J Intell ManufDOI 10.1007/s10845-009-0244-3

A new bottleneck detecting approach to productivityimprovement of knowledgeable manufacturing system

Hong-Sen Yan · Yu-Wei An · Wen-Wu Shi

Received: 24 May 2008 / Accepted: 26 January 2009© Springer Science+Business Media, LLC 2009

Abstract Manufacturing systems are usually restricted byone or more bottlenecks. Identification of bottleneck is akey factor to improving the throughput of a production sys-tem. However, locating the bottleneck is no easy task. Thispaper proposes a novel method of bottleneck detecting forknowledgeable manufacturing system (KMS). Presented anet-like model of the knowledgeable manufacturing systemfor bottleneck detection, which adapts well to the bottleneckanalysis in flexible product lines. Based on the model, theconcept of entire production net is defined and an approach toidentifying bottlenecks in entire production nets is developedand proven effective theoretically. A self-learning method isintroduced for storing the knowledge of bottlenecks and theirconditions in the knowledge base to detect which cells arerequired to upgrade their capacity. Validity of the approachis verified by the numerical experiment.

Keywords Knowledgeable manufacturing system ·Production bottleneck · Self-learning · Knowledgeablemanufacturing cell

Introduction

From the rapid development of manufacturing technology,emerge various innovative manufacturing modes, amongwhich are concurrent engineering, agile manufacturing, intel-ligent manufacturing, knowledgeable manufacturing, etc.The knowledgeable manufacturing system (KMS) is highlyintelligent, characterized by its self-adaptation, self-learning,

H.-S. Yan (B) · Y.-W. An · W.-W. ShiKey Laboratory of Measurement and Control of CSE, Schoolof Automation, Southeast University, Ministry of Education,Nanjing, Jiangsu 210096, Chinae-mail: hsyan@seu.edu.cn

self-evolution, self-reconfiguration, self-training and self-maintenance (Yan and Liu 2001; Yan 2006). Self-learning isone of KMS’s key merits, applicable to many problems, suchas selection of production style selection,production bottle-neck analysis, price forecasting, fault diagnosis, inventorymanagement and so on (Shi 2007). This paper proposes anovel approach to KMS shifting bottleneck analysis alongwith a self-learning method for storing the knowledge ofbottlenecks and their conditions in the knowledge base.

In light of the theory of constraints, in the process of pro-duction, if the capacity of one production cell is weak enough,it will become a bottleneck to restrict the full utilization of theother production cells (Goldratt 1990; Verma 1997). There-fore, it is necessary to locate and improve the bottlenecks inthe system for the enhancement of production performance.Bottleneck detecting plays an important role in the design ofa complex production system (Jacobs et al. 1993; Banaszak1997; Radovilsky 1998), distributed planning and controlsystems for the virtual enterprise (Soares et al. 2000), andevaluation of manufacturing process analysis (Hernandez-Matias et al. 2006), simulation improvement of productionmanagement at appliance manufacturer (Kale et al. 2007),etc.

To cope with rapid product demand changes, enterprisesprefer to employ more intelligent and flexible manufactur-ing systems (Kusiak 2000). And with the changes of productstyles, batches and sequences, bottlenecks tend to shift fromone manufacturing cell to another (Moss and Yu 1999).

Different methods are also proposed to detect the bottle-necks in serial production lines. Generally, a definition ofbottleneck (BN) in serial production lines is presented andidentification rules are developed, with the general assump-tion that a machine is the bottleneck if the sensitivity of thesystem’s performance index to its production rate in iso-lation is the greatest, as compared to all other machines

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(Kuo et al. 1996). This definition captures the system natureof BN, but it is confined to Bernoulli production lines.

One common method is set to analyze the queue lengthsof the manufacturing cells (Zhuang et al. 1998). In this case,with either the queue length or waiting time determined, theentity with the longest queue length or waiting time is con-sidered to be bottleneck. This method is suitable for ana-lyzing production systems with infinite buffers or at leastof very large buffer sizes. For systems containing only lim-ited buffers (i.e., limited buffer sizes) or systems withoutbuffers, the results are of insufficient accuracy. Furthermore,this approach is incapable of analyzing other elements but theprocessing machines. For example, the AGV system itself isa type of queue, yet it does not have a queue length or waitingtime in the classical sense, so we cannot rely upon the queuelength for bottleneck detection. But in the AGV system, bot-tlenecks still exist. For example, according to Roser’s defi-nition (Roser et al. 2002), the AGV with the longest activeperiod (the period of moving to a pickup location, movingto a drop off location, recharging, and being repaired whilewaiting time is referred to inactive period) can be consideredas bottleneck.

Chiang et al. (1998) propose a system-theoretic approachto bottlenecks in Markovian production lines, which is basedon the sensitivity of the system production rate to machines’reliability parameters. This bottleneck detecting methodrelies either on the data available on the factory floor throughreal time measurements or on the data that can be construc-tively calculated using the machines and buffers parameter,but the definition is not applicable to systems with arbitrarynumber of machines or to those in the automotive industry.

Chiang et al. (2000) also come up with a method fordown-time bottleneck identification which is developed inthe framework of serial production lines with unreliablemachines and finite buffers, and the identification tool isdesigned in light of the probabilities of machine blockagesand starvations; this paper provides a practical tool for BNidentification in large volume manufacturing environment.

Li and Meerkov (2000) present a definition of due-timeperformance bottlenecks and propose a method for their iden-tification. The identifying tool of the method is based on thedata available on the factory floor through real time mea-surements. The defect of the method is that it is not provedanalytically and only numerical justification is obtained.

In the papers written by Kuo et al. (1996), Chiang et al.(1998), Chiang et al. (2000), and Li and Meerkov (2000),production systems with linear topology (i.e., serial produc-tion lines) are analyzed, in which single-product is processed.In this model, parts are processed sequentially on machines:pieces flow from the first machine to the last machine wherethe process cycle is completed. When a machine is not avail-able, parts wait in the buffer immediately upstream themachine. However, modern production systems, such as

flexible manufacturing systems, are assuming more and morecomplex non-linear structures, and they are capable of man-ufacturing a broad range of products, and of changing theircharacteristics quickly and frequently by machines execut-ing split and merge operations or modifying the route that theproduct follows through a set of system resources (such pro-duction systems will be introduced in the following section).It is thus difficult to apply methods mentioned above to flex-ible manufacturing systems, quite significant theoreticallyand practically to develop an effective bottleneck detectingmethod for these systems.

Roser et al. (2001) come up with another method for shift-ing bottleneck detection based on the duration during whicha manufacturing cell is active without interruption, but theirmethod has not been strictly proved. Roser et al. (2002) pro-pose a method for shifting bottleneck detection by measur-ing the utilizations of the different manufacturing cells, thismethod is targeted directly at the machine, and the bottle-neck can be detected for any given period of time. The shift-ing bottleneck detection method can also be employed todetect short term or instantaneous bottlenecks or the non-bottlenecks reliably. However, it flaws in being slightly moredifficult to implement than other methods.

In summary, there are various methods presented in liter-ature, most of which, however, fail to address the productionconditions concerning bottlenecks and KMS’s great flexibil-ity, falling far short of the requirements to be met.

To solve the existing problems, this paper proposes a net-like model of KMS for shifting bottleneck detection, on thebasis of which, the concept of entire production net and amethod for identifying bottlenecks in entire production netsare proposed. Furthermore, a self-learning method is intro-duced for storing the knowledge of shifting bottlenecks andtheir conditions in the knowledge base to determine whatcells require to be improved under varying conditions.

Net-like model of KMS

In order to meet the ever-changing market demands, the flexi-ble manufacturing system suitable for small batch productionis adopted by more and more enterprises. KMS is known to bea highly flexible and intelligent manufacturing system. Dueto the changes of product styles, batches and sequences, bot-tlenecks tend to shift from one manufacturing cell to another.Thus it is impossible to design an ideal manufacturing systemthat can eliminate all production bottlenecks. As mentionedbefore, the existing methods of shifting bottleneck analysisfail to take production condition changes and KMS’s greatflexibility into consideration, thus incapable of adjusting theproduction capacity and production process to attain the opti-mal objective. As a feasible solution to the defect, the net-like model of KMS is proposed here in Fig. 1. It is actually a

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. . . input

input

input

output

output . . .

1 c

2 c

3 c 4 c 5 c

6 c

7 c

n c

Fig. 1 Net-like model of KMS

front buffer rear bufferprocessor

input output

Fig. 2 Knowledgeable manufacturing cell (KMC)

generalization of the complex manufacturing system, inwhich manufacturing cells perform different productionoperations and are decoupled by intermediate buffers. Man-ufacturing cells take parts from the front buffer, execute aspecific operation on them, and place the worked parts inthe rear buffer. Though KMS is similar to the flexible manu-facturing system (FMS), the former is of higher intelligencethan the latter. The net-like model can be established by deter-mining the layout of nodes in the model, and relationshipsbetween them, according to the real layout of manufacturingcells, product assembly structure and part processing routes.

As is shown in Fig. 1, the KMS is a multiple-input andmultiple-output (MIMO) dynamic system running in rules.The KMS consists of β knowledgeable manufacturing cellscu, u = 1, . . . , β, and each cell cu has a front buffer bf

u , arear buffer br

u and a processor (also denoted as cu), as shownin Fig. 2. Two buffers in different knowledgeable manufac-turing cells can overlap. For instance, the rear buffer br

a ofmanufacturing cell ca can overlap the front buffer bf

b of man-ufacturing cell cb, i.e., br

a = bfb. If a part is transferred from

ca to cb directly, the virtual buffers bra and bf

b are set up formanufacturing cells ca and cb. In this case,when ca is pro-ducing, the buffers br

a and bfb are considered to be empty; and

when ca completes the process and waits to transfer the partto cell cb, the buffers br

a and bfb are considered to contain the

part. In the following, four types of cells relative to a part,the entire production net, and bottleneck cells are defined inbrief:

• Front and rear cells: If KMC cm transfers part p j toKMC cn , and then cm is the front cell of cn relative top j , and cn is the rear cell of cm relative to p j , denoted

as NodeCm

Pj−→ NodeCn . One KMC may have a numberof front and rear cells.

• Start and finish cells: If KMC ci has no front cell rela-tive to part p j , then the KMC ci is a start cell relative topart p j , i.e., S(ci , p j ); if KMC ci has no rear cell rela-tive to p j , the KMC ci is a finish cell relative to p j , i.e.,F(ci , p j ).

• The entire production net: In KMS shown in Fig. 1, underthe condition of F(ci , p j ), all the manufacturing cells(which transfer parts to KMC ci directly or indirectlyfor manufacturing part p j ) plus the flow of these partsform the entire production net relative to p j , denoted byNet (ci , p j ), and the set of nodes (i.e., vertexes com-posed of the manufacturing cells described above) inNet (ci , p j ) is denoted by VNet(ci , p j ).

• Bottleneck cells and bottlenecks: To improve the produc-tion capacity of the entire production net Net (ci , p j ),the production capacities of manufacturing cells in theset CB = {cb1, cb2, . . . , cbγ } and CB ⊆ VNet(ci , p j )

must be improved simultaneously, while the others needno change. CB is the set of bottleneck cells, that is,B(ci , p j ), and its corresponding processors cb1, cb2, . . . ,

and cbγ are referred to as bottlenecks.

Many production systems, such as assembly/disassemblynetworks system and non-linear multi-product multi-stagelines system (Helber 1998; Colledani et al. 2008) can be con-sidered as the net-like model of KMS. An illustration (takingfrom Colledani et al. 2008) is depicted in Fig. 3, where thesquares represent machines and the circles denote buffers. Tothe best of our knowledge, no bottleneck detecting methodfor such systems has yet been reported.

Let Cc denote the intersection of VNet(ci , pl) and VNet(c j , pg), i.e., Cc = VNet(ci , pl) ∩ VNet(c j , pg). If Cc �=φ, Cc is called coupling set of VNet(ci , pl) and VNet(c j , pg),and Cc = φ, VNet(ci , pl) and VNet(c j , pg) are considered tobe independent. In the period of time ts, if none of the struc-tures of entire production nets changes in the whole KMS,i.e., neither new product is produced nor product structurechanges, ts is called a stable period. Any unstable period canbe divided into a number of stable periods.

The bottleneck cell B(ci , p j ) is changeable in differenttime slots of one stable period or in different stable periods.So the bottleneck cell may shift from one KMC to another indifferent time periods. In this paper, the shifting bottleneckcell is investigated in one stable period.

The general description about shifting bottleneck is(according to Roser et al. 2002): bottleneck shift may be dueto a system random failure event or due to a gradual change

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Fig. 3 Five-stage non-linearsystem 1M 4M

5M

6M

3M

2M

7M

8M

9M

1B

2B

3B

4B

5B

6,1B

6,2B

7B

8B

of the manufacturing system with time. At a given moment,the capacity of one production cell will be too weak to work,causing production delays, whereas only several hours ordays later non-bottleneck production cell may become a newbottleneck. Frequently, upon controlling a system bottleneck,new ones keep emerging one by one, that is to say the bot-tleneck is shifting from cell to cell. We now present a simpleexample to illustrate the aforementioned shifting bottleneck.Production cell which has the longest process time to pro-duce unit product is defined as theoretical production bottle-neck. The physical layout of production system is shown inFig. 4, and the flow of material follows the arrows. In thiscase, production cells c2, c3, c4 and c5 are dedicated cells,i.e., each can produce assemble only one type of parts. Onthe contrary, c1 is a flexible cell and can process assemblesimultaneously both part types p11 and p21. The productstructure is shown in Fig. 5. Processing time of each partand its corresponding production cell are given in Table 1,where the expectation value of processing time in the thirdrow is required for performing only one operative task inits corresponding cell, and in the last row, it is given forperforming simultaneously two operative tasks in its corre-sponding cell. Assuming that planning horizon consists oftwo time periods, in the first period, two products p1 and p2

are processed at the same time, while only p1 is processedin the second period. The theoretical production bottleneckscan be determined by the expectation of production time.In the first period, due to max{2t11, 5t12, t1, 3t22, 3t21, t2} =max{2 × 12, 5 × 3, 20, 3 × 28, 3 × 34, 75} = 3t21 (whereti j and ti denote processing times of part pi j and product pi

respectively), product p21 is processed in the cell c1, so c1 canbe determined as theoretical production bottleneck, while inthe second period, max{2t11, 5t12, t1} = t1 and product p1 isprocessed in the cell c3, so c3 can be determined as theoret-ical production bottleneck. Bottleneck cell shifts from c3 toc1 in different periods.

For the application of group technology and numericalcontrol technology, one knowledgeable manufacturing cellcan process parts of different types. Let index α representthe quantity of part types in KMS shown in Fig. 1, index β

the quantity of knowledgeable manufacturing cells, and p =

1c

2c3c

4c

5c

Fig. 4 Physical layout of production system

3c

1c

1c

4c

5c

2c 22p11p 12p 202p

21p

1p 2p

2 5

1

3 5

Fig. 5 Product structure with product processing (rectangle represent-ing component)

Table 1 Expectation value of processing time and corresponding pro-duction cell

Production cell: c1 c2 c3 c4 c5

Parts and products: p11 p21 p12 p1 p22 p2

Sole part or productprocessed

8 25 3 20 28 75

Multi-parts orproductssimultaneouslyproduced

12 34

[p1 . . . pw . . . pα]T (w = 1, . . . , α) denote the part vector,c = [c1 . . . cu . . . cβ ]T (u = 1, . . . , β) the vector of manufac-turing cells. Let vector bf

u(t) = [bfu(p1, t) . . . bf

u(pw, t) . . .

bfu(pα, t)]T denote the state of front buffer of cu, br

u(t) =[br

u(p1, t) . . . bru(pw, t) . . . br

u(pα, t)]T the state of rear bufferof cu , where bf

u(pw, t) represents the quantity of part pw in

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the front buffer of cu at time t , and bru(pw, t) the quantity

of part pw in the front buffer of cu at time t . Let pmu (t) =[

pmu (p1, t) . . . pm

u (pw, t) . . . pmu (pα, t)

]Tdenote the state of

production of cu , where pmu (pw, t) represents the quantity

of part pw being processed in cu at time t . Let brifu

(t) =[bri

fu(p1, t) . . . bri

fu(pw, t) . . . bri

fu(pα, t)

]Tdenote the parts

transferred from the rear buffer of ci to the front one of cu ,where bri

fu(pw, t) represents the quantity of pw transferred

from the rear buffer of ci to the front one of cu .The state of front buffers is

Bf(t) = [bf1(t) . . . bf

u(t) . . . bfβ(t)]

=

⎡⎢⎢⎢⎢⎢⎣

bf1(p1, t) . . . bf

u(p1, t) . . . bfβ(p1, t)

. . . . . . . . . . . . . . .

bf1(pw, t) . . . bf

u(pw, t) . . . bfβ(pw, t)

. . . . . . . . . . . . . . .

bf1(pα, t) . . . bf

u(pα, t) . . . bfβ(pα, t)

⎤⎥⎥⎥⎥⎥⎦

. (1)

The state of rear buffers is

Br(t) = [br1(t) . . . br

u(t) . . . brβ(t)]

=

⎡⎢⎢⎢⎢⎢⎣

br1(p1, t) . . . br

u(p1, t) . . . brβ(p1, t)

. . . . . . . . . . . . . . .

br1(pw, t) . . . br

u(pw, t) . . . brβ(pw, t)

. . . . . . . . . . . . . . .

br1(pα, t) . . . br

u(pα, t) . . . brβ(pα, t)

⎤⎥⎥⎥⎥⎥⎦

. (2)

The state of production is

Pm(t) = [pm1 (t) . . . pm

u (t) . . . pmβ (t)]

=

⎡⎢⎢⎢⎢⎣

pm1 (p1, t) . . . pm

u (p1, t) . . . pmβ (p1, t)

. . . . . . . . . . . . . . .

pm1 (pw, t) . . . pm

u (pw, t) . . . pmβ (pw, t)

. . . . . . . . . . . . . . .

pm1 (pα, t) . . . pm

u (pα, t) . . . pmβ (pα, t)

⎤⎥⎥⎥⎥⎦

. (3)

The state of parts transferred form rear buffers to frontbuffers is

Brf (t) =

[br

f1(t) . . . br

fu(t) . . . br

fβ (t)]

=[

β∑i=1

brif1(t) . . .

β∑i=1

brifu

(t) . . .β∑

i=1b

rifβ

(t)

]

=

⎡⎢⎢⎢⎢⎢⎢⎢⎣

β∑i=1

brif1

(p1, t) . . .β∑

i=1bri

fu(p1, t) . . .

β∑i=1

brifβ

(p1, t)

. . . . . . . . . . . . . . .β∑

i=1bri

f1(pw, t) . . .

β∑i=1

brifu

(pw, t) . . .β∑

i=1bri

fβ(pw, t)

. . . . . . . . . . . . . . .β∑

i=1bri

f1(pα, t) . . .

β∑i=1

brifu

(pα, t) . . .β∑

i=1bri

fβ(pα, t)

⎤⎥⎥⎥⎥⎥⎥⎥⎦

.

(4)

As the quantity of parts is unlikely to be negative in Eqs. 1–4,so we obtain

bfu(pw, t) ≥ 0, br

u(pw, t) ≥ 0,

pmu (pw, t) ≥ 0, bri

fu(pw, t) ≥ 0.

Suppose the stable period is t0 − t1. Then according to Eqs. 1–4, the following formula can be established:

Bf(t1) = Bf(t0) + ∑e∈Ebb(t0,t1)

Brf (t)

− ∑e∈Em(t0,t1)

Pm(t) + ∑e∈E in(t0,t1)

Γ (t)(5)

where e denotes event, for instance, parts sent from rear bufferto front buffer, parts to be processed, and input parts sent fromoutside to the KMS buffers, etc. Ebb(t0, t1) denotes that partsare transferred between two buffers in the period of t0 − t1;Em(t0, t1), the set of events that parts are to be processed inthe period of t0 − t1; E in(t0, t1), the set of events that partsare sent from outside to the KMS buffers in the period oft0 − t1;Γ(t), which is a α ×β matrix, represents the quantityof parts, arriving from outside of the entire production net attime t in the period t0 − t1.

The left side of Eq. 5 is a α×β matrix, in which the vectorof u rank is

bfu(t1) = bf

u(t0) +∑

e∈Ebbu (t0,t1)

brfu

(t)

−∑

e∈Emu (t0,t1)

pmu (t) +

∑

e∈E inu (t0,t1)

γ u(t)

= bfu(t0) +

β∑i=1

∑

e∈Ebbu (t0,t1)

brifu

(t)

−∑

e∈Emu (t0,t1)

pmu (t) +

∑

e∈E inu (t0,t1)

γu(t) (6)

where Ebbu (t0, t1) denotes the set of events that parts are trans-

ferred from the rear buffers of the other manufacturing cellsto the front ones of cell cu in the period of t0 − t1; Em

u (t0, t1),the set of events that parts begin to be processed in the man-ufacturing cell cu in the period of t0 − t1; E in

u (t0, t1), theset of events that parts are sent from outside to the frontbuffer of cu in the period of t0 − t1; γu(t) = [γu(p1, t) . . .

γu(pw, t) . . . γu(pα, t)]T denotes the quantity of parts sentfrom outside of the entire production net to cu at time t in theperiod t0 − t1.

Buffer states of KMS

Due to the product diversity and small batch production ofKMS, its modes of material stream are not fixed. Moreover,products are processed simultaneously in a large number oftypes, leading to the fact that many manufacturing cells alter-

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nate between idle state and working state frequently. As aresult, it is hard to detect the shifting bottlenecks. Moreover,most of the existing bottleneck detection methods fall farshort of the expectation (see section “Introduction”). There-fore, this paper sets up a net-like model of KMS, in whichthe knowledgeable manufacturing cells are classified intofour types according to the four different states: (a) frontand rear buffers both enabled; (b) front buffer enabled whilerear buffer non-enabled; (c) front buffer non-enabled whilerear buffer enabled; (d) front and rear buffers both non-enabled. “Buffer enabled” means that the adequate quantityof parts must be stored in the corresponding buffers in orderto keep manufacturing cells working continuously. If a bufferis enabled, processor can be operative or prepared (maybe inworking or not in working). If a buffer is non-enabled, pro-cessor can also be operative or prepared, but not in workingfor unprepared parts because there is not enough quantityof parts to meet production requirements. Buffer capacityis measured by the numbers of items stored in buffers. Inthis paper, buffer capacities are finite, so that they may beconstraints of production system. Based on the four states, amethod for detecting bottlenecks in an entire production netis proposed. The present bottleneck detecting method deter-mines whether to generate a bottleneck through estimatingwork in process

Method for acquiring the entire production nets

Knowledgeable manufacturing cells include assembly cells,machining cells, etc. If a KMC is an assembly cell, it usu-ally has multi-inputs and single output. The input and outputinvolve many different parts (finished product is viewed asa special part). For instance, let M(ci , {p j , pk, . . . , pr }, pl)

denote that the parts {p j , pk, . . . , pr } are assembled to partpl by cell ci ; and then M(ci , {p j , pk, . . . , pr }, pl) is calledone of the task modes of ci . The inputs of task modeM(ci , {p j , pk, . . . , pr }, pl) are {p j , pk, . . . , pr }, and its out-put is pl . Let M(cq , {pm}, pn) denote that part pm ismachined to part pn by cq (for the sake of convenience, thedenotations pm and pn represent the different process stepsof the same part). And M(cq , {pm}, pn) is called a task modeof cell cq , in which the input is pm and output pn . As trans-port tools and production cells may become bottlenecks andboth have the same flow of material, if we suppose that thefront buffers of transport tools overlap with the rear ones ofmanufacturing cells and the rear buffers of transport toolsoverlap with the front ones of manufacturing cells, trans-port tools can be regarded as special manufacturing cells.Thus, transport tools and production cells go by the gen-eral name of general manufacturing cells. For example, letM(ct , {pw}, pw) denote that pw is transferred by transporttool ct , and M(ct , {pw}, pw) is called a task mode of ct . Thebuffers of transport tools consist of those of manufacturing

cells; for instance, ct transfers pw from the rear buffers ofc1, c2 and c3 to the front buffers of c4 and c5. Then the fol-lowing formulas can be obtained:

bft (pw, t) = br

1(pw, t) + br2(pw, t) + br

3(pw, t),

brt (pw, t) = bf

4(pw, t) + bf5(pw, t).

Let Ui denote the set of the task modes of KMC ci , and Uthe set of task modes of all manufacturing cells, i.e., U =U1, . . . , Uβ . Let adjoining function brv

fi(p j , t0, t1) denote

whether or not the part p j is transferred from the rear bufferof KMC cv to the front buffer of ci in the stable period oft0 − t1. If

∑e∈Ebb

i (t0,t1)

brvfi

(p j , t)=0, then brvfi

(p j , t0, t1) = 0;

If∑

e∈Ebbi (t0,t1)

brvfi

(p j , t) > 0, then brvfi

(p j , t0, t1) = 1.

According to the set of task modes U and adjoining func-tions, the entire production nets can be detected by the fol-lowing method (assuming the stable period is t0 − t1):

Algorithm 1 The method for obtaining the entire produc-tion nets

Step 1: Get the set of finish cells according to their defini-tion.

Step 2: Select a finish cell F(ci , pl) from the set of finishcells, and set NodeCi as the root node of an entireproduction net of pl .

Step 3: Detect the input set {p j , pk, . . . , pr } of task modeM(ci , {p j , pk, . . . , pr }, pl) of ci , and the value ofadjoining function brv

fi(p j , t0, t1), v = 1 . . . β. If

brvfi

(p j , t0, t1) = 1, then set cv as the child node of

NodeCi , that is, NodeCv

pj→ NodeCi , and examinethe values of adjoining function brv

fi(pk, t0, t1),…,

brvfi

(pr , t0, t1) in sequence, to require all child nodesof NodeCi .

Step 4: If the child node ck of NodeCi serves for the pro-duction of part p j in the entire production netNet (ci , pl), and ck is not a start cell relative top j , then obtain the child nodes of NodeCk viathe method presented in Step 3, and repeat Steps3–4 until all start cells in Net (ci , pl) have beendetected.

Step 5: Repeat Steps 2–4 until all entire production netshave been obtained.

The aim of Algorithm 1 is to find the structure of productmanufactured in the whole production system. In this pro-cess, it is irrelevant to buffers capacity.

Buffer states of KMC

Suppose NodeCv

p j→ NodeCi in the stable period t0 − t1,and

[t ′0, t ′1

] ⊆ [t0, t1]. Let the front buffer state function

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bfi (p j , t0, t1) denote the state of part p j in the front buffer

of general manufacturing cell ci . If ∀t∈[t ′0, t ′1], ∃bfi (p j , t) <

ni (p j ) and pmi (p j , t)= 0, then set bf

i (p j , t0, t1)= 0, whereni (p j ) denotes the least quantity of p j required by ci tostart or maintain production. If ∀, ∃bf

i (p j , t) ≥ ni (p j ) orpm

i (p j , t) > 0, then set bfi (p j , t0, t1)= 1. If bf

i (p j , t0, t1)=0,it means that the part p j in the front buffer of ci is discontinu-ously enabled in the stable period [t0, t1]; if bf

i (p j , t0, t1)= 1,it means that the part p j in the front buffer of ci is continu-ously enabled in [t0, t1].

Let brv(ci , p j , t) denote the state of p j in the rear buffer

of general manufacturing cell. If cv is a manufacturing cell,then set br

v(ci , p j , t) = brv(p j , t); if cv is a transport tool,

set brv(ci , p j , t) = bf

i (p j , t). Similarly, let rear buffer statefunction br

v(ci , p j , t0, t1) denote the state of p j in the rearbuffer of ci . If ∀t ∈ [

t ′0, t ′1]

and ∃brv(ci , p j , t) < ni (p j ),

then brv(ci , p j , t0, t1) = 0; If ∀t ∈ [

t ′0, t ′1], ∃br

v(ci , p j , t) ≥ni (p j ), then br

v(ci , p j , t0, t1) = 1. If brv(ci , p j , t0, t1) = 0,

which means that the part p j in the rear buffer of cv is dis-continuous enabled for the requirement of ci in the stableperiod [t0, t1]; if br

v(ci , p j , t0, t1) = 1, it means that the partp j in rear buffer of general manufacturing cell cv is con-tinuously enabled for the requirement of general manufac-turing cell ci in the stable period [t0, t1]. Due to the statesof front and rear buffers, the general manufacturing cell hasfour states to one task mode. If one task mode has multi-inputs, then the general manufacturing cell is continuouslyenabled on condition that all inputs of the task mode are con-tinuously enabled. For example, if one task mode of ci isM(ci , {p j , pk, . . . , pr }, pl) under the condition of NodeCipl→ NodeCk , then the following results can be obtained

(where “&” denotes logic operator “and”):

State 1. Front and rear buffers are both continuously ena-bled, marked by M(ci , {p j , pk, . . . , pr }, pl)= FF,i.e., bf

i (p j , t0, t1)& bfi (pk, t0, t1)& … & bf

i (pr , t0,t1) = 1 and br

i (ck, pl , t0, t1) = 1.State 2. Front buffer is continuously enabled, but rear buffer

discontinuously enabled, marked by

M(ci , {p j , pk, . . . , pr }, pl) = FE, i.e.,

bfi (p j , t0, t1)&bf

i (pk, t0, t1)&. . .&bfi (pr , t0, t1)

= 1 and bri (ck, pl , t0, t1) = 0.

State 3. Front buffer is discontinuously enabled, but rearbuffer continuously enabled, marked by

M(ci , {p j , pk, . . . , pr }, pl) = EF, i.e.,

bfi (p j , t0, t1)&bf

i (pk, t0, t1)&. . .&bfi (pr , t0, t1)

= 0 and bri (ck, pl , t0, t1) = 1.

State 4. Front and rear buffers are both discontinuouslyenabled, marked by M(ci , {p j , pk, . . . , pr }, pl) =EE, i.e.,

bfi (p j , t0, t1) & bf

i (pk, t0, t1) & . . . & bfi (pr , t0, t1)

= 0 and bri (ck, pl , t0, t1) = 0.

If ci is a finish cell relative to pl , then its output is fin-ished goods, as a result of which the upper limitation of fin-ished goods warehouse can be allowed to be infinite andbr

i (. . . , pl , t0, t1) can be set to zero, that is, bri (. . . , pl , t0, t1)

= bri (0, pl , t0, t1) = 0.

Method for shifting bottleneck analysis

As the changes of production conditions, such as the changeof manufacturing products, may lead to the change of entireproduction nets, and some manufacturing cells are coupled indifferent entire production nets, so when a new entire produc-tion net comes out or an old entire production net disappears,many other entire production nets may change as well. Theknowledge of production bottlenecks under different condi-tions exists in production data. In this paper, a new methodis proposed to detect the shifting bottlenecks under differentproduction conditions according to the buffer states of gen-eral manufacturing cells. And a machine learning methodis adopted to store the knowledge of bottlenecks and theirconditions in the knowledge base.

Relations between buffer states and bottlenecks

The states of one task mode are FF, FE, EF and EE. If the stateof KMC cv is FF, then the rear buffer of cv is continuouslyenabled for the rear general manufacturing cell ci , whichmeans the production capacity of cv is large enough, that is,cv cannot be bottleneck; if the state of cv is FE, it meanscv is continuously busy for the lack of production capacity;but it depends on the front buffer state of ci to determinewhether cv is bottleneck; when the states of direct or indirectrear general manufacturing cell are FF, FE or EF, cv is notconsidered to be a bottleneck; cv is a bottleneck only whenthe states of direct or indirect rear general manufacturing cellis EE. Therefore, when the state of cv is FE, it only meets anecessary condition but not a sufficient one for cv to becomea bottleneck.

Proposition 1 SupposeNodeCv

p j→ NodeCi . If bfi (p j , t0,

t1) = 0 in the stable period t0 − t1, then M(cv, {. . .}, p j ) =EE or M(cv, {. . .}, p j ) = FE.

Proof For bfi (p j , t0, t1) = 0, it can be concluded that

∃ [t ′0, t ′1

] ⊆ [t0, t1] ,∀t ′ ∈ [t ′0, t ′1

], ∃bf

i (p j , t ′) < ni (p j ) andpm

i (p j , t ′) = 0. According to Eq. 6, we obtain

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bfi (p j , t ′) = bf

i (p j , t ′0) +β∑

k=1

∑

e∈Ebbi (t ′0,t ′)

brkfi

(p j , t)

−∑

e∈Emi (t ′0,t ′)

pmi (p j , t)

+∑

e∈E ini (t ′0,t ′)

γi (p j , t) (7)

As ∀t ∈ [t ′0, t ′1

]and ∃pm

i (p j , t) = 0, we get∑

e∈Emi (t ′0,t ′)

pmi

(p j , t) = 0. As bfi (p j , t ′0) ≥ 0, γi (p j , t) ≥ 0, brk

fi(p j , t) ≥

0, k = 1, . . . , β, soβ∑

k=1

∑e∈Ebb

i (t ′0,t ′)brk

fi(p j , t) ≥ 0 and

∑e∈E in

i (t ′0,t ′)γi (p j , t) ≥ 0. Thus Eq. 8 can be derived from

Eq. 7:

bfi (p j , t ′) = bf

i (p j , t ′0) +β∑

k=1

∑

e∈Ebbi (t ′0,t ′)

brkfi

(p j , t)

+∑

e∈E ini (t ′0,t ′)

γi (p j , t) < ni (p j ), (8)

⇒β∑

k=1

∑

e∈Ebbi (t ′0,t ′)

brkfi

(p j , t) < ni (p j ), (9)

⇒∑

e∈Ebbi (t ′0,t ′)

brvfi

(p j , t) < ni (p j ). (10)

Because the transport tool is regarded as a special manufac-turing cell, as has been said before, the front buffers of trans-port tools overlap with the rear ones of manufacturing cellsand the rear buffers of transport tools overlap with the frontones of manufacturing cells. Thus, if cv is a manufacturingcell, then br

v(ci , p j , t ′) = brv(p j , t ′) = ∑

e∈Ebbi (t ′0,t ′)

brvfi

(p j , t);

if cv is a transport tool, then brv(ci , p j , t ′) = bf

i (p j , t ′),and br

v(ci , p j , t ′) < ni (p j ), as a result of which brv(ci , p j ,

t0, t1) = 0, no matter whether cv is a manufacturing cell ortransport tool, that is, M(cv, {. . .}, p j ) = EE or M(cv, {. . .},p j ) = FE under the condition of NodeCv

p j→ NodeCi . ��If the capacity of the transport tool is large enough, then

its part-carrying time can be omitted. On this condition, onlymanufacturing cells are considered for production bottleneckdetection, and Proposition 1 is still applicable.

In an entire production net, there are many routes from afinish cell to start cells, the first manufacturing cell, whosestate is FE in each route, called busy node.

The present bottleneck detecting method determineswhether to generate a bottleneck through estimating workin process. Processing times and machine failures may prop-agate through the entire production net. For example, if the

processor (or machine) i has been in bottleneck (or failure)for a long enough time, its upstream machines will be blockedand product capacity of its downstream machines will beweakened or starved. In this case, state of cell ci (whichconsists of processor i and its front and rear buffers) is FE,while states of other cells may be FF or EE. Algorithm 2 canidentify bottleneck cells based these state of all cells.

Algorithm 2 Method for busy nodes detection in Net (ci , pl)

in t0 − t1

Step 1: Initialize the set of busy nodes CS=φ, and tempo-rary stack variable St = φ.

Step 2: If M(ci , {p j , pk, . . . , pr }, pl) = FE, then setCS = ci and go to Step 5; otherwise go to Step 3.

Step 3: If M(ci , {p j , pk, . . . , pr }, pl) = EE, then exam-ine the front buffer state function bf

i (pφ, t0, t1), φ =j, k,…,r ; if bf

i (pφ, t0, t1) = 0, then St = St∪{(ci , pφ)}.

Step 4: Select (ci , pφ) from stack St , St = St\{(ci , pφ)},search the child nodes of manufacturing cell ci on

conditionofNodecu

pφ→ NodeCi . (According to Pro-position 1, M(cu, {. . .}, pφ) = EE or FE.) IfM(cu, {. . .}, pφ) = FE, then set CS = CS ∪ {cu};if M(cu, {. . .}, pφ) = EE, then examine front bufferstate function bf

u(. . . , t0, t1); if bfu(. . . , t0, t1) = 0,

set St=St∪{(cu, . . .)},andrepeatStep4until St=φ.Step 5: Output the set of busy nodes CS.

Proposition 2 General manufacturing cells in the set of busynodes CS in entire production net Net (ci , pl), obtained byAlgorithm 2, are the production bottlenecks in the period[t ′0t1

], that is CS ⊆ B(ci , pl).

Proof 1) As we know, the finish cell has only two states:FE and EE. If M(ci , {p j , pk, . . . , pr }, pl)= FE, cv ∈ Net(ci , pl), v �= i , one task mode of cv is M(cv, {. . .},p j ), and pfm

v (p j , t)′ > pfmv (p j , t) in the period

[t ′0t1

], then

brv(p j , t)′ ≥ br

v(p j , t) ⇒ brvfi

(p j , t)′ ≥ brvfi

(p j , t), where

pfmv (p j , t) denotes the quantity of p j output by cv at time t

before the capacity of cv is improved, pfmv (p j , t)′ the quan-

tity of p j output by cv at time t after the capacity of cv isimproved, br

v(p j , t)′ that of p j in the rear buffer of cv at time tafter the capacity of cv is improved, and brv

fi(p j , t)′ that of p j

transferred from the rear buffer of cv to the front buffer of ci

at time t after the capacity of cv is improved. Then we obtain

bfi (p j , t1) = bf

i (p j , t0) +β∑

k=1

∑

e∈Ebbi (t0,t1)

brkfi

(p j , t)

−∑

e∈Emi (t0,t1)

pmi (p j , t)

+∑

e∈E ini (t0,t1)

γi (p j , t)

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bfi (p j , t1)

′ = bfi (p j , t0)

+ν−1∑k=1

∑

e∈Ebbi (t0,t1)

brkfi

(p j , t)

+∑

e∈Ebbi (t0,t1)

brvfi

(p j , t)′

+β∑

k=v+1

∑

e∈Ebbi (t0,t1)

brkfi

(p j , t)

−∑

e∈Emi (t0,t1)

pmi (p j , t)

+∑

e∈E ini (t0,t1)

γi (p j , t) (11)

So, we have b fi (p j , t1)′ − b f

i (p j , t1) = ∑e∈Ebb

i (t0,t1)

brvfi

(p j , t)′ − ∑e∈Ebb

i (t0,t1)

brνfi

(p j , t) ≥ 0. As bfi (p j , t0, t1) = 1,

i.e., ∀t ∈ [t0, t1] , ∃bfi (p j , t) ≥ ni (p j ) or pm

i (p j , t)> 0, so bf

i (p j , t1)′ ≥ ni (p j ) or pmi (p j , t) > 0, and

bfi (p j , t0, t1)′ = 1. It means that the capacity improvement

of the other general manufacturing cells cannot change theFE state of ci , and only that of ci can upgrade the capacityof Net (ci , pl), that is, ci ∈ B(ci , pl) or CS ⊆ B(ci , pl).

It can also be concluded that the other general manufac-turing cells in the set of busy nodes CS = {cs1, cs2, . . . ,

csd} are bottlenecks, i.e., CS ⊆ B(ci , pl), the proof of whichis similar to Step 1. and thus omitted. ��

Machine learning method for shifting bottleneck analysis

Proposition 2 describes how to detect the production bottle-neck in terms of the ideal buffer states (see the subsection“Buffer states of KMC”). But in practice, if bf

i (p j , t0, t1) =0, with a short total non-enabled time, then p j in the frontbuffer of ci is approximately continuously enabled in [t0, t1].Thus, the definition of the front buffer state function forapplication needs to be modified. First, divide the stableperiod [t0, t1] into ξ equal time slots, [t10, t11], [t11, t12], …,[t1(ξ−1), t1ξ ], where t10 = t0, t1ξ = t1. Then the front bufferstate in practice can be obtained by Algorithm 3 (wherea ∈ (0, 1], and n1

ξis called buffer ready percentage). The rear

buffer state function can also be acquired in a similar way.

Algorithm 3 Determination of front buffer state function inpractice

Step 1: Initialize the variables n1 = 0, q = 1, and v = a.Step 2: Detect the value of bf

i (p j , t1(q−1), tq) in time slot(t1(q−1), tq ]. If bf

i (p j , t1(q−1), tq)=1, then

n1 = n1 + 1.

Step 3: If q < ξ, q = q + 1, then go to Step 2; otherwise goto Step 4.

Step 4: If n1ξ

≥ v, then bfi (p j , t0, t1) = 1, else bf

i (p j ,

t0, t1) = 0.

Since production conditions change frequently, which maycause bottleneck shifts, it is necessary to adopt a machinelearning method for shifting bottleneck analysis. The com-monly used ones include: rote learning, learning from exam-ples, learning by analogy, inductive learning, explanation-based learning, etc (Mitehell 1997; Michalski 1998). To setup a shifting bottleneck analysis subsystem for KMS, therote learning method is employed in our study. The subsys-tem structure is shown in Fig. 6. The stable period detectionagent detects the change of stable periods according to pro-duction plans, by which the new stable period will be put inthe list of stable periods. The entire production net acquire-ment agent acquires the entire production nets by Algorithm1 according to the list of stable periods and the states of gen-eral manufacturing cells, and then puts new entire produc-tion nets in the list of entire production nets. The productionbottleneck detection agent divides stable period into manyequal time slots, and acquires the values of front and rearbuffer state function in every little time slot by Algorithm 3according to the list of entire production nets and the states ofgeneral manufacturing cells, and then pinpoint the shiftingproduction bottlenecks in each stable period by Algorithm2. The rote-learning agent stores the information of stableperiods, entire production nets, buffer states, and produc-tion bottlenecks in the database. The rote-learning process isdepicted by Algorithm 4. By the cooperative work of the sta-ble period detection agent, entire production net acquirementagent, production bottleneck detection agent and rote learn-ing agent, the KMS shifting bottleneck analysis subsystemcan update the knowledge of production bottleneck in time.Consequently, the capacity and efficiency of production canbe upgraded on basis of the knowledge of shifting productionbottlenecks, stable periods and entire production nets in thedatabase.

Algorithm 4 The rote-learning for storing knowledge ofproduction bottleneck

Step 1: Let Bottleneck data set BDB = ∅.Step 2: Acquire the stable periods knowledge (SPK), the

entire production nets knowledge (EPNK) and theproduction bottleneck knowledge (PBK) fromrespective agents.

Step 3: Construct knowledge data KD = {SPK, EPNK,PBK}.

Step 4: If KD/∈BDB, then BDB=BDB∪KD; otherwiseBDB = BDB.

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Fig. 6 Structure of KMSshifting bottleneck analysissubsystem

Algorithm 1

Are new stable periods

obtained ?

List of new stable periods

List of entire production nets

Obtain stable periods according to

production task

Is the up list empty ?

Is the right list empty ?

Agent of detecting stable periods

Agent of obtaining entire production net

Agent of obtaining temporary bottlenecks

Rote-learning

Agent of obtaining shifting bottlenecks

Algorithm 2

Yes

Yes

No

NoYes

No

Production data

Bottleneck data

The knowledge of shifting production bottlenecks isexploited mainly for two ends. First, when the capacity ofan entire production net needs to be improved, it is necessaryto enhance that of the shifting production bottlenecks in theentire production net gradually until the entire production netcapacity meets the requirement. Second, when the efficiencyof an entire production net needs upgrading, it is necessaryto reduce the capacities of non-bottleneck general manufac-turing cells gradually, provided that the bottleneck positionsdo not change in the entire production net.

Computational experiment

The algorithms proposed in this paper are implemented viaDelphi7.0. And the KMS shifting bottleneck analysis subsys-tem is developed accordingly. Our experiments are made ona Pentium 2.4 G personal computer (PC) with 512 MB mem-ory under Win 2000. It takes 0.120 sec. to obtain the entireproduction nets in the following example by Algorithm 1,and 0.235 sec. to detect the shifting production bottlenecksby Algorithm 2.

In this experiment, the aforementioned method has beenapplied at the KMS originating from an automobile enter-prise. The layout of the system is shown in Fig. 7 (the squaresindicate buffers and the circles represent processors), and itconsists of five knowledgeable manufacturing cells{c1, c2, c3, c4, c5}, each of which perform complex process-ing or assembly manipulation; in the period of t0 − t8, thetask modes of the five cells are shown below (suppose the

transport tools in this example have capacities large enough,to be neglected in the entire production nets):

Cell 1: M(c1, {p101}, p11), M(c1, {p201}, p21)

Cell 2: M(c2, {p102}, p12), M(c2, {p301}, p31)

Cell 3: M(c3, {p11, p12}, p1)

Cell 4: M(c4, {p21}, p22), M(c4, {p302}, p32)

Cell 5: M(c5, {p202, p22}, p2), M(c5, {p31, p32}, p3)

The production tasks in the period of t0 − t8 are shown inTable 2.

Table 3 shows the adjoining functions of the five cells.The three entire production nets are obtained by Algo-

rithm 1 in light of the data of Tables 2 and 3: Net (c3, p1),

Net (c5, p2) and Net (c5, p3), corresponding respectively to(t0 − t1, t4 − t5, t5 − t6, t7 − t8), (t1 − t2, t4 − t5, t6 − t7, t7 − t8)and (t2 − t3, t5 − t6, t6 − t7, t7 − t8), as shown in Figs. 8– 10.

The product structures are: one p1 is composed of two p11

and five p12; one p2 is composed of three p22 and two p202;and one p3 is composed of four p31 and two p32.

The processing times of c1, c2, c3, c4, c5 are subject toexponential distributions, as shown in Table 4.

To verify the validity of the algorithms proposed in thispaper, a simulated production system is built up with the cellcapacities shown in Table 4. The ready buffer percentagesof general cell buffers are obtained, as shown in Fig. 11.With the threshold variable v = 0.95 in Algorithm 3, the cellbuffer states are given in Fig. 12. Via Proposition 2, shiftingbottlenecks in various production conditions are identified(see Fig. 13 and Table 5).

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Fig. 7 System layout

2c

1c 5c4c

301p

32p

31p

22p

3p

101p

11p

102p

2p

1p

3c

12p

202p

201p

21p

201p

Table 2 Production tasks

Production Stable period

t0 − t1 t1 − t2 t2 − t3 t3 − t4 t4 − t5 t5 − t6 t6 − t7 t7 − t8

p1 Yes No No No Yes Yes No Yes

p2 No Yes No No Yes No Yes Yes

p3 No No Yes No No Yes Yes Yes

Notes: “Yes” refers to product being processed, “No” means product not being processed

Table 3 Adjoining functions of five cells in different stable periods

Adjoiningfunction

Stable period

t0 − t1 t1 − t2 t2 − t3 t3 − t4 t4 − t5 t5 − t6 t6 − t7 t7 − t8

ts = t0 ts = t1 ts = t2 ts = t3 ts = t4 ts = t5 ts = t6 ts = t7

te = t1 te = t2 te = t3 te = t4 te = t5 te = t6 te = t7 te = t8

br1f3

(p11, ts, te) 1 0 0 0 1 1 0 1

br1f4

(p21, ts, te) 0 1 0 0 1 0 1 1

br2f3

(p12, ts, te) 1 0 0 0 1 1 0 1

br2f5

(p31, ts, te) 0 0 1 0 0 1 1 1

br4f5

(p22, ts, te) 0 1 0 0 1 0 1 1

br4f5

(p32, ts, te) 0 0 1 0 0 1 1 1

… 0 0 0 0 0 0 0 0

Fig. 8 Entire production netNet (c3, p1)

1c

2c

3c

101p 11p

102p 12p

11p

12p

1pf

1br

1b

f3b

r3b

f2b r

2b

The results of bottleneck detection by a simulated experi-ment are presented in Fig. 13. For example, for the first rowand the first column, “T0-Tl-N31” shows that the productp1 is produced in the entire production net Net (c3, p1)(con-sists of cells c1, c2 and c3) in the stable period [t0, t1]. Based

on the results shown in Fig. 13, running states of produc-tion cells and results of bottleneck detection are recorded inTable 6. The results given in Table 6 indicate that productionbottleneck is shifting from one cell to another in differentperiods.

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Fig. 9 Entire production netNet (c5, p2)

4c

5c

21p 22p

22p 202p2p

1c

201p 21p

202p

f1b

r1b f

4br4b

f5b r

5b

Fig. 10 Entire production netNet (c5, p3)

2c

4c

5c

301p 31p

302p 32p

31p 32p3pf

2br2b

f4b

r4b

f5b

r5b

Table 4 Parameters of negative exponential distributions under different production state

Processing cell:Parts and products:

c1 c2 c3 c4 c5

p11 p21 p12 p31 p1 p22 p32 p2 p3

Sole part or product processed 0.1250 0.0400 0.3333 0.0833 0.0500 0.0357 0.0667 0.0133 0.0313

Multi-parts or products simultaneously produced 0.0833 0.0294 0.2000 0.0714 0.0286 0.0385 0.0122 0.0263

Fig. 11 Ready bufferpercentages of generalmanufacturing cells.Note: “Tk-Tl-Nij,Front, Rear,Cm” shows the entireproduction net Net (ci , p j ) inthe stable period [tk , tl ], thefront buffer, the rear buffer, andmanufacturing cell cmrespectively

Based on the product structures and parameters of negativeexponential distributions for parts and products being pro-cessed in different production periods shown in Table 4, thetheoretical production bottlenecks can be determined through

computing the expectation of production time (see Table 7).For instance, in the whole system, there is only one productp1 being processed in the cell c3 in the first stable period,one product p1 consisting of two parts p11 and five parts p12

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Fig. 12 Buffer states ofknowledgeable manufacturingcells. Note: “Tk-Tl-Nij” meansthe same as that in Fig.11.,“FF”, “FE”, “EF”, “EE”represent the states of whetheror not the front and rear bufferbeing continuously enabled, and“E”, “F” refer to the state of thebuffer not being continuouslyenabled and continuouslyenabled respectively

Fig. 13 Shifting productionbottlenecks under differentproduction state. (Note: “Yes”means correspondingproduction net beingmanufacturing, “No” denotescorresponding production netbeing not working)

are produced in the cells c1 and c2 respectively. Since onlyone part or product is processed in each cell in this period,parameters of negative exponential distributions of process-ing time are µ11 = 0.1250, µ12 = 0.3333, and µ1 = 0.0500(let µi j and µi denote parameters of negative exponentialdistributions of processing time of part pi j and product pi

respectively). The processing time expectation of one p11 isE(t11) = 1/µ11 = 8minutes, that of one p12 is E(t12) =

1/µ12 = 3minutes, and that of one p1 is E(t1) = 1/µ1 =20minutes. Thus, as max{2E(t11), 5E(t12), E(t1)} = E(t1)and product p1 is processed in the cell c3,so c3 can be deter-mined as theoretical production bottleneck. For the stableperiod, the system simultaneously processes two products p2

and p3, and related product cells are c1, c2, c4 and c5. Sinceparts p21 and p31 are processed solely in the cells c1 and c4

respectively, parameters of negative exponential distributions

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Table 5 Results of shifting bottleneck analysis in different stable peri-ods

Stable period Entire production net Shifting bottleneck

t0 − t1 Net (c3, p1) c3

t1 − t2 Net (c5, p2) c4

t2 − t3 Net (c5, p3) c2

t3 − t4 None None

t4 − t5 Net (c3, p1) c1

Net (c5, p2) c1

t5 − t6 Net (c3, p1) c2

Net (c5, p3) c2

t6 − t7 Net (c5, p2) c4

Net (c5, p3) c4

t7 − t8 Net (c3, p1) c2

Net (c5, p2) c4

Net (c5, p3) c2

of processing time, taken from the third row in Table 4, areµ21 = 0.0400 and µ31 = 0.0833, and the processing timeexpectations of one p21 and p31 are E(t21) = 1/µ21 =25minutes and E(t31) = 1/µ31 = 12 minutes, while partsp22 and p32 are simultaneously processed for products p2

and p3 in the cell c4, so that parameters of negative exponen-tial distributions of processing time, taken from the fourthrow in Table 4, are µ22 = 0.0286 and µ32 = 0.0385. Theprocessing time expectations of one p22 and p32 are E(t22) =1/µ22 = 35minutes and E(t32) = 1/µ32 = 26minutes. Alsowe can obtain E(t2) = 82minutes and E(t3) = 38minutesto produce one p2 and p3 respectively, while p202 is a com-ponent and its processing time expectation E(t202) = 0.Seen from the product structures and processing time expec-tations as presented above, as max{3E(t22), 2E(t202), E(t2),4E(t31), 2E(t32), E(t3)} = 3E(t22) and part p22 is pro-cessed in the cell c4, so c4 can be determined as theoreticalproduction bottleneck.

Comparing Tables 5 and 7 the consistency of shifting bot-tlenecks with theoretical ones is confirmed, which proves theeffectiveness of our method.

Table 7 Theoretical production bottlenecks

Production statement of corresponding cell Theoreticalproductionbottlenecks

p1 p2 p3

c1 c2 c3 c1 c4 c5 c2 c4 c5

Yes No No c3

No Yes No c4

No No Yes c2

Yes Yes No c1

Yes No Yes c2

No Yes Yes c4

Yes Yes Yes c2 c4

Notes: “Yes” refers to product being processed, “No” means productbeing not processed

The shifting bottlenecks represent the manufacturing cellsthat impede the capacity of the production system in thestrongest manner. The requirement for the capacity upgradeof the entire production net calls for the capacity improve-ment of shifting bottlenecks. On the other hand, when thecapacity of non-bottleneck cell is reduced, the efficiency ofthe entire production net will be improved. The proposedshifting bottleneck analysis approach works well in throw-ing light upon rational allocation of manufacturing cellcapacity.

Conclusions

Since most existing bottleneck detection methods mainly suitserial production line, a net-like model for KMS shifting bot-tleneck detection is proposed in this paper, which adaptswell to highly flexible and intelligent manufacturing sys-tems. Based on the model, the concept of entire produc-tion net and a method for identifying bottlenecks in suchnets are presented. As the transport tools are taken as spe-cial manufacturing cells, such a model can be applied toalmost all kinds of production systems, for example, as hasbeen already pointed out, assembly/disassembly networks

Table 6 Running states of production cells and results of bottleneck detection

Production cells Stable period

t0 − t1 t1 − t2 t2 − t3 t3 − t4 t4 − t5 t5 − t6 t6 − t7 t7 − t8

c1 r-n r-n stopping stopping r-b r-n r-n r-n

c2 r-n stopping r-b stopping r-n r-b r-n r-b

c3 r-b stopping stopping stopping r-n r-n r-n r-n

c4 stopping r-b r-n stopping r-n r-n r-b r-b

c5 stopping r-n r-n stopping r-n r-n r-n r-n

Note: “r-n” denotes “running and non-bottleneck”, “r-b” means “running and bottleneck”

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system, and non-linear multi-product multi-stage lines sys-tem, etc. The expected benefit from the proposed approachis to redound greatly to the throughput of a production sys-tem by detecting and eliminating production bottlenecksefficiently.

One manufacturing cell may process a variety of prod-ucts, and the same manufacturing cell can be a bottleneck toproduct A but non-bottleneck to product B. Since the entireproduction net is constructed according to product structureand if products are manufactured one kind after another, thebottlenecks detected in the entire production net correspondto the product or products being manufactured. As a result,this method can not only detect the bottleneck cell, but alsodetermine what capacity of the manufacturing cell may resultin the bottleneck.

Because several kinds of products are usually manufac-tured in KMS at the same time and some manufacturing cellsmay be simultaneously connected with different entire pro-duction nets, a bottleneck may not correspond to a single kindof product, but to the joint effects of many kinds of productsbeing simultaneously manufactured. Since the buffer statesof manufacturing cells in Algorithm 2 result from many kindsof products being simultaneously manufactured, it can detectsuch a bottleneck.

In a real manufacturing system, the production conditionsmay change frequently. As most existing bottleneck detectionmethods have one implicit assumption that the bottlenecksdetected are under the constant production condition, so itis hard to apply them in practice. In this paper, a self-learn-ing approach is applied to the storage of shifting bottleneckknowledge and their conditions in the knowledge base formining knowledge to determine what cells need improvingunder such conditions.

Since the rear and front buffer state functions are mod-ified in Algorithm 3, the bottleneck detection method canstill work when the capacity of one manufacturing cell isstochastic. The examples cited and analyzed help to con-firm the effectiveness of this shifting bottleneck analysismethod.

However, with the quantitative information in Algorithm 3neglected, only the qualitative results (i.e., those pinpointingproduction bottleneck cells) are obtained by the proposedapproach to shifting bottleneck detection. Thus, for futurestudy we will probe further into the way the bottleneck cellimpedes the capacity of production system, based upon suchthe quantitative information by which the optimal capacityadjustment is calculated and made.

Acknowledgements This work is supported in part by the NationalNatural Science Foundation of China under Grants 60574062 and 50875046 and by the National High Tech R & D Program of China underGrant 2007AA04Z112. We thank the anonymous referees and Profes-sor Li Lu very much for their valuable comments.

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