Identify Formula for Throughput with Multi-Variate Regression

DECISION SCIENCES INSTITUTE Using multi-variate regression and simulation to identify a generic formula for throughput of flow manufacturing lines with identical stations Samrawi Berhanu Gebermedhin and Xiaojun Shan University of Houston Clear Lake Email: gebermedhins8380@uhcl.edu and shan@uhcl.edu ABSTRACT Estimating throughput of a manufacturing line is important for planning. This paper use simulation and multivariate regression to estimate a generic equation to determine the throughput. Minitab was used to create random data that follow normal distributions: station speed, mean time between failure, mean time to repair, buffer capacity and line length. one thousand different random manufacturing lines were developed and modelled using plantsim. The outputs were used to run a multi-variate regression analysis to find an equation that best fits the data. A regression equation with R equal 9% was found. Station speed has the largest impact on throughput. KEYWORDS: Simulation, Regression, PlantSim, Manufacturing throughput INTRODUCTION Setting up a manufacturing line requires significant planning to ensure its success. The manufacturing line should be capable of yielding the minimum required throughput. An efficiently planned manufacturing line can save a company a significant amount of money in the long run. The initial step of setting up a manufacturing line is to determine the best layout. The best layout should ) minimize labor cost, ) minimize product damage, 3) maximize the throughput speed, 4) easily incorporate new processes, and 5) maximize safety and morale. The throughput of a manufacturing line is the number of jobs that a manufacturing line can complete in a certain amount of time. The throughput is measured in the number of products completed per hour. This is important because a manufacturing line should be capable to satisfy the anticipated demand for that product in order not to lose sales and customer satisfaction. However, this is very difficult to calculate because many factors affect the throughput of a manufacturing line. To simplify the problem of calculating the throughput of a manufacturing line, Alden (00) developed a general two-station line. Each of the stations has three variables with a buffer in the middle. The variables are ) the speed of a station, S (jobs per hour), ) the failure rate of a station, λ (failures per m), 3) the repair rate of station, µ (repairs per hour), and 4) buffer size, B (number of jobs). The station speed measures how many jobs per hour that a station completes. The failure rate of a station measures how often a machine breaks down per hour. The repair rate of station measures how long it takes to repair the machine. It is measured in hours. The buffer size measures how many jobs can be stored in the buffer. The buffer is used to store products processed by station before moving to station. Alden calculates the throughput of the model based on the buffer. The buffer can be in the following states: ) full or blocked (station is blocked if the buffer is full because it has no place to store its processed parts), and ) empty or starved (station is starved because it has no parts coming from station to process). 9464

Alden s model is based on the following assumptions: ) The buffer does not fail, and jobs flow through it with zero transit time; ) A station does not fail if it is blocked or starved; 3) Operating times between failures at a station are exponentially distributed with mean ( ); 4) Repair times at a station are exponentially distributed with mean ( ; 5) The first station is never starved and the second station is never blocked. The two stations can be in the following states: ) U (up): Both stations are running; ) F (filling): Station is down while station is running; 3) E (emptying): station is down while station is running; 4) FB (fail blocked): station is blocked because station has failed and the buffer is full; 5) FS (fail starved): station is starved because station has failed; 6) SB (speed blocked) : station is blocked because station has slower speed; and 7) SS (speed starved): station is starved because station has a slower speed. By applying the renewal theory, Alden [] finds the expected renewal period. The renewal period is the total amount of time the system is running by finding the fraction of time spent in each state. () The fraction of time that the system is Up and Empty found as these are the times when the system is still producing: The throughput can then be calculated by: is the station speed of station and. We use the to account for its stand-alone availability. Finding the throughput of the two-station manufacturing line is complex because of the many situations that have to be considered. Increasing the number of stations would add many more situations and factors that have to be modelled in order to accurately estimate throughput. To simplify this problem, the stations in this paper are considered to be similar; that is, they have the same set of parameter (S, λ, µ and B). This implies that it will be possible to find one equation which will be able to determine the throughput of a manufacturing line with multiple identical stations. Data collected from a simulation software will be input into a multiple regression analysis to determine an equation that models the throughput most accurately. This equation can then be easily used to find the effects of the independent variable without setting up multiple simulations. A simplified general equation for a manufacturing line with identical station is useful. It will provide a simple equation to accurately estimate the throughput. Furthermore, the equation can be used to study the effects of changing the factors on the throughput. The regression model also tests how accurate the equation is compared to the given data. Having an equation is much easier and faster to manipulate and analyze than to set up and run a simulation. The equation can also show us which factor(s) have the greatest impact on the throughput. The remainder of the paper is organized as follows. The section on literature review discusses incubators and AHP. The section on research methodology and results describes the methods in details and presents the findings of this paper. Finally, the section on conclusion draws some conclusions. () (3) (4) 9464

LITERATURE REVIEW To the best of the authors knowledge, there is few studies using statistical methods to estimate throughput of a manufacturing line. However, many research studies have been conducted on different optimization techniques of assembly lines. According to Rekiek et al. (00), the design of a production system comprises of sub problems: ) selection of pieces of equipment from a set of candidate solutions for each manufacturing operation, ) balancing and dimensioning of workstations (assignment of operations to workstations), 3) dimensioning storage areas, i.e. buffers, 4) dimensioning transportation systems, and 5) layout. The main objective of setting up a manufacturing/production system is to increase the efficiency of the line by maximizing the ratio between throughput and required costs. This makes it important to find the optimal values for different variables that affect throughput. Blumenfeld & Li (005) took an analytical approach to identify a general equation for throughput of a similar station manufacturing line. Their equation was based on Alden s model. This equation was used to produce and analyze many different manufacturing lines. Askin & Standridge (993) took a similar approach to identify a general equation for manufacturing lines with stations that have variable operation times. Equations such as these are used to increase throughput by easily analyzing the effects of each variable on the throughput. However, a statistical analysis is advantageous because it has the ability to produce a single linear equation. The coefficients of each variable can be easily compared to determine which factors have a greater effect on throughput. Fechet & Nedelcu (0) used multiple regression to find the relationship between incomes, number of employees, product price and total revenue. Chatterjee & Hadi (0) stated that to successfully implement a regression analysis requires a balance of theoretical results, empirical rules, and subjective judgment. RESEARCH METHODOLOGY Manufacturing throughput is an important factor that a company needs to know before starting manufacturing. The information on throughput is needed so that the company can estimate its production capacity. This capacity is then compared to the sales demand to determine if any changes are needed before beginning manufacturing. There are four independent factors that affect the throughput of a manufacturing line: ) the speed of a station, S (jobs per hour), ) mean Time Between Failure (MTBF),, 3) Mean Time to Repair (MTTR),, and 4) buffer size, B (number of jobs). A multi-variate regression analysis can be used to find the relationship between several independent variables and a dependent variable. In this case, multi-variate regression is used to find the relationship between throughput, ρ (jobs per hour) and the four independent variables. Manufacturing lines have many different stations with different throughputs. Finding exact throughput for such a line is complex. To simplify the problem, it is assumed that all the stations are identical. This assumption makes it possible to identify a general formula to estimate throughputs of a manufacturing line with M identical stations. Data Description PlantSim is a simulation software produced by Siemens. PlantSim can be used to model any type of manufacturing line. A simulation can be run in PlantSim to determine throughout and bottleneck of the whole system or other statistics of independent stations. Minitab Random function was used to generate random data with a normal distribution. Table shows the variables and the lower and upper limits. 9464 3

Table : Key system parameters of a manufacturing line Independent Variable Station Speed MTBF MTTR Buffer size Line length Simulated Throughput Units Jobs/Hour Min Min Jobs/Hour Number Jobs/Hour Range 0-60 0-00 -0 0-40 -0 The distribution of the random data was analyzed. The graphs of the distributions curve which best fit the random data for each variable are presented in Figures -5. They all follow the Johnson transformation and they have very high P value and low AD values. The data matches this distribution very closely. Figure Generated data for station speed Figure Generated data for MTBF Figure 3 Generated data for MTTR Figure 4 Generated data for buffer size 9464 4

Figure 5 Generated data for line length The MTBF and MTTR are given in minutes for the following reasons. The MTBF is converted to percentage of time that the station is available in PlantSim. Therefore, by using both values in minutes will not affect this percentage. The MTTR is input in minutes into the simulation software; however, the system automatically converts its units into hours while running the simulation. Therefore, the unit of the simulation throughput is still be in jobs per hour. A variety of manufacturing lines were examined in this study to find the most realistic limits for production lines. The two special cases are buffer size and line length. The buffer size includes 0 for some manufacturing lines that do not use buffer zones and line length starts at because there must be at least stations to make a manufacturing line. Using this method, a total of 000 random manufacturing lines were created in PlantSim. The randomly generated manufacturing lines were then sorted in ascending order of line length. This made it easier to run the model for each situation before adding a station to run the next set of models. PlantSim was then used to find the throughput of each situation. Table A. in the Appendix shows the variables used and the simulation throughput. RESULTS A multi-variate regression model was run using throughput as the dependent variable and the four independent variables as the predicators. The output from Minitab is shown in Table and Figure 5-8. Table : Outputs of the Multi-Variate Regression Analysis Analysis of Variance Source DF j SS Adj F-Value P-Value MS Regression 5 4448 6890 37.09 0.000 Station Speed (S) 3800 380 0 9806.8 0.000 MTBF 4 764 658.48 0.000 MTTR 86 86 08.0 0.000 Buffer size (B) 7 37 78.04 0.000 Line length (M) 5 475 40.95 0.000 Error 995 535 Total 000 45983 9464 5

Model Summary S R-sq R-sq(adj) R-sq(pred) 3.40659 9.0% 9.06% 9.98% Coefficients Term Coef SE Coef T-Value P-Value VIF Constant.9 0.55.34 0.00 Station Speed (S) 0.7456 0.00753 99.03 0.000.00 MTBF 0.05368 0.0009 5.66 0.000.0 MTTR -0.6679 0.00-33.9 0.000.0 Buffer size (B) 0.54 0.0094 6.67 0.000.00 Line length (M) -0.3 0.005-6.4 0.000.0 Regression Equation Throughput =.9 + 0.7456 S + 0.05368 MTBF - 0.6679 MTTR + 0.54 B - 0.3 M Table A. shows the values of unusual outliers of the data and their fit. Figure 6: Residential plot (throughput versus fits) of the multi-variate regression analysis 9464 6

Figure 6: Residential plot (throughput versus order) of the multi-variate regression analysis Figure 7: Normal probability plot of the multi-variate regression analysis 9464 7

Frequency Gebermedhin & Shan Figure 8: Throughput residual histogram 00 Histogram (response is Throughput) 50 00 50 0-0 -6 - -8 Residual -4 0 4 DISCUSSION The results of the regression analysis from Minitab are discussed in three aspects: ) the statistical significance of the relation between the dependent variable and each independent variable, ) the fit between the model and the data; 3) the assumptions of the model. To determine the statistical significance of the relation between the dependent variable, the p- values needs to be compared to the significance level. The null hypothesis is that the coefficient of the independent variable is equal to 0, which implies that there is no relation between the dependent variable and the four independent variables. In the case, we use the significance value (α) of 0.05, The p-values for all our independent variable are less than our significance level. This implies that we can reject the null hypothesis Since a continuous predicator was used in this analysis, we can conclude that the coefficients of the independent variables are not equal to 0. In order to determine how well the model fits the data, we analyze the S and R values from the regression analysis. The S value is measured in the unit of the dependent variable and represents the standard deviation of how far the data values fall from the fitted values. The S value for the regression analysis is 3.4 which indicates that the models results are +/- 7% of actual throughput. There are three different R values: ) R_sq = 9.%, shows the variation in the response compared in the model; ) R_sq (adj) = 9.06%, adj stands for adjusted which mean this r-square is adjusted for the number of independent variables used to run the regression analysis; and 3) R_sq (pred) = 9.98%, pred stands for predicted which means this value shows how accurately the model can predict responses for new observations. 9464 8

The R-squared values are all above 9%, which implies that the model fits the data well. However, the residual plots still need to be analyzed to determine if the analysis meets the assumption. The regression analysis has outputted four different residual plots (Figures 5-8). Figure 5 shows the residual versus fit plot and evaluate whether the residuals are randomly distributed with a constant variance. The points in Figure 5 seem to randomly lie on either side of the x-axis. Figure 6 shows the residual versus order plot and determines whether the residuals are independent from one another. The points are randomly scattered around the x-axis until observation number 750. After this point, the points are skewed more towards the upper part of the graph. There is no clear pattern in Figure 6 so it suggests that the residuals are independent of each other. Figure 7 shows the normal probability plot and determines whether the residuals are normally distributed or not. Figure 7 shows that most of the points lie around the straight line with several outliers. Figure 8 show the throughput residual histogram and suggests that the residues are normally distributed. CONCLUSION Multi-variate regression analysis is used to find a general formula, which provide an accurate estimation of the throughput of a manufacturing line with identical stations. This formula is easier and much faster to provide an estimation of manufacturing throughput than simulation. This formula also makes it easy to change any of the variables and analyze the effects on throughput immediately. The equation that was found using regression analysis is: Throughput =.9 + 0.7456 S + 0.05368 MTBF - 0.6679 MTTR + 0.54 B - 0.3 M We can also rank the five independent variables in terms of how much they affect the throughput. We can see that increasing station speed has the most effect on throughput. Also, having a lower mean time to repair a broken machine affects the throughput significantly. It should also be noted that having a shorter line length increases the throughput. The effect of the buffer would be negligible once it surpasses the total number of jobs each station produces in a third of an hour. This is because the maximum time considered for mean time to repair is twenty minutes, which implies that a machine will be fixed within twenty minutes after it fails. Therefore, a buffer supply of twenty minutes is sufficient to feed the station down the line to continue production process. This equation can also be used for non-identical stations if a suitable relationship between factors of the manufacturing line and factors in the equation can be found. This will make this equation much more practical and applicable to many different situations. Appendix Table A.: Randomly generated independent variables in ascending order of line length Note that the columns below represent the following Observation # () Station Speed () MTBF (3) MTTR (4) Buffer size (5) Line length (6) Simulation Throughput (7) Jobs/Hour Min Min Jobs/Hour Number Jobs/Hour Jobs/Hour 3 4 5 6 7 39 8 3 6 4 9 9 8 3 55 78 6 53 4 7 38 5 8 4 5 37 00 8 3 33 6 60 09 7 6 55 7 54 8 9 53 8 35 8 6 9 9 40 5 39 0 44 00 4 7 4 4 90 3 34 9464 9

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38 8 9 4 0 9 6 38 46 09 7 5 9 3 383 3 65 0 9 384 48 4 0 3 9 385 54 38 8 9 46 386 5 7 38 9 4 387 50 8 3 9 46 388 38 05 5 9 9 389 7 7 4 0 9 0 390 8 54 33 9 7 39 60 67 3 0 55 39 3 30 8 0 0 393 54 56 5 36 0 45 394 3 03 8 3 0 395 3 50 9 0 396 9 5 5 0 0 8 397 0 78 3 0 8 398 6 9 3 0 399 0 73 4 38 0 0 400 35 3 9 0 6 40 59 35 6 3 0 54 40 6 7 0 9 0 0 403 4 47 9 35 0 37 404 5 46 38 0 37 405 4 70 5 0 0 0 406 33 6 3 0 0 3 407 8 64 8 0 0 6 408 53 89 6 6 0 48 409 38 97 8 36 0 8 40 5 84 6 38 0 4 4 8 7 5 30 0 7 4 45 7 4 7 0 37 43 4 85 3 0 0 3 44 4 43 4 30 0 36 45 3 94 3 4 0 6 46 50 79 0 3 0 9 47 4 4 7 7 0 34 48 7 50 3 38 49 46 87 0 38 40 6 95 4 5 4 57 8 37 4 43 9 6 34 43 54 7 6 6 44 3 93 7 0 45 40 4 6 9 46 55 36 0 7 47 58 80 5 7 48 54 4 7 3 49 59 69 0 3 430 43 69 4 3 43 6 70 9 34 43 7 8 4 433 40 53 7 4 434 6 4 435 6 6 9 436 85 6 3 437 59 35 3 3 438 7 5 0 38 439 40 33 7 0 440 56 96 6 0 44 4 05 7 3 44 47 3 5 39 443 54 86 7 444 74 0 0 445 8 69 4 8 446 55 8 0 447 6 46 4 3 448 7 97 9 3 449 6 84 6 3 450 7 69 7 36 45 9 80 8 35 45 3 3 7 453 7 85 8 5 0 4 0 39 0 5 0 5 0 35 0 50 0 8 0 33 0 9 0 55 0 37 0 36 0 40 5 0 7 4 30 5 36 48 3 40 40 0 6 0 4 5 4 5 7 9 5 454 3 8 4 3 455 68 3 34 456 56 79 8 457 34 98 6 458 4 33 0 459 60 73 8 34 460 30 95 0 39 46 34 69 5 9 46 30 5 5 37 463 4 5 5 33 464 5 96 0 5 465 54 54 5 466 54 99 7 30 467 5 95 0 3 468 0 8 7 469 4 89 9 3 470 49 77 5 47 4 56 4 8 47 7 5 6 4 473 48 77 3 3 474 4 8 7 7 475 4 0 5 476 3 33 8 33 477 3 98 6 8 478 4 55 6 3 479 6 8 7 480 45 68 5 48 3 63 4 48 53 4 6 5 483 46 8 484 48 60 5 485 3 09 3 4 486 46 00 4 3 487 33 9 0 0 488 5 89 3 8 489 35 49 7 5 7 48 4 3 56 8 9 7 38 5 46 5 5 9 47 4 3 4 9 39 5 3 35 5 37 3 6 35 38 9 5 5 9464 3

490 9 55 8 37 49 44 48 3 49 37 63 3 7 493 3 9 9 494 6 85 6 6 495 49 36 6 496 45 6 9 6 497 46 0 7 7 498 3 7 7 499 44 47 7 9 500 0 6 0 6 50 34 47 4 50 4 50 6 6 503 8 59 6 504 6 89 6 7 505 3 83 8 8 506 8 66 9 37 507 0 47 5 0 508 47 67 4 5 509 34 69 6 40 50 6 0 8 3 5 36 48 6 5 44 35 9 4 53 3 3 4 4 54 4 66 9 7 55 54 07 8 9 56 3 53 3 8 57 33 04 0 7 58 7 93 5 36 59 08 9 39 50 45 54 35 5 5 47 4 5 30 4 53 38 60 5 8 54 3 3 55 5 83 6 38 4 43 3 8 4 38 36 0 0 36 8 6 3 0 4 3 6 45 5 5 3 39 9 33 36 9 8 7 0 40 3 9 3 0 47 56 9 40 8 3 57 59 66 0 36 58 49 99 5 4 59 5 44 8 8 530 8 8 6 53 6 39 39 53 40 9 8 35 533 37 8 39 534 46 35 9 3 535 56 53 9 3 536 53 60 3 35 537 4 9 7 538 47 7 5 539 9 48 8 3 540 7 3 9 33 54 38 37 6 0 54 8 9 8 543 3 5 4 40 544 5 49 9 0 545 43 59 8 546 58 49 7 547 5 88 7 6 548 48 9 0 549 35 09 6 3 550 43 93 7 4 55 59 6 3 4 55 96 9 5 553 57 3 5 7 554 53 50 9 3 555 49 60 8 4 556 5 6 8 7 557 49 70 5 5 558 5 64 5 559 33 46 8 3 560 5 58 6 7 56 44 4 40 53 44 4 9 9 36 4 3 5 3 5 3 30 3 9 3 3 3 3 3 6 3 4 3 7 3 3 40 3 49 3 4 3 8 3 33 3 7 3 57 3 9 3 44 3 4 3 44 3 9 3 45 3 9 3 3 4 3 0 56 48 04 3 6 563 9 85 3 564 33 43 6 565 37 6 3 566 49 09 7 6 567 5 5 3 568 6 5 37 569 44 99 7 570 7 66 4 3 57 8 90 5 7 57 5 63 0 38 573 39 6 0 5 574 98 3 8 575 30 70 4 30 576 44 56 0 4 577 3 53 6 7 578 37 3 579 4 90 7 8 580 37 65 8 34 58 54 50 8 3 58 3 44 8 583 5 40 7 38 584 9 40 7 37 585 64 5 3 586 5 99 0 0 587 34 0 0 588 37 7 8 589 4 87 7 0 590 37 76 0 59 43 3 6 8 59 3 46 4 6 593 46 88 4 594 59 07 6 595 3 8 5 596 38 3 36 597 37 75 5 38 3 36 3 4 3 8 3 8 3 8 3 4 3 0 3 33 3 4 3 7 3 4 3 9 3 9 3 7 3 4 3 9 3 3 3 3 3 3 3 4 3 9 3 3 8 3 3 3 6 3 30 3 3 8 3 33 3 3 34 3 5 3 6 3 3 34 9464 4

598 5 75 5 7 599 33 6 9 0 600 36 79 8 5 60 9 37 0 0 60 30 4 3 5 603 3 04 7 604 53 99 7 605 5 7 8 606 46 96 9 5 607 57 8 4 5 608 35 97 4 6 609 54 03 9 5 60 40 78 8 8 6 60 46 8 5 6 30 6 4 9 63 50 06 8 0 64 53 54 0 6 65 3 9 8 3 66 6 00 4 67 68 9 4 6 69 47 6 3 7 60 44 49 9 4 6 53 0 5 3 6 9 50 9 9 63 74 9 64 7 03 8 7 65 8 3 6 34 66 5 4 8 35 67 5 53 6 35 68 40 5 9 36 69 33 07 0 630 49 5 5 63 40 5 5 39 63 90 8 36 633 8 4 0 3 3 4 3 34 3 5 3 5 3 3 47 3 0 3 39 3 33 3 33 3 6 3 7 3 40 4 6 4 34 4 47 4 0 4 5 4 9 4 0 4 46 4 34 4 7 4 4 4 4 3 4 6 4 4 4 44 4 3 4 4 4 4 8 4 0 4 5 634 4 37 5 0 635 35 73 3 7 636 9 9 637 6 58 7 0 638 0 6 6 3 639 54 73 9 9 640 6 3 8 64 4 33 3 64 3 8 8 3 643 47 86 8 0 644 4 40 8 3 645 38 5 646 3 84 9 7 647 4 57 5 8 648 40 3 4 649 57 08 3 650 58 45 3 5 65 5 6 8 7 65 30 9 7 36 653 46 65 8 4 654 37 34 7 7 655 33 8 0 7 656 5 50 5 0 657 39 33 7 33 658 5 58 4 4 659 8 06 5 5 660 54 4 6 66 47 8 0 66 4 70 7 8 663 7 8 8 6 664 55 30 7 30 665 47 79 9 8 666 49 85 9 9 667 3 4 4 8 668 4 83 6 0 669 4 35 6 4 9 4 3 4 7 4 5 4 8 4 43 4 6 4 38 4 7 4 4 0 4 0 4 0 4 37 4 39 4 56 4 53 4 35 4 3 4 8 4 30 4 7 4 4 4 3 4 3 4 5 4 4 9 4 3 4 4 7 4 40 4 36 5 6 5 8 5 670 3 38 7 3 67 54 6 3 38 67 49 93 5 673 57 77 3 36 674 7 84 4 675 59 0 5 0 676 40 85 3 8 677 5 3 6 30 678 4 33 4 34 679 39 75 7 38 680 7 67 5 38 68 3 40 0 68 7 59 6 3 683 3 84 6 37 684 9 03 4 6 685 3 37 5 686 59 85 4 4 687 33 38 33 688 4 54 689 9 99 9 690 0 5 9 4 69 3 77 9 38 69 54 6 7 6 693 9 33 4 694 44 70 9 8 695 3 6 7 4 696 3 04 7 9 697 39 0 8 9 698 6 8 699 57 03 30 700 38 06 8 37 70 37 93 3 35 70 9 57 9 39 703 60 03 3 5 704 4 9 8 5 705 8 6 37 5 8 5 53 5 47 5 56 5 6 5 36 5 39 5 40 5 3 5 8 5 4 5 8 5 4 5 5 7 5 5 55 5 4 5 6 5 3 5 8 5 5 50 5 0 5 8 5 5 9 5 9 5 0 5 46 5 34 5 36 5 6 5 56 5 0 5 8 9464 5

706 8 4 8 707 0 39 0 7 708 30 44 3 709 6 85 7 36 70 8 46 7 5 8 4 9 7 6 33 5 0 73 5 79 7 7 74 3 68 4 5 75 3 0 5 76 44 6 77 43 88 8 5 78 9 44 3 0 79 53 54 5 7 70 58 6 5 7 44 0 3 36 7 48 64 5 36 73 5 36 3 35 74 53 34 0 39 75 34 77 40 76 88 35 77 9 56 7 36 78 37 88 7 3 79 44 6 0 0 730 43 88 7 73 5 54 9 39 73 8 93 4 733 56 7 8 734 56 08 9 3 735 46 88 6 9 736 4 6 5 3 737 57 05 9 39 738 7 04 7 3 739 36 74 7 30 740 6 0 5 33 74 40 00 4 33 5 0 5 6 5 9 5 3 5 3 5 4 5 9 5 45 5 9 5 8 6 0 6 37 6 8 6 49 6 48 6 3 6 34 6 7 6 6 6 3 6 6 7 6 5 6 39 6 34 6 30 6 5 6 0 6 48 6 40 6 3 6 50 6 4 6 30 6 9 6 38 74 5 4 8 0 743 37 3 6 744 47 68 5 745 39 45 9 0 746 56 44 6 36 747 4 8 6 39 748 39 86 8 35 749 59 07 6 750 7 93 8 40 75 39 78 4 8 75 44 75 4 753 4 4 8 754 44 70 8 34 755 0 67 6 36 756 97 9 7 757 57 5 8 5 758 3 99 9 759 4 39 0 3 760 53 84 5 4 76 4 8 3 7 76 6 9 8 3 763 50 97 3 9 764 47 83 7 765 3 65 4 36 766 9 79 8 767 9 6 3 768 9 4 3 769 36 6 9 37 770 5 98 7 9 77 49 3 6 77 5 7 4 0 773 54 55 4 39 774 3 3 0 5 775 37 53 3 5 776 46 09 8 5 777 46 3 9 6 33 6 6 6 44 6 35 6 5 6 6 33 6 38 6 5 6 33 6 43 6 6 37 6 9 6 9 6 8 6 0 6 6 37 6 3 6 5 6 49 6 36 6 7 6 6 6 0 6 3 6 3 6 45 6 6 46 6 5 7 9 7 6 7 40 7 45 778 5 57 4 9 779 4 45 7 5 780 4 4 39 78 39 50 8 78 6 33 9 783 53 74 6 4 784 4 6 7 785 9 53 0 4 786 3 34 5 5 787 0 0 788 7 88 6 6 789 4 6 9 9 790 56 35 7 79 37 46 4 38 79 33 99 8 0 793 4 33 8 8 794 37 8 9 795 57 40 8 6 796 0 797 4 96 3 4 798 44 57 6 7 799 48 9 9 5 800 38 36 5 3 80 8 8 3 9 80 5 6 6 803 54 45 6 8 804 3 3 6 6 805 5 03 5 6 806 50 46 0 6 807 83 7 6 808 54 76 6 809 50 54 5 4 80 37 30 4 8 47 69 7 8 58 9 5 83 39 75 7 0 7 44 7 4 7 8 7 3 7 3 7 3 7 5 7 6 7 6 7 9 7 4 7 7 36 7 3 7 3 7 8 7 6 7 4 7 9 7 3 7 34 7 9 7 3 7 7 7 9 7 8 7 3 7 3 7 36 7 9 7 47 7 7 7 30 7 36 7 3 7 33 9464 6

84 40 3 7 30 85 38 07 8 39 86 5 6 0 87 39 37 6 38 88 37 97 9 6 89 45 53 4 3 80 37 33 3 8 36 3 8 3 86 3 83 3 5 6 84 0 68 8 4 85 30 37 6 86 54 8 3 87 58 56 0 3 88 69 4 6 89 8 90 7 830 44 9 7 8 83 0 30 6 38 83 50 38 6 8 833 57 94 0 834 0 8 30 835 0 6 6 836 39 39 39 837 34 9 5 8 838 5 43 0 5 839 4 7 5 3 840 4 78 9 38 84 7 7 3 84 3 40 8 7 843 56 55 0 37 844 3 86 6 3 845 3 80 8 4 846 47 3 6 6 847 44 4 5 848 7 77 7 34 849 7 33 5 7 9 7 9 7 4 7 3 7 34 7 3 7 9 7 9 7 3 7 7 9 7 4 7 7 48 7 0 7 4 7 40 7 9 7 5 8 43 8 7 8 9 8 7 8 6 8 8 8 3 8 7 8 4 8 0 8 43 8 8 8 8 8 30 8 4 8 4 850 9 65 7 7 85 39 5 6 3 85 5 88 3 0 853 8 95 4 36 854 4 44 3 4 855 30 5 0 8 856 6 8 7 6 857 3 94 0 35 858 59 67 7 4 859 56 6 4 9 860 39 4 7 86 9 39 3 0 86 55 65 5 4 863 5 4 7 39 864 0 9 9 38 865 36 5 4 866 40 86 7 4 867 9 5 3 0 868 7 89 9 4 869 3 04 7 870 40 47 3 30 87 39 90 6 9 87 3 75 8 873 5 7 6 0 874 4 80 0 35 875 37 3 0 3 876 66 9 5 877 39 00 0 4 878 49 9 7 35 879 38 43 9 8 880 4 89 3 5 88 0 99 9 3 88 4 5 4 883 49 8 884 33 50 9 5 885 53 3 0 6 8 7 8 0 8 50 8 5 8 4 8 5 8 9 8 8 35 8 5 8 34 8 6 8 35 8 4 8 0 8 3 8 30 8 3 8 3 8 3 8 8 8 3 8 9 8 46 8 0 8 30 8 9 8 7 8 4 8 9 8 3 8 5 8 0 8 8 9 8 3 886 59 36 7 9 887 9 83 3 888 5 3 0 36 889 8 04 8 40 890 30 63 8 89 4 88 5 9 89 45 46 6 35 893 43 76 7 6 894 4 07 5 3 895 3 94 6 9 896 54 98 0 36 897 53 70 5 8 898 54 9 3 899 9 9 8 900 50 38 4 9 90 44 36 8 9 90 4 83 0 5 903 3 8 6 4 904 55 9 3 8 905 49 5 3 5 906 43 36 6 907 5 65 0 33 908 8 99 8 3 909 0 5 9 5 90 50 75 0 4 9 4 0 5 0 9 34 7 5 5 93 43 30 0 38 94 3 7 6 9 95 4 00 6 96 3 5 9 97 53 5 5 30 98 55 6 8 99 3 8 6 3 90 40 7 9 43 88 7 8 8 8 6 8 35 8 4 8 4 8 3 8 38 8 33 8 33 8 8 49 8 48 8 38 8 7 8 40 8 3 8 0 8 3 8 4 8 48 8 3 8 4 8 5 9 8 9 6 9 37 9 3 9 9 9 9 9 8 9 6 9 9 9 8 9 4 9 36 9464 7

9 7 6 4 7 93 34 7 6 94 5 66 7 34 95 30 57 9 3 96 68 6 8 97 8 47 4 3 98 8 65 6 5 99 54 49 6 0 930 5 97 0 93 9 56 5 34 93 44 5 8 933 9 86 8 3 934 9 0 7 935 5 3 33 936 6 99 0 3 937 4 5 0 9 938 46 05 8 939 4 3 5 38 940 5 66 6 9 94 59 3 8 4 94 54 5 9 943 7 43 4 944 6 4 3 945 4 76 7 9 946 5 9 3 5 947 35 68 8 4 948 3 60 5 3 949 39 78 5 0 950 38 43 8 7 95 55 86 9 95 55 50 9 6 953 5 40 5 7 954 58 46 0 34 955 48 39 3 8 956 55 4 957 5 40 6 5 9 6 9 8 9 43 9 0 9 9 9 6 9 5 9 36 9 9 9 4 9 30 9 7 9 9 4 9 3 9 5 9 3 9 9 6 9 6 9 49 9 0 9 7 9 3 9 3 9 9 9 9 4 9 9 9 7 9 8 9 3 9 4 9 34 9 8 9 0 958 34 5 7 34 959 57 7 960 47 38 6 35 96 33 4 3 3 96 4 5 3 963 5 7 0 0 964 45 5 9 965 7 76 6 3 966 04 6 33 967 48 05 4 4 968 3 05 8 969 54 57 7 970 40 89 9 33 97 0 87 7 4 97 7 00 36 973 55 70 5 6 974 40 38 6 39 975 6 37 976 34 48 8 38 977 43 70 6 978 49 4 8 3 979 4 89 4 6 980 37 49 3 3 98 40 5 4 98 4 5 3 9 983 7 5 4 9 984 30 65 9 36 985 8 5 3 8 986 59 75 9 3 987 34 93 8 9 988 46 69 0 989 5 78 4 33 990 49 05 7 7 99 3 99 9 3 99 39 44 6 33 993 8 90 8 8 9 5 9 9 9 38 9 3 9 8 9 6 9 39 9 6 9 9 45 9 6 9 4 0 33 0 0 7 0 36 0 33 0 0 3 0 4 0 5 0 39 0 6 0 9 0 0 5 0 0 5 0 34 0 9 0 3 0 49 0 44 0 0 37 0 6 994 44 5 6 995 46 7 4 30 996 60 6 7 4 997 6 43 4 34 998 59 98 8 3 999 43 58 0 00 0 3 09 3 3 0 30 0 44 0 3 0 3 0 47 0 34 0 5 9464 8

Table A.: Fits and Diagnostics for Unusual Observations Minitab Obs # Simulated Throughput Fit Resid Std Resi d 7.49 4.875-7.66 -.5 R 47 5.96 6.49-0.33-3.04 R 6 5.8 39.48-3.659-4.03 R 67 5.64 33.00-7.737 -.8 R 93 54.573 47.70 6.870.0 R 98 7.05 35.38-8.77 -.44 R 37.545 9.8-7.737 -.8 R 44 3.06 39.5-7.045 -.08 R 63 4.430.60-8.90 -.4 R 69 4.0.89-7.797 -.30 R 80 7.6 6.503-9.4 -.73 R 06 3.640.697-8.058 -.37 R 07.396 9.47-6.85 -.0 R 9 6.893 35.330-8.437 -.49 R 39 3.35 30.85-6.834 -.0 R 70 3.8 33.377-9.564 -.8 R 85 6.699 5.79-8.480 -.50 R 94 9.935 8.45-8.58 -.5 R 309 9.934 37.9-7.85 -. R 30.03 33.68 -.54-3.69 R 347 55.583 48.56 7.37.6 R 380 5.80.989-7.809 -.30 R 395 30.664 40.683-0.09 -.96 R 398 8.35 35.660-7.30 -.5 R 46 7.8 5.84-8.0 -.36 R 49.68.356-9.88 -.7 R 48.659 30.854-8.96 -.4 R 59 0.880 9.00-8.30 -.39 R 50 30. 39.6-9.039 -.66 R 530 5.950 6.554-0.605-3. R 580 0.554 9.00-8.456 -.49 R 589 0.376 3.88 -.505-3.69 R 6.870 30.769-7.899 -.33 R 67 6.950 6.680-9.730 -.87 R 69 8.47 3.74-3.53-6.86 R 63 9.605.480 7.5.0 R 645 7.478 9.7 -.44-3.6 R 656 8.078 37.403-9.34 -.75 R 67 0.367 8.46-8.050 -.37 R 688 3.948.599-8.65 -.55 R 689 6.66 33.038-6.87 -.03 R 705 5.79 33.87-8.04 -.36 R 77 5. -.70 6.8.0 R 746 3.468 0.630-7.6 -. R 75 53.53 46.439 6.83.0 R 790 49.530 4.705 6.85.0 R 88 8.86 35.567-7.80 -.5 R 87 9.07 9.933-0.86-3.0 R 9464-9 -

83 3.889 4.5-9.6 -.73 R 860 4.35 3.848-9.53 -.8 R 904 0.749 35.38-4.489-4.7 R 9 3.47 37.94-4.46-4.7 R 9 6.75 3.388-5.638-4.6 R 93.675 3.576-8.90 -.6 R 945.635 0.93-7.558 -.3 R 988 9.08 37.480-8.398 -.47 R 997 6.905 4.543-7.638 -.5 R 999 6.68 34.53-8.085 -.38 R REFERENCES Alden J. M. (00). Estimating performance of two workstations in series with downtime and unequal speeds, Research Publication R&D-9434, General Motors R&D Center, Warren, Michigan. Askin R. G. & Standridge, C. R. (993). Modeling and analysis of manufacturing systems, Wiley, New York. Blumenfeld, D. E. (990). A simple formula for estimating throughput of serial production lines with variable processing times and limited buffer capacity, International Journal of Production Systems, 8, 63 8. Blumenfeld D. E. & Li J. (005). An analytical formula for throughput of a production line with identical stations and random failures, Mathematical Problems in Engineering, 3, 93-308. Enginarlar E., Li J., & Meerkov S. M. (005). How lean can lean buffers be? IIE Transactions 37, 333 34. Fechete F. & Nedelcu A. (0). Analysis of the economic performance of an organization using multiple regression, Procedia Economics and Finance, 3, 509-54 Chattergee S. & Hadi A. S. (0). Regression analysis by example, Wiley, New York. Colledani, M., M. Matta, T. Grasso, & Tolio T. (003). A new analytical method for optimal buffer space allocation in production lines. In 37 CIRP International Seminar on Manufacturing Systems, Budapest, 3 37. Enginarlar E., Li J., Meerkov S. M., & Zhang, R. Q. (00). Buffer capacity for accommodating machine downtime in serial production lines, International Journal of Production Research, 40(3), 60 64 Gershwin, S. B. & Schor J. E. (000). Efficient algorithms for buffer space allocation. Annals of Operations Research 93, 7 44. Gershwin, S. B. & Fallah-Fini S. (007). A general model and analysis of a discrete two-machine production line. In Sixth Conference on the Analysis of Manufacturing Systems, Lunteren, Netherlands, 39 44. 9464-0 -

Di Mascolo, M., David, R., & Dallery, Y. (99). Modelling and analysis of assembly systems with unreliable machines and finite buffers, IIE Transactions, 3(4), 35-330. Nimon K. F. & Oswald F. L., (03). Understanding the Results of Multiple Linear Regression Beyond Standardized Regression Coefficients, Organizational Research Method, 6(4), 650-674. Rekiek B., Dolgui A., Delchambre A., & Bratcu. A. (00). State of the art of optimization method for assembly line, Annual Reviews in Control, 6(), 63-74. Shi C. (009). Efficient buffer design algorithms for production line profit maximization, International Journal of Production Economics, (), December, 75 740 9464 - -