Flow Shop Production

1
Flow Shop Production
http//business.mrwood.com.au/unit3/opstrat/opstra
t1.asp
2
Flow shop layout
cf. Heizer, J., Render, B., Operations
Management, Prentice Hall, 2006, Chapter 9 cf.
Francis, R., McGinnis, L., White, J., Facility
Layout and Location An Analytical Approach,
Prentice Hall, 1992
3
Flow shop production

• Object-oriented
• Assignment is derived from the items work plans.
  
• Uniform material flow
• Linear assignment (in most cases)
• Useful if (and only if) only one kind of product
  or a limited amount of different kinds of
  products is manufactured (i.e. low variety high
  volume)

4
Flow shop production

• According to time-dependencies we distinguish
  between
• Flow shop production without fixed time
  restriction for each workstation
  (Reihenfertigung)
• Flow shop production with fixed time restriction
  for each workstation (Assembly line balancing,
  Fließbandabgleich)

5
Flow shop production

• No fixed time restriction for the workload of
  each workstation
• Intermediate inventories are needed
• Material flow should be similiar for all products
• Some workstations may be skipped, but going back
  to a previous department is not possible
• Processing times may differ between products

6
Flow shop production

• Fixed time restricition (for each workstation)
• Balancing problems
• Cycle time (Taktzeit) upper bound for the
  workload of each workstation.
• Idle time if the workload of a station is
  smaller than the cycle time.
• Production lines, assembly lines
• automated system (simultaneous shifting)

7
Assembly line balancing

• Production rate Reciprocal of cycle time
• The line proceeds continuously.
• Workers proceed within their station parallel
  with their workpiece until it reaches the end of
  the station afterwards they return to the
  beginning of the station.
• Further possibilites
• Line stops during processing time
• Intermittent transport workpieces are
  transported between the stations.

8
Assembly line balancing

• Fließbandabstimmung, Fließbandaustaktung,
  Leistungsabstimmung, Bandabgleich
• The mulit-level production process is
  decomomposed into n operations/tasks for each
  product.
• Processing time tj for each operation j
• Restrictions due to production sequence of
  precedences may occur and are displayed using a
  precedence graph
• Directed graph witout cyles G (V, E, t)
• No parallel arcs or loops
• Relation i lt j is true for all (i, j)

9
Example
Operation j Predecessor tj
1 - 6
2 - 9
3 1 4
4 1 5
5 2 4
6 3 2
7 3, 4 3
8 6 7
9 7 3
10 5, 9 1
11 8,1 10
12 11 1
Precedence graph
10
Flow shop production

• Machines (workstations) are assigned in a row,
  each station contains 1 or more operations/tasks.
  
• Each operation is assigned to exactly 1 station
• i before j , (i, j) ? E
• i and j in same station or
• i in an earlier station than j
• Assignment of operations to stations
• Time- or cost oriented objective function
• Precedence conditions
• Optimize cycle time
• Simultaneous determination of number of stations
  and cycle time

11
Single product problems

• Simple assembly line balancing problem
• Basic model with alternative objectives

12
Single product problems

• Assumptions
• 1 homogenuous product is produced by performing n
  operations
• given processing times ti for operations j
  1,...,n
• Precedence graph
• Same cycle time for all stations
• fixed starting rate (Anstoßrate)
• all stations are equally equipped (workers and
  utilities)
• no parallel stations
• closed stations
• workpieces are attached to the line

13
Alternative1

• Minimization of number of stations m (cycle time
  is given)
• Cycle time c
• lower bound for number of stations
• upper bound for number of stations

14
Alternative 1

• Derivation of upper bound
• t(Sk) workload of station k Sk, k 1, ..., m
  
• Integer property
• Sum of inequalities
• and integer property of m

? tmax t(Sk) gt c i.e. t(Sk) ? c 1 - tmax
? k 1,...,m-1
?
? upper bound
15
Alternative 2

• Minimization of cycle time
• (i.e. maximization of prodcution rate)
• lower bound for cycle time c
• tmax max tj ? j 1, ... , n processing
  time of longest operation ? c ? tmax
• Maximum production amount qmax in time horizon T
  is given
• ?
• Given number of stations m ?

16
Alternative 2

• lower bound for cycle time
• upper bound for cycle time

17
Alternative 3

• Maximization of efficiency (Bandwirkungsgrad)
• Determination of
• Cycle time c
• Number of stations m
• ? Efficiency (BG)
• BG 1 ? 100 efficiency (no idle time)

18
Alternative 3

• Lower bound for cycle time see Alternative 2
• Upper bound for cycle time cmax is given
• Lower bound for number of stations
• Upper bound for number of stations

19
ExampIe

• T 7,5 hours
• Minimum production amount qmin 600 units
• seconds/unit

20
ExampIe
Arbeitsgang j Vorgänger tj
1 - 6
2 - 9
3 1 4
4 1 5
5 2 4
6 3 2
7 3, 4 3
8 6 7
9 7 3
10 5, 9 1
11 8,1 10
12 11 1
Summe   55
?tj 55 ? No maximum production amount ?
Minimum cycle timecmin tmax 10 seconds/unit
21
ExampIe
Combinations of m and c leading to feasible
solutions.
22
ExampIe

• maximum BG 1(is reached only with invalid
  values m 1 and c 55)
• Optimal BG 0,982(feasible values for m and c
  10 ? c ?45 und m ? 2)? m 2 stations? c 28
  seconds/unit

23
Example

• Possible cycle times c for varying number of
  stations m

Stationen m theoretisch min Taktzeit minimale realisierbare Taktzeit c Bandwirkungsgrad 55/c?m
1 55 nicht möglich da c ? 45 -
2 28 28 0,982
3 19 19 0.965
4 14 15 0,917
5 11 12 0.917
6 10 10 0,917
Increasing cycle time ? Reduction of BG
(increasing idle time) until 1 station can be
omitted. BG has a local maximum for each number
of stations m with the minimum cycle time c where
a feasible solution for m exists.
24
Further objectives

• Maximization of BG is equivalent to
• Minimization of total processing time
  (Durchlaufzeit) D m ? c
• Minimization of sum of idle times
• Minimization of ratio of idle time LA
  1 BG
• Minimization of total waiting time

25
LP formulation

• We distinguish between
• LP-Formulation for given cycle time
• LP-Formulation for given number of stations
• Mathematical formulation for maximization of
  efficiency

26
LP formulation for given cycle time

• Binary variables
• number of station, where operation j
  is assigned to
• Assumption Graph G has only 1 sink, which is
  node n

? j 1, ..., n ? k 1, ..., mmax
27
LP formulation for given cycle time

• Objective function
• Constraints

• ? j 1, ... , n ... j on exactly 1 station
• k 1, ... , mmax ... Cycle time
• Precedence cond.
• ... Binary variables

? j and k
28
Notes

• Possible extensions
• Assignment restrictions (for utilities or
  positions)
• elimination of variables or fix them to 0
• Restrictions according to operations
• Operations h and j with (h, j) ? ? are not
  allowed to be assigned to the same station.

29
LP formulation for given number of stations

• Replace mmax by the given number of stations m
• c becomes an additional variable

30
LP formulation for given number of stations

• Objective function Minimize Z(x, c) c
  cycle time
• Constraints
• ? j 1, ... , n ... j on exactly 1 station
• ? k 1, ... , m ... cycle time
• ? ... precedence cond.
• ? j und k ... binary variables

c ? 0 and integer
31
LP formulation for maximization of BG

• If neither cycle time c nor number of stations m
  is given ? take the formulation for given cycle
  time.
• Objective function (nonlinear)
• Additional constraintsc ? cmax
• c ? cmin

32
LP formulation for maximization of BG

• Derive a LP again ? Weight cycle time and number
  of stations with factors w1 and w2
• Objective function (linear)
• Minimize Z(x,c) w1?(?k?xnk) w2?c
• ? Large Lp-models!
• ? Many binary variables!

33
Heuristic methods in case of given cycle time

• Many heuristic methods(mostly priorityrule
  methods)
• Shortened exact methods
• Enumerative methods

34
Priorityrule methods

• Determine a priortity value PVj for each
  operation j
• Prioritiy list
• A non-assigned operation j can be assigned to
  station k if
• all his precedessors are already assigned to a
  station 1,..k and
• the remaining idle time in station k is equal or
  larger than the processing time of operation j

35
Priorityrule methods

• Requirements
• Cycle time c
• Operations j1,...,n with processing times tj ? c
  
• Precedence graph, defined by a set of
  precedessors
• Variables
• k number of current station
• idle time of current station
• Lp set of already assigned operations
• Ls sorted list of n operations in respect to
  priority value

36
Priorityrule methods

• Operation j ? Lp can be assigned, if tj ?
  and h ? Lp is true for all h ? V(j)
• Start with station 1 and fill one station after
  the other
• From the list of operations ready to be assigned
  to the current station the highest prioritized is
  taken
• Open a new station if the current station is
  filled to the maximum

37
Priorityrule methods

• Start determine list Ls by applying a prioritiy
  rule k 0 LP lt ... No operations
  assigned so far
• Iteration
• repeat
• k k1 c
• while there is an operation in list Ls that
  can be assigned to station k do
• begin
• select and delete the first operation j (that
  can be assigned to) from list Ls
• Lp lt Lp,j - tj
• end
• until Ls lt
• Result Lp contains a valid sorted list of
  operations with m k stations.

Single-pass- vs. multi-pass-heuristics
(procedure is performed once or several times)
38
Priorityrule methods

• Rule 1 Random choice of operations
• Rule 2 Choose operations due to monotonuously
  decreasing (or increasing) processing time PVj
  tj
• Rule 3 Choose operations due to monotonuously
  decreasing (or increasing) number of direct
  followers PVj ??(j)?
• Rule 4 Choose operations due to monotonuously
  increasing depths of operations in GPVj
  number of arcs in the longest way from a source
  of the graph to j

39
Priorityrule methods

• Rule 5 Choose operations due to monotonuously
  decreasing positional weight (Positionswert)
  
• Rule 6 Choose operations due to monotonuously
  increasing upper bound for the minimum number of
  stations needed for j and all its predecessors
• Rule 7 Choose operations due to monotonuously
  increasing upper bound for the latest possible
  station of j

40
Example Rule 5
S1 1,3,2,4,6 S2 7,8,5,9,10,11 S3 12
j 1 2 3 4 5 6 7 8 9 10 11 12
tj 6 9 4 5 4 2 3 7 3 1 10 1
PVj(5)
42
31
23
16
20
18
11
15
25
18
1
12
m 3 stations
Cycle time c 28 -gt
BG ?tj / (328) 0,655
41
Example Regel 7, 6 und 2

• 3

j 1 2 3 4 5 6 7 8 9 10 11 12
PVj(7)
PVj(6)
PVj(2)
1
2
1
2
2
2
2
2
2
2
2
2
1
1
1
1
1
1
1
1
1
2
2
2
1
1
10
3
2
6
9
4
5
4
3
7
Apply rule 7 (latest possible station) at
first If this leads to equally prioritized
operatios -gt apply rule 6 (minimum number of
stations for j and all predecessors) If this
leads to equally prioritized operatios -gt apply
rule 2 (decreasing processing times tj)
Solution c 28 ? m 2 BG 0,982 S1
1,3,2,4,5 S2 7,9,6,8,10,11,12
42
More heuristic methods

• Stochastic elements for rules 2 to 7
• Random selection of the next operation (out of
  the set of operations ready to be applied)
• Selection probabilities proportional or
  reciprocally proportional to the priority value
• Randomly chosen priority rule
• Enumerative heuristics
• Determination of the set of all feasible
  assignments for the first station
• Choose the assignment leading to the minimum idle
  time
• Proceed the same way with the next station, and
  so on (greedy)

43
Further heuristic methods

• Heuristics for cuttingpacking problems
• Precedence conditions have to be considered as
  well
• E.g. generalization of first-fit-decreasing
  heuristic for the bin packing problem.
• Shortest-path-problem with exponential number of
  nodes
• Exchange methods
• Exchange of operations between stations
• Objective improvement in terms of the
  subordinate objective of equally utilized stations

44
Worst-Case analysis of heuristics

• Solution characteristics for integer c and tj
• (j 1,...,n) for Alternative 2
• ? Total workload of 2 neigboured stations has to
  exceed the cycle time
• Worst-Case bounds for the deviation of a solution
  with m
• Stations from a solution with m stations

m/m ? 2 - 2/m for even m and m/m ? 2 - 1/m
for odd m m lt c?m/(c - tmax 1) 1
45
Determination of cyle time c

• Given number of stations
• Cycle time unknown
• Minimize cycle time (alternative 1) or
• Optimize cycle time together with the number of
  stations trying to maximize the systems
  efficiency (alternative 3).

46
Iterative approach for determination of minimal
cycle time

1. Calculate the theoretical minimal cycle
   time(or cmin tmax if this is larger) and
   c cmin
2. Find an optimal solution for c with minimum m(c)
   by applying methods presented for alternative 1
3. If m(c) is larger than the given number of
   stations increase c by ? (integer value) and
   repeat step 2.

47
Iterative approach for determination of minimal
cycle time

• Repeat until feasible solution with cycle time ?
  c and number of stations ? m is found
• If ? gt 1, an interval reduction can be applied
  if for c a solution with number of stations ? m
  has been found and for c-? not, one can try to
  find a solution for c-?/2 and so on

48
Example rule 5

• m 5 stations
• Find maximum production rate, i.e. minimum
  cycle time

j 1 2 3 4 5 6 7 8 9 10 11 12
tj 6 9 4 5 4 2 3 7 3 1 10 1
PVj(5) 42 25 31 23 16 20 18 18 15 12 11 1
cmin ?tj/m 55/5 11 (11 gt tmax 10)
49
Example rule 5

• Solution c 11
• 1,3, 2,6, 4,7,9, 8,5, 10,11, 12
• Needed 6 gt m 5 stations
• ? c 12, assign operation 12 to station 5
• ? S5 10,11,12

For larger problems usually, c leading to an
assignment for the given number of stations, is
much larger than cmin. Thus, stepwise increase of
c by 1 would be too time consuming -gt increase by
? gt 1 is recommended.
50
Classification of complex line balancing problems

• Parameters
• Number of products
• Assignment restrictions
• Parallel stations
• Equipment of stations
• Station boundaries
• Starting rate
• Connection between items and transportation
  system
• Different technologies
• Objectives

51
Number of products

• Single-product-models
• 1 homogenuous product on 1 assembly line
• Mass production, serial production
• Multi-product models
• Combined manufacturing of several products on 1
  (or more) lines.
• Mixed-model-assembly Products are variations
  (models) of a basic product ? they are processed
  in mixed sequence
• Lot-wise multiple-model-production Set-up
  between production of different products is
  necessary ? Production lots (the line is
  balanced for each product separately) ?
  Lotsizing and scheduling of products ? TSP

52
Assignment restrictions

• Restricted utilities
• Stations have to be equipped with an adequate
  quantity of utilities
• Given environmental conditions
• Positions
• Given positions of items within a station? some
  operation may not be performed then (e.g.
  underfloor operations)
• Operations
• Minimum or maximum distances between 2 operations
  (concerning time or space)
• ? 2 operations may not be assigned to the same
  station
• Qualifications
• Combination of operations with similiar complexity

53
Parallel stations

• Models without parallel stations
• Heterogenuous stations with different operations
  ? serial line
• Models with parallel stations
• At least 2 stations performing the same operation
• Alternating processing of 2 subsequent operations
  in parallel stations
• Hybridization Parallelization of operations
• Assignment of an operation to 2 different
  stations of a serial line

54
Equipment of stations

• 1-worker per station
• Multiple workers per station
• Different workloads between stations are possible
• Short-term capacity adaptions by using jumpers
• Fully automated stations
• Workers are used for inspection of processes
• Workers are usually assigned to several stations

55
Station boundaries

• Closed stations
• Expansion of station is limited
• Workers are not allowed to leave the station
  during processing
• Open stations
• Workers my leave their station in (rechtsoffen)
  or in reversed (linksoffen) flow direction of
  the line
• Short-term capacity adaption by under- and
  over-usage of cycle time.
• E.g. Manufacturing of variations of products

56
Starting rate

• Models with fixed starting rate
• Subsequent items enter the line after a fixed
  time span.
• Models with variable starting rate
• An item enters the line once the first station of
  the line is idle
• Distances between items on the line may vary (in
  case of multiple-product-production)

57
Connection between items and transportation
systems

• Unmoveable items
• Items are attached to the transportation system
  and may not be removed
• Maybe turning moves are possible
• Moveable items
• Removing items from the transportation system
  during processing is allowed
• Post-production
• Intermediate inventories
• Flow shop production without fixed time
  constraints for each station

58
Different technologies

• Given production technologies
• Schedules are given
• Different technologies
• Production technology is to be chosen
• Different alternative schedules are given
  (precedence graph) and/or
• different processing times for 1 operation

59
Objectives

• Time-oriented objectives
• Minimization of total cycle time, total idle
  time, ratio of idle time, total waiting time
• Maximization of capacity utilization (systems
  efficieny) most relevant for (single-product)
  problems
• Equally utilized stations
• Further objectives
• Minimization of number of stations in case of
  given cycle time
• Minimization of cycle time in case of given
  number of stations
• Minimization of sum of weighted cycle time and
  weighted number of stations

60
Objectives

• Profit-oriented approaches
• Maximization of total marginal return
• Minimization of total costs
• Machines- and utility costs (hourly wage rate of
  machines depends on the number of stations)
• Labour costs often identical rates of labour
  costs for all workers in all stations
• Material costs defined by output quantity and
  cycle time
• Idle time costs Opportunity costs depend on
  cycle time and number of stations

61
Multiple-product-problems

• Mixed model assemblySeveral variants of a basic
  product are processed in mixed sequence on a
  production line.
• Processing times of operations may vary between
  the models
• Some operations may not be necessary for all of
  the variants
• ? Determination of an optimal line balancing and
  of an optimal sequence of models.

62

• multi-model
• Lot-wisemixed-model
• production
• With machine set-up

63

• mixed-model
• Without set-up
• Balancing for a theoretical average model

64
Balancing mixed-model assembly lines

• Similiar models
• Avoid set-ups and lot sizing
• Consider all models simultaneously
• Generalization of the basic model
• Production of p models of 1 basic model with up
  to n operations production method is given
• Given precedence conditions for operations in
  each model j 1,...,n ? aggregated precendence
  graph for all models
• Each operation is assigned to exactly 1 station
• Given processing times tjv for each operation j
  in each model v
• Given demand bv for each model v
• Given total time T of the working shifts in the
  planning horizon

65
Balancing mixed-model assembly lines

• Total demand for all models in planning horizon
• Cumulated processing time of operation j over
  all models in planning horizon

66
LP-Model

• Aggregated model
• Line is balanced according to total time T of
  working shifts in the planning horizon.
• Same LP as for the 1-product problem, but cycle
  time c is replaced by total time T

67
LP-Model

• Objective function

number of the last station (job n)
Constraints for all j 1, ... , n ... Each
job in 1 station for all k 1, ... , n ...
Total workload in station k for all ...
Precedence conditions for all j and k
68
Example
v 1, b1 4 v 2, b2 2
v 3, b3 1 aggregated model
69
Example

• Applying exact method
• given T 70
• Assignment of jobs to stations with m 7
  stationsS1 1,3S2 2 S3 4,6,7 S4
  8,9 S5 5,10 S6 11 S7 12

70
Parameters

• ... Workload of station k for model v in T
• ... Average workload of m stations for model v
  in T
• Per unit
• ... Workload of station k for 1 unit of
  model v
• ... Avg. workload of m stations for 1 unit of
  model v
• Aggregated over all models
• ... Total workload of station k in T

71
Example parameters per unit
?kv       Station k       Avg.
Model v 1 2 3 4 5 6 7 ?v
1 10 7 11 10 6 10 1 7,86
2
3
x 4
x 2
7
8
4
0
7,43
11
11
11
8
13
12
14
3
8
3
8,71
x 1
72
Example - Parameters
?kv       Station k       Avg.
Model v 1 2 3 4 5 6 7 ?v
1 40 28 44 40 24 40 4 31,43
2
3
t(Sk) 70 63 70 70 35 70 7 55
22
8
22
14
16
0
14,86
22
8
12
13
14
3
8
3
8,71
73
Conclusion

• Station 5 and 7 are not efficiently utilized
• Variation of workload ?kv of stations k is higher
  for the models v as for the aggregated model
  t(Sk)
• Parameters per unit show a high degree of
  variation for the models. Model 3, for example,
  leads to an high utilization of stations 2, 3,
  and 4.
• If we want to produce several units of model 3
  subsequently, the average cycle time will be
  exceeded -gt the line has to be stopped

74
Avoiding unequally utilized stations

• Consider the following objectives
• Out of a set of solutions leading to the same
  (minimal) number of stations m (1st objective),
  choose the one minimizing the following 2nd
  objective
• ...Sum of absolute deviation in utilization
• Minimization by, e.g., applying the following
  greedy heuristic

75
Thomopoulos heuristic

• Start Deviation ? 0, k 0
• Iteration until non-assigned jobs are available
• increase k by 1
• determine all feasible assignments Sk for the
  next station kchoose Sk with the minimum sum of
  deviation
• ? ? ?(Sk)

76
Thomopoulos example

• T 70
• m 7
• Solution
• 9 stations (min. number of stations 7)
• S1 1, S2 3,6, S3 4,7, S4 8, S5
  2,
• S6 5,9, S7 10, S8 11, S9 12
• Sum of deviation ? 183,14

77
Thomopoulos heuristic

• Consider only assignments Sk where workload t(Sk)
  exceeds a value ? (i.e. avoid high idle times).
• Choose a value for ?
• ? small
• well balanced workloads concerning the models
• Maybe too much stations
• ? large
• Stations are not so well balanced
• Rather minimum number of stations very large ? ?
  maybe no feasible assignment with t(Sk) ? ?

78
Thomopoulos heuristic Example

• ? 49
• Solution
• 7 stations
• S1 2, S2 1,5, S3 3,4, S4
  7,9,10, S5 6,8, S6 11, S7 12
• Sum of deviation ? 134,57

79
Exact solution

• 7 stations
• S1 1,3, S2 2, S3 4,5, S4 6,7,9 ,
  S5 8,10, S6 11, S7 12
• Sum of deviation ? 126

?kv       Station k       Avg.
Modelv 1 2 3 4 5 6 7 ?v
1 40 28 40 36 32 40 4 31,43
2 22 22 16 12 10 22 0 14,86
3 8 13 7 8 14 8 3 8,71
t(Sk) 70 63 63 56 56 70 7 55
80
Further objectives

• Line balancing depends on demand values bj
• Changes in demand ? Balancing has to be reivsed
  and further machine set-ups have to be considered
• Workaround
• Objectives not depending on demand
• sum of absolute deviations in utilization
  per unit

81
Further objectives

• Disadvantages of this objective
• Large deviations for a station (may lead to
  interruptions in production). They may be
  compensated by lower deviations in other stations
• ? ... Maximum deviation in utilization per
  unit