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Chapter 5 Inventory Control Subject to Uncertain Demand

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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

Decisions

Continuous Decisions

Intermittent-Time

Decisions

Continuous Review

System

Periodic Review

Systems

EOQ, EPQ

EOQ

(S, T) System

(Q, R) System

(s, S) System

Base Stock

Optional

Replenishment

Two Bins

Timing Decisions

Structure of timing decisions

or

“Christmas tree” model

One-Time DecisionSituation 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).

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

The Concept of Marginal Analysis

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

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)

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.

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.

=

u

F

(

Q

)

+

c

c

u

o

c

=

³

*

u

Q

min{

Q

:

P

(

Q

)

}

£

X

+

c

c

u

o

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:

C(Q+1)-C(Q)=(cu+ co)PX(Q) – cu ≥ 0

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 $8.50. Each lobster costs L.L. $14.50
- unit cost of a L.L. stockout

Cu = 7.50 = lost profit

- unit cost of having a left-over lobster

Co = 14.50 - 8.50 = cost – salvage value = 6

- target L.L. service level

CR = Cu/(Cu + Co) = 7.5 / (7.5 + 6) = .56

- Demand follows a discrete (relative frequency) distribution as given on next page

=

³

*

u

Q

min{

Q

:

P

(

Q

)

}

£

X

+

c

c

u

o

Lively LobstersDemand 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.

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

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 = 11.73 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* ?

Determination of the Optimal Order Quantity for Newsvendor Example

Using table A-4 find z = 0.74, with μ = 11.73 and σ = 4.74

Newsvendor has to order 15 copies every week.

Q=600

LT=3 wk

Reorder

Point=300

Time

Place Order

Order

arrives

Place Order

Order

arrives

If Demand is deterministic (EOQ)If Demand is stochastic

Inv

Q=600

LT=3 wk

LT=3 wk

Reorder

Point=300

Time

Order

arrives

Place Order

Order

arrives

Place Order

If Demand is stochastic

Inv

Q=600

LT=3 wk

LT=3 wk

Reorder

Point=400

Place Order

Time

Order

arrives

Order

arrives

Place Order

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

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

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.

100 —

90 —

80 —

70 —

60 —

50 —

40 —

30 —

20 —

10 —

0 —

Class B

Class A

Percentage of dollar value

10

20

30

40

50

60

70

80

90

100

Percentage of items

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

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)

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

Comparison of Type 1 and Type 2 Services

Order Cycle Demand Stock-Outs

1 180 0

2 75 0

3 235 45

4 140 0

5 180 0

6 200 10

7 150 0

8 90 0

9 160 0

10 40 0

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.

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

Order Point, Order Quantity Model (R,Q)

R + Q

- 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

Inventory Level

R

Safety Stock (SS)

Place order

Receive order

Time

Lead Time

Probabilistic ModelsWhen to Order

Expected Demand

Inventory Level

P(Stockout)

Freq

Optimal

Order

Quantity

SS

s

Reorder Point , R

Safety Stock (SS)

Place order

Receive order

Time

Lead Time

IP

Order

received

Order

received

Order

received

Order

received

Q

Q

Q

On

Hand

On-hand inventory

R

Order

placed

Order

placed

Order

placed

Time

L1

L2

L3

TBO1

TBO2

TBO3

Uncertain Demand

Expected cost function

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

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 :

Expected Cost Function:

Partial Derivatives:

(1)

(2)

Cost MinimizationThis is the first equation

we will use to determine

optimal values Q and R

Partial Derivatives:

(2)

Cost MinimizationThis is the second equation

we will use to determine

optimal values Q and R

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.

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.

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)

Solution

- Partial derivative outcomes:

Solution

- From Uniform U(0,200) distribution:

Solution

- Iteration 2:

Solution

- Iteration 3:

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 =190-800(2/52)=159

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 $0.0225 per week
- Find (Q,R)

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

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.

Solution

- Iteration 1:
- From the standard normal table:

Solution

- Iteration 2:

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.

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.

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

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:

Service Constraints: Type 2

- Result:

s

R+Q

(s, S) system

R

(Q,,R) system

Other Continuous Review System: Order-Up-To-Level (s, S) vs (Q, R) SystemS

s

R + Q

(s,S) system

R

(Q,R) system

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

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 +

Periodic Review

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

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

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

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

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 =

Expected cost function

- Include expected:

holding, setup, penalty and ordering (per unit) costs

- Average Holding Cost:
- Average Set-up Cost:

Expected cost function

- Expected Shortage per Cycle:
- f(x)dx = P(demand in T + is between x and x + dx)
- Expected Penalty Cost :

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

Solution

- Holding cost is:

h = Ic = .30 (122.50) = 36.75 / 52 = $.7067 per week

- Compute:
- Demand Distribution is Normal
- mean = 125 variance = 104.17
- Z = 2.455 from Normal table
- S = 125+(2.455)(104.17)1/2 = 150.06

Solution

- If penalty cost drops to $10 per unit:
- Compute:
- S = 125+(1.47)(104.17)1/2 = 140
- p = 1? S = 125+(-.54)(104.17)1/2 = 119.49

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

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

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.

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