1. Flow shop production Object-oriented
Assignment is derived from the item´s 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)
(c) Prof. Richard F. Hartl
Layout and Design

2. 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“)
(c) Prof. Richard F. Hartl
Layout and Design

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 products
Some workstations may be skipped, but going back to a previous department is not possible
Processing times may differ between products
(c) Prof. Richard F. Hartl
Layout and Design

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)
(c) Prof. Richard F. Hartl
Layout and Design

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 beginning of the station.
Further possibilites:
Line stops during processing time
Intermittent transport: workpieces are transported between the stations.
(c) Prof. Richard F. Hartl
Layout and Design

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 < j is true for all (i, j)
(c) Prof. Richard F. Hartl
Layout and Design

7. Example Precedence graph Operation j Predecessor tj 1 - 6 2 9 3 4 5 7
3, 4 8 10
5, 9 11
8,1 12
Precedence graph
(c) Prof. Richard F. Hartl
Layout and Design

8. 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
(c) Prof. Richard F. Hartl
Layout and Design

9. Single product problems
Simple assembly line balancing problem
Basic model with alternative objectives
(c) Prof. Richard F. Hartl
Layout and Design

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
(c) Prof. Richard F. Hartl
Layout and Design

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
(c) Prof. Richard F. Hartl
Layout and Design

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) > c i.e. t(Sk)  c tmax  k =1,...,m-1 
 upper bound
(c) Prof. Richard F. Hartl
Layout and Design

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 
(c) Prof. Richard F. Hartl
Layout and Design

14. Alternative 2 lower bound for cycle time: upper bound for cycle time
(c) Prof. Richard F. Hartl
Layout and Design

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)
(c) Prof. Richard F. Hartl
Layout and Design

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
(c) Prof. Richard F. Hartl
Layout and Design

17. ExampIe T = 7,5 hours Minimum production amount qmin = 600 units
seconds/unit
(c) Prof. Richard F. Hartl
Layout and Design

18. ExampIe tj = 55  No maximum production amount
Arbeitsgang j
Vorgänger tj 1 - 6 2 9 3 4 5 7
3, 4 8 10
5, 9 11
8,1 12
Summe 55
tj = 55 
No maximum production amount
 Minimum cycle time cmin = tmax = 10 seconds/unit
(c) Prof. Richard F. Hartl
Layout and Design

19. ExampIe Combinations of m and c leading to feasible solutions.
(c) Prof. Richard F. Hartl
Layout and Design

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
(c) Prof. Richard F. Hartl
Layout and Design

21. minimale realisierbare Taktzeit c
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
0,982 3 19
0.965 4 14 15
0,917 5 11 12
0.917 6 10
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.
(c) Prof. Richard F. Hartl
Layout and Design

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:
(c) Prof. Richard F. Hartl
Layout and Design

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
(c) Prof. Richard F. Hartl
Layout and Design

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
(c) Prof. Richard F. Hartl
Layout and Design

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
(c) Prof. Richard F. Hartl
Layout and Design

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.
(c) Prof. Richard F. Hartl
Layout and Design

27. LP formulation for given number of stations
Replace mmax by the given number of stations m
c becomes an additional variable
(c) Prof. Richard F. Hartl
Layout and Design

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
(c) Prof. Richard F. Hartl
Layout and Design

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 constraints: c  cmax
c  cmin
(c) Prof. Richard F. Hartl
Layout and Design

30. LP formulation for maximization of BG
Derive a LP again  Weight cycle time and number of stations with factors w and w2
Objective function (linear):
Minimize Z(x,c) = w1(kxnk) + w2c
 Large Lp-models!
 Many binary variables!
(c) Prof. Richard F. Hartl
Layout and Design

31. Heuristic methods in case of given cycle time
Many heuristic methods (mostly priorityrule methods)
Shortened exact methods
Enumerative methods
(c) Prof. Richard F. Hartl
Layout and Design

32. 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
(c) Prof. Richard F. Hartl
Layout and Design

33. Priorityrule methods Requirements: Variables Cycle time c
Operations j=1,...,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
(c) Prof. Richard F. Hartl
Layout and Design

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
(c) Prof. Richard F. Hartl
Layout and Design

35. Priorityrule methods
Start: determine list Ls by applying a prioritiy rule; k := 0; LP := <]; ... No operations assigned so far
Iteration:
repeat
k := k+1; := 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:= < Lp,j]; :=- tj
end;
until Ls = <];
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)
(c) Prof. Richard F. Hartl
Layout and Design

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 G: PVj : = number of arcs in the longest way from a source of the graph to j
(c) Prof. Richard F. Hartl
Layout and Design

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 it´s predecessors:
Rule 7: Choose operations due to monotonuously increasing upper bound for the latest possible station of j:
(c) Prof. Richard F. Hartl
Layout and Design