IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS

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IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS: SYSTEMS 1

Modeling and Analysis of Multiproduct MultistageManufacturing System for Quality Improvement

Shichang Du, Member, IEEE, Rui Xu, and Lin Li, Member, IEEE

Abstract—The ability to produce multiple types of productsusing the same manufacturing system plays an essential rolein the success of a manufacturing enterprise. Meanwhile, mostcomplex manufacturing systems include many stages, called mul-tistage manufacturing systems (MMSs). MMS has a cascadeproperty, which means the product quality is not only affectedby the current stage, but is also related to the outputs of theupstream stage. Multiproduct MMSs (M3Ss) have been widelyapplied in industry. Thus, this paper is devoted to modelingand analyzing steady-state M3Ss for quality improvement. Thediscrete Markov model for single-product-multistage system isextended to novel Markov models for multiproduct-two-stage sys-tems and multiproduct-multistage systems by calculating statetransition probabilities of six key manufacturing parameters toobtain an acceptable product quality probability. Based on thedeveloped models, product sequence analysis is carried out toobtain the best product sequence under Bernoulli conditions andBernoulli relaxation conditions, and bottleneck analysis underBernoulli conditions is conducted to identify the machine and/orparameters whose improvement can lead to the largest qualityimprovement. The applicability of the proposed models is val-idated through numerical experiments and a case study usingreal-world data.

Index Terms—Bottleneck, Markov model, multiple products,multistage manufacturing system (MMS), quality control.

NOMENCLATURE

Mi ith stage in multistage manufacturing sys-tems (MMSs).

M′i Stage merged by the first i stages in MMSs.

kj Batch size of product j.K Total amount of products produced in one batch.gi Mi or M′

i is producing a good product.di Mi or M′

i is producing a defective product.Sl One possible sequence when producing multiple

products.

Manuscript received July 23, 2016; accepted September 20, 2016. This workwas supported in part by the Major Program of the National Natural ScienceFoundation of China under Grant 51535007 and in part by the NationalNatural Science Foundation of China under Grant 51275558. This paperwas recommended by Associate Editor A. Trappey. (Corresponding author:Shichang Du.)

S. Du and R. Xu are with the Department of Industrial Engineeringand Management, Shanghai Jiao Tong University, Shanghai 200240, China(e-mail: [email protected]; [email protected]).

L. Li is with the Department of Industrial and Manufacturing Engineering,University of Illinois at Chicago, Chicago, IL 60607 USA (e-mail:[email protected]).

Color versions of one or more of the figures in this paper are availableonline at http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TSMC.2016.2614766

Slm mth type of product in sequence Sl.

Slm,j jth product internal of Sl

m.gi,sl

m,j Mi or M′i is in good state gi when producing

product Slm,j.

di,slm,j Mi or M′

i is in defective state di when producingproduct Sl

m,j.α1 Probability for M1 to transit from state g1 to

state d1.α1,Sl

p,Slr

Probability for M1 to transit from state g1 to

state d1 when switching from producing Slp to

producing Slr.

β1 Probability for M1 to transit from state d1 tostate g1.

β1,Slp,S

lr

Probability for M1 to transit from state d1 to

state g1 when switching from producing Slp to

producing Slr.

α′i Probability for Mi to transit from state gi to

state di(i ≥ 2).α′

i,Slp,S

lr

Probability for M′i to transit from state gi to

state di(i ≥ 2) when switching from producingSl

p to producing Slr.

β ′i Probability for M′

i to transit from state gi tostate gi(i ≥ 2).

β ′i,Sl

p,Slr

Probability for M′i to transit from state gi to

state gi(i ≥ 2) when switching from producingSl

p to producing Slr.

γi In case of good incoming parts, the probabilityfor Mi to transit from state gi to state di.

γi,Slp,S

lr

In case of good incoming parts, the probabilityfor Mi to transit from state gi to state di whenswitching from producing Sl

p to producing Slr.

μi In case of good incoming parts, the probabilityfor Mi to transit from state di to state gi.

μi,Slp,S

lr

In case of good incoming parts, the prob-ability for Mi to transit from state di tostate gi when switching from producing Sl

p toproducing Sl

r.ηi In case of defective incoming parts, the prob-

ability for Mi to transit from state gi tostate di.

ηi,Slp,S

lr

In case of defective incoming parts, the probabil-ity for Mi to transit from state gi to state di whenswitching from producing Sl

p to producing Slr.

θi In case of defective incoming parts, the proba-bility for Mi to transit from state di to state gi.

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2 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS: SYSTEMS

θi,Slp,S

lr

In case of defective incoming parts, the prob-ability for Mi to transit from state di tostate gi when switching from producing Sl

p toproducing Sl

r.Ai Matrix of state transition probabilities for the

system with i stages.Am,j Matrix of state transition probabilities for the

system with i stages when producing the jthproduct internal of Sl

m.Xi Matrix of steady-state probabilities for the sys-

tem with i stages.Xm,j Matrix of steady-state probabilities for the sys-

tem with i stages when producing the jth productinternal of Sl

m.P Probability of the system in one certain

steady state.P(gi) Probability of the system with i stages in

a good state.P(gl

bp) Probability that the system produces goodproducts under batch production (BP) withsequence Sl.

P(glsp) Probability that the system produces good

products under single production (SP) withsequence Sl.

Sγi Sensitivity of P(gi) with respect to γi.Sμi Sensitivity of P(gi) with respect to μi.Sηi Sensitivity of P(gi) with respect to ηi.Sθi Sensitivity of P(gi) with respect to θi.Sα1 Sensitivity of P(gi) with respect to α1.Sβ1 Sensitivity of P(gi) with respect to β1.δ1 Maximum difference between 1 and (α1,sl

i,slj+

β1,sli,s

lj).

δ2 Maximum difference between 1 and (γ2,sli,s

lj+

μ2,sli,s

lj).

δ3 Maximum difference between 1 and (η2,sli,s

lj+

θ2,sli,s

lj).

I. INTRODUCTION

THE ABILITY to produce multiple types of products usingthe same manufacturing system plays an essential role in

the success of a manufacturing enterprise. Product flexibil-ity increases the rapid responsiveness of a system; it takesfull advantage of available system resources to produce mul-tiple types of products using the same manufacturing systemthat deals with internal and external production uncertaintieswith time. Meanwhile, most complex manufacturing systemsinvolve many stages, called MMSs. As products move throughthese stages, the variations in product quality are usuallyintroduced and propagated. Multiproduct MMSs (M3Ss) arebecoming more and more popular and necessary and havebeen widely applied in automobile vehicles, engine, aerospace,electronics, and appliance industry.

In M3Ss, the final product quality is dependent on notonly product flexibility but also product quality propagation.For example, flexible fixtures play an important role to enableflexibility of the whole manufacturing system, and they are

Fig. 1. Two-product two-stage manufacturing system.

programmable in order to hold and clamp different types ofproducts in a same manufacturing system. The fixtures locat-ing accuracy determine the product quality. When the productis changed from one type to another one, the fixtures need toadapt themselves to the desired corresponding positions. Sincethere exist relocation errors of the fixtures, their conditionscould be changed from “good” to “bad.” Assume there areproducts A and B machined in two-stage system (see Fig. 1).If the fixture is in good condition for product A and the subse-quent product is also product A at stage 1, then a good qualityproduct would be produced. However, if the subsequent prod-uct is changed from products A to B, then the fixture needsto readjust its position (there exists a transition probabilityfrom good position to bad position) and either good qualityor defective products at stage 1 may be produced. The defec-tive product B may be produced at stage 1. Thus, the productquality at stage 1 is dependent on not only the quality of theprevious product but also the product type, namely, the productflexibility can affect the quality. Meanwhile, if the charac-teristic of defective product B (caused by product flexibility)produced at stage 1 is used as the locating datum to machineother characteristics of product B at stage 2, then the qualityvariation of product B would propagate from stages 1 to 2.Flexibility can affect the quality propagation. Therefore, bothproduct flexibility and quality propagation could significantlyaffect the final product quality in M3Ss. However, there hasbeen no analytical model to investigate and improve productquality considering both product flexibility and product qualitypropagation in M3Ss.

Modeling and analysis of MMSs for quality propagationhas received intensive investigation. One of the most popularanalytical models used for quality improvement is state spacemodel [1]. After that, a great deal of extended research onquality propagation based on state space model has been con-ducted. Detailed descriptions of existing research on the statespace model are provided in a monograph [2] and a survey [3].In another research line of analytical methods, a Markovmodel has been widely used as an analytical tool to inves-tigate the interactions between the manufacturing system andproduct quality [4]. The related literatures are reviewed in [5].However, in spite of above efforts, the current Markov modelsare explored only for single-product-multistage systems with-out product flexibility [6], [7] or multiproduct-single-stagesystem without quality propagation [8]–[11]. To the bestknowledge of the authors, there lacks an efficient methodto model and analyze M3Ss which features both qualitypropagation and product flexibility.

The main contribution of this paper is to propose novelMarkov models for steady-state M3Ss for quality improve-ment, which take both product flexibility and product qualitypropagation into consideration.

The remainder of this paper is organized as follows.Section II reviews the related literature. Section III introduces


DU et al.: MODELING AND ANALYSIS OF M3Ss FOR QUALITY IMPROVEMENT 3

M3Ss and formulates the problem. Two Markov models formultiproduct-two-stage system and multiproduct-multistagesystem are developed, respectively, in Section IV. Productsequence and quality improvement analysis are carried out inSections V and VI. A case study is presented to illustrate theproposed models and their applicability in Section VII. Finally,the conclusion is given in Section VIII.

II. LITERATURE REVIEW

Product flexibility has been widely recognized as a crit-ical component to achieve a competitive advantage in themarket place. Flexible manufacturing systems (FMSs) havebeen widely designed and applied [12]–[14]. Detailed descrip-tions of existing research on manufacturing flexibility analysismethods and applications were reviewed in [15]–[18].

Modeling and analysis of product quality propagationin MMSs have attracted substantial research attention inrecent decades. Data-driven methods focus on investigatingpatterns from massive historical quality datasets to modelthe relationships between product quality and manufacturingsystems [19]–[22].

Unlike data-driven models, analytical models employ off-line analysis of MMSs based on fundamental physical laws.The state space model is one of the most popular ana-lytical models used for quality propagation analysis [1]. Itis further investigated in 3-D assembly systems [23]–[26]and machining systems [27]–[33]. Detailed descriptions ofthe existing research on state space model are provided ina monograph [2] and a survey [3]. However, analysis of com-plex MMSs using the state space model based on physicallaws is often intractable [3], and such analysis either relieson complicated kinematics models of manufacturing systems,or is only applicable to deal with dimensional errors and theapplication area is limited [34], [35].

In another research concentration, Markov models havebeen widely used as analytical tools to investigate the productquality. The related literature and empirical evidences showthat manufacturing system design has a significant impacton product quality [5]. Modeling and analysis of manufac-turing systems using Markov models for product qualityimprovement have received more and more research attention.

Some Markov models are developed to analyze prod-uct quality propagation in single-product-multistage systems.Markov-chain-based quality propagation models are developedto evaluate quality propagation in automotive paint shops [6]and battery assembly process [7]. The analytical methodsusing Markov models are proposed to evaluate three casesof long manufacturing lines [36]. The impact of manufactur-ing system design on product quality is investigated througha case study at an automotive paint shop [37]. Also, analyti-cal methods are proposed for the joint design of quality andmanufacturing parameters [38] and investigate joint produc-tion and quality control in manufacturing systems with randomdemand [39].

Some other Markov models are derived to analyze the qual-ity of multiproduct-single-stage systems. The early Markovmodel is developed to study how production system design,

quality and productivity are inter-related [40]. Then a Markovmodel to evaluate quality performance of multiproduct-single-stage systems is presented in [41]. Based on Markov chainprocesses, some analytical methods are developed to evaluatequality performance in multiproduct manufacturing systemswith BP in [8].

Several research studies on product sequencing and bot-tlenecks using Markov models in multiproduct-single-stagesystems to improve product quality have been conducted.The impact of product sequencing and batch policies onproduct quality is investigated and some insights to achievebetter quality are presented [9]. Quality bottleneck analysisis carried out based on data from a factory floor [10], [11].An arrow-based bottleneck identification method is presentedin [6]. A Markov model is applied to characterize a furni-ture assembly system in [42]. A Markov model is explored tocontrol dynamic energy for energy efficiency improvement ofsustainable manufacturing systems [43].

In spite of the above efforts, M3Ss with quality propagationand product flexibility still lacks an in-depth study, and thereis no efficient Markov model to analyze these systems forquality improvement. The goal of this paper is to contributeto this end.

III. MODEL ASSUMPTIONS AND

PROBLEM FORMULATION

Consider an n-product-r-stage system; the followingassumptions define the processing stages, product types, andtheir interactions in the Markov models.

1) The manufacturing system consists of r stages whichcan produce n different types of products. The corre-sponding batch size for product j is kj(1 ≤ j ≤ n).The total amount of products produced in one batch isK = ∑n

i=1 ki.2) We only study the working or production period of the

system. Machine breakdowns are not considered.3) Define the stage Mi(i = 1, 2, . . . , r) in a good state gi

or a defective state di if it is producing a product withgood quality or defective quality at time t. The qualityof incoming product is characterized by good state gi

or defective state di with probabilities P(gi) and P(di),respectively.

4) There exist quality degradation and quality correction inthe system. The quality might get worse or better aftera certain stage. The quality of the incoming parts forMi(i ≥ 2) at time (t + 1) depends on the state of Mi−1at time t. The states gi−1 and di−1 for Mi−1 at time tmeans good and defective parts coming for Mi at time(t + 1), respectively.

5) When M1 is in good state g1, it has probability α1 totransit to defective state d1 and probability (1 − α1) togood state g1. When M1 is in defective state d1, it hasprobability β1 to transit to good state g1 and probability(1 − β1) to defective state d1.

6) With good incoming parts, when Mi(i ≥ 2) is in goodstate gi, it has probability γi to transit to defective state di

and probability (1 − γi) to good state gi. When Mi is in



Fig. 2. Framework of the proposed method.

defective state di, it has probability μi to transit to goodstate gi and probability (1 − μi) to defective state di.With defective incoming parts, when Mi(i ≥ 2) is ingood state gi, it has probability ηi to transit to defectivestate di and probability (1 − ηi) to good state gi. WhenMi is in defective state di, it has probability θi to transitto good state gi and probability (1 − θi) to defectivestate di.

7) For the system producing n different types of prod-ucts, there are (n − 1)! possible production sequences.For a certain sequence Sl = {Sl

1, Sl2, . . . , Sl

n}(l =1, 2, . . . , (n − 1)!), Sl

m denotes the mth type of productin this sequence, where m ∈ {1, 2, . . . , n}.

8) For a certain product Slm in sequence Sl, the system

will not transit to processing another type of productSl

n(n �= m) until it finishes processing the last one kSlm

of product Slm.

9) When M1 or Mi(i = 2, 3, . . . , r) is processing Slm,j at

time t, it may maintain processing the same type ofproduct Sl

m,j+1 or switch to processing another typeof product Sl

m+1,1 at (t + 1). Then the correspond-ing transition probabilities for M1 and Mi are denotedas α1,Sl

p,Slr, β1,Sl

p,Slr

and γ2,Slp,S

lr, μ2,Sl

p,Slr, η2,Sl

p,Slr, and

θ2,Slp,S

lr

(p, r = 1, 2, . . . , n). When p = r, these prob-abilities imply the internal transition probabilities of thesame type of product. When p �= r, they indicate theexternal transition probabilities between different typesof products.

We refer to α1, γi, ηi(i ≥ 2) as quality failure probabilitiesand (i ≥ 2)β1, μi, θi(i ≥ 2) as quality repair probabilities.Similar to throughput analysis and in accordance with somequality analysis work based on Markov model [6], [7], [41],we assume that all these transition probabilities are constant.Actually in real manufacturing systems, machines have stable

Fig. 3. Iterative procedure for multistage systems.

production periods during which the state transitions can beseen as stable.

The problem addressed is then formulated as follows.Problem: Under the above assumptions 1)–9), develop

a proper method to evaluate the steady-state quality perfor-mance of M3Ss as a function of system parameters, investi-gate the sequence properties, perform sensitivity analysis andidentify quality bottlenecks.

The solutions to the problem are given in Sections IV–VI.The framework of the proposed method is illustrated inFig. 2. Based on the Markov model for single-product-two-stage systems in [44], a Markov model for multiproduct-two-stage systems is developed. Then this model is extended tomultiproduct-multistage systems. Product sequence analysisand quality improvement analysis are conducted based on thedeveloped models.

IV. MARKOV MODEL FOR M3SS

A. Review of the Markov Model forSingle-Product-Multistage-Systems

Based on the assumptions 1)–9) and the work of [44], fora two-stage system producing a given type of product Sl

m,j,the transition probabilities and the matrix of state transitionprobabilities are presented (1), as shown at the top of the nextpage.

The matrix of steady state probabilities is denoted as

X2 = [ P(g1g2) P(d1g2) P(g1d2) P(d1d2) ]. (2)

Based on the Markov model, we have

X2A2 = X2 (3)

P(g1g2) + P(d1g2) + P(g1d2) + P(d1d2) = 1 (4)

and

P(g2) = P(g1g2) + P(d1g2). (5)

Equation (5) represents the probability that the system isproducing a good product. This probability can be seen as anindicator to evaluate the quality performance of the system.

The model for single-product-two-stage systems can beextended to single-product-r-stage systems (r ≥ 3) byapplying the iteration method in [44] which has been validated



A2 =

⎡

⎢⎢⎢⎢⎢⎢⎣

(1 − α1,Sl

m,j,Slm,j

)(1 − γ2,Sl

m,j,Slm,j

)α1,Sl

m,j,Slm,j

(1 − γ2,Sl

m,j,Slm,j

) (1 − α1,Sl

m,j,Slm,j

)γ2,Sl

m,j,Slm,j

α1,Slm,j,S

lm,j

γ2,Slm,j,S

lm,j

β1,Slm,j,S

lm,j

(1 − η2,Sl

m,j,Slm,j

) (1 − β1,Sl

m,j,Slm,j

)(1 − η2,Sl

m,j,Slm,j

)β1,Sl

m,j,Slm,j

η2,Slm,j,S

lm,j

(1 − β1,Sl

m,j,Slm,j

)η2,Sl

m,j,Slm,j(

1 − α1,Slm,j,S

lm,j

)μ2,Sl

m,j,Slm,j

α1,Slm,jS

lm,j

μ2,Slm,jS

lm,j

(1 − α1,Sl

m,j,Slm,j

)(1 − μ2,Sl

m,j,Slm,j

) (1 − α1,Sl

m,j,Slm,j

)μ2,Sl

m,j,Slm,j

β1,Slm,j,S

lm,j

θ2,Slm,j,S

lm,j

(1 − β1,Sl

m,j,Slm,j

)θ2,Sl

m,j,Slm,j

β1,Slm,j,S

lm,j

(1 − θ2,Sl

m,j,Slm,j

) (1 − β1,Sl

m,j,Slm,j

)(1 − θ2,Sl

m,j,Slm,j

)

⎤

⎥⎥⎥⎥⎥⎥⎦

(1)

by extensive numerical experiments. The general iterative pro-cedure is illustrated in Fig. 3 and the main steps are presentedas follows.

1) Merge M1 and M2 to one stage M′2, and gain the quality

of the new two-stage system M′2 − M3 based on the

model for one-product-two-stage system.2) Merge M′

2 and M3 to one stage M′3, and gain the quality

of the new two-stage system M′3 − M4. Continue this

iterative process until the first (r − 1) stages are mergedto one stage M′

r−1 and gain the quality of the final two-stage system M′

r−1 − Mr.During the iterative process of single-product-r-stage sys-

tems, any two-stage system M′i − Mi+1 has six basic

parameters. They are γi+1,Slm,j,S

lm,j

, μi+1,Slm,j,S

lm,j

, ηi+1,Slm,j,S

lm,j

,and θi+1,Sl

m,j,Slm,j

describing the characteristics of Mi+1 and

α′i,Sl

m,j,Slm,j

, β ′i,Sl

m,j,Slm,j

describing the characteristics of the

merged machine M′i . α′

i,Slm,j,S

lm,j

, β ′i,Sl

m,j,Slm,j

can be calculated by

α′i,Sl

m,j,Slm,j

=P(gi−1gi)γi,Sl

m,j,Slm,j

+ P(di−1gi)ηi,Slm,j,S

lm,j

P(gi−1gi) + P(di−1gi)(6)

β ′i,Sl

m,j,Slm,j

=P(gi−1di)μi,Sl

m,jSlm,j

+ P(di−1di)θi,Slm,j,S

lm,j

P(gi−1di) + P(di−1di). (7)

The matrix of state transition probabilities (8), as shown at thetop of the next page.

The corresponding matrix of the steady-state probabilities is

Xi+1 = [ P(gigi+1) P(digi+1) P(gidi+1) P(didi+1) ]. (9)

According to the definition of a Markov chain, we obtain

Xi+1Ai+1 = Xi+1 (10)

P(gigi+1) + P(digi+1) + P(gidi+1) + P(didi+1) = 1. (11)

The final probability of producing a product with goodquality for the merged two-stage system is

P(gi+1) = P(gigi+1) + P(digi+1). (12)

B. Proposed Markov Model forMultiproduct-Two-Stage Systems

Based on the Markov model for single-product-two-stagesystems, by further considering product flexibility, we canextend the model to multiproduct-two-stage systems. It canbe seen from assumption 9) that different positions of a partin the batch may lead to different transition probabilities.

1) When M2 is processing the first part of a cer-tain product sequence Sl

m,1(2 ≤ m ≤ n),the corresponding probabilities are α1,Sl

m,Slm

, β1,Slm,Sl

m,

γ2,Slm−1,S

lm

, μ2,Slm−1,S

lm

, η2,Slm−1,S

lm

, and θ2,Slm−1,S

lm

. Thematrix of steady state probability Xm,1 and tran-sition probabilities Am,1 are (13) and (14), asshown at the top of the next page. When m =1, i.e., M2 is processing the first part of thewhole batch cycle, the corresponding probabilities areα1,Sl

1,Sl1, β1,Sl

1,Sl1, γ2,Sl

n,Sl1, μ2,Sl

n,Sl1, η2,Sl

n,Sl1, θ2,Sl

n,Sl1,

and (15) and (16), as shown at the top of thenext page.

2) When M2 is processing the jth part of a certain productSl

m,j, m = 1, 2, . . . , n, j = 2, 3, . . . , kSli− 1, the cor-

responding probabilities are α1,Slm,Sl

m, β1,Sl

m,Slm

, γ2,Slm,Sl

m,

μ2,Slm,Sl

m, η2,Sl

m,Slm

, θ2,Slm,Sl

m, and (17) and (18), as shown

at the top of the next page.3) When M2 is processing the last part of a certain prod-

uct sequence Slm,k

Slm

(m = 1, 2, . . . , n), the correspond-

ing probabilities are α1,Slm,Sl

m+1, β1,Sl

m,Slm+1

, γ2,Slm,Sl

m,

μ2,Slm,Sl

m, η2,Sl

m,Slm

, θ2,Slm,Sl

m, and (19) and (20), as shown

at the top of the next page. When m = n, it means onebatch cycle is finished. Then (m + 1) would be 1 asa new batch cycle starts.

Based on the above analysis, we have the matrix of steadystate probabilities and matrix of state transition probabilitiesin the form of a block matrix

X =[

X1,1, X1,2, . . . , Xm,kSlm

,Xm+1,1, . . . , Xn−1,kSln−1

,

Xn,1, . . . , Xn,kSln−1, Xn,k

Sln

]T(21)

A =

⎡

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

0 · · · 0 0 0 · · · 0 A1,1A1,2 · · · 0 0 0 · · · 0 0

0. . . 0 0 0 · · · 0 0

0 · · · Am,kSlm

0 0 · · · 0 0

0 · · · 0 Am+1,1 0 · · · 0 00 · · · 0 0 Am+1,2 · · · 0 0

0 · · · 0 0 0. . . 0 0

0 · · · 0 0 0 · · · An,kSln

0

⎤

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

.

(22)

The sum of all the steady state probabilities equals 1

n∑

m=1

kslm∑

j=1

P(

g2,slm,j

)+

n∑

m=1

kslm∑

j=1

P(

d2,slm,j

)= 1 (23)

and

AX = X. (24)



Ai+1 =

⎡

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

(

1 − α′1,Sl

m,j,Slm,j

)(1 − γ2,Sl

m,j,Slm,j

)α′

1,Slm,j,S

lm,j

(1 − γ2,Sl

m,j,Slm,j

) (1 − α1,Sl

m,j,Slm,j

)γ2,Sl

m,j,Slm,j

α′1,Sl

m,j,Slm,j

γ2,Slm,j,S

lm,j

β ′1,Sl

m,j,Slm,j

(1 − η2,Sl

m,j,Slm,j

) (

1 − β ′1,Sl

m,j,Slm,j

)(1 − η2,Sl

m,j,Slm,j

)β ′

1,Slm,j,S

lm,j

η2,Slm,j,S

lm,j

(

1 − β ′1,Sl

m,j,Slm,j

)

η2,Slm,j,S

lm,j(

1 − α′1,Sl

m,j,Slm,j

)

μ2,Slm,j,S

lm,j

α′1,Sl

m,jSlm,j

μ2,Slm,jS

lm,j

(

1 − α′1,Sl

m,j,Slm,j

)(1 − μ2,Sl

m,j,Slm,j

) (

1 − α′1,Sl

m,j,Slm,j

)

μ2,Slm,j,S

lm,j

β ′1,Sl

m,j,Slm,j

θ2,Slm,j,S

lm,j

(

1 − β ′1,Sl

m,j,Slm,j

)

θ2,Slm,j,S

lm,j

β ′1,Sl

m,j,Slm,j

(1 − θ2,Sl

m,j,Slm,j

) (

1 − β ′1,Sl

m,j,Slm,j

)(1 − θ2,Sl

m,j,Slm,j

)

⎤

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(8)

Xm,1 =[

P(

g1,Slm,2

g2,Slm,1

)P(

d1,Slm,2

g2,Slm,1

)P(

g1,Slm,2

d2,Slm,1

)P(

g1,Slm,2

d2,Slm,1

) ](13)

Ai,1 =

⎡

⎢⎢⎢⎢⎢⎢⎣

(1 − α1,Sl

m,Slm

)(1 − γ2,Sl

m−1,Slm

)β1,Sl

m,Slm

(1 − η2,Sl

m−1,Slm

) (1 − α1,Sl

m,Slm

)μ2,Sl

m−1,Slm

β1,Slm,Sl

mθ2,Sl

m−1,Slm

α1,Slm,Sl

m

(1 − γ2,Sl

m−1,Slm

) (1 − β1,Sl

m,Slm

)(1 − η2,Sl

m−1,Slm

)α1,Sl

m,Slmμ2,Sl

m−1,Slm

(1 − β1,Sl

m,Slm

)θ2,Sl

m−1,Slm(

1 − α1,Slm,Sl

m

)γ2,Sl

m−1,Slm

β1,Slm,Sl

mη2,Sl

m−1,Slm

(1 − α1,Sl

m,Slm

)(1 − μ2,Sl

m−1,Slm

)β1,Sl

m,Slm

1 − θ2,Slm−1,Sl

m

)

α1,Slm,Sl

mγ2,Sl

m−1,Slm

(1 − β1,Sl

m,Slm

)η2,Sl

m−1,Slm

α1,Slm,Sl

m

(1 − μ2,Sl

m−1,Slm

) (1 − β1,Sl

m,Slm

)(1 − θ2,Sl

m−1,Slm

)

⎤

⎥⎥⎥⎥⎥⎥⎦

(14)

X1,1 =[

P(

g1,Sl1,2

g2,Sl1,1

)P(

d1,Sl1,2

g2,Sl1,1

)P(

g1,Sl1,2

d2,Sl1,1

)P(

g1,Sl1,2

d2,Sl1,1

) ](15)

A1,1 =

⎡

⎢⎢⎢⎢⎢⎢⎣

(1 − α1,Sl

1,Sl1

)(1 − γ2,Sl

n,Sl1

)β1,Sl

1,Sl1

(1 − η2,Sl

n,Sl1

) (1 − α1,Sl

1,Sl1

)μ2,Sl

n,Sl1

β1,Sl1,S

l1θ2,Sl

n,Sl1

α1,Sl1,S

l1

(1 − γ2,Sl

n,Sl1

) (1 − β1,Sl

1,Sl1

)(1 − η2,Sl

n,Sl1

)α1,Sl

1,Sl1μ2,Sl

n,Sl1

(1 − β1,Sl

1,Sl1

)θ2,Sl

n,Sl1(

1 − α1,Sl1,S

l1

)γ2,Sl

n,Sl1

β1,Sl1,S

l1η2,Sl

n,Sl1

(1 − α1,Sl

1,Sl1

)(1 − μ2,Sl

n,Sl1

)β1,Sl

1,Sl1

(1 − θ2,Sl

n,Sl1

)

α1,Sl1,Sl

1γ2,Sl

n,Sl1

(1 − β1,Sl

1,Sl1

)η2,Sl

n,Sl1

α1,Sl1,S

l1

(1 − μ2,Sl

n,Sl1

) (1 − β1,Sl

1,Sl1

)(1 − θ2,Sl

n,Sl1

)

⎤

⎥⎥⎥⎥⎥⎥⎦

(16)

Xm,j =[

P(

g1,Slm,j+1

g2,Slm,j

)P(

d1,Slm,j+1

g2,Slm,j

)P(

g1,Slm,j+1

d2,Slm,j

)P(

g1,Slm,j+1

d2,Slm,j

) ](17)

Am,j =

⎡

⎢⎢⎢⎢⎢⎢⎣

(1 − α1,Sl

m,Slm

)(1 − γ2,Sl

m,Slm

)β1,Sl

m,Slm

(1 − η2,Sl

m,Slm

) (1 − α1,Sl

m,Slm

)μ2,Sl

m,Slm

β1,Slm,Sl

mθ2,Sl

m,Slm

α1,Slm,Sl

m

(1 − γ2,Sl

m,Slm

) (1 − β1,Sl

m,Slm

)(1 − η2,Sl

m,Slm

)α1,Sl

m,Slmμ2,Sl

m,Slm

(1 − β1,Sl

m,Slm

)θ2,Sl

m,Slm(

1 − α1,Slm,Sl

m

)γ2,Sl

m,Slm

β1,Slm,Sl

mη2,Sl

m,Slm

(1 − α1,Sl

m,Slm

)(1 − μ2,Sl

m,Slm

)β1,Sl

m,Slm

1 − θ2,Slm,Sl

m

)

α1,Slm,Sl

mγ2,Sl

m,Slm

(1 − β1,Sl

m,Slm

)η2,Sl

m,Slm

α1,Slm,Sl

m

(1 − μ2,Sl

m,Slm

) (1 − β1,Sl

m,Slm

)(1 − θ2,Sl

m,Slm

)

⎤

⎥⎥⎥⎥⎥⎥⎦

(18)

Xm,kSlm

=[

P

(

g1,Slm+1,1

g2,Slm,k

Slm

)

P

(

d1,Slm+1,1

g2,Slm,k

Slm

)

P

(

g1,Slm+1,1

d2,Slm,k

Slm

)

P

(

g1,Slm+1,1

d2,Slm,k

Slm

)]

(19)

Am,kSlm

=

⎡

⎢⎢⎢⎢⎢⎢⎣

(1 − α1,Sl

m,Slm+1

)(1 − γ2,Sl

m,Slm

)β1,Sl

m,Slm+1

(1 − η2,Sl

m,Slm

) (1 − α1,Sl

m,Slm+1

)μ2,Sl

m,Slm

β1,Slm,Sl

m+1θ2,Sl

m,Slm

α1,Slm,Sl

m+1

(1 − γ2,Sl

m,Slm

) (1 − β1,Sl

m,Slm+1

)(1 − η2,Sl

m,Slm

)α1,Sl

m,Slm+1

μ2,Slm,Sl

m

(1 − β1,Sl

m,Slm+1

)θ2,Sl

m,Slm(

1 − α1,Slm,Sl

m+1

)γ2,Sl

m,Slm

β1,Slm,Sl

m+1η2,Sl

m,Slm

(1 − α1,Sl

m,Slm+1

)(1 − μ2,Sl

m,Slm

)β1,Sl

m,Slm+1

1 − θ2,Slm,Sl

m

)

α1,Slm,Sl

m+1γ2,Sl

m,Slm

(1 − β1,Sl

m,Slm+1

)η2,Sl

m,Slm

α1,Slm,Sl

m+1

(1 − μ2,Sl

m,Slm

) (1 − β1,Sl

m,Slm+1

)(1 − θ2,Sl

m,Slm

)

⎤

⎥⎥⎥⎥⎥⎥⎦

(20)

By summing up the elements in Xi,j, we find that

P(

g1,Slm,j+1

g2,Slm,j

)+ P

(d1,Sl

m,j+1g2,Sl

m,j

)+ P

(g1,Sl

m,j+1d2,Sl

m,j

)

+ P(

g1,Slm,j+1

d2,Slm,j

)

= P(

g1,Slm,j+2

g2,Slm,j+1

)+ P

(d1,Sl

m,j+2g2,Sl

m,j+1

)

+ P(

g1,Slm,j+2

d2,Slm,j+1

)+ P

(g1,Sl

m,j+2d2,Sl

m,j+1

)(25)

which leads to

P(

g1,Slm,j+1

g2,Slm,j

)+ P

(d1,Sl

m,j+1g2,Sl

m,j

)+ P

(g1,Sl

m,j+1d2,Sl

m,j

)

+ P(

g1,Slm,j+1

d2,Slm,j

)= 1

K. (26)

The final quality can be obtained from (23)–(26)

P(

gl)

=n∑

m=1

ksli∑

j=1

P(

g2,slm,j

). (27)



P(gbp

) =∑n

m=1

(kSl

m− 1

)(1 − γ2,Sl

m,Slm

)+∑n

m=2

(1 − γ2,Sl

m−1,Slm

)+(

1 − γ2,Sln,S

l1

)

∑nm=1 kSl

m

(28)

α′i,Sl

m,Slm

=P(

g′i−1,Sl

mgi,Sl

m, j)γi,Sl

m,Slm

+ P(

d′i−1,Sl

mgi,Sl

m, j)ηi,Sl

m,Slm

P(

g′i−1,Sl

mgi,Sl

m, j)

+ P(

d′i−1,Sl

mgi,Sl

m, j) (29)

α′′i,Sl

m,Slm

=P(

g′i−1,Sl

mgi,Sl

m, 1)γi,Sl

m,Slm

+ P(

d′i−1,Sl

mgi,Sl

m, 1)ηi,Sl

m,Slm

P(

g′i−1,Sl

mgi,Sl

m, 1)

+ P(

d′i−1,Sl

mgi,Sl

m, 1) (30)

α′i,Sl

m,Slm+1

=P

(

g′i−1,Sl

m+1gi−1,Sl

m, kSl

m

)

γi,Slm,Sl

m+1+ P

(

d′i−1,Sl

m+1gi,Sl

m, kSl

m

)

ηi,Slm,Sl

m+1

P

(

g′i−1,Sl

m+1gi−1,Sl

m, kSl

m

)

+ P

(

d′i−1,Sl

m+1gi,Sl

m, kSl

m

) (31)

β ′i,Sl

m,Slm

=P(

g′i−1,Sl

mdi,Sl

m, j)μi,Sl

m,Slm

+ P(

d′i−1,Sl

mdi,Sl

m, j)θi,Sl

m,Slm

P(

g′i−1,Sl

mdi,Sl

m, j)

+ P(

d′i−1,Sl

mdi,Sl

m, j) (32)

β ′′i,Sl

m,Slm

=P(

g′i−1,Sl

mdi,Sl

m, 1)μi,Sl

m,Slm

+ P(

d′i−1,Sl

mdi,Sl

m, 1)θi,Sl

m,Slm

P(

g′i−1,Sl

mdi,Sl

m, 1)

+ P(

d′i−1,Sl

mdi,Sl

m, 1) (33)

β ′i,Sl

m,Slm+1

=P(

gi−1,Slm+1

di,Slm, kSl

m

)μi,Sl

m,Slm+1

+ P(

di−1,Slm+1

di,Slm, kSl

m

)θi,Sl

m,Slm+1

P(

gi−1,Slm+1

di,Slm, kSl

m

)+ P

(di−1,Sl

m+1di,Sl

m, kSl

m

) (34)

If we ignore quality propagation in the system, namely,η2,Sl

m,Slm

= γ2,Slm,Sl

m, η2,Sl

m−1,Slm

= γ2,Slm−1,S

lm

, and η2,Sln,S

l1

=γ2,Sl

n,Sl1, then the probability of producing a product with good

quality could be (28), as shown at the top of this page, whichis consistent with conclusion (26) made in [9].

C. Proposed Markov Model forMultiproduct-Multistage Systems

Based on the model for single-product-multistage sys-tems that focuses on quality propagation and the model formultiproduct-two-stage systems that focuses on product flexi-bility, we derive the general model for M3Ss. The correspond-ing probabilities are: α′

i,Slm,Sl

m, α′′

i,Slm,Sl

m, α′

i,Slm,Sl

m+1, β ′

i,Slm,Sl

m,

β ′′i,Sl

m,Slm

, β ′i,Sl

m,Slm+1

, γi,Slm,Sl

m, γi,Sl

m,Slm+1

, ηi,Slm,Sl

m, ηi,Sl

m,Slm+1

,

μi,Slm,Sl

m, μi,Sl

m,Slm+1

, θi,Slm,Sl

m, and θi,Sl

m,Slm+1

, where m =1, 2, . . . , r − 1.

Especially when m = n, it means a batch cycle has beenfinished and the system will enter another production cycle.Under this situation, α′

i,Slm,Sl

m+1, for example, will be denoted

as α′i,Sl

n,Sl1

and so as for β ′i,Sl

m,Slm+1

, γi,Slm,Sl

m+1, ηi,Sl

m,Slm+1

,

μi,Slm,Sl

m+1, and θi,Sl

m,Slm+1

. α′i,Sl

m,Slm

, α′′i,Sl

m,Slm

, α′i,Sl

m,Slm+1

, β ′i,Sl

m,Slm

,

β ′′i,Sl

m,Slm

, and β ′i,Sl

m,Slm+1

are obtained by applying iteration

method and they can be expressed as follows, (29)–(34), asshown at the top of this page.

Similar to the matrices of transition probabilities that havebeen stated in Section IV-B, the transition probability matrices

for the products in the first, middle, and last positions are(35)–(37), as shown at the top of the next page.

According to the definition of a Markov chain, wecan get the probability of producing a product with goodquality for M3Ss. Especially when the probabilities sat-isfy Bernoulli distribution, we have α1,i,j + β1,i,j = 1,γ1,i,j + μ2,i,j = 1, η1,i,j + θ2,i,j = 1(i = 1, 2, . . . , n, j =1, 2, . . . , n) [8], [40], [45], [46], then the expression ofthe final quality for a n-product-r-stage system with sequenceSl can be obtained (38), as shown at the top of the next page.

V. PRODUCT SEQUENCE ANALYSIS

As indicated in the models for multiproduct-two-stage sys-tems and multiproduct-multistage systems, different sequencesinvolve different transition probabilities, which meansthe final quality might differ from each other underdifferent sequences. Therefore, an appropriate sequenceof multiple products could improve the final prod-uct quality. Furthermore, different sequence strategies,BP as a, · · · , a, b, · · · b, c, · · · c, · · · , a, · · · , or SP asa, b, c, · · · a, b, c, · · · also have impact on the quality.

A. Bernoulli Case

In order to simplify the analysis and make conclusionsmore explicit, we first focus on the quality properties regard-ing the sequence of multiproduct-two-stage systems underthe Bernoulli case. The Bernoulli case is often employed in



ASlm,1 =

⎡

⎢⎢⎢⎢⎢⎣

(1 − α′′

i,Slm,Sl

m

)(1 − γi+1,Sl

m−1,Slm

)β ′′

i,Slm,Sl

m

(1 − ηi+1,Sl

m−1,Slm

) (1 − α′′

i,Slm,Sl

m

)μi+1,Sl

m−1,Slm

β ′′i,Sl

m,Slmθi+1,Sl

m−1,Slm

α′′i,Sl

m,Slm

(1 − γi+1,Sl

m−1,Slm

) (1 − β ′′

i,Slm,Sl

m

)(1 − ηi+1,Sl

m−1,Slm

)α′′

i,Slm,Sl

mμi+1,Sl

m−1,Slm

(1 − β ′′

i,Slm,Sl

m

)θi+1,Sl

m−1,Slm(

1 − α′′i,Sl

m,Slm

)γi+1,Sl

m−1,Slm

β ′′i,Sl

m,Slmηi+1,Sl

m−1,Slm

(1 − α′′

i,Slm,Sl

m

)(1 − μi+1,Sl

m−1,Slm

)β ′′

i,Slm,Sl

m

(1 − θi+1,Sl

m−1,Slm

)

α′′i,Sl

m,Slmγi+1,Sl

m−1,Slm

(1 − β ′′

i,Slm,Sl

m

)ηi+1,Sl

m−1,Slm

(1 − α′′

i,Slm,Sl

m

)μi+1,Sl

m−1,Slm

(1 − β ′′

i,Slm,Sl

m

)(1 − θi+1,Sl

m−1,Slm

)

⎤

⎥⎥⎥⎥⎥⎦

(35)

ASlm,j =

⎡

⎢⎢⎢⎢⎢⎣

(1 − α′

i,Slm,Sl

m

)(1 − γi+1,Sl

m,Slm

)β ′

i,Slm,Sl

m

(1 − ηi+1,Sl

m,Slm

) (1 − α′

i,Slm,Sl

m

)μi+1,Sl

m,Slm

β ′i,Sl

m,Slmθi+1,Sl

m,Slm

α′i,Sl

m,Slm

(1 − γi+1,Sl

m,Slm

) (1 − β ′

i,Slm,Sl

m

)(1 − ηi+1,Sl

m,Slm

)α′

i,Slm,Sl

mμi+1,Sl

m,Slm

(1 − β ′

i,Slm,Sl

m

)θi+1,Sl

m,Slm(

1 − α′i,Sl

m,Slm

)γi+1,Sl

m,Slm

β ′i,Sl

m,Slmηi+1,Sl

m,Slm

(1 − α′

i,Slm,Sl

m

)(1 − μi+1,Sl

m,Slm

)β ′

i,Slm,Sl

m

(1 − θi+1,Sl

m,Slm

)

α′i,Sl

m,Slmγi+1,Sl

m,Slm

(1 − β ′

i,Slm,Sl

m

)ηi+1,Sl

m,Slm

(1 − α′

i,Slm,Sl

m

)μi+1,Sl

m,Slm

(1 − β ′

i,Slm,Sl

m

)(1 − θi+1,Sl

m,Slm

)

⎤

⎥⎥⎥⎥⎥⎦

(36)

ASlm,k

Slm

=

⎡

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

(

1 − α′i,Sl

m,Slm+1

)(1 − γi+1,Sl

m,Slm

)β ′

i,Slm,Sl

m+1

(1 − ηi+1,Sl

m,Slm

) (

1 − α′i,Sl

m,Slm+1

)

μi+1,Slm,Sl

mβ ′

i,Slm,Sl

m+1θi+1,Sl

m,Slm

α′i,Sl

m+1,Slm

(1 − γi+1,Sl

m,Slm

) (

1 − β ′i,Sl

m,Slm+1

)(1 − ηi+1,Sl

m,Slm

)α′

i,Slm,Sl

m+1μi+1,Sl

m,Slm

(

1 − β ′i,Sl

m,Slm+1

)

θi+1,Slm,Sl

m(

1 − α′i,Sl

m,Slm+1

)

γi+1,Slm,Sl

mβ ′

i,Slm,Sl

m+1ηi+1,Sl

m,Slm

(

1 − α′i,Sl

m,Slm+1

)(1 − μi+1,Sl

m,Slm

)β ′

i,Slm,Sl

m+1

(1 − θi+1,Sl

m,Slm

)

α′i,Sl

m,Slm+1

γi+1,Slm,Sl

m

(

1 − β ′i,Sl

m,Slm+1

)

ηi+1,Slm,Sl

m

(

1 − α′i,Sl

m,Slm+1

)

μi+1,Slm,Sl

m

(

1 − β ′i,Sl

m,Slm+1

)(1 − θi+1,Sl

m,Slm

)

⎤

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(37)

P(gbp

) =∑n

i=1

(kSl

i− 1

)[1 − γr,Sl

i,Sli− γr−1,Sl

i,Sli

(ηr,Sl

i,Sli− γr,Sl

i,Sli

)− · · · − γ3,Sl

i,Sli

∏rt=4

(ηt,Sl

i,Sli− γt,Sl

i,Sli

)]

∑ni=1 kSl

i

−∑n

i=1

(kSl

i− 1

)[(1 − α1,Sl

i,Sli

)γ2,Sl

i,Sli

∏rt=3

(ηt,Sl

i,Sli− γt,Sl

i,Sli

)+ α1,Sl

i,Sliη2,Sl

i,Sli

∏rt=3

(ηt,Sl

i,Sli− γt,Sl

i,Sli

)]

∑ni=1 kSl

i

+∑n

i=2

[1 − γr,Sl

i−1,Sli− γr−1,Sl

i−1,Sli

(ηr,Sl

i−1,Sli− γr,Sl

i−1,Sli

)− · · · − γ3,Sl

i−1,Sli

∏rt=4

(ηt,Sl

i−1,Sli− γt,Sl

i−1,Sli

)]

∑ni=1 kSl

i

−∑n

i=2

[(1 − α1,Sl

i−1,Sli

)γ2,Sl

i−1,Sli

∏rt=3

(ηt,Sl

i−1,Sli− γt,Sl

i−1,Sli

)+ α1,Sl

i−1,Sliη2,Sl

i−1,Sli

∏rt=3

(ηt,Sl

i−1,Sli− γt,Sl

i−1,Sli

)]

∑ni=1 kSl

i

+[1 − γr,Sl

n,Sl1− γr−1,Sl

n,Sl1

(ηr,Sl

n,Sl1− γr,Sl

n,Sl1

)− · · · − γ3,Sl

n,Sl1

∏rt=4

(ηt,Sl

n,Sl1− γt,Sl

n,Sl1

)]

∑ni=1 kSl

i

−[(

1 − α1,Sln,S

l1

)γ2,Sl

n,Sl1

∏rt=3

(ηt,Sl

n,Sl1− γt,Sl

n,Sl1

)+ α1,Sl

n,Sl1η2,Sl

n,Sl1

∏rt=3

(ηt,Sl

n,Sl1− γt,Sl

n,Sl1

)]

∑ni=1 kSl

i

(38)

P(gbp

) =∑n

m=1

(kSl

m− 1

)(1 − α1,Sl

m,Slmη2,Sl

m,Slm

+ α1,Slm,Sl

mγ2,Sl

m,Slm

− γ2,Slm,Sl

m

)

∑nm=1 kSl

m

+∑n

m=2

(1 − α1,Sl

m−1,Slmη2,Sl

m−1,Slm

+ α1,Slm−1,S

lmγ2,Sl

m−1,Slm

− γ2,Slm−1,S

lm

)

∑nm=1 kSl

m

+(

1 − α1,Sln,S

l1η2,Sl

n,Sl1+ α1,Sl

n,Sl1γ2,Sl

n,Sl1− γ2,Sl

n,Sl1

)

∑nm=1 kSl

m

(39)

P(gsp) =

∑nm=2

(1 − α1,Sl

m−1,Slmη2,Sl

m−1,Slm

+ α1,Slm−1,S

lmγ2,Sl

m−1,Slm

− γ1,Slm−1,S

lm

)

n

+(

1 − α1,Sln,S

l1η2,Sl

n,Sl1+ α1,Sl

n,Sl1γ2,Sl

n,Sl1− γ1,Sl

n,Sl1

)

n(40)

quality analysis and assumes that the system quality followsa Bernoulli distribution: α1,i,j+β1,i,j = 1, γ1,i,j+μ2,i,j = 1, andη1,i,j+θ2,i,j = 1(i = 1, 2, . . . , n, j = 1, 2, . . . , n). Accordingly,

we obtain the probability that the system produces goodproducts under BP P(gbp) and SP P(gsp), respectively,(39) and (40), as shown at the top of this page.



Pbp(gS1

) = (ka − 1)[1 − γ2,a,a + α1,a,a(γ2,a,a − η2,a,a

)]

ka + kb + kc

+ (kb − 1)[1 − γ2,b,b + α1,b,b(γ2,b,b − η2,b,b

)]

ka + kb + kc

+ (kc − 1)[1 − γ2,c,c + α1,c,c(γ2,c,c − η2,c,c

)]

ka + kb + kc

+ 1 − γ2,c,a + α1,c,a(γ2,c,a − η2,c,a

)+ 1 − γ2,b,c + α1,b,c(γ2,b,c − η2,b,c

)

ka + kb + kc

+ 1 − γ2,a,b + α1,a,b(γ2,a,b − η2,a,b

)

ka + kb + kc(41)

Pbp(gS2

) = (ka − 1)[1 − γ2,a,a + α1,a,a

(γ2,a,a − η2,a,a

)]

kA + kB + kC

+ (kb − 1)[1 − γ2,b,b + α1,b,b

(γ2,b,b − η2,b,b

)]

kA + kB + kC

+ (kc − 1)(1 − γ2,c,c + α1,c,c

(γ2,c,c − η2,c,c

)]

kA + kB + kC

+ 1 − γ2,b,a + α1,b,a(γ2,b,a − η2,b,a

)+ 1 − γ2,c,b + α1,c,b(γ2,c,b − η2,c,b

)

ka + kb + kc

+ 1 − γ2,a,c + α1,a,c(γ2,a,c − η2,a,c

)

ka + kb + kc(42)

Psp(gS1

) = 1 − γ2,c,a + α1,c,a(γ2,c,a − η2,c,a

)+ 1 − γ2,b,c + α1,b,c(γ2,b,c − η2,b,c

)

3

+ 1 − γ2,a,b + α1,a,b(γ2,a,b − η2,a,b

)

3(43)

Psp(gS2

) = 1 − γ2,b,a + α1,b,a(γ2,b,a − η2,b,a

)+ 1 − γ2,c,b + α1,c,b(γ2,c,b − η2,c,b

)

3

+ 1 − γ2,a,c + α1,a,c(γ2,a,c − η2,a,c

)

3(44)

Here we first take a three-product-two-stage system as anexample to study the quality properties of product sequenceand then extend these properties to more general cases.Assume that three different products a, b, c are produced.There exist two different sequences: one is S1: a → b → c,the other is S2: a → c → b. The corresponding probabilitiesare

α1,a,b, α1,b,c, α1,c,a, α1,a,c, α1,c,b, α1,b,a, α1,a,a, α1,b,b, α1,c,c

γ2,a,b, γ2,b,c, γ2,c,a, γ2,a,c, γ2,c,b, γ2,b,a, γ2,a,a, γ2,b,b, γ2,c,c

η2,a,b, η2,b,c, η2,c,a, η2,a,c, η2,c,b, η2,b,a, η2,a,a, η2,b,b, η2,c,c.

According to (39), the probabilities of producing a good prod-uct for sequence S1 and S2 under BP are expressed as P(g1

bp)

and P(g2bp), respectively, (41) and (42), as shown at the top of

this page.And the probabilities of producing a good product under SP

for the two sequences can be obtained from (40) and expressedas P(g1

sp) and P(g2sp), respectively, (43) and (44) as shown at

the top of this page.Comparing (41) with (42), we can draw Conclusion 1.Conclusion 1: When BP is applied in M3Ss, the best product

sequence depends on the external rather than internal transitionprobabilities. By comparing the difference among the external

transition probabilities under different sequences, we can findout the best product sequence.

Comparing (41) with (43) [or comparing (42) with (44)],we have Conclusion 2.

Conclusion 2: Best sequences under BP and SP in M3Ssare consistent with each other, which means if sequence 1 isthe best one under BP, then it’s also the best one under SP.

Conclusions 1 and 2 can be extended to Conclusion 3 whenn types of products are manufactured in the system.

Conclusion 3: Under the Bernoulli case, for an M3S pro-cessing n different types of products, the best sequences underBP and SP are consistent with each other, which meansP(gl

bp) > P(gmbp) ⇔ P(gl

sp) > P(gmsp). And the sequence

satisfying the condition

max

{n∑

m=2

(1 − α1,Sl

m−1,Slmη2,Sl

m−1,Slm

+ α1,Slm−1,S

lmγ2,Sl

m−1,Slm

− γ2,Slm−1,S

lm

)

+(

1 − α1,Sln,S

l1η2,Sl

n,Sl1+ α1,Sl

n,Sl1γ2,Sl

n,Sl1− γ2,Sl

n,Sl1

)}

is the best sequence.Proof of Conclusion 3 can be found in the Appendix.



B. Bernoulli Relaxation Case

Although the Bernoulli case is very similar to the real man-ufacturing conditions, it still seems strict to some extent. Thus,we slightly relax the Bernoulli case and extend the model tomore general cases in practice. Under the Bernoulli relaxationcase, the summation of failure probability and repair probabil-ity does not have to equal 1 but just to be close to 1. In otherwords

δ1 = max{∣∣∣1 − α1,sl

i,slj− β1,sl

i,slj

∣∣∣}

δ2 = max{∣∣∣1 − γ2,sl

i,slj− μ2,sl

i,slj

∣∣∣}

δ3 = max{∣∣∣1 − η2,sl

i,slj− θ2,sl

i,slj

∣∣∣}

δmax = max{δ1, δ2, δ3}where i = 1, 2, . . . , n, j = 1, 2, . . . , n, and 0 ≤ δ1, δ2,

δ3, δmax 1.Conclusion 4: The probability of producing a good product

for M3Ss in a certain interval is decided by δmax and

1

1 + δmaxχ l ≤ P

(gl)

≤ 1

1 − δmaxχ l (45)

where for BP

χ l = χ lbp = P

(gl

bp

)

=∑n

i=1

(kSl

i− 1

)(1 − α1,Sl

i,SliηSl

i,Sli+ α1,Sl

i,Sliγ2,Sl

i,Sli− γ2,Sl

i,Sli

)

∑ni=1 kSl

i

+∑n

i=2

(1 − α1,Sl

i−1,Sliη2,Sl

i−1,Sli+ α1,Sl

i−1,Sliγ2,Sl

i−1,Sli− γ2,Sl

i−1,Sli

)

∑ni=1 kSl

i

+(

1 − α1,Sln,Sl

1η2,Sl

n,Sl1+ α1,Sl

n,Sl1γ2,Sl

n,Sl1− γ2,Sl

n,Sl1

)

∑ni=1 kSl

i

(46)

and for SP

χ l = χ lsp = P

(gl

ss

)

=∑n

i=2

(1 − α1,Sl

i−1,Sliη2,Sl

i−1,Sli+ α1,Sl

i−1,Sliγ2,Sl

i−1,Sli− γ2,Sl

i−1,Sli

)

n

+(

1 − α1,Sln,Sl

1η2,Sl

n,Sl1+ α1,Sl

n,Sl1γ2,Sl

n,Sl1− γ2,Sl

n,Sl1

)

n. (47)

Proof of Conclusion 4 can be found in the Appendix.Conclusion 5: Under Bernoulli relaxation case,

when 0 < δmax < [(χ lsp − χm

sp)/(χlsp + χm

sp)] =[(χ l

bp − χmbp)/(χ

lbp + χm

bp)], we still have P(glbp) > P(gm

bp) ⇔P(gl

sp) > P(gmsp).

Proof of Conclusion 5 can be found in the Appendix.Actually not only when 0 < δmax <

[(χ lsp − χm

sp)/(χlsp + χm

sp)] = [(χ lbp − χm

bp)/(χlbp + χm

bp)]but also under most Bernoulli relaxation cases, the conclusionP(gl


sp) > P(gmsp) holds. In order to verify

it, extensive numerical experiments have been carried out.We assume that δmax ≤ 0.2 and α ∈ [0, 0.2], β ∈ [0.8, 1],γ ∈ [0, 0.2], μ ∈ [0.8, 1], η ∈ [0, 1], and θ ∈ [0, 1] [9, 11].Corresponding transition probabilities under certain sequencesare randomly generated within their intervals and the resultsof the final quality are estimated by applying the model formultiproduct-two-stage systems. When there are more than

Fig. 4. Probability that conclusion holds under Bernoulli relaxation case.

three types of products, alternative sequences can be chosento improve quality. Here we increase the number of producttypes from 3 to 6 and the number of possible sequencesincreases from 2 to 120. For the given number of producttypes, make comparisons between the results of final qualityunder different sequences and each comparison is based on1000 times of numerical experiments. The results show thatthe probability that P(gl


sp) > P(gmsp) holds

is around 97% when the number of product types rangesfrom 3 to 6.

When there are three types of products, the probabilitythat the conclusion P(gl


sp) > P(gmsp)

holds is about 91%. But when the types of products increaseto 4–6, the probabilities are all between 96% and 97%(Fig. 4).

So it is reasonable to conclude that in practice, the bestsequences under BP and SP in M3Ss are consistent witheach other.

VI. QUALITY IMPROVEMENT ANALYSIS

A. Sensitivity Analysis

The product quality for a multistage system depends on thequality failure probabilities γi, ηi and quality repair proba-bilities μi, θi. Changes of theses parameters can lead to theimprovement of quality P(gi). It is necessary to find out whichparameter could bring about the largest quality improvement.Sensitivity analysis of P(gi) with respect to γi, ηi, μi, and θi

can help figure out this question.For the sensitivity analysis, we change only one parame-

ter and while the others remain unchanged. Accordingly, thechanged parameters and probabilities are γ ′

i , μ′i, η′

i, θ ′i , and

Pγi(gi), Pμi(gi), Pηi(gi), Pθi(gi), respectively. Then the sen-sitivity of P(gi) with respect to γi, μi, ηi, and θi could bewritten as

Sγi =∣∣Pγi(gi) − P(gi)

∣∣/

P(gi)∣∣γ ′

i − γi∣∣/γi

(48)

Sμi =∣∣Pμi(gi) − P(gi)

∣∣/

P(gi)∣∣μ′

i − μi∣∣/μi

(49)

Sηi =∣∣Pηi(gi) − P(gi)

∣∣/

P(gi)∣∣η′

i − ηi∣∣/ηi

(50)

Sθi =∣∣Pθi(gi) − P(gi)

∣∣/

P(gi)∣∣θ ′

i − θi∣∣/θi

. (51)



P(g4) =(

kSl1− 1

)[1 − γ4,Sl

1,Sl1− γ3,Sl

1,Sl1

(η4,Sl

1,Sl1− γ4,Sl

1,Sl1

)]

kSl1+ kSl

2

−(

kSl1− 1

)[(1 − α1,Sl

1,Sl1

)γ2,Sl

1,Sl1

(η3,Sl

1,Sl1− γ3,Sl

1,Sl1

)(η4,Sl

1,Sl1− γ4,Sl

1,Sl1

)]

kSl1+ kSl

2

−(

kSl1− 1

)[α1,Sl

1,Sl1η2,Sl

1,Sl1

(η3,Sl

1,Sl1− γ3,Sl

1,Sl1

)(η4,Sl

1,Sl1− γ4,Sl

1,Sl1

)]

kSl1+ kSl

2

+(

kSl2− 1

)[1 − γ4,Sl

2,Sl2− γ3,Sl

2,Sl2

(η4,Sl

2,Sl2− γ4,Sl

2,Sl2

)]

kSl1+ kSl

2

−(

kSl2− 1

)[(1 − α1,Sl

2,Sl2

)γ2,Sl

2,Sl2

(η3,Sl

2,Sl2− γ3,Sl

2,Sl2

)(η4,Sl

2,Sl2− γ4,Sl

2,Sl2

)]

kSl1+ kSl

2

−(

kSl2− 1

)[α1,Sl

2,Sl2η2,Sl

2,Sl2

(η3,Sl

2,Sl2− γ3,Sl

2,Sl2

)(η4,Sl

2,Sl2− γ4,Sl

2,Sl2

)]

kSl1+ kSl

2

+[1 − γ4,Sl

2,Sl1− γ3,Sl

2,Sl1

(η4,Sl

2,Sl1− γ4,Sl

2,Sl1

)]

kSl1+ kSl

2

−[(

1 − α1,Sl2,S

l1

)γ2,Sl

2,Sl1

(η3,Sl

2,Sl1− γ3,Sl

2,Sl1

)(η4,Sl

2,Sl1− γ4,Sl

2,Sl1

)]

kSl1+ kSl

2

−[α1,Sl

2,Sl1η2,Sl

2,Sl1

(η3,Sl

2,Sl1− γ3,Sl

2,Sl1

)(η4,Sl

2,Sl1− γ4,Sl

2,Sl1

)]

kSl1+ kSl

2

+[1 − γ4,Sl

1,Sl2− γ3,Sl

1,Sl2

(η4,Sl

1,Sl2− γ4,Sl

1,Sl2

)]

kSl1+ kSl

2

−[(

1 − α1,Sl1,S

l2

)γ2,Sl

1,Sl2

(η3,Sl

1,Sl2− γ3,Sl

1,Sl2

)(η4,Sl

1,Sl2− γ4,Sl

1,Sl2

)]

kSl1+ kSl

2

−[α1,Sl

1,Sl2η2,Sl

1,Sl2

(η3,Sl

1,Sl2− γ3,Sl

1,Sl2

)(η4,Sl

1,Sl2− γ4,Sl

1,Sl2

)]

kSl1+ kSl

2

(55)

Especially, when i = 1, the sensitivity analysis of P(g1)

with respect to α1 and β1 is needed. According to [41], wecan obtain

P(g1) = β1

α1 + β1. (52)

Assume the changed parameters are α′1, β ′

1 and Pα1(g1),Pβ1(g1). Then the sensitivity of P(g1) with respect to α1 andβ1 would be

Sα1 =∣∣Pα1(g1) − P(g1)

∣∣/

P(g1)∣∣α′

1 − α1∣∣/α1

(53)

Sβ1 =∣∣Pβ1(g1) − P(g1)

∣∣/

P(g1)∣∣β ′

1 − β1∣∣/β1

. (54)

The parameter with max(Sγi , Sμi , Sηi , Sθi) when i ≥ 2 ormax(Sα1 , Sβ1) when i = 1 is the most sensitive and has thelargest impact on P(gi).

B. Bottleneck Analysis

For M3Ss, the final quality is the function of inter-nal and external failure probabilities and repair probabili-ties. Because of product variety, a certain set of failure orrepair probabilities contains both internal and external prob-abilities. For example, for quality failure probabilities withrespect to good incoming parts γi, there are internal prob-abilities γi,Sl

m,Slm(m = 1, 2, . . . , n) and external probabilities

γi,Slm−1,S

lm(m = 2, 3, . . . , n). Then there are several kinds of

quality bottlenecks, namely, internal and external quality bot-tleneck for γ (IQB − γ and EQB − γ ), internal and externalquality bottleneck for μ (IQB−μ and EQB−μ), internal andexternal quality bottleneck for η (IQB − η and EQB − η), andinternal and external quality bottleneck for θ (IQB − θ andEQB − θ ). They are defined as follows.

Definition 1: For i �= j, if |[∂P(gn)/∂γi,Slm,Sl

m]| >

|[∂P(gn)/∂γj,Slm,Sl

m]|, then under sequence Sl, γi,Sl

m,Slm

is theIQB − γ for product Sl

m, which means when producing prod-uct Sl

m, stage i is the quality bottleneck with respect to failureprobabilities for good incoming parts.

Definition 2: For i �= j, if |[∂P(gn)/∂μi,Slm,Sl

m]| >

|[∂P(gn)/∂μj,Sl,m,Sl

m]|, then under sequence Sl, μi,Sl

m,Slm

is the

IQB − μ for product Slm, which means when producing prod-

uct Slm, stage i is the quality bottleneck with respect to repair

probabilities with defective incoming parts.Definition 3: For i �= j, if |[∂P(gn)/∂ηi,Sl

m,Slm

]| >

|[∂P(gn)/∂ηj,Sl,m,Sl

m]|, then under sequence Sl, ηi,Sl

m,Slm

is the

IQB − η for product Slm, which means when producing prod-

uct Slm, stage i is the quality bottleneck with respect to failure

probabilities with defective incoming parts.Definition 4: For i �= j, if |[∂P(gn)/∂θi,Sl

m,Slm

]| >

|[∂P(gn)/∂θj,Sl,m,Sl

m]|, then under sequence Sl, θi,Sl

m,Slm

is the

IQB − θ for product Slm, which means when producing



∣∣∣∣∣

∂P(g4)

∂γ4,Sl1,S

l1

∣∣∣∣∣=∣∣∣∣∣∣

(kSl

1− 1

){[(1 − α1,Sl

1,Sl1

)(η2,Sl

1,Sl1− γ2,Sl

1,Sl1

)+ 1 − η2,Sl

1,Sl1

](η3,Sl

1,Sl1− γ3,Sl

1,Sl1

)+ 1 − η3,Sl

1,Sl1

}

kSl1+ kSl

2

∣∣∣∣∣∣

(56)

∣∣∣∣∣

∂P(g4)

∂γ4,Sl2,S

l2

∣∣∣∣∣=∣∣∣∣∣∣

(kSl

2− 1

){[(1 − α1,Sl

2,Sl2

)(η2,Sl

2,Sl2− γ2,Sl

2,Sl2

)+ 1 − η2,Sl

2,Sl2

](η3,Sl

2,Sl2− γ3,Sl

2,Sl2

)+ 1 − η3,Sl

2,Sl2

}

kSl1+ kSl

2

∣∣∣∣∣∣

(57)

∣∣∣∣∣

∂P(g4)

∂γ3,Sl1,S

l1

∣∣∣∣∣=∣∣∣∣∣∣

(kSl

1− 1

)[(1 − α1,Sl

1,Sl1

)(η2,Sl

1,Sl1− γ2,Sl

1,Sl1

)+ 1 − η2,Sl

1,Sl1

](η4,Sl

1,Sl1− γ4,Sl

1,Sl1

)

kSl1+ kSl

2

∣∣∣∣∣∣

(58)

∣∣∣∣∣

∂P(g4)

∂γ3,Sl2,S

l2

∣∣∣∣∣=∣∣∣∣∣∣

(kSl

2− 1

)[(1 − α1,Sl

2,Sl2

)(η2,Sl

2,Sl2− γ2,Sl

2,Sl2

)+ 1 − η2,Sl

2,Sl2

](η4,Sl

2,Sl2− γ4,Sl

2,Sl2

)

kSl1+ kSl

2

∣∣∣∣∣∣

(59)

∣∣∣∣∣

∂P(g4)

∂γ2,Sl1,S

l1

∣∣∣∣∣=∣∣∣∣∣∣

(kSl

1− 1

)[(1 − α1,Sl

1,Sl1

)(η3,Sl

1,Sl1− γ3,Sl

1,Sl1

)(η4,Sl

1,Sl1− γ4,Sl

1,Sl1

)]

kSl1+ kSl

2

∣∣∣∣∣∣

(60)

∣∣∣∣∣

∂P(g4)

∂γ2,Sl2,S

l2

∣∣∣∣∣=∣∣∣∣∣∣

(kSl

2− 1

)[(1 − α1,Sl

2,Sl2

)(η3,Sl

2,Sl2− γ3,Sl

2,Sl2

)(η4,Sl

2,Sl2− γ4,Sl

2,Sl2

)]

kSl1+ kSl

2

∣∣∣∣∣∣

(61)

∣∣∣∣∣

∂P(g4)

∂γ4,Sl2,S

l1

∣∣∣∣∣=∣∣∣∣∣∣

[(1 − α1,Sl

2,Sl1

)(η2,Sl

2,Sl1− γ2,Sl

2,Sl1

)+ 1 − η2,Sl

2,Sl1

](η3,Sl

2,Sl1− γ3,Sl

2,Sl1

)+ 1 − η3,Sl

2,Sl1

kSl1+ kSl

2

∣∣∣∣∣∣

(62)

∣∣∣∣∣

∂P(g4)

∂γ4,Sl1,S

l2

∣∣∣∣∣=∣∣∣∣∣∣

[(1 − α1,Sl

1,Sl2

)(η2,Sl

1,Sl2− γ2,Sl

1,Sl2

)+ 1 − η2,Sl

1,Sl2

](η3,Sl

1,Sl2− γ3,Sl

1,Sl2

)+ 1 − η3,Sl

1,Sl2

kSl1+ kSl

2

∣∣∣∣∣∣

(63)

∣∣∣∣∣

∂P(gbp

)

∂γ3,Sl2,S

l1

∣∣∣∣∣=∣∣∣∣∣∣

[(1 − α1,Sl

2,Sl1

)(η2,Sl

2,Sl1− γ2,Sl

2,Sl1

)+ 1 − η2,Sl

2,Sl1

](η4,Sl

2,Sl1− γ4,Sl

2,Sl1

)

kSl1+ kSl

2

∣∣∣∣∣∣

(64)

∣∣∣∣∣

∂P(gbp

)

∂γ3,Sl1,S

l2

∣∣∣∣∣=∣∣∣∣∣∣

[(1 − α1,Sl

1,Sl2

)(η2,Sl

1,Sl2− γ2,Sl

1,Sl2

)+ 1 − η2,Sl

1,Sl2

](η4,Sl

1,Sl2− γ4,Sl

1,Sl2

)

kSl1+ kSl

2

∣∣∣∣∣∣

(65)

∣∣∣∣∣

∂P(g4)

∂γ2,Sl2,S

l1

∣∣∣∣∣=∣∣∣∣∣∣

(1 − α1,Sl

2,Sl1

)(η3,Sl

2,Sl1− γ3,Sl

2,Sl1

)(η4,Sl

2,Sl1− γ4,Sl

2,Sl1

)

kSl1+ kSl

2

∣∣∣∣∣∣

(66)

∣∣∣∣∣

∂P(g4)

∂γ2,Sl1,S

l2

∣∣∣∣∣=∣∣∣∣∣∣

(1 − α1,Sl

1,Sl2

)(η3,Sl

1,Sl2− γ3,Sl

1,Sl2

)(η4,Sl

1,Sl2− γ4,Sl

1,Sl2

)

kSl1+ kSl

2

∣∣∣∣∣∣

(67)

product Slm, stage i is the quality bottleneck with respect to

repair probabilities with defective incoming parts.Definition 5: For i �= j, if |[∂P(gn)/∂γi,Sl

m−1,Slm

]| >

|[∂P(gn)/∂γj,Slm−1,S

lm

]|, then under sequence Sl, γi,Slm−1,S

lm

isthe EQB − γ for transition m, which means when transitingfrom product Sl

m−1 to Slm, stage i is the quality bottleneck with

respect to failure probabilities with good incoming parts.Definition 6: For i �= j, if |[∂P(gn)/∂μi,Sl

m−1,Slm

]| >

|[∂P(gn)/∂μj,Slm−1,S

lm

]|, then under sequence Sl, μi,Slm−1,S

lm

isthe EQB − μ for transition m, which means when transitingfrom product Sl


respect to repair probabilities with good incoming parts.Definition 7: For i �= j, if |[∂P(gn)/∂ηi,Sl

m−1,Slm

]| >

|[∂P(gn)/∂ηj,Slm−1,S

lm

]|, then under sequence Sl, ηi,Slm−1,S

lm

is

the EQB − η for transition m, which means when transitingfrom product Sl


respect to failure probabilities with defective incoming parts.Definition 8: For i �= j, if |[∂P(gn)/∂θi,Sl

m−1,Slm

]| >

|[∂P(gn)/∂θj,Slm−1,S

lm

]|, then under sequence Sl, θi,Slm−1,S

lm

isEQB − θ for transition m, which means when transiting fromproduct Sl


respect to repair probabilities with defective incoming parts.For instance, according to (45), the final quality P(g4) for

a two-product-four-stage system is (55), as shown at the topof the previous page.

From (55), we have (56)–(67), as shown at the top of thispage.

By comparing (56), (58), and (60), we findthe IQB − γ for product Sl

1 which is the stage



Fig. 5. Four types of products manufactured by the multistage system.

with max{[∂P(g4)/∂γ4,Sl1,S

l1] [∂P(g4)/∂γ3,Sl

1,Sl1]

[∂P(g4)/∂γ2,Sl1,S

l1]}. Comparing (58), (60), and (62), we

obtain the IQB − γ for product Sl2 which is the stage

with max{[∂P(g4)/∂γ4,Sl2,S

l2] [∂P(g4)/∂γ3,Sl

2,Sl2] [∂P(g4)/

∂γ2,Sl2,S

l2]}. We define (56)–(61) as IQB indicators for

quality failure probabilities for good incoming parts.Similarly, (62), (64), and (66) indicate the EQB − γ forproduct Sl

1 and (63), (65), and (67) indicate the EQB − γ

for product Sl2. We define (62)–(67) as EQB indicators for

quality failure probabilities for defective incoming parts.Two conclusions can be made from (55)–(67).Conclusion 6: IQB indicators are related to the transition

probabilities of both upstream and downstream stages as wellas the batch size of the product.

Conclusion 7: EQB indicators are only related to the tran-sition probabilities of both upstream and downstream stages.

VII. CASE STUDY

To validate the applicability of the proposed method, a casestudy has been conducted. To ensure the confidentiality of thedata, all the parameters introduced below have been modifiedand only used for illustration.

A. Manufacturing System Description

The model is applied to a four-product-five-stage manu-facturing system to evaluate the quality performance of thesystem. The four types of products (valve shells) that need tobe manufactured in this system are shown in Fig. 5. There arefive dependent stages (OP 10, 20, 30, 40, and 50) that the fourtypes of products will go through. The relationships amongthese five stages and the quality propagation are discussed indetail in [32].

B. Results and Discussions

1) Model Prediction Error: Taking product 1 for an exam-ple. According to historical data, the internal transition prob-abilities of product 1 are calculated as α1 = 0.05, β1 = 0.94,γi = [0.05, 0.1, 0.07, 0.04], μi = [0.92, 0.87, 0.91, 0.95],ηi = [0.52, 0.55, 0.43, 0.57], and θi = [0.45, 0.45, 0.55, 0.45].Based on the developed model, the quality changes of product1 along the five-stage system can be estimated. The result isshown in Fig. 6.

The probability of producing a good product estimatedfrom the model is 89.46%. The actual final quality based on

Fig. 6. Estimated probabilities of producing good product 1 through thefive-stage system.

historical data is 89.71%, and the prediction error is 0.25%.The result demonstrates the effectiveness and practicability ofthe proposed model. For all of these four products, the averageprediction error is about 0.24%.

2) Production Sequence Analysis: According to the his-torical data, the failure probabilities and repair probabilitiesapproximately follow Bernoulli distribution, which allowsus to study the system under Bernoulli case. According toConclusion 3, to obtain the best sequence, we need to knowthe various external transition probabilities shown in Table I.They are obtained through the following process.

For a certain part j processed by Mi, either good or defective,there exist four possible statuses.

1) Both parts (j − 1) and j are good; thus, Mi maintainsa good state gi.

2) Part (j − 1) is good but part j is defective; thus, Mi

transits from gi to di.3) The part (j − 1) is defective but part j is good; thus, Mi

transits from di to gi.4) Both parts (j−1) and j are defective; thus, Mi maintains

a defective state di.The transition probabilities are estimated from historical

data by calculating the proportions of the status: proportionof status 2) represents α, γ , and η while proportion of sta-tus 3) refers to β, μ, and θ . If part (j − 1) and part j belongto the same type of product, then the estimated probabil-ities represent internal transition probabilities. Otherwise, ifpart (j − 1) and part j belong to different types of products,then the estimated probabilities represent external transitionprobabilities.

According to (38), the sequence with the largest valuefor four-product-five-stage systems is the best sequence.



TABLE IEXTERNAL TRANSITION PROBABILITIES OF THE SYSTEM

The optimization model can be expressed as

max4∑

i=1

4∑

j=1

yi,j

[(1 − γ5,i,j − γ4,i,j

(η5,i,j − γ5,i,j

)

− γ3,i,j(η4,i,j − γ4,i,j

)(η5,i,j − γ5,i,j

)

+ (1 − α1,i,j

)γ2,i,j

(η3,i,j − γ3,i,j

)(η4,i,j − γ4,i,j

)

× (η5,i,j − γ5,i,j

)− α1,i,jη2,i,j(η3,i,j − γ3,i,j

)

× (η4,i,j − γ4,i,j

)(η5,i,j − γ5,i,j

)]

s.t.4∑

i=1

yi,j = 1, (i �= j)

4∑

j=1

yi,j = 1, (i �= j)

ϕi − ϕj + Nyi,j ≤ N − 1, (i �= j; i = 2, 3, 4; j = 2, 3, 4)

where all yi,j = 0 or 1 and all ϕi > 0 ; when yi,j = 1 prod-uct i is processed right before product j in the batch cycle,otherwise, yi,j = 0, ϕi > 0. By solving the model, we obtainy1,4 = 1, y2,1 = 1, y3,2 = 1, y4,3 = 1, and other yi,j = 0. Thisresult indicates that product 1 is processed before product 4,and product 2 is processed before product 1 while product 3 isprocessed before product 2, and product 4 is processed beforeproduct 3. Then in this case, the best sequence for the four-product-two-stage system would be 1 → 4 → 3 → 2. Underthis sequence, the corresponding probabilities are α1,1,4, α1,4,3,α1,3,2, α1,2,1, γi,1,4, γi,4,3, γi,3,2, γi,2,1, ηi,1,4, ηi,4,3, ηi,3,2, andηi,2,1 (i = 2, 3, 4, 5).

Based on the above analysis, we can obtain the EQB foreach kind of parameters. Similar to (62), (64), and (66), wecan obtain the derivatives of P(g5) with respect to each exter-nal transition probabilities. As the comparison of derivativesis independent of the batch size, thus, it can be neglected.Therefore as for γi,1,4 the quality failure probability with goodincoming parts when transiting from products 1–4, the relatedderivatives under sequence 1 → 4 → 3 → 2 are calculatedas [∂P(g5)/∂γ5,1,4] = 0.8034, [∂P(g5)/∂γ4,1,4] = 0.7695,[∂P(g5)/∂γ3,1,4] = 0.2856, and [∂P(g5)/∂γ2,1,4] = 0.1671.Since [∂P(g5)/∂γ5,1,4] has the largest value, the fifth stageOP50 is the EQB for the quality failure probability with goodincoming parts when transiting from products 1–4 in this case.The improvement of γ5,1,4 can bring the largest benefit to thefinal quality.

3) Bottleneck Analysis: Also, take product 1 for an exam-ple. The quality has the biggest decrease in the third stage(Fig. 6). Taking the third stage into consideration, we cando the sensitivity analysis to find out which transition prob-ability P(g3)is most sensitive. In this case, γ3, μ3, η3, andθ3 are increased or decreased by given percentages and thesensitivities with respect to the four parameters at 10% areSγ3 = 9.43%, Sμ3 = 12.13%, Sη3 = 4.21%, and Sθ5 = 0.61%,respectively, which indicates that the quality P(g3) is mostsensitive to the quality failure probability for good incomingparts μ3. Based on the model and Definitions 1–4, we can findthe IQB for this system. For the repair probability with goodincoming parts μi, the relationships between μi and P(g4) areshown in Fig. 7.



Fig. 7. Relationships between μi and P(g4).

From Fig. 7, we can see that changes of μ5 have thegreatest impact on the final quality P(g5). The derivatives ofP(g5) also show the same result. The corresponding deriva-tives are [∂P(g5)/∂μ2] = 0.0065, [∂P(g5)/∂μ3] = 0.0255,[∂P(g5)/∂μ4] = 0.0570, and [∂P(g5)/∂μ5] = 0.0938. As[∂P(g5)/∂μ5] is the largest among these derivatives, the fifthstage OP50 is the IQB for repair probability with good incom-ing parts μi in this case. This means changes of μ5 can leadto the largest improvement to the final quality.

VIII. CONCLUSION

M3Ss have been widely applied in industry. It is veryimportant to develop a proper method to evaluate the qual-ity performance of M3Ss. This paper is devoted to modelingand analyzing steady-state M3Ss for quality improvement andfilling the gap between FMSs and quality propagation amongmultiple stages. Two novel Markov models for multiproduct-two-stage systems and multiproduct-multistage systems aredeveloped for quality improvement. The models take bothproduct flexibility and quality propagation into consideration.Several important quality properties including production strat-egy, product sequence, and quality bottleneck identification areanalyzed and a few practical conclusions are noted to providesome insights on quality improvement. Finally, a case study onvalve shells is conducted to illustrated the effectiveness of theproposed models. The average prediction error of the modelsis about 0.24%.

Based on the proposed models in this paper, future workcan be conducted as follows.

1) The Markov models can be extended to the one charac-tering the transient-state quality performance. A Markovmodel is desirable to be developed for both steady-stateand transient-state quality performance.

2) Some other properties in M3Ss for quality improvementcan be investigated, such as monotonicity, settling time,and quality loss.

3) The future work also can be directed to modeling andanalysis of serial-parallel M3Ss.

APPENDIX

PROOFS

A. Proof of Conclusion 3

According to (39), when there are n types of products, underBP, the difference between the probabilities of producing goodproducts with sequence Sl and Sm is (A1), as shown at the topof the next page, where

n∑

i=1

kSli=

n∑

i=1

kSmi

(A2)

n∑

i=1

(kSl

i− 1

)(1 − αSl

i,SliηSl

i,Sli+ αSl

i,SliγSl

i,Sli− γSl

i,Sli

)

=n∑

i=1

(kSm

i− 1

)(1 − αSm

i ,SmiηSm

i ,Smi

+ αSmi ,Sm

iγSm

i ,Smi

− γSmi ,Sm

i

). (A3)

Thus, A(1) can be simplified as (A4), as shown at the topof the next page.

On the other hand, under SP, the difference between prob-abilities of producing good product with sequence Sl and Sm

is (A5), as shown at the top of the next page.Comparing A(1) with A(5), it can be seen that they have

the same signature, which means the best sequences under BPand SP are consistent with each other: P(gl

bp) − P(gmbp) ⇔

P(glss) − P(gm

ss).

B. Proof of Conclusion 4

Under Bernoulli case, assume that δ1i,j = 1−α1,Sl

i,Sli−β1,Sl

i,Sli,

δ2i,j = 1 − γ2,Sl

i,Sli− μ2,Sl

i,Sli, δ3

i,j = 1 − η2,Sli,S

li− θ2,Sl

i,Sli

and the



P(

glbp

)− P

(gm

bp

)

=∑n

i=1

(kSl

i− 1

)(1 − α1,Sl

i,Sliη2,Sl

i,Sli+ α1,Sl

i,Sliγ2,Sl

i,Sli− γ2,Sl

i,Sli

)

∑ni=1 kSl

i

+∑n

i=2

(1 − α1,Sl

i−1,Sliη2,Sl

i−1,Sli+ α1,Sl

i−1,Sliγ2,Sl

i−1,Sli− γ2,Sl

i−1,Sli

)+(

1 − α1,Sln,S

l1η2,Sl

n,Sl1+ α1,Sl

n,Sl1γ2,Sl

n,Sl1− γ2,Sl

n,Sl1

)

∑ni=1 kSl

i

−∑n

i=1

(kSm

i− 1

)(1 − α1,Sm

i ,Smiη2,Sm

i ,Smi

+ α1,Smi ,Sm

iγ2,Sm

i ,Smi

− γ2,Smi ,Sm

i

)

∑ni=1 kSm

i

−∑n

i=2

(1 − α1,Sm

i−1,Smiη2,Sm

i−1,Smi

+ α1,Smi−1,S

miγ2,Sm

i−1,Smi

− γ2,Smi−1,S

mi

)+(

1 − α1,Smn ,Sm

1η2,Sm

n ,Sm1

+ α1,Smn ,Sm

1γ2,Sm

n ,Sm1

− γ2,Smn ,Sm

1

)

∑ni=1 kSm

i

(A1)

P(

glbp

)− P

(gm

bp

)

=∑n

i=2

(1 − α1,Sl

i−1,Sliη2,Sl

i−1,Sli+ α1,Sl

i−1,Sliγ2,Sl

i−1,Sli− γ2,Sl

i−1,Sli

)+(

1 − α1,Sln,S

l1η1,Sl

n,Sl1+ α1,Sl

n,Sl1γ2,Sl

n,Sl1− γ2,Sl

n,Sl1

)

∑ni=1 kSl

i

−∑n

i=2

(1 − α1,Sm

i−1,Smiη1,Sm

i−1,Smi

+ α1,Smi−1,S

miγ1,Sm

i−1,Smi

− γ1,Smi−1,S

mi

)+(

1 − α1,Smn ,Sm

1η2,Sm

n ,Sm1

+ α2,Smn ,Sm

1γ2,Sm

n ,Sm1

− γ1,Smn ,Sm

1

)

∑ni=1 kSl

i

(A4)

P(

glss

)− P

(gm

ss

)

=∑n

i=2

(1 − α1,Sl

i−1,Sliη2,Sl

i−1,Sli+ α1,Sl

i−1,Sliγ2,Sl

i−1,Sli− γ2,Sl

i−1,Sli

)+(

1 − α1,Sln,S

l1η2,Sl

n,Sl1+ α1,Sl

n,Sl1γ2,Sl

n,Sl1− γ2,Sl

n,Sl1

)

n

−∑n

i=2

(1 − α1,Sm

i−1,Smiη2,Sm

i−1,Smi

+ α1,Smi−1,S

miγ2,Sm

i−1,Smi

− γ2,Smi−1,S

mi

)+(

1 − α1,Smn ,Sm

1η2,Sm

n ,Sm1

+ α1,Smn ,Sm

1γ2,Sm

n ,Sm1

− γ2,Smn ,Sm

1

)

n(A5)

matrix of state transition probabilities will be

C =

⎡

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

0 C1,2 · · · 0 0 0 · · · 0

0 0. . . 0 0 0 · · · 0

0 0 · · · Ci,kSli

0 0 · · · 0

0 0 · · · 0 Ci+1,1 0 · · · 00 0 · · · 0 0 Ci+1,2 · · · 0

0 0 · · · 0 0 0. . . 0

0 0 · · · 0 0 0 · · · Cn,kSln

C1,1 0 · · · 0 0 0 · · · 0

⎤

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(A6)

where (A7), as shown at the top of the next page.Ci,j can be written as the summation of C1

i,j and C2i,j,

and (A8) and (A9), as shown at the top of the next page.According to (26)

B = [ B1 B2 · · · Bn ] (A10)

B1 = B2 = · · · = Bn = [0 0 0 1

K

]. (A11)

Then for a general case (A12), as shown at the top of thenext page.

In the Bernoulli case where C2i,j = 0, (A13), as shown at

the top of the page following the next pageDefining function F as the summation of the first and second

column of the fourth row in a 4 × 4 matrix, we have

P(

gl)

= 1

K

n∑

i=1

kSl

i∑

j=1

F

[(C1

i,j + C2i,j

)−1]

(A14)

χ l = 1

K

n∑

i=1

kSl

i∑

j=1

F

[(C1

i,j

)−1]

(A15)

P(

gl)

− 1

1 + δmaxχ l

=∑n

i=1∑k

Sli

j=1

{

(1 + δmax)F

[(C1

i,j + C2i,j

)−1]

− F

[(C1

i,j

)−1]}

K(1 + δmax).

(A16)

Calculate the block matrixes (C1i,j + C2

i,j)−1 and (C1

i,j)−1,

we obtain that when δmax 1(1 + δmax)F[(C1i,j + C2

i,j)−1] −

F[(C1i,j)

−1] ≥ 0 which means P(gl) − [1/(1 + δmax)]χ l ≥ 0.Similarly, P(gl) − [1/(1 − δmax)]χ l ≤ 0 can be proved.



Ci,j =

⎡

⎢⎢⎢⎢⎢⎢⎣

(1 − α1,Sl

i,Sli

)(1 − γ2,Sl

i,Sli

)− 1 α1,Sl

i,Sli

(1 − γ2,Sl

i,Sli

) (1 − α1,Sl

i,Sli

)γSl

i,Sli

1

βSli,S

li

(1 − ηSl

i,Sli

) (1 − βSl

i,Sli

)(1 − ηSl

i,Sli

)− 1 βSl

i,SliηSl

i,Sli

1(

1 − αSli,S

li

)μSl

i,Sli

αSli,S

liμSl

i,Sli

(1 − αSl

i,Sli

)(1 − μSl

i,Sli

)− 1 1

βSli,S

liθSl

i,Sli

(1 − βSl

i,Sli

)θSl

i,Sli

βSli,S

li

(1 − θSl

i,Sli

)1

⎤

⎥⎥⎥⎥⎥⎥⎦

=

⎡

⎢⎢⎢⎢⎢⎢⎣

(1 − αSl

i,Sli

)(1 − γSl

i,Sli

)− 1 αSl

i,Sli

(1 − γSl

i,Sli

) (1 − αSl

i,Sli

)γSl

i,Sli

1(

1 − αSli,S

li

)(1 − ηSl

i,Sli

)αSl

i,Sli

(1 − ηSl

i,Sli

)− 1

(1 − αSl

i,Sli

)ηSl

i,Sli

1(

1 − αSli,S

li

)(1 − γSl

i,Sli

)αSl

i,Sli

(1 − γSl

i,Sli

) (1 − αSl

i,Sli

)γSl

i,Sli− 1 1

(1 − αSl

i,Sli

)(1 − ηSl

i,Sli

)αSl

i,Sli

(1 − ηSl

i,Sli

) (1 − αSl

i,Sli

)ηSl

i,Sli

1

⎤

⎥⎥⎥⎥⎥⎥⎦

+

⎡

⎢⎢⎢⎢⎢⎣

0 0 0 0

−δ1i,j

(1 − ηSl

i,Sli

)δ1

i,j

(1 − ηSl

i,Sli

)−δ1

i,jηSli,S

li

0

−δ2i,j

(1 − αSl

i,Sli

)−δ2

i,jαSli,S

li

δ2i,j

(1 − αSl

i,Sli

)0

δ3i,j

(1 − αSl

i,Sli

)− δ1

i,j

(1 − ηSl

i,Sli− δ3

i,j

)−δ3

i,jαSli,S

li+ δ1

i,j

(1 − ηSl

i,Sli− δ3

i,j

)δ3

i,j

(1 − αSl

i,Sli− δ1

i,j

)− δ1

i,jηSli,S

li

0

⎤

⎥⎥⎥⎥⎥⎦

(A7)

C1i,j =

⎡

⎢⎢⎢⎢⎢⎢⎣

(1 − αSl

i,Sli

)(1 − γSl

i,Sli

)− 1 αSl

i,Sli

(1 − γSl

i,Sli

) (1 − αSl

i,Sli

)γSl

i,Sli

1(

1 − αSli,S

li

)(1 − ηSl

i,Sli

)αSl

i,Sli

(1 − ηSl

i,Sli

)− 1

(1 − αSl

i,Sli

)ηSl

i,Sli

1(

1 − αSli,S

li

)(1 − γSl

i,Sli

)αSl

i,Sli

(1 − γSl

i,Sli

) (1 − αSl

i,Sli

)γSl

i,Sli− 1 1

(1 − αSl

i,Sli

)(1 − ηSl

i,Sli

)αSl

i,Sli

(1 − ηSl

i,Sli

) (1 − αSl

i,Sli

)ηSl

i,Sli

1

⎤

⎥⎥⎥⎥⎥⎥⎦

(A8)

C2i,j =

⎡

⎢⎢⎢⎢⎢⎣

0 0 0 0

−δ1i,j

(1 − ηSl

i,Sli

)δ1

i,j

(1 − ηSl

i,Sli

)−δ1

i,jηSli,S

li

0

−δ2i,j

(1 − αSl

i,Sli

)−δ2

i,jαSli,S

li

δ2i,j

(1 − αSl

i,Sli

)0

δ3i,j

(1 − αSl

i,Sli

)− δ1

i,j

(1 − ηSl

i,Sli− δ3

i,j

)−δ3

i,jαSli,S

li+ δ1

i,j

(1 − ηSl

i,Sli− δ3

i,j

)δ3

i,j

(1 − αSl

i,Sli− δ1

i,j

)− δ1

i,jηSli,S

li

0

⎤

⎥⎥⎥⎥⎥⎦

(A9)

X = BC−1 = B(

C1 + C2)−1 =

[B1 B2 · · · Bn

]

×

⎡

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

0 · · · 0 0 0 · · · 0(

C11,1 + C2

1,1

)−1

(C1

1,2 + C21,2

)−1 · · · 0 0 0 · · · 0 0

0. . . 0 0 0 · · · 0 0

0 · · ·(

C1i,k

Sli

+ C2i,k

Sli

)−1

0 0 · · · 0 0

0 · · · 0(

C1i+1,1 + C2

i+1,1

)−10 · · · 0 0

0 · · · 0 0(

C1i+1,2 + C2

i+1,2

)−1 · · · 0 0

0 · · · 0 0 0. . . 0 0

0 · · · 0 0 0 · · ·(

C1n,k

Sln

+ C2n,k

Sln

)−1

0

⎤

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(A12)

C. Proof of Conclusion 5

According to (39) and (40), χ lbp, χm

bp, χ lsp, and χm

sp can be

obtained, (A17)–(A20), as shown at the top of the next page.

Then according to (A2) and (A3)

χ lss − χm

ss

χ lss + χm

ss= χ l

bp − χmbp

χ lbp + χm

bp

.



X′ = BC−1 = B(

C1)−1

= [B1 B2 · · · Bn

]

⎡

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

0 · · · 0 0 0 · · · 0(

C11,1

)−1

(C1

1,2

)−1 · · · 0 0 0 · · · 0 0

0. . . 0 0 0 · · · 0 0

0 · · ·(

C1i,k

Sli

)−1

0 0 · · · 0 0

0 · · · 0(

C1i+1,1

)−10 · · · 0 0

0 · · · 0 0(

C1i+1,2

)−1 · · · 0 0

0 · · · 0 0 0. . . 0 0

0 · · · 0 0 0 · · ·(

C1n,k

Sln

)−1

0

⎤

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(A13)

χ lbp = P

(gl

bp

)=∑n

i=1

(kSl

i− 1

)(1 − α1,Sl

i,Sliη2,Sl

i,Sli+ α1,Sl

i,Sliγ1,Sl

i,Sli− γ2,Sl

i,Sli

)

∑ni=1 kSl

i

+∑n

i=2

(1 − α1,Sl

i−1,Sliη1,Sl

i−1,Sli+ α1,Sl

i−1,Sliγ2,Sl

i−1,Sli− γ2,Sl

i−1,Sli

)+(

1 − α1,Sln,S

l1η2,Sl

n,Sl1+ α1,Sl

n,Sl1γ2,Sl

n,Sl1− γ2,Sl

n,Sl1

)

∑ni=1 kSl

i

(A17)

χmbp = P

(gm

bp

)=∑n

i=1

(kSm

i− 1

)(1 − α1,Sm

i ,Smiη2,Sm

i ,Smi

+ α1,Smi ,Sm

iγ2,Sm

i ,Smi

− γ2,Smi ,Sm

i

)

∑ni=1 kSm

i

+∑n

i=2

(1 − α1,Sm

i−1,Smiη2,Sm

i−1,Smi

+ α1,Smi−1,S

miγ2,Sm

i−1,Smi

− γ2,Smi−1,S

mi

)+(

1 − α1,Smn ,Sm

1η2,Sm

n ,Sm1

+ α1,Smn ,Sm

1γ2,Sm

n ,Sm1

− γ2,Smn ,Sm

1

)

∑ni=1 kSm

i

(A18)

χ lsp = P

(gl

ss

)

=∑n

i=2

(1 − α1,Sl

i−1,Sliη2,Sl

i−1,Sli+ α1,Sl

i−1,Sliγ2,Sl

i−1,Sli− γ2,Sl

i−1,Sli

)+(

1 − α1,Sln,S

l1η2,Sl

n,Sl1+ α1,Sl

n,Sl1γ2,Sl

n,Sl1− γ2,Sl

n,Sl1

)

n(A19)

χmsp = P

(gm

ss

)

=∑n

i=2

(1 − α1,Sm

i−1,Smiη2,Sm

i−1,Smi

+ α1,Smi−1,S

miγ2,Sm

i−1,Smi

− γ2,Smi−1,S

mi

)+(

1 − α1,Smn ,Sm

1η2,Sm

n ,Sm1

+ α1,Smn ,Sm

1γ2,Sm

n ,Sm1

− γ2,Smn ,Sm

1

)

n

(A20)

Based on Conclusion 4

1

1 + δmaxχ l < P

(gl)

<1

1 − δmaxχ l

(χ l

ss − χmss

)− δmax(χ l

ss + χmss

)

1 − δ2max

< P(

glss

)− P

(gm

ss

)<

(χ l

ss − χmss

)+ δmax(χ l

ss + χmss

)

1 − δ2max

(A21)

(χ l

bp − χmbp

)− δmax

(χ l

bp + χmbp

)

1 − δ2max

< P(

glbp

)− P

(gm

bp

)<

(χ l

bp − χmbp

)+ δmax

(χ l

bp + χmbp

)

1 − δ2max

(A22)

when 0 < δmax < [(χ lss − χm

ss)/(χlss + χm

ss)] =[(χ l

bp − χmbp)/(χ

lbp + χm

bp)], we have (χ lss − χm

ss) − δmax(χlss +

χmss) > 0 and (χ l

bp − χmbp) − δmax(χ

lbp + χm

bp) > 0, then



P(glss) − P(gm

ss) > 0 and P(glbp) − P(gm

bp) > 0, which meansP(gl


ss) > P(gmss) in the Bernoulli case.

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Shichang Du (M’16) received the B.S. and M.S.E.degrees in mechanical engineering from the HefeiUniversity of Technology, Hefei, China, in 2000 and2003, respectively, and the Ph.D. degree in industrialengineering and management from Shanghai JiaoTong University, Shanghai, China, in 2008.

He was a Visiting Scholar with the Universityof Michigan, Ann Arbor, MI, USA, from 2006 to2007. He is an Associate Professor with ShanghaiJiao Tong University. His current research interestsinclude quality and reliability engineering, qual-

ity control with analysis of error flow, and monitoring and diagnosis ofmanufacturing process.

Rui Xu received the B.S. degree from the Schoolof Mechanical Engineering, Shanghai Jiao TongUniversity, Shanghai, China, in 2013, where she iscurrently pursuing master’s degree in industrial engi-neering and management.

Her current research interests include Markovmodeling and analysis of multistage manufacturingsystems.

Lin Li (M’12) received the B.S. degree in mechani-cal engineering from Shanghai Jiao Tong University,Shanghai, China, in 2001, and the M.S.E. degreesin mechanical engineering and industrial and opera-tions engineering and the Ph.D. degree in mechan-ical engineering from the University of Michigan,Ann Arbor, MI, USA, in 2003, 2005, and 2007,respectively.

He is currently an Associate Professor withthe Department of Mechanical and IndustrialEngineering and the Director of Sustainable

Manufacturing Systems Research Laboratory, University of Illinois atChicago, Chicago, IL, USA. His current research interests include sustain-able manufacturing systems, reliability engineering, intelligent maintenancesystems, and healthcare systems.

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