FIFTH INTERNATIONAL AMSPM May, 2005 - Zakynthos Island, Greece Systems & Industrial Engineering University of Arizona Selection of Inventory Control Points

FIFTH INTERNATIONAL AMSPMMay, 2005 - Zakynthos Island, Greece

Systems & Industrial Engineering

University of Arizona

Selection of Inventory Control Points in Multistage Pull Systems

Ronald G. AskinShravan Krishnan

Systems & Industrial EngineeringUniversity of ArizonaTucson, AZ 85721




Overview

• Problem Introduction • Brief Literature Review• Model 1 – Known Container Size• Model 2 – Selecting the Container Size• Model 3 – Stage Dependent Containers• Summary and Conclusions




Tucson: Sonoran Desert




Kanban Controlled Pull System




Kanban Uses & Advantages

• Low – Moderate Variety• Moderate – High Volume, Low Variability• Reliable Processes (Predictable Lead Time)

• Low Information System Requirement• Self-adjusting (to minor variation/uncertainty)• Minimal Inventory Accumulation




Kanban Control with Distant Workstations




Background Literature

General Texts:• Y. Monden, TPS, 1998 (+ T. Ono)• Askin & Goldberg, Lean Production

Systems, 2002• R. Schoenberger, Japanese Mfg.

Tech., 1982

Research:

•Askin et al. IIE Trans., 1993

•Mitra & Mitrani, Mgmt Sci., 1990,

•Wang & Wang, IJPR, 1990,

•Spearman et al., IJPR, 1990 (CONWIP)

•Philipoom eta al, IJPR, 1987




Selecting the Control Points

1a

2n 1n …. 1b ….

2a 2b

Information flow (kanban)

Material flow

….

Control Section i Control Section i+1

Material and information flow




Model 1: Container Size Known

Notation:a = setup cost plus MH cost/n at iC = collection time at stage iD = Demand (mean/time)f = Fixed buffer cost/timeM = # stagesh = holding cost per unit/time at iL = Production lead time at it = transport time from i α = Service rate = Standard dev. demand/time

Variables:

1

1

j j

ij j r rr i r i

c L t

1 if stage is a control point that serves stage

0 otherwise ij

j iY

1 if stage is a control point

0 otherwise i

iX

ij = lead time i thru j




Known Container Size n

Minimize Costs (Fixed, Setup, Cycle, SS)

Subject to:

All stages assigned;

Identify Control Points;

Continuous Sections;

Last Stage has Buffer

1

( 1)( )

2i m

i i j j j ij i ii M i M j M i j

a D h nMin f X z h X c Y L t

n

1ijj i

Y i M

, , ij jY X i j M i j

1, ,ij i jY Y i j M

1mX

[0,1]jX [0,1]ijY




Shortest Path Analogy

0 1 2 m ……………

M02

M01

M12

M1m

M2m

M0m

= Relevant Cost if j and k are consecutive control points

1, 0,..., 1; 1,...,jk k k j kM f z h j m k j m




Single Control Section Result

Theorem 1: For the cost structure defined in model 1, a single control point is always optimal if for all stages j=1,…,m-1,

1, 1,j m m j ij m j mf z h h h .

Note: Sufficient condition almost always holds since for a, b >0,

a b a b




Model 2: Selecting n

• Case 1: Fixed Processing time

• Case 2:Variable Processing time

max 1 1/ 2min ,[2 ]j m ii M

n n Dh a

( )i i iL k s n p

i ii M

D h L

Add WIP cost to objective function




Model 2 Case 2

1

1

11

2*

2

m

ii

mj m

m i iim

D an

h Ph z k k D h p

1. Theorem 1 still holds for any n

2. Shortest Path Problem given n

j

ij rr i

P p

whereNonlinear! 1 ( )m f n




Model 2: Computational Results• Case 1:

– f = $0, $1000 (two configurations)– a = [0.1,0.12,0.13,0.08,0.15,0.22]– h = [1,2,3,4,5,6], [1,1,1,1,1,1] (2 configurations)– D = 100 units per day– α = 0.95– σ = 5 units– c = 0.2 days for each stage– p = 0.1 days for each stage– Number of stages = 6.

f h Control Points n Cost 0 1,…,.1 6 13 $1,848.

1000 1,…,1 6 13 $2,848 0 1,…,6 1, 6 6 $6,840

1000 1,…,6 6 6 $7,843




Model 3: Stage Dependent Container

• Nesting property:

• Objective function:

11j j jn r n

Integer r

11

1

(1 ) ( 1)

2 2

( )

j j j ji m mj j

j M i M j Mij jj i

i i i j j ij iji M j M i j

h X n ra D h nMin f X

Y n

h L t D z h X Y

Subject to




Heuristic

1. Estimate container sizes (working backwards from m to 1)

1/ 2

0 01

1/ 20

01

2

2max , }

kk k

k

m

kl

l kk

k

Daif n n

h

n D an otherwise

h




Heuristic cont.

* 01 1

1 1,*( , ]

( )max{0, } ( )

2

0,..., 1; 1,...,

k

ll j k jk k

jk k l l l k j kl j kjk

a Dh n n

M f h L t D z hn

j m k j m

1* max max1

2

min ,...., ,

k

ll j

jk j kk

D a

n n nh

2. Compute heuristic flow costs for shortest path algorithm

Case 1




Case 2

* 01 1 *

1,*1

( )( )

2

0,1,..., 1; 1,...,

k

l kl j k jk k

jk k l l jk k j kl jjk

a Dh n n

M f h L n D z hn

j m k j m

'1 1

1 1,1

( )( ) ( ( ) ) 0,1,..., 1; 1,...,

2

k

l kl j k jk k

jk l l jk l k j kl jjk

a Dh n n

f n h L n t D z h j m k j mn

( ) ( )l jk l l jkL n k s p n

'1 1mn 1 ' *

1max , 1,...,k k kkn n n k m

*jkn can be determined by minimizing




Model f h a D Control

Points n Cost

2 0 1,1,1,1,1,1 1,1,1,1,1,1 100 6 35 1885 2 0 1,1,1,1,1,1 32,16,8,4,2,1 100 6 100 1963 2 0 1,2,3,4,5,6 1,1,1,1,1,1 100 6,1 15 6983 2 0 1,2,3,4,5,6 32,16,8,4,2,1 100 6 46 7175 2 1000 1,1,1,1,1,1 1,1,1,1,1,1 100 6 35 2885 2 1000 1,1,1,1,1,1 32,16,8,4,2,1 100 6 100 2963 2 1000 1,2,3,4,5,6 1,1,1,1,1,1 100 6 15 7985 2 1000 1,2,3,4,5,6 32,16,8,4,2,1 100 6 46 8175 3 0 1,1,1,1,1,1 1,1,1,1,1,1 100 6 35 1888 3 0 1,1,1,1,1,1 32,16,8,4,2,1 100 6 100 1967 3 0 1,2,3,4,5,6 1,1,1,1,1,1 100 6,1 15,13 6991 3 0 1,2,3,4,5,6 32,16,8,4,2,1 100 6,1 80,33 7158 3 1000 1,1,1,1,1,1 1,1,1,1,1,1 100 6 35 2888 3 1000 1,1,1,1,1,1 32,16,8,4,2,1 100 6 100 2967 3 1000 1,2,3,4,5,6 1,1,1,1,1,1 100 6 15 7997 3 1000 1,2,3,4,5,6 32,16,8,4,2,1 100 6 46 8187

* 01 1

1 1,*( , ]

( )( )

2

0,..., 1; 1,...,

k

ll j k jk k

jk k l l l k j kl j kjk

a Dh n n

M f h L t D z hn

j m k j m

`

Some model 3 costs are higher because model 3 contains this extra term




Summary and Future

• Single control point often optimal for simple system

• Expression for container size

• Multiple control points for highly varying costs (high value added)

• Multiple products with limited processor time

• Assembly and General product structures

• Discrete (Poisson) demand

• Batch vs. Unit processors (eg. Ovens)

Documents

FIFTH INTERNATIONAL AMSPM May, 2005 - Zakynthos Island, Greece Systems & Industrial Engineering University of Arizona Selection of Inventory Control Points