Chaotic Behavior in a Deterministic Model of Manufacturing

Transcription

1 Chaotic Behavior in a Deterministic Model of Manufacturing The Supply Chain & Logistics Institute School of Industrial and Systems Engineering Georgia Institute of Technology Atlanta, GA USA August 8, 2007

2 Imagine... A logistics system with tens of thousands of workers Operates at near optimality No management, no IT department, no industrial engineers, no consultants

3 Imagine... A logistics system with tens of thousands of workers Operates at near optimality No management, no IT department, no industrial engineers, no consultants

4 How should workers coördinate to share work?

5 Bucket brigades Local operating rule Work forward until someone takes your work; then go back and take work from a slower worker.

6 Bucket brigades 1 2 3

7 Bucket brigades 1 2 3

8 Bucket brigades 1 2 3

9 Bucket brigades 1 2 3

10 Bucket brigades 1 2 3

11 Bucket brigades 1 2 3

12 Behavior of bucket brigades Theorem Under bucket brigades, balance emerges spontaneously. In other words, the line balances itself and better than any engineering department could do it!

13 Some users of bucket brigades Anderson Merchandisers (+25%) Harcourt-Brace Blockbuster Music (+27%) McGraw-Hill CVS Drugstores, Inc. (+34%) Radio Shack Dell Computer Readers Digest (+8%) Ford Parts Distribution Centers (+50%) Wawa (+50%) The Gap (+27%) Walgreen s

14 Order-picking in a warehouse

15 Assembling sandwiches

16 Assembling televisions

17 2-worker bucket brigades 0 1 Time until next completion: t = (1 x) /v 2 Distance travelled by worker 1 in that time: v 1 t Location of next hand-off: f (x) = (v 1 /v 2 ) (1 x).

18 The dynamics function 1 f (x) = (v 1 /v 2 ) (1 x)

19 Convergence to fixed point 1 f (x) = (v 1 /v 2 ) (1 x)

20 Convergence to fixed point 1 f (x) = (v 1 /v 2 ) (1 x)

21 Convergence to fixed point 1 f (x) = (v 1 /v 2 ) (1 x)

22 Convergence to fixed point 1 f (x) = (v 1 /v 2 ) (1 x)

23 Convergence to fixed point 1 f (x) = (v 1 /v 2 ) (1 x)

24 Convergence to fixed point 1 f (x) = (v 1 /v 2 ) (1 x)

25 Undesirable self-organization

26 Convergence and chaos Time Time Location Location

27 Deterministic chaos x k+1 = f (x k ) Definition A map is chaotic iff there exists x 0 such that the orbit O(x 0 ) = {x 0, x 1,...} is both dense and unstable in [0, 1].

28 Chaotic dynamics Worker 1 = (1, 1/3) Worker 2 = (1, 1)

29 Chaotic dynamics

30 Chaotic dynamics

31 Chaotic dynamics

32 Chaotic dynamics

33 Sensitive dependence on initial conditions x (n+1) = 1 2x (n) mod 1 There is no least-significant bit!

34 Sensitive dependence on initial conditions x (n+1) = 1 2x (n) mod 1 There is no least-significant bit! ?

35 Sensitive dependence on initial conditions x (n+1) = 1 2x (n) mod 1 There is no least-significant bit! ? ??

36 Sensitive dependence on initial conditions x (n+1) = 1 2x (n) mod 1 There is no least-significant bit! ? ?? ???

37 Sensitive dependence on initial conditions x (n+1) = 1 2x (n) mod 1 There is no least-significant bit! ? ?? ??? ????

38 Sensitive dependence on initial conditions x (n+1) = 1 2x (n) mod 1 There is no least-significant bit! ? ?? ??? ???? 0. 0?????

39 Sensitive dependence on initial conditions x (n+1) = 1 2x (n) mod 1 There is no least-significant bit! ? ?? ??? ???? 0. 0????? 0.??????......

40 Symptoms of chaos Sensitive dependence on initial conditions Periodic orbits are unstable Dense orbits Cannot be reliably simulated Time Seemingly random behavior Location

41 Seemingly random start/intercompletion times Cumulative Percentage Intercompletion Time

42 Strange attractors Position v

43 Variability Process variability Fluctuations in processing time Unforeseen outages Setups Worker availability

44 Variability Process variability Fluctuations in processing time Unforeseen outages Setups Worker availability Flow variability Interarrival time of work

45 Variability Process variability Fluctuations in processing time Unforeseen outages Setups Worker availability Flow variability Interarrival time of work Deterministic chaos

46 For more information Web page: jjb Post: Professor Supply Chain & Logistics Institute School of Industrial and Systems Engineering Georgia Institute of Technology Atlanta, GA USA