Stochastic Inventory Modeling
Chapter 16
Assumptions in Deterministic Models
1.
2.
3.
4.
Demand is known and constant
Lead time is known and constant
Order quantity does not depend on price
Order quantity arrives all at once when
needed
5. Planned shortages are not allowed
– Relaxed the 3rd, 4th and 5th assumption
Independent Demand
• Inventory exists to meet the demand of customers.
• Customers can be external (purchasers of products) or
internal (workers using material).
• Management needs accurate forecast of demand.
• Items that are used internally to produce a final product are
referred to as dependent demand items.
• Items that are final products demanded by an external
customer are independent demand items.
Deterministic and Stochastic
Models
• If demand and lead time are known
(constant), they are called deterministic
models
• If they are treated as random (unknown),
they are stochastic
• Each random variable can have a
probability distribution
• Attention is focused on the distribution of
demand during the lead time
Inventory Control Systems
• Three basic types of systems:
• continuous model (fixed-order quantity), periodic
model (fixed-time), and Single-Period model
• Continuous system: an order is placed for the
same constant amount when inventory decreases
to a specified level.
• Periodic system: an order is placed for a variable
amount after a specified period of time.
• Single-period system: an order is placed just for
one period
Continuous Inventory Systems
• Continual record of inventory level is maintained.
• Whenever inventory decreases to a predetermined
level, the reorder point, an order is placed for a
fixed amount
• Management must be aware of status of inventory
level
• System is relatively expensive to maintain
Continuous System: Deterministic Model
Reorder Point
• The reorder point is the inventory level at which a new
order is placed.
• Order must be made while there is enough stock in place to
cover demand during lead time.
• Formulation:
•R = dL
•where d = demand rate per time period
•
L = lead time
Ideal Deterministic Model
Q+S
R
S
Continuous System: Stochastic Model
• Inventory level might be depleted at slower or faster rate
during lead time.
• When demand is uncertain, safety stock is added as a
hedge against stockout.
• Focus must be on demand distribution during the lead time
Reorder Point and Safety Stock
Reorder Point Quantity
• Under deterministic conditions, when both
demand and lead time are constant, the
reorder point is set equal to lead time
demand.
• Under probabilistic conditions, when
demand and/or lead time varies, the reorder
point often includes safety stock
• Safety stock is the amount by which the
reorder point exceeds the expected
(average) lead time demand.
Safety Stock and Service Level
• Safety stock determines the chance of a stockout
during lead time
• The complement of this chance is called the service
level
• Service level is defined as the probability of not
incurring a stockout during any one lead time
• The higher the probability inventory will be on hand,
the more likely customer demand will be met.
• Service level of 90% means there is a .90
probability that demand will be met during lead time
and .10 probability of a stockout.
Safety Stock and Service Level
S
Service Level
0.5
0.5
1.0
Reorder Point
• Assumptions
– Lead-time demand is normally distributed
with mean µ and standard deviation .
– Approximate optimal order quantity: EOQ
– Service level is defined in terms of the
probability of no stockouts during lead time
and is reflected in z.
– Shortages are not backordered.
– Inventory position is reviewed continuously.
Reorder Point with Variable Demand
Reorder Point with Variable Demand
R  d L  Z d L
where:
R  reorder point
d  average daily demand
L  lead time
 d  the standard deviation of daily demand
Z  number of standard deviations correspond ing
to service level probability
Z d L  safety stock
Reorder Point with Variable Demand Example
• For following data, determine reorder point and safety stock
for service level of 95%.
d  30 yd per day
L  10 days
d  5 yd per day
For 95% service level, Z  1.65 (Table A -1, appendix A )
R  d L  Zd L  30(10)  (1.65)(5)( 10 )  300  26.1
 326.1 yd
Safety stock is second term in reorder point formula : 26.1.
Reorder Point with Variable Lead Time
• For constant demand and variable lead time:
R  d L  Zd L
where:
d  constant daily demand
L  average lead time
 L  standard deviation of lead time
d L  standard deviation of demand during lead time
Zd L  safety stock
Reorder Point with Variable Lead Time Example
• Carpet Discount Store:
d  30 yd per day
L  10 days
 L  3 days
Z  1.65 for a 95% service level
R  d L  Zd L  (30)(10)  (1.65)(30)(3)  300 148.5  448.5 yd
Reorder Point
Variable Demand and Lead Time
• When both demand and lead time are variable:
2
2 2
R  d L  Z ( d ) L  ( L) d
where:
d  average daily demand
L  average lead time
2
2 2
( d ) L  ( L) d  standard deviation of demand during lead time
2
2 2
Z ( d ) L  ( L) d  safety stock
Reorder Point
Variable Demand and Lead Time Example
• Carpet Discount Store:
d  30 yd per day
 d  5 yd per day
L  10 days
 L  3 days
Z  1.65 for 95% service level
2
2 2
R  d L  Z ( d ) L  ( L ) d
 (30)(10)  (1.65) (5)(5)(10)  (3)(3)(30)(30)
 300  150.8
 450.8 yds
Periodic Inventory Systems
• Inventory level (on hand) is counted at specific time
intervals
• An order placed that brings inventory up to a specified level
• Less costly to track of inventory level
• Requires a new order quantity each time an order is placed
• Used in smaller retail stores, drugstores, grocery stores and
offices
Periodic Review Order Quantity
• Assumptions
– Inventory position is reviewed at constant intervals.
– Demand during review period plus lead time period
is normally distributed with mean µ and standard
deviation .
– Service level is defined in terms of the probability of
no stockouts during a review period plus lead time
period and is reflected in z.
– On-hand inventory at ordering time: H
– Shortages are not backordered.
– Lead time is less than the review period length.
Order Quantity for Variable Demand
• For normally distributed variable daily demand:
Q  d (tb  L)  Z d tb  L  I
where:
d  average demand rate
tb  the fixed time between orders
L  lead time
 d  standard deviation of demand
Z d tb  L  safety stock
I  inventory in stock
Order Quantity for Variable Demand Example
• Corner Drug Store with periodic inventory system.
• Order size to maintain 95% service level:
d  6 bottles per day
 d  1.2 bottles
tb  60 days
L  5 days
I  8 bottles
Z  1.65 for 95% service level
Q  d (tb  L)  Z d tb  L  I
 (6)(60  5)  (1.65)(1.2) 60  5 8
 398 bottles
Example: Ace Brush
• Joe Walsh is a salesman for the Ace Brush Company.
Every three weeks he contacts Dollar Department
Store so that they may place an order to replenish
their stock. Weekly demand for Ace brushes at Dollar
approximately follows a normal distribution with a
mean of 60 brushes and a standard deviation of 9
brushes.
• Once Joe submits an order, the lead time until Dollar
receives the brushes is one week. Dollar would like
at most a 2% chance of running out of stock during
any replenishment period. If Dollar has 75 brushes in
stock when Joe contacts them, how many should
they order?
Example: Ace Brush
• The review period plus the lead time totals 4 weeks
• This is the amount of time that will elapse before the
next shipment of brushes will arrive
• Weekly demand is normally distributed with:
Mean weekly demand, µ
= 60
Weekly standard deviation,  = 9
Weekly variance,  2
= 81
• Demand for 4 weeks is normally distributed with:
Mean demand over 4 weeks, µ = 4 x 60 = 240
Variance of demand over 4 weeks,  2 = 4 x 81= 324
Standard deviation over 4 weeks,  = (324)1/2 = 18
Single-Period Order Quantity
• A single-period order quantity model
(sometimes called the newsboy problem)
deals with a situation in which only one order
is placed for the item and the demand is
probabilistic.
• If the period's demand exceeds the order
quantity, the demand is not backordered and
revenue (profit) will be lost.
• If demand is less than the order quantity, the
surplus stock is sold at the end of the period
(usually for less than the original purchase
price).
Single-Period Order Quantity
• Assumptions
– Period demand follows a known probability
distribution:
• normal: mean is µ, standard deviation is 
• uniform: minimum is a, maximum is b
– Cost of overestimating demand: $co
– Cost of underestimating demand: $cu
– Shortages are not backordered.
– Period-end stock is sold for salvage (not
held in inventory).
Single-Period Order Quantity
• Formulas
Optimal probability of no shortage:
P(demand < Q *) = cu/(cu+co)
Optimal probability of shortage:
P(demand > Q *) = 1 - cu/(cu+co)
Optimal order quantity, based on demand
distribution:
normal:
Q * = µ + z
uniform:
Q * = a + P(demand < Q *)(b-a)
Example: McHardee Fashion
• McHardee Fashion produces a jacket and
wishes to determine how many units to
manufacture. There is a fixed cost of $5,000
to produce the item and the incremental
profit per unit is $0.45. Any unsold items can
be sold at salvage at a $.55 loss. Sales for
this item are estimated to be normally
distributed. The most likely sales volume is
12,000 units and they believe there is a 5%
chance that sales will exceed 20,000. How
many units should be printed?
Solution
• µ = 12,000
• To find , note that z = 1.65 corresponds to a 5%
tail probability
• Therefore, (20,000 - 12,000) = 1.65 or  = 4848
• Co=.55 and Cu=.45, (Cu/(Cu+Co))=.45/(.45+.55)=.45
• Find Q * such that P(D < Q *) = .45.
• The probability of 0.45 corresponds to z = -.12.
Thus, Q * = 12,000 - .12(4848) = 11,418 Jackets
Solution-Cont.
• If any unsold copies can be sold at salvage at a
$.65 loss, how many units should be produced?
• Co=.65, (Cu/(Cu+Co))=.45/(.45+.65) = .4091
• Find Q * such that P(D < Q *) = .4091. z = -.23
gives this probability.
• Thus, Q * = 12,000 - .23(4848) = 10,885 units
• However, since this is less than the breakeven
volume of 11,111 jackets (= 5000/.45), no item
should be produced because if the company
produced only 10,885 units it will not recoup its
$5,000 fixed cost.