Petri Net Models for Supply Chain Operational Management and Performance Evaluation

(1)

Mariagrazia Dotoli,

Maria Pia Fanti,

Giorgio

Iacobellis

, Agostino M.

Mangini

DEE

-

Politecnico di Bari

[email protected]

Petri Net Models for Supply Chain

Operational Management and

Performance Evaluation

(2)

Presentation Outline

Presentation

Outline

•

• IntroductionIntroduction toto SupplySupply ChainsChains (SC)(SC) •

• AimAim ofof the the researchresearch •

• SC SC operationaloperational modelmodel byby First First OrderOrder HybridHybrid Petri Petri NetsNets (FOHPN)(FOHPN) •

• SC SC operationaloperational modelmodel byby ColouredColoured TimedTimed Petri Petri NetsNets (CTPN)(CTPN) •

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Presentation Outline

Presentation

Outline

•

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Introduction to Supply Chains (SC)

Suppliers Manufacutrers Distributors Retailers Customers Coordination center

Information Flow

Material Flow

Recyclers

SC (Supply Chain):

SC (Supply Chain): collectioncollection of of independent

independent companiescompanies withwith complementary

complementary skyllsskylls integratedintegrated with

with transportationtransportation and and storagestorage systems

systems

Aims of SC:

converting raw material into

finished products

improving profits

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SC definition: Operative level

•

• EntitiesEntities ofof a SC:a SC: •

• 1 1 -- SuppliersSuppliers:: provideprovide rawraw material, componentsmaterial, components and semiand semi--finishedfinished products.products. •

• 2 2 –– ManufacturersManufacturers//assemblersassemblers: : transformtransform input material/componentsinput material/components intointo desired

desired productsproducts. . •

• 3 3 -- DistributorsDistributors: : are intermediate are intermediate nodesnodes ofof materialsmaterials flow.flow. •

• 4 4 –– RecyclersRecyclers//dede--manufacturersmanufacturers:: theythey feedfeed recoveredrecovered material. material. •

• 5 5 –– CustomersCustomers or or retailersretailers:: are sinkare sink nodesnodes ofof material flow.material flow. •

• 6 6 -- LogisticsLogistics and and transporterstransporters:: connectconnect the facilitiesthe facilities. . The SC

The SC dynamicsdynamics dependsdepends on the management methodologieson the management methodologies specifyingspecifying the the business

business modelmodel, the information and material flow., the information and material flow.

• Make-to-Stock (MTS): A push strategy so that customers are satisfied from stocks of inventory of finished goods.

• (R.Q) inventory control rule for MTS: fixed part quantities Q are ordered any time the stock level drops below the reorder level R.

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Presentation Outline

Presentation

Outline

•

• AimAim ofof the researchthe research •

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Aim of the research

Aim

of the

research

Models

Models based on PNsbased on PNs proposed in the related literature:proposed in the related literature:

1)

1) ViswadhamViswadham 2000, Dotoli 2000, Dotoli etet al. 2005 -al. 2005 - GeneralizedGeneralized StochasticStochastic Petri Petri NetsNets, , resource

resource orientedoriented model : model : placesplaces are SC are SC facilitiesfacilities, , tokenstokens are are lotslots of of productsproducts.. Limits:

Limits: state space of the SC model is excessively largestate space of the SC model is excessively large, the , the differentdifferent typestypes of of products

products in the system are notin the system are not modeled.modeled. 2)

2) ChenChen etet al. 2005 -al. 2005 - Modeling and performance evaluation of supply chains using Modeling and performance evaluation of supply chains using batch deterministic and stochastic Petri nets

batch deterministic and stochastic Petri nets ..

Limits:

Limits: the paper does not face the basic aspect of the SC management and the paper does not face the basic aspect of the SC management and optimization problems at the operative level

optimization problems at the operative level. .

We

We propose two effective and efficient modular models for SC propose two effective and efficient modular models for SC management and control at the operative level:

management and control at the operative level:

•

• 1) First Order1) First Order HybridHybrid Petri Petri NetsNets (FOHPN). (FOHPN). •

• 2) Coloured2) Coloured TimedTimed Petri NetsPetri Nets (CTPN).(CTPN). •

• The first may be used for testing /selection of SC operational control strategies.The first may be used for testing /selection of SC operational control strategies. •

• The second may be employed for performance evaluation and verification of the The second may be employed for performance evaluation and verification of the SC operation, enlightening the SC profitability, productivity, a

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Presentation Outline

Presentation

Outline

•

• SC operationalSC operational modelmodel byby First OrderFirst Order HybridHybrid Petri Petri NetsNets (FOHPN)(FOHPN) •

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A First

Order

Hybrid

Petri Net Model

for

Supply

Chain

Management

IEEE Trans. Autom. Science and Engineering, 2009

Mariagrazia Dotoli, Maria Pia Fanti, Giorgio Iacobellis, Agostino Marcello Mangini

Dipartimento di Elettrotecnica ed Elettronica, Politecnico di Bari

First

-

order hybrid Petri nets.

An application to distributed manuf. systems

Nonlinear Analysis: Hybrid Systems, 2008

Mariagrazia Dotoli, Maria Pia Fanti, Alessandro Giua, Carla Seatzu

Dipartimento di Elettrotecnica ed Elettronica, Politecnico di Bari

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Model with FOHPN (I):

Model

with

FOHPN (I):

FOHPN are introduced by

• Balduzzi, Giua, Menga, (2000). Modelling and Control with First order Hybrid Petri Nets. IEEE Trans. Robotics and Automation, 4 (16), pp. 382 – 399.

Key features of FOHPNs:

• increasing in computational efficiency with respect to place/transition models;

• reducing the dimension of the state space due to fluid approximation;

• the possibility of using gradient information to speed up optimization;

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• We use a hybrid model to describe the dynamics of the SC that is traced by the flow of products between facilities and transporters. • The continuous dynamics models the flow of material, the

manufacturing, the assembling of different products and the storage of products in the buffers.

• The discrete event dynamics models the occurrence of stochastic events:

the blocking of the raw material supply and transport operations due to strikes, shifts accidents, etc..

Model with FOHPN (I):

Model

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•

• A FOHPN is a bipartite digraph described by the sevenA FOHPN is a bipartite digraph described by the seven--tupletuple

PN

=

(

P,

P

,

T

,

Pre,

Pre

,

Post,

Post

,

Δ

,

F, RS

).

Model with FOHPN (II):

Definition of PN

Model

with

FOHPN (II):

Definition

of PN

•

P

==

P

_c_c∪∪

P

_d_d is_is the set of the set of placesplaces thatthat modeling modeling resourcesresources:: places

places

P

_c_c modelmodel flow of material (flow of material (buffersbuffers and and relatedrelated capacity

capacity)) places

places

P

_d_d modelmodel preferencespreferences, , constraintsconstraints and state and state (operative,

(operative, failedfailed) of SC ) of SC entitiesentities

T

==

T

_I_I∪∪

T

_c_c∪∪

T

_S_S∪∪

T

_D_Dtransitions_transitions setset •

• transitionstransitions

T

_I_I modelsmodels preferencespreferences •

T

_c_c modelsmodels the the continouscontinous flow of materialflow of material (production,

(production, assemblingassembling, , recyclingrecycling, , loadload//unloadunload operationsoperations)) •

T

_S_S modelmodel the discrete flow of the discrete flow of materialsmaterials ((transporttransport, , distribution

distribution), ), stochasticsstochastics eventsevents ((blockingblocking, , requestsrequests) ) •

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Model with FOHPN (III):

Definition of PN

Model

with

FOHPN (III):

Definition

of PN

•

• FunctionFunction ΔΔ specifies the timing associated to timed transitionsspecifies the timing associated to timed transitions (zero (zero for

for immediate immediate transitionstransitions, , exponentialexponential or or triangulartriangular distributiondistribution forfor stochastic

stochastic transitionstransitions, , constantconstant valuesvalues forfor deterministicdeterministic transitionstransitions).). •

• FunctionFunction F F specifiesspecifies VV_mj_mj and and VV_Mj_Mj the minimum and the minimum and maximummaximum firingfiring speeds

speeds associatedassociated toto jj--thth continuous transitioncontinuous transition .. •

• FunctionFunction RS RS associates a probability value called random switch to associates a probability value called random switch to conflicting discrete transitions

conflicting discrete transitions •

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A

A macro eventmacro event occurs when: occurs when:

• a discrete transition fires;

• a continuous place becomes empty;

• a continuous place, whose marking is increasing (decreasing), reaches a flow level that enables (disables) one or more discrete transitions

Any admissible IFS (Instantaneous Firing

Speeds, IFS) vector v at m is a feasible solution of the following set of linear set S(PN,m): :

Model with FOHPN (IV):

The net dynamics: time driven and event driven

Model

with

FOHPN (IV):

The net

dynamics: time

dynamics

: time driven

driven

and event

and

event

driven

( )

(

,

)

( )

j C i i j j t T

m

C p t v

∈

τ =

∑

τ

The equation governing the dynamics of the

discrete transitions is:

The equation governing the dynamics of the

continuous transitions: continuous transitions: ( ) ( ) ( ) ( ) ( ) ( ) k k k k k k − − τ = τ + τ τ = τ + τ c c cd d d dd m m σ m m σ C C

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•

• The FOHPN is described by a discreteThe FOHPN is described by a discrete--time state equation:time state equation:

1

(τ_k₊ ) = A(τ_k) (τ + τ_k) B( _k) (τ_k)

x x u

Model with FOHPN (V):

The net dynamics: time driven and event driven

Model

with

FOHPN (V):

The net

dynamics

: time

driven

and

event

driven

1 1 1 1 1 ( ) 0 0 ( ) ( ) ( ) 0 0 ( ) 0 ( ) ( ) ( ) 0 0 ( ) ( ) ( ) 0 k k cc k dc k k k k dd k k k k k k + + + + + ⎡ τ ⎤ ⎡ ⎤⎡ τ ⎤ ⎡ τ ⎤ τ − τ ⎡ ⎤ ⎢ _τ ⎥ ₌ _⎢ _⎥⎢ _τ ⎥₊_⎢ _τ _⎥ ⎢ ⎥ ⎢ ⎥ _⎢ _⎥⎢ ⎥ _⎢ _⎥_⎣ _τ _⎦ ⎢ _τ ⎥ _⎢_⎣ _τ _⎥_⎦⎢ _τ ⎥ _⎢_⎣ _τ _⎥_⎦ ⎣ ⎦ ⎣ ⎦ c c d d v σ ν m I m C C m I m C D v f σ state input •

• The behavior of the system is described in the The behavior of the system is described in the macroperiodmacroperiod [[ττ_k_k,_,ττ_k+1_k+1) by _{) by} the following equations:

the following equations:

y

(

τ

_k

) =

Ex

(

τ

_k

)

•

• The net state is given by the net marking and by the vector The net state is given by the net marking and by the vector νν(₍ττ_k_k) of timers _{) of timers} associated to discrete timed transitions at time

associated to discrete timed transitions at time ττ_k_k.. •

• The net input includes the macro-The net input includes the macro-period duration and the firing vector period duration and the firing vector σσ((ττ_k+1_k+1) ) associated to the firing of transition

associated to the firing of transition tt_j_j at at ττ_k_k ._. •

• The elements of matrix The elements of matrix DD((ττ_k_k) and vector _{) and vector}ff((ττ_k_k) are equal to 0/1 and depend on the _{) are equal to 0/1 and depend on the} macro

macro--event occurring at the sampling instant.event occurring at the sampling instant. •

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p

_B_B: the buffer of finite capacity: the buffer of finite capacity

p

p’

’

_B_B: the capacity : the capacity

m

_B_B++

m’

m

’

_B_B=C=C_B_B

t

TT

i

i : the : the trasporttrasport

t

DD

i

i: the demand: the demand

Q

Q_i_i, , QQ’’_i_i: fixed order quantity : fixed order quantity R

R_B_B: reorder level: reorder level p’B tTn tT1 tD1 tD_m pC pB Q1 Qn t1 Q’1 Q’m Q’m Q’1 CB - RB CB 0 CB - RB -Q1 C_B- R_B -Q_n

…

Input buffer

module

by

the (R,Q) policy

Model with FOHPN (VI):

Subsystems model of the SC

Model

with

FOHPN (VI):

Subsystems

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p

_B_B: : the buffer of finite capacitythe buffer of finite capacity

p

p’

’

_B_B: : the capacity the capacity

m

_B_B++

m

’

_B_B=C=C_B_B

t

₁₁ : the arrival of material: the arrival of material

p

_k_k: the supplier in operative state: the supplier in operative state

p

p’

’

_k_k:: the supplier blocked the supplier blocked

t

_k_k

:

the blocking of blocking of materilmateril supplysupply

t

t’

’

_k_k: the : the restoretionrestoretion of of materilmateril supplysupply

t

_T_T: : trasporttrasport operationoperation p’k pk t’k tk _p B p’B t1 tT 0 CB Q Q Q Q

Supplier

module

Model with FOHPN (VII):

Model

with

FOHPN (VII):

Subsystems

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p

_Bi: the input buffer of finite capacitythe input buffer of finite capacity

p’

_Bi: avaibleavaible capacitycapacity

m

_Bi_Bi++

m

’

_Bi_Bi==CC_Bi_Bi

p

_B1: the output buffer of finite capacitythe output buffer of finite capacity

p’

_B1: avaibleavaible capacity capacity

m

_B1_B1++

m

’

_B1_B1=C=C_B1_B1

Manufacturer

/assembler

module

Model with FOHPN (VIII):

Model

with

FOHPN (VIII):

Subsystems

model of the SC

CB2 – RB2 CB3 – RB3 CBn – RBn Q2 Q3 Qn CB2 – RB2 –Q2 CB3 – RB3 –Q3 CBn – RBn –Qn t_j p’B2 0 pB2 CB2 p’B1 CB1 pB1 0 Q1 p’B3 0 pB3 CB3 … CBn 0 pBn Q2 Q2 Q1 Q3 Q3 Qn Qn p’Bn

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p

p_Bi_Bi: downstream facility’: downstream facility’s input s input buffer

buffer

p

p’’_Bi_Bi: downstream facility: downstream facility’’s s avaibleavaible capacity

capacity mm_Bi_Bi++mm’’_Bi_Bi=C=C_Bi_Bi

p

p_k_k: transporter in the operative : transporter in the operative state

state

p

p’’_k_k: transporter in the blocking : transporter in the blocking state

state

t

t_k_k: transporter bloking: transporter bloking

t

t’’_k_k: transporter becomes : transporter becomes operative

operative

p

p_ci_ci: control places that : control places that determines the

determines the choisechoise of of material to transport

material to transport

p

p₁₁: it indicates if the transporter : it indicates if the transporter is

is avaibleavaible or not avaibleor not avaible

t

tTT i

i : transporters of i items, : transporters of i items,

triangular distribution triangular distribution

Transport

module

Model with FOHPN (IX):

Model

with

FOHPN (IX):

Subsystems

model of the SC

p’k pk t’k tk tT1 pC1 . . . pCn tTn t1 tn p1 p’B1 pB1 p’Bn pBn Q Q CB1-R1 CB1-R1-Q 0 0 CB1 CBn CBn-Rn CBn-Rn-Q Q Q Q Q . . . . . .

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p

p_B_B: : buffer of finite capacitybuffer of finite capacity

p

p’’_B_B: : the the capacitycapacity mm_B_B++mm’’_B_B=C=C_B_B

t

tTT i

i : transporter of i items: transporter of i items

t

tDD i

i: request of i items: request of i items

P

P_c_c: control place : control place

Distributor

module

Model with FOHPN (X):

Model

with

FOHPN (X):

Subsystems

model of the SC

p’_B tT_n tT₁ tD₁ _tD m p_C p_B Q1 Qn t₁ Q’1 Q’m Q’m Q’1 CB - RB C_B ₀ CB - RB -Q1 C_B- R_B -Q_n

…

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•

p

_B_B: : the buffer of finite the buffer of finite capacitycapacity •

•

p

’

_B_B: the : the capacitycapacity

m

_B_B++

m’

m

’

_B_B=C=C_B_B. . •

•

t

₁₁ : product request, exponential : product request, exponential distribution

distribution •

•

p

_F_F: retailer: retailer •

•

t

_L_L

:

detereoretiondetereoretion of finished products of finished products •

•

ps

: products to be de: products to be de--manufacturedmanufactured

•

p

_D_D: : goods to be goods to be descerdeddescerded

•

t

_T_T

:

transport operationtransport operation

pB p’B t1 pF Q CB – RB - Q CB – RB CB 0 Q1 Q1 Q1 pS Q2 tL µQ2 tT Q3 pD (1-µ)Q₂

Retailer

module

Model with FOHPN (XI):

Model

with

FOHPN (XI):

Subsystems

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De

-

manufacturer

module

Model with FOHPN (XII):

Model

with

FOHPN (XII):

Subsystems

model of the SC

pB2 p’B2 t1 pB1 CB2 ₀ pBn p’Bn 0 CBn

…

Q2 p’B1 _CB1 ₀ Q2 Qn Qn pB p’B3 0 CB3 Q3 Q3 Q’2 Q’2 Q’₃ Q’3 Q’_n Q’n CB1 – RB1 Q1 CB1 – RB1 –Q1 p

p_Bi_Bi: the input buffer of finite : the input buffer of finite capacitycapacity

p

p’’_Bi_Bi: the input buffer : the input buffer capacitycapacity mm_B1_B1++mm’’_B1_B1=C=C_B1_B1

p

p_Bi_Bi: the output buffer of finite : the output buffer of finite capacitycapacity

p

p’’_B_B_i_i: the output buffer : the output buffer capacitycapacity mm_Bi_Bi++mm’’_Bi_Bi=C=C_Bi_Bi

t

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Model with FOHPN (XIII):

SC control

Model

with

FOHPN (XIII):

SC control

1

(τ_k₊ ) = A(τ_k) (τ + τ_k) B( _k) (τ_k)

x x u •• AA((ττ_k_k) and ) and BB((

τ

_k_k) describe the SC in [) describe the SC in [ττ_k_k,,ττ_k+1_k+1) ) and depend on the event occurring in t= and depend on the event occurring in t=ττ_k_k

•

• The vector IFS The vector IFS vv((ττ_k_k)₎∈∈

S

(PN,_(PN,mm((

τ

_k_k)) affects the matrix )) affects the matrix BB((ττ_k_k)₎

Ö

Ö Choice of Choice of v(v(ττ_k_k)) in each macroin each macro--period on the basis of the knowledge of period on the basis of the knowledge of

the SC state in order to optimize a particular objective functio

the SC state in order to optimize a particular objective function with n with

y

y((ττ_k_k)=)=EmEmcc₍₍_τ_τ k

k) and with ) and with mmrra reference vector a reference vector

Ö

Ö Input vector Input vector uu((ττ_k_k) determined by a decision maker) determined by a decision maker

1 1 ( ) ( ) k k k k + + τ − τ ⎡ ⎤ τ = ⎢ _τ ⎥ ⎣ σ ⎦ u Control Action Decision Maker B(τk) zk E A(τk) x(τk) y(τk) x(τk+1) + + - +v(τk) u(τk) mr

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Model with FOHPN (XIV):

SC control

Model

with

FOHPN (XIV):

SC control

•

• Controller 1 (C1): flow maximization.Controller 1 (C1): flow maximization.

m

m_r_r=0=0, y, y=m=mcc_{, max}_{, max}

v

vJJ11=max=maxvv((11TT··vv) s.t) s.t. . vv∈∈S(PN,S(PN,m)m)

•

• Controller 2 (C2): stored materials minimization.Controller 2 (C2): stored materials minimization.

m

m_r_r=0=0, y, y, min, min_v_vJJ₂₂ s.t. s.t. vv∈∈S(PN,_S(PN,m) withm) with 2 ( ) ( , )v ( ))

i j c c i _k _i _j _j _k p Y t T J m p t ∈ ∈ ⎡ ⎤ = _⎢ τ + τ _⎥ ⎢ ⎥ ⎣ ⎦

∑

C •

• Controller 3 (C3): buffer inventory control.Controller 3 (C3): buffer inventory control.

m

m_ri_ri reference value, reference value, yy,, min

min_v_vJJ₃₃ s.t. s.t. vv∈∈S(PN,_S(PN,m) withm) with

2 3 m ( ( ) ( , )v ( )) i j c ri i k i j j k p Y t T J m p t ∈ ∈ ⎡ ⎤ = _⎢ − τ + τ _⎥ ⎢ ⎥ ⎣ ⎦

∑

C Control Action Decision Maker B(τk) zk E A(τk) x(τk) y(τk) x(τk+1) + + - +v(τk) u(τk) mr

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s7 s4 s5 s3 s2 l1 l3 l4 l5 l2 l6 l9 l7 l8 l10 l11 s1 l12 s6 u1 M, K M, K C, H C, H, M PC PC PC PC PC PC C si lk Sito Trasportatore Cliente u2 uj H 3 suppliers 2 manufacturers 2 retailers 1 de-manufacturer 1 distributor

Case Study A (I):

Desktop Computer production

Case

Study

A (I):

Desktop Computer production

1 product (PC)

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3 suppliers

2 manufacturers 2 retailers

1 de-manufacturer 1 distributor

Case Study A (II):

FOHPN model

Case

Study

A (II):

FOHPN model

The model selects the production rates by solving the controller optimization problem

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• T= Throughput average number of products obtained in a time unit

•

SI= System Inventory

average number of products storedstored in in allall the the system

system buffersbuffers duringduring the the runrun time TPtime TP

• LT=SI/T =Lead Time a measure of the time spent by the SC to

convert raw material into final products

Case Study A (III):

Performance indices and Controller

Case

Study

A (III):

Performance

indices

and Controller

• Controller C1 (flow maximization)

• Controller C2 (stored materials minimization)

P

_Y={

p

₁,

p

₃,…,

p

₁₃,

p

₃₁,

p

₃₃}.

• Controller C3-1 (buffer inventory control)

P

_Y={

p

₃₁,

p

₃₃},

m

_r31=

m

_r33=80

• Controller C3-2 (buffer inventory control)

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Case Study A (IV):

Data

Case

Study

A (IV):

Data

Buffers and transporters capacity, reorder level and

fixed order quantities

Transitions continuous (discrete) firing speed

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Case Study A (V):

Simulation

Case

Study

A (V):

Simulation

•

• The model The model isis implementedimplemented and and simulatedsimulated in MATLAB in MATLAB environmentenvironment..

•

• 50 50 indipendentindipendent replicationsreplications of TP=600 h, of TP=600 h, withwith a a transienttransient periodperiod of TR=100 h.

of TR=100 h. WeWe obtainobtain a a halfhalf widthwidth thatthat in the in the worstworst case case equalsequals 4.7%

4.7% forfor a 95% a 95% confidenceconfidence intervalinterval.. •

• At the beginning of each macro_{At the beginning of each macro}-_-period (t=_{period (t=}ττ_k_k) the program describes _{) the program describes} and solves the optimization problem on the basis of the knowledg and solves the optimization problem on the basis of the knowledge e

of

x

((ττ_k_k) and fixes _{) and fixes}vv((ττ_k_k), _),AA((ττ_k_k), _),BB((ττ_k_k), _),ττ_k+1_k+1 (next macro_{(next macro}--event). After this event). After this it determines the next state

it determines the next state

x

((ττ_k+1_k+1) by the state equation and the _{) by the state equation and the} procedure is re

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Case Study A (VI):

Simulation results

Case

Study

A (VI):

Simulation

results

•

• ThroughputThroughput maxmax withwith C1, min withC1, min with C2,

C2, hitghthitght withwith C3-C3-1 e C31 e C3--22

•

• System System InventoryInventory min withmin with C1, C1, max

max withwith C1 and intermediate C1 and intermediate with

with C3-C3-1 e C31 e C3--2 (2 (lessless thanthan C3-C3-2) 2)

1.75 1.41 1.72 1.74 0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40 1.60 1.80 2.00 C1 C2 C3-1 C3-2 Controller Th roug hput [ P ar ts per hour ] 1278.20 925.93 1194.60 1026.30 200.00 400.00 600.00 800.00 1000.00 1200.00 1400.00 S y s tem In ven to ry [P a rts]

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Case Study A (VII):

Simulation results

Case

Study

A (VII):

Simulation

results

731.29 657.28 694.22 588.61 0.00 100.00 200.00 300.00 400.00 500.00 600.00 700.00 800.00 C1 C2 C3-1 C3-2 Controller Lead Ti m e [Hour s ] 1000 200 300 400 500 600 50 100 150 Time [hours] inve ntory le vel [p arts] C1 C2 C3-1 C3-2 •

• LeadLead timetime min withmin with C3-C3-2.2. •

• LeadLead timetime higherhigher withwith C3-C3-1 1 that

that withwith C3-C3-22

•

• buffer buffer markingmarking forfor final final productsproducts pp₃₁₃₁ into

into manufacturermanufacturer M1 .M1 . •

• C1: buffer alwaysC1: buffer always fullfull •

• C2: buffer alwaysC2: buffer always emptyempty •

• C3-C3-1 and C3-1 and C3-2 2 tendtend toto keepkeep the stock the stock levels

levels aroundaround the imposedthe imposed set-set-pointpoint (80

(32)

Presentation Outline

Presentation

Outline

•

• SC operationalSC operational modelmodel byby ColouredColoured TimedTimed Petri NetsPetri Nets (CTPN)(CTPN) •

(33)

A Coloured Timed Petri Net Model

for Supply Chain Management

and Performance Evaluation

“Modeling, Control, Simulation and Diagnosis of

Complex Industrial and Energy Systems”,

ISA/O3NEDIA, Luca Ferrarini and Carlo Veber,

Editors, 2009, ISBN 978-1-934394-90-8 2009

Mariagrazia Dotoli, Maria Pia Fanti, Giorgio Iacobellis Dipartimento di Elettrotecnica ed Elettronica, Politecnico di Bari

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•

• Compact and Compact and resourceresource orientedoriented modelmodel toto describedescribe the SC the SC dynamicsdynamics byby the flow the flow of

of batchesbatches ofof productsproducts amongamong SC SC facilitiesfacilities byby transporterstransporters.. •

• The discrete The discrete eventevent dynamicsdynamics modelsmodels stochasticstochastic events: client events: client demansdemans, , block/

block/restartrestart ofof supplies, block/supplies, block/restartrestart ofof transportationstransportations due todue to strikes, strikes, accidents

accidents.. •

• The The tokentoken color color definesdefines the the productproduct characteristicscharacteristics (e.g., type(e.g., type, , quantityquantity))

Model with CTPN (I):

Characteristics

Model

with

CTPN (I):

Characteristics

• Jensen, Colored Petri Nets: Basic Concepts, Analysis Methods and Practical Use,

EATS Monography and Theoretical Computer Science, Springer Verlag, 1992.

• Formalism using timed PN in which a color is associated to any token

Advantages:

• Modularity

• Model suitable for simulation • Applicability of control policies

• Possibility to distinguish the SC products: different dimensions and quantities, different manufacturing/transportation paths

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•

A bipartite digraph described by the 9

-

tuple

CTPN

CTPN=

=

(

P,

P

,

T

, Co,

Pre,

Pre

,

Post,

Post

,

Ω

,

F, RS,M

₀₀

).

•

P

set of places modelling activities: transportation,

storage in manufacturers, buffers availability,

presence of orders

•

T

=

T

_F_F

∪

T

_I_I

set of transitions where

Transitions in

T

_F_F

model the flow of jobs

Transitions in

T

_I_I

model the input of jobs in the

system

Model with CTPN (III):

PN Definition

Model

with

CTPN (III):

PN

(36)

<

λ

j,

θ

j

>

•

CTPN=

CTPN

=

(

P,

P

,

T

, Co,

Pre,

Pre

,

Post,

Post

,

Ω

,

F, RS,M0

)

Coloured tokens represent jobs and orders.

•

• Each token represents a batch of parts or of orders Each token represents a batch of parts or of orders ((rawraw material, material, semi

semi--finishedfinished partsparts).).

•

• The marking M denotes the state of the CTPN:The marking M denotes the state of the CTPN: M(p

M(p) gives the colours associated with each token in p.) gives the colours associated with each token in p. M0 is the initial marking

M0 is the initial marking •

• The colour of each token j is a couple:The colour of each token j is a couple: <<λλj_j,,θθj_j> , > , where where λλj_j is the batch is the batch type and

type and θθj_j the corresponding assigned quantity the corresponding assigned quantity θθj_j.. •

•

Co

mapsmaps eacheach placeplace

p

∈∈

P

into_into the set of the set of admissibleadmissible coulorcoulor

Co(p)

and and eacheach transition

transition tt∈∈

T

into_into the set of the set of admissibleadmissible fireablefireable coulorcoulor

Co

(t). (t). LetLet ΩΩ

defined as

defined as: : ΩΩ =₌∪∪x_x∈∈P_P∪∪T{Co(x)}. _T{Co(x)}.

Model with CTPN (IV):

PN Definition

Model

with

CTPN (IV):

PN

(37)

•

CTPN=(

CTPN

=(

P,

P

,

T, Co,

T

, Co,

Pre,

Pre

,

Post

,

Ω

,

F, RS,M

₀₀

).

+ ⎧ × → ⎪ ⎨ × → ⎪⎩ \ ` c d P T Pre,Post : P T

•

F

: specifies the firing time associated to each

transition (exponential distribution or null value).

•

Matrix

Pre

gives the enabling condition:

t

∈

T

is enabled at marking M with respect to a colour

c

∈

Co(t

Co

(t

)

iff

for each p in input of t, it holds

M(p)

≥

Pre

(p,t)(c

),

•

Matrices

Pre

and

Post

update the marking

: upon firing,

t leads to a new marking M

’

:

M

’

(p

)=

M(p)+

Post

(p,t)(c)

-

Pre

(p,t)(c

).

Model with CTPN (V):

PN Definition

Model

with

CTPN (V):

PN

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The model of a

general

purpose

site

(manufacturer(manufacturer/assembler, /assembler,

distributor

distributor, de, de--manufacturermanufacturer, , suppliersupplier))

The CTPN modelling the SC sites

The CTPN

modelling

the SC

sites

Inventory control rule: (R,Q)

•

p

_i_i: the product is in the : the product is in the input bufferinput buffer •

•

p

’

_ic_ic: the : the available capacityavailable capacity. . •

•

t

_t_t :the transport of material.:the transport of material. •

•

t

_I_I :the uploading in the production buffer._{:the uploading in the production buffer.} •

•

p

_P_P: the : the product is in productionproduct is in production •

•

p’

p

’

_Pc_Pc: the capacity of the production facility: . . •

•

t

_p_p

:

the production operation the production operation •

•

p

_o_o: : the product is in the output bufferthe product is in the output buffer •

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The input buffer can accomodate

n1

lots

of

type

λ

1 and of

θ

1

pieces

Capacities

,

reorder

functions

and

reorder

quantities

can

be

different

for

each

type

of

product

:

Cap

(

p

_Ic_Ic

)= (n

₁₁

<

λ

1,

θ

1>,n

₂₂

<

λ

2,

θ

2>

…

)

The model of a

general

purpose

site

distributor

The CTPN modelling

the SC sites (II)

The CTPN

modelling

the SC

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UP

UP isis a a functionfunction thatthat updatesupdates the the tokentoken colors

colors withwith the new color the new color obtainedobtained afterafter the

the operationoperation.. No

No labellabel standsstands forfor ““the the functionfunction makes

makes no no transformationtransformation in the in the elementselements””..

The model of a

general

purpose

site

distributor

The CTPN modelling

the SC sites (III)

The CTPN

modelling

the SC

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The model of a

general

purpose

site

distributor

The output

The output inventoryinventory of of productproduct λλ1 in _{1 in}pp_O_O is

is higherhigher thanthan the output the output reorderreorder pointpoint,, i.e. M(

i.e. M(pp_O_O)>R()>R(pp_Oc_Oc)(<)(<λλ1,1,θθ1>):1>): Transition

Transition tt_I_I isis inhibitedinhibited butbut tt_P_P continue continue to

to produce.produce. The output

The output inventoryinventory of of productproduct λλ1 in _{1 in}pp_O_O drops

drops belowbelow the output the output reorderreorder pointpoint,, i.e. M(

i.e. M(pp_O_O)< R()< R(pp_Oc_Oc)(<)(<λλ1,_1,θθ1>):_1>): Transition

Transition tt_I_I isis enabledenabled withwith respectrespect toto the color <

the color <λλ1,_1,θθ1>: _1>: Q(

Q(pp_Oc_Oc)(<)(<λλ1,_1,θθ1>) _1>)productsproducts of of typetype λλ1₁ will be produced.

The CTPN modelling

the SC sites (IV)

The CTPN

modelling

the SC

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•

p

_T_T

: batch under transport

•

p

_Tc_Tc

:

transport capacity

.

•

t

_T_T

: transport operation

•

t

_O_O

:

the transport uploading

•

p

_U_U

: available products

•

p

_D_D

:

outstanding orders

.

•

t

_oD_oD

: arrival of customer demand

•

t

_o_o

:

marketing of product of a

•

recycler

The model of a

transporter

The model of a

customer

The CTPN modelling

the SC sites (V)

The CTPN

modelling

the SC

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3 suppliers

2 manufacturers

2 retailers

Case Study B (I):

Case

Study

B (I):

• Two connections from suppliers

to retailers are added

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3 suppliers

2 manufacturers

2 retailers

Case Study B (II):

CTPN model

Case

Study

B (II):

CTPN model

The model is compact and is able to distinguish among

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• T= Throughput The average number of products per time unit

•

Inventory=INin+INout

inventory level of buffers

• INin Average nr of products in the input buffers • INout Average nr. of product in the output buffers

• CTij=

Order fulfilment cyle time of the product j-th for the

i-th customer,

i.e. the average time elapsing from the i-th customer request and the j-th product delivery.

• CT=

Order fulfilment cyle time,

i.e. the average time elapsing from a customer request and the product delivery.

Case Study B (III):

Performance indices

Case

Study

B (III):

Performance

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Case Study B (IV):

Data

Case

Study

B (IV):

Data

Data buffers and transporters Average firing time (hours) of exponential transitions of CTPN

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Case Study B (V):

Simulation results

Case

Study

B (V):

Simulation

results

•

• The CTPN The CTPN isis simulatedsimulated in the ARENA in the ARENA environmentenvironment •

• PlacesPlaces model model resourceresource withwith limitadedlimitaded capacitycapacity , timed, timed transitionstransitions model model delay

delay, , tokenstokens modelmodel entitiesentities and coulorsand coulors model model attributsattributs of entitiesof entities..

20

20 independentindependent replicationsreplications of of TP=11856 h,

TP=11856 h, withwith a transienta transient period

period of TR=3025 h.theof TR=3025 h.the halfhalf width

width of the of the confidanceconfidance interval

interval, , beingbeing 2.7% in the 2.7% in the worst

worst casecase Since

Since customerscustomers havehave the the same

same orderorder frequencyfrequency, , throughputs

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Case Study B (VI):

Simulation results

Case

Study

B (VI):

Simulation

results

•

• The CT are The CT are aboutabout 7h 7h consistentconsistent withwith the

the firingfiring timestimes •

• The CT42 and CT52 The CT are lowerare lower forfor the

the lowerlower valuesvalues of the firingof the firing times. times.

•

• The input buffersThe input buffers tend

tend toto accommodateaccommodate more

more partsparts thanthan the the output

(49)

Presentation Outline

Presentation

Outline

•

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Conclusions and Future Research

Conclusions

and Future

Research

We represent SC using the PN discrete event systems formalism

We propose two alternative SC modular models, devoted to the efficient description, management, and performance evaluation of SC at the

operational level. The FOHPN model

¾ describes material as continuous flows and the stochastic events occurring

in the system as discrete events;

¾ is resource oriented, modular and efficient;

¾ enables the designer to impose an optimal SC dynamics according to

objective functions and the knowledge of the system state.

The CTPN model

¾ describes in detail the SC products flow; ¾ is resource oriented, modular and efficient; ¾ is suitable for implementing control strategies.

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Mariagrazia Dotoli,

Maria Pia Fanti,

Giorgio

Iacobellis

, Agostino M.

Mangini

DEE

-

Politecnico di Bari

[email protected]

Petri Net Models for Supply Chain

Operational Management and

Performance Evaluation