Professor Arne Thesen, University of Wisconsin-Madison

1

An Efficient Real Time Scheduling Scheme for Certain Flexible Manufacturing Systems

Arne ThesenDepartment of Industrial EngineeringUniversity of Wisconsin-MadisonMadison, WI USA

Professor Arne Thesen, University of Wisconsin-MadisonProfessor Arne Thesen, University of Wisconsin-Madison Slide 2

1 This talk Problem is to develop simple but efficient control scheme for a given

class of production systems Pre-production analysis

– Rule independent bounds on performance– Introduce three state-independent schemes– Optimal state-dependent scheduling

Evaluation– Both analytic, and simulation results

Conclusion– Very good simple scheduling rules can be found

Professor Arne Thesen, University of Wisconsin-MadisonProfessor Arne Thesen, University of Wisconsin-Madison Slide 3

1.1 Example Type 1 parts are processed on the cell and on machine 1, Type 2 parts are processed on the cell and on machine 2. Type 3 parts are processed on the cell and on machine 3.

Mean proces-sing time is 2Cell

Meanprocessing

time ism1=0.24m2=0.48m3=0.72

Additional stations

Buffer, Capacity = 3

Machine 1

Machine 2 Additional stations

Buffer, Capacity = 3Mean proces-sing time is 3

Machine 3 Additional stations

Buffer, Capacity = 3Mean proces-sing time is 1

Professor Arne Thesen, University of Wisconsin-MadisonProfessor Arne Thesen, University of Wisconsin-Madison Slide 4

1.2 The Real-Time Scheduling Problem Determine in real time what part should be processed next at a cell

– A number of different parts are available for processing– Processing times are not known with certainty

The cell feeds a number of machines– Information about current and future states is limited

The expected production rate for the overall system should be maximized.

The best control system is no control system

Professor Arne Thesen, University of Wisconsin-MadisonProfessor Arne Thesen, University of Wisconsin-Madison Slide 5

1.3 Four Real Time Decision Rules Random (Push)

– Next part is selected at random, probabilities reflect product mix– Cell often blocked

Rotation (Push)– Parts produced in fixed sequence– Sequences for some mixes may be difficult to develop

Circulating tokens (Pull)– A fixed number of part-specific tokens rotate in a FIFO manner– Token mix established from part routing and product mix

SMDP (Push)– Optimal state-dependent rule if Markov assumptions hold

Professor Arne Thesen, University of Wisconsin-MadisonProfessor Arne Thesen, University of Wisconsin-Madison Slide 6

1.4 The Problem: More Details

Parts Parts of M different types are produced.There is an unlimited supply of raw materials for all parts.All parts produced by the system can be sold.

Facilities All parts are first processed at a cellThey then continue to separate assembly lines for each part type.The first station in each line is the bottleneck; hence subsequent stationsneed not be analyzed.Buffer capacity (bi) in front of each line i is restricted.

Processing Processing times are exponentially a distributed random variables with the following parameters:mi = Mean processing time at the machining center for parts of type i.ai = Mean processing time at station i for parts of type i.

Control Cell: Parts may be processed in any sequence. The identity of the part tobe processed next is determined at the time processing start.Assembly lines: Any part in the input buffer may be selected for processing.

AvailableInformation

Three types of information is available:Processing times (actual or expected)System states (Buffer full/ not full, expected completion times time)Product mix (Target and/or actual)

Objective Maximize parts produced per hour.

Professor Arne Thesen, University of Wisconsin-MadisonProfessor Arne Thesen, University of Wisconsin-Madison Slide 7

1.5. The Problem: Previous Work1984 Yao and Buzacott PSQ heuristic: predictable performance

1988 Seidmann Introduced Semi-Markov Decision Processes(SMDPs) for control policies for manufacturingsystems.

1991 Yih & Thesen Extend SMDP to applications where transitionprobabilities were difficult to establish analytically

1992 Chen Extended Seidmann’s work to non-exponentials

1991 Chen and Thesen Reviews other policies for real time control.1993 Stidham and

WeberReviews other policies for real time control

1996 Thesen & Chen Unequal buffer allocations

1996 Thesen State independent rules

Professor Arne Thesen, University of Wisconsin-MadisonProfessor Arne Thesen, University of Wisconsin-Madison Slide 8

2. Pre-Production Analysis Bound on Performance Three state independent schemes Optimization

Professor Arne Thesen, University of Wisconsin-MadisonProfessor Arne Thesen, University of Wisconsin-Madison Slide 9

2.1 A Bound on PerformanceIgnoring issues of blocking and queuing delays, linear programming can be used to establish

an upper bound on expected profit:Maximize z= x1 + x2 + x3 (Production per unit time)

Subject to: 0.24 x1 + 0.48 x2 + 0.72 x3 <= 1 (Capacity of cell)

2 x1 <= 1 (Capacity of machine 1) 3 x2 <= 1 (Capacity of machine 1) 1 x3 <= 1 (Capacity of machine 1)

Where:xi = Parts of type i produced per unit time

The optimal production rate is: z= 110 parts per hour, and x1 = 30 pph, x2 = 20 pph, x3 = 60 pph

The corresponding product mix is: x1 = 27.3%, x2 = 18.2%, x3 = 54.5%

Professor Arne Thesen, University of Wisconsin-MadisonProfessor Arne Thesen, University of Wisconsin-Madison Slide 10

2.2 The Rotation schedule The bound suggests that we produce parts in the following proportions

– 30/110 of Part 1, 20/110 of Part 2 and 60/110 of Part 3. Thus the following sequence is feasible;

– 1, 1, 1, 2, 2, 3, 3, 3, 3, 3, 3,… However, to avoid blocking parts should be evenly spaced:

– 3, 1, 3, 2, 3, 1, 3, 2, 3, 1, 3, … The resulting product mix is

– 27.3%, 18.2% and 54.2% of parts 1, 2 and 3.

Professor Arne Thesen, University of Wisconsin-MadisonProfessor Arne Thesen, University of Wisconsin-Madison Slide 11

2.3 Circulating tokens Must arrive at Cell at a rate equal to the desired production rate Round trip times depends on token count and processing times Queuing theory man be used to estimate proper # of tokens Optimal initial token sequence is : 3, 1, 2, 3 , 1 , 2

Cell

Machine 1

Machine 2

Machine 3

Professor Arne Thesen, University of Wisconsin-MadisonProfessor Arne Thesen, University of Wisconsin-Madison Slide 12

2.4 A heuristic for allocation of tokens

Step Tokens Throughput Add1 Add 2 Add 3 Maxstates

0 0, 0, 0 0 29.6 19.6 46.1 3

1 0, 0, 1 46.1 72.6 62.4 51.4 13

2 1, 0, 1 72.6 74.0 87.3 76.7 79

3 1, 1, 1 87.3 89.1 87.8 90.5 168

4 1, 1, 2 90.5 93.1 91.2 91.7 427

5 2, 1, 2 93.1 94.0 94.0 94.9 1282

6 2, 1, 3 94.9 96.1 95.9 89.3 3057

7 3, 1, 3 96.1 78.6 91.1 91.8 8207

Throughput estimated from steady-state Markov balance equations

Professor Arne Thesen, University of Wisconsin-MadisonProfessor Arne Thesen, University of Wisconsin-Madison Slide 13

2.5 Optimization: Semi-Markov Decision Processes

Assuming that– All system states can be enumerated (next slide)– Decisions in a given state are always made the same way , and,– Processing times are exponentially distributed.

Then we can compute steady state probabilities for– being in each state,– making any state transition.

If rewards are given for some transitions (e.g. “make part”),– expected profit for given set of decisions can be computed,– dynamic programming can be used to find optimal set of decisions.

Resulting decisions can form “rule-base” for optimal state dependent scheduling system

Optimal decisions for state space with 50,000 states easily obtained

Professor Arne Thesen, University of Wisconsin-MadisonProfessor Arne Thesen, University of Wisconsin-Madison Slide 14

2.5 State transition diagram for case with two machines, each with one buffer space

Probabilistic states

(?;--;bb)

Decision States

Blocked State

(? ; - - ; - -)(a;--;--)

(?;-a;--)

(a;-a;--) (b;-a;--) (a;--;-b) (b;--;-b)

(?;aa;--)

(b;aa;--)

(?;-a;-b)

(?;--;-b)

(b;--;--)

(a;--;bb)(b;-a;-b)(a;-a;-b)

(?;aa;-b)

(?;aa;bb)

(b;aa;-b) (a;-a;bb)

(?;-a;bb)

Cell Buffer 1 Machine 1 Buffer 2 Machine 2Part of type a in cellMachines empty

Professor Arne Thesen, University of Wisconsin-MadisonProfessor Arne Thesen, University of Wisconsin-Madison Slide 15

2. 5. The optimal rule-base Thesen and Chen found the following optimal policy

L.P. BOUND 110 Parts/Hour OPTIMAL (No blocking) 109 Parts/Hour OPTIMAL (Blocking) 95 Parts/Hour

Space Available In Start Production of Part of

Line 1? Line 2? Line 3? Type 1 Type 2 Type 3

Y I I Y

N Y I Y

N N I Y

Professor Arne Thesen, University of Wisconsin-MadisonProfessor Arne Thesen, University of Wisconsin-Madison Slide 16

3. Evaluation Example problem

– Simulation– Markov process

Other scenarios Blocking avoidance

Simulation results are averages for 10,000,000 partsAnalytic results are obtained for statespaces of up to 100,000 states

Professor Arne Thesen, University of Wisconsin-MadisonProfessor Arne Thesen, University of Wisconsin-Madison Slide 17

3.1 The example proble

Rule Parameters Throughput Product mix

Random 0.273, 0.182, 0.545 69.6 pph 27.3, 18.2, 54.5

Rotation 3, 1, 3, 2, 3, 1, 3, 2,3, 1, 3, …

80.1 pph 27.3, 18.2, 54.5

Tokens t1=3, t2=1, t3=3 96.1 pph 29.3, 17.0, 53.7

SMDP if space then 1, else if space then 2 else 3

94.9 pph 30.6, 20.5, 48.9

Bound 110.0 pph 27.3, 18.2, 54.5

Professor Arne Thesen, University of Wisconsin-MadisonProfessor Arne Thesen, University of Wisconsin-Madison Slide 18

3.2 Additional Scenarios

Mean ProcessingTimes at Cell

Ideal ProductMix (%)

Expected MachineUtilization (%)

m1 m2 m3 p1 p2 p3 Cell m1 m 2 m 3

1 0.12 .24 .36 27 18 55 50 100 100 1002 0.24 .48 .72 27 18 55 100 100 100 1003 .375 .75 1.13 37 25 38 100 100 100 504 0.60 1.20 1.80 50 33 17 100 100 100 175 1.00 2.00 3.00 67 33 0 100 100 75 06 .36 .24 0.12 27 18 55 38 100 100 1007 .72 .48 0.24 27 18 55 76 100 100 1008 1.13 .75 .375 20 20 60 100 67 100 1009 1.8 1.20 0.6 0 25 75 100 0 100 100

10 3.00 2.00 1.00 0 0 100 100 0 0 100

Mean processing times at machines 1, 2 and 3 are 2, 3, and 1 minutes

Professor Arne Thesen, University of Wisconsin-MadisonProfessor Arne Thesen, University of Wisconsin-Madison Slide 19

3. 2 Rules for scenarios 1 - 10Scenario

Ideal Mix Rotation Sequence Tokens

12

27.3, 18.2, 54.5 3, 1, 3, 2, 3, 1, 3, 2, 3, 1, 3, 2-2-33-1-3

3 37.5,25,37.5 1, 3, 2, 1, 3, 1, 3, 2, . 3-3-24 50, 33.3, 16.7 1, 2, 1, 2, 1, 3, ... 3-3-15 67, 33, 0 1, 1, 2, ... 3-2-067

27.3, 18.2, 54.5 3, 1, 3, 2, 3, 1, 3, 2, 3, 1, 3, 2-3-32-2-3

8 20, 20, 60 3, 1, 3, 1, 3, ... 1-3-39 0, 25, 75 3, 3, 3, 2 0-1-310 0, 0, 100 3 0-0-3

Professor Arne Thesen, University of Wisconsin-MadisonProfessor Arne Thesen, University of Wisconsin-Madison Slide 20

3.2 Simulation Analysis: Simple rules

0.0

20.0

40.0

60.0

80.0

100.0

120.01 2 3 4 5 6 7 8 9 10

Scenario

Thro

ughp

ut (p

ph)

randomrotationtokensSMDP

Professor Arne Thesen, University of Wisconsin-MadisonProfessor Arne Thesen, University of Wisconsin-Madison Slide 21

3.3 Blocking avoidance

0.0

20.0

40.0

60.0

80.0

100.0

120.0

1 2 3 4 5 6 7 8 9 10

Scenario

Thro

ughp

ut (p

ph)

randomABRotationABtokensSMDP

Professor Arne Thesen, University of Wisconsin-MadisonProfessor Arne Thesen, University of Wisconsin-Madison Slide 22

3.4 Observations Good rules must

– Produce parts in proper mix– Avoid delays due to blocking

The token rule achieves this by– Using a small number of tokens– Using proper combination of tokens

The optimality of the token assignment heuristic must be proven Extensions to other distributions and unequal buffer sizes yield similar

results

Professor Arne Thesen, University of Wisconsin-MadisonProfessor Arne Thesen, University of Wisconsin-Madison Slide 23

4. Conclusion Our goal was to find a simple control scheme for a production system Three state-independent schemes were developed Their performance was compared to an optimal control scheme The token based scheme was found to give near optimal performance A benefit of this scheme is its lack of need for real-time information Future work include

– Analytic estimators of expected throughput for this rule– Proof of optimality for token allocation heuristic