# A Systematic Approach to Quality Oriented Product Sequencing for Multistage Manufacturing Systems

Authors: Faping Zhang; Shahid Ikramullah Butt
Journal: Mathematical Problems in Engineering
 (2016)

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## Abstract

Product sequencing is one way to reduce cost and improve product quality for multistage manufacturing systems (MMS). However, systematically evaluating the influence of product sequence on quality performance for MMS is still a challenge. By considering the rate of incoming conforming product, manufacturing system quality transition between batch to batch, and quality propagation along stages, this paper investigates the appropriate batch policies and product sequencing for MMS so that satisfied quality performance can be achieved. A model to analyze the relationship between the product sequencing and quality performance is conducted just by using the quality inspection data and the complex engineering knowledge used in the variation method is avoided. Based on Markov Chain processes methodology, quality performance is modeled as a function of transition states jointly determined by multistage condition, product sequencing, incoming part quality, and propagation of the rate of conforming products among multistage. Quality related batch strategies are discussed for optimal quality performance. Two kinds of quality efficiency are put forward to facilitate the modeling and the discussion. The results of the model will lead to guidelines for quality management in multistage manufacturing systems.


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## Body

## 1. Introduction

(A) Motivation. The quality performance of a multistage manufacturing system reflects the capability of the system delivering conforming quality features during manufacturing (i.e., to meet process specification) at each stage. However, the product quality is determined not only by the stability of the manufacturing processes, but also by the layout of process and product sequencing. It is desirable to conduct a systematic research that addresses quality assurance in all phases of product realization in terms of rate of confirming parts, which is the result of daily production quality inspection.Production sequence and batch size are the components of production operation mostly concerned in production scheduling. Improvement in these components is one way to reduce cost and also improve product quality for multistage manufacturing systems (MMS). For instance, during machining processes, fixtures are used to locate workpiece in a given location. Uncertainty would increase when adjusting fixture locator in bidirection than always in one direction during batch to batch production, which is affected by the change of product types from batch to batch. Hence different process variability and risk of product quality are introduced during production operation. Furthermore, for multistage machining processes, between the machining stages, there may be different adjustment requirements for fixtures at each stage, which impact the success of the fixtures adjustment and the product quality transition during the batch production transitions. Another example of the case is the double color (or multiple color) car painting process, where at least two painting stages will be used (depending on how colorful a car is). Each stage may have different color painting sequence according to production operation between the batch to batch changeovers. As we know, at each stage, the paint quality is strongly correlated to the number of available paint colors [1] and it may temporarily fluctuate with color change [2] and propagate to next stage.Although product quality is one of important issues in MMS due to changeover, the impact of production sequencing and batch policies on quality for a multistage manufacturing system with batch production has not been addressed. It is still a challenge to evaluate quality performance with production operation for MMS without the intensive engineering knowledge. On the other hand, various types of readily available data create tremendous opportunities for developing an evaluation methodology. For example, the first part of a batch will be totally inspected to ensure the successful transition during product changeover. Hence, this data can be used to evaluate transition concerning the product quality. This paper investigates the appropriate batch size and production sequence, so that the quality requirement is satisfied.The remainder of the paper is organized as follows: in Section1(B), a brief review of relevant literature is presented. In Sections 2 and 3, the problem statement is discussed in detail and quality performance modeling is formulated. Section 4 provides intensively evaluation of the MMS quality performance. In Section 5 an example and discussion are given. The conclusions are drawn in Section 6.(B) Literature Review. The evaluation of the performance of manufacturing system’s quality has been focused as an important research issue in quality control area. It has been approached from two perspectives. The first is to study the variation propagation along stages by means of the intensive usage of engineering knowledge for multistage manufacturing system. Lots of researchers have developed the methodology of Stream of Variation (SoV) to model the multistage variation accumulation [3–8] for machining and assembly process and the state space model is one of the frequently used tools. The SoV based quality modeling not only emphasizes on describing the forming of variation along the stages but also curbs the variation. Hence, intensive engineering knowledge is involved. Moreover, not enough research is carried out to study the quality issues related to the rate of incoming and outgoing conforming product with systematic approach, which is the actual requirement in quality performance evaluation of manufacturing system while there is abundance of historical production data which has not been given much attention previously.Secondly, full utilization of the measured data during manufacturing process inspection for quality control and then evaluating the quality performance of manufacturing system has also been studied. The measured data is usually used to quantify the rate of nonconformance of the outgoing units of the manufacturing system. The attribute control chart used for the statistical process control of the manufacturing process is one of the traditional approaches in this regard. To evaluate the quality performance of the manufacturing system with different setting, series of studies have been done [9–14]. Quality parameters of machines in serial production lines have been analyzed to trade off the machine efficiency [9, 10]. A revised learning model with time-varying learning rates was proposed for embedding learning effects into a multistage assembly/production setting for suppliers with the rate of nonconformance units along with the inspection sample rates [11]. Learning curve was also used to reduce the percentage of defective items per lot produced by manufacturing system with 100 per cent screening [12]. The above research is mainly concerned about the setting up of the machining system. The impact of manufacturing system configuration including the inspection policies, buffer dimensioning on the performance of the system in terms of reliability and productivity, product quality, capacity, scalability, and cost has also been discussed by some researchers [13–15].Actually, the input factors, such as the product sequencing and batching of the manufacturing system, also have a significant influence on the system quality performance. Being one of the most effective ways to improve production system efficiency, product sequencing has been studied extensively in the field of production scheduling. The scheduling problem considers a batch capacity constraint, sequence-dependent processing times, incompatible product families, additional resources, machine capability, and product sequence to achieve scheduling aim [16, 17]. Quality issue has not been considered much as constraint factor or the scheduling aim, although few researchers have combined quality issue with production scheduling [18, 19] for a special kind of the production process only. Series of researches [20, 21] have been conducted and a quantitative model based on Markov Chain to evaluate the quality performance of a flexible machining system was presented which is considered as a single manufacturing system. The work is mainly focused on the state transition properties of the single-stage between batch to batch operations. The research is primarily focused on one single-stage manufacturing system without incoming part quality. So this gives us an opportunity to conduct a thorough study on quality performance evaluation for MMS under different production operation.With the emergence of new manufacturing systems such as reconfiguration system with the dynamic and highly competitive marketplace demands, evaluation of quality performance from multistage aspect is becoming more and more crucial. Another aspect is to evaluate it from the system level to predict quality performance according to the rate of incoming conforming product to that of outgoing product in a statistical way, where the detailed engineering knowledge along the multistage manufacturing system is not concerned.


## 2. Problem Formulation

For a given multistage manufacturing system producing variety of products, the following assumptions are used to address the issues of the manufacturing system, product types, production sequence, and product quality.(1)
The multistage manufacturing system hasm serial stages and processes n different types of products. Each product type i is manufactured in a batch with batch size bi, bi≥1 and i=1,2,…,n.(2)
Forn different types of parts to be processed in a manufacturing system, there are n-1! kinds of product sequences, denoted as sl, l=1,…,n-1!. The products are introduced into the system with sequence sl=s1l,s2l,…,snl, where stl denotes the tth product type in the sequence sl, and t∈1,2,…,n, stl∈1,2,…,n.(3)
Every product enters the manufacturing system with a certain rate of conforming productγstl,j which will change along the stages.(4)
For each sequencesl, the multistage manufacturing system will work on product type stl for bstl parts before switching to product type st+1l. It is assumed that product type s1l is processed again after processing type snl.(5)
Thekth stage state of a multistage manufacturing system is in normal state osil,j,k or abnormal state fsil,j,k, i=1,…,n, j=1,…,bsil, when it is processing the jth part in the batch of the product type sil with good quality or with defects, respectively.(6)
The output part quality of thekth stage of a multistage manufacturing system is in good state gsil,j,k or defective state dsil,j,k, i=1,…,n, j=1,…,bsil, if it is processing the jth part in the batch of the product type sil with good quality or with defects, respectively. Thus, there are 2B, B=∑i=1nbi, states in the system for a given sequence, defined by the quality status, product type processed, and its position within a batch.(7)
When thekth stage is in normal state osil,j,k, sil=1,…,n, j=1,…,bsil-1, k=1,…,m, it has probabilities λsil,sil,k to transit to abnormal state fsil,j+1,k and 1-λsil,sil,k to normal state osil,j+1,k. Similarly, when the stage is in abnormal state fsil,j,k, sil=1,…,n, j=1,…,bsil-1, k=1,…,m, it can transit to normal state osil,j+1,k with probability μsil,sil,k and to abnormal state fsil,j+1,k with 1-μsil,sil,k probability.(8)
When thekth stage is processing the last part within a batch and in normal state osil,bsil,k, sil=1,…,n, j=1,…,bsil-1, k=1,…,m, it has probabilities λsi+1l,sil,k and 1-λsi+1l,sil,k to transit to states fsi+1l,1,k and osi+1l,1,k, respectively. Similarly, when the stage is in abnormal stage fsil,bsil,k, it has probabilities μsi+1i,sil,k and 1-μsi+1l,sil,k to transit to states osi+1l,1,k and fsi+1l,1,k, respectively.(9)
When thekth stage is in state osnl,bsnl,k, it has probabilities λs1l,snl,k and 1-λs1l,snl,k to transit to states fs1l,1,k and os1l,1,k, respectively. Similarly, when the kth stage is in state fsnl,bsnl,k, it has probabilities μs1l,snl,k and 1-μs1l,snl,k to transit to states os1l,1,k and fs1l,1,k, respectively. It is sure that all 0<λij<1, 0<μij<1, ∀i,j.(10)
Another assumption including the inspection policy is one hundred percent inspection which is error-free.LetPosil,j,k and Pfsil,j,k, sil=1,…,n, j=1,…,bsil-1, k=1,…,m, denote the probabilities that the kth stage is in state osil,j,k or fsil,j,k, respectively. Pgsil,j,k and Pdsil,j,k, sil=1,…,n, j=1,…,bsil-1, k=1,…,m, denote the probabilities that the kth stage is to output part quality state in gsil,j,k and dsil,j,k respectively. So Pgsil,j,m and Pdsil,j,m should be the probabilities of the system’s output part quality state: good or defective, which can also be denoted as Pgsil,j and Pdsil,j. Thus,(1)Pgsil,j=Pgsil,j,m,Pdsil,j=Pdsil,j,m.Then the overall quality performance of a multiple stage manufacturing system for a given sequencesil, that is, the probability to produce a good or a defective part in batch production, is defined as Pgbtl and Pdbtl, respectively and expressed mathematically as(2)Pgbtl=∑sil=1n∑j=1bsilPgsil,jPdbtl=∑sil=1n∑j=1bsilPdsil,j.


## 3. Quality Performance Modeling

Literature [22] represents two categories influencing factors of the output part quality of a multistage manufacturing system, that is, incoming part quality and manufacturing system conditions related to output quality such as the machine reliability. The Bernoulli quality model can be used to consider both conditions [23]. We can employ Bernoulli quality model to establish the relationship among incoming part quality, manufacturing system conditions, and outgoing product quality, from which we can also fulfill the quality performance evaluation of the multistage manufacturing system.


### 3.1. General Modeling

Consider a serial, multistage manufacturing process producing one type of product at a time. At each stage there are two factors contributing to the output quality of the machining stage, namely, incoming part quality and the stage transition rate as shown in Figure1.Figure 1
Schematic of single-stage qualities.So for thekth stage, we have (3)Pgsil,j,k=Pgsil,j,k-1·Posil,j,k,(4)Pdsil,j,k=Pgsil,j,k-1·Pfsil,j,k+Pdsil,j,k-1·Posil,j,k+Pdsil,j,k-1·Pfsil,j,k.Analogously, for thej-1 product in batch sil at the kth stage, we have(5)Pgsil,j-1,k=Pgsil,j-1,k-1·Posil,j-1,k,(6)Pdsil,j-1,k=Pgsil,j-1,k-1·Pfsil,j-1,k+Pdsil,j-1,k-1·Posil,j-1,k+Pdsil,j-1,k-1·Pfsil,j-1,k.From the assumptions (1)–(9), when 1<j≤bsil, that is, transition occurs within the batch without change product type, we have(7)Posil,j,k=Posil,j-1,k·1-λsil,sil,k+Pfsil,j-1,k·μsil,sil,k,(8)Pfsil,j,k=Posil,j-1,k·λsil,sil,k+Pfsil,j-1,k·1-μsil,sil,k.Then combining (3) to (8), we have(9)Pgsil,j,k=Pgsil,j,k-1·esil,j-1,k1-λsil,sil,k+1-esil,j-1,kμsil,sil,k.For the detailed conduction of (9), see Appendix A, where esil,j-1,k=Pgsil,j-1,k/Pgsil,j-1,k-1 denotes the stage to stage quality efficiency of the kth stage at processing sil batch and j-1 part. So we can get (10)esil,j,k=esil,j-1,k1-λsil,sil,k+1-esil,j-1,kμsil,sil,k.As Figure 2 shows the whole manufacturing system outgoing product quality; namely, the outgoing quality of the system for the sequence sil of jth part should be(11)Pgsil,j=Pgsil,j,m=1∑i=1nbsil∏k=1mesil,j-1,k1-λsil,sil,k+1-esil,j-1,kμsil,sil,k·γsil=1∑i=1nbsil∏k=1mesil,j,k·γsil=1∑i=1nbsilesil,j·γsil,where esil,j=∏k=1mesil,j,k is the manufacturing system’s stage to stage quality efficiency for jth part in the production sequence sil and Pgsil,j,m is outgoing quality of the final stage of the system for the sequence sil.Figure 2
Schematic of multistage qualities.Similarly, whenj=1 and i>1, transition is from the last part in previous batch to the first part in current batch. A product switch from type si-1l to type sil is involved. Hence, we can get(12)Pgsil,1=Pgsil,1,m=1∑i=1nbsilesil,1·γsil,where esil,1=∏k=1mesil,bsi-1l,k1-λsil,si-1l,k+1-esil,bsi-1l,kμsil,si-1l,k. Finally, if i=j=1, a product switch from type snl to type s1l occurs. Thus, it follows that(13)Pgs1l,1=Pgs1l,1,m=1∑i=1nbsiles1l,1·γs1l,where es1l,1=∏k=1mes1l,bsnl,k1-λs1l,snl,k+1-es1l,bsnl,kμs1l,snl,k. So, the product quality is denoted as Pgsil,j; (14)Pgbtl=∑i=1n∑j=1bsilPgsil,j=1∑i=1nbsil∑i=1n∑j=1bsilesil,j·γsil=1∑i=1nbsil∑i=1nbsil-1esil,2·γsil+∑i=2nesil,1·γsil+es1l,1·γs1l.In addition, the total probability is equal to 1:(15)∑i=1n∑j=1bsilPgsil,j+Pdsil,j=1.We assume that(16)δsil,sil,k=1-λsil,sil,k-μsil,sil,ksil=1,…,n,j=1,…,bsil-1,k=1,…,m.So (10) can be rewritten as(17)esil,j,k=esil,j-1,k1-λsil,sil,k+1-esil,j-1,kμsil,sil,k=δsil,sil,kesil,j-1,k+μsil,sil,k.Similarly,(18)esil,1,k=esi-1l,bsi-1l,k1-λsil,si-1l,k+1-esi-1l,bsi-1l,kμsil,si-1l,k=δsil,si-1l,kesi-1l,bsi-1l,k+μsil,si-1l,k,(19)es1l,1,k=esnl,bsnl,k1-λs1l,snl,k+1-esnl,bsnl,kμs1l,snl,k=δs1l,snl,kesnl,bsnl,k+μs1l,snl,k.VectorEk, Φk, and matrix Ψk are introduced as follows:(20)Ek=es1l,1,k,…,es1l,bs1l,k,es2l,1,k,…,es2l,bs2l,k,…,esnl,1,k,…,esnl,bsnl,kT,Φk=μs1l,snl,k,μs1l,s1l,k,…,μs1l,s1l,k,μs2l,s1l,k,μs2l,s2l,k,…,μsnl,sn-1l,k,μsnl,snl,k,…,μsnl,snl,kT,Ψk=0⋯000⋯0δs1l,snl,kδs1l,s1l,k⋯000⋯000⋱000⋯000⋯δs1l,s1l,k00⋯000⋯0δs2l,s1l,k0⋯000⋯00δs2l,s2l,k⋯000⋯000⋱000⋯000⋯δsnl,snl,k0.So (16), (17), and (18) can be rewritten as (21)Ek=ΨkEk+Φk.It follows that(22)Ek=I-Ψk-1Φk.Note that the inverse always exists due to the fact that an irreducible Markov Chain with finite number of states has a unique solution. Equation (19) can be used to calculate stage quality efficiency at each stage. We can reorganize the Ek as diagonal matrix as follows:(23)Ek=es1l,1,k0000es1l,2,k0000⋱0000esnl,bsnl,k,Ek-1=es1l,1,k-10000es1l,2,k-10000⋱0000esnl,bsnl,k-1.The rate of incoming conforming quality of each part is denoted as(24)Γsil=γs1l,10000γs1l,20000⋱0000γsnl,bsnl.Introduce vectors Xk as follows to represent the good part probability of the kth stage. (25)Xk=Pgs1l,1,k,…,Pgs1l,bs1l,k,Pgs2l,1,k,…,Pgs2l,bs2l,k,…,Pgsnl,1,k,…,Pgsnl,bsnl,kT.SoXk=1/∑i=1nbsilEkΓsil=1/∑i=1nbsilI-Ψk-1ΦkΓsil.The final probability of good parts should be equal to the outgoing good parts of the lastmth stage:(26)X=1∑i=1nbsil∏k=1mEkΓsil=1∑i=1nbsil∏k=1mI-Ψk-1ΦkΓsil.Therefore, the probability of good partsPgbtl is evaluated as follows.Theorem 1.
Under assumptions(1)–(9), the probability of good parts Pgbtl is calculated by (27)Pgbtl=∑i=1Bxi,where B=∑i=1nbi and xi is element of X and can be solved from (21) and thus(28)Pgbtl=∑i=1nbsil-1∏k=1mδsil,sil,kesil,j-1,k+μsil,sil,k·γsil∑i=1nbsil+∑i=2n∏k=1mδsil,si-1l,kesi-1l,bsi-1l,k+μsil,si-1l,k·γsil+∏k=1mδs1l,snl,kesnl,bsnl,k+μs1l,snl,k·γs1l∑i=1nbsil.Many manufacturing plans accumulate a lot of production data including all transition probabilities. Then, using these probabilities, Theorem1 can be used to evaluate the product quality.


## 3.1. General Modeling

Consider a serial, multistage manufacturing process producing one type of product at a time. At each stage there are two factors contributing to the output quality of the machining stage, namely, incoming part quality and the stage transition rate as shown in Figure1.Figure 1
Schematic of single-stage qualities.So for thekth stage, we have (3)Pgsil,j,k=Pgsil,j,k-1·Posil,j,k,(4)Pdsil,j,k=Pgsil,j,k-1·Pfsil,j,k+Pdsil,j,k-1·Posil,j,k+Pdsil,j,k-1·Pfsil,j,k.Analogously, for thej-1 product in batch sil at the kth stage, we have(5)Pgsil,j-1,k=Pgsil,j-1,k-1·Posil,j-1,k,(6)Pdsil,j-1,k=Pgsil,j-1,k-1·Pfsil,j-1,k+Pdsil,j-1,k-1·Posil,j-1,k+Pdsil,j-1,k-1·Pfsil,j-1,k.From the assumptions (1)–(9), when 1<j≤bsil, that is, transition occurs within the batch without change product type, we have(7)Posil,j,k=Posil,j-1,k·1-λsil,sil,k+Pfsil,j-1,k·μsil,sil,k,(8)Pfsil,j,k=Posil,j-1,k·λsil,sil,k+Pfsil,j-1,k·1-μsil,sil,k.Then combining (3) to (8), we have(9)Pgsil,j,k=Pgsil,j,k-1·esil,j-1,k1-λsil,sil,k+1-esil,j-1,kμsil,sil,k.For the detailed conduction of (9), see Appendix A, where esil,j-1,k=Pgsil,j-1,k/Pgsil,j-1,k-1 denotes the stage to stage quality efficiency of the kth stage at processing sil batch and j-1 part. So we can get (10)esil,j,k=esil,j-1,k1-λsil,sil,k+1-esil,j-1,kμsil,sil,k.As Figure 2 shows the whole manufacturing system outgoing product quality; namely, the outgoing quality of the system for the sequence sil of jth part should be(11)Pgsil,j=Pgsil,j,m=1∑i=1nbsil∏k=1mesil,j-1,k1-λsil,sil,k+1-esil,j-1,kμsil,sil,k·γsil=1∑i=1nbsil∏k=1mesil,j,k·γsil=1∑i=1nbsilesil,j·γsil,where esil,j=∏k=1mesil,j,k is the manufacturing system’s stage to stage quality efficiency for jth part in the production sequence sil and Pgsil,j,m is outgoing quality of the final stage of the system for the sequence sil.Figure 2
Schematic of multistage qualities.Similarly, whenj=1 and i>1, transition is from the last part in previous batch to the first part in current batch. A product switch from type si-1l to type sil is involved. Hence, we can get(12)Pgsil,1=Pgsil,1,m=1∑i=1nbsilesil,1·γsil,where esil,1=∏k=1mesil,bsi-1l,k1-λsil,si-1l,k+1-esil,bsi-1l,kμsil,si-1l,k. Finally, if i=j=1, a product switch from type snl to type s1l occurs. Thus, it follows that(13)Pgs1l,1=Pgs1l,1,m=1∑i=1nbsiles1l,1·γs1l,where es1l,1=∏k=1mes1l,bsnl,k1-λs1l,snl,k+1-es1l,bsnl,kμs1l,snl,k. So, the product quality is denoted as Pgsil,j; (14)Pgbtl=∑i=1n∑j=1bsilPgsil,j=1∑i=1nbsil∑i=1n∑j=1bsilesil,j·γsil=1∑i=1nbsil∑i=1nbsil-1esil,2·γsil+∑i=2nesil,1·γsil+es1l,1·γs1l.In addition, the total probability is equal to 1:(15)∑i=1n∑j=1bsilPgsil,j+Pdsil,j=1.We assume that(16)δsil,sil,k=1-λsil,sil,k-μsil,sil,ksil=1,…,n,j=1,…,bsil-1,k=1,…,m.So (10) can be rewritten as(17)esil,j,k=esil,j-1,k1-λsil,sil,k+1-esil,j-1,kμsil,sil,k=δsil,sil,kesil,j-1,k+μsil,sil,k.Similarly,(18)esil,1,k=esi-1l,bsi-1l,k1-λsil,si-1l,k+1-esi-1l,bsi-1l,kμsil,si-1l,k=δsil,si-1l,kesi-1l,bsi-1l,k+μsil,si-1l,k,(19)es1l,1,k=esnl,bsnl,k1-λs1l,snl,k+1-esnl,bsnl,kμs1l,snl,k=δs1l,snl,kesnl,bsnl,k+μs1l,snl,k.VectorEk, Φk, and matrix Ψk are introduced as follows:(20)Ek=es1l,1,k,…,es1l,bs1l,k,es2l,1,k,…,es2l,bs2l,k,…,esnl,1,k,…,esnl,bsnl,kT,Φk=μs1l,snl,k,μs1l,s1l,k,…,μs1l,s1l,k,μs2l,s1l,k,μs2l,s2l,k,…,μsnl,sn-1l,k,μsnl,snl,k,…,μsnl,snl,kT,Ψk=0⋯000⋯0δs1l,snl,kδs1l,s1l,k⋯000⋯000⋱000⋯000⋯δs1l,s1l,k00⋯000⋯0δs2l,s1l,k0⋯000⋯00δs2l,s2l,k⋯000⋯000⋱000⋯000⋯δsnl,snl,k0.So (16), (17), and (18) can be rewritten as (21)Ek=ΨkEk+Φk.It follows that(22)Ek=I-Ψk-1Φk.Note that the inverse always exists due to the fact that an irreducible Markov Chain with finite number of states has a unique solution. Equation (19) can be used to calculate stage quality efficiency at each stage. We can reorganize the Ek as diagonal matrix as follows:(23)Ek=es1l,1,k0000es1l,2,k0000⋱0000esnl,bsnl,k,Ek-1=es1l,1,k-10000es1l,2,k-10000⋱0000esnl,bsnl,k-1.The rate of incoming conforming quality of each part is denoted as(24)Γsil=γs1l,10000γs1l,20000⋱0000γsnl,bsnl.Introduce vectors Xk as follows to represent the good part probability of the kth stage. (25)Xk=Pgs1l,1,k,…,Pgs1l,bs1l,k,Pgs2l,1,k,…,Pgs2l,bs2l,k,…,Pgsnl,1,k,…,Pgsnl,bsnl,kT.SoXk=1/∑i=1nbsilEkΓsil=1/∑i=1nbsilI-Ψk-1ΦkΓsil.The final probability of good parts should be equal to the outgoing good parts of the lastmth stage:(26)X=1∑i=1nbsil∏k=1mEkΓsil=1∑i=1nbsil∏k=1mI-Ψk-1ΦkΓsil.Therefore, the probability of good partsPgbtl is evaluated as follows.Theorem 1.
Under assumptions(1)–(9), the probability of good parts Pgbtl is calculated by (27)Pgbtl=∑i=1Bxi,where B=∑i=1nbi and xi is element of X and can be solved from (21) and thus(28)Pgbtl=∑i=1nbsil-1∏k=1mδsil,sil,kesil,j-1,k+μsil,sil,k·γsil∑i=1nbsil+∑i=2n∏k=1mδsil,si-1l,kesi-1l,bsi-1l,k+μsil,si-1l,k·γsil+∏k=1mδs1l,snl,kesnl,bsnl,k+μs1l,snl,k·γs1l∑i=1nbsil.Many manufacturing plans accumulate a lot of production data including all transition probabilities. Then, using these probabilities, Theorem1 can be used to evaluate the product quality.


## 4. Quality Oriented Product Sequencing

Product sequencing does impact on quality performance of manufacturing systems. Some researchers have addressed this issue for individual stage production [24]. They approached this issue for car body painting of color changeover [25]. They assumed that these works imply that color changes do impact paint quality but their work is mainly focused on the individual stage production.For multistage manufacturing systems research is not conducted in the same way. Actually it is also true that product sequencing has a certain impact on quality performance of multistage manufacturing system.


### 4.1. Optimal Product Sequences for General Cases

For product sequencing, we want to know which sequence will have the best quality performance under the known transition data and the rate of incoming conforming product. From the above assumptions, we have the following corollary.Corollary 2.
Under assumptions(1)–(9), the optimal production sequence is the sequence that satisfies(29)minSil∑i=1n∑j=1bsilesil,j-γsil.It is easy to prove that if two pieces of data have identical sum, then the smaller the range of the two data, the bigger the multiple of the two data. We also know that the optimal product sequence is the sequence to make thePgbtl largest. Namely, the bigger the rate of incoming product conformance γsil is, the larger the corresponding system quality transition esil,j=∏k=1mej-1,k1-λsil,sil,k+1-ej-1,kμsil,sil,k is and both of them should have the same sequence, namely, for small to large or from large to small.It is easy to see that the transition probabilities within batches do not play a role in searching for optimal value because the value ofesil,j-γsil is identical in one batch for j≠1; thus only every esil,1-γsil is different. Hence, we have the following corollary.Corollary 3.
Under assumptions(1)–(9), the optimal production sequence is the sequence that satisfies(30)minSil∑i=1nesil,1-γsil.


### 4.2. Sequence and Quality Efficiency

Quality efficiency, a very important quality performance index for multistage manufacturing system, is not only determined by system properties but also affected by the products sequence. Although quality efficiency has been studied intensively but not in multistage manufacturing system and at the same time the effect on sequencing is not considered [20]. The aim of this section is to investigate the relationship between sequence and quality efficiency.Definition 4 (batch to batch quality transition efficiency).
At thekth stage, it is defined as(31)Tsil,sil,k=μsil,sil,kλsil,sil,k+μsil,sil,ksil=1,…,n,j=1,…,bsil-1,k=1,…,m.Corollary 5.
Under assumptions(1)–(9), the following statement holds(32)esil,j,k>esil,j-1,kiffTsil,sil,k>esil,j-1,k.Proof.
See AppendixB.This corollary indicates that if we know the former part’s quality efficiency at the stage and the stage quality transition efficiency between batch to batch is higher than that, then the outgoing part quality efficiency should be larger than the former part quality efficiency at this stage. This can be used to predict product quality at thekth stage if we know the former batch quality efficiency at the kth stage and the stage quality transition efficiency.


## 4.1. Optimal Product Sequences for General Cases

For product sequencing, we want to know which sequence will have the best quality performance under the known transition data and the rate of incoming conforming product. From the above assumptions, we have the following corollary.Corollary 2.
Under assumptions(1)–(9), the optimal production sequence is the sequence that satisfies(29)minSil∑i=1n∑j=1bsilesil,j-γsil.It is easy to prove that if two pieces of data have identical sum, then the smaller the range of the two data, the bigger the multiple of the two data. We also know that the optimal product sequence is the sequence to make thePgbtl largest. Namely, the bigger the rate of incoming product conformance γsil is, the larger the corresponding system quality transition esil,j=∏k=1mej-1,k1-λsil,sil,k+1-ej-1,kμsil,sil,k is and both of them should have the same sequence, namely, for small to large or from large to small.It is easy to see that the transition probabilities within batches do not play a role in searching for optimal value because the value ofesil,j-γsil is identical in one batch for j≠1; thus only every esil,1-γsil is different. Hence, we have the following corollary.Corollary 3.
Under assumptions(1)–(9), the optimal production sequence is the sequence that satisfies(30)minSil∑i=1nesil,1-γsil.


## 4.2. Sequence and Quality Efficiency

Quality efficiency, a very important quality performance index for multistage manufacturing system, is not only determined by system properties but also affected by the products sequence. Although quality efficiency has been studied intensively but not in multistage manufacturing system and at the same time the effect on sequencing is not considered [20]. The aim of this section is to investigate the relationship between sequence and quality efficiency.Definition 4 (batch to batch quality transition efficiency).
At thekth stage, it is defined as(31)Tsil,sil,k=μsil,sil,kλsil,sil,k+μsil,sil,ksil=1,…,n,j=1,…,bsil-1,k=1,…,m.Corollary 5.
Under assumptions(1)–(9), the following statement holds(32)esil,j,k>esil,j-1,kiffTsil,sil,k>esil,j-1,k.Proof.
See AppendixB.This corollary indicates that if we know the former part’s quality efficiency at the stage and the stage quality transition efficiency between batch to batch is higher than that, then the outgoing part quality efficiency should be larger than the former part quality efficiency at this stage. This can be used to predict product quality at thekth stage if we know the former batch quality efficiency at the kth stage and the stage quality transition efficiency.


## 5. An Example and Discussion

### 5.1. An Example

The method put forward in previous sector has been used to evaluate paint quality at an automotive paint shop. Three types of double color cars are painted in one production line with two painting booths, which can be seen as a two-stage manufacturing system. The main color is always painted in the first painting booth and the minor color in the second. The color of the three double color cars is listed in Table1.Table 1
The color of the three double color cars.
Colors Car 1 Car 2 Car 3 Main color White Black Gray Minor color Gray White BlackOf course the quality transition probabilities from the different color differ. Based on historical data, transition probabilitiesλij and μij are calculated. Table 2 lists the transition probabilities of the two booths. It has been shown that λij is usually small and μij is large. Since the operation has no concern about quality performance of the system, it can be seen as the random organized production sequencing. Under that situation, numerical experiments are carried out to simulate the quality performance by randomly choosing the product sequence. Here we assume that the rate of incoming conforming car white body is identical. We obtain that the average quality ratio is only about 0.75. There is only 0.57% difference compared with the actual data (the data has been changed because of the confidentiality).Table 2
The transition parameters.
Transition Booth 1 Booth 2 λ μ λ μ White Black 0.08 0.85 0.04 0.91 White Gray 0.08 0.89 0.01 0.92 Black White 0.1 0.9 0.07 0.86 Black Gray 0.1 0.87 0.09 0.88 Gray White 0.1 0.7 0.07 0.9 Gray Black 0.03 0.95 0.03 0.92Then, the transition data has been used to investigate the impact of production operation on quality performance of the system. Using the above method, an optimal sequence car 2-car 1-car 3 is obtained. The quality rate is 0.866, and 15.9% quality improvement is achieved. Here although the improvement is not too large but it should be noted that this improvement is achieved only by changing the production sequence, which is almost with no cost.There are two kinds of the product sequencing to produce the three types of cars in total. They are sequence car 1-car 2-car 3 and car 1-car 3-car 2, respectively. Quality performances of the two sequences arePgbt1=0.747 and Pgbt2=0.866, respectively.It is easy to see that the product sequence car 1-car 3-car 2 has a better quality performance than that of the sequence of car 1-car 2-car 3 even for batch policy and this point has been verified from the actual production data. We wanted to know why there is a big difference of the rate of conforming output product quality efficiency and this change is the result of different transaction state from batch to batch. The average stage quality efficiency is 0.817 and 0.921 for the sequence one and 0.91 and 0.961 for the second sequence, respectively. The second sequence has 11.4% and 4.3% higher stage efficiency than that of the first sequence because of the rate of the conforming input product which is identical for the two sequences. The second sequence has a smaller difference between the rate of the conforming input product and the stage efficiency which is the same as Corollary2 showing that the second sequence should have a high rate of the conforming output product.


### 5.2. Discussion

In this research, we used the rate of conforming product to represent the incoming and outgoing quality and probability of system state as good or bad. The rate of conforming product is referred to as the probability of the good part. Another factor concerned for MMS is the cost; furthermore quality related cost should be involved in the model. For example, dissimilar part has different cost; optimal production operation should minimize the quality related cost defined as the quality related cost risk. This goal can be achieved by the following equation:(33)minsilRq=∑i=1n∑j=1bsilPdsil,jwsil,jsil=1,…,n,j=1,…,bsil-1,k=1,…,m.Here Rq denotes the quality related cost risk and wsil,j denotes the one part cost of the product type i in sil sequence.


## 5.1. An Example

The method put forward in previous sector has been used to evaluate paint quality at an automotive paint shop. Three types of double color cars are painted in one production line with two painting booths, which can be seen as a two-stage manufacturing system. The main color is always painted in the first painting booth and the minor color in the second. The color of the three double color cars is listed in Table1.Table 1
The color of the three double color cars.
Colors Car 1 Car 2 Car 3 Main color White Black Gray Minor color Gray White BlackOf course the quality transition probabilities from the different color differ. Based on historical data, transition probabilitiesλij and μij are calculated. Table 2 lists the transition probabilities of the two booths. It has been shown that λij is usually small and μij is large. Since the operation has no concern about quality performance of the system, it can be seen as the random organized production sequencing. Under that situation, numerical experiments are carried out to simulate the quality performance by randomly choosing the product sequence. Here we assume that the rate of incoming conforming car white body is identical. We obtain that the average quality ratio is only about 0.75. There is only 0.57% difference compared with the actual data (the data has been changed because of the confidentiality).Table 2
The transition parameters.
Transition Booth 1 Booth 2 λ μ λ μ White Black 0.08 0.85 0.04 0.91 White Gray 0.08 0.89 0.01 0.92 Black White 0.1 0.9 0.07 0.86 Black Gray 0.1 0.87 0.09 0.88 Gray White 0.1 0.7 0.07 0.9 Gray Black 0.03 0.95 0.03 0.92Then, the transition data has been used to investigate the impact of production operation on quality performance of the system. Using the above method, an optimal sequence car 2-car 1-car 3 is obtained. The quality rate is 0.866, and 15.9% quality improvement is achieved. Here although the improvement is not too large but it should be noted that this improvement is achieved only by changing the production sequence, which is almost with no cost.There are two kinds of the product sequencing to produce the three types of cars in total. They are sequence car 1-car 2-car 3 and car 1-car 3-car 2, respectively. Quality performances of the two sequences arePgbt1=0.747 and Pgbt2=0.866, respectively.It is easy to see that the product sequence car 1-car 3-car 2 has a better quality performance than that of the sequence of car 1-car 2-car 3 even for batch policy and this point has been verified from the actual production data. We wanted to know why there is a big difference of the rate of conforming output product quality efficiency and this change is the result of different transaction state from batch to batch. The average stage quality efficiency is 0.817 and 0.921 for the sequence one and 0.91 and 0.961 for the second sequence, respectively. The second sequence has 11.4% and 4.3% higher stage efficiency than that of the first sequence because of the rate of the conforming input product which is identical for the two sequences. The second sequence has a smaller difference between the rate of the conforming input product and the stage efficiency which is the same as Corollary2 showing that the second sequence should have a high rate of the conforming output product.


## 5.2. Discussion

In this research, we used the rate of conforming product to represent the incoming and outgoing quality and probability of system state as good or bad. The rate of conforming product is referred to as the probability of the good part. Another factor concerned for MMS is the cost; furthermore quality related cost should be involved in the model. For example, dissimilar part has different cost; optimal production operation should minimize the quality related cost defined as the quality related cost risk. This goal can be achieved by the following equation:(33)minsilRq=∑i=1n∑j=1bsilPdsil,jwsil,jsil=1,…,n,j=1,…,bsil-1,k=1,…,m.Here Rq denotes the quality related cost risk and wsil,j denotes the one part cost of the product type i in sil sequence.


## 6. Conclusion

Quality performance is one of the most important aspects of the manufacturing systems. With the emergence of new manufacturing systems such as reconfiguration of a system and the dynamic and highly competitive marketplace demands, quality performance evaluation from multiaspect is becoming more and more vital. One of the aspects is to evaluate it from system level, namely, to predict quality performance from the rate of incoming conforming product to that of outgoing product in a statistical manner, where the detailed engineering knowledge along the multistage manufacturing system is not concerned. Based on the Markovian Chain model this paper approaches quality performance evaluation of multistage manufacturing system by considering the rate of incoming conforming product, production sequencing, and quality transition between batch to batch and quality propagation along the multistage. Different production policy is compared and optimal production sequencing is given. Two kinds of quality efficiency are put forward to facilitate the modeling and the discussion. Research shows optimal sequences should be such that the rate of incoming conforming product and the corresponding system quality transition have the same sequence, whereas batch size has minor influence on quality performance. The results of the research lead to the guidelines of quality management in multistage manufacturing systems.The future research works in this direction areas are as follows.(1)
There could be situation that under bad machine state and defective incoming part stage can still produce a good part with small probability. Another considerable factor should be that under good machine state and good incoming part quality, there will be a small probability to produce bad quality part according to the stage process capability although the probabilities may be very small but those situations should be considered in the model.(2)
Configuration of manufacturing system has definitely a significant effect on system quality performance. It is needed to conduct research to unveil the relationship between manufacturing system configuration and quality performance, to know what kind of the configuration has a better performance than others.(3)
Only considering the rates of nonconforming units is not good enough for the research of multistage manufacturing system, which is one limitation of the proposal. Since the variation value of the quality character is more sensitive to the assignable causes of the system, the next study is to combine the variation value with the Markovian Chain to depict the quality performance of the manufacturing system.



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*Source: 9270345-2016-12-22.xml*