1. Production and Operation Managements
Inventory Control Subject to Unknown Demand
Prof. JIANG Zhibin
Dr. GENG Na
Department of Industrial Engineering & Management
Shanghai Jiao Tong University

2. Inventory Control Subject to Unknown Demand
Contents
Introduction
The newsboy model
Lot Size-Reorder Point System;
Service Level in (Q, R) System;

3. Introduction Sources of Uncertainties
In consumer preference and trends in the market;
In the availability and cost of labor and resources;
In vendor resupply times;
In weather and its effects on operations logistics;
Of financial variables such as stock prices and interest rates;
Of demand for products and services.

4. Introduction
Uncertainty of a quantity means that we cannot predicate its value in advance.
A department store cannot exactly predicate the sales of a particular item on any given day;
An airline cannot exactly predicate the number of people that will choose to fly on any given flight.
How can these firms choose the number of items to keep in inventory or the number of flights to schedule on any given route?
Based on the past experience for planning;
Probability distribution is estimated based on historical data;
Minimize expected cost or maximize the expected profit when uncertainty is present.

5. Introduction Some Examples
In the economic recession of the early 1990s, some business that relied on direct consumer spending, suffered severe losses.
Sears and Macy’s department stores, long standing successes in American retail market made poor earning in 1991.
Several retailers enjoyed dramatic successes.
Both The Gap and Limited in the fashion business did very well.
Wal-Mart Stores continues its ascendancy and surpassed Sear as the largest retailer in the United State.
Intelligent inventory management in the face of uncertainty certainly played a key role in the success of these firms.

6. Introduction
As almost all inventory management refers to some level of uncertainty, what is the value of the deterministic inventory control model?
Provide a basis for understanding the fundamental trade-offs encountered in inventory management;
May be good approximations depending on the degree of uncertainty in the demand.

7. In many cases DDDet even DRam0:
Introduction
Let D be the demand for an item over a given period of time. We express it as the sum of two parts DDet and Dran: D=DDet+DRam
In many cases DDDet even DRam0:
When the variance of the random component, DRam is small relative to the magnitude of DDet；
When the predictable variation is more important than random variation;
When the problem is too complex to include an explicit representation of randomness in the model.

8. Introduction
In many situations the random component of the demand is too important to ignore.
As long as the expected demand per unit times is relatively constant, and the problem structure is not too complex, explicit treatment of demand uncertainty is desirable.

9. Introduction
Two basic inventory control models subject to uncertainty:
Periodic review-the inventory level is known at discrete points in time only;
For one planning period-the objective is to balance the costs of overage and underage; useful for determining run sizes for items with short useful lifetimes (Fashions, foods, newspaper)-newsboy model.
For multiple planning period-Complex, topics of research, and rarely implemented.
Continuous review-the inventory level is known at all times.
Extensions of the EOQ model to incorporate uncertainty, service level approaches are frequently implemented
Easy to compute and implement
Accurately describe most systems in which there is ongoing replenishment of inventory items under uncertainty

10. The newsboy model
Example 5.1-Mac wishes to determine the number of copies of the Computer Journal he should purchased each Sunday. The demand during any week is a random variable that is approximately normally distributed, with mean and standard deviation Each copy is purchased for 25 cents and sold for 75 cents, and he is paid for 10 cents for each unsold copy by his supplier.
Discussion:
One obvious solution is to buy enough copies to meet the demand, which is 12 copies.
Wrong: If he purchase a copy that does not sell, his out-of-pocket expense is only 25-10=15 cents. However, if he is unable to meet the demand of a customer, he loses 75-25=50cents.
Suggestion: He should buy more than the mean. How many?

11. The newsboy model Notation-the newsboy model
A single product is to be ordered at the beginning of a period and can be used only to satisfy the demand during that period.
Assume that all relevant costs can be determined on the basis of ending inventory. Define:
c 0=Cost per unit of positive inventory remaining at the end of the period (overage cost);
cu=Cost per unit of unsatisfied demand, which can be thought as a cost per unit of negative ending inventory (underage cost).
Assume that the demand D is a continuous nonnegative random variable with density function f(x) and cumulative distribution function F(x).
The decision variable Q is the number of units to be purchased at the beginning of the period.
The goal is to determine Q to minimize the expected costs incurred at the end of the period.

12. The newsboy model
A general outline for analyzing most stochastic inventory problems is as follows:
Develop an expression for cost incurred as a function of both the random variable D and the decision variable Q.
Determine the expected value of this expression with respect to the density function or probability function of demand.
Determine the value of Q such that the expected cost function is minimized.
Development of Cost Function
Define G(Q, D) as the total overage and underage cost incurred at the end of the period when Q units are ordered at the start of the period and D is the demand.
Q-D is the demand units left at the end of the period as long as QD;
If Q<D, then Q-D is negative and the number of units remaining on hand at the end of the period is 0.

13. The newsboy model
max{Q-D, 0} represents the units left at the end of the period.
max {D-Q, 0} indicates the excess demand over supply, or unsatisfied demand.
G(Q, D)=c0max{Q-D, 0}+cumax{D-Q, 0}
The expected cost function is defined as:
G(Q)=E(G{Q, D))

14. The newsboy model Determining the Optimal Policy
Determine the value of Q that minimizes the expected cost G(Q).
G(Q) is convex such that Q(Q) has minimal value
Since the slope is negative at Q=0, G(Q) is decreasing at Q=0.

15. The newsboy model Optimal solution, Q*, such that The critical ratio.
Fig5-3 Expected Cost Function for Newsboy Model
Optimal solution, Q*, such that
The critical ratio.
The critical ratio is strictly between 0 and 1, meaning that for a continuous demand, this equation is always solvable.

16. The newsboy model
Since F(Q*) is defined as the probability that the demand does not exceed Q*, the critical ratio is the probability of satisfying all the demand during the period if Q* units are purchased at the beginning of the period.
Fig. 5-4 Determination of the Optimal Order Quantity for Newsboy Example
Q*=z+=4.74 =15.2415
Example Mac’s newsstand
Suppose that the demand for the Journal is approximately normally distributed with mean =11.73 and standard deviation =4.74. c0=25-10=15, and cu=75-25=50 cents. The critical ratio is cu/(co+cu)=0.50/( )=0.77. Hence, he ought to purchase enough copies to satisfy all of the weekly demand with probability The optimal Q* is the 77th percentile of the demand distribution.
F(Q*)

17. The newsboy model- Optimal Policy for Discrete Demand
In some cases, accurate representation of the observed pattern of demand in term of continuous distribution is difficult or impossible.
In the discrete case, the critical ratio will generally fall between two values of F(Q).
The optimal solution procedure is to locate the critical ratio between two values of F(Q) and choose the Q corresponding to the higher value.

18. The newsboy model- Optimal Policy for Discrete Demand
Example Mac’s newsstand
f(4) =3/52 is obtained by dividing frequencies 4 (the numbers of times 3 that a given weekly demand 4 occur during a year, i.e. 52 weeks) by 52;
The critical ratio is 0.77, which corresponds to a value of F(Q) between Q=14 and Q=15.

19. The newsboy model- Extension to Include Starting Inventory
Suppose that the starting inventory is some value u and u>0.
The optimal policy is simply to modify that for u=0.
The same ideal is that we still want to be at Q* after ordering.
If u<Q*, order Q*-u; If u>Q*, do not order.
Note that Q* should be understood as order-up-to point rather than the order quantity when u>0.
Example 5.2 (Cont.)-Suppose that Mac has received 6 copies of the Journal at the beginning of the week from other supplier. The optimal policy still calls for having 15 copies on hand after ordering, thus he would order the difference 15-6=9 copies.

20. The newsboy model- Extension to Multiple Planning Periods
The ending inventory in any period becomes the starting inventory in the next period.
If excess demand is back-ordered, interpret cu as the loss-of-goodwill cost and co as the holding cost.
If excess demand is lost, interpret cu as the loss-of-goodwill cost plus the lost profit and co as the holding cost.
However, the multi-period newsboy model was unrealistic for two reasons: it did not include a setup cost for placing an order and it did not allow for a positive lead time.

21. The newsboy model- Extension to Multiple Planning Periods
Example 5.3: Suppose that Mac is considering how to replenish the inventory of a very popular paperback thesaurus that is ordered monthly. Copies of the thesaurus unsold at the end of a month are still kept on the shelves for future sales. Assume that customers who request copies of the thesaurus when they are out of stock will wait until the following month. Mac buys the thesaurus for $1.25 and sells it for Mac estimates a loss-of-goodwill cost of 80 cents each time a demand for a thesaurus must be back-ordered. Monthly demand for the book is fairly closely approximated by a normal distribution with mean 20 and standard deviation 10. Mac uses a 20 percent annual interest rate to determine his holding cost. How many copies of the thesaurus should be purchased at the beginning of each month?

22. The newsboy model- Extension to Multiple Planning Periods
Answer for Example 5.3:
=20 and standard deviation =10
c0=1.25*0.2/12=0.208 holding cost
cu=80 cents.
The critical ratio is cu/(co+cu)=0.80/( )=0.74
Hence, he ought to purchase enough copies to satisfy all of the monthly demand with probability The optimal Q* is the 74th percentile of the demand distribution.
Q*=z+=10 =26.426

23. Lot Size-Reorder Point System
For random demand, Q and R are regarded as independent decision variables;
Assumptions
Continuous review-demands are recorded as they occur;
Random and stationary demand-the expected value of demand over any time interval of fixed length is constant; the expected demand rate is  unite/year.
Fixed positive lead time for placing an order;
Assume the following costs
Setup cost $K per order;
Holding cost at $h per unit held per year;
Proportional order cost of $c per item;
Stock-out cost $p per unit of unsatisfied demand, or shortage cost or penalty cost;

24. Lot Size-Reorder Point System
Demand Description
The response time is the amount of time required to effect a change in the on-hand inventory level.
The response time is the reorder lead time 
The demand during the lead time is the random variable of interest.
It is assumed that demand during lead time is continuous random variable D with probability density function (pdf) f(x) and cumulative distribution function (cdf) F(x).
is the mean of the demand during the lead time.
is the standard deviation.

25. Lot Size-Reorder Point System
Two Independent Decision Variables
Q = the lot size or order quantity;
R=the reorder level in units of inventory.
The policy is that when the level of on-hand inventory drops to R, an order for Q units is placed such that it will arrive in  units of time.
Fig 5-5 Changes in Inventory Over Time for Continuous-Review (Q, R) System

26. Lot Size-Reorder Point System
Derivation of the Expected Cost Function-develop an expression for the expected average annual cost in terms of the decision variables (Q, R) and search for the optimal values of (Q, R) to minimize this cost.
Holding cost
Assume that the mean rate of demand is  units per year;
The expected inventory level varies between s and Q+s, where s is the safety stock, defined as the expected level of on-hand inventory just before an order arrives, s=R-.
The average inventory is s+Q/2=R- +Q/2.
The holding cost should not be charged against the inventory level when it is negative.

27. Lot Size-Reorder Point System
Penalty Cost
Occurs only when the system is subject to shortage.
The number of units of excess demand is simply the amount by which the demand over the lead time, D, exceeds the reorder level, R.
The expected number of shortages that occurs in one cycle is determined by
n(R)
As n(R) represents the expected number of stock-outs incurred in a cycle, the expected number of stock-outs incurred per unit time is n(R)/T=n(R)/Q.

28. Lot Size-Reorder Point System
Proportional Ordering Cost Component.
The expected proportional order cost per unit of time is c;
Since this item is independent of variables Q and R, it does not affect the optimization, and thus may be ignored.
The Cost Function:
G(Q, R)=h(Q/2+R-)+K/Q+pn(R)/Q.
The objective is to choose Q and R to minimize G(Q, R).
The solution procedure requires iterating between (1) and (2) until the two successive values of Q and R are the same.

29. Lot Size-Reorder Point System
When the demand is normally distributed, n(R) is computed by using the standardized loss function L(z).
where (x) is standardized normal density
If lead time demand is normal with mean  and standard deviation , then
Calculations of the optimal policy are carried out using Table A-4 at Page 528 of the book.

30. Lot Size-Reorder Point System
The procedure of computing Q and R:

31. Lot Size-Reorder Point System
Example 5.4 Harvey’s Specialty Shop sells a popular mustard that purchased from English company. The mustard costs $10 a jar and requites a six-month lead time for replenishment stock. The holding cost is computed on basis 20% annual interest rate; the lost-of-goodwill cost is $25 a jar; and bookkeeping expenses for placing an order amount to about $50. During the six-month lead time, average 100 jars are sold, but with substantial variation from one six-month period to the next. The demand follows normal distribution and the standard deviation of demand during each six-month period is 25. How should Harvey control the replenishment of the mustard?

32. Lot Size-Reorder Point System
Solution to Example 5.4
To find the optimal values of R and Q
The mean lead time demand in six-month lead time is 100, the mean yearly demand is 200, giving =200;
h=100.20=2; K=50; P=25;

33. Lot Size-Reorder Point System

34. Lot Size-Reorder Point System
Results for Example 5.4: The optimal values of (Q, R)=(111, 143), that is, when Harvey’s inventory of this type mustard hits 143 jars, he should place an order for 111 jars.
Example 5.4 (Cont.): determine the following
Safety stock;
The average annual holding, setup, and penalty costs associated with the inventory control of the mustard;
The average time between placement of orders;
The proportion of order cycles in which no stock-outs occur>Among given number of order cycles, how many order cycles do not have stock-outs?
The proportion of demands that are not met.

35. Lot Size-Reorder Point System
Solution to Example 5.4 (Cont.)
The safety stock is s=R-= =43 jars;
Three costs:
The holding cost is h(Q/2+s)=2(111/2+43)=$197/jar;
The setup cost is K/Q=50200/111=$90.09/jar;
The penalty cost is p n(R)/Q=25  2000.4575/111=$20.61/jar
Hence, the total average cost under optimal inventory control policy is $307.70/jar.
The average time between placement of orders:
T=Q/ =111/200=0.556 yr=6.7months;
Compute the probability that no stock-out occurs in the lead time, which is the same as that the probability that the lead time demand does not exceeds the reorder point: P(DR)=F(R)=1-Qh/p = =0.956;
The expected demand per cycle must be Q; the expected number of stock-outs per cycle is n(R). Hence, the proportion of demand that stock out is n(R)/Q=0.4575/111=0.004.

36. Service Levels in (Q, R) System
In reality, it is difficult to determine the exact value of stock-out cost p. A common substitute for a stock-out cost is a service level.
Service level generally refers to the probability that a demand or a collection of demand is met.
Service level can be applied both to periodic review and continuous review systems, that is, (Q, R) system.
Two types of service levels for continuous review system : Type 1 and Type 2
Type 1 Service
Specify the probability of not stocking out in the lead time, denoted as .
As the value of R can be completely specified by , computation of R and Q can be decoupled.
The computation of the optimal (Q, R) values subject to Type 1 service constraint is straightforward:
Determine R to satisfy the equation F(R)= ;
Set Q=EOQ

37. Service Levels in (Q, R) System
Discusses on Type 1 Service
 is interpreted as the proportion of cycles in which no stock-out occurs;
A Type 1 service objective is suitable when a shortage occurrence has the same consequence independent of its time or amount. For example, a production line is stopped whether 1 unit or 100 units are short.
However, Type 1 service does not illustrate how does the shortage occur.
Usually, when we say we would like provide 95% service, we mean that we would like to be able to fill 95% of the demand when they occur, rather than fill all of the demands in 95% of the order cycles. –not be specified by Type 1 Service.
In addition, different items have different cycle lengths, this measure will not be consistent among different products, making the proper choice of  difficult.

38. Service Levels in (Q, R) System
Type 2 Service
Measures the proportion of demands that are met from stock, denoted by .
Since n(R)/Q is the average fraction of demands that stock out each cycle, then specification of  results in constraint n(R)/Q =1- .
This constraint is more complex than that arising from Type 1 service, because it involves both Q and R.
Although EOQ is not optimal in this case, it usually gives pretty good results.
If EOQ is used to estimate the lot size, then we would find R to solve n(R)=EOQ(1- ).

39. Service Levels in (Q, R) System
Example 5.5
Harvey feels uncomfortable with assumption that the stock-out cost is $25 and decide to use a service level criterion instead. Suppose that he chooses to use 98%.
Type 1 service: =0.98, find R to solve F(R)=0.98. From Table A-4, z=2.05, R= z+=25 =151.
Type 2 service: =0.98, n(R)=EOQ(1- ), which corresponds to L(z)= EOQ(1- )/ =100(1-0.98)/25=0.08. From Table A-4, z=1.02, then R= z+=25 =126.
The same values of  and  gives considerably different values of R.

40. Service Levels in (Q, R) System
Optimal (Q, R) Polices Subject to Type 2 Constraints
EOQ is only an approximation of the optimal lot size.
A more accurate value of optimal Q can be obtained as follows
Solving for p in Equation (2) gives
Substituting p in Equation (1) results in
Service level order quantity, SOQ (Service level order quantity) formula

41. Service Levels in (Q, R) System

42. Service Levels in (Q, R) System
The optimal values of Q and R satisfying a 98 percent fill rate constraint are (Q, R)=(114, 124).
The cost is $252, only $2 higher than that for (100, 126). Therefore, EOQ is good approximation.

43. Additional Discussion of Periodic-review Systems
(s, S) Policies
It is difficult to implement a continuous-review solution in a periodic-review environment because the inventory level is likely to overshoot the reorder point R during a period, which makes it impossible to place an order the instant the inventory reaches R.
Define two numbers, s and S, to be used as follows: When the level of on-hand inventory is less than or equal to s, an order for the difference between the inventory and S is placed.
If u is the starting inventory in any period, then the (s, S) policy is
If u≤s, order S-u;
Else, do not order.
Approximation: to compute a (Q,R) policy using the methods described earlier, and set s=R and S=R+Q.
This approximation will give reasonable results in many cases, and is probably the most commonly used.

44. Additional Discussion of Periodic-review Systems
Service Level in Periodic-Review Systems
Type 1 service objective-find the order-up-to point Q so that all of the demand is satisfied in a given percentage of the periods, which can be determined by F(Q)=, where F(Q) is the probability that the demand during the period does not exceed Q.
Type 2 service objective
To find the Q to satisfy the Type 2 service objective , it is necessary to obtain an expression for the fraction of demand that stock out each period.
Define n(Q), the expected number of demands that stock out at the end of period.
Since the demand per period is , then the proportion of demand that stock out each period is n(Q)/=1-, giving n(Q) =(1-).

45. Additional Discussion of Periodic-review Systems
Example 5.9: Mac, the owner of the newsstand described in Example 5.1, wishes to use a Type 1 service level of 90 percent to control his replenishment of the Computer Journal. The z value corresponding to the 90th percentile of the unit normal is z=1.28. Hence,
Q*=σz+μ=(4.74)(1.28)+11.73=17.8≈18
Using a Type 2 service of 90 percent, we obtain
n(Q)=(1-β) μ=(0.1)(11.73)=1.173
It follows that L(z)=n(Q)/ σ=1.173/4.74=0.2475; From Table A-4, we find z ≈0.35
Then Q*= σz+μ =(4.74)(0.35)+11.73=13.4 ≈13

46. Multiproduct Systems ABC Analysis
One issue that we have not discussed is the cost of implementing an inventory control system and the trade-offs between the cost of controlling the system and the potential benefits that accrue from that control.
In multiproduct inventory systems, not all products are equally profitable. Control cost may be reasonable in some cases and not in others.
It is important to differentiate profitable from unprofitable items.
Borrow a concept from economics: Pareto effect
The economist Vilfredo Pareto(1848～1923) , studying the distribution of wealth in the 19th century, noted that a large portion of wealth was owned by a small segment of the population
Pareto effect in inventory control: a large portion of the total dollar volume of sales is often accounted for by a small number of inventory items.

47. Multiproduct Systems
Assume that items are ranked in decreasing order of the dollar value of annual sales.
The cumulative value of sales generally results in a curve in the right side.
Typically, the top 20 percent of the items account for about 80 percent of the annual dollar volume of sales, the next 30 percent of the items for the next 15 percent of sales, and the remaining 50 percent for the last 5 percent of dollar volume. The three item groups are labeled A, B, and C, respectively.
A items should be watched most closely. Inventory levels for A items should be monitored continuously. More complicated forecasting procedures is needed.
For B items inventories could be reviewed periodically, items could be ordered in groups rather than individually, and somewhat less sophisticated forecasting methods could be used.
The minimum degree of control would be applied to C items. For very inexpensive C items with moderate levels of demand, large lot sizes are recommended to minimize the frequency that these items are ordered. For expensive C items with very low demand, the best policy is generally to order these items as they are demanded.

48. Multiproduct Systems
Example 5.10, Performance of 20 Stock Items Selected at Random

49. Multiproduct Systems
Twenty Stock Items Ranked in Decreasing Order of Annual Dollar Volume

50. Homework for Chapter 5
P235 Q8
P251, Q13
P251, Q20
P253, Q23

51. The End!