Directory UMM :Data Elmu:jurnal:I:International Journal of Production Economics:Vol70.Issue1.Mar2001:

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*Corresponding author. Tel.:#972-48649860; fax:# 972-48649780.

E-mail address:[email protected] (G. Levitin).

Structure optimization for continuous production systems

with bu

_!

ers under reliability constraints

G. Levitin

!,

*

, L. Meizin

"

!Reliability Department, Planning Dev.&Technology Division, The Israel Electric Corporation Ltd., P.O. Box 10, Bait Amir, Haifa 31000, Israel

"Center for Technological Education, Holon, Israel Received 3 January 1999; accepted 12 April 2000

Abstract

A problem of structure optimization of a series}parallel production system is considered where redundant producing elements (machines) and in-process bu!ers are included in order to provide a desired level of reliability. A procedure which determines the minimal cost system con"guration is proposed. In this procedure, producing elements and bu!ers are chosen from a list of products available in the market. The elements are characterized by their cost, estimated average up and down times and productivity (machines) or capacity (bu!ers). System reliability is de"ned as the ability to satisfy consumer demand which is represented by a piecewise cumulative load curve. A genetic algorithm is used as an optimization technique. An example of the redundancy optimization of a power station coal feeding system is presented. ( 2001 Elsevier Science B.V. All rights reserved.

Keywords: Series}parallel production system; Bu!ers; Genetic algorithm; Reliability

1. Introduction

The need to face requirements for increasing pro-ductivity and reducing cost of system makes the design phase one of the most important phases in life-cycle of a manufacturing system. One of the most important problems in system design is struc-ture optimization, and one of natural objectives of this problem is system cost minimization subject to requirement of meeting the demand with the desired level of system reliability.

To help smooth out variations in production and to increase availability of the system, a temporary storages between the stages of manufacturing pro-cess are usually introduced as shown in Fig. 1. In this case, if stageBfails,Ais not blocked andCis not necessarily starved. Note that each stage may in its turn have a sophisticated structure with raw materials and"nished parts' #ows. The determina-tion of the bu!ers size and the structure of produ-cing components a!ects the cost of the system and its availability.

Some partial analytical solutions for the problem of structure optimization may be found, for example, in [1}3]. Two areas of a continuous pro-duction system with unreliable equipment and an intermediate storage are considered in [1,2]. The

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Nomenclature

N number of components in the system

i maximal permissible number of produ-_{cing elements in}_i_{th system component} H

i number of di_{elements of type}!erent versions of system_i _{available on the}

market

i number of di_type_i_available!erent versions of bu!ers of h(i,j) version ofjth producing element included

intoith system component

f(i) version of bu!er chosen for ith system component

h cost of producing element of versionh G

h productivity of producing element of ver-_sion_h ;

h estimated average up time of producing_{element of version}_h D

h estimated average down time of produ-_{cing element of version}_h <

f capacity of bu!er of versionf CB_f cost of bu!er of versionf

M number of the demand levels considered W vector of demand levels, W"M=

iN,

1)i)M

q vector of probabilities, corresponding to demand levels,q"_Mq

iN, 1)i)M G

505 total system productivity g

i total potential productivity of compon-_ent_i p

ij(w) probability that demand_{because of unavailability of element}wis not satis"_jed_of

componenti

ij(w) productivity de_{bility of element}"cit caused by unavaila-_j_{of component}_i t

ij(w) time of bu_{availability of element}!ers unloading caused by un-_j_{of component}_i C

4:4 expected total system cost P

6$ total unsatis"ed demand probability E system availability index

EH minimal allowable system availability

Fig. 1. Series of production stages.

times to failure and the times to repair are assumed in [1] to be exponentially distributed. The produc-tion loss in [2] is supposed to have a gamma distribution. The optimal size of the bu!er storage and the optimal stock in the bu!er are found to minimize the total inventory cost and average loss of upstream and downstream sections of the system due to faults in process.

An approximate solution of a bu!er design prob-lem in a queuing system with"nite waiting spaces is presented in [3], where the bu!er design problem is to determine the smallest bu!er capacity so that the proportion of lost customer is below an acceptable level. These solutions do not consider reliability and structure of upstream and downstream parts of the system.

The zero-bu!er production rates of unreliable production system can be predicted by using method that was introduced in [4]. The series-connected machines are characterized by their capacity and reliability.

An e$cient algorithm for design of a production line that has a bottleneck was presented in [5]. The machines have been selected, and the only decision remaining is the amount of space to allocate for in-process inventory.

A method of performance evaluation for tree-structured assembly/disassembly (AD) systems

with"nite bu!er capacity is given in [6]. The times

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Fig. 2. Structure of series}parallel production system with bu!ers.

times are assumed to be exponentially distributed. The method decomposes a K-machine tree-struc-tured AD system to a system ofK!1 two-machine lines and then "nds the processing rates, failure rates and repair rates of the decomposed system that makes performance of the two systems close to each other. In papers of Gershwin and Goldis [5] and Jeong and Kim [6], the cost of inventory and equipment is not considered.

Design of zero-bu!er production system subject to reliability constraints is examined by Levitin et al. [7,8]. The objective is to minimize the total cost of equipment while providing a required level of reliability. Algorithms presented in these works allow engineers to compose system components using equipment with di!erent productivity, relia-bility and cost.

Queuing network models with "nite capacity queues and blocking are used to represent systems

with "nite capacity resources and with resource

constraints. Such system includes production and manufacturing systems, computer and com-munication systems. A review of the models and analytical results for manufacturing#ow line sys-tems is presented in by Dallery and Gershwin [9]. Application examples of this class of models to distributed computer systems, communication networks and production systems can be found in [10].

In this paper an optimal production system de-sign problem is considered. The system includes

in-process inventory and has series}parallel con" g-uration. The objective is to minimize the total cost of the system with regard to the selected level of the system reliability. The following assumptions are made:

1. The probability of simultaneous unavailability of elements is negligibly small.

2. Because of demand variation there is enough time to replenish bu!ers between their unload-ing. That means that all bu_!ers are full when they have to be used to compensate the system productivity de"cit.

3. System elements downtimes are shorter than intervals of demand constancy.

2. Problem formulation and description of system model

A system consisting ofNcomponents connected in series is considered (Fig. 2). Each component of type i contains a number of di!erent producing elements. Because of technical limitations (space, communication, etc.) each componentican contain no more thanK

i elements. Di!erent versions and

number of elements may be chosen for any given system component. Elements are characterized by their cost, productivity and estimated average up and down times according to their version. The element capacity may be measured as a percentage of maximal demand.

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For each component i there are a number of element versions available in the market. A vector of parameters G

h,Ch,;h,Dh, can be speci"ed for

each versionhof equipment. The structure of sys-tem componentican be de"ned by the numbers of versions of elements the component containsh(i, j) for 1)j)K

i. The value of h(i, j) can vary in the

range 0}H

i, whereHi is a total number of versions

available for the element of typei. In order to allow the total number of elements in the component to vary,h(i,j) can obtain the value of 0 which corres-ponds to an absence of elements.

Each component n(1)n)N!1) can also contain a bu!er chosen from the list of available bu!ers. The bu!ers of di!erent versionsfdi!er by their capacity <

f and cost CBf. The vector

f"Mf(n)N, where 0)f(n))F

n, de"nes versions of

bu!ers chosen for each component. HereF n is the

total number of di!erent bu!er versions available fornth component. Note thatf(n)"0 means that no bu!ers are installed at componentj.

The vectors f"_Mf(n)N and h

i"Mh(i,j)N

(1)n)N!1, 1)i)N, 1)j)K

i) de"ne the

entire system structure. For a given set of vectors

f, h

1,h2,2,hN, the total cost of the system can be

calculated as C 4:4" N~1 + n/1 C_Bf

(n)#

N + i/1 Ki + j/1 C

h(i,j). (1)

Following assumption 1, the entire system reliabil-ity estimation can be based on the consideration of the unavailability of single elements. Therefore, the number of di!erent system states that can contrib-ute to the reduction of total productivity of a system is equal to the total number of elements in the systemS,

S"₊N

i/1 Ki + j/1 a ij, where a

ij"

G

0, h(i,j)"0, 1, h(i,j)O0.

Each state corresponds to the unavailability of element j of component i and is characterized by the total system productivity G

505(i,j) and by the

probabilityp

ij(w) that demandwwill not be

satis-"ed because of the unavailability of this element.

In order to de"ne the variable demand for the production system, the demand variation cycle is divided into M intervals, with duration

i (i"1,2,M). Each interval has demand level =

ide"ned as a percentage of the maximal demand.

The probability q

i of demand level =i, can be

estimated as

q i"¹i

N

j/1 ¹

Vectors W"_M=

jN and q"MqjN(1)j)M)

de-"ne the cumulative demand curve, which is usually

known for every system.

In continuous production systems with variable demand, the overall probability that the demand will not be met is used as a measure of system unreliability. (For example, in electric power sys-tems, loss of load probability is usually estimated [7,11]). To obtain this index, the total unsatis"ed demand probability should be calculated as

P 6$" M + m/1 N + i/1 Ki + j/1 Pr(G

505(i, j)(=m)

" ₊M

m/1 N + i/1 Ki + j/1 q

mpij(=m). (2)

The reliability of the entire system requires an availability indexE"1!P

6$that is not less than

some preliminary speci"ed levelEH.

Now we can formulate the problem of produc-tion system structure optimizaproduc-tion as follows:"nd the minimal cost system con"guration

f,h

1, h2,2, hNthat provides the required

reliabil-ity levelEH: (f, h

1, h2,2,hN)"argMC(f, h1, h2,2,hN)P

minDE(f, h

1,h2,2, hN, W, q)*EHN. (3)

3. Determination of system unsatis5ed demand probability

Consider the production system with a given structure (determined vectorsf,h

1, h2,2, hN). In

this system for each elementnof componenti, the versionh(i,n) is chosen, and for each componenti, the version of bu!erf(i) is chosen.

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Under assumption 1 one can consider separately each system component. Indeed if one of elements of componentiis unavailable at the moment, the rest of components have their maximal productiv-ity (which should be no less than maximal demand). Only a component with an unavailable element can become the bottleneck of the system and, therefore, only the in#uence of this component on the entire system productivity should be estimated.

Let theith component has total productivity

g i"

j/1

G h(i,j).

If elementnof this component is unavailable, the potential productivity of the entire system is re-duced to g

i!Gh(i,n). The productivity de"cit in

component i due to the unavailability of element

ndepends on the system demandwand is de"ned as follows:

d_in(w)"w!g

i#Gh(i,n). (4)

If d_in(w))0, the element unavailability does not cause the entire system productivity reduction and

in(w)"0. If productivity de"cit exists (din(w)'0),

it can be compensated by unloading downstream located bu!ers. The total capacity of these bu!ers is

N~1 +

j/i <

f(j).

If the downtime of the unavailable element isD h(i,n),

the time of bu!ers unloading is

in(w)"

G

h(i,n),

N~1 +

j/i <

f(j)/din(w)*Dh(i,n),

N~1 +

j/i <

f(j)/din(w), N~1

j/i <

f(j)/din(w)(Dh(i,n).

(5) The probability that demandwwill not be satis"ed because of unavailability of elementnis

p in(w)"

G

0, d_in(w))0,

h(i,n)!tin(w))/(;h(i,n)#Dh(i,n)), din(w)'0.

(6) Applying this equation to all the demand levels

m (1)j)M) and all system elements and then

using expression (2), we can obtain P

6$ for the

entire system.

4. Optimization technique

Eq. (3) formulates a complicated combinatorial optimization problem. An exhaustive examination of all possible solutions is not realistic, considering reasonable time limitations. As in most combina-torial optimization problems, the quality of a given solution is the only information available during the optimal solution search. Therefore, a heuristic search algorithm is needed which uses only esti-mates of the solution quality and which does not require derivative information to determine the next direction of the search.

The family of genetic algorithms (GAs) is based on a simple principle of the evolutionary search in the solution space. GAs have been proven to be e!ective optimization tools for a large number of applications. Successful applications of GAs to re-liability engineering problems are reported in [7,8,12_}17]. The bibliography of their applications in manufacturing and industrial engineering is available in [18].

It is recognized that GAs have theoretical prop-erty of global convergence [19]. Despite the fact that its convergence reliability and convergence velocity are contradictory, for most practical, mod-erately sized combinatorial problems the proper choice of GA parameters allows optimal solutions to be obtained in a short time.

4.1. Genetic algorithm

Basic notions of the GA are originally inspired by biological genetics. The GA operates with

`chromosomalarepresentation of solutions, where crossover, mutation and selection procedures are applied. Unlike various constructive optimization algorithms that use sophisticated methods to ob-tain a good singular solution, the GA deals with a set of solutions (population) and tends to manipu-late each solution in the most simple manner.

`Chromosomala representation requires the solution to be coded as a"nite length string.

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A brief introduction to genetic algorithms is pre-sented in [20]. More detailed information on GA can be found in Goldberg's comprehensive book [21] and the recent developments in GA theory and practice can be found in book of BaKck [19]. The basic structure of the version of GA, referred to as GENITOR [22], is as follows.

First, the initial population ofN

4randomly

con-structed solutions (strings) is generated. Within this population, new solutions are obtained during the genetic cycle by using crossover and mutation op-erators. The crossover produces a new solution (o!spring) from a randomly selected pair of parent solutions, facilitating the inheritance of some basic properties from the parents to the o!spring. Muta-tion results in slight changes to the o!springs' struc-ture and maintains a diversity of solutions. This procedure avoids premature convergence to a local optimum and facilitates jumps in the solution space.

Each new solution is decoded and its objective function ("tness) values are estimated. These values, which are a measure of quality, are used to compare di!erent solutions.

The comparison is accomplished by a selection procedure that determines which solution is better: the newly obtained solution or the worst solution in the population. The better solution joins the population, while the other is discarded. If the population contains equivalent solutions following selection, redundancies are eliminated and the population size decreases as a result.

After new solutions are producedN

3%1times, new

randomly constructed solutions are generated to replenish the shrunken population, and a new genetic cycle begins.

The GA is terminated after N

# genetic cycles.

The "nal population contains the best solution

achieved. It also contains di!erent near-optimal solutions which may be of interest in the decision-making process.

4.2. Solution representation and decoding procedure

To apply the genetic algorithm to a speci"c prob-lem, the solution representation and the decoding procedure must be de"ned.

To choose a combination of bu!ers and elements with di!erent versions, our GA deals with¸length

integer strings. ¸ is the total number of versions available

¸"N!1#₊N

i/1

K i.

Each solution is represented by string Z"Mz

1,z2,2,zLN, where for each 1)j)N!1, z

jde"nes the number of the bu!er's version chosen

for componentjand for each

N#i~1₊

n/1

n)j)N!1# i~1

n/1

K n#Ki, z

jde"nes version of element included into

compon-enti.

To obtain the random solutions all the numbers

j (0)j)¸) are generated in the range

0)z

j)max

1xixN

i,KiN (7)

and are decoded in the following manner:

h(i, j)"mod

Ki(zj) for versions of producing_{element belonging to}_i_th

component and

f(i)"mod

Fi(zj), for versions of bu_{belonging to} _i_{th component,}!ers element

where mod

xy"y![y/x]x([y/x] is the maximal

integer that does not exceedy/x).

The decoding procedure allows any arbitrary integer string with elements belonging to range (7) to represent a feasible solution in the form of vectorsf, h

1,2,hN.

For each given set of vectors, the decoding pro-cedure"rst calculatesC

4:4using expression (1), then

for each demand level=

m (1)m)M) and each

system element estimates the probability that this demand will not be satis"ed using Eqs. (4)}(6). Finally, the total unsatis"ed demand probability is evaluated using Eq. (2) and theEindex is obtained. In order to let the genetic algorithm search for the solution with minimal total cost, whileEis not less than the required valueEH, the solution quality

("tness) is evaluated as follows:

F(f,h

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Table 1

Characteristics of the system elements available in the market

Component Description K Version G ; _D _C

no. number (%) (h) (h) (mln$)

1 Primary feeder 5 1 120 755.0 16.0 0.590

2 100 841.0 19.0 0.535

3 85 710.0 20.0 0.470

4 85 705.0 15.0 0.420

5 48 720.0 18.0 0.400

6 31 320.0 12.0 0.180

7 26 677.0 12.0 0.220

2 Primary conveyor 5 1 100 1200.0 11.0 0.205

2 53 880.0 3.0 0.091

3 28 1140.0 7.0 0.056

3 Stacker-reclaimer 3 1 100 70.0 2.0 7.525

2 60 73.0 2.0 4.720

3 40 66.0 2.0 3.590

4 20 70.5 1.5 2.420

4 Secondary feeder 6 1 72 530.0 12.0 0.102

2 72 400.0 6.0 0.096

3 55 490.0 12.0 0.071

4 25 410.0 8.0 0.049

5 25 380.0 10.0 0.044

where

X(x)"

G

ux, x*0,

0, x(0,

anduis a su$ciently large penalty.

For solutions that meet the requirementE*EH

the"tness of the solution is equal to its total cost.

4.3. Crossover and mutation procedures

The crossover operator for given parent strings

P1, P2 and the o!spring string O is de"ned as follows:"rstP1 is copied toO, then all numbers of elements belonging to the fragment betweenkand

m positions of the string P2 (where k and m are random values, 1)k(m)¸) are copied to the corresponding positions ofO. The following example illustrates the crossover procedure for¸"6:

P1"z

1z2z3z4z5z6,

P2"z₁z₂z₃z₄z₅z₆,

O"z

1z2z3z4z5z6.

The mutation procedure swaps elements initially located in two randomly chosen positions.

5. Illustrative example

5.1. Description of the system to be optimized

A power station coal feeding system supplies a boiler. It consists of four basic components (types of element).

1. Primary feeder which loads the coal to the pri-mary conveyor.

2. Primary conveyor which transports the coal to the stacker-reclaimer.

3. Stacker-reclaimer which lifts the coal up to the burner level.

4. Secondary feeder which loads the burner feeding system of the boiler.

The characteristics of products available on the market for each type of equipment are presented in Table 1. The table shows average up and down

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Table 4

Parameters of the optimal solutions for di!erent reliability requirements

0 With bu!ers Without bu!ers

E C

4:4 Component Elements Bu!ers E C4:4 Component Elements

0.950 0.952 8.535 1 2 3 1 2

2 2, 2 * 0.951 8.546 2 2, 2, 2

3 1 * 3 1

4 3, 3, 3 * 4 3, 3, 3

0.960 0.960 8.630 1 4, 6 2 1 4, 6, 6

2 2, 2 * 0.961 8.700 2 2, 2

3 1 * 3 1

4 3, 3, 3 * 4 3, 3, 3

0.970 0.972 8.740 1 6, 6, 6 * 1 4, 4, 6

2 2, 2 2 0.970 8.996 2 2, 2, 3

3 1 * 3 1

4 3, 3, 3 * 4 3, 3, 3

0.980 0.980 10.100 1 4, 6 3 1 6, 6, 6, 6, 6

2 2, 2 2 0.980 12.116 2 2, 2, 2

3 1 2 3 2, 3, 4

4 3, 3, 3 * 4 3, 3, 3

0.990 0.991 10.925 1 6, 6, 6, 6 _* 1 6, 6, 6, 6

2 2, 2 2 0.990 13.251 2 2, 2, 3

3 2, 3 2 3 2, 3, 3

4 3, 3, 3 * 4 3, 3, 3

0.999 0.999 13.755 1 6, 6, 6, 6 * 1 6, 6, 6, 6, 6

2 2, 2 * 0.999 14.401 2 2, 3, 3, 3

3 2, 2 1 3 2, 3, 3

4 3, 3, 3 * 4 3, 3, 3

Table 3

Parameters of the cumulative load demand curve

i(%) 100 80 50 20

i(h) 0.29 0.13 0.41 0.17

Table 2

Characteristics of the bu!ers that can be installed

Component Version < C

no. number (%_*h) (mln$)

1 1 320 0.200

2 200 0.110

3 150 0.080

2 1 180 0.160

2 120 0.100

3 1 100 3.200

2 50 1.400

times, nominal productivity and cost per unit. The nominal productivities are given as a percentage of the maximal boiler demand. For example, if the productivity required to provide maximal power

generating capacity of the boiler is 125 tons per hour and the nominal productivity of the stacker-reclaimer is 100 tons per hour, the capacity of the latter is 80%. Table 1 also contains limitations on the total number of parallel units of the same type. These limitations are caused by space and mainten-ance constraints.

Each system component can have bu!ers (coal bins) with di!erent capacity. The cost of di!erent bu!ers depends on their size and location. Capaci-ties and costs of bu!ers are presented in Table 2.

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Fig. 3. Total system cost as function of system reliability.

Table 3 contains the data of piecewise cumulat-ive boiler demand curve.

5.2. Discussion of the solutions obtained

The optimal solutions obtained by the proposed algorithm for di!erent reliability constraints

speci-"ed by index EH are presented in Table 4. Table

4 also shows the calculated reliability indexE, the cost of the systemC(solution"tness) and its struc-ture, presented by a vector of elements version numbershand vector of bu!er numbersffor each component i. Six di!erent optimal solutions for

EH varying from 0.95 to 0.999 are presented. To estimate the in#uence of feeders on the system cost and reliability, the optimal solutions are compared to solutions in which no bu!ers are available. One can see that with bu!ers the same reliability level is achieved with lower cost.

The optimization algorithm proposed in this paper allows the reliability}cost tradeo! depend-ence to be evaluated. Such a dependdepend-ence for the system considered is shown in Fig. 3. Each point on this curve represents the optimal structure with minimum cost.

The number of versions of elements to be considered as candidates for "nal solution from among optimal solutions for di!erent EH

is much smaller than initial number of versions available. It makes the decision making process much easier. In our example (see Table 4) only the third version should be considered for the secondary feeder and only the second, fourth and sixth versions for the primary feeder. Only the bu!er of version 2 should be considered for installa-tion between primary conveyor and stacker-reclaimer.

When design is performed under uncertainty conditions it is important to estimate in#uence of variable parameters. In the example presented the main uncertainty is in the evaluation of cost of ground needed for coal bins allocation. In order to evaluate the sensitivity of the optimal solution to this factor, a penalty is introduced which is added to the total system cost proportionally to the num-ber of bu!ers used in the system. Fig. 4 represents the dependence of the optimal system cost for

0"0.98 on the value of the penalty. The cost of

system with "xed structure which is optimal for zero penalty case is given for comparison. One

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Fig. 4. Total system cost as function of penalty on bu!ers inclusion.

Table 5

Optimal solutions for di!erent penalties on bu!ers use Penalty range Component Elements Bu!ers (mln$)

0}0.02 1 4, 6 3

2 2, 2 2

3 1 2

4 3, 3, 6 *

0.01}0.1 1 6, 6, 6, 6 3

2 2, 2 *

3 1 *

4 3, 3, 3

0.1_}1.896 1 6, 6, 6, 6 _*

2 2, 2 *

3 1 2

4 3, 3, 3 *

'1.896 1 6, 6, 6, 6, 6 *

2 2, 2, 2 *

3 2, 3, 4 *

4 3, 3, 3 *

can see that as the penalty increases, the optimal solution changes in order to reduce the total num-ber of bu!ers (the optimal structures are represent-ed in Table 5). To maintain the requirrepresent-ed reliability level the additional producing elements are added to the system, thus compensating for the removal of bu!ers. The cost of resulting solutions is lower than the cost of solutions which were optimal in the low penalty case.

The C language realization of the algorithm was tested on a DEC station 5000/240. Numerous tests on a set of di!erent randomly generated problems with N)15, K

i)10, Fi)5 and Hi)15 show

that the best times of convergence to the optimal solutions are obtained when GA parameters vary in the following ranges: 100)N

S)150,

2000)N

3%1)3000. The running times for each

problem did not exceed 130 s and the number of genetic cyclesN

Ctill obtaining the optimal solution

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References

[1] L.K. Meizin, Choosing the optimal bu!er stock between two sections of a continuous process, Automation and Remote Control 45 (8) (1984) 1086}1089.

[2] L.K. Meizin, Optimizing contingency reserves, Automa-tion and Remote Control 48 (2) (1987) 268}271. [3] T. Kimura, Optimal bu!er design of an M/G/s queue with

"nite capacity, Stochastic Models 12 (1) (1996) 165}180. [4] J.A. Buzacott, Prediction of the e$ciency of production

systems without internal storage, International Journal of Production Research 6 (3) (1968) 173}188.

[5] S.B. Gershwin, Y. Goldis, E$cient algorithms for transfer line design, Report No. LMP-95-005, MIT Laboratory for Manufacturing and Productivity, Cambridge, MA, 1995. [6] K.-C, Jeong, Y.-D. Kim, Performance analysis of

assem-bly/disassembly systems with unreliable machines and random processing times, IIE Transactions 30 (1) (1998) 41}53.

[7] G. Levitin, A. Lisnianski, D. Elmakis, Structure optimiza-tion of power system with di!erent redundant elements, Electric Power Systems Research 43 (1997) 19}27. [8] G. Levitin, A. Lisnianski, H. Ben-Haim, D. Elmakis,

Re-dundancy optimization for series-parallel multistate sys-tems, IEEE Transactions Reliability 47 (2) (1998) 165}172. [9] Y. Dallery, S.B. Gershwin, Manufacturing#ow line sys-tems: A review of models and analytical results, Queueing Systems Theory and Applications 12 (1992) 3}94. [10] S. Balsamo, Queueing networks with "nite capacity

queues and blocking, Proceedings of the International Conference on Performance Theory, Measurement and Evaluation of Computer and Communication Systems PERFORMANCE'96, Lausanne, 1996, pp. 7}11. [11] R. Billinton, R. Allan, Reliability Evaluation of Power

Systems, Plenum Press, New York, 1994.

[12] A. Munoz, S. Martorell, V. Serdarell, Genetic algorithms in optimizing surveilance and maintenance of components, Reliability Engineering & System Safety 57 (2) (1997) 107}120.

[13] L. Painton, J. Campbell, Genetic algorithm in optimiza-tion of system reliability, IEEE Transacoptimiza-tions on Reliability 44 (1995) 172}178.

[14] A. Kumar, R. Pathak, Y. Gupta, Genetic-algorithm-based reliability optimization for computer network expancion, IEEE Transactions on Reliability 44 (1995) 63}72. [15] M. Gen, R. Cheng, Optimal design of system reliability

using interval programming and genetic algorithm, Computers and Industrial Engineering 31 (1/2) (1996) 237}240.

[16] D. Coit, A. Smith, Reliability optimization for series} par-allel systems using genetic algorithm, IEEE Transactions on Reliability 45 (1996) 254}266.

[17] D. Coit, A. Smith, Penalty guided genetic search for reliability design optimization, Computers and Industrial Engineering 30 (4) (1996) 895}904.

[18] J. Alander, An indexed bibliography of genetic algorithms in manufacturing, Report Series No. 94-1-MANU, Univer-sity of Vaasa, Finland, 1998. (Available in the Internet at http://www.uwasa."/Ijal.)

[19] T. BaKck, Evolutionary Algorithms in Theory and Practice. Evolution Strategies. Evolutionary Programming. Genetic Algorithms, Oxford University Press, Oxford, 1996. [20] S. Austin, An introduction to genetic algorithms, AI

Ex-pert 5 (1990) 49}53.

[21] D. Goldberg, Genetic Algorithms in Search, Optimization and Machine Learning, Addison-Wesley, Reading, MA, 1989.

[22] D. Whitley, J. Kauth, GENITOR: A di!erent Genetic Algorithm, Technical Report CS-88-101, Colorado State University, Fort Collins, USA, 1998.

(1)

First, the initial population ofN

4randomly con-structed solutions (strings) is generated. Within this population, new solutions are obtained during the genetic cycle by using crossover and mutation op-erators. The crossover produces a new solution (o!spring) from a randomly selected pair of parent solutions, facilitating the inheritance of some basic properties from the parents to the o!spring. Muta-tion results in slight changes to the o!springs' struc-ture and maintains a diversity of solutions. This procedure avoids premature convergence to a local optimum and facilitates jumps in the solution space.

Each new solution is decoded and its objective function ("tness) values are estimated. These values, which are a measure of quality, are used to compare di!erent solutions.

After new solutions are producedN

3%1times, new randomly constructed solutions are generated to replenish the shrunken population, and a new genetic cycle begins.

The GA is terminated after N

# genetic cycles. The "nal population contains the best solution achieved. It also contains di!erent near-optimal solutions which may be of interest in the decision-making process.

4.2. Solution representation and decoding procedure To apply the genetic algorithm to a speci"c prob-lem, the solution representation and the decoding procedure must be de"ned.

To choose a combination of bu!ers and elements with di!erent versions, our GA deals with¸length

integer strings. ¸ is the total number of versions available

¸"N!1#+N

i/1

Each solution is represented by string Z"Mz

1,z2,2,zLN, where for each 1)j)N!1,

jde"nes the number of the bu!er's version chosen for componentjand for each

N#i~1₊ n/1

n)j)N!1# i~1

n/1

n#Ki,

jde"nes version of element included into compon-enti.

To obtain the random solutions all the numbers

j (0)j)¸) are generated in the range 0)z

j)max 1xixN

i,KiN (7)

and are decoded in the following manner:

h(i, j)"mod

Ki(zj) for versions of producing_{element belonging to}_i_th component

and

f(i)"mod

Fi(zj), for versions of bu_{belonging to} _i_{th component,}!ers element where mod

xy"y![y/x]x([y/x] is the maximal integer that does not exceedy/x).

The decoding procedure allows any arbitrary integer string with elements belonging to range (7) to represent a feasible solution in the form of vectorsf, h

1,2,hN.

For each given set of vectors, the decoding pro-cedure"rst calculatesC

4:4using expression (1), then for each demand level=

m (1)m)M) and each system element estimates the probability that this demand will not be satis"ed using Eqs. (4)}(6). Finally, the total unsatis"ed demand probability is evaluated using Eq. (2) and theEindex is obtained. In order to let the genetic algorithm search for the solution with minimal total cost, whileEis not less than the required valueEH, the solution quality ("tness) is evaluated as follows:

F(f,h

(2)

Table 1

Characteristics of the system elements available in the market

Component Description K Version G ; _D _C

no. number (%) (h) (h) (mln$)

1 Primary feeder 5 1 120 755.0 16.0 0.590

2 100 841.0 19.0 0.535

3 85 710.0 20.0 0.470

4 85 705.0 15.0 0.420

5 48 720.0 18.0 0.400

6 31 320.0 12.0 0.180

7 26 677.0 12.0 0.220

2 Primary conveyor 5 1 100 1200.0 11.0 0.205

2 53 880.0 3.0 0.091

3 28 1140.0 7.0 0.056

3 Stacker-reclaimer 3 1 100 70.0 2.0 7.525

2 60 73.0 2.0 4.720

3 40 66.0 2.0 3.590

4 20 70.5 1.5 2.420

4 Secondary feeder 6 1 72 530.0 12.0 0.102

2 72 400.0 6.0 0.096

3 55 490.0 12.0 0.071

4 25 410.0 8.0 0.049

5 25 380.0 10.0 0.044

where

X(x)"

G

ux, x*0, 0, x(0,

anduis a su$ciently large penalty.

For solutions that meet the requirementE*EH the"tness of the solution is equal to its total cost.

4.3. Crossover and mutation procedures

The crossover operator for given parent strings

P1, P2 and the o!spring string O is de"ned as follows:"rstP1 is copied toO, then all numbers of elements belonging to the fragment betweenkand

m positions of the string P2 (where k and m are random values, 1)k(m)¸) are copied to the corresponding positions ofO. The following example illustrates the crossover procedure for¸"6:

P1"z

1z2z3z4z5z6,

P2"z₁z₂z₃z₄z₅z₆,

O"z

1z2z3z4z5z6.

The mutation procedure swaps elements initially located in two randomly chosen positions.

5. Illustrative example

5.1. Description of the system to be optimized A power station coal feeding system supplies a boiler. It consists of four basic components (types of element).

1. Primary feeder which loads the coal to the pri-mary conveyor.

2. Primary conveyor which transports the coal to the stacker-reclaimer.

3. Stacker-reclaimer which lifts the coal up to the burner level.

4. Secondary feeder which loads the burner feeding system of the boiler.

The characteristics of products available on the market for each type of equipment are presented in Table 1. The table shows average up and down

(3)

Table 4

Parameters of the optimal solutions for di!erent reliability requirements

0 With bu!ers Without bu!ers

E C

4:4 Component Elements Bu!ers E C4:4 Component Elements

0.950 0.952 8.535 1 2 3 1 2

2 2, 2 * 0.951 8.546 2 2, 2, 2

3 1 * 3 1

4 3, 3, 3 * 4 3, 3, 3

0.960 0.960 8.630 1 4, 6 2 1 4, 6, 6

2 2, 2 * 0.961 8.700 2 2, 2

3 1 * 3 1

4 3, 3, 3 * 4 3, 3, 3

0.970 0.972 8.740 1 6, 6, 6 * 1 4, 4, 6

2 2, 2 2 0.970 8.996 2 2, 2, 3

3 1 * 3 1

4 3, 3, 3 * 4 3, 3, 3

0.980 0.980 10.100 1 4, 6 3 1 6, 6, 6, 6, 6

2 2, 2 2 0.980 12.116 2 2, 2, 2

3 1 2 3 2, 3, 4

4 3, 3, 3 * 4 3, 3, 3

0.990 0.991 10.925 1 6, 6, 6, 6 _* 1 6, 6, 6, 6

2 2, 2 2 0.990 13.251 2 2, 2, 3

3 2, 3 2 3 2, 3, 3

4 3, 3, 3 * 4 3, 3, 3

0.999 0.999 13.755 1 6, 6, 6, 6 * 1 6, 6, 6, 6, 6

2 2, 2 * 0.999 14.401 2 2, 3, 3, 3

3 2, 2 1 3 2, 3, 3

4 3, 3, 3 * 4 3, 3, 3

Table 3

Parameters of the cumulative load demand curve

i(%) 100 80 50 20

i(h) 0.29 0.13 0.41 0.17 Table 2

Characteristics of the bu!ers that can be installed Component Version < C

no. number (%_*h) (mln$)

1 1 320 0.200

2 200 0.110

3 150 0.080

2 1 180 0.160

2 120 0.100

3 1 100 3.200

2 50 1.400

times, nominal productivity and cost per unit. The nominal productivities are given as a percentage of the maximal boiler demand. For example, if the productivity required to provide maximal power

Each system component can have bu!ers (coal bins) with di!erent capacity. The cost of di!erent bu!ers depends on their size and location. Capaci-ties and costs of bu!ers are presented in Table 2.

(4)

Fig. 3. Total system cost as function of system reliability. Table 3 contains the data of piecewise

cumulat-ive boiler demand curve.

5.2. Discussion of the solutions obtained

The optimal solutions obtained by the proposed algorithm for di!erent reliability constraints

speci-"ed by index EH are presented in Table 4. Table 4 also shows the calculated reliability indexE, the cost of the systemC(solution"tness) and its struc-ture, presented by a vector of elements version numbershand vector of bu!er numbersffor each component i. Six di!erent optimal solutions for

The number of versions of elements to be considered as candidates for "nal solution from among optimal solutions for di!erent EH is much smaller than initial number of versions available. It makes the decision making process much easier. In our example (see Table 4) only the third version should be considered for the secondary feeder and only the second, fourth and sixth versions for the primary feeder. Only the bu!er of version 2 should be considered for installa-tion between primary conveyor and stacker-reclaimer.

0"0.98 on the value of the penalty. The cost of system with "xed structure which is optimal for zero penalty case is given for comparison. One

(5)

Fig. 4. Total system cost as function of penalty on bu!ers inclusion.

Table 5

Optimal solutions for di!erent penalties on bu!ers use Penalty range Component Elements Bu!ers (mln$)

0}0.02 1 4, 6 3

2 2, 2 2

3 1 2

4 3, 3, 6 *

0.01}0.1 1 6, 6, 6, 6 3

2 2, 2 *

3 1 *

4 3, 3, 3

0.1_}1.896 1 6, 6, 6, 6 _*

2 2, 2 *

3 1 2

4 3, 3, 3 *

'1.896 1 6, 6, 6, 6, 6 *

2 2, 2, 2 *

3 2, 3, 4 *

4 3, 3, 3 *

The C language realization of the algorithm was tested on a DEC station 5000/240. Numerous tests on a set of di!erent randomly generated problems with N)15, K

i)10, Fi)5 and Hi)15 show that the best times of convergence to the optimal solutions are obtained when GA parameters vary in the following ranges: 100)N

S)150, 2000)N

3%1)3000. The running times for each problem did not exceed 130 s and the number of genetic cyclesN

Ctill obtaining the optimal solution did not exceed 4000.

(6)

References

[1] L.K. Meizin, Choosing the optimal bu!er stock between two sections of a continuous process, Automation and Remote Control 45 (8) (1984) 1086}1089.

[2] L.K. Meizin, Optimizing contingency reserves, Automa-tion and Remote Control 48 (2) (1987) 268}271. [3] T. Kimura, Optimal bu!er design of an M/G/s queue with

"nite capacity, Stochastic Models 12 (1) (1996) 165}180. [4] J.A. Buzacott, Prediction of the e$ciency of production

systems without internal storage, International Journal of Production Research 6 (3) (1968) 173}188.

assem-bly/disassembly systems with unreliable machines and random processing times, IIE Transactions 30 (1) (1998) 41}53.

Systems, Plenum Press, New York, 1994.

[12] A. Munoz, S. Martorell, V. Serdarell, Genetic algorithms in optimizing surveilance and maintenance of components, Reliability Engineering & System Safety 57 (2) (1997) 107}120.

[13] L. Painton, J. Campbell, Genetic algorithm in optimiza-tion of system reliability, IEEE Transacoptimiza-tions on Reliability 44 (1995) 172}178.

using interval programming and genetic algorithm, Computers and Industrial Engineering 31 (1/2) (1996) 237}240.

[16] D. Coit, A. Smith, Reliability optimization for series} par-allel systems using genetic algorithm, IEEE Transactions on Reliability 45 (1996) 254}266.

[17] D. Coit, A. Smith, Penalty guided genetic search for reliability design optimization, Computers and Industrial Engineering 30 (4) (1996) 895}904.

Ex-pert 5 (1990) 49}53.

[21] D. Goldberg, Genetic Algorithms in Search, Optimization and Machine Learning, Addison-Wesley, Reading, MA, 1989.

[22] D. Whitley, J. Kauth, GENITOR: A di!erent Genetic Algorithm, Technical Report CS-88-101, Colorado State University, Fort Collins, USA, 1998.

Directory UMM :Data Elmu:jurnal:I:International Journal of Production Economics:Vol70.Issue1.Mar2001:

Structure optimization for continuous production systems

with bu

_!

ers under reliability constraints

G. Levitin

!,

*

, L. Meizin

"

G

N

G

G

G

G

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