OPTIMIZED FUZZY SCHEDULING OF MANUFACTURING

SYSTEMS

N. C. Tsourveloudis, L. Doitsidis

Department of Production Engineering and Management, Technical University of Crete,

Chania 73100,Greece

S. Ioannidis

Department of Mathematics, University of the Aegean, Samos, Greece

Keywords: Work-In-Process, Backlog, Fuzzy Scheduling, Evolutionary Algorithms, Manufacturing Systems.

Abstract: An Evolutionary Algorithm (EA) strategy for the optimization of generic Work-In-Process (WIP)

scheduling fuzzy controllers is presented. The EA is used to tune a set of fuzzy control modules which are

used for distributed and supervisory WIP scheduling. The distributed controllers objective is to control the

rate in each production stage so that satisfies the demand for final products while reducing WIP within the

system. The EA identifies the parameters for which the fuzzy controller performs optimal with respect to

WIP and backlog minimization. The proposed strategy is compared to known heuristically tuned fuzzy

control approaches. Simulation results show that the EA strategy improves system’s performance.

1 INTRODUCTION

The Work-In-Process (WIP) inventory is measured

by the number of unfinished parts in the buffers

throughout the manufacturing system. For various

reasons reported in (Conway et al, 1998) and

elsewhere, the in-process inventories should stay as

small as possible. The important question in WIP

management is: what is the minimum necessary

WIP? The answer, which is not straightforward, is

that WIP is highly associated with the fluctuations of

demand. WIP is accumulated when the actual

production rate is higher than the demand. However,

when WIP is very low, unpredicted phenomena,

such as machine failures, may lead the actual

production behind the demand and thus to delayed

deliveries and unsatisfied customers. Obviously,

product demands of constant level and pattern make

scheduling easier than randomly changing demands.

Control policies that tend to keep WIP in low

levels have drawn a great deal of attention from

researchers and practitioners (Gershwin et al, 1994),

(Bai et al, 1994). Recently, artificial intelligence-

based methodologies for the WIP control of realistic

(in terms of modelling assumptions) manufacturing

systems have been presented ((Tsourveloudis et al

2000),(Ioannidis et al, 2004), (Custodio et al, 1994)).

In previous work ((Tsourveloudis et al 2000),

(Ioannidis et al, 2004)), distributed and supervisory

schemes for the control of WIP where introduced. In

both approaches presented the controllers performed

better from traditional and surplus-based policies.

However, neither this approach has adopted a

systematic methodology that ensures optimal design

of the in – process inventory controllers.

In this paper we present an Evolutionary

Algorithm (EA) strategy for optimization of generic

WIP scheduling fuzzy controllers introduced in

(Tsourveloudis et al 2000), (Ioannidis et al, 2004).

The scheduling problem objective is to control the

production rate in a way that satisfies the demand for

final products while keeping minimum WIP within

the production system. During the evolution, the EA

identifies those set of parameters for which the fuzzy

controller performs optimal with respect to WIP

minimization for several demand patterns.

2 FUZZY SCHEDULING

A production system can be viewed as a network of

machines and buffers. Items are received at each

machine and wait for the next operation in a finite

capacity buffer. The machines break down randomly

and may be incapable of producing more parts

196

C. Tsourveloudis N., Doitsidis L. and Ioannidis S. (2005).

OPTIMIZED FUZZY SCHEDULING OF MANUFACTURING SYSTEMS.

In Proceedings of the Second International Conference on Informatics in Control, Automation and Robotics, pages 196-201

DOI: 10.5220/0001184401960201

 SciTePress

j,i

i,l

because of starvation and/or blocking phenomena.

Due to a failed machine with operational neighbors,

the level of the downstream buffer decreases, while

the upstream increases. If the repair time is big

enough, then the broken machine will either block

the next station or starve the previous one. This

effect will propagate throughout the system.

Clearly, production scheduling of realistic

manufacturing plants must satisfy multiple

conflicting criteria and also cope with the dynamic

nature of such environments. Fuzzy logic offers the

mathematical framework that allows for simple

knowledge representations of the production control

/ scheduling principles in terms of IF-THEN rules.

Two approaches of production scheduling have

been identified, the distributed and the supervisory

fuzzy scheduling. The advantage of the fuzzy

controllers used in the distributed approach is that

computationally simple and therefore facilitate

application to real time control/scheduling.

In the distributed fuzzy scheduling system

presented in (Tsourveloudis et al, 2000), three basic

subsystems have been introduced, namely transfer

line, assembly and disassembly module. The

majority of the real production networks can be

decomposed into these subsystems. Each subsystem

can be seen as a distributed fuzzy logic controller.

The inputs of the control modules (Table 1) are

the buffer levels B

, B

, the state s

of the

machine M

, the production surplus x

of M

and the

sole output is the processing rate r

of M

Table 1: Control Modules

Module Schema

Line

Assembly

Disassembly

The control objective of the distributed

scheduling approach, is to satisfy the demand keep

WIP as low as possible. This is attempted by

regulating the processing rate r

at every time instant.

The expert knowledge that describes the control

objective can be summarized as follows:

If the surplus level is satisfactory then try to

prevent starving or blocking by increasing or

decreasing the production rate accordingly.

If the surplus is either too low or too high then

produce at maximum or zero rate respectively.

The above knowledge is formally represented,

for the control modules of Table 1, by fuzzy rules. In

the case of the transfer line rule has the form:

IF b

j,i

is LB

(k)

AND b

i,l

is LB

(k)

AND s

is LS

(k)

AND x

is LX

(k)

THEN r

is LR

(k)

where k is the rule number, i is the number of

machine or workstation, LB is a linguistic value of

the variable buffer level b with term set B= {Empty,

Almost Empty, OK, Almost Full, Full}, s

denotes the

state of machine i, which can be either 1 (operative)

or 0 (stopped); consequently S= {0, 1}. LX

represents the value that surplus x takes and it is

chosen from the term set X= {Negative, OK,

Positive}. The production rate r takes linguistic

values LR from the term set R= {Zero, Low, Almost

Low, Normal, Almost High, High}. The processing

rate r

of each machine at every time instant is

⎪

⎩

⎪

⎨

⎧

′

∑

∗

)(

),,,(

iRi

iiliijisi

sif

sxbbfr

, (1)

where,

),,,(

,, iiliijis

sxbbf represents a fuzzy

inference system ([7], [8]) that takes as inputs the

level b

j,i

of the upstream buffer, the downstream

buffer level b

i,l

, x

is the surplus (cumulative

production minus demand) and s

is a non fuzzy

variable denoting the state of the machine, which

can be either 1 (operative) or 0 (stopped). In (1),

)(

∗

is the membership function of the aggregated

production rate, which is given by

)],,,(

),,,([minmax)(

)(

iiliij

iliijAND

xbb

rxbb

xbbr

iliij

µµ

∗∗

, (2)

where

),,(

iliijAND

xbb

is the membership function

of the conjunction of the inputs and

),,,(

)(

iiliij

rxbb

is the membership function of

the k-th activated rule. That is

)()()(),,(

,,,, iXliBijBiliijAND

xbbxbb

∗∗∗∗

∧∧=

µµµµ

, (3)

]

[

)(),(

),(),(),,,(

)()(

)()()(

,,,,

iiliij

bbrxbb

kkk

µµ

µµµ

→

, (4)

In equations (3), (4),

)(

,ijB

∗

and )(

,liB

∗

are

the membership functions (MFs) of the actual

upstream and downstream buffer levels and

)(

∗

is the membership function of production surplus.

In the supervised fuzzy scheduling approach, the

supervisory controller utilizes macroscopic data of

higher hierarchies to adjust the overall system's

behavior. Potentially, this may happen by modifying

the lower level controllers in a way to ultimately

achieve desired specifications. The supervisory

controller’s task, introduced in (Ioannidis et al,

2004) and its optimization discussed in the next

paragraph, is the tuning of the previously presented

j,i

k,i

i,l

j,i

i,l

i,k

OPTIMIZED FUZZY SCHEDULING OF MANUFACTURING SYSTEMS

197

distributed fuzzy controllers, in a way that improves

certain performance measures without causing a

dramatic change in the control architecture. The

overall scheduling approach remains modular since

the production control modules are not modified but

tuned by the additional supervisory controller.

In the supervisory scheduling scheme it is

assumed that the demand and the cumulative

production are known. This is important for the

production surplus monitoring and control and for

scheduling decisions based on production surplus.

The expert knowledge that describes the supervisory

control objective builds on the assumption that

adaptive surplus bounds may improve the

production systems performance and can be

summarized in the following statements:

If the upper surplus bound is reduced there is an

immediate reduction of WIP.

If the upper surplus bound is increased there is

an increase of WIP and the total production rate

leading to a small reduction of backlog.

If the lower surplus bound is increased a

substantial reduction of backlog and an increase in

WIP is achieved.

If there is a reduction of lower surplus bound as

a result we have a deterioration of backlog with an

improvement of WIP.

Surplus bounds are decided by the output of IF-

THEN rules of the following form:

IF mx

is LMX

(k)

AND e

is LE

(k)

AND e

(k)

THEN u

is LU

(k)

AND l

is LL

(k)

where, k is the rule number, mx

is the mean surplus

of the end product, LMX is a linguistic value of the

with term set MX= {Negative Big, Negative

Small, Zero, Positive Small, Positive Big}, e

is the

error of end product surplus (the difference between

surplus x

and the lower bound of surplus), with

linguistic value term set E

= {Negative, Zero,

Positive}. The relative deviation of WIP is denoted

and LE

is the linguistic value chosen from the

term set E

= {Negative, Zero, Positive}. The upper

surplus bound correction factor u

takes linguistic

values LU

from U

= {Negative, Negative Zero,

Zero, Positive Zero, Positive} and the lower surplus

bound correction factor l

takes linguistic values LL

from the term set L

= {Negative, Negative Zero,

Zero, Positive Zero, Positive}.

The crisp arithmetic values,

and

, of the

corrections of the upper and lower surplus bounds,

respectively, are given by the following

defuzzification formulas:

∑

⋅

)(

cUc

, (5a)

∑

⋅

)(

cLc

, (5b) (5a) (5a)

where

)(

and

)(

are the MFs of the upper

and lower surplus bounds, respectively. These MFs

represent the aggregated outcome of the fuzzy

inference procedure. The correct selection of input

and output membership functions characterizes the

performance of the overall scheduling task.

Since the form of the fuzzy rules of both the

distributed and supervised approach for fuzzy

scheduling have been identified, a crucial point is

the selection of the MFs. The correct choice of the

MFs is by no means trivial but plays a crucial role in

the success of an application. Consequently, the

selection of MFs if not based on a systematic

optimization procedure cannot guarantee minimum

WIP level. This is the main drawback of the

heuristic selection of MFs in case of known (or

almost known) demand patterns. The evolutionary

algorithm developed and explained in the next

section, creates MFs that fit best to scheduling

objectives. In this context, the design of the fuzzy

controllers (distributed or supervisory) can be

regarded as an optimization problem in which the set

of possible MFs constitutes the search space.

3 EVOLUTIONARY FUZZY

SCHEDULING

The use of evolving genetic structures for the

production scheduling problem, has recently gained

a lot of acceptance for the automated and optimal

design of fuzzy logic systems (Tedford et al, 2001).

Here, we consider the application of an evolutionary

algorithm for the optimal selection of MFs.

The MF defined in the previous paragraphs are

used to construct the chromosome. The basic idea is

to represent the complete set of MFs by an

individual (chromosome) and to evolve shape and

location of the MFs. This is shown in Figure 1 for

the case of trapezoidal and triangular MFs. An initial

population is derived from the first chromosome by

repeated application of the mutation operator. The

objective is to optimize a performance measure

which in the EAs context is called fitness function.

In each generation, the fitness of every chromosome

is first evaluated based on the performance of the

production network system which is controlled

through the membership functions represented in the

chromosome. A specified percentage of the better

fitted chromosomes, is retained for next generation.

Parents are selected repeatedly from the current

chromosomes generation, and new chromosomes are

generated from the parents. One generation ends

when the number of chromosomes for the next

ICINCO 2005 - INTELLIGENT CONTROL SYSTEMS AND OPTIMIZATION

198

generation has reached the quota. This process is

repeated for a pre-selected number of generations.

1−n

2−n

Figure 1: Chromosome created by the MFs

The structure of the distributed fuzzy logic

controllers as far as it concerns the rule base and the

linguistic variables remains the same with those

described in Section 2. The controllers used for

training have randomly created membership

functions. The initial population is consisted of

individuals which have the same initial chromosome

which contains the points a

, (i=1,…, n) that define

the membership functions of the inputs and the

output of the controllers. In case of more than one

controller the chromosome consists of the points that

define all membership functions of these controllers.

the membership functions, which correspond to the

linguistic variables, are randomly created in the

begging of the evolution process.

The evolutionary algorithm maintains a

population of individuals in each generation /

iteration. Individuals represent a different set of

distributed fuzzy controllers. In every generation the

individuals are sorted from the best to the worst

based on their fitness score. As far as it concerns the

fitness function, in the case of the distributed fuzzy

control evolution concept, it has the following form:

))()(()(

−

⎥

⎦

⎤

⎢

⎣

⎡

−=

∑

jji

tPRtDxF

, (6)

where, t is the current simulation time, T is the total

simulation time and D(t) is the overall demand and

PR(t) is the cumulative production of the system.

The architecture of the distributed evolution scheme

is presented in Figure 2.

The best individual is considered the one with

the biggest fitness. The fittest individuals are

selected and undergo mutations. The fittest

controllers and their mutated offsprings are forming

the new population. After some generations the

algorithm converges and the best individuals

represent near optimal solutions. After the evolution

process the membership functions shape is altered.

In case of the supervisory fuzzy evolutionary

scheduling, the procedure is similar. In the lower

control level were used the heuristic fuzzy

distributed controllers introduced in (Tsourveloudis

et al, 2000). The parameters were chosen as in the

distributed case, that is, population number is 40 and

mutation rate is 0.1. From the overall population the

20 fittest individuals are qualified for the next

generation while the rest are replaced by mutation of

the fittest. Each individual is evaluated by the results

of a simulation run of 200 time units. The

architecture of the supervisor evolution scheme is

shown in Figure 3.

In the case of the fuzzy supervisory evolutionary

concept the fitness function is:

BLcWIPc

, (7)

where,

WIP

and

are the mean work-in-process

and mean backlog, respectively. The c

, c

are

weighting factors that represent the unit costs of

inventory and backlog respectively. By taking into

account these costs in the fitness function we may

adjust the importance of

WIP

and

Figure 2: Distributed fuzzy evolutionary concept

Figure 3: The fuzzy supervisory evolutionary concept

4 EXPERIMENTAL RESULTS

We have used the evolutionary algorithm presented

to optimize the performance of the unsupervised /

distributed and the supervised production control

schemes. The evolutionary fuzzy approaches are

tested and compared with the heuristic approaches

introduced in (Tsourveloudis et al, 2000), (Ioannidis

et al, 2004). We assume continuous parts flow

within the system. In the continuous-flow simulation

the discrete production is approximated by the

production of a liquid item (Kouikoglou et al, 1997).

OPTIMIZED FUZZY SCHEDULING OF MANUFACTURING SYSTEMS

199

Several assumptions were made for all simulations.

Machines fail randomly with a failure rate p

and are

repaired randomly with rate rr

. Unlimited repair

personnel is assumed. Time to failure and time to

repair are exponentially distributed. Demand is

either constant or stochastic with rate d. In stochastic

case it follows the Poisson distribution. Machines

operate at known, but not necessarily equal rates.

Each machine produces in a rate r

≤ µ

, where µ

the maximum processing rate of machine M

. The

initial buffers are infinite sources of raw material

and so the initial machines are never starved. Buffers

between adjacent machines M

, M

have finite

capacities. Set-up times or transportation times are

negligible or are included in the processing times.

In order to test both the distributed and the

supervised evolutionary fuzzy approach the systems

presented in figures 4, 5 were used.

Figure 4: Production line

Figure 5: Production Network

4.1 Distributed evolutionary fuzzy

approach

Several scenarios have been studied for the case of

the Evolutionary Distributed Fuzzy (EDF) approach

and the results were compared with the ones

produced from the Heuristic Distributed Fuzzy

approach (HDF).

For the case of the production line, the system

under consideration consists of five machines

producing one product type. The failure and repair

rates are equal for all machines. The repair rates are

=0.5 and the failure rates are p

= 0.1. The

processing rates are also equal for all machines and

are equal to µ

= 2 (i=1,...,5).

In Figure 6 the evolution

WIP

for both

evolutionary and heuristic systems in a simulation

run of 10000 time units is presented.

Comparative results for the

WIP

and

for

various demand patterns are shown in Table 2. All

buffer capacities are equal to BC

= 10.

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000

Time

Mean WIP

Evolutionary

Heuristic

Figure 6: Evolution of WIP in the production line with

stochastic demand (

d = 1)

In Figure 7 the evolution of mean backlog

for the same case is presented.

0 1000 2000 3000 4000 5000 6000 7000 8000 9000

Time

Mean Backlog

Evolutionary

Heuristic

Figure 7: Evolution of

in the production line with

stochastic demand (

d = 1)

The production cost consists of inventory and

backlog costs. Inventory costs are due to the capital

invested for the purchase of raw material and the

handling of material during the production process.

It is assumed that inventory cost is independent from

the stage of process. Thus, the mean production cost

C is given by:

BLcWIPcC

, (8)

where c

, c

are the unit costs of inventory and

backlog respectively.

The cost analysis results for the production line

examined in test case for stochastic demand are

presented in Table 3, where the production cost of

the EDF control approach is compared with the HDF

approach for various values of c

and c

. The

distributed approach was also tested in the

production network presented in Figure 6. The

production system under consideration consists of

five machines producing one part type. The failure

and repair rates of all machines are equal. The repair

rates are rr

= 0.5 and the failure rates are p

= 0.1.

The processing rates are also equal for all machines

and are equal to µ

= 5 (i=1,...,5). All buffer

capacities are equal to BC

= 10. Comparative

results for the

WIP

and

for various demand

patterns are shown in Table 4.

ICINCO 2005 - INTELLIGENT CONTROL SYSTEMS AND OPTIMIZATION

200

Table 2: Results for the test case of the production line

HDF

EDF

Demand

PIW

Constant 1 11.492 1.72 6.371 1.567

0.5 19.393 0.057 6.417 0.438

Stochastic

1 12.719 2.496 8.07 2.427

Table 3: Cost analysis

Cost C

Demand c

HDF EDF

0.99

0.01

19.2

6.357

0.75

0.25

14.56

4.922

0.5

9.725

3.428

0.25

0.75

4.891

1.933

0.5

0.01

0.99

0.25

0.498

0.99

0.01

12.617

8.014

0.75

0.25

10.163

6.659

0.5

7.608

5.249

0.25

0.75

5.052

3.834

0.01

0.99

2.598

2.483

Table 4: Results for the production network test case

HDF

EDF

Demand

PIW

Constant 1 21.356 0.078 16.542 0.737

0.5

20.097

0.049

17.34

0.136

Stochastic

20.496

0.087

10.046

0.673

4.2 Supervised fuzzy evolutionary

approach

The Evolutionary Supervised Fuzzy approach (ESF)

was tested in the case of the production line of the

Figure 4 and was compared with the Heuristic

Supervised Fuzzy approach. Comparative results for

the

WIP

and production cost C, when c

and c

are equal to 0.5, for various stochastic demand

patterns are shown in Table 5. The supervised

approach was also tested in the production network

presented in Figure 5. Table 6 shows comparative

results of

WIP

and C, when c

and c

are equal to

0.5, for various stochastic demand patterns.

Table 5: Comparative results for the test case of the

production line

HSF ESF

Demand

PIW

0.5

19.886

0.167

10.027

6.446

0.32

3.383

0.5

2.584

7.153

9.17

3.853

6.51

Table 6: Results for the production network test case

HSF ESF

Demand

PIW

0.5

5.43

0.09

2.76

1.406

1.888

1.647

7.162

0.505

3.834

3.036

2.681

2.859

4.55

2.777

8.666

8.914

5.55

7.232

5 CONCLUSIONS

An evolutionary algorithm strategy for the

optimization of already established fuzzy production

control architectures ((Tsourveloudis et al, 2000),

(Ioannidis et al, 2004)) has been presented. The EA

strategy selects the membership functions of the

fuzzy controllers in a way that WIP and backlog

values minimize fitness function based on

production surplus. Simulation results, for a number

of taste cases, have shown an important

improvement of performance and production related

costs, with the use of EA strategies. More

specifically the EA strategies manage to reduce

substantially the weighted sum of WIP and backlog

and thus improving the inventory and backlog costs.

Evolutionary algorithms clearly represent a

successful approach towards the optimization of

fuzzy production control approaches.

In the future it would be very interesting to

consider the case of seasonal demand. Another

interesting extension would be the use of EA

strategies in more complex production systems such

as multiple-part-type and/or re-entrant systems.

ACKNOWLEDGMENT

This work was supported from a grant from the Greek

Secretariat for Research and Technology/European Union,

E.P.A.N. M.4.3.6.1, 2004

ΣΕ 01330013. L.Doitsidis was

supported from “Herakleitos” fellowships for research

from the Technical University of Crete EPEAEK II 88727.

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