Inventory Control and Management
Bonnie L. Robeson, Ph.D., M.A.S.
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Course Roadmap
1. Operations Strategy
2. Process Analysis and Design
3. Managing Flows
4. Managing Variability
5. Inventory Management
Newsvendor Model
ROP Model ≈ EOQ + Newsvendor
6. Toyota Production System
7. Operations in a Networked World
What’s your inventory experience
at your most recent work?
We did not have inventory of any kind.
We had inventory, but I had nothing to do with it.
I interacted with inventory (e.g., used supplies of spare parts) but was not responsible for managing it.
I was directly involved in managing inventory (e.g., through procurement, IT, stocking policies, etc.).
Inventory: A stock of materials used to facilitate production or satisfy customer demands.
Types of inventory
Raw materials, purchased parts (RM)
Work in process (WIP)
Finished goods (FG)
Definitions
Working stock: inventory being actively processed
Congestion stock: inventory that builds up unintentionally due to variability
Cycle stock: inventory resulting from batch operations
Safety stock: inventory intentionally held to buffer variability
Anticipation stock: inventory built up
in expectation of future demand
Types of Inventory
To protect against uncertainties – demand, supply, lead times, schedule changes
Safety stock
To allow economic production and purchase
Cycle inventory
To cover anticipated changes in demand/supply
Anticipation inventory
To provide for transit
Pipeline inventory
Purpose of Inventory
Rank items according to annual dollar usage
Class A: top 5-10% of items that constitute 50% or more of total annual usage (in dollars)
Use most sophisticated inventory tools, plus possibly individualized attention
Class B: next 50-70% of items that account for most of the remaining dollar usage
Use formal tools, but not individualized management
Class C: remaining 20-40% of items that represent
only a minor portion of total dollar usage
Use simple tools (e.g., 2-bin system); don’t permit
disruptions due to stockouts
ABC Classification
Based on Pareto concept (80/20 rule) and total usage in dollars of each item.
Classification of A, B, and C items based on usage.
Purpose is to set effort priorities to manage different SKUs, i.e., to allocate scarce management resources.
ABC Inventory Analysis
A items: 20% of SKUs, 80% of dollars
B items: 30% of SKUs, 15% of dollars
C items: 50% of SKUs, 5% of dollars
Three classes is arbitrary; could be any number.
Percentages are approximate.
Dollar use may not reflect importance of a particular SKU! Some critical but low value items may be classified as A.
A B C Classification
Annual Usage of Items by Dollar Value
Cost vs Service
Basic Inventory Tradeoff
Service (speed)
Make to Stock System
Make to Order System
Variability in demand and/or production degrades service
Inventory can buffer that variability … at a cost
Say you are shopping at Trade Joe’s, and are deciding between
What are key considerations?
Inventory Management, OR
Bananas, Bananas, Bananas: Bananas, Bananas
Inventory Models
Newsvendor
When to order?
How much safety stock to carry?
When to use pooling/postponement?
What service level to set?
Inventory Models
All models are wrong. Some are useful.
Introduced in 1913 by Ford W. Harris, “How Many Parts to Make at Once”
Think about grocery shopping
Each trip costs time and gas, so you do not want to shop 7 times a week
Now, if you shop once a month, you save on gas and time, but need a huge refrigerator. Plus, your grocery items could spoil.
Tradeoff: Fixed Costs versus Inventory Costs
Question: Compared to a normal time, have you shopped for groceries more or less frequently during the pandemic?
The EOQ Model
Fixed Costs:
The fixed costs of systems and labor required to place, received, and audit a replenishment order. Transport cost is included only if it is fixed per order
Inventory Costs:
The per-unit cost of holding an item in inventory for a specific period, including costs to finance the inventory, labor to maintain storage facilities, rent, utilities, taxes, and insurance.
Cost of Inventory Policy
What is the optimal inventory policy if there is no fixed cost?
Always order one unit of product
What is the optimal inventory policy if there is no inventory cost?
Order the maximum allowable quantity!
Extreme Cases
Ordering handle caps for Xootr Scooters
$0.85 = cost to Nova Cruz to purchase each handle cap from its supplier in Taiwan.
$300 = customs fee for each shipment, independent of the amount ordered.
700 = demand for handle caps per week.
Note, each “handle cap” is actually a pair, so one is needed per Xootr.
40% = Nova Cruz’s annual inventory holding cost.
How many handle caps should they order each time they order from their supplier?
The inventory “saw-tooth” pattern
Shipment arrives
Shipment arrives
Assume we can time when the shipments arrive so that they arrive when we have zero inventory.
Q = Quantity in each order (what we need to choose)
R = Flow Rate of demand (700 per week)
Q / R = Time between shipments
R / Q = Frequency of placing orders
Expressed as cost per unit or SKU
Quantity discounts possible
Ordering (or setup) cost
Paperwork, electronic entry, worker time for ordering
Worker time for setup, downtime
Transportation costs
Typically a fixed cost per order (or setup)
Cost of Inventory
Carrying (or holding) cost
Cost of capital (market rate or internal rate of return)
Cost of storage (space, insurance, taxes)
Cost of obsolescence, deterioration, and loss (shrinkage)
Estimated U.S. average is 35% of SKU cost per year.
Businesses often use cost of capital (understated).
Stockout cost
Back order costs (expressed as a fixed cost per backorder or as a function of aging of backorders)
Lost income
Customer dissatisfaction; loss of future sales
Cost of Inventory
Cost of capital: 9-20%
Obsolescence: 2-5%
Storage: 2-5 percent
Material handling: 1-3%
Shrinkage: 1-3%
Taxes & insurance: 1-3%
Distribution of Carrying Cost
Answers question: “How much should we order?”
Used for independent demand items.
Objective is to find order quantity (Q) that minimizes total cost (TC) of managing inventory.
Must calculate for each SKU.
Widely used and very robust (i.e., works well in a variety of situations, even when its assumptions do not perfectly hold).
Economic Order Quantity
Demand rate is constant, recurring, and known.
Lead time is constant and known.
No stockouts allowed.
Items are ordered or produced in a lot or batch, and the lot is received all at once.
Costs are constant:
Unit cost (no quantity discounts).
Carrying cost is constant per unit.
Ordering (setup) cost per order.
Item is a single product or SKU; demand not influenced by other items.
EOQ Assumptions
Purchase costs:
$0.85 per unit × 700 per week = $595 per week
Q cannot influence our weekly purchase cost!
h = Inventory holding cost per unit time:
40% annual holding cost, so …
0.4 × $0.85 = $0.34 = cost to hold a unit for one year…
h = $0.34 / 52 = $0.006538 = cost to hold a unit for one week.
Average inventory = Q / 2
Average inventory cost per unit time = h × Q / 2
K = Setup cost:
This is the cost per order and it is independent of the amount ordered.
R / Q = frequency of placing orders, so …
Setup cost per unit time = K × R / Q
Independent demand (this chapter)
Finished goods, spare parts
Based on market demand, independent of other items
Requires forecasting
Dependent demand
Components/parts of the finished (parent) products
Demand is a known function of (parent) independent demand items
Calculate instead of forecast
Types of Demand
Trade-off between ordering frequency (i.e., order size) and inventory level.
Frequent orders (small lot sizes) lead to lower average inventory level, i.e., higher total ordering costs and lower total holding costs.
Less frequent orders (large lot sizes) lead to higher average inventory level, i.e., lower total ordering costs and higher total holding costs.
EOQ Lot size intuition
D = Demand rate, units per year
S = Cost per order placed or setup cost, dollars per order
C = Unit cost, dollars per unit
i = Carrying rate, percent of dollar value per year
Q = Lot size, units
TC= Total of ordering cost plus carrying cost, dollars per year
Notations in EOQ Calculations
Ordering cost per year =
(cost per order) x (orders per year)
Carrying cost per year =
(annual carrying rate) x (unit cost) x (average inventory level)
Total annual cost (TC) =
ordering cost per year + carrying cost per year
= SD/Q + iCQ/2
Cost equations in EOQ
Total Cost of Inventory
Objective and solution
Objective:
Choose Q to minimize the average (setup and holding) cost per unit time, C(Q):
Order the Economic Order Quantity (EOQ) = Q*
Setup costs
Holding costs
Costs for Xootr handle caps
Heavy Metal Industries uses quarter inch stainless steel bolts, 1½ inches long, at a steady rate of 100 per week (50 weeks/year)
Bolts cost $0.02 each
HMI uses a carrying cost rate of 25% per year
It costs $12.50 to place an order of bolts.
How many bolts should HMI order at a time?
Practice Problem
Please indicate your answer
none of the above
R = 100 units/week
h = 25% × 0.02 / 50 = $0.0001 per week
R = 100 units/week = 5,000 units/year
h = 25% × 0.02 = $0.005 per year
Solution: A Different Way
What’s the ordering frequency?
What’s the total setup costs per year?
What’s the total inventory costs per year?
Assumptions:
1. There is one single period to consider
2. Demand is random but its distribution is known
3. Overage and underage costs are linear
4. The firm wants to minimize the total expected cost
newspapers or other items with rapid obsolescence
Christmas trees or other seasonal items
fashion items
capacity for short-life products
The Newsvendor (aka Newsboy) Model
Newsvendor Model Notation
unit cost of underage
Marginal Analysis
P (MP) >= ML –MP
P (MP) + P (ML) >= ML
P (MP + ML) >= ML
P >= ML
Donuts sell for $6 per carton
Purchase Donuts for $4 per carton
There is no salvage value
P >= 4/ (4+2) = .66
Calculation
Daily Cartons sold Probability Cumulative Prob
4 .05 1.00 >= .66
5 .15 .95 >=.66
6 .15 .80 >=.66
10 .10 .10
Café du Donut’s
Objective: Minimize
expected costs of overage
expected costs of underage
Solution Approach:
Look at marginal cost of xth unit of inventory.
If expected overage cost exceeds expected underage cost, then we should order less than x
If expected shortage cost exceeds expected
overage cost, we should order more than x
Optimum order quantity occurs when
expected cost of overage = expected cost of underage
Solution Approach
Newsvendor Problem
Rocky pays $0.5 for each paper, and sells them for $1.50
Daily newspaper demand distribution:
Demand: 87 88 89 90 91 92 93
Probability: 0.03 0.07 0.2 0.4 0.2 0.07 0.03
If Rocky buys 87 papers, profit = $___
Should Rocky buy 88? Let’s see…
Will sell 87 papers for sure, profit = $87
With probability _____, the 88th paper will not be sold, and it costs Rocky $_____
With probability _____, the 88th paper will be sold and brings Rocky profit of $_____
Cost < > Benefit?
Should Rocky buy the 88th paper?
Marginal Analysis
0.03 × $0.5 = $0.015 < 0.97 × $1 = $0.97
Absolutely!
With probability _____, the 89th paper will not be sold, and it costs Rocky $_____
With probability _____, the 89th paper will be sold and brings Rocky profit $_____
Cost < > Benefit?
Should Rocky buy the 89th paper?
Marginal Analysis
0.1 × $0.5 = $0.050 < 0.9 × $1 = $0.90
Definitely!
With probability _____, the 90th paper will not be sold, and it costs Rocky $_____
With probability _____, the 90th paper will be sold and brings Rocky profit $_____
Cost < > Benefit?
Should Rocky buy the 90th paper?
Marginal Analysis
0.3 × $0.5 = $0.15 < 0.7 × $1 = $0.70
Of course!
With probability _____, the 91st paper will not be sold, and it costs Rocky $_____
With probability _____, the 91st paper will be sold and brings Rocky profit $_____
Cost < > Benefit?
Should Rocky buy the 91th paper?
Optimal quantity = _______ newspapers
Marginal Analysis
0.7 × $0.5 = $0.35 > 0.3 × $1 = $0.30
Pays 0.5 sells for 1.5 Payoff Table
Buy 87 88 89 90 91 92 93
87 0.03 87 $86.50 $86.00 $85.50 $85.00 $84.50 $84.00
88 0.07 87 $88.00 $86.50 $87.00 $86.50 $86.00 $85.50
89 0.2 87 $88.00 $89.00 $88.50 $88.00 $87.50 $87.00
90 0.4 87 $88.00 $89.00 $90.00 $89.50 $89.00 $88.50
91 0.2 87 $88.00 $89.00 $90.00 $91.00 $90.50 $90.00
92 0.07 87 $88.00 $89.00 $90.00 $91.00 $92.00 $91.50
93 0.03 87 $88.00 $89.00 $90.00 $91.00 $92.00 $93.00
Expected Profit 87 87.955 88.735 $89.34 $62.01
Payoff Table
Use Cumulative Probability
Probability >= ML/ MP +ML
Probability = .5/ 1 + .5 = 3.33
Marginal Analysis
Co = Cost of over-stocking a unit
Cu = Cost of under-stocking a unit
Buy the x+1st unit, as long as its cost < benefit:
P(D x) Co < P(D > x) Cu or equivalently
P(D x) < Cu/(Cu + Co)
Optimal order quantity, Q* is the smallest value for which
P(D Q*) Cu/(Cu + Co)
Q* is the smallest quantity that service level is at least Cu/(Cu + Co)
Marginal Analysis: Generalized
The “Critical Ratio”
P(DQ) Co < (1 – P(DQ)) Cu
0.03 × ($–0.5) + 0.97 × $1 = $ 0.985
0.1 × ($–0.5) + 0.9 × $1 = $ 0.85
0.3 × ($–0.5) + 0.7 × $1 = $ 0.55
0.7 × ($–0.5) + 0.3 × $1 = $ –0.05
Optimal Order Quantity
If we approximate demand with a continuous distribution G(x), then
The optimum order quantity makes these two costs equal
Sensitivity of Order Quantity
to Cost Parameters
Optimal order quantity increases in underage cost (Cu)
Optimal order quantity decreases in overage cost (Co)
JIT is a form of inventory management that requires working closely with suppliers so that raw materials arrive as production is scheduled to begin, but no sooner. The goal is to have the minimum amount of inventory on hand to meet demand.
Just in time inventory
https://www.youtube.com/watch?v=cAUXHJBB5CM
Just in Time
Finish individual and team registration by Tuesday (3/1) 4:30pm
Follow the instructions on Blackboard under the folder for the game
Suggested team names: using your team initials (lower case).
E.g., Abesamis-Hill-Hoaglund-Laber ahhl
Littlefield Registration
Note: Q* increases in both and if z is positive (i.e., if ratio is greater than 0.5)
If demand is normally distributed with mean and standard deviation , then the critical ratio formula reduces to:
Newsvendor Model with Normal Demand
Probability there is enough stock to meet demand (i.e., service level)
Demand for T-shirts is normal with mean 1000 and standard deviation 250.
Cost of shirts is $10.
Selling price is $15.
Unsold shirts can be sold off at $8.
Model Parameters:
Cu = 15 – 10 = $5
Co = 10 – 8 = $2
Newsvendor Example – T Shirts
T-Shirt Example: Solution
We can compute the optimal number of T-shirts to order example as follows:
Since shortages are worse than overages, we want the probability of meeting demand to be 71.4% (i.e., greater than 50-50). To achieve this, we must buy more than the expected demand quantity.
If leftover T-shirts must be discarded, then Co=$10, and
T-Shirt Example: Sensitivity
Since we lose more money when we over-buy, we should purchase fewer T-shirts. To make the probability of having enough inventory to meet demand only equal 1/3, we purchase less than the expected demand.
If We Double Standard Deviation of Demand
a system with Cu=1, Co=1
a system with Cu=1, Co=2
a system with Cu=2, Co=1
a system with Cu=2, Co=2
Which system will experience the largest increase in the optimal order quantity?
Probability of stockout during lead time
Probability of no stockout during lead time
How Much Safety Stock to Hold?
Distribution of demand during lead time
Cycle service (CS) = fraction of order cycles that do not stock out
Probability of stockout during lead time
Probability of no stockout during lead time
How Much Safety Stock to Hold?
Distribution of demand during lead time
Note: z measures distance from mean in terms of standard deviation, .
This will allow us to use z-tables!
In this example, z=5.
If lead time demand is normally distributed with a mean of = 1,200 units and std dev of = 300 units, what should ROP be in order to achieve CS = 0.5?
Practice Problem
ROP = + z, where (z)=CS
Since (0) = 0.5, we need z=0 to achieve CS =0.5.
Hence, ROP = + z = 1200 + 0(300) = 1200
If lead time demand distribution is normal(,), then
ROP* = + z
where (z)=Cu/(Cu + Co).
Note: ROP* increases in and also increases in provided z>0.
Reorder Point Formula
when Demand is Normal
z is the “safety factor”
z is the “safety stock”
How would you estimate b in practice?
In contrast to the Newsvendor model in which a backorder cost was clearly the cost of a lost sale (because the items were perishable), in this case we might have customers that stick around and get the product later. In such a case, what is the cost of not having the item? The cost of their waiting for the order to come in?
COVID-19 Vaccination Challenge writeup due one hour before next class
Littlefield Game
Starting time: 3/3 (Thursday) 12pm
Ending time: 3/11 (Thursday) 11:59am
Will last 7 days and 24 hours every day
A practice run?
Let’s plan to start it Tuesday (3/1)
Group writeup due one hour before Class 7 (3/10)
Until Next Class
UnitsUnit Cost
% of Total
15,000 1.50$ 7,500$ 2.9%
21,500 8.00 12,000 4.7%
310,000 10.50 105,000 41.2%
46,000 2.00 12,000 4.7%
57,500 0.50 3,750 1.5%
66,000 13.60 81,600 32.0%
75,000 0.75 3,750 1.5%
84,500 1.25 5,625 2.2%
97,000 2.50 17,500 6.9%
103,000 2.00 6,000 2.4%
Total254,725$ 100.0%
Item Annual Usage in Units Unit Cost Dollar Usage % of Total Dollar Usage
1 5,000 $ 1.50 $ 7,500 2.9%
2 1,500 8.00 12,000 4.7%
3 10,000 10.50 105,000 41.2%
4 6,000 2.00 12,000 4.7%
5 7,500 0.50 3,750 1.5%
6 6,000 13.60 81,600 32.0%
7 5,000 0.75 3,750 1.5%
8 4,500 1.25 5,625 2.2%
9 7,000 2.50 17,500 6.9%
10 3,000 2.00 6,000 2.4%
Total $ 254,725 100.0%
Item Annual Usage in Units Unit Cost Dollar Usage Percentage of Total Dollar Usage
1 5,000 $ 1.50 $ 7,500 2.9%
2 1,500 8.00 12,000 4.7%
3 10,000 10.50 105,000 41.2%
4 6,000 2.00 12,000 4.7%
5 7,500 0.50 3,750 1.5%
6 6,000 13.60 81,600 32.0%
7 5,000 0.75 3,750 1.5%
8 4,500 1.25 5,625 2.2%
9 7,000 2.50 17,500 6.9%
10 3,000 2.00 6,000 2.4%
Total $ 254,725 100.0%
Order Quantity
C(Q) = cost per week
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