Abstract - In this study, the effects of time delays in the

stability of a production line system are investigated. The time delay, which may be caused by whether machine failure, human error or product change, is very important parameter in the production line. Production period can be also considered as pure time delay in the modeling of production systems. Continuous flow models are used represent the dynamics of production lines incorporated time delays in the model. The reference model for the study consists of three independent lines of which two of them are parallel to each other and the last one is tandem to those, with each having their own limited capacity buffers. Our goal is to determine the system's stability regions that are drawn in the space of time delays based on production rates in order to give decision makers a guideline how to balance production lines facing time delays.

Keywords - Production line, Stability, Time-Delay,

Kronecker Sum, CTCR

I. INTRODUCTION The competition in the market compels manufacturing business become more cost oriented. Due to the pressure of decrease in cost, the control and automation of the entire production system holds a crucial role. Utilizing automation systems and control structures decision makers have necessary tools to optimize production rates, buffer levels, inventory, transportation structures and countless other aspects. Researchers struggle to optimize all those different aspects and make them work together in a coherent way but the structure of the systems always keeps changing and makes it almost impossible to control the entire production system. Production systems are commonly modeled using various modeling techniques. These techniques include continuous time differential modeling, discrete event simulation, operation research and so on. Most of these are also used in modeling of logistics including supply chains and networking problems. [1,2]. Continuous time differential modeling, which is also preferred as the modeling method in this study, has been used since 1960’s. It is easier to be built and modified, and requires moderate computing power. It is satisfactory in throughput and real-time solutions, but it is deficient in modeling of changes in the system due to machine-human faults, lead times and uncertain production durations[1,2]. In the study, we utilized variable time-delays to incorporate these changes in the model. We determined

the values of delays, which sustains the stability of the system and then construct a map of time delays according to production rate. In the paper, we propose a case study model that consists of three production lines of which two of them are parallel to each other and the third one is tandem to those with each of them having their own limited capacity buffers. Lines working together in parallel are noted as the first and the second lines throughout the paper, which are the upper stream of the system. The last one is named as third line, and it is downstream (tandem) of our production system. First and second lines have infinite material supply and buffer of third line has unlimited demand. These kinds of production lines are well known in the industry, especially in automotive factories. The stability of the production line is compromised when there is a certain amount delay in the production of each line due to malfunction of the machines or unexpected error which may be either machine or human oriented. In this paper, we used Cluster Treatment of Characteristic Roots CTCR with Extended Kronecker Summation Method (EKSM) to investigate time delay effects on the stability of the production line. We noticed that stability switches occur at frequencies that is determined by the production speed of the line and sizing of the buffers placed in between the production lines can be achieved depending on the production speed and amount of the mid-products. In this study, our goal is to construct the map of the stability regions of the time delays which are inherently part of a production systems during the normal and anormal operation. In Section 2, we presented the general methods used when constructing a production line model with continuous flow models and present our production system model and how we construct it. In Section 3, we present the method to analyze the systems with time delays. In Section 4, the results of the implemented method on time delayed systems. Finally in last section, conclusions and discussions are revelaed.

II. MODELING AND PROBLEM STATEMENT In this section modeling of a production line (assembly line) with time delays using continuous flow models is given.The mathematical model is constructed by employing fluid flow models. This type of modeling has been widely used in various types of network systems particularly in supply chain-transportation researches. The physical parts of fluid flow system consist of two

Stability of Production Lines with Multiple Delays

Narthan Cemal Saadet1, Ali Fuat Ergenç 2 1 Mechatronics Research Center, Istanbul Technical University, Istanbul, Turkey

2Department of Control Engineering, Istanbul Technical University, Istanbul, Turkey (ali.ergenc@itu.edu.tr )

978-1-4577-0739-1/11/$26.00 ©2011 IEEE 1218

pipelines. One of them is for feed and the other one is for the discharge. There is a tank in the middle of these pipelines. The analogy between the assembly line and fluid flow is that feed pipe is considered as material flow into the system while discharge is demand and tank is the stock (buffer) level[1].

(1)

In (1) represent the production rate, the demand and the buffer (inventory) level respectively [3,4]. In our case study model, our material flows are supplied by production lines to their own buffers and discharge is the last assembly line. The model stated by (1) does not involve all the required variables, because we examine a system that is dynamic not only due to the input variable but also by means of the components of the system. A production line consists of humans, machines, unfinished-goods moving through the conveyor belts which affect the production. In addition to these, manager's decisions also affect the entire production. All those aspects of the production have their own dynamics and they changes during the period of time. Even (1) is sufficient for a system that has no interference during the process, it doesn’t include effects of the disturbances. All the disturbances coming from components of the system can be modeled as a time delay effect in the production.

(2)

(3)

(4)

In (2) the mathematical representation of a single production line with time delay effects of components is given. Laplace transformation of (2) is in (3)[5,3,4]. Here

is the controller to schedule production in terms of the level of the buffer, which transformation is shown in (4) where is our reference variable[6]. Modeling of network problems using continuous time modeling has been a subject for researches for a long time and there are plenty of works, which have been made by wide variety of disciplines. Studies on manufacturing systems and also traffic models focused on the effects of failures with single stage single buffer systems and serial production systems where machine and buffers are tandem to each other[7,8]. Meanwhile, the effects of time delay on stability of single stage serial supply chains and production inventory systems are studied[9,4,1]. Several control techniques such as PID(Proportional–Integral–Derivative), MPC (Model Predictive Control) are implemented to control the buffer levels and to stabilize the delayed systems. Although there are studies in tandem networks, to the authors' knowledge there is no example of parallel network studies on manufacturing or supply chains.

Fig. 1. Fluid Analogy of the Production System. General equation of is given as the following.

(5)

System equations of the proposed production system are established as:

(6)

In Fig.1, the fluid flow analogy of production systems is displayed. Fig.2 exhibits the corresponding continuous model of the analogy of the system in Fig.1. Both machines and production lines can be seen in both figures. The first and second lines, which are parallel to each other, have two machines per line. For the observation of the effects of delays, time delays are placed at the end of the lines after the second machines. This is appropriate for our purpose, to examine the overall effects of the delays on the entire line. For the sake of easiness of model, time delays in the first two lines are considered as similar and the delay in the third line different. Thus, the system has two independent delays both of which determine the stability zone of the production.

Fig. 2. Continuous Model of the Proposed Production System.

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The continuous model of the system, all the constants and reference variables are depicted in Fig.2. Here, and are the reference inputs for the buffer levels, K1 and K2 are the constants for setting reference values for each machine-buffer system. The final product of the total system is assemblage of three parts. The first piece is produced in the first line and used one of them was used per final product. The second piece is produced in the second line and used two of them per final product, so K2 is equal to 2*K1. In the model we have two complementary assumptions 1) it is assumed that all the pieces being produced are intact and no discard pieces are produced 2) If there is a faulty piece it is assumed as a time delay, which is determined by the speed of the production. We use two different reference values, because is for buffer levels of the system and is for stock level of finished goods. Even though the third line has a buffer, it is considered as finished goods stock. First machines of the first and second lines are assumed to be never starved and demand is always above zero . The characteristic equation of the production system which has governing equations (5) to (6) is stated below.

(7) The mathematical model of the production line stated above is classified as a linear time invariant time delayed system. The main goal of this study is to determine the safe operation zones of the system depending on the delays on the each line. As mentioned before, the source of the delay can be either machine or man oriented or just a faulty piece. We aim to resolve the stability problem of the production line depending on the delays in production process. In the following section a brief explanation for Extended Kronecker Summation Method for delay interval determination is given.

III. EXTENDED KRONECKER SUMMATION METHOD

Extended Kronecker Summation Method (EKSM) is a procedure utilized in CTCR method [10,11,12] to determine stability switching hypersurfaces , which are functions of the time delay values and parametric uncertainties of the system. The method is essentially based on the properties of the Kronecker summation of matrices. The systems we deal with are linear time invariant, retarded multiple time delayed systems (LTI-MTDS), general form of which is given as,

(8)

where , , , are matrices in , and the vector of time delays =

the elements of which are rationally independent from each other. Formally, we use boldface capital notation for vector and matrix quantities in the text. In the text, open unit disc, unit circle and outside of unit circle are referred as , respectively. Therefore, represents the entire complex plane. In addition to these notifications, the complex domain can be separated into to open left and open right half planes and imaginary axis. The characteristic equation of the system in (8) is

(9)

where is an degree polynomial in , 's

are quasi-polynomials in with the parametric uncertainties 's and all the delays except .

is the highest order of commensuracy of delay in the dynamics . contains terms with the highest degree of and they are the factors multiplying the representative exponential of the highest commensuracy of , i.e., . Definition 1: The stability posture of the system in (8) is determined by the number of characteristic roots of (9) in . This number is naturally a function of the delays and the parametric uncertainties, which are the parameters in (8). Wherever the number is equal to 0, the system is labeled as "stable''. For any stability switching a characteristic root must exist on . This is a direct result of the "root continuity'' argument in the parametric space [13,14]. Kronecker sum of two square matrices and induces the "eigenvalue addition'' character to the matrices [15,16]. We take advantage of this feature as discussed next in a definition and the theorem: Definition 2: Auxiliary Characteristic Equation ( ) of the system in (8), with is defined in [12] as follows :

C

(10)

Theorem 1: For the system given in (8) the following findings are equivalent: 1) A vector of -dimensional unitary complex numbers

, satisfies . 2)There exists at least one pair of imaginary characteristic roots, , of (9). 3)There exists a corresponding delay vector

and a parameter vector , where holds. Proof : The proof can be found in [12].

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Equation (10) is both necessary and sufficient condition for a point generating an

for a certain . Since this equation is completely free from the delays, and a function of and

, the procedure is now considerably simplified to find solutions of (10) with respect to certain . To

determine the imaginary characteristic roots of (9) one simply plugs such a and into (9) and solves for . These roots reveal the crossing frequencies we are interested in and form the set, i.e., (11) One then uses the individual components of to determine the respective delays which are

(12)

where implies the the delay value for various values.

IV. MAIN RESULTS The stable buffer level, which means limited deviation of the referans level, is the indicator of the stability of the production system. The stability is essential in production system so that product per unit time is constant and mid-buffer levels are kept in a certain range. In this study, where we modeled a production system as a flow of materials in a process, the stability of the system is determined by the number of the roots of the characteristic equation [7] those lie on the right half plane. Since the mathematical model (7) includes delay terms as the production time and possible malfunctions, detemination of the stability of the system is not a trivial task. We utilized CTCR method with EKSM to assert the stability switchings of the system w.r.t. delays in the system as well as the stability map variations due to a change of the production speed ( ). The stability of proposed system is investigated in a case study. A. Case Study The state space representation of the production system stated in (6) is

(13) where,

(14)

The characteristic equation (CE) of a production system with two feeder lines and an assembly line derived from (14) is printed below .

(15) where and represents production speed. Using canonical form a polynomial and Definition 2, (10). Notice that the characteristic equation of the system has commensurate delays as well as crosstalks between them. In order to achieve the stability map of the system we utilize Auxiliary Characteristic Equation ( ACE) of the system, defined in Definition 2 (10)

(16) The stability map of the system, which has characteristic equation (15), is depicted in Fig.3 using EKSM . As it is seen in the Fig.3 when production speed ( ) increases the stability region shrinks. This is an expected outcome of the stability determination. The system is more susceptible to the delays in production line when the speed is higher. The relation between the delay terms and the production speed is that delay value of stability switching is function of reciprocal of the production speed. The relation is presented in Fig.4.

Fig. 3 Stability Map of the System w.r.t Delays and variation.

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On the other hand, an interesting feature, the stability switching frequency ( crossing frequency) of the system is equal to unit production speed (5). This property of the system is due to the value imaginary axis solution of which is independent of the delay value that generates crossings.

Fig. 4. The Delay values of the Stability Switching w.r.t. variation.

V. CONCLUSION In the study, stability of a production line system with two feeder lines and one assembly line is investigated. The dynamics of the system represented using continuous flow models with time delays to incorporate production time, malfunction and unpredicted operation delays. The stability of the system is examined using CTCR with EKSM. The result of our investigation reveals that, the stability boundaries of the system w.r.t time delays are inversely proportional to production speed. Meanwhile, the oscillatory behaviour of the system when it is unstable is depent on the production speed as well. Thus, stability of the system can be sustained by controlling production speed in case of unavoidable delays in the production line. On the other hand, the levels of the buffers of the system presents oscillatory behavior when stability is lost. It is observed that the amplitude of the oscillations are important for determining the buffer size for the operation. Since determination of buffer size is an important issue when the decision makers design the

Fig. 5. Stability Switching Frequency of the System w.r.t. variation.

production line, the outcome of this study can be used as a guide for production planners.

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