Home > 6 Sigma, Lean, Theory of Constraint > The Dice Game of “Velocity” – Part 1

The Dice Game of “Velocity” – Part 1

I have just finished reading “Velocity: Combining Lean, Six Sigma and the Theory of Constraints to Achieve Breakthrough Performance – A Business Novel” with my Kindle. The author Jeff Cox is the co-author of  “The Goal“. This time the story is about Amy, the newly named president of Hi-T Composites Company could not get any bottom line improvement after implementing Lean Six Sigma for a year. In the end, she convinced her team to combine TOC with LSS approach in order to achieve and exceed the bottom line goal.

A critical piece of the story is a dice game. It is this dice game that has finally got everyone on the same page, including the stubborn LSS guy Wayne to change his approach. A key insight is to abandon the balanced line approach at which Wayne has been working. The team finally has agreed on changing to an unbalanced production with everything synchronized to the bottleneck.

In the book, Amy was betting her career on this dice game to convince her staffs as well as to generate the same results in actual production. It worked out that way in the novel. But in practice, would you bet your career on a dice game? I cannot held to ask the following questions:

  • How repeatable are the results of the dice game  described in the novel? How sound is the statistics behind it?
  • How close is the game in resemblance to the reality of a production line? What are the limitations? Under what conditions would the TOC approach (Drum-Buffer-Rope) work better or worse?
  • Under what conditions does a balanced line with takt time work better or worse than an unbalanced line? How to quantify the variability in order to determine which approach to use?

The book has left these questions unanswered. That means these theories may or may not work at your reality. In order to better understand these questions, I intend to use simulation and analytic techniques to explore further. Stay tuned.

In Scenario 1, a balanced line is simulated with everyone starts with a single dice (same capacity) and the same 4 pennies (Initial buffer size).

In this simulation, WIP has increased from 20 to 26 by the 20th round and the total output is 62 pennies. This “throughput” number can be compared to the 70 pennies, which is the average dice point (3.5) times 20 rounds. 62 is in general less than 70 because of throughout lost as a result of variability.

In order to improve the performance of throughput, it was suggested to unbalance the line and create a constraint. Murphy is given only 1 dice while everyone else is then given 2 dices. The results look like the following:

This time WIP has increased from the initial 20 to 42 by te 20th round and total output is 81 pennies. This is significant throughput improvement but with a high WIP, especially around the bottleneck in front of Murphy.

In order to further improve the performance, a DBR (Drum-Buffer-Rope) method is introduced. In this case, Amy’s dices are being taken and she only releases pennies to the line according to the signal given by Murphy on what he rolls. In addition, Murphy is given a higher initial inventory buffer of 12 pennies.

This time WIP has actually decreased from 28 to 23 by the 20th round and the total output is at 91.

In the final case, the team discussed about improving the yield of at the bottleneck through Lean and Six Sigma. In order to simulate this, the dice roll of Murphy is mapped to number betweens 4 to 6.

The results indicated that WIP stayed low at 21 after 20 rounds, the throughput has been further improved 110.

It is shown that the simulation described in the book is generally repeatable. The logic behind these calculations can be nicely summarized with a G/G/1 queue and solved with Markov Chain analysis. We will discussed how practical are these results in application to real production line next time.

  1. David Gerschick
    November 23, 2010 at 6:14 am

    please send me any information on the dice game that you may have.

  2. Michael Pitcher
    November 24, 2010 at 3:05 am

    Nice article. I would appreciate a link to download the excel spreadsheets.

  3. bryan McNamara
    November 25, 2010 at 9:44 am

    I would also like the link to the simulation

    • Insighter
      November 25, 2010 at 12:29 pm

      Thanks for your interest. Please subscribe to the blog through email or RSS feed and the excel program will be sent to you upon confirmation of the subscription. In order to subscribe through email, simply check the box “Notify me of new posts via email.” below your comment input space.

  4. Tap
    November 25, 2010 at 11:37 pm

    Thank you for this contribution. I would like to receive the file also.

  5. November 26, 2010 at 12:01 am

    please send me the download url for excel

  6. Carlos Mejia
    November 26, 2010 at 7:13 am

    INterested in the book and also interested in downloading the excel spreadsheet.


  7. November 29, 2010 at 1:26 am


    Great article! Would love to receive a link to the simulation in Excell.
    I am a TOC practitioner myself, would come in handy during courses I facilitate to ‘newbees’ to TOC.

    Keep on being the change,


  8. Joe
    December 17, 2010 at 2:56 am

    Very interesting! I’d like to receive the excel simulation if possible. Thnaks!

    • Insighter
      December 17, 2010 at 3:28 am

      Thanks for your interest. Please subscribe to the blog through email or RSS feed and the excel program will be sent to you upon confirmation of the subscription. In order to subscribe through email, simply check the box “Notify me of new posts via email.” below your comment input space. A email will then sent to you for confirmation.

  9. osama youssef
    March 16, 2011 at 11:06 am

    very good interested article

    • Insighter
      March 16, 2011 at 12:17 pm

      THanks for your interest.


  10. Arjen Huizinga
    March 27, 2011 at 8:49 am

    Incorrect numbers: can anyone confirm?

    Apart from the questions raised by Insigther I myself wanted to confirm the results as described in the book. To that purpose I have made my own simulation in Excel. BTW this does not mean that I doubt the theory…

    Although I am in the middle of testing so far the statements in the book “Velocity” seem on the positive side to say the least. According to Tom while doing some explaining to Ami during the first game: “55 pennies output in 20 turns is pretty good for a Balanced Line. Most Rounds ended up in the 40’s. One time we hit 60, but never the target of 65 was hit.”

    My results in 12 Simulations of 20 turns:
    Average output: 61
    Output >60: 7 times i.e. 58%
    Minimum Output: 51
    Maximum Output: 70
    I didn’t do more sim’s because if I have these results in the first 12 then it can not be coincidence. Or is it Murphy?

    Can anyone confirm these results?

    • Insighter
      March 27, 2011 at 1:01 pm

      I have sent you an excel sheet that run the every scenario for 100 times and plot the results. The result can be compare to Little’s Law as well. This should answer your question on the validity of the results.


      • Arjen Huizinga
        March 27, 2011 at 1:42 pm

        Thanks James, your sheet confirms my results. My simulation seems ok.
        Very nice sheet: compliments!



    April 11, 2011 at 9:14 pm

    Thanks a lot for this information. I’ve just finished reading this article and I’d like to have the excel sheet that simulates the dice game.

    Best regards,

    Industrial Engineer – Six Sigma Green Belt

    • Insighter
      April 14, 2011 at 9:44 am

      thanks for your interest. The spreadsheet will be sent to you upon the confirmation of your subscriptions to the blog.

  12. Yury
    April 14, 2011 at 4:50 am

    Realy useful article.

  13. April 16, 2011 at 11:54 am

    Thanks very much – best description I’ve seen so far!

  14. Bruce
    June 26, 2011 at 2:16 pm

    Nicely written summary. Thank You!

  15. Candace McHatton
    July 12, 2011 at 6:56 am

    Great information here. I just finished reading both The Goal and Velocity and think you have an ideal summarization of the theory. I would love to have the spreadsheets to do the Dice game. Would you please send them to me?
    Thank you!

  16. Mike
    July 25, 2011 at 6:23 am

    Great Article!

  17. Shaun
    November 16, 2011 at 3:48 pm


    • Insighter
      January 20, 2012 at 8:49 am

      Please confirm your subscription tot the blog and the spreadsheet will be sent to you. Thanks for your interest.


  18. anthony
    December 9, 2011 at 2:00 am

    Great game.

  19. January 18, 2012 at 5:17 pm

    Excellent thinking here

  20. Benito Cuezva
    January 19, 2012 at 6:46 am

    Please send me sheet

    • Insighter
      January 20, 2012 at 8:48 am


      Thanks for your interest. Please confirm your subscription to the blog and the spreadsheet will be sent to you.


  21. January 27, 2012 at 2:49 am

    Excellent summary! Saved some time from me to have the rules written in here. The spreadsheet would be usefull…

  22. October 2, 2012 at 2:27 pm

    In scenario two, it is said they are adding a constraint. In fact, what was done was that every other station doubled capacity and Murphy’s station remained at same capacity as before. What would total throughput be if Murphy’s capacity were doubled as well. I would assume that WIP would not be as high.
    If anyone has indepth knowledge of TOC, please let me know if my assumptions are correct.

    • Insighter
      October 3, 2012 at 8:03 am

      The amount of WIP depends on the variability (measured by standard deviation) of the performance at each work station. Increasing the capacity by allowing each station to cast 2 dice (varing from 1-12 instead of varying from 1-6 uniformly) will increase the overall variability and henece will increase not only the throughput but the average amount of WIP in the system.

  23. ndiarka mbodji
    October 7, 2012 at 11:13 am


    can you please send me the spreadsheet for the dice game simulation?

    • Insighter
      October 11, 2012 at 6:41 am

      thanks for you interest. Please subscrib to the blog in order to have to the spreadsheet sent to you.


  24. carles guillamon
    October 16, 2012 at 6:26 am

    excelent article. I “discovered” TOC a few years ago and I’m still enthusiasted with it. Velocity was also my last reading on TOC.

  25. Duane Meckelborg
    November 22, 2012 at 8:32 pm

    I have started work as a material specialist, raw material, steel and forms. The company started by 6 brothers in small town Saskatchewan Canada has now grown to 3 facilities with approximately 450 staff. The growth has become logistically challenging with supply and coordination of materials and meeting output targets as the major focus of our Strategic Planning Team. I have listened to Velocity and The Goal on audio books on cd during my commute. I would like this spreadsheet to make a presentation to the Team. I am very impressed with this style of presentation on the subject. Thank You. Duane Meckelborg

    • Insighter
      November 27, 2012 at 11:52 pm

      PLease subscribe to the blog. the spreadsheet will then be sent to you. thanks for your interest.

  26. December 4, 2012 at 6:10 am

    We’re in the process of developing a continuos improvement program utilizing Lean and 6 Sigma, I have a T.O.C back ground and am reading Velocity. I want to play the dice game with our Department Managers. Thanks

  27. Sebastien Levesque
    December 11, 2012 at 7:13 am

    I would love a copy of the Velocity Penny/Dice Game Excel File! The article is really well crafted and practical!

  28. April 17, 2013 at 1:55 pm

    I would like to subscribe to the blog. I had great success implementing TOC in my last position. I am looking for a way to show the entire velocity idea to my new team. This spreadsheet would save me hours. Thank you.

    • Insighter
      May 23, 2013 at 11:31 pm

      Lost track on whether I have sent it to you. Let me know if I have not.

  29. Feraldo
    May 23, 2013 at 12:34 pm

    Could you do one for a CONWIP system

    • Insighter
      May 23, 2013 at 11:30 pm

      sounds worth trying. thanks.

  30. Josh
    May 28, 2013 at 12:24 pm

    Loved ‘Velocity.’ Especially excited to demonstrate the simulation game to others.

  31. Brian Burns
    June 7, 2013 at 5:01 am

    Great article

  32. Brian Burns
    June 7, 2013 at 5:20 am

    We have started a tranformation based on the principles of this book.

    • Insighter
      June 10, 2013 at 6:32 pm

      Interesting. Share you transformation story when you have a chance. Hope it was successful.

  33. Dave Beaudet
    June 23, 2013 at 4:52 am

    We are going through a similar problem and solution now where we balanced our production line, but variation is causing us to fall off schedule. I am attempting to identify how to find the constraint, buffer it and use six sigma to minimize variation in it. I will definetly use this simulation next week to help explain my concepts. Thanks for the breakdown from the book.

  34. Barbara Taylor
    October 9, 2013 at 6:37 pm

    Very interesting approach and well written. I look forward to having the spreadsheets to use.

  35. Fuat
    July 16, 2014 at 10:41 am

    A very nice Illustration of the theory.
    Thank you!

  36. September 3, 2014 at 6:26 pm

    Thanks for putting this together. We are using this for an Operations meeting during our Lean discussion.

  37. Dave
    April 21, 2016 at 4:46 pm

    I would like a copy of this spreadsheet. Thanks!

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