1 / 18

Approximate Three-Stage Model: Active Learning – Module 3

Approximate Three-Stage Model: Active Learning – Module 3. Dr. Cesar Malave Texas A & M University. Background Material. Any Manufacturing systems book has a chapter that covers the introduction about the transfer lines and general serial systems. Suggested Books:

sian
Download Presentation

Approximate Three-Stage Model: Active Learning – Module 3

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. Approximate Three-Stage Model:Active Learning – Module 3 Dr. Cesar Malave Texas A & M University

  2. Background Material • Any Manufacturing systems book has a chapter that covers the introduction about the transfer lines and general serial systems. • Suggested Books: • Chapter 3(Section 3.4) of Modeling and Analysis of Manufacturing Systems, by Ronald G.Askin and Charles R.Stanridge, John Wiley & Sons, 1993. • Chapter 3 of Manufacturing Systems Engineering, by Stanley B.Gershwin, Prentice Hall, 1994.

  3. Lecture Objectives • At the end of the lecture, each student should be able to • Evaluate the effectiveness (availability) of a three-stage transfer line given the • Buffer capacities • Failure rates for the work stations • Repair rates for the work stations

  4. Time Management • Introduction - 5 minutes • Readiness Assessment Test (RAT) - 5 minutes • Lecture on Three Stage Model - 15 minutes • Team Exercise - 15 minutes • Homework Discussion - 5 minutes • Conclusion - 5 minutes • Total Lecture Time - 50 minutes

  5. Approximate Three-Stage Model • Introduction • Markov chains can be used to model transfer lines with any number of stages • The number of states to be considered increases with the number of stages, say M stages with intermediate buffers of capacity Z require 2M(Z+1)M-1 states

  6. Readiness Assessment Test (RAT) • Consider a three-stage line with two buffers • Assume that a maximum of one station is down at a time. • Determine the probability for station i to be down where xi = αi / bi

  7. Three-Stage Model (Contd..) Deeper analysis into the model: • Consider a line without buffers • For every unit produced, station i is down for xi cycles • xiis the ratio of average repair time to uptime • From stations i = 1,…,M, all the other stations are operational except station i • Considering the pseudo workstation 0 with cycle failure and repair rates α0 and β0,we have

  8. 2 2 • Model Analysis: • Let us consider the station 2 and the three types of states it can produce • Production is there when all stations are up • Production is there when station 1 is down, but station 2 operates because of storage utilization from the buffer 1 • Production is there when station 3 is down, but station 2 operates because of storage utilization from the buffer 2 1 3 1 # # Workstation # Buffer #

  9. Model Analysis (contd..): • Let us define hij(Z1,Z2) as the proportion of time station j operates when i is under repair for the specified buffer limits EZ1Z2 = E00 + P1h12(Z1,Z2) + P3h32(Z1,Z2) - Eq 1 • Effectiveness of the line can be calculated by converting the three stage model into a two-stage model with the help of a pseudo work station • Case 1: From buffer 1, there are two possibilities: • Line is down when station 2 is down with failure rateα2 • when station 3 is down and buffer 2 is full - with a failure rate {α3[1 – h32(Z1,Z2)]}

  10. Model Analysis (contd..): • Stations 2 and 3 along with the connecting buffer are replaced by a pseudo station 2’ with a failure rate α2’= α2 +α3[1 – h32(Z1Z2)] - Eq 2 ~ α2 +α3[1 – h32(Z2)] as h32() will not depend on Z1 • Hence for a two-stage line, effectiveness can be written as EZ = E0 + P1h12(Z) - Eq 3 = E0 + P2h21(Z) where • Pi is the probability that station i is down as referred before • h12(Z) is nothing but P1h12(Z1Z2)

  11. Model Analysis (contd..): • Had we known h32(Z1Z2), we could have solved the two pseudo station line using the equations defined for estimating the effectiveness of two-staged lines with buffers and calculated the effectiveness of the three-stage line by substituting the values obtained in Eq 1. • The question is do we know the value of h32(Z1Z2) ? The answer is no ! • Case 2: From buffer 2, there are two possibilities – • Line is down when station 2 is down with a failure rate α2 • when station 1 is down and buffer 1 is empty - with a failure rate {α1[1 – h12(Z1,Z2)]}

  12. Model Analysis (contd..): • Stations 1, 2 and the connecting buffer can be replaced by a pseudo workstation 1’with a failure rate α1’= α2 +α1[1 – h12(Z1Z2)] - Eq 4 ~ α2 +α1[1 – h12(Z1)] as h12() will not depend on Z2 • Station 3 will have a failure rate α3 • The two-stage pseudo line can be solved by estimating h32(Z1Z2) from h21(Z) ofEq 3 • Solvingfor Case 1, i.e. estimating hij() factor is involved as an input for Case 2 and vice versa. Thus by utilizing these two cases, the effectiveness of the three-stage model can be found.

  13. Solution Procedure: 1. Initialize h12(Z1,Z2) at, say, 0.5. Denote stages 1and 2 in any pseudo two-stage approximation as 1’ and 2’, respectively. Calculate E00 , the effectiveness for the unbuffered line. 2. Solve the two-stage line with α1’ given by - Eq 4.Estimate h32(Z1,Z2) = h2’1’(Z) from Eq 3. α2’ = α3 3. Solve the two-stage line with α2’ given by - Eq 2.Estimate h12(Z1,Z2) = h1’2’(Z) from Eq 3. α1’ = α1 If suitable convergence criteria is satisfied, go tostep 4, otherwise go to step 2. 4. Finally, effectiveness for a three-stage line is estimated by EZ1Z2 = E00 + P1h12(Z1,Z2) + P3h32(Z1,Z2)

  14. Team Exercise • A 20-stage transfer line with two buffers is being considered. Tentative plans place buffers of size 15 after workstations 10 and 15. The first 10 workstations have a cumulative failure rate of α = 0.005. Workstations 11 through 15 have a cumulative failure rate of α = 0.01 and workstations 16 through 20 together yield an α = 0.005. Repair of any station would average 10 cycles in length. Estimate the effectiveness of this line design.

  15. Solution Step 1: Set h12(15,15) = 0.5 Step 2: Combine stations 1 and 2 α1’= α2 +α1[1 – h12(15,15)] = 0.01 + 0.005[.5] = 0.0125 α2’= α3 = 0.005.Hence, we find that x1’= 0.125, x2’= 0.05, s = x2’/x1’= 0.4. Using Buzacott’s expression with s ≠ 1,we find C = 0.951898 and E15 = 0.8751. Now, using Eq 3with P2 =x2’/(1+x1’+x2’), we find h32(15,15) ≈ (E15–E0)/P2 = 0.564

  16. Solution (contd..) Step 3: Combine stations 2 and 3 α2’= α2 +α3[1 – h32(15,15)] = 0.01 + 0.005[.436] = 0.01218 α1’= α1 = 0.005.Hence, we find that x1’= 0.05, x2’= 0.1218, s = x2’/x1’= 2.436. Using Buzacott’s expression with s ≠ 1,we find C = 1.04916 and E15 = 0.87740. Now, using the result E15 = E0 + P1h12 , estimate h12(15,15). NowP1 =x1’/(1+x1’+x2’) = 0.04267, we find h12(15,15) ≈ (E15–E0)/P1 = 0.563 As our new estimate of 0.563 differs from our initial guess of 0.5, we return to step 2. As we continue the process, we find that h12(15,15) = h12(15,15) = 0.563 Step 4: Estimate 3-stage effectiveness E15 15 = E00 + P1h12(15,15) + P3h32(15,15) ≈ 0.08333 + [0.05/(1+0.05+0.1+0.05)]*0.563 + [0.05/(1+0.05+0.1+0.05)]*0.563 = 0.88

  17. Homework • Consider a three-stage transfer line with buffers between each pair of stages. Stage I has a failure rate αi and repair rate bi. The maximum buffer sizes are Z1 and Z2 , respectively. Assume geometric failure and repair rates and ample repair workers. • How many states are there for the system? • Consider state (RWWz10) where 0<z1< Z1. Write the balance equation for this state.

  18. Conclusion • Markov chain models can be used to determine the increase in output for a single buffer. • Accurate output determination for a general line with many buffers is a difficult problem.

More Related