Kapitel 4 / 1

1
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)

2
Flow shop production

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

3
Flow shop production

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

4
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)

5
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 begin
  of the station.
• Further possibilites
• Line stops during processing time
• Intermittent transport workpieces are
  transported between the stations.

6
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)

7
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
8
Flow shop production

• Machines (workstations) are assigned in a row,
  each station containing 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 staions
• Time- or cost oriented objective function
• Precedence conditions
• Optimize cycle time
• Simultaneous determination of number of stations
  and cycle time

9
Single product problems

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

10
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

11
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

12
Alternative 1

• 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
13
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 ?

14
Alternative 2

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

15
Alternative 3

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

16
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

17
ExampIe

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

18
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
19
ExampIe
Combinations of m and c leading to feasible
solutions.
20
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

21
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.
22
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

23
LP formulation

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

24
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
25
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
26
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.

27
LP formulation for given number of stations

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

28
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 integer
29
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

30
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!

31
Heuristic methods in case of given cycle time

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

32
Priorityrule methods

• Determine a priortity value PVj for each
  operations 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.

33
Priorityrule methods

• Requirements
• Cycle time c
• Operations j1,...,n with processing times tj ? c
  
• Precedence graph, defined by a sets 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

34
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

35
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)
36
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

37
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