Statistical Inventory Models F Newsperson Model: –Single order in the face of uncertain demand...
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Transcript of Statistical Inventory Models F Newsperson Model: –Single order in the face of uncertain demand...
Statistical Inventory Models Newsperson Model:
– Single order in the face of uncertain demand– No replenishment
Base Stock Model: – Replenish one at a time– How much inventory to carry
(Q, r) Model– Order size Q– When inventory reaches r
Issues
How much to order– Newsperson problem
When to order– Variability in demand during lead-time– Variability in lead-time itself
Newsperson Problem
Ordering for a One-time market– Seasonal sales– Special Events
How much do we order?– Order more to increase revenue and
reduce lost sales– Order less to avoid additional
inventory and unsold goods.
Newsperson Problem
Order up to the point that the expected costs and savings for the last item are equal
Costs: Co
– cost of item less its salvage value– inventory holding cost (usually small)
Savings: Cs
– revenue from the sale– good will gained by not turning
away a customer
Newsperson Problem
Expected Savings:– Cs *Prob(d < Q)
Expected Costs:– Co *[1 - Prob(d < Q)]
Find Q so that Prob(d < Q) is
Co
Cs + Co
Example
Savings:– Cs = $0.25 revenue
Costs:– Co = $0.15 cost
Find Q so that Prob(d < Q) is 0.375
0.15
0.25 + 0.15
Finding Q (An Example)
Normal Distribution (Upper Tail)
z0
z 0.00 0.01 0.02 0.03 0.04 0.05 0.060.0 0.50000 0.49601 0.49202 0.48803 0.48405 0.48006 0.476080.1 0.46017 0.45620 0.45224 0.44828 0.44433 0.44038 0.436440.2 0.42074 0.41683 0.41294 0.40905 0.40517 0.40129 0.397430.3 0.38209 0.37828 0.37448 0.37070 0.36693 0.36317 0.35942
Example Continued
If the process is Normal with mean and std. deviation , then (X- )/ is Normal with mean 0 and std. dev. 1
If in our little example demand is N(100, 10) so = 100 and .– Find z in the N(0, 1) table: z = .32– Transform to X: (X-100)/10 = .32
X = 103.2
Extensions
Independent, periodic demands All unfilled orders are backordered No setup costs
Cs = Cost of one unit of backorder one period
Co = Cost of one unit of inventory one period
Extensions
Independent, periodic demands All unfilled orders are lost No setup costs
Cs = Cost of lost sale (unit profit)
Co = Cost of one unit of inventory one period
Base Stock Model
Orders placed with each sale– Auto dealership
Sales occur one-at-a-time Unfilled orders backordered Known lead time l No setup cost or limit on order
frequency
Different Views
Base Stock Level: R– How much stock to carry
Re-order point: r = R-1– When to place an order
Safety Stock Level: s– Inventory protection against variability in lead
time demand– s = r - Expected Lead-time Demand
Different Tacks
Find the lowest base stock that supports a given customer service level
Find the customer service level a given base stock provides
Find the base stock that minimizes the costs of back-ordering and carrying inventory
Finding the Best Trade-off
As with the newsperson– Cost of carrying last item in inventory =– Savings that item realizes
Cost of carrying last item in inventory– h, the inventory carrying cost $/item/year
Cost of backordering– b, the backorder carrying cost $/item/year
Finding Balance
Cost the last item represents:– h*Fraction of time we carry inventory– h*Probability Lead-time demand is less than R– h*P(X < R)
Savings the last item represents:– b*Fraction of time we carry backorders– b*Probability Lead-time demand exceeds R– b*(1-P(X < R))
Choose R so that P(X < R) = b/(h + b)
Customer Service Level
What customer service level does base stock R provide?
What fraction of customer orders are filled from stock (not backordered)?
What fraction of our orders arrive before the demand for them?
What’s the probability that lead time demand is smaller than R?
P(X < R)
Smallest Base Stock
What’s the smallest base stock that provides desired customer service level? e.g. 99% fill rate.
What’s the smallest R so that P(X < R) > .99?
Control Policies Periodic Review
– eg, Monthly Inventory Counts– order enough to last till next review + cushion– orders are different sizes, but at regular intervals
Continuous Review– constant monitoring– (Q, R) policy– orders are the same
size but at irregular intervals
Continuous Review
Time
Inve
ntor
y
Reorder Level
Order Quantity
Safety Stock
Safety Stock
Inventory used to protect against variability in Lead-Time Demand
Lead-Time Demand: Demand between the time the order to restock is placed and the time it arrives
Reorder Point is:
R = Average Lead-Time Demand
+ Safety Stock
Order Quantity
Trade-off – fixed cost of placing/producing order, A– inventory carrying cost, h
A Model
Choose Q and r to minimize sum of– Setup costs– holding costs– backorder costs
Approximating the Costs
Setup Costs– Setup D/Q times per year
Average Inventory is – cycle stock: Q/2 – safety stock: s – Total: Q/2+s
Q/2 + r - Expected Lead-time Demand Q/2 + r -
Estimating The Costs
Backorder Costs– Number of backorders in a cycle
0 if lead-time demand < r x-r if lead-time demand x, exceeds r n(r) = r
(x-r)g(x)dx
– Expected backorders per year n(r)D/Q
The Objective
minimize Total Variable CostAD/Q (Setup cost)h(Q/2 + r - ) (Holding cost)bn(r)D/Q (Backorder cost)
An Answer
Q = Sqrt(2D(A + bn(r))/h) P(XŠ r) = 1 - hQ/bD Compute iteratively:
– Initiate: With n(r) = 0, calculate Q– Repeat:
From Q, calculate r With this r, calculate Q
Another Tack
Set the desired service level and figure the Safety Stock to Support it.
Use trade-off in Inventory and Setups to determine Q (EOQ, EPQ, POQ...)
Variability in Lead-Time Demand
Variability in Lead-Time Variability in Demand X = Xt: period t in lead-time)
Var(X) = Var(Xt)E(LT) + Var(LT)E(Xt)2
s = z*Sqrt(Var(X)) Choose z to provide desired level
of protection.
Safety Stock
Analysis similar to Newsperson problem sets number of stockouts:– Savings of Inventory carrying cost– Cost of One more item short each time we
stocked out
Co =Stockouts/period* Cs
Stockouts/period = Co / Cs
Example
Safety Stock of Raw Material X– Cost of Stocking out?
Lost sales Unused capacity Idle workers
– Cost of Carrying Inventory Say, 10% of value or $2.50/unit/year
– Number of times to stock out:
2.50/2,500,000 or 1 in a million (exaggerated)
Example Assuming:
– Average Demand is 6,000/qtr (~ 92/day)– Variance in Demand is 100 units2/qtr (1.5/day)– Average Lead Time is 2 weeks (10 days)– Variance in Lead-Time is 4 days2
– Lead-Time Demand is normally distributed E(X) = 92*10 = 920 Var(X) = 1.5*10 + 4*(8464)
~ 34,000
Example
Look up 1 in a million on the Normal Upper Tail Chart– z ~ 4.6
Compute Safety Stock– s = 4.6*Sqrt(34,000) = 4.6*184 = 846
Compute Reorder Point– r = 920 + 846 = 1,766
Other Issues
Why Carry Inventory? How to Reduce Inventory? Where to focus Attention?
Why Carry Inventory?
Buffer Production Rates From:– Seasonal Demand– Seasonal Supplies
“Anticipation Inventory”
Other Types of Inventory
“Decoupling Inventory”– Allows Processes to Operate Asynchronously– Examples:
DC’s “decouple” our distribution from individual
customer orders Holding tanks “decouple” 20K gal. syrup mixes
from 5gal. bag-in-box units.
Other Types of Inventory
“Cycle Stock”– Consequence of Batch Production– Used to Reduce Change Overs:
8 hours and 400 tons of “red stripe” to change Pulp Mill from Hardwood to Pine Pulp
4 hours to change part feeders on a
Chip Shooter
Reduce Setup Time!
Other Types of Inventory
“Pipeline Inventory”– Goods in Transit – Work in Process or WIP– Allows Processes to be in Different Places– Example:
Parts made in Mexico, Taurus
Assembled in Atlanta
Other Types of Inventory
“Safety Stock”– Buffer against Variability in
Demand Production Process Supplies
– Avoid Stockouts or Shortages
Using Inventory
Inventory Finished Goods or Raw Materials? Inventory at Central Facility or at DCs? Extremes:
– High Demand, Low Cost Product– Low Demand, High Cost Product
Reducing Inventory
Reducing Anticipation Inventories– Manage Demand with Promotions, etc.– Reduce overall seasonality through product mix– Expand Markets
Reducing Inventory
Reducing Cycle Stock– Reduce the length of Setups
Redesign the Products Redesign the Process
– Move Setups Offline
– Fixturing, etc.
– Reduce the number of Setups Narrow Product Mix Consolidate Production
Reducing Inventory
Reducing Pipeline Inventory– Move the Right Products, eg, Syrup not Coke– Consolidate Production Processes– Redesign Distribution System– Use Faster Modes
Reducing Inventory
Reducing Safety Stock– Reduce Lead-Time– Reduce Variability in Lead-Time– Reduce the Number of Products– Consolidate Inventory
ABC Analysis
Where to focus Attention:Dollar Volume = Unit Price * Annual Demand
– Category A: 20% of the Stock Keeping Units (SKU’s) account for 80% of the Dollar Volume
– Category C: 50% of the SKU’s with
lowest Dollar Volume– Category B: Remaining 30% of
the SKU’s