L08_SCTools_042215

I’m indebted to Dr. Mark Wilkinson, Senior Industrial Engineer and Intel

Technologist, Logic Technology Development, Intel Corporation for initial

organization of this PPT deck.

Supply Chain Analysis Tools

MS&E 262

Supply Chain Management

© Hausman and Wilkinson 2

Inventory/Service Trade-off Curve

Poor

Low

High

Good

Service

Inventory

Analytical tools help lower the curve!

Motivation


Outline

• Analytical Tools from MS&E 260/261

– EOQ

– Newsvendor

– Lot Size Reorder (Q,R) Model

– Periodic Review (T,S) Model

• Supply Chain Improvement Tools


Standard Inventory Models

Demand: Known

Supply: Known

Demand: Unknown

Supply: Known

Demand: Unknown

Supply: Unknown

Trade-off:

Ordering Cost

Holding Cost

Penalty Costs:

None

Ordering vs. Holding Ordering vs. Holding

Underage/Overage Underage/Overage

Case I Case II Case III


Standard Inventory Models

• Economic Order Quantity (EOQ) – Deterministic; determine order quantity

• Newsvendor

– Captures demand variability; determine one-time-shot quantity

• Lot Size-Reorder Model

– Includes variability in an on-going setting

– Determine order quantity (Q) when an trigger level (R) is met

• Periodic Review Model

– Includes variability in an on-going setting

– Determine order-up-to level (S) when ordering every T time units

Case I

Case II

Case II/III

Case II/III


Economic Order Quantity Model

• How much to order/produce?

– Fixed order cost of $ K

– Inventory holding cost of $ h = $ Ic

– Shortages prohibited

– Deterministic (constant) demand rate per year, D

T

Q

Time, t

Slope = -D

(T = Q/D)

Inventory Level

Analytical Tools


EOQ Model - Derivation

Total = G(Q)-cD

Holding = IcQ/2

Setup = KD/Q

Q Q*

Annual

Holding +

Setup Cost

Ic

KDQ

2*

G(Q) = cD + KD/Q + IcQ/2

Purchasing Setup Holding

Analytical Tools


EOQ Model – Sensitivities/Shortcomings

• “40 – 20 – 02” rule – 40 % error in an input parameter results in 20 % error in

Q

– The result is a 2 % increase in the costs, G(Q)

•Shortcomings of EOQ Model?

Analytical Tools

Cost function is relatively insensitive to errors in Q

• Zero lead time (easily extended to fixed lead time)

• Infinite production rate (finite production rate, P, if P > D)

• No shortages allowed (easily extendable)

• Constant, deterministic demand rate


Newsvendor Model

•Motivation: At the start of each day, a newspaper vendor must decide

on the number of papers to purchase and sell. Daily sales

cannot be predicted exactly, and are represented by a

random variable, D.

•Relevant Costs: Co = unit cost of overage (not enough demand)

Cu = unit cost of underage (too much demand)

•Example: Fashion (Ralph Lauren, Ann Taylor)

Retail: Stanford Shopping Center, Web Sites

Overage: Gilroy Outlet Mall

Analytical Tools


• Shortcomings of Newsvendor Model?

Newsvendor Model (cont.)

ou

u

cc

cQF

)( *

Analytical Tools

• No consideration of positive lead times

• One shot model (can be extended to multiple periods)

• No setup cost for placing orders included


What is the Expected Profit?

Expected Demand = Expected Sales + Expected Lost Sales

Expected Lost Sales = L(z) s when demand follows normal distribution

L(z) is the standardized loss function for a Normal(0,1) distribution Excel: L(z) = NORMDIST(z,0,1,False) – z(1-NORMSDIST(z)) *

Order Quantity (Q) = Expected Sales + Expected Overage

Expected Overage = max (0, Q – Expected Sales) … no back orders

Expected Profit = (p - c) Expected Sales – (c - s) Expected Overage

= (p - c) (m – L(z) s) – (c - s) (Q – (m - L(z) s))

Analytical Tools

* Nahmias 6th ed, p. 278


Lot Size – Reorder Point (Q, R) Model

• We need to decide two things:

– How much to order each time we place an

order (Q)?

– At which inventory position (R) do we order?

Q

t= Lead Time

t

Inventory

Position

Time

R

s

Safety Stock


Safety Stock

• Since demand (and possibly supply) is uncertain, the

system builds in safety stock to protect against uncertainty

during the reorder lead time.

Safety Stock = Insurance associated with uncertainty

• Safety stock in the (Q,R) model is a function of:

– Demand distribution (standard deviation of demand)

– Desired availability (e.g., 99% probability demand satisfied)

– Supplier’s lead time duration

– Supplier’s availability and delivery randomness (Case III)


(Q,R) Model – Notation:

• Average demand rate/year l

• Setup cost K

• Variable cost c

• Holding cost h=Ic

• Order quantity Q

• Reorder point R

• Lead time t

• Safety stock s

Analytical Tools


(Q,R) Model – Unit Shortage Cost p

• The expected total annual cost (excluding lc) is

R

dxxfRxQ

pRQ

IcQ

KRQG )()(

2),(

llt

l

l

l

p

QIcRF

Ic

RpnKQ

1)(

))((2

Defining n(R) = E [# of units short in a cycle], we obtain

Setup Holding Shortage

Analytical Tools


(Q,R) Model – Unit Shortage Cost p (cont.)

• For normally distributed demand, define

where z = (R-mL)/sL.

• Hence n(R) = sL L(z)

• Use the approximate solution:

– Q = EOQ

– Obtain z from tables for L(z)

– R = mL + zsL

z

dttztzL )()()(

Analytical Tools

mL: mean lead time demand; sL: standard deviation of lead time demand


(Q,R) Model – Service Level Approaches

• Type 1:

– a = Prob(no stockout in lead time)

• Type 2:

– b = Proportion of demand met from on-hand stock

L

QzL

s

b )1()(

Better

Bad

bs

1)()(

Q

zL

Q

Rn L

• Recall n(R) = E[# of units short in cycle]

Analytical Tools


Response Time = T + t

Periodic Review – (T, S) Model

• Review every T time units

• Order such that system inventory position

reaches S units at every review

Time

Inventory

S

t

Lead Time

T t


(T,S) Model (cont.)

Where

mt+T = mean demand over t+T periods

st+T = standard deviation of demand over t+T periods

z see (Q,R) model

Hence

Safety Stock = z st+T

T = EOQ/l

S = mt+T + z st+T

Analytical Tools


Additional Tools/Methods

• ABC Analysis

• Exchange Curves

• Forecasting

• Scheduling

• Aggregate Planning

• Capacity Expansion

• Optimization (LP, IP, MIP, DP)

Analytical Tools


Supply Chain Improvements

• Uniform vs. Non-uniform Service Levels

• Risk Pooling/Consolidation

• Multi-Echelon Analysis

• Postponement

• Leadtime Reduction

• Review Period Reduction

• Variable Lead-time


Example 1: Risk Pooling/Consolidation

• What is meant by “Risk Pooling”?

• Example

– Laser Printer Supply Chain


© Hausman and Wilkinson 23 http://www.ups.com/maps

Penang, Malaysia

Long Beach

CA, USA

Memphis

TN, USA

Represents a DC location for distributor “D1”

Laser Printer: Finished Goods Logistics


© Hausman and Wilkinson 24 http://www.ups.com/maps

UPS Ground Map for “Memphis, TN”



• Assume the following distributor network:

– 5 Independent Distributors (D1, D2, D3, D4, D5)

– Each distributor operates 8 DCs across the US

• Who are the distributors’ customers?

• Relevant metrics for a DC?

Laser Printer’s Distributor Network



Discuss opportunities for risk pooling:

– For any particular distributor?

– For any particular location (e.g., Memphis,

TN)?

– For the original equipment mfr (OEM)?

Active Learning



Assume that:

• 1. Demands at the multiple DCs are statistically

independent.

• 2. The means and standard deviations of demand for

the multiple product DCs are identical.

• 3. The leadtimes for the multiple DCs are

identical/constant.

• 4. The review periods at the DCs are identical.

• 5. The safety factors for the DCs are identical.

• 6. All DCs have the same inventory value.




Let: si = standard deviation of

demand per period at

DC i;

ti = lead time for DC i;

Ti = review period for DC i;

zi = safety factor for DC i;

n = number of DCs in a region

(e.g., DCs in Memphis, TN).




ts Tnz

ts Tzn

Safety Stock at DCi =

Total System Safety Stock =

System Safety Stock, Unpooled =

System Safety Stock, Pooled =

iiTiz ts

iiTi

n

iz ts 1






E.g., Reduction Effect in Memphis for n = 5: 55.3%

Reduction Effect through Pooling :

SafetyStock , Unpooled Total SafetyStock , Pooled

Total

SafetyStock , Unpooled Total

-

=

ts Tnzts Tznts Tnz

n

11

-


Example 2: Lead Time Reduction


Example 3: Postponement

• Concept

– Delay/postpone product differentiation until as late as possible in the production process


Blanks

Manufacturer

Intel

Mask FacilitySF

Warehouse

Ba

Bb

B

B

a

b

Ba

Bb


Before Postponement

Ba

Bb

Engineering

Production

Engineering

Production

Supplier-coated blanks

B

a

B

b

t = 4 weeks

t = 4 weeks

Supplier-coated blanks

Dapi

Daei

Dbpi

Dbei

Delivery

Demand for blank B in week i = resist user ruiD


After Postponement

Ba

Bb

Engineering

Production

Engineering

Production

B

a

b

t = 1-2 days

Uncoated blanks

Dapi

Daei

Dbpi

Dbei

t = 3 weeks

In-house

Coating

In-house

Coating

Delivery


Summary

• Use known tools as needed in projects

• Understand underlying assumptions

• Perform sensitivity analysis on different

parameters

• At worst, simulate the system!

Analytical tools can help significantly improve

supply chain performance!


Details:


Lead Time Demand Variability

• Expectation of Sum = Sum of Expectations

• General Variance Formula (Lead time = t periods, demand in

period i = di)

• Variance of Sum = Sum of Variances (for independent variables)

tt

ss1

LT

1

22 where,),(2i

i

ji

ji

i

dd ddddCOViLT

Analytical Tools

* assuming independence between periods

• Example: Lead time demand*

Mean mL = t m

Variance sL2 = t s 2


Random Lead Times

• If lead time is random, with mean t and variance s2

• And demand in time t has mean mt and variance s 2t

Then the demand during (random) lead time has *

Mean mL = tm

Variance sL2 = m 2 s2 +t s 2

*Assuming orders do not cross and successive lead times are independent

Analytical Tools

Documents

L08_SCTools_042215