2.852 Lecture 25: An Integrated Quality/Quantity Model of a Transfer Line Jongyoon Kim Stanley B. Gershwin Copyright c 2003 Jongyoon Kim, Stanley B. Gershwin All right reserved. AGENDA 1. Motivation 2. Research Objectives 3. Research Directions 4. Model Descriptions 5. Preliminary Results 6. Future Research Plan Copyright c 2003 Jongyoon Kim, Stanley B. Gershwin All rights reserved MOTIVATION Quality “Effective implementation of quality program leads to significant wealth creation.” (38% to 46% of higher stock value)* “At 3 sigma level, quality cost accounts for Revenue.”** 25 – 40% of “A dissatisfied customer will tell 9 to 10 people about an unhappy experience, even if the problem is not serious.”*** Goal: Perfect Quality! Copyright c 2003 Jongyoon Kim, Stanley B. Gershwin All rights reserved *Hendricks and Singhal, Management Science **Harry and et al., “6 Sigma” ***Pande, Holpp, “What is 6 Sigma?” MOTIVATION System Effect on Quality Design and control of production system have significant impact on product quality”* • Jaguar** After Ford acquired Jaguar, Jaguar’s quality improved rapidly. Product design wasn’t changed, but production system changed. • Toyota*** Numerous study show that Toyota’s particular attention to the production system’s impact on quality Copyright c 2003 Jongyoon Kim, Stanley B. Gershwin All rights reserved *Inman et al., General Motors R&D center ** Smith, Wards Auto World, July 2001 *** Monden, Toyota Production System MOTIVATION Quality, Quantity and Production System Design *Womack, et al. (1990) Machine that changed the world Is there any relationships among quality, productivity, and production system design? If then, can we quantify them? Copyright c 2003 Jongyoon Kim, Stanley B. Gershwin All rights reserved MOTIVATION Toyota Production System Does reduction of inventory lead to higher quality? Does JIDOKA (stopping the lines whenever abnormalities occur) improve quality and productivity in every case? Existing arguments about this are based on anecdotal experience or qualitative reasoning that lack sound scientific foundation. Scientific conceptual and computational model is needed. Copyright c 2003 Jongyoon Kim, Stanley B. Gershwin All rights reserved RESEARCH OBJECTIVE Mission Gain in-depth understanding to investigate how manufacturing system design and operations simultaneously influence quality and productivity. Copyright c 2003 Jongyoon Kim, Stanley B. Gershwin All rights reserved MOTIVATION System Yield System yield is the fraction of production that is of acceptable quality. System yield depends on •Individual operation yields, •Inspection strategies, •Operation policies, •Environmental conditions in a very complex way. •Buffer sizes, and Comprehensive approaches are needed to manage system yield effectively. Copyright c 2003 Jongyoon Kim, Stanley B. Gershwin All rights reserved Limit of Partial Approach MOTIVATION Focusing on the yield of individual operations gives limited influence on the system yield. M1 B1 M2 B8 M9 B9 M10 For typical mfg. operations, Cp =1.3, which means yield = 99% Probability of non-defective output is •For one operation – 99% •For 10 operations – 90.4% •For 100 operations – 36.6% Copyright c 2003 Jongyoon Kim, Stanley B. Gershwin All rights reserved Limit of Robust Processes MOTIVATION Hypoid gear set factory in Michigan • Lead time : KASPER BORING LATHES (8) KASPER CNC SPINDLE LATHES (12) GLEASON #116 ROUGHERS (57) INCOMING PINION FORGINGS INCOMING RING FORGINGS 3 weeks KASPER TURNING LATHES (8) BARNES DRILLS (4) SNYDER DRILL STANDARD DRILL • Chronic Scrap: 5 - 6 %(50,000 ppm) • Machine capability: GLEASON 606/607 GEAR CUTTERS (43) • Inventory : HEAT TREAT GLEASON #960 (12)ANNEAL CELL GLEASON #116 FINISHERS (64) GLEASON #514 GEAR SET MATCHING PRATT & WHITNEY GRINDERS (14) GLEASON #506 HEAT TREAT 36 LAPPERS 300,000 units ID HONING MACHINES (6) OERLIKON GLEASON #19 24 LAPPERS Cp = 1.67 (233 ppm) GLEASON 17A ROLL TESTERS (21) 36 LAPPERS WHEELABRATOR SHOT-PEEN (7) Assembly Copyright c 2003 Jongyoon Kim, Stanley B. Gershwin All rights reserved MOTIVATION Influence of Material Flow Control A material control scheme can affect the performance of a factory dramatically.* Various practices are used to control material flow. MRP/ERP Kanban Conwip *Bonvik Couch and Gershwin, “A comparison of production line control mechanism”, International Journal of Production Research, 1997 Copyright c 2003 Jongyoon Kim, Stanley B. Gershwin All rights reserved MOTIVATION What has been done? - Quality Control Managerial Practices Statistical Quality Control (SQC) Poka-Yoke -> Early detection of defective parts Total Quality Management (TQM) Six Sigma -> Root cause elimination Academic Approaches Inspection Location Problem -> Only heuristic solutions using simulation have been proposed. Optimal Quality Control Chart Design Copyright c 2003 Jongyoon Kim, Stanley B. Gershwin All rights reserved MOTIVATION Managerial Practices Kanban CONWIP Base Stock MRP ERP Academic Approaches Control Point Policy Hybrid Control Generalized Kanban Extended Kanban Copyright c 2003 Jongyoon Kim, Stanley B. Gershwin All rights reserved What has been done? - Material Flow Control MOTIVATION Missing Links Practices in Quality Control have focused on one operation; each machine is treated separately. But quality is a system-wide problem. Practices in material flow control assume that each machine produces non-defective parts. But capacity is wasted if machines are working on already defective parts. Quality control and material flow control are interrelated and need to be treated together. Copyright c 2003 Jongyoon Kim, Stanley B. Gershwin All rights reserved Ideal Inspections MOTIVATION Production Inspection M1 M1 B1 M2 B8 M9 B9 M10 Ideally, inspection is ubiquitous. Bad parts are caught and scrapped or reworked immediately. No downstream capacity is wasted on parts that will be scrapped. Problems are identified and corrected immediately. Copyright c 2003 Jongyoon Kim, Stanley B. Gershwin All rights reserved Actual Inspections MOTIVATION M1 B1 M2 B8 M9 B9 M10 However, ubiquitous inspection is expensive. Inspection is often done at inspection stations. Question: What is the best distribution of inspection stations? Copyright c 2003 Jongyoon Kim, Stanley B. Gershwin All rights reserved RESEARCH OBJECTIVE Mission Find the best way to achieve high quality with low cost by bringing quality control and material flow control together. Develop a conceptual and computational tool which avoids conflict between productivity and quality. The tool will efficiently assess quality and throughput simultaneously. Copyright c 2003 Jongyoon Kim, Stanley B. Gershwin All rights reserved Research Direction Analytic or Simulation? We choose a continuous material analytical model because Considerably less computation required time than simulation -> Larger searchable design space Same inputs always give same outputs ->Easy to evaluate reliable direction for improvement Continuous optimization is much easier than discrete optimization. But… Tough to develop. Approximate. Copyright c 2003 Jongyoon Kim, Stanley B. Gershwin All rights reserved Research Plan Big picture 1. Immediate Research Plan Characterization 2M 1B Long line decomposition 2. Long term possible research tasks Optimization Case study for PSA 3. Ultimate Goal Redesign factory layouts. Improving inspection policies Revamping material flow controls SAVE BIG MONEY Copyright c 2003 Jongyoon Kim, Stanley B. Gershwin All rights reserved Immediate Research Plan Research Plan Characterization Simulator Construction – For a comparison purpose Zero buffer/ Infinite buffer case Transition Equations 2M-1B case Finding probability density function Boundary conditions Performance measures evaluations Development of a new decomposition method for long line Copyright c 2003 Jongyoon Kim, Stanley B. Gershwin All rights reserved Quality Model Description Two sources of process variation Assignable Variation: variation due to a specific, identifiable cause which changes the process mean. Random Variation: variation that is inherent in the design of the process and cannot be removed. d Tool Change d Mean Assignable Variation Random Variation Copyright c 2003 Jongyoon Kim, Stanley B. Gershwin All rights reserved In this research, we focus on Assignable Variation. REVIEW Quality Failures Two kinds of process variation Common Cause Variation: random variation that is inherent in the design of the process and cannot be removed. Assignable Cause Variation: variation due to a specific, identifiable cause. Two types of quality failures Bernoulli-type: quality failure due to common cause of variation-> quality of each part is independent of others. Markovian-type: quality failure due to assignable cause of variation.-> Once a bad part is produced, all the subsequent parts will be bad until the operation is repaired. Copyright c 2003 Jongyoon Kim, Stanley B. Gershwin All rights reserved Model Description Characterization State of One Machine Objective – Capture important features but try to be simple! •Each machine can produce ‘good’ or ‘bad’ parts •Each machine has inspection 0 p 0’’ r 1 g p p -1 r 1 g -1 f 0 f r r 0’ • How many states do we need? -> as few as possible! Copyright c 2003 Jongyoon Kim, Stanley B. Gershwin All rights reserved Model Description Characterization State of One Machine Each machine has 3 states •State 1:Machine is producing non-defective parts. p g 1 f -1 0 r Copyright c 2003 Jongyoon Kim, Stanley B. Gershwin All rights reserved •State –1: Machine is producing defective parts but the operator doesn’t know it. •State 0: Machine is not operating REVIEW Single Machine Analysis = a speed at which a machine processes material while it is operating and not constrained by the other machine or buffer. p= probability rate that machine fails (=1/MTTF) r= probability rate that machine is repaired (= 1/MTTR) f = rate of transition from state –1 to state 0 (=1/Mean Time to Detect and Stop) -> more inspection leads smaller MTDS & larger f g = rate of transition from state 1 to state –1 (=1/Mean Time to Quality Failure) -> more stable operation leads larger MTQF & smaller g Copyright c 2003 Jongyoon Kim, Stanley B. Gershwin All rights reserved Model Description Characterization State of 2M1B M1 B1 M2 x : total amount of material at B1 y : amount of defective material at B1 State Definition 1: status of machine 1 (1,-1,0) ( x, y , 1 , 2 ) 2 : status of machine 2 (1,-1,0) •Blockage of machine 1 and starvation of machine 2 are dependent on x and independent of y. •Inspection at Machine 2 can trigger state change at machine 1. Copyright c 2003 Jongyoon Kim, Stanley B. Gershwin All rights reserved REVIEW 2M1B Special Cases Production Rate Infinite Buffer Case Infinite Buffer Finite Buffer Zero Buffer Case # 1 2 3 4 5 6 7 8 9 10 Buffer Size Develop formulas with special cases: zero buffer and infinite buffer cases. Results are very good. Copyright c 2003 Jongyoon Kim, Stanley B. Gershwin All rights reserved Pe(A) 0.762 0.708 0.657 0.577 0.527 0.745 0.762 1.524 0.762 1.524 Pe(S) 0.761 0.708 0.657 0.580 0.530 0.745 0.760 1.522 0.762 1.526 % Difference 0.17% 0.00% -0.06% -0.50% -0.42% 0.01% 0.30% 0.14% 0.01% -0.13% Zero Buffer Case Case # 1 2 3 4 5 6 7 8 9 10 Pe(A) 0.657 0.620 0.614 0.529 0.480 0.647 0.706 1.377 0.706 1.377 Pe(S) 0.662 0.627 0.621 0.534 0.484 0.651 0.712 1.526 0.711 1.380 % Difference -0.73% -1.15% -1.03% -0.99% -0.77% -0.57% -0.91% -9.79% -0.77% -0.22% Model Description 2M1B Internal Transition Equations 9 transition equations are derived f (1,1, x, y) f (1,1, x, y) 2 y f (1,1, x, y) (2 1 ) ( p1 g1 p2 g 2 ) f (1,1, x, y) r2 f (1,0, x, y) r1 f (0,1, x, y) t x x y f (1,0, x, y) f (1,0, x, y) p2 f (1,1, x, y) 1 ( p1 g1 r2 ) f (1,0, x, y) f 2 f (1,1, x, y) r1 f (0,0, x, y) t x f (1,1, x, y) f (1,1, x, y) 2 y f (1,1, x, y) g 2 f (1,1, x, y) (2 1 ) ( p1 g1 f 2 ) f (1,1, x, y) r1 f (0,1, x, y) t x x y f (0,1, x, y) f (0,1, x, y) 2 y f (0,1, x, y) p1 f (1,1, x, y) 2 (r1 p2 g 2 ) f (0,1, x, y) r2 f (0,0, x, y) f1 f (1,1, x, y) t x x y f (0,0, x, y) p1 f (1,0, x, y) p2 f (0,1, x, y) (r1 r2 ) f (0,0, x, y) f 2 f (0,1, x, y) f1 f (1,0, x, y) t f (0,1, x, y) f (0,1, x, y) 2 y p1 f (1,1, x, y) g 2 f (0,1, x, y) (r1 f 2 ) f (0,1, x, y) 2 (0,1, x, y) f1 f (1,1, x, y) t x x f (1,1, x, y) f (1,1, x, y) 2 y f (1,1, x, y) g1 f (1,1, x, y) ( p2 g 2 f1 ) f (1,1, x, y) (2 1 ) ( 1 ) r2 f (1,0, x, y) t x x y f (1,0, x, y) f (1,0, x, y) f (1,0, x, y) g1 f (1,0, x, y) 1 1 (r2 f1 ) f (1,0, x, y) p2 f (1,1, x, y) f 2 f (1,1, x, y) t x y f (1,1, x, y) f (1,1, x, y) 2 y f (1,1, x, y) g1 f (1,1, x, y) g 2 f (1,1, x, y) (2 1 ) ( 1 ) ( f1 f 2 ) f (1,1, x, y) t x x y Copyright c 2003 Jongyoon Kim, Stanley B. Gershwin All rights reserved Model Description 2M1B Internal Transition Equations The transition equations are linear partial differential equations in t,x,y with coefficients that are nonlinear functions or x and y. Unlikely to be solved Facts • Starvation and Blockage of machines are independent of y. • Average value of y can be calculated from other formulations. Redefine the 2M1B state as ( x,1 , 2 ) Copyright c 2003 Jongyoon Kim, Stanley B. Gershwin All rights reserved 2M1B REVIEW M1 State Definition ( x,1 , 2 ) B M2 x : total amount of material at B 1 : status of machine 1 (1,-1,0) 2: status of machine 2 (1,-1,0) No part is scrapped: defective parts are marked and reworked later. Copyright c 2003 Jongyoon Kim, Stanley B. Gershwin All rights reserved REVIEW 2M1B Finite Buffer Case Finite buffer case: Approach: Develop Markov process model; write and solve transition equations and boundary equations. When buffer B is neither empty nor full the behavior is described by differential equations with probability density function f(x,1,2). When buffer B is either empty or full, the behavior is described by boundary equations with probability mass functions P(0,1,2), P(N,1,2). Copyright c 2003 Jongyoon Kim, Stanley B. Gershwin All rights reserved REVIEW 2M1B Internal Transition Equations 9 transition equations for f(x,1,2) are derived f ( x,1,1) f ( x,1,1) ( 2 1 ) ( p1 g1 p2 g 2 ) f ( x,1,1) r2 f ( x,1,0) r1 f ( x,0,1) t x f ( x,1,0) f ( x,1,0) p2 f ( x,1,1) 1 ( p1 g1 r2 ) f ( x,1,0) f 2 f ( x,1, 1) r1 f ( x,0,0) t x f ( x,1, 1) f ( x,1, 1) g 2 f ( x,1,1) ( 2 1 ) ( p1 g1 f 2 ) f ( x,1, 1) r1 f ( x,0, 1) t x f ( x,0,1) f ( x,0,1) p1 f ( x,1,1) 2 (r1 p2 g 2 ) f ( x,0,1) r2 f ( x,0,0) f1 f ( x, 1,1) t x f ( x,0,0) p1 f ( x,1,0) p2 f ( x,0,1) (r1 r2 ) f ( x,0,0) f 2 f ( x,0, 1) f1 f ( x, 1,0) t f ( x,0, 1) f ( x,0, 1) p1 f ( x,1, 1) g 2 f ( x,0,1) (r1 f 2 ) f ( x,0, 1) 2 f1 f ( x, 1, 1) t x f ( x, 1,1) f ( x, 1,1) g1 f ( x,1,1) ( p2 g 2 f1 ) f ( x, 1,1) ( 2 1 ) r2 f ( x, 1,0) t x f ( x, 1,0) f ( x, 1,0) g1 f ( x,1,0) 1 (r2 f1 ) f ( x, 1,0) p2 f ( x, 1,1) f 2 f ( x, 1, 1) t x f ( x, 1, 1) f ( x, 1, 1) g1 f ( x,1, 1) g 2 f ( x, 1,1) ( 2 1 ) ( f1 f 2 ) f ( x, 1, 1) t x Copyright c 2003 Jongyoon Kim, Stanley B. Gershwin All rights reserved REVIEW Solution to Internal Transition Equations x Assume a solution in a form of f ( x,1 , 2 ) e G1 (1 )G2 ( 2 ). After much mathematical manipulation, the equations are simplified into 2 equations and 2 unknowns. i x i i f ( x , , ) e G ( ) G It turns out that there are multiple roots i 1 2 1 1 2 ( 2 ) depending on machine parameters. (3 to 7 roots) A general solution to the transition equations is a linear RN combination of the roots. f ( x, 1 , 2 ) ci ei x G1i (1 )G2i ( 2 ) i 1 Copyright c 2003 Jongyoon Kim, Stanley B. Gershwin All rights reserved Model Description 2M1B Internal Transition Equations Assume that f ( x,1 , 2 ) e x G1 (1 )G2 ( 2 ) Then, the transition equations in steady state become {( 2 1 ) ( p1 g1 p 2 g 2 )G1 (1)G2 (1)} r2 G1 (1)G2 (0) r1G1 (0)G2 (1) 0 {1 ( p1 g1 r2 )}G1 (1)G2 (0) p 2 G1 (1)G2 (1) f 2 G1 (1)G2 (1) r1G1 (0)G2 (0) 0 {( 2 1 ) ( p1 g1 f 2 )}G1 (1)G2 (1) g 2G1 (1)G2 (1) r1G1 (0)G2 (1) 0 { 2 (r1 p2 g 2 )}G1 (0)G2 (1) p1G1 (1)G2 (1) r2G1 (0)G2 (0) f1G1 (1)G2 (1) 0 p1G1 (1)G2 (0) p2G1 (0)G2 (1) (r1 r2 )G1 (0)G2 (0) f 2G1 (0)G2 (1) f1G1 (1)G2 (0) 0 { 2 (r1 f 2 )}G1 (0)G2 (1) p1G1 (1)G2 (1) g 2G1 (0)G2 (1) f1G1 (1)G2 (1) 0 {( 2 1 ) ( p2 g 2 f1 )}G1 (1)G2 (1) g1G1 (1)G2 (1) r2G1 (1)G2 (0) 0 {1 (r2 f1 )}G1 (1)G2 (0) g1G1 (1)G2 (0) p2G1 (1)G2 (1) f 2G1 (1)G2 (1) 0 {( 2 1 ) ( f1 f 2 )}G1 (1)G2 (1) g1G1 (1)G2 (1) g 2 G1 (1)G2 (1) 0 Copyright c 2003 Jongyoon Kim, Stanley B. Gershwin All rights reserved 2M1B Transition Equations (5) Model Description If we define i pi Gi (1) G (1) ri f i i Gi (0) Gi (0) i f i gi Gi (1) Gi (1) i pi g i ri Gi (0) Gi (1) 1 2 0 2 1 2 Then, 9 transition equations become Copyright c 2003 Jongyoon Kim, Stanley B. Gershwin All rights reserved (1 2 ) 1 2 1 2 1 (1 2 ) 1 2 1 2 1 2 1 2 (1 2 ) 2 1 (1 2 ) 1 2 Model Description 2M1B Internal Transition Equations Gi (1) Gi (1) Yi , Zi Gi (0) Gi (0) Define And introduce two parameters: M, N p1Y1 r1 f1Z1 ( p2Y2 r2 f 2 Z 2 ) M 1 2 0 1 1 (1 1 1 ) 1 (1 )N 2 Y1 Z1 Y2 Z 2 In addition to that, we already have p1 g1 r1 Y1 f1 g1 Y1 Z1 , p2 g 2 Copyright c 2003 Jongyoon Kim, Stanley B. Gershwin All rights reserved r2 Y2 f2 g2 Y2 Z2 Model Description 2M1B Internal Transition Equations After multiple mathematical manipulation we get {( M r1 )( 1 N 1) f1}2 {( p1 g1 f1 ) r1 ( 1 N 1)}{(M r1 )( 1 N 1) f1} r1 0 (red ) ( f1 p1 )( 1 N 1) ( f1 p1 )( 1 N 1) {( M r2 )( 2 N 1) f 2 }2 {( p2 g 2 f 2 ) r2 ( 2 N 1)}{( M r2 )( 2 N 1) f 2 } r2 0 (blue) ( f 2 p2 )( 2 N 1) ( f 2 p2 )( 2 N 1) It turns out that this formulation is easy to be solved because • Curves only have asymptotes perpendicular to M or N • Locations of gaps at red curves and blue curves are easily calculated • Locations and number of roots are easily estimated Copyright c 2003 Jongyoon Kim, Stanley B. Gershwin All rights reserved Model Description Copyright c 2003 Jongyoon Kim, Stanley B. Gershwin All rights reserved 2M1B Internal Transition Equations Model Description 2M1B Transition Equations (10) Root finding From the characterization of red curves and blue curves what we found are: •There are 3 roots at the top •There are no roots at the bottom •There are maximum 4 roots at RHS or LHS •If there are any roots at RHS, then there are no roots at LHS and vice versa Based on the characterization of the curves, a special algorithm to find all the roots is developed. Copyright c 2003 Jongyoon Kim, Stanley B. Gershwin All rights reserved REVIEW 2M1B Boundary Equations How to determine coefficients ci in probability density functions and probability masses P(0,1 , 2 ), P( N ,1 , 2 ) ? 22 Boundary equations are derived for 1 2 case and solved. ( p1 g1 p2b g2b ) P(0,1,1) r1P(0,0,1) 0 g2b P(0,1,1) ( p1 g1 f 2b ) P(0,1, 1) r1P(0,0, 1) 0 p1 P(0,1,1) r1 P(0,0,1) 2 f (0,0,1) f1P (0, 1,1) r2 P (0,0,0) 0 p1 P(0,1, 1) r1 P(0,0, 1) 2 f (0,0, 1) f1P(0, 1, 1) 0 g1P(0,1,1) ( f1 p2b g2b ) P(0, 1,1) 0 g1P(0,1, 1) g 2b P(0, 1,1) ( f1 f 2b ) P(0, 1, 1) 0 … Copyright c 2003 Jongyoon Kim, Stanley B. Gershwin All rights reserved REVIEW 2M1B Performance Measures After finding all of probability densities and masses, Total production rate is calculated from PT P 1 T N 2 1,0,1 2 1[ { f ( x, 1, 2 ) f ( x,1, 2 )}dx P(0,1, 2 ) P(0, 1, 2 )] 0 1 1,0,1 {P( N , 1 ,1) P( N , 1 , 1)} Average inventory is expressed as N x xf ( x,1 , 2 )dx NP ( N ,1 , 2 ) 1 1,0,1 2 1,0,1 0 Copyright c 2003 Jongyoon Kim, Stanley B. Gershwin All rights reserved 2M1B Performance Measures REVIEW After finding all of probability densities and masses, Effective production rate is calculated from PE where PE2 PT PE2 PT PT N 1[ f ( x,1, 2 )dx P(0,1, 2 )] 2 {P( N ,1, 1) P( N ,1,1)} 2 [ f ( x,1 ,1)dx P( N ,1 ,1)] 1{P(0, 1,1) P(0,1,1)} P 1 E PE1 2 1,0,1 1 1,0,1 0 N 0 Copyright c 2003 Jongyoon Kim, Stanley B. Gershwin All rights reserved Preliminary Result 2M1B Validation Solution for 2M1B with equal production rate case is found and validated through comparison with simulation. Case # 1 2 3 4 5 6 7 8 9 10 PR(Analytic) 0.806 0.855 0.936 0.944 0.909 0.922 0.909 0.925 0.840 0.763 PR(Sim) 0.808 0.858 0.938 0.946 0.911 0.924 0.910 0.926 0.843 0.767 %Difference Inv(Analytic) -0.25% 2.500 -0.37% 25.000 -0.23% 4.709 -0.22% 12.654 -0.19% 2.781 -0.24% 9.213 -0.07% 2.220 -0.18% 7.242 -0.38% 20.020 -0.49% 4.983 Copyright c 2003 Jongyoon Kim, Stanley B. Gershwin All rights reserved Inv(Sim) 2.619 24.883 4.989 12.757 2.832 9.318 2.321 7.080 20.149 5.110 %Difference -4.53% 0.47% -5.60% -0.81% -1.81% -1.13% -4.39% 2.30% -0.64% -2.48% 2M1B Validation (1) REVIEW Analytic solution for 2M1B with 1 2 case is found and validated through comparison with simulation. Average inventory estimation 10.00% 8.00% 8.00% 6.00% 6.00% 4.00% 4.00% -2.00% -4.00% -2.00% -4.00% -6.00% -6.00% -8.00% -8.00% -10.00% -10.00% Case Number Average absolute error = 0.73% Copyright c 2003 Jongyoon Kim, Stanley B. Gershwin All rights reserved Case Number Average absolute error = 2.75% 49 46 43 40 37 34 31 28 25 22 19 16 13 7 10 0.00% 4 49 46 43 40 37 34 31 28 25 22 19 16 13 10 7 4 0.00% 2.00% 1 2.00% % error of Inv 10.00% 1 % error of PE Effective production rate estimation 2M1B with 1 2 Boundary Equations 26 Boundary equations are derived and solved. 1 f (0,1, 0) p2b P(0,1,1) f 2b P(0,1, 1) 1 f (0, 1, 0) p2b P(0, 1,1) f 2b P(0, 1, 1) 2 f ( N ,0,1) 0 2 f ( N , 0, 1) 0 ( p1 g1 p2b g2b ) P(0,1,1) (2 1 ) f (0,1,1) r1P(0,0,1) 0 (2 1 ) f ( N ,1,1) r2 P( N ,1,0) ( 2 1 ) f ( N , 1,1) r2 P( N , 1,0) … Copyright c 2003 Jongyoon Kim, Stanley B. Gershwin All rights reserved 2M1B with 1 2 2M1B Validation (2) Analytic solution for 2M1B with 1 2 case is found and validated through comparison with simulation. Average inventory estimation 10.00% 8.00% 8.00% 6.00% 6.00% 4.00% 4.00% -2.00% 2.00% -4.00% -6.00% -6.00% -8.00% -8.00% -10.00% -10.00% Average absolute error = 0.68% Copyright c 2003 Jongyoon Kim, Stanley B. Gershwin All rights reserved Case Number Average absolute error = 3.41% 49 46 43 40 37 34 31 28 25 22 19 16 13 10 -2.00% -4.00% Case Number 7 0.00% 4 49 46 43 40 37 34 31 28 25 22 19 16 13 10 7 4 0.00% 1 2.00% % error of Inv 10.00% 1 % error of PE Effective production rate estimation Preliminary Result Quality Information Feedback Quality Information Feedback Inspection @ M2 notify M1 is producing bad parts M1 M2 Inspection @ M1 notify M1 is producing bad parts Downstream machines can detect defective parts made by an upstream machine and notify the operator at the machine. Copyright c 2003 Jongyoon Kim, Stanley B. Gershwin All rights reserved Preliminary Result Quality Information Feedback If there is quality information feedback, the yield of the system depends on the time gap between making a defect and identifying the defect. System yield is a function of buffer size: A smaller buffer increases system yield since lower inventory level leads to a smaller the time gap. Copyright c 2003 Jongyoon Kim, Stanley B. Gershwin All rights reserved Quality Information Feedback Preliminary Results Quality feedback slightly decrease total production rate but increases *effective production rate significantly. 0.74 0.79 0.73 0.78 0.72 Effective Production Rate Production Rate 0.8 0.77 0.76 Pr w/o feedback Pr w/ feedback 0.75 0.74 0.71 0.7 0.69 0.68 ePr w/o feedback ePr w/ feedback 0.67 0.73 0.66 0.72 0 5 10 15 20 Buffer Size 25 0.65 0 5 10 15 20 25 Buffer Size *The effective production rate is the rate of production of good parts. Copyright c 2003 Jongyoon Kim, Stanley B. Gershwin All rights reserved How to Increase Quality Preliminary Results Effect of maintenance (MTTD) on quality monotonically decreases. Effect of inspection (MTTI) on quality is monotonically decreases 1 0.8 0.9 0.7 Effective Production Rate Effective Production Rate There is a combination of maintenance and inspection policies that achieve target quality with minimum cost. 0.8 0.7 0.6 0.5 0.5 0.4 0.3 0.4 0.3 0 0.6 0.2 50 100 150 200 250 300 350 Mean Time To Defect 400 450 Copyright c 2003 Jongyoon Kim, Stanley B. Gershwin All rights reserved 500 0.1 0 20 40 60 80 100 120 Mean Time To Identify 140 160 180 How to increase productivity Preliminary Results In some situations, increasing inspection reliability is more effective than increasing buffer size to boost productivity. 0.66 0.93 0.925 0.655 Effective Production Rate Effective Production Rate 0.92 0.65 0.645 0.64 0.915 0.91 0.905 0.9 0.895 0.635 0.89 0.63 0 5 10 15 20 25 Buffer Size 30 35 Copyright c 2003 Jongyoon Kim, Stanley B. Gershwin All rights reserved 40 0.885 0 5 10 15 20 Buffer Size 25 30 35 40 How to increase productivity? Preliminary Result In some situations, increasing machine stability is more effective than increasing buffer size to enhance productivity. 0.33 0.85 0.32 0.84 Effective Production Rate Effective Production Rate 0.31 0.3 0.29 0.28 0.27 0.82 0.81 0.8 0.79 0.26 0.78 0.25 0.24 0 0.83 5 10 15 20 25 Buffer Size 30 35 Copyright c 2003 Jongyoon Kim, Stanley B. Gershwin All rights reserved 40 0.77 0 5 10 15 20 25 Buffer Size 30 35 40 Quality Information Feedback REVIEW Inspection @ M2 M2 M1 Downstream machines can detect defective parts made by an upstream machine and notify the operator at the machine. Inspection @ M1 System yield is a function of buffer size: A smaller buffer increases system yield since lower inventory level leads to a early detection of quality failures. Quality information feedback is captured by modifying f. Copyright c 2003 Jongyoon Kim, Stanley B. Gershwin All rights reserved Quality Information Feedback MFG. SYSTEM BEHAVIOR Quality feedback results in more effective production rate and less total production rate. Increase of buffer size is beneficial contrary to TPS. 0.8 Effective Production Rate Total production Rate 0.8 0.75 Without feedback With feedback 0.7 0.65 0 5 10 15 20 Buffer Size Copyright c 2003 Jongyoon Kim, Stanley B. Gershwin All rights reserved 25 Without feedback With feedback 0.75 0.7 0.65 0 5 10 15 Buffer Size 20 25 MFG. SYSTEM BEHAVIOR Harmful Buffer Case The effective production rate may decrease as the buffer size increases when •M1 is faster than M2 and quality feedback exists. •M1 produces bad parts frequently. •Inspection at M1 is poor and inspection at M2 is good. This is a case when inventory reduction is good as TPS advocates. 0.8 Effective Production Rate 0.75 0.7 Without feedback With feedback 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3 0 10 5 Copyright c 2003 Jongyoon Kim, Stanley B. Gershwin All rights reserved 15 20 25 Buffer Size 30 35 40 Effectiveness of Inspection MFG. SYSTEM BEHAVIOR Effect of inspection (MTDS) on effective production rate decreases as f increases. increases as a line gets longer. B1 M1 M2 B1 0.9 0.9 0.8 0.8 0.7 0.6 0.398 0.5 0.4 0.3 M2 B2 M3 B3 M4 0.7 0.6 0.5 0.560 0.4 0.3 0.2 0.2 0.1 Effective Production Rate Effective Production Rate M1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 f Copyright c 2003 Jongyoon Kim, Stanley B. Gershwin All rights reserved 0.1 0 0.1 0.2 0.3 0.4 0.5 f 0.6 0.7 0.8 0.9 1 Effectiveness of Operation Stabilization MFG. SYSTEM BEHAVIOR Effect of machine stabilization (MTQF) on effective production rate decreases as MTQF increases. increases as a line gets longer. B1 M2 M1 B1 0.9 0.9 0.8 0.8 0.7 0.7 0.6 0.609 0.5 0.4 0.3 0.2 0.1 0 Effective Production Rate Effective Production Rate M1 M2 B2 M3 B3 0.6 0.5 0.701 0.4 0.3 0.2 0.1 0 50 100 150 200 250 300 350 400 450 500 MTQF Copyright c 2003 Jongyoon Kim, Stanley B. Gershwin All rights reserved M4 0 0 50 100 150 200 250 300 350 400 MTQF 450 500 How to increase productivity MFG. SYSTEM BEHAVIOR In some situations, increasing inspection reliability is more effective than increasing buffer size to boost productivity. 0.8 Effective Production Rate 0.75 MTDS = 20 MTDS = 10 MTDS = 2 0.7 0.65 0.6 0.55 0.5 0 5 10 15 Copyright c 2003 Jongyoon Kim, Stanley B. Gershwin All rights reserved 20 25 Buffer Size 30 35 40 How to increase productivity MFG. SYSTEM BEHAVIOR In some situations, increasing machine stability is more effective than increasing buffer size to enhance productivity. 1 Effective Production Rate 0.9 0.8 0.7 MTQF = 20 MTQF = 100 MTQF = 500 0.6 0.5 0.4 0.3 0 5 10 15 Copyright c 2003 Jongyoon Kim, Stanley B. Gershwin All rights reserved 20 Buffer Size 25 30 35 40 LONG LINE Extension Longer line increases the dimension of partial differential equations used at internal transitions equations. No good exact method known for longer lines. It is reasonable to use approximation methods to obtain solutions for transfer lines with more than two machines. Decomposition techniques have been successfully used for various kind of long line analysis. Tandem long line Assembly/disassembly line Closed loop Copyright c 2003 Jongyoon Kim, Stanley B. Gershwin All rights reserved LONG LINE Copyright c 2003 Jongyoon Kim, Stanley B. Gershwin All rights reserved Decomposition Technique Decomposition Technique LONG LINE Decomposition Technique Decompose the line (L) into a set of two-machine lines L(i) in a way that performance measures of the L(i)s are close to those of the original line L. Pseudo-machine Mu(i) models the part of the line upstream of Bi and Md(i) models the part of line downstream from Bi. Decomposition techniques work well even though no mathematical proof is available. Procedures Develop equations for 10(k-1) pseudo-machine parameters for kmachine line. Develop algorithm to solve the equations efficiently. Copyright c 2003 Jongyoon Kim, Stanley B. Gershwin All rights reserved Ubiquitous Inspection Case LONG LINE M1 B1 M2 B2 M3 B3 M4 Each machine has both of operational failures and quality failures Each operation works on different features. Inspection at machine Mi can detect defective features made by Mi not others. For each decomposed line L(i), incoming parts from upstream are viewed as non-defective ones. -> gi is independent of other machine parameters: gu(i) = gi , gd(i) = gi+1 Outgoing defective parts from L(i) are not checked from inspection downstream -> fi is independent of others: fu(i) = fi , fd(i) = fi+1 Copyright c 2003 Jongyoon Kim, Stanley B. Gershwin All rights reserved 3-state-machine / 2-state-machine LONG LINE How to determine u (i), ru (i), pu (i), d (i), rd (i) and pd (i ) ? Strategy: approximate 3-state machine with 2-state machine. The strategy will work well if the transition time from up states (1 or -1) to down state closely follow to the exponential distribution. 0 10 p -1 g 1 f -1 0 1' 0 Frequency distribution 10 p' -2 10 -3 10 -4 10 Operating r Stopped r -5 10 0 100 200 300 400 500 Transition time from 1 to 0 Copyright c 2003 Jongyoon Kim, Stanley B. Gershwin All rights reserved 600 700 3-state-machine / 2-state-machine LONG LINE 10.00% 8.00% 8.00% 6.00% 6.00% 4.00% 4.00% -2.00% -4.00% -6.00% -6.00% -8.00% -8.00% -10.00% Case Number Average absolute error = 0.68% Copyright c 2003 Jongyoon Kim, Stanley B. Gershwin All rights reserved Case Number Average absolute error = 1.07% 97 91 85 79 73 67 61 55 49 43 37 31 25 19 -2.00% -4.00% -10.00% 13 0.00% 7 97 91 85 79 73 67 61 55 49 43 37 31 25 19 13 7 0.00% 2.00% 1 2.00% % error of Inv 10.00% 1 % error of PE A 2-state-machine with parameter adjustments closely approximates the corresponding 3-statemachine. Long Line Validation LONG LINE B1 M2 M3 B2 20.00% 20.00% 15.00% 15.00% 10.00% 10.00% 5.00% 0.00% -5.00% 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 -10.00% % error of Inv 1 % error of PE M1 -15.00% B3 5.00% 0.00% -5.00% 1 2 3 4 5 6 7 8 -10.00% -20.00% Case Number Case Number Average absolute error = 0.25% Average absolute error = 4.21% 20.00% 20.00% 15.00% 15.00% 10.00% 10.00% 5.00% 0.00% 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 -10.00% -15.00% % error of Inv 3 % error of Inv 2 9 10 11 12 13 14 15 16 17 18 19 20 -15.00% -20.00% -5.00% M4 5.00% 0.00% -5.00% 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 -10.00% -15.00% -20.00% -20.00% Case Number Average absolute error = 3.66% Copyright c 2003 Jongyoon Kim, Stanley B. Gershwin All rights reserved Case Number Average absolute error = 2.54% UPCOMING TASK Effective of Jidoka Does JIDOKA (stopping the lines whenever abnormalities occur) improve quality and productivity in every case? Hypothesis The effectiveness of Jidoka on productivity depends on which type of quality failures (Markovian or Bernoulli) is dominant. The effective production rate may decrease by adopting Jidoka practice when quality failures are mixture of Bernoulli and Markovian Copyright c 2003 Jongyoon Kim, Stanley B. Gershwin All rights reserved Next Long Line Analysis Task FUTURE PLAN Operational Failures + Quality Failures M1 B1 Inspection M2 B2 M3 B3 M4 Operational Failures M1 undergoes both of quality failures and operational failures. Other machines (M2, M3 , M4 ) have only operational failures. Inspection takes place only at the final machine (M4) Copyright c 2003 Jongyoon Kim, Stanley B. Gershwin All rights reserved Next Long Line Analysis Task FUTURE PLAN Inspection M1 B1 M2 B2 M3 B3 M4 Operational Failures + Quality Failures Each machine has both of quality failures and operational failures. Inspection is only at the final machine (M4) and detect bad parts made by any of upstream machines. Copyright c 2003 Jongyoon Kim, Stanley B. Gershwin All rights reserved Copyright c 2003 Jongyoon Kim, Stanley B. Gershwin All rights reserved RESEARCH PROGRESS Term 1999 - Spring 2002 Fall 2002 Spring 2003 Summer 2003 Fall 2003 Research Progress Research on Toyota Production Systems . Chacterization. 2M1B model formulation. Simulation building. Infinite buffer and zero buffer validations. Internship at General Motors Proposal to General Motors for collaboration. Thesis proposal finalization. 2M1B model completion. 1st committee meeting (Dec, 15th) Long line decomposition without quality feedback. Validation of the decomposition technique. Spring 2003 Summer 2004 2nd committee meeting (April 7th) Study on Jidoka practice. Long line decomposition with quality feedback. Numerical experimentations and intuition building. 3rd committee meeting (late May/ early June) Possible case study at GM Finalize the research Finish thesis write up Defense (Early September) Copyright c 2003 Jongyoon Kim, Stanley B. Gershwin All rights reserved Supplementary Notes Input Parameters for 2M1B Special Cases Infinite Buffer Case Case # 1 2 3 4 5 Mu1 1.0 1.0 1.0 1.0 1.0 Mu2 1.0 1.0 1.0 1.0 1.0 r1 0.400 0.100 0.100 0.100 0.100 r2 0.100 0.100 0.100 0.100 0.100 p1 0.010 0.010 0.010 0.010 0.010 p2 0.010 0.010 0.005 0.005 0.005 g1 0.001 0.001 0.001 0.005 0.001 g2 0.001 0.001 0.001 0.001 0.001 f1 0.100 0.100 0.100 0.100 0.100 f2 0.500 0.500 0.500 0.500 0.100 g2 0.001 0.001 0.001 0.001 0.001 f1 0.100 0.100 0.100 0.100 0.100 f2 0.100 0.100 0.100 0.100 0.300 Zero Buffer Case Case # 1 2 3 4 5 Mu1 1.0 1.0 1.0 1.0 1.0 Mu2 1.0 1.0 1.0 1.0 1.0 r1 0.400 0.100 0.100 0.100 0.400 r2 0.100 0.100 0.100 0.100 0.100 Copyright c 2003 Jongyoon Kim, Stanley B. Gershwin All rights reserved p1 0.010 0.010 0.010 0.010 0.010 p2 0.010 0.010 0.000 0.010 0.010 g1 0.001 0.001 0.001 0.010 0.001 Supplementary Notes Input Parameters for 2M1B Validation Intermediate Buffer Size Case Case # 1 2 3 4 5 6 7 8 9 10 Mu1 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 Mu2 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 r1 0.100 0.100 0.300 0.300 0.300 0.300 0.200 0.200 0.100 0.100 r2 0.100 0.100 0.300 0.300 0.200 0.200 0.300 0.300 0.100 0.100 p1 0.005 0.005 0.010 0.010 0.010 0.010 0.010 0.010 0.010 0.010 Copyright c 2003 Jongyoon Kim, Stanley B. Gershwin All rights reserved p2 0.005 0.005 0.010 0.010 0.010 0.010 0.010 0.010 0.010 0.010 g1 0.010 0.010 0.005 0.005 0.005 0.005 0.005 0.005 0.005 0.001 g2 0.010 0.010 0.005 0.005 0.005 0.005 0.005 0.005 0.005 0.001 f1 0.100 0.100 0.500 0.500 0.500 0.500 0.500 0.500 0.200 0.200 f2 0.100 0.100 0.500 0.500 0.500 0.500 0.500 0.500 0.200 0.200 N 5 50 10 25 5 15 5 20 40 10 FB? N N N N N N N N Y Y Supplementary Notes Input Parameters for Preliminary Result Quality feedback Mu1 1 Mu2 1 r1 0.1 r2 0.1 p1 0.01 p2 0.01 g1 0.01 g2 0.01 f1 0.1 f2 0.9 Increase Quality Mu1 1 Mu2 1 r1 0.1 r2 0.1 p1 0.01 p2 0.01 g1 0.01 g2 0.01 f1 0.1 f2 0.1 Mu1 1 Mu2 1 r1 0.1 r2 0.1 p1 0.005 p2 0.005 g1 0.001 g2 0.001 Mu1 1 Mu2 1 r1 0.1 r2 0.1 p1 0.01 p2 0.01 f1 0.1 f2 0.1 How to increase productivity (1) How to increase productivity (2) Copyright c 2003 Jongyoon Kim, Stanley B. Gershwin All rights reserved SUPPLEMENTARY NOTES Simplified Internal Transition Equations After much mathematical manipulation. the 9 equations and 7 unknowns are simplified to {( M r1 )( 1 N 1) f1}2 {( p1 g1 f1 ) r1 ( 1 N 1)}{(M r1 )( 1 N 1) f1} r1 0 (red ) ( f1 p1 )( 1 N 1) ( f1 p1 )( 1 N 1) {( M r2 )( 2 N 1) f 2 }2 {( p2 g 2 f 2 ) r2 ( 2 N 1)}{( M r2 )( 2 N 1) f 2 } r2 0 (blue) ( f 2 p2 )( 2 N 1) ( f 2 p2 )( 2 N 1) where p1 1 1 G1 (1) G (1) G (1) G (1) r1 f1 1 ( p2 2 r2 f 2 2 )M G1 (0) G1 (0) G2 (0) G2 (0) (1 1 1 1 ) (1 )N G1 (1) G1 (1) G (1) G ( 1) 2 2 2 G1 (0) G1 (0) 1 G2 (0) G2 (0) Copyright c 2003 Jongyoon Kim, Stanley B. Gershwin All rights reserved SUPPLEMENTARY NOTES Copyright c 2003 Jongyoon Kim, Stanley B. Gershwin All rights reserved Shape of SITEs SUPPLEMENTARY NOTES Case # 1 2 3 4 5 6 7 8 9 10 Mu1 1 1 1 1 1 1 2 3 1 2 Mu2 1 1 1 1 1 1 1 2 2 3 r1 0.1 0.3 0.2 0.1 0.1 0.1 0.1 0.1 0.1 0.1 r2 0.1 0.3 0.05 0.1 0.1 0.1 0.1 0.1 0.1 0.1 Copyright c 2003 Jongyoon Kim, Stanley B. Gershwin All rights reserved Input Parameters for 2M1B Special Cases p1 0.01 0.005 0.01 0.05 0.01 0.01 0.01 0.01 0.01 0.01 p2 0.01 0.005 0.01 0.005 0.01 0.01 0.01 0.01 0.01 0.01 g1 0.01 0.05 0.01 0.01 0.05 0.01 0.01 0.01 0.01 0.01 g2 0.01 0.05 0.01 0.01 0.005 0.01 0.01 0.01 0.01 0.01 f1 0.2 0.5 0.2 0.2 0.2 0.5 0.2 0.2 0.2 0.2 f2 0.2 0.5 0.2 0.2 0.2 0.1 0.2 0.2 0.2 0.2 Input Parameters for Preliminary Results SUPPLEMENTARY NOTES Mu1 1 Mu2 1 Quality Feedback Harmful Buffer Case Effectiveness… Mu1 2 Mu2 1 r1 0.3 r2 0.1 Mu1 1 Mu2 1 Copyright c 2003 Jongyoon Kim, Stanley B. Gershwin All rights reserved r1 0.1 r2 0.1 p1 0.005 p2 0.01 r1 0.1 r2 0.1 p1 0.01 p2 0.01 g1 0.05 g2 0.01 p1 0.01 p2 0.01 g1 0.01 g2 0.01 f1 0.05 f2 0.9 g1 0.01 g2 0.01 f1 0.1 f2 0.9 Byp1 1 Byp2 1 f1 0.2 f2 0.2 Byn1 0 Byn1 0 SAT1 1.832 SAT2 0.835 N 30 Input Parameters for Preliminary Results SUPPLEMENTARY NOTES Intermediate buffer case: same machine speed Case # Mu1 Mu2 r1 r2 p1 p2 g1 g2 f1 f2 N r1 r2 p1 p2 g1 g2 f1 f2 N Yp1 1 1 1 0.1 0.1 0.01 0.01 0.01 0.01 0.2 0.2 30 Yp1 Yp2 Yn1 Yn2 Case # Mu1 Mu2 1 1 0 0 26 1 1 0.1 0.1 0.01 0.1 0.01 0.01 0.2 0.2 30 1 1 0 0 2 1 1 0.1 0.1 0.01 0.01 0.01 0.01 0.2 0.2 5 1 1 0 0 27 1 1 0.1 0.1 0.01 0.001 0.01 0.01 0.2 0.2 30 1 1 0 0 3 1 1 0.1 0.1 0.01 0.01 0.01 0.01 0.2 0.2 20 1 1 0 0 28 1 1 0.1 0.1 0.01 0.01 0.1 0.01 0.2 0.2 30 1 1 0 0 4 1 1 0.1 0.1 0.01 0.01 0.01 0.01 0.2 0.2 50 1 1 0 0 29 1 1 0.1 0.1 0.01 0.01 0 0.01 0.2 0.2 30 1 1 0 0 5 0.5 0.5 0.1 0.1 0.01 0.01 0.01 0.01 0.2 0.2 30 1 1 0 0 30 1 1 0.1 0.1 0.01 0.01 0.01 0.1 0.2 0.2 30 1 1 0 0 6 2 2 0.1 0.1 0.01 0.01 0.01 0.01 0.2 0.2 30 1 1 0 0 31 1 1 0.1 0.1 0.01 0.01 0.01 0.001 0.2 0.2 30 1 1 0 0 7 3 3 0.1 0.1 0.01 0.01 0.01 0.01 0.2 0.2 30 1 1 0 0 32 1 1 0.1 0.1 0.01 0.01 0.01 0.01 0.9 0.2 30 1 1 0 0 8 1 1 0.01 0.01 0.01 0.01 0.01 0.01 0.2 0.2 30 1 1 0 0 33 1 1 0.1 0.1 0.01 0.01 0.01 0.01 0.05 0.2 30 1 1 0 0 9 1 1 0.05 0.05 0.01 0.01 0.01 0.01 0.2 0.2 30 1 1 0 0 34 1 1 0.1 0.1 0.01 0.01 0.01 0.01 0.2 0.9 30 1 1 0 0 10 1 1 0.5 0.5 0.01 0.01 0.01 0.01 0.2 0.2 30 1 1 0 0 35 1 1 0.1 0.1 0.01 0.01 0.01 0.01 0.2 0.05 30 1 1 0 0 11 1 1 0.1 0.1 0.001 0 0.01 0.01 0.2 0.2 30 1 1 0 0 36 1 1 0.1 0.1 0.01 0.01 0.01 0.01 0.2 0.2 5 1 1 0 0 12 1 1 0.1 0.1 0.05 0.05 0.01 0.01 0.2 0.2 30 1 1 0 0 37 1 1 0.1 0.1 0.01 0.01 0.01 0.01 0.2 0.2 20 1 1 0 0 13 1 1 0.1 0.1 0.1 0.1 0.01 0.01 0.2 0.2 30 1 1 0 0 38 1 1 0.1 0.1 0.01 0.01 0.01 0.01 0.2 0.2 50 1 1 0 0 14 1 1 0.1 0.1 0.01 0.01 0.001 0.001 0.2 0.2 30 1 1 0 0 39 1 1 0.2 0.05 0.01 0.01 0.01 0.01 0.2 0.2 5 1 1 0 0 15 1 1 0.1 0.1 0.01 0.01 0.05 0.05 0.2 0.2 30 1 1 0 0 40 1 1 0.2 0.05 0.01 0.01 0.01 0.01 0.2 0.2 20 1 1 0 0 16 1 1 0.1 0.1 0.01 0.01 0.1 0.1 0.2 0.2 30 1 1 0 0 41 1 1 0.2 0.05 0.01 0.01 0.01 0.01 0.2 0.2 50 1 1 0 0 17 1 1 0.1 0.1 0.01 0.01 0.01 0.01 0.02 0.02 30 1 1 0 0 42 1 1 0.05 0.2 0.01 0.01 0.01 0.01 0.2 0.2 5 1 1 0 0 18 1 1 0.1 0.1 0.01 0.01 0.01 0.01 0.5 0.5 30 1 1 0 0 43 1 1 0.05 0.2 0.01 0.01 0.01 0.01 0.2 0.2 20 1 1 0 0 19 1 1 0.1 0.1 0.01 0.01 0.01 0.01 0.95 0.95 30 1 1 0 0 44 20 1 1 0.5 0.1 0.01 0.01 0.01 0.01 0.2 0.2 30 1 1 0 0 45 1 1 1 1 0.05 0.1 0.2 0.1 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.2 0.2 0.2 0.2 50 30 1 0.9 1 0.9 0 0 0 0 21 1 1 0.01 0.1 0.01 0.01 0.01 0.01 0.2 0.2 30 1 1 0 0 46 1 1 0.1 0.1 0.01 0.01 0.01 0.01 0.2 0.2 30 1 1 0.2 0.2 22 1 1 0.1 0.5 0.01 0.01 0.01 0.01 0.2 0.2 30 1 1 0 0 47 1 1 0.1 0.1 0.01 0.01 0.01 0.01 0.2 0.2 30 0.8 1 0 0 23 1 1 0.1 0.01 0.01 0.01 0.01 0.01 0.2 0.2 30 1 1 0 0 48 1 1 0.1 0.1 0.01 0.01 0.01 0.01 0.2 0.2 30 1 0.8 0 0 1 0.1 0.1 0.01 0.01 0.01 0.01 0.2 0.2 30 1 1 0.2 0 1 0.1 0.1 0.01 0.01 0.01 0.01 0.2 0.2 30 1 1 0 0.2 24 1 1 0.1 0.1 0.1 0.01 0.01 0.01 0.2 0.2 30 1 1 0 0 49 1 25 1 1 0.1 0.1 0.001 0.01 0.01 0.01 0.2 0.2 30 1 1 0 0 50 1 Copyright c 2003 Jongyoon Kim, Stanley B. Gershwin All rights reserved Yp2 Yn1 Yn2