1. Stochastic Inventory Modeling
Chapter 16

2. Assumptions in Deterministic Models
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
Planned shortages are not allowed
Relaxed the 3rd, 4th and 5th assumption

3. 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.

4. 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

5. 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

6. 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

7. Continuous System: Deterministic Model

8. 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

9. Ideal Deterministic Model
Q+S R S

10. 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

11. Reorder Point and Safety Stock

12. 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.

13. 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.

14. Safety Stock and Service Level
0.5
1.0
0.5

15. Reorder Point Assumptions with mean µ and standard deviation .
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.

16. Reorder Point with Variable Demand

17. Reorder Point with Variable Demand

18. Reorder Point with Variable Demand Example
For following data, determine reorder point and safety stock for service level of 95%.

19. Reorder Point with Variable Lead Time
For constant demand and variable lead time:

20. Reorder Point with Variable Lead Time Example
Carpet Discount Store:

21. Variable Demand and Lead Time
Reorder Point
Variable Demand and Lead Time
When both demand and lead time are variable:

22. Variable Demand and Lead Time Example
Reorder Point
Variable Demand and Lead Time Example
Carpet Discount Store:

23. 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

24. 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.

25. Order Quantity for Variable Demand
For normally distributed variable daily demand:

26. Order Quantity for Variable Demand Example
Corner Drug Store with periodic inventory system.
Order size to maintain 95% service level:

27. 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?

28. 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,  = 81
Demand for 4 weeks is normally distributed with:
Mean demand over 4 weeks, µ = 4 x = 240
Variance of demand over 4 weeks,  2 = 4 x 81= 324
Standard deviation over 4 weeks,  = (324)1/2 = 18

29. 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).

30. 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).

31. 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)

32. 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 $ 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?

33. Solution
µ = 12,000
To find , note that z = 1.65 corresponds to a 5% tail probability
Therefore, (20, ,000) = 1.65 or  = 4848
Co=.55 and Cu=.45, (Cu/(Cu+Co))=.45/( )=.45
Find Q * such that P(D < Q *) = .45.
The probability of 0.45 corresponds to z = Thus, Q * = 12, (4848) = 11,418 Jackets

34. 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/( ) = .4091
Find Q * such that P(D < Q *) = z = -.23 gives this probability.
Thus, Q * = 12, (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.