1. Chapter 5 Inventory Control Subject to Uncertain Demand

2. Timing Decisions Quantity decisions made together with decision
When to order?
One of the major decisions in management of the inventory systems.
Impacts: inventory levels, inventory costs, level of service provided
Models:
One time decisions
Continuous review systems
Periodic review systems

3. Timing Decisions Intermittent-Time Decisions Continuous Decisions
One-Time
Continuous Review
System
Periodic Review
Systems
EOQ, EPQ
(Q, R) System
Base Stock
Two Bins
(s, S) System
Optional
Replenishment
(S, T) System
EOQ
Structure of timing decisions

4. One-Time Decision
Situation is common to retail and manufacturing environment
Consider seasonal goods, which are in demand during short period only.
Product losses its value at the end of the season. The lead time can be longer than the selling season  if demand is higher than the original order, can not rush order for additional products.
Example
newspaper stand
Christmas ornament retailer
Christmas tree
finished good inventory
Trivial problem if demand is known (deterministic case), in practical situations demand is described as random variable (stochastic case).
“newsvendor” model or
“Christmas tree” model

5. Example: One-Time Decision
Mrs. Kandell has been in the Christmas tree business for years. She keeps track of sales volume each year and has made a table of the demand for the Christmas trees and its probability (frequency histogram).
Solution:
Q – order quantity; Q* - optimal
D – demand: random variable with probability density function f(D)
F(D) – cumulative probability function:
F(D) = Pr (demand ≤ D)
co – cost per unit of positive inventory
cu – cost per unit of unsatisfied demand
Economics marginal analysis:
overage and underage costs are balanced
Demand, D
Probabilityf(D) 22
0.05 24
0.10 26
0.15 28
0.20 30 32 34 36

6. The Concept of Marginal Analysis
finding the expected profit of ordering one more unit.
Probability of not selling
Your Last item in stock and having extra inventory on hand at the end on the period
P(X < Q)
Probability of
Selling everything, and facing shortage
P(X ≥ Q) Q m

7. Critical ratio for the newsvendor problem
P(X<Q)
(Co applies)
P(X>Q)
(Cu applies)
Single Period Inventory Model Marginal Analysis:
E (revenue on last sale) = E (loss on last sale)

8. Example: One-Time Decision (cont)
Shortages = lost profit + lost of goodwill
Overage = unit cost + cost of disposal of the overage
Either ignore the purchase cost, because it does not impact the optimal solution or implicitly consider it in the overage and underage costs.
Expected overage cost of the order Q* is
P(Demand < Q*) co = F(Q*)co
Expected shortage cost is
P(Demand > Q*) cu = (1-F(Q*)) cu
For order Q* those two costs are equal: F(Q*)co = (1-F(Q*))cu
- probability of satisfying demand during
the period, also is known as critical ratio
To calculate Q* we must use cumulative probability distribution.

9. Example: One-Time Decision (cont.)
Mrs. Kandell estimates that if she buys more trees than she can sell, it costs about $40 for the tree and its disposal. If demand is higher than the number of trees she orders, she looses a profit of $40 per tree.
Demand D
Probability f(D)
Cum Probability
F(D) 22
0.05 24
0.10
0.15 26
0.30 28
0.20
0.50 30
0.70 32
0.85 34
0.95 36
1.00

10. Critical fractile for the newsvendor problem
When the demand is a discrete random variable, the condition
may not be satisfied at equality (jumps, due to discreteness). Here’s the appropriate condition to use: o u c Q F + = ) ( } ) ( :
min{ * o u X c Q P + =
C(Q+1)-C(Q)=(cu+ co)PX(Q) – cu ≥ 0

11. Single-Period & Discrete Demand: Lively Lobsters
Lively Lobsters (L.L.) receives a supply of fresh, live lobsters from Maine every day. Lively earns a profit of $7.50 for every lobster sold, but a day-old lobster is worth only $ Each lobster costs L.L. $14.50
unit cost of a L.L. stockout
Cu = = lost profit
unit cost of having a left-over lobster
Co = = cost – salvage value = 6
target L.L. service level
CR = Cu/(Cu + Co) = 7.5 / ( ) = .56
Demand follows a discrete (relative frequency) distribution as given on next page

12. Lively Lobsters } ) ( : min{ c Q P + ³ = Demand follows a* o u X c Q P + =
Demand follows a
discrete (relative
frequency)
distribution:
Result:
order 25 Lobsters,
because that is the
smallest amount
that will serve at
least 56% of the
demand on a
given night.

13. The Nature of Uncertainty
Suppose that we represent demand as
D = Ddeterministic + Drandom
If the random component is small compared to the deterministic component, the models used in chapter 4 will be accurate. If not, randomness must be explicitly accounted for in the model.
In chapter 5, assume that demand is a random variable with cumulative probability distribution F(D) and probability density function f(D).
D - continuous random variable, N(μ, σ)
estimated from history of demand
seems to model many demands accurately
Objective: minimize the expected costs – law of large numbers

14. The Newsvendor Model
The critical ration can also be derived mathematically.
At the start of each day, a newsvendor must decide on the number of papers to purchase. Daily sales cannot be predicted exactly, and are represented by the random variable D with normal distribution N(μ, σ), where μ = and σ = 4.74
It can be shown that the optimal number of papers to purchase is given
by F(Q*) = cu / (cu + co),
where cu = 75 – 25 = 50, and c0 = 25 – 10 =15
unrealized profit per unit = (selling price – purchase price);
loss per excess = (purchase price – disposal price);
F(Q*) = cu / (cu + co) = 0.77  Pr ( D < Q* ) = 0.77
How to find Q* ?

15. Determination of the Optimal Order Quantity for Newsvendor Example
Using table A-4 find z = 0.74, with μ = and σ = 4.74
Newsvendor has to order 15 copies every week.

16. Describing Demand

17. If Demand is deterministic (EOQ)
Inv
Time
Q=600
Order
arrives
Place Order
Reorder
Point=300
LT=3 wk

18. If Demand is stochastic
Inv
Q=600
LT=3 wk
LT=3 wk
Reorder
Point=300
Time
Order
arrives
Place Order
Place Order
Order
arrives

19. If Demand is stochastic
Inv
Q=600
LT=3 wk
LT=3 wk
Reorder
Point=400
Place Order
Order
arrives
Time
Order
arrives
Place Order

20. Average Inventory Profile
Q=600
Avg. Inv
=400
Avg. Cycle Inv=300
Safety Stock=100
Time

21. Terminology On-hand stock: Stock that is physically on the shelf
Net stock = On-hand stock – Backorders
Inventory Position =
On-hand + On-order - Backorders - Committed
Complete backordering: backordered demand is filled as soon as an adequate-size replenishment arrives
Complete lost sales: when out of stock, demand is lost, customers go somewhere else

22. Why Safety Stock?
Safety Stock: Avg level of the net stock just before a replenishment arrives
Pressure for higher safety stocks
Increased product variety and customization
Increased demand uncertainty
Increased pressure for product availability
Pressure for lower safety stocks
Short product life cycles

23. The ABC Inventory Classification System
The ABC classification, devised at General Electric during the 1950s, helps a company identify a small percentage of its items that account for a large percentage of the dollar value of annual sales. These items are called Type A items.
Adaptation of Pareto’s Law
20% of the people have 80% of the wealth (in 1897 Italy)
Since most of our inventory investment is in Type A items, high service levels will result in huge investments in safety stocks.
Tight management control of ordering procedures is essential for Type A items.

24. The ABC Inventory Classification System
For Type B items inventories can be reviewed periodically
Items can be ordered in small groups, rather than individually.
Type C items require the minimum degree of control
Parameters are reviewed twice a year. Demand for Type C items may be forecasted by simple methods. The most inexpensive items of type C can be ordered in large lot, to minimize number of orders. An expensive type C items ordered only as they are demanded 10 20 30 40 50 60 70 80 90
100
Percentage of items
Percentage of dollar value
100 —
90 —
80 —
70 —
60 —
50 —
40 —
30 —
20 —
10 —
0 —
Class C
Class A
Class B

25. Replenishment Policies
When and how much to order
Continuous review with (Q,R) policy:
Inv. is continuously monitored and when it drops to R, an order of size Q is placed
Periodic review with (R,S) policy:
Inv. is reviewed at regular periodic intervals (R), and an order is placed to raise the inv. to a specified level (order-up-to level, S)

26. Measures of Customer Service: (Average) Product availability measures
Fill rate (P2 ):
Percentage of demand that is satisfied from inventory
Compute Expected Shortage per Replenishment Cycle (ESPRC)
P2 = 1-ESPRC/Q
Cycle Service Level (P1 ): Prob. of no stockout per replenishment cycle, or percentage of cycles without stockouts

27. Comparison of Type 1 and Type 2 Services
Order Cycle Demand Stock-Outs
For a type 1 service objective there are two cycles out of ten in which a stock-out occurs, so the type 1 service level is 80%. For type 2 service, there are a total of 1,450 units demand and 55 stockouts (which means that 1,395 demand are satisfied). This translates to a 96% fill rate.

28. Continuous Review Systems
The EOQ, production lot size, and planned shortage models assume continuous review
(Q, R) Policies
These models call for policies prescribing an order point (R) and order quantity (Q)
Such policies can be implemented by
A point-of-sale computerized system
The two-bin system: use inventory from bin 1 until empty which triggers reorder… replenishment fills bin 2 and remainder goes into bin 1

29. Order Point, Order Quantity Model (R,Q)
Inventory Level R
Safety Stock (SS)
Place order
Receive order
Time
Lead Time
Inventory position constantly monitored
Inventory position used to place an order, and not the net stock
Stockout occurs if demand during lead time exceeds the reorder point
Simple system… suppliers like the predictability of constant order quantities

30. Probabilistic Models When to Order
Expected Demand
Inventory Level
P(Stockout)
Freq
Optimal
Order
Quantity SS
Reorder Point , R s
Safety Stock (SS)
Place order
Receive order
Time
Lead Time

31. Uncertain Demand R On-hand inventory Time Order received Q On Hand
placed IP R
TBO1
TBO2
TBO3 L1 L2 L3

32. Expected Inventory (Assumptions)
I(t) Q
Slope -l SS
Time

33. Expected cost function
Include expected: holding, setup, penalty and ordering (per unit) costs
Average Holding Cost:
Average Set-up Cost:

34. Expected cost function
Expected Shortage per Cycle:
Interpret n(R) as the expected number of stockouts per cycle given by the loss integral formula.
The unit normal loss integral values appear in Table A-4.
Expected Penalty Cost :

35. Cost Minimization Expected Cost Function: Partial Derivatives: (1) (2)
This is the first equation
we will use to determine
optimal values Q and R

36. Cost Minimization Partial Derivatives: (2) This is the second equation
we will use to determine
optimal values Q and R

37. Solution Procedure
The optimal solution procedure requires iterating between the two equations for Q and R until convergence occurs
A cost effective approximation is to set Q=EOQ and find R from the second equation.

38. Finding Q and R, iteratively
1. Compute Q = EOQ.
2. Substitute Q in to Equation (2) and compute R.
3. Use R to compute n(R) in Equation (1).
4. Solve for Q in Equation (1).
5. Go back to Step 2, continue until convergence.

39. Example A company purchases air filters at a rate of 800 per year
$10 to place an order
Unit cost is $25 per filter
Inventory carry cost is $2/unit per year
Shortage cost is $5
Lead time is 2 weeks
Assume demand during lead time follows a uniform distribution from 0 to 200
Find (Q,R)

40. Solution
Partial derivative outcomes:

41. Solution
From Uniform U(0,200) distribution:

42. Solution
Iteration 1:
F(R)
200 R

43. Solution
Iteration 2:

44. Solution
Iteration 3:

45. Solution R didn’t change => CONVERGENCE (Q*,R*) = (94,190) I(t) 253
Slope
-800
190
159
With lead time equal to 2 weeks:
SS = R – lt = (2/52)=159

46. Example
Demand is Normally distributed with mean of 40 per week and a weekly variance of 8
The ordering cost is $50
Lead time is two weeks
Shortages cost an estimated $5 per unit short to expedite orders to appease customers
The holding cost is $ per week
Find (Q,R)

47. Solution Demand is per week.
Lead time is two weeks long. Thus, during the lead time:
Mean demand is 2(40) = 80
Variance is (2*8) = 16
Demand observed in one week is independent from demand observed in any other week:
E(demand over 2 weeks) = E (2*demand over week 1)
= 2 E(demand in a single week) = 2 μ = 80
Standard deviation over 2 weeks is σ = (2*8)0.5 = 4

48. Finding Q and R, iteratively
1. Compute Q = EOQ.
2. Substitute Q in to Equation (2) and compute R.
Use R to compute average backorder level, n(R) to use in Equation (1).
4. Solve for Q using Equation (1).
5. Go to Step 2 until convergence.

49. Solution
Iteration 1:
From the standard normal table:

50. Solution
Iteration 2:
This is the unit normal loss expression. Table A - 4 gives values.

51. Solution
Iteration 2:

52. Service Levels in (Q,R) Systems
In many circumstances, the penalty cost, p, is difficult to estimate.
For this reason, it is common business practice to set inventory levels to meet a specified service objective instead.
Type 1 service: Choose R so that the probability of not stocking out in the lead time is equal to a specified value.
Appropriate when a shortage occurrence has the same consequence independent of its time and amount.
Type 2 service: Choose both Q and R so that the proportion of demands satisfied from stock equals a specified value.
In general,  is interpreted as the fill rate.

53. Solution to (Q,R) Systems with Type 1 Service Constraint
For type 1 service, if the desired service level is α then one finds R from F(R)= α and Q=EOQ
Specify a, which is the proportion of cycles in which no stockouts occur.
This is equal to the probability that demand is satisfied.

54. Solution to (Q,R) Systems with Type 2 Service Constraint
Type 2 service requires a complex iterative solution procedure to find the best Q and R
However, setting Q=EOQ and finding R to satisfy n(R) = (1-β)Q (which requires Table A-4) will generally give good results

55. Service Constraints: Type 2
May specify fill rate b, and use EOQ for Q to compute R
Or, solve for p :
and substitute into the equation:

56. Service Constraints: Type 2
Result:

57. Other Continuous Review System: Order-Up-To-Level (s, S) vs (Q, R) System
R+Q R s S
(Q,,R) system
(s, S) system S s
R + Q
(s,S) system R
(Q,R) system

58. Periodic Review System: Order-Up-To-Level (R, S) System

59. Periodic Review Systems
Continuous Review Systems
Always knew level of on-hand inventory
Could place an order at any time
Often, we are constrained by WHEN we can order -- and it may be periodically
Train dispatched once a week
Delivery truck arrives each morning
Thus, we do not need to continuously review inventory, just check periodically

60. Periodic Review Systems
If demand were known and constant, we would just resort to our EOQ solution, possibly modifying it to meet shipping date
Now: demand is random variable
Setting:
Place an order every T periods
Policy: Order up to S
The value of Q (order quantity) will now change periodically
Previous concern: demand exceeding supply during the lead time
Now: demand exceeding supply during the period and lead time, or T + 

61. Periodic Review I(t) Order arrives. Cycle continues. S Q
Order up to S
every T periods of time.
I(t)
Order arrives. Cycle continues. S
Demand Q
Lead Time passes… t t
Time T

62. Expected cost function
Include expected holding, setup, penalty and ordering (per unit) costs
Average Inventory Level: S
At level R*,on average, order Q = S-R* units.
τ periods later, units arrive.
Inventory level? R* τ T

63. Expected cost function
Include expected: holding, setup, penalty and ordering (per unit) costs
Average Inventory Level: S
S-lt units present when
Q arrive (expected) as lt units consumed over
leading time. T

64. Expected cost function
Include expected: holding, setup, penalty and ordering (per unit) costs
Average Inventory Level: S
S - lt
lT units removed
(expected) from
inventory over
time T. T

65. Expected cost function
Include expected: holding, setup, penalty and ordering (per unit) costs
Average Inventory Level: S
S-lt lT
S-lt-lT T
Average Inventory Level =

66. Expected cost function
Include expected:
holding, setup, penalty and ordering (per unit) costs
Average Holding Cost:
Average Set-up Cost:

67. Expected cost function
Expected Shortage per Cycle:
f(x)dx = P(demand in T +  is between x and x + dx)
Expected Penalty Cost :

68. Cost Minimization Expected Cost Function: Derivative:
Recall that T and  are given:

69. Cost Minimization
Derivative:
or:

70. Example
A special control board is used in a version of a product on the production line
The board cost is $122.50
The holding cost rate is 30% per year
Reorders are placed at the start of each week, and the supplier delivers these parts in one week
The shortage cost is $100 per board due to worker downtime
Weekly demand is N (μ=125, δ2=104.17)
Set up cost (K) is $120
Find S

71. Solution Holding cost is:
h = Ic = .30 (122.50) = / 52 = $.7067 per week
Compute:
Demand Distribution is Normal
mean = variance =
Z = from Normal table
S = 125+(2.455)(104.17)1/2 =

72. Solution If penalty cost drops to $10 per unit: Compute:

73. Alternative Periodic Review Policy: (s, S) Policy
Define two levels, s < S, and let u be the starting inventory at the beginning of a period.
Then, if u is less or equal to s, order (S – u),
if u is more than s, do not place an order.
In general, computing the optimal values of s and S is extremely difficult, so few systems operate using optimal (s,S) values.
Approximation:
Compute optimal values for (Q,R) model and set s = R and S = R + Q

74. Homework Assignment Read Ch. 5 (5.1 – 5.8)
5.3, 5.6 – 5.8, 5.12, 5.15, 5.19
5.25 – 5.27

75. References Presentations by McGraw-Hill/Irwin and by Wilson,G.R.
“Production & Operations Analysis” by S.Nahmias
“Factory Physics” by W.J.Hopp, M.L.Spearman
“Inventory Management and Production Planning and Scheduling” by E.A. Silver, D.F. Pyke, R. Peterson
“Production Planning, Control, and Integration” by D. Sipper and R.L. Bulfin Jr.