~ Pergamon Computers ind. Engng Vol. 30, No. 2, pp. 185-192, 1996 Copyright © 1996 Elsevier Science Ltd

0360-8352(95)00174-3 Printed in Great Britain. All rights reserved 0360-8352/96 $15.00 + 0.00

A S T A T I S T I C A L A N A L Y S I S T O D E S I G N A N D O P E R A T E

A N U N R E L I A B L E T R A N S F E R L I N E W I T H E X O G E N O U S

R A N D O M U N I T D E M A N D

ABRAHAM MEHREZ and MICHAEL Y. HU Graduate School of Management, Kent State University, Kent, OH 44242-0001, U.S.A.

(Received 5 September 1995)

Abstract--The purpose of this paper is to employ statistical methods such as cluster, discriminant, and other multivariate techniques to classify discrete unreliable transfer lines and to further evaluate how design and operational factors affect their performance. The empirical analysis to illustrate the statistical approach is based on a model and an experimental analysis derived by Mehrez and Patuwo (Computers ind. Engng 26, 307-320, 1994).

1. INTRODUCTION

Recently Mehrez and Patuwo [1] developed a model for a transfer line with exogenous demand. Their model was motivated by an observation on design and operational issues in a chemical plant in Israel. For the convenience of the reader the features of the plant are described subsequently. (For further details see Mehrez and Patuwo [1]). The plant consists of five facilities/stations processing a chemical raw material in a fixed sequence.

The raw material, which is potassium and sodium salt from the Dead Sea, is transformed at each station into an intermediate compound through a set of operations and into a final product (fertilizer) at the last station through a push mechanism, and then is shipped abroad. The processing stations are subject to breakdown due to mechanical failures. The push system is used instead of the pull system because of an agreement with the labor union to avoid worker strikes. The plant is located near the Dead Sea and the warehouse is located at the port, about 50 km away. It is assumed that the supply of raw material is infinite and hence there is always raw material available to be processed at the first station and the demand per period for the finished product is quite frequent. Actually, this demand rate is measured by the number of dedicated ships per period used to transport the finished unit abroad. At most, a single unit leaves the port per period (per day). Figure 1 illustrates this system.

This system is called a push-transfer line production system [2]. The literature on transfer lines primarily focuses on optimizing various measures of performance under restricted structures. Most of the analytical models consider only two-station or three-station lines. The push system is used in many companies in the chemical industry. Managers in these companies have to cooperate with labor unions. Unions are concerned with premiums and norms. A pull system leads to idle time and reduces the salaries of workers. Although a push system may increase inventory levels and total costs, it is a solution to avoid labor strikes. Furthermore, these companies are increasing inventories to take advantage of unexpected demand opportunities that are not considered in the overall strategy of a firm. A review and development of analytical transfer line models are provided by Buzacott and Hanifin [3], Jafari and Shanthikumar [4], Hillier and So [5, 6], Altiok and Stidham [7], Gershwin and Berman [2], Gershwin and Schick [8]. Analytical models typically focus on optimiz- ing a specific structure and ignore the possibilities of using statistical methods to identify the effect of each factor or a combination of factors across structures on outcome performance measures. The objectives of statistically classifying, discriminating and evaluating alternative transfer lines in particular and production systems in general are significant in practice to improve design and operational engineering decisions (see Ref. [9]).

185

186 Abraham Mehrez and Michael Y. Hu

To illustrate the considered approach, we present in the next section the main features of the Mehrez and Patuwo system. In Section 3, the results of the experiment derived by Ref. [1] are provided. In Section 4, the statistical results of this approach are presented. Finally, directions toward future research are suggested.

2. THE M O D E L

The notations are borrowed from Ref. [1]. Consider an M-station serial production system as depicted in Fig. 1. At the design level, a decision may be made to determine capacities, stations and their reliabilities. At the operational level, work-in-process, capacity utilization as well as other decision variables are determined. The model presented here considers the case where an off-line inspection of the machines is accomplished at the beginning of each production period and, depending upon the condition of the machine, a decision to operate or not to operate is made by the production line controller. In order for the mth station to produce one unit t, the following three conditions must be met:

(i) at least one unit of work-in-process inventory is available at the preceding buffer at the end of period t - 1;

(ii) the ruth station is declared to be reliable to operate in period t; (iii) The ruth buffer is capable of absorbing one unit produced at the mth station at the end of

period t. This implies that the ruth buffer is not full or, if it is full, the (m + 1)st station is not blocked [it can process one unit and transfer it down-stream or in the case of the last station (m = M), an exogenous demand of one unit occurs in period t].

Given that the above three conditions are satisfied, we assume the m th station operates without interruption throughout the period t.

The discrete structure of the problem does not restrict the analysis since the length of the period can be selected to be arbitrarily small. A station is said to be starved if there is no unit available to be processed. So station 1 is never starved. Since the demand is exogenously determined, the blocking and the starving probabilities depend on the probability distribution function of the demand. Clearly, both blocking and starving depend the output of a production line. For prescribed buffer capacities and station reliabilities, various performance measures are evaluated by first calculating the steady state probabilities of a Markov chain representing the states of the buffers. These probabilities are calculated under the assumption that in a given period, the failure probability of a station is independent of the states of the other stations.

There is an infinite supply of raw material that feeds into the first station. Let C~, C2 . . . . . Cu be the buffer capacities after stations 1, 2 . . . . . M, respectively. Furthermore, let XI m), m = 1 . . . . . M, be the state of the ruth buffer (the content of the mth buffer) at the beginning of period t. In Ref. [1] a computational procedure to compute:

P[X m = j ] = lira P(XI m) = j ) t ~

is provided. Based on the calculations of the steady state probabilities the following six commonly used performance measures of the system are considered in our study. (Rationale for using these m e a s u r e s is provided on p. 311 of Ref. [1].)

C m

1. Yi = E[ XM] = ~=JP[XN=J], expected content of the last buffer (warehouse). j=J

2. Y2 = p[xM>~ 1], probability of meeting the demand.

3. Y3 = X = ~ E[Xm]/M, average work-in-process and inventory. m = |

Fig. 1. Serial production line with exogenous demand.

S t a t i s t i c a l a n a l y s i s to d e s i g n a n d o p e r a t e a n u n r e l i a b l e t r a n s f e r l ine 187

M

4. Y4 = ~ P[ Xm= O]/M, average starving probability. m = l

M

5. ]I5 = 2 p [ x m = Cm]/M, a v e r a g e blocking probability. m ~ l

M

6. Y6 = ~. ]E(Xm) - XI/M, mean absolute deviation (MAD) of the work-in-process. m ~ l

In all the test cases performed, Ref. [1] chose d = P[D = 1] = 0.9 and P[D = 0] = 0.1, i.e. the expected demand rate is 0.9 units per period. I12 = P(XM>~ 1), is o f pr imary concern, since it measures the long-run ability o f the product ion line to meet the external demand.

In the reported experiment, we will consider only product ion line (configurations) with prescribed station reliabilities and capacities which results in P[X M >t 1] >/0.9. The cut-off probabil i ty o f 0.9 is chosen on the basis o f trying to match the long-run probabil i ty o f having at least one unit to meet the demand to the probabil i ty o f having a demand. Again, choosing P[X M >t 1] > / P { D = 1] does not guarantee there will be no shortages. In fact, the choice o f P[X M >1 1] >/0.9 results in E[shortage] ~< 0.09.

3. T H E R E S U L T S O F T H E E X P E R I M E N T

We used in our statistical study the eight different structures (configurations) identified by Ref. [1]. They are:

• Structure 1: five-stations (M = 5) in series with station reliabilities:

r I = 0.95, r 2 = r 3 = r 4 = 0.90, r5 = 0.95.

• Structure 2: five-stations (M = 5) in series with station reliabilities:

r I = 0.95, r 2 = 0.85, r 3 = 0 . 9 0 , r 4 ---- 0.9529, r5 = 0.95.

• Structure 3: five-stations (M = 5) in series with station reliabilities:

r 1 = 0.95, r 2 = 0.9529, r 3 = 0.90, r 4 = 0 . 8 5 , r 5 = 0.95.

• Structure 4: five-stations (M --- 5) in series with station reliabilities:

rl = 0.90, r2 = r 3 = r4, r5 = 0.90.

• Structure 5: five-stations (M = 5) in series with station reliabilities:

r 1 = 0.85, r2 = r3 = r4, r5 = 0.85.

• Structure 6: three-stations (M = 5) in series with station reliabilities:

r I = 0.95, r 2 = 0.729, r 3 ----- 0.95.

• Structure 7: three-stations (M = 5) in series with station reliabilities:

rj = 0.95, r2 = 0.999, r 3 = 0.95.

• Structure 8: five-semi-parallel stations (M = 5, see Fig. 2) with station reliabilities:

rl = 0.95, r 2 = r 3 = r 4 = 0.90, r 5 = 0.95.

Mot ivat ion for considering these structures stems from consulting experience gathered by the first au thor o f this study in the chemical industry. Most companies in this industry use a push-transfer line system and raw materials are being extracted from mines, under the assumption o f sufficiently large supply. These companies produce salt and other chemicals in a flow shop system. They are located in various countries in the world, such as Canada, China, Israel, Chile and South Africa. The first au thor ' s consulting experience with these companies leads us to formulate and analyze the model f rom a statistical point of view.

The reliabilities o f the middle stations in structures 1, 2 and 3 are chosen such that r 2 . r 2 . r 3 = 0 . 9 3 = 0.729. In structure 2, the middle stations are arranged in ascending order o f their station reliabilities, whereas in structure 3, they are arranged in descending order of their station

188 Abraham Mehrez and Michael Y. Hu

Fig. 2. Semi-parallel production line with exogenous demand.

reliabilities. In structures 4 and 5, the two end stations are less reliable than those of structure 1. In addition, structure 4 captures the case where all stations are identical, and structure 5 has the two end stations as bottlenecks.

The three-station line in structure 6 corresponds to the five-station line case where there is no buffer storage after stations 2 and 3. In this case, we can combine the three middle stations into one station with reliability 0.729 (see e.g. Ref. [1]). Structure 8 is a semi-parallel structure with the three middle stations arranged in parallel and a buffer storage after each station (see Fig. 2). Finally, structure 7 is like structure 8 with one common buffer storage after stations 2, 3 and 4. In this case, the three middle stations can be treated as one station with reliability of 1 - ( 1 - 0 . 9 ) 3 = 0.999.

In Table 1 we present the disaggregate results for those line configurations (80) with probabilities of having at least one unit on-hand to meet the expected demand of 0.9 units per period (with the exception of structure 6). Here, we chose the best 10 cases for each structure, where "best" means that P[X~>~ 1] is as close as possible to 0.900 and, in the case of ties, we chose those cases with smaller design capacity. As noted before, the choice of 0.900 probability does not imply that there will be no shortages.

4. STATISTICAL ANALYSIS

The eight structures can be defined by the following parameters: P1 = number of stations--3 or 5; P2 = series or semi-parallel; P3 = reliabilities are arranged in either ascending, descending or stable order; P4 = first station has low, average or high reliability; and P5 = end station has low, average or high reliability. Performance of these eight structures can be gauged by YI, expected content of the last buffer (warehouse); 112, the probability of having one unit in the last buffer to meet the demand; Y3, average work-in-process and inventory; Y4, average starving probability; I"5, average blocking probability; and Y6, mean absolute deviation of work-in-process.

In order to facilitate the comparison of the eight production structures along each of the six dimensions, YJ-Y6, cluster analysis was first used to form groups of structures. The eight structures will be grouped by the five defining parameters. Cluster analysis primarily is composed of a number of algorithms designed to group objects with respect to some similarity measures (Dillon and Goldstein [10]). The formation of groups may sometimes be sensitive to the particular algorithm used. It is thereby advisable to use several clustering algorithms to ensure the existence of natural groupings. After the groups are formed, two multivariate statistical procedures will be employed to establish the relationship between the groups and performance measures. Linear discriminant analysis will be used to capture the differences among the groups with respect to linear combination/s of the performance measures. A stepwise routine will indicate linear combination/s of a subset of performance measures that will maximally distinguish among the groups. Moreover, the stepwise discriminant procedure will provide insights on the extent of correlation among the performance measures. In addition, the stepwise discriminant analysis approach will be sup- plemented by multivariate analysis of variance, which will then provide information as to along

Sta t i s t ica l ana lys i s to des ign a n d ope ra t e a n unre l i ab le t r ans fe r line 189

which performance dimension/s the groups are different. Wilks' lambda will be used as a measure of the amount of variability in the dependent performance measures being explained by differences in groups. The overall F-statistic for Wiiks' lambda shows the statistical significance of the differences in any of the performance measures that can be attributed to groups. Univariate analysis of variance will also be used to investigate the group differences for each dependent measure. Lastly, Tukey's [10] pairwise comparison will be employed to identify the pair/s of means are significantly different from each other.

Table I. Observations with P [X u I> 1] ~ 0.900

Legend

Yl: E [X M] = expected content of the last buffer (warehouse)

1"2: P [XM>~ 1] = probability of having one unit in the last buffer to meet the demand

Y3: ~" ~ ~ E[Xm]/M = average work-in-process and inventory

g4: ~ P[ Xm = O]/M = average starving probability m-t

Y3: ~. P[X" = c~]/M = average blocking probability

Y6: ~ IE[Xm[-'~I/M = m e a n absolute deviation of work-in-process.

Structure 1 P [demand = l] = 0.9 Number of stations in series, M = 5 Reliability of the stations = 0.9500 0.9000 0.9000 0.9000 0.9500 Obs. Capacities Yi Y2 Y3 Y4 r5

I. 1 2 2 3 4 1,538 0.900 1.235 0.092 2. 1 4 3 2 2 1.227 0.900 1.384 0.077 3. 2 4 2 2 2 1.227 0,900 1.548 0.065 4, I 2 5 3 2 1.227 0.900 1.316 0.083 5. 2 3 3 I 4 1.541 0.900 1.547 0.069 6, 4 3 2 2 2 1.226 0.900 1.835 0.064 7. 2 1 4 4 3 1.421 0.900 1,380 0.087 8. 2 2 4 1 5 1.608 0.900 1.512 0.073 9. 5 3 2 2 2 1.227 0.900 2,030 0.063

10. 2 I 4 5 3 1.423 0.900 1.382 0.087

0.359 0.369 0.381 0.357 0.437 0.394 0,385 0.445 0.393 0.383

Structure 2 P [demand = 1] = 0.9 Number of stations in series, M = 5 Reliability of the stations = 0.9500 0.8500 0.9000 0.9529 0.9500 Obs. Capacities Yi Y2 Y3 Y4 r5

1. 2 2 2 2 4 1.541 0.900 1.336 0.097 2. 2 2 3 2 3 1.419 0,900 1.331 0,095 3. 1 3 3 2 5 1.611 0.900 1,228 0.107 4. I 3 3 4 3 1,420 0.900 1.216 0.105 5. 1 4 2 2 5 1.609 0.900 1.249 0.106 6. I 5 2 2 4 1.537 0,900 1.257 0.104 7. 1 5 2 3 3 1.422 0.900 1.252 0,103 8. 4 3 3 2 2 1.229 0.900 1.757 0.081 9. 4 4 2 2 2 1.229 0.900 1.789 0.080

10. I 3 3 5 3 1.422 0.900 1.218 0.105

0.316 0.305 0.269 0.256 0.282 0.282 0.269 0.308 0.319 0.254

Strecture 3 P [demand = 1] = 0.9 Number of stations in series, M = 5 Reliability of the stations = 0.9500 0.9529 0.9000 0.8500 0.9500 Obs. Capacities Yi Y2 Y3 Y4 r~

1. I 1 4 3 4 1.536 0,900 1.357 0.083 2. 2 1 3 4 3 1.422 0.900 1.450 0.068 3. 2 2 5 2 2 1.228 0.900 1.831 0,053 4. 2 3 3 3 2 1.227 0.900 1.723 0.053 5. l 4 2 2 5 1.610 0.900 1.555 0.070 6. l 4 2 4 3 1.421 0,900 1.543 0.069 7, I 5 2 2 4 1.538 0.900 1.663 0.069 8. I 5 2 3 3 1.423 0.900 1.659 0,067 9. 3 l 5 3 2 1.226 0.900 1.830 0.058

10. 3 2 3 4 2 1.227 0.900 1,762 0.053

0.436 0.438 0.438 0.407 0,407 0.393 0.406 0.393 0.457 0.417

r~

0.157 0.348 0,384 0.226 0.278 0.731 0.273 0.303 0.984 0,271

r~

0.265 0.221 0.236 0.170 0.288 0.294 0.251 0,761 0.778 0.169

r~

0.429 0.323 0.687 0,438 0.460 0.443 0.634 0,636 0.845 0.466

continued overeaf

190 A b r a h a m Mehrez and Michael Y. H u

Table l--continued

Structure 4 P [demand = 1] = 0.9 Number of stations in series, M = 5 Reliability of the stations = 0.9000 0.9000 0.9000 0.9000 0.9000 Obs. Capacities Yi Y2 Y3 Y4

1. 2 2 3 3 3 1.421 0.900 1.472 0.076 2. 2 2 2 4 4 1.537 0.900 1.457 0.080 3. 2 4 4 3 2 1.227 0.900 1.651 0.065 4. 3 2 5 2 3 1.424 0.900 1.681 0.065 5. 5 2 2 3 3 1.423 0.900 1.887 0.065 6. 1 3 3 4 5 1.608 0.900 1.410 0.096 7. 1 3 3 5 4 1.540 0.900 1.417 0.096 8. 1 4 4 4 3 1.423 0.900 1.465 0.091 9. 3 1 4 4 4 1.536 0.900 1.615 0.081

10. 3 4 2 5 2 1.227 0.900 1.757 0.061

0.290 0.292 0.260 0.295 0.310 0.259 0.259 0.254 0.334 0.271

Structure 5 P [demand = 1] = 0.9 Number of stations in series, M = 5 Reliability of the stations = 0.8500 0.9000 0.9000 0.9000 0.8500 Obs. Capacities Y1 Y2 Y3 Y4

1. 3 2 4 5 5 1.606 0.900 1.804 0.073 2. 3 4 3 5 4 1.539 0.900 1.845 0.072 3. 4 2 3 5 5 1.607 0.900 1.872 0.069 4. 5 2 3 5 4 1.538 0.900 1.963 0.065 5. 3 3 5 4 5 1.612 0.900 1.828 0.071 6. 3 4 4 4 5 1.611 0.900 1.834 0.071 7. 3 4 5 4 4 1.536 0.900 1.856 0.070 8. 4 2 5 4 5 1.607 0.900 1.883 0.067 9. 4 4 3 4 5 1.604 0.900 1.869 0.068

10. 5 3 3 4 5 1.611 0.900 1.916 0.065

0.181 0.139 0.188 0.191 0.143 0.130 0.129 0.179 0.134 0.151

Structure 6 P [demand = 1] = 0.9 Number of stations in series, M = 3 Reliability of the stations = 0.9500 0.7290 0.9500 Obs. Capacities Yi Y2 Y3 r4

1. 4 4 2 1.007 0.810 1.937 0.136 2. 4 4 3 1.093 0.810 1.947 0.140 3. 4 5 2 1.007 0.810 1.939 0.136 4. 5 4 2 1.007 0.810 2.271 0.136 5. 4 4 4 1.130 0.810 1.954 0.141 6. 4 5 3 1.094 0.810 1.948 0.140 7. 4 6 2 1.007 0.810 1.939 0.136 8. 5 4 3 1.094 0.810 2.280 0.140 9. 5 5 2 1.007 0.810 2.272 0.136

10. 6 4 2 1.007 0.810 2.604 0.136

0.353 0.305 0.352 0.353 0.292 0.304 0.352 0.305 0.352 0.353

Structure 7 P [demand = 1[ = 0.9 Number of stations in series, M = 3 Reliability of the stations = 0.9500 0.9990 0.9500 Obs. Capacities Yi Y,, ]I3 v4

1. 1 1 2. 2 1 3. 1 2 4. 1 3 5. 3 1 6. 4 1 7. 2 2 8. 1 4 9. 1 5

10. 5 I

r,

0.075 0.128 0.253 0.383 0.752 0.200 0.203 0.239 0.323 0.383

0.332 0.329 0.463 0.585 0.230 0.195 0.231 0.335 0.219 0.348

r6 1.265 1.259 1.264 1.709 1.254 1.258 1.264 1.702 1.708 2.153

v~

0.921 0.92t 0.946 0.054 0.946 0.017 0.946 0.946 1.223 0.025 0.894 0.341 0.947 0.947 1.219 0.036 0.889 0.357 0.952 0.952 1.519 0.032 0.875 0.753 0.952 0.952 1.522 0.018 0.882 0.731 0.953 0.953 1.838 0.017 0.879 1.150 0.954 0.954 1.533 0.019 0.870 0.386 0.954 0.954 1.834 0.031 0.870 1.172 0.954 0.954 2.152 0.030 0.866 1.596 0.954 0.954 2.163 0.016 0.877 1.583

0.9000 0.9500 r,

Structure 8 P [demand = 1] = 0.9 Number of stations in series, M = 5 Reliability of the stations = 0.9500 0.9000 0.9000 Obs. Capacities Yt Y: Y3 Y5

1. 1 1 1 t 2. 2 1 1 1 3. 1 1 1 2 4. 1 1 2 1 5. 1 2 1 1 6. 3 1 1 1 7. 4 1 1 1 8. 5 1 1 1 9. 1 1 3 1

10. 1 3 1 1

0.948 0.948 0.850 0.150 0.850 0.092 0.952 0.952 1.039 0.107 0.842 0.274 0.953 0.953 1.030 0.119 0.841 0.264 0.953 0.953 1.015 0.096 0.844 0.160 0.953 0.953 0.985 0.108 0.798 0.166 0.953 0.953 1.226 0.098 0.842 0.555 0.953 0.953 1.418 0.095 0.842 0.858 0.953 0.953 1.614 0.094 0.842 1.171 0.954 0.954 1.188 0.071 0.845 0.414 0.954 0.954 1.104 0.087 0.786 0.322

Statist ical analys is to design and opera te an unrel iable t ransfer l ine

Table 2. Centroid hierarchical cluster analysis

Normalized No. of centroid clusters Clusters joined distance

7 6 7 0.240523 6 2 3 0.340151 5 1 CL6 0.564076 4 CL5 8 0.626181 3 4 5 0.680301 2 CL4 CL7 0.772393 1 CL2 CL3 1.119573

191

The five defining parameters PI-P5 were used to form groups of production structure. Due to the fact that these five parameters were largely nonmetric, e.g. increasing, decreasing or stable reliabilities, a distance matrix was set up by calculating the number of mismatches between any pair of structures along P1-P5. That is, for example, for structures 1 and 3, four out of five parameters have the same characteristics. Thus the distance measure takes on a value of 1/5. The 8 x 8 distance matrix was used as input into several clustering algorithms (SAS [11]). The average linkage, centroid hierarchical and Ward's minimum variance approaches were used and all three clustering routines provided basically the same results, indicating the existence of three natural clusters. As shown in Table 2, three clusters were identified with normalized centroid distance of 0.68. Beyond this level, there is a substantial increase in the distance measure. The three clusters are composed of cluster 1--structures 1, 2, 3 and 8; cluster 2--structures 4 and 5; and cluster 3--structures 6 and 7. Intuitively, when one considers the composition of the structures within each cluster, cluster 1 contains structures with five stations, and with high reliability for both the first and end stations. Cluster 2 can be distinguished by the presence of stable reliabilities in the middle three stations. Cluster 3 corresponds to the three station structures. As argued earlier, the fact that different clustering algorithms yield the same grouping of structures, provides evidence for natural groupings among the structures.

Table 3 contains the final set of stepwise discriminant analysis results for using YI-Y6 to discriminate among the clusters. All six performance measures were instrumental in discriminating among the three groups. Yt entered into the discriminant function at stage 1 and was later removed at the final stage, indicating potentially the high correlation between Y~ and the other five variables. Y~ by itself accounted for the largest amount of the differences among the three groups. Yet, as the other five variables were introduced into the discriminant function, the total amount of difference among the three groups explained by Y2-Y 6 was greater than that being explained by Y~. As shown in Table 2, all variables contributed significantly to explaining the differences among the groups. The p-values in all cases were less than 0.01 level.

In order to gain more insights into the differences among the performance measures with respect to the groups of production structure, multivariate analysis of variance was conducted with the six performance measures as dependent variables and groups as independent variables. Results are given in Table 4. Wilks' lambda took on a value of 0.07 and was significant at the 0.0001 level. That is to say, roughly 93% of the variability in the dependent measures can be explained by the group differences. The significance implies that along some, if not all, of the six dependent variables, differences among the three groups can be detected.

The univariate analysis of variance results in Table 4 indicates significant difference among groups for Y,, Y2, Y3, I15 and Y6. It can be concluded that high reliability in for first and end stations as in the case of structures in cluster 1, leads to larger average value, 1.287 for Y~, expected content of the last buffer, and Y2, 0.913, the probability of having one unit in the last buffer to

Table 3. Stepwise discriminant results

Step Entered Removed F-STAT p-Value

I Yi 41.841 0.0001 2 Y3 19.023 0.0001 3 Y~ 26.144 0.0001 4 Y4 2.629 0.0789 5 ~ 37.408 O.O001 6 Y5 7.898 0.0008 7 Yi 0.177 0.8379

192 Abraham Mehrez and Michael Y. Hu

Table 4. MANOVA results Tukey's pairwise

Cluster I Cluster 2 Cluster 3 F - S T A T comparison ~ Y~ 1.287 1.152 0.997 41.84 (1,2)(1,3)(2,3)

(0.0001) Y2 0.913 0.900 0.879 4.99 (I,3)

(0.0092) Y3 1.416 1.724 1.852 16.24 (I,3)(1,2)

(o.ooo l) Y4 0.085 0.073 0.083 0.92

(0.4020) Y5 0.482 0.219 0.608 18.23 (1,2)(2,3)

(o.oool) ]"6 0.426 0.310 1.146 38.13 (I,3)(2,3)

(0.0001) aSignificant Tukey pairwise differences at ct = 0.05 are reported here. Wilks' lambda = 0.071. F-statistic = 32.99, p-value = 0.0001.

meet the demand ; smal ler average value, 1.416 in Y3, average work- in-process and inventory; m e d i u m average value for I15, average b locking p robab i l i t y and small average value, 0.426 for Y6, mean abso lu te dev ia t ion o f work- in-process . Clus ter 2, as charac ter ized by stable rel iabi l i ty in the middle three s ta t ions, on average, has low average value for Ys, 0.219, average b locking p robab i l i ty and Y6, 0.310, mean abso lu te devia t ion o f work- in-process . Clus ter 3 with the three-s ta t ion s t ructure as c o m p a r e d to f ive-stat ions in the o ther clusters, has the smallest averages for Yl, 0.997, expected con ten t o f last buffer and Y2, 0.879, the p robab i l i t y o f having one unit in the last buffer; highest average value in Y3, 1.852, average work- in-process and Ys, 0.608, average b locking p robab i l i t y and unusual ly large value o f 1.146 for Y6, mean abso lu te devia t ion o f work- in-process .

5. FUTURE RESEARCH

To the best o f our knowledge this s tudy is the first a t t e mp t to employ stat ist ical tools for the classification, d i s semina t ion and eva lua t ion o f p roduc t ion systems. The results are self exp lana to ry and they can assist the design on the p roduc t ion engineer and dec i s ion-maker to test conf igura t ions that are charac te r ized by a mul t i -d imens ion set o f pa rame te r s and measures o f per formance . Fu tu re research m a y be conduc ted t oward the fol lowing direct ions:

I. Extend the s tudy to deal with a l ternat ive p roduc t ion systems. 2. Employ a l ternat ive s tat is t ical me thods which explore t ime dependen t behav ior o f the system,

detect out l iers on o ther features o f the system. 3. C o m b i n e d a t a pol icy analysis me thods used by Mehrez and Pa tuwo [l] in their s tudy with

s tat is t ical inference analysis to provide more insight into systems design and opera t ion .

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