Analysis and Approximation of a JIT Production Line* Scott Jordan El~~wiml Engineering and Computer Science Uniwrsity of colifornia, k k e l q . CA W720

ABSTRACT Production lines arc often modeled as queueing networks with finite inventory between each stage. Little is known, however, about the average produdion rate and inventory levels when the service distribution at each stage is normal. This paper approximates the service distribution using iterative methods rather than simulation. The results show that iterative methods are useful when the problem is small and that approximation of the service distribution, by another distribution with the same mean and variance. is valid for steady-state results such as average production rate or average inventory level.

Subject Amzc Inventory Management and Queueing Networks

INTRODUCTION

Recently, industrial engineers and management have been looking at the Jap- anese production technique known as “just-in-time with kanban” (JIT). This tech- nique, thought to account in part for the Japanese manufacturing success, attempts to limit inventory to minimum levels by allowing a fixed maximum inventory level between each production stage This system is implemented by using a bin for each allowed unit of inventory with a card attached called a kanban (pronounced kahn- bahn). The typical American system, in contrast, has no such maximum produc- tion stage level; instead each stage operates on its own schedule.

A JIT system is based on the notion of maximizing profit in a manufacturing setting by balancing inventory costs with production revenues: i n m i n g the number of kanbans allows a higher average production rate by reducing the number of idle periods at each stage, but results in a higher average inventory level. Of interest to JIT. then, is the optimal number of kanbans.

A QUEUEING MODEL

A queueing network model (see Figure 1) of the JIT system must incorporate two features: a maximum inventory level between each stage and randomness in the production time at each stage The number of kanbans is determined by adding a “waiting room” to the front of each queue. At most this waiting room can hold the number of items equal to the number of kanbans. I f this waiting room is full, the previous work station must halt production until a space becomes available. The randomness of production times is accomplished by specifying a distribution of these times, known as the service distribution of the queue This paper will inves- tigate production lines with normal service distributions.

Figure 2 shows a queueing model for a two-line, two-stage system. Room 1 represents one kanban and Room 2 represents two kanbans. Station 1 must cease production if Room 1 is full, and Station 2 must cease production if Room 2 is full. Station 3 takes one item from Room 1 (an “A” part) and two items from Room 2 (“B” parts) and assembles them into the finished product. This assembly pro- cess, and the service distributions, will be incorporated into the model in the n u t section. (See [3] or [7] for more information on queueing models.)

‘Tlus paper was supported by the Fannie and John Hertz Foundation.

672

19881 Jordan 673

Figure 1: An example of a queueing model for a three kanban system.

semi-finished products finished

products

I l l - - - - @

Station I 3 kanbans Stdtlon 2

Figure 2 A queueing model for a multiline system.

raw pans “A“ P M S

IlIll5lled

Stailon 3 raw pans

Station 2 I b m n 2

To narrow the scope to a reasonable degree. two assumptions will be made about the queueing network. First, the supply of raw parts is assumed to be inexhaustable; or, in technical terms, the first queue is assumed to be in saturation. Second, the service distributions at each stage are assumed to be equal. This is admittedly an ideal situation, but these assumptions can be relaxed in future work. Most, but not all, of the work to follow also deals with the relatively simple case of two stages.

Approaches: Simulation and Markov Chain Models

There are many approaches to analyzing this queueing network. Huang, Rees, and mylor [6] simulated a multiline, multistage process. Here a Markov chain model will be used to analyze the queueing network. The purposes are twofold: to see how far such inherently more analytical methods can be extended and to check on some questionable results in [6].

The queueing network can be modeled as a Markov chain by including in the state space all we need to know to predict the evolution of the network: the number of units in inventory at each stage and the amount of service provided on the cur- rent unit at each stage. Figure 3 shows this information for the two-line, two-stage example introduced in the previous section. We must record five pieces of infor- mation: the number of A and I3 parts, and the service provided on the current unit at Stations I , 2, and 3. These first two are shown in the first two rows in Figure 3, and include work-in-progress in Station 3. For instance, the number of B parts may be any integer from zero (Room 2 empty and Station 3 idle) to four (Room 2 full and Station 3 busy). Allowed transitions are indicated by arrows; for example, the number of “A” parts must decrease by two since Station 3 uses two at a time. The last three pieces of information, the service provided at the three stations, are shown in the last three rows of Figure 3. Discrete time rather than continuous time is used for the service time distribution so that there are a finite number of “states” in any one row. ( I f the real-time interval represented by a discrete time step is kept

614 &cision Sciences [Vol. 19 ,

Figurn 3: A Markov chain model for a multiline system.

small, as compared to the mean and standard deviation of the service distribution, using discrete time presents no problems.) Arrows again show allowable transitions.

With these five pieces of information, the multiline, multistage structure, as well as the assembly process, is now built into the model. The randomness is incor- porated by specifying the probability of each allowable transition in the last three rows in Figure 3-that is, the probability of finishing given the amount of service already provided. The synchronization inherent in the kanban system (and especially in the assembly stage) is incorporated by logical restrictions on the allowable t m - sitions in the first two rows. For example, a decrease in the number of A parts must accompany a decrease in the number of B parts and a transition to State I in Row 5 , thus signifying completion of assembly at Station 3. (See (21 or (131 for more infor- mation on Markov chain models.)

Markov chain models suffer from a "state space" explosion problem. In the example above, we already have 3(5)(48)(48)(48) states, if max=48, and this number would grow exponentially with the number of stages. To decrease the number of states, the service distribution is often approximated by an aponential distribution with the same mean service time, or in discrete time by a geometric distribution. These distributions have the unique property that, at any time, the time needed to finish the service on a unit is independent of the amount of service provided to that unit so far. This allows us not to keep track of the amount of service pro- vided so far, and the number of states thus decreases dramatically. (In our exam- ple, we would now m u i r e only 3(5) states.) Furthermore, some closed-form results are available for queueing networks with aponential distributions (see [7] and [12]).

Another distribution that reduces the number of states, but may approximate the true service distribution more accurately, is the Erlang distribution, with param- eters chosen to match the true service distribution's mean and variance. (The discrete counterpart is the negative binomial distribution.) For this distribution too. there are some closed-form results (see [ I I D .

19881 Jordan 675

Four different distributions will be used in the following work. Each has a mean of 48 minutes, so ideally we would produce ten units per day. The first two are normal distributions, with standard deviations equal to one-tenth of the mean and to one-half of the mean, respectively called narrow normal and wide normal. The third is an erlang distribution, with four stages so that it has standard devia- tion equal to one-half its mean, meant to approximate the wide normal. (To obtain a standard deviation of one-tenth of the mean with an erlang would require 100 stages and many more states than the narrow normal it would be meant to approx- imate.) The fourth is an exponential distribution, meant to approximate both of the normal.

The state space for all but the exponential distribution are large enough to restrict analysis to a one-line, two-stage production system. With an exponential distribution, however, we can analyze the multiline, multistage system simulated by Huang et al. [a].

Approaches: Analysis of the Markov Chain

To calculate the expected daily production or the average inventory level. given some number of kanbans, backward iteration is applied to the Markov chain state space (see [2]). The actual expectation of the number of units to be produced (or the inventory level) during a specified period of time is thus calculated, given that the Markov chain is initially in some specified state. I f the distribution of the Markov chain at the beginning of the day is known, the average over the states c a n be used to obtain the expected daily production or the average inventory level. This result can be obtained in one iteration, whereas simulation requires enough runs to guaran- tee an acceptable confidence interval.

The initial distribution of the Markov chain, that is, the status of the produc- tion line at the beginning of each day, can also be calculated by iteration (see [3)). Ideally, this distribution could be obtained by solving a linear matrix equation, but the size of the state space makes this unreasonable.

RESULTS

The expected daily production without overtime from the Markov analysis, for a one-line, two-stage system and using the four distributions, is shown in Figure 4. Expected production per day increases toward a limit of 10.0 units as the number of kanbans increases. Average production is always higher for the narrow normal distribution than for the wide normal, since variation in service times increases the probability that the queue can become empty. Although the average daily pro- duction for an exponential distribution can be calculated in closed form 171 [12], it is a very bad approximation to the average daily production for either normal distribution. The daily production for the erlang distribution looks like a promising appmximation to that for the wide normal, and it saves considerable computing time

Similarly, we can calculate expected inventory using these methods. I t will be shown, however, that for any finite number of kanbans, n, the average inventory in steady state is equal to ( n + 1)/2.

The trade-off between production revenues and inventory costs are shown in Figure 5. To maximize profit, one would choose the number of kanbans where the slopes of the two curves are equal-that is, where marginal rwenue equals marginal cost. The cases of cheap and expensive, but constant, inventory costs are shown. As expected, it is desirable to increase the number of kanbans used as inventory costs decrease.

676 Decision Sciences [Vol. 19

mum 1: Expected daily production without ovcrtime for a one-line, two-stage pro- duction system.

0 1 1 3

Numba of b b u u

Fi8om 5: Production vs. inventory trade-off.

4

m

B .y

i - 0

n

n I 3

Number of I(rmbuu

4

19881 Jodan 677

Distribution of the Queue Length

Much of the character of JIT production lines can be seen by examining the steady-state distribution of the intermediate queue lengths. Figure 6 shows, for one through four kanban systems, the amount of time the kanbans are filled. Since the actions taken if all the kanbans are full or all are empty are similar, these distribu- tions are symmetric in a steady state, therefore, average inventory is simply (n+ 1)/2.

Expected production in steady state is a function of the percentage of time a stage is idle-that is, the percentage of time the queue is empty. We see that, as claimed above, increasing the variance in the service time distribution increases the probability that the queue is empty.

The similarity of the steady-state distributions of the queue length for wide normal and erlang service distributions (as seen in Figure 6) indicates that we can obtain good approximations to results that are in steady state with any service dis- tribution that has the correct mean and variance. Indeed, the steady-state Markov chain behavior is almost completely determined by the first two moments of the service distribution. This is of great use in the modeling of real systems; one can merely find the mean and variance of the amount of time required to produce at each stage in a production line and essentially ignore the actual shape of the service distribution. It is important to note, however, that this applies only to steady-state results; some forms of oKmme depend heady on the shape of the service distribution.

COMPARISON WITH SIMULATION RESULTS

Huang, Rees, and lhylor [6] have simulated a multiline, multistage produc- tion system. The advantage of simulation, as demonstrated in (61, is that it can handle problems with very large state spaces, which would be difficult to solve using iterative methods of analysis. Simulation, however, also has its drawbacks.

Figure 7(a-c) displays the simulation results of Huang et al. [a], the results obtained here by iteration for a one-line, two-stage system (labeled Iteration), and Lee's simulation results [9] for the same one-line. two-stage system. In Figure 7(c), there also appears the expected daily production for the multiline system used by Huang et al. [6], calculated iteratively assuming the queue is full at the beginning of the day.

We should expect a similar increase in average daily production (from using one kanban to two) for the single-line and multiline systems, but the multiline system always has a lower average production. The results in 161 show a similar increase when the service distribution is wide normal but not when it is narrow normal. Simulation can produce such inaccurate results if the number of trials simulated is not large enough. In addition, the average daily production estimates produced by Huang et al. are too high. In fact, when the service distribution is exponential, we know from analysis that the expected daily production for a one-line, two-stage system is equal to 10(n+I) / (n+2) , i f n is the number of kanbans. This analysis a g m s with both the iterative and simulation results for the single-line system; these results and the iterative results shown for the multiline system should provide an upper bound on expected daily production for the system simulated by Huang et al. This demonstrates the danger of truncating a service distribution in simula- tion; exponential distributions have extremely long tails and truncation can decrease the mean substantially.

In summary, both simulation and ilerative methods have limitations. Simula- tion requires a long enough run to guarantee an acceptable confidence interval.

678 Decision Sciences [Vol. 19

Figure 6 Steady-state distributions of queue length for various kanban systems.

n a n w normal

nmm normpl

!I C

Erlwq

0

!:

i:. n 0

0 0

exmnential

P O

0- L-.-

*Means that a station a n work only if the n q t station has no work-h-progras. This is obviously impractical. but it is useful to display as a limiting case

19881 Jottian 679

?I -OWmm* .I -- -

0 1 1 8

Qu-

n, 1 1 8

Q--w

2:m : O l ? >

Qu 1lp.l

)I -- :I nuro*rm**

Figu

re 7

: Si

mul

atio

n re

sults

of

Hua

ng e

t al

. [a].

m

00 0

c L

Wls

l I

t J

1 2

Num

ber

of K

anba

ne

Fig.

7(c

) Pr

odud

ion

rate

s lo

r Ex

pone

ntla

l Dis

tribu

tion

1988) Jordan 68 I

The required length is not always apparent, and the result is not exact. Iterative Markov chain methods do not suffer from this problem, but they do suffer from state-space explosion, and they are limited to steady-state results such as average production rate or average inventory level.

FURTHER WORK In addition to the results presented here, we have studied the effects of allow-

ing overtime and of using a more complicated method of controlling the inventory level. Even so, this study is far from a definitive investigation into the problem. Further work can be done on the effects of unequal service distributions, addi- tional stages, different controls, and inventory costs that vary by unit or by stage, to name just a few. [Received: May 19, 1986. Accepted: March 2, 1987.1

REFERENCES

Barten, K. A queueing simulator for determining optimum inventory levels in a sequential pro- cess. Journal of lndustriol Engineering 1962, 13. 245-252. Derman, C. Finite stale markovian dcrision pmsses. New York: Academic Press. 1970. Feller. W. An introduction to probability theory and its applications. New York: Wily. 1957. Hatcher. J. M. The effect of internal storage on the production rate of a series of stages having aponential service t ima. A I I E 7h?nsactions 1969, 1. ISO-lS6. Hillier. F. S.. & Bollia. R. W. The effect of some design factors on the efficiency of production l i n a with variable operation t ima. Journal of Industrial Engineering, 1966. 17. 651458. H u m & P. Y., Rccs. L. P., dLPylor, B. W.. 111. A simulation analysis of the Japanese just-in-time technique (with kanbans) for a multiline, multistage production system. &cision Sciences, 1983. 14. 326-343. Weinrock. L. Queueing systems, Vol. I: nmry New York: Wily. 1976. K n o n A. D. The inefficiency of a series of work smtions-A simpk formula. Internotional Journal of Production Research. 1970. 8. 109-119. Lee, M. C. Response o/a basic just-in-time production system under dgfeerent stochastic condi- tions. Unpublished B.S. thais. University of Califomia-Berkcly. 1986. Muth, E. J. The production rate of a scria of work stations with variable service t ima. Interna- tional Journal of P d u c t i o n Research. 1973, 11. 155-169. Muth. E. J. Numerical methods applicable to a production line with stochastic servers. Algorithmic Methods in Probability: TIMS Studies in the Monagement Sciences. 1977. 143-159. Neuts. M. F. Matrix geometric solutions in stochastic modek An algorithmic a p p m h . Baltimore MD: Johns Hopkins University Press. 1981. Ross. S. M. Introduction to probability models. New York: Academic Press, 1980.

Scott Jordan is a graduate student in the Department of Electrical Engineering and Computer Science at the University of California at Berkeley. His research interals are in stochastic systems.