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Multi-Echelon Inventory Management. Prof. Larry Snyder Lehigh University Dept. of Industrial & Systems Engineering OR Roundtable, June 15, 2006. Outline. Introduction Overview Network topology Assumptions Deterministic models Stochastic models Decentralized systems. Overview.

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multi echelon inventory management

Multi-Echelon Inventory Management

Prof. Larry Snyder

Lehigh University

Dept. of Industrial & Systems Engineering

OR Roundtable, June 15, 2006

outline
Outline
  • Introduction
    • Overview
    • Network topology
    • Assumptions
    • Deterministic models
  • Stochastic models
  • Decentralized systems

Multi-Echelon Inventory – June 15, 2006

overview
Overview
  • System is composed of stages (nodes, sites, …)
  • Stages are grouped into echelons
  • Stages can represent
    • Physical locations
    • BOM
    • Processing activities

Multi-Echelon Inventory – June 15, 2006

overview4
Overview
  • Stages to the left are upstream
  • Those to the right are downstream
  • Downstream stages face customer demand

Multi-Echelon Inventory – June 15, 2006

network topology
Network Topology
  • Serial system:

Multi-Echelon Inventory – June 15, 2006

network topology6
Network Topology
  • Assembly system:

Multi-Echelon Inventory – June 15, 2006

network topology7
Network Topology
  • Distribution system:

Multi-Echelon Inventory – June 15, 2006

network topology8
Network Topology
  • Mixed system:

Multi-Echelon Inventory – June 15, 2006

assumptions
Assumptions
  • Periodic review
    • Period = week, month, …
  • Centralized decision making
    • Can optimize system globally
    • Later, I will talk about decentralized systems
  • Costs
    • Holding cost
    • Fixed order cost
    • Stockout cost (vs. service level)

Multi-Echelon Inventory – June 15, 2006

deterministic models
Deterministic Models
  • Suppose everything in the system is deterministic (not random)
    • Demands, lead times, …
    • Possible to achieve 100% service
  • If no fixed costs, explode BOM every period
  • If fixed costs are non-negligible, key tradeoff is between fixed and holding costs
    • Multi-echelon version of EOQ
    • MRP systems (optimization component)

Multi-Echelon Inventory – June 15, 2006

outline11
Outline
  • Introduction
  • Stochastic models
    • Base-stock model
    • Stochastic multi-echelon systems
    • Strategic safety stock placement
    • Supply uncertainty
  • Decentralized systems

Multi-Echelon Inventory – June 15, 2006

stochastic models
Stochastic Models
  • Suppose now that demand is stochastic (random)
    • Still assume supply is deterministic
    • Including lead time, yield, …
  • I’ll assume:
    • No fixed cost
    • Normally distributed demand: N(,2)
  • Key tradeoff is between holding and stockout costs

Multi-Echelon Inventory – June 15, 2006

the base stock model
The Base-Stock Model
  • Single stage (and echelon)
  • Excess inventory incurs holding cost of h per unit per period
  • Unmet demand is backordered at a cost of p per unit per period
  • Stage follows base-stock policy
    • Each period, “order up to” base-stock level, y
    • aka order-up-to policy
    • Similar to days-of-supply policy: y /  DOS

Multi-Echelon Inventory – June 15, 2006

the base stock model14
The Base-Stock Model
  • Optimal base-stock level:where z comes from normal distribution and
  •  is sometimes called the “newsboy ratio”

Multi-Echelon Inventory – June 15, 2006

interpretation
Interpretation
  • In other words, base-stock level = mean demand + some # of SD’s worth of demand
  • # of SD’s depends on relationship between h and p
  • Ash  z   y* 
  • Asp  z   y* 
  • If lead time = L:

Multi-Echelon Inventory – June 15, 2006

stochastic multi echelon systems
Stochastic Multi-Echelon Systems
  • Need to set y at each stage
  • Could use base-stock formula
    • But how to quantify lead time?
    • Lead time is stochastic
    • Depends on upstream base-stock level and stochastic demand
  • For serial systems, exact algorithms exist
    • Clark-Scarf (1960)
    • But they are cumbersome

Multi-Echelon Inventory – June 15, 2006

an approximate method
An Approximate Method
  • Assume that each stage carries sufficient inventory to deliver product within S periods “most of the time”
    • Definition of “most” depends on service level constant, z
    • S is called the committed service time (CST)
  • We simply ignore the times that the stage does not meet its CST
    • For the purposes of the optimization
    • Allows us to pretend LT is deterministic

Multi-Echelon Inventory – June 15, 2006

net lead time
Net Lead Time

S3

S2

S1

  • Each stage has a processing time T and a CST S
  • Net lead time at stage i = Si+1 + Ti – Si

3

2

1

T3

T2

T1

“bad” LT

“good” LT

Multi-Echelon Inventory – June 15, 2006

net lead time vs inventory
Net Lead Time vs. Inventory
  • Suppose Si=Si+1 + Ti
    • e.g., inbound CST = 4, proc time = 2, outbound CST = 6
    • Don’t need to hold any inventory
    • Operate entirely as pull (make-to-order, JIT) system
  • Suppose Si=0
    • Promise immediate order fulfillment
    • Make-to-stock system

Multi-Echelon Inventory – June 15, 2006

net lead time vs inventory20
Net Lead Time vs. Inventory
  • Precise relationship between NLT and inventory:
  • NLT replaces LT in earlier formula
  • So, choosing inventory levels is equivalent to choosing NLTs, i.e., choosing S at each stage
  • Efficient algorithms exist for finding optimal S values to minimize expected holding cost while meeting end-customer service requirement

Multi-Echelon Inventory – June 15, 2006

key insight
Key Insight
  • It is usually optimal for only a few stages to hold inventory
    • Other stages operate as pull systems
  • In a serial system, every stage either:
    • holds zero inventory (and quotes maximum CST)
    • or quotes CST of zero (and holds maximum inventory)

Multi-Echelon Inventory – June 15, 2006

case study

14

PART 2

CHARLESTON ($7)

8

14

45

PART 5

CHICAGO ($155)

PART 3

AUSTIN ($2)

14

PART 1

DALLAS ($260)

0

6

5

14

15

45

7

PART 6

CHARLESTON ($2)

PART 4

BALTIMORE($220)

55

32

8

32

5

8

PART 7

CHARLESTON($30)

14

14

Case Study

(Adapted from Simchi-Levi, Chen, and Bramel, The Logic of Logistics, Springer, 2004)

  • # below stage = processing time
  • # in white box = CST
  • In this solution, inventory is held of finished product and its raw materials

Multi-Echelon Inventory – June 15, 2006

a pure pull system

14

PART 2

CHARLESTON ($7)

8

14

45

PART 5

CHICAGO ($155)

PART 3

AUSTIN ($2)

14

PART 1

DALLAS ($260)

77

6

5

14

15

45

7

PART 6

CHARLESTON ($2)

PART 4

BALTIMORE($220)

55

32

8

32

5

8

PART 7

CHARLESTON($30)

14

14

A Pure Pull System
  • Produce to order
  • Long CST to customer
  • No inventory held in system

Multi-Echelon Inventory – June 15, 2006

a pure push system

14

PART 2

CHARLESTON ($7)

8

14

45

PART 5

CHICAGO ($155)

PART 3

AUSTIN ($2)

14

PART 1

DALLAS ($260)

0

6

5

14

15

45

7

PART 6

CHARLESTON ($2)

PART 4

BALTIMORE($220)

55

32

8

32

5

8

PART 7

CHARLESTON($30)

14

14

A Pure Push System
  • Produce to forecast
  • Zero CST to customer
  • Hold lots of finished goods inventory

Multi-Echelon Inventory – June 15, 2006

a hybrid push pull system

7

PART 2

CHARLESTON ($7)

8

14

45

PART 5

CHICAGO ($155)

PART 3

AUSTIN ($2)

9

PART 1

DALLAS ($260)

30

6

5

14

15

45

7

PART 6

CHARLESTON ($2)

PART 4

BALTIMORE($220)

8

32

8

32

5

8

PART 7

CHARLESTON($30)

14

14

push/pull boundary

A Hybrid Push-Pull System
  • Part of system operated produce-to-stock, part produce-to-order
  • Moderate lead time to customer

Multi-Echelon Inventory – June 15, 2006

cst vs inventory cost
CST vs. Inventory Cost

Push System

Push-Pull System

Pull System

Multi-Echelon Inventory – June 15, 2006

optimization shifts the tradeoff curve
Optimization Shifts the Tradeoff Curve

Multi-Echelon Inventory – June 15, 2006

supply uncertainty
Supply Uncertainty
  • Types of supply uncertainty:
    • Lead-time uncertainty
    • Yield uncertainty
    • Disruptions
  • Strategies for dealing with demand and supply uncertainty are similar
    • Safety stock inventory
    • Dual sourcing
    • Improved forecasts
  • But the two are not the same

Multi-Echelon Inventory – June 15, 2006

risk pooling
Risk Pooling
  • One warehouse, several retailers
    • Who should hold inventory?
  • If demand is uncertain:
    • Smaller inventory req’t if warehouse holds inv.
    • Consolidation is better
  • If supply is uncertain (but demand is not):
    • Disruption risk is minimized if retailers hold inv.
    • Diversificaitonis better

Multi-Echelon Inventory – June 15, 2006

inventory placement
Inventory Placement
  • Hold inventory upstream or downstream?
  • Conventional wisdom:
    • Hold inventory upstream
    • Holding cost is smaller
  • Under supply uncertainty:
    • Hold inventory downstream
    • Protects against stockouts anywhere in system

Multi-Echelon Inventory – June 15, 2006

outline31
Outline
  • Introduction
  • Stochastic models
  • Decentralized systems
    • Suboptimality
    • Contracting
    • The bullwhip effect

Multi-Echelon Inventory – June 15, 2006

decentralized systems
Decentralized Systems
  • So far, we have assumed the system is centralized
    • Can optimize at all stages globally
    • One stage may incur higher costs to benefit the system as a whole
  • What if each stage acts independently to minimize its own cost / maximize its own profit?

Multi-Echelon Inventory – June 15, 2006

suboptimality
Suboptimality
  • Optimizing locally is likely to result in some degree of suboptimality
  • Example: upstream stages want to operate make-to-order
    • Results in too much inventory downstream
  • Another example:
    • Wholesaler chooses wholesale price
    • Retailer chooses order quantity
    • Optimizing independently, the two parties will always leave money on the table

Multi-Echelon Inventory – June 15, 2006

contracting
Contracting
  • One solution is for the parties to impose a contracting mechanism
    • Splits the costs / profits / risks / rewards
    • Still allows each party to act in its own best interest
    • If structured correctly, system achieves optimal cost / profit, even with parties acting selfishly
  • There is a large body of literature on contracting
    • In practice, idea is commonly used
    • Actual OR models rarely implemented

Multi-Echelon Inventory – June 15, 2006

bullwhip effect bwe
Bullwhip Effect (BWE)
  • Demand for diapers:

Multi-Echelon Inventory – June 15, 2006

irrational behavior causes bwe
Irrational Behavior Causes BWE
  • Firms over-react to demand signals
    • Order too much when they perceive an upward demand trend
    • Then back off when they accumulate too much inventory
  • Firms under-weight the supply line
  • Both are irrational behaviors
  • Demonstrated by “beer game”

Multi-Echelon Inventory – June 15, 2006

rational behavior causes bwe
Rational Behavior Causes BWE
  • Famous paper by Hau Lee, et al. (1997)
  • BWE can be cause by rational behavior
    • i.e., by acting in “optimal” ways according to OR inventory models
  • Four causes:
    • Demand forecast updating
    • Batch ordering
    • Rationing game
    • Price variations

Multi-Echelon Inventory – June 15, 2006