An absorbing Markov chain model for production systems with rework and scrapping

https://doi.org/10.1016/j.cie.2008.02.009Get rights and content

Abstract

This work models the flow of material through the production system as an absorbing Markov chain characterising the uncertainty due to scrapping and reworking. A realistic estimate of material requirement computed by the model enables better system design, MRP, capacity requirement planning, and inventory control. The work identifies production system parameters under scrapping and reworking, and accurately estimates the quantity of raw materials required. This being a probabilistic model can handle the problem even when data on some intangible costs are not available.

Introduction

This paper models the production flow of components as an absorbing Markov chain. Deterministic modelling is unrealistic, as all materials that start from raw material store do not reach the finished component stage due to scrapping and reworking. Hence, a stochastic process of a type called absorbing Markov chain has been adopted. The data required for such a model are (i) the relative frequency with which the item goes from one stage of production to another, and (ii) relative frequency of rework and scrap at various stages.

Markov models are widely used for manufacturing system modelling, in a variety of environment such as system repair and failure aspect (Foster & Garcia-Diaz, 1983); exponential service, failure, and repair process of machines (Gershwin and Berman, 1981, Hong and Seong, 1993); exponential failure and repair process for machines and material handling robot (Suliman, 2000); and performability analysis of flexible manufacturing systems under machine failure, and spare inventory and repair processes (Rupe & Kuo, 2001).

A discrete manufacturing system, where a work-part moves through the system and comes out as a component is modelled as Absorbing Markov Chain (AMC). The work-part is a raw material or semifinished part before the production operations. At every stage of production the part is subjected to inspection; if it does not conform to specifications, it is either scrapped or reworked. The reworked component undergoes inspection again. Fig. 1 shows a manufacturing process that requires three serial operations modelled as an AMC.

It can be seen that observations are made at ‘epochs’2 when the part transits from one state to another. Here, a state may represent raw material, particular normal operation, particular rework operation, scrap or finished goods. At any observation epoch, the part occupies a state, which is a discrete random variable X. As the parameter t (representing time) changes, the random variable X generates a random process {X(t):t  0}. This stochastic process with discrete state space and discrete values of the parameter t becomes a discrete time (first order) Markov chain when the transition from one state to the next depends only on the current state. Among the states some are transient and the others absorbing. A Markov chain with one or more absorbing states is known as absorbing Markov chain. An absorbing state is, as the name implies, one that endures. In other words, when a work-part reaches such a state, it never leaves the state. A scrapped work-part remains scrapped, and a finished work-part remains with no further changes.

The states along with the transition probability matrix constitute the Markov chain model. The transition probability pij is the probability that a work-part transits from state i to state j in one step. The statistical data available from the manufacturing system can be used for developing the transition probability matrix (P). The hypothetical statistical data related with scrap rate and rework rate for the above example are given in the Table 1 and the transition probability matrix generated from the data in Fig. 2. The model may also be represented as a graph (transition diagram) as shown in Fig. 3.

The probabilities shown in the transition probability matrix and in the transition diagram can be explained with the help of Table 1. The first data of the table shows the scrap rate of the incoming material after inspection, which is 0.3 percent. This indicates that, an item from the incoming material state (state 1) transits to the scrap state (state 10) with a probability of 0.003 and to the state representing normal turning (state 2) with a probability of 0.997. The data related with normal turning indicates that from state 2 there are three transitions to states 3, 5 and 10 with probabilities 0.96, 0.02 and 0.02, respectively, the sum of probabilities adding up to 1. The rework turning data indicates that there are two transitions from rework turning state (state 5), to reworked scrap state (state 9) and to normal drilling state (state 3) with probabilities 0.02 and 0.98, respectively. Similarly the other elements in the table can be interpreted.

The AMC of the production process can be modelled with two or three absorbing states. In an AMC with two absorbing states, one is for finished goods and the other for scrap. In an AMC with three absorbing states (as in the above example), one stands for finished goods and the others for scrap generated from normal and rework operations. These two types of models have been used extensively in literature (Davis and Kennedy, 1987, Viswanadham and Narahari, 1992, White, 1970). Models of AMC with three absorbing states is superior to that of two absorbing states as the former provides more information such as the amount of material reworked that became scrap. This information is useful for decisions related to inspection and quality control.

White (1970) presents a two absorbing states model for the production system and provides expected value and variance for the resources consumed by the process. These results are used to determine the amount of raw material to be scheduled through the production system to meet a known demand. This is a profit maximisation model.

Davis and Kennedy (1987) describe models of serial production system, which are developed in an evolutionary process, and they explain the usefulness when the production process is modelled as AMC. Each step in the evolution requires more data yielding additional useful information. They use models of both two and three absorbing states. The results derived include expected quantity of raw material required, estimation of equipment requirement, over-time calculation, and work-in-process inventory, which are useful for production planning and control applications.

The present work identifies an anomaly in the estimation of equipment requirement and presents another method of estimation. The present work concerns manufacturing lead-time estimation and the amount of raw material to be scheduled through the production system to meet finished component demand with certain service level. It also presents a cost minimisation approach.

Section snippets

Notations

The symbols used in the study are given below.

    s

    number of transient states

    r

    number of absorbing states

    Q

    matrix of transition probabilities between transient states

    R

    matrix of transition probabilities from transient states to absorbing states

    E

    matrix of expected number of times in (transient) state j, given starting state i

    A

    matrix of probabilities of absorption in (absorbing) state j, given starting state i

    G

    matrix of expected resource consumed in state j before absorption, given starting state i

    T

Properties of absorbing Markov chain

As an AMC has a mix of absorbing states and transient states, it will be advantageous to rearrange the transition probability matrix into the following form to obtain certain useful results (Ravindran et al., 1987, White, 1970).p=QR0I,whereQ=s×smatrixR=s×rmatrix0=r×szero matrixI=r×ridentity matrix

The results provided by White (1970) are as shown below.E=[I-Q]-1A=ERG=ETB=GZ

In a general AMC any one of the transient states can be the starting state. In the matrices E, A and G the rows represent

Some results for production system

When a production system is modelled as an AMC, there is little or no advantage in obtaining values for the above results for states other than the first transient state as starting state. This is due to the property that always a work-part starts from first transient state, which is usually raw material state. The results to be derived in this paper consider this property.

Illustration

The solution procedure is illustrated through the example given in Fig. 1. The matrices E and A are shown in Fig. 4, Fig. 5, respectively. An element eij of matrix E represents the mean number of times a transient state j is occupied for the initial state i. An element aij of the matrix A is the absorption probability which shows the fraction of the work-part that starts in the state i and ends in the absorbing state j. In the case of production system, the element a11 is sufficient to estimate

Service level criteria

As already stated, the production process can be considered as a Binomial experiment. Probability of success of the Binomial experiment is the probability that the work-part that started from the raw material state gets absorbed at finished work-part (component). (Raw material state represents the first transient state of the Markov chain of the production system). The probability of success is denoted in the previous section as a.

For a given number of products, the quantity of components

Closure

The Markov model adequately describes the production system under uncertainties due to scrapping and reworking. The analysis shows clearly the interaction between design and control decisions, and it provides an opportunity for the management to analyse quality-related problems. It reveals the necessity of collecting more data on quality control and operational issues. The information for planning and design such as MRP, capacity requirement planning and system design could be obtained from the

Acknowledgement

The authors are thankful to anonymous referees for their valuable suggestions to improve this version of the paper.

References (13)

There are more references available in the full text version of this article.

Cited by (26)

  • Reliability evaluation for a manufacturing network with multiple production lines

    2012, Computers and Industrial Engineering
    Citation Excerpt :

    The failure rate of each machine influences the capability of a manufacturing network and a machine failure leads to defective products, which might be reworked or scrapped. ( Liu, Kim, & Hwang, 2009; Pillai & Chandrasekharan, 2008). Thus, an important issue to be considered is how the reworking action affects the amount of output products in a manufacturing system.

  • Continuous materials requirements planning (CMRP) approach when order type is lot for lot and safety stock is zero and its applications

    2011, Applied Soft Computing Journal
    Citation Excerpt :

    In this paper, some other advantages and disadvantages of this approach will be pointed out. Almost in all of related researches, MRP system has been considered as a DMRP; since orders, demands, scheduled receipts etc., are defined in discrete time or distinguished time periods [4,5,9,13]. In this paper, we explain the weaknesses of a DMRP approach and the manner of applying CMRP approach in some problems.

View all citing articles on Scopus
1

Tel.: +91 422 2652226; fax: +91 422 2656274 (M.P. Chandrasekharan).

View full text