Service level Probability of no stockout ROP
Expected demand
ROP
Time
Quantity
Safety stock
0
Safety Stock LT Place Receive order order
Risk of a stockout
z
z-scale
ROP=Expected demand during lead time+ Safety Stock =Expected demand during lead time+ zσ dLT
ROP Example 1 Suppose that the manager of a construction supply house determined from historical record that demand for sand during lead time average 50 tons. In addition, suppose the manager determined that demand during lead time could be described by a normal distribution that has a mean of 50 tons and a standard deviation of 5 tons. Assuming that the manager is willing to accept a stockout risk of no more than 3%.
3% 5 50 0 What value of z is appropriate?
zz Z=1.88
EXCEL: =NORMSINV(0.97)
How much safety stock should be held?
zσ dLT = 1.88 * 5 = 9.40 What reorder point should be used? ROP=50+9.40=59.40
Uniform Distribution
Levers for Reducing Safety Inventory
A random variable ( X ) between some minimum (a) and maximum (b) value are equally likely.
Reduce demand variability through improved forecasting. Reduce replenishment lead time. Pool safety inventory for multiple locations or products through physical or virtual centralization. Exploit product substitution. Postpone product-differentiation processing until closer to the point of actual d emand
X ~ u ( a, b)
1 b −a
SL
a
ROP
b
Newsboy Problem
Newsboy Model
The Single-period Model (Newsboy problem)
Used to handle ordering of perishables and items that have a limited useful life. (fashion and seasonal apparel, hotel rooms, airline tickets) Example:
Overbooking of airline flights Ordering of fashion items Any type of one-time order.
Example: On consecutive Sunday, Mac, the owner of a local newsstand, purchases a number of copies of the The Computer Journal . He pays 25 cents for each copy and sell each for 75 cents. Copies he has not sold during the week can be returned to his supplier for 10 cents each. The supplier is able to salvage the paper for printing futu re issues. Mac has kept careful records of the d emand each week for the Journal. (This includes the number of copies actually sold plus the number of cust omer requests that could not be satisfied.)
Underage cost & Overage cost
Underage cost (Shortage cost ): the unrealized profit per unit.
C u
C o
=
Revenue per unit – Cost per unit
Overage cost (Excess cost): cost of overstocking =
Original cost per unit – Salvage vale per unit
The Goal of Newsboy Model
Continuous Stocking Level
To identify the order quantity, or stocking level, that will minimize the long-run excess and shortage costs. Demand could be continuous distribution or discrete distribution. Expected marginal benefit from raising order size=Expected marginal cost ) Cu = SL * × Co (1-SL *
The service level is the probability that demand will not exceed the stocking level. C u
∗
Service level (SL ) =
C u
+
C o
Optimal stocking quantity (S o )is then determined
C u
∗
Service level ( SL ) =
C u
+
C o
Example
Sweet cider is delivered weekly to Cindy’s Cider Bar. Demand varies uniformly between 300 and 500 liters per week. Cindy pays 20 cents per liter for the cider and charges 80 cents per liter for it. Unsold cider has no salvage value and can not be carried over into the next week due to spoilage. Find the optimal stocking level and its stockout risks for that quantity.
C o C u
= =
∗
SL
C u C u
+
C o
=
0. 6 = 0 .75 0.6 + 0.2
75% 300
S o
500
S o =300+0.75(500-300)=450 liters
$0.20-$0=0.20 per unit $0.80-$0.20=0.60 per unit
If demand is normal distribution with a mean of 200 liters per week and a standard deviation of 10 liters per week.
Discrete Stocking Level
S o
zσ
= µ +
=200+0.675*10 =206.75 liters
75% 200
=
S o
Computer service level Round-up rule: Whenever you are looking up a target value in a table and the target value falls between two entries, choose the entry that leads to the larger order quantity.
Example
Demand for long-stemmed red roses at a small flower shop can be approximated using a Poisson distribution that has a mean of four dozen per day. Profit on the roses is $3 per dozen. Leftover flowers are marked down and sold the next day at a loss of $2 per dozen. Assume that all marked-down flowers are sold. What is the optimal stocking level? What is the expected profit?
Demand(dozen/day) 0 1 2 3 4 5
Relative Frequency 0.018 0.074 0.146 0.196 0.195 0.156
Cumulative Frequency
…
∗
SL
=
C u C u
+
C o
3
=
3+ 2
=
0.6
0.018 0.092 0.238 0.434 0.629 0.785 …
S o =4 dozens
Expected Profit= -4*2*0.018+ [1*3-3*2]*0.074+
C u
=
$3
C o
=
[2*3-2*2]*0.146+ [3*3-1*2]*0.196+
$2
4*3*(1-0.434) =8.09
Example
A hotel near the university always fills up on the evening before football games. History has sh own that when the hotel is fully booked, the number of last minute cancellations(no shows) is as follows.The average room rate is $80. When the hotel is overbooked, policy is to find a room in a near hotel and to pay for the room for the customer. This usually costs the hotel approximately $200 since rooms booked on such late notice are exp ensive. How many rooms should the hotel overbooked?
Number of No-Shows 0 1 2 3 4 5 6 7 8 9 10 ∗
C u
=
$80
C o
=
$200
SL
=
C u C u
+
C o
Probability 0.05 0.08 0.1 0.15 0.2 0.15 0.11 0.06 0.05 0.04 0.01
=
S o =3 rooms
Cumulative Probability 0.05 0.13 0.23 0.38 0.58 0.73 0.84 0.9 0.95 0.99 1
80 = 0.2857 200 + 80
