1. Layout and Design Kapitel 4 / 1 (c) Prof. Richard F. Hartl 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)

2. Layout and Design Kapitel 4 / 2 (c) Prof. Richard F. Hartl 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. Layout and Design Kapitel 4 / 3 (c) Prof. Richard F. Hartl 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. Layout and Design Kapitel 4 / 4 (c) Prof. Richard F. Hartl 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. Layout and Design Kapitel 4 / 5 (c) Prof. Richard F. Hartl 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. Layout and Design Kapitel 4 / 6 (c) Prof. Richard F. Hartl Assembly line balancing „Fließbandabstimmung“, „Fließbandaustaktung“, „Leistungsabstimmung“, „Bandabgleich“ The mulit-level production process is decomomposed into n operations/tasks for each product. Processing time t j 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)

7. Layout and Design Kapitel 4 / 7 (c) Prof. Richard F. Hartl Example Operation jPredecessortjtj 1-6 2-9 314 415 524 632 73, 43 867 973 105, 91 118,110 12111 Precedence graph

8. Layout and Design Kapitel 4 / 8 (c) Prof. Richard F. Hartl 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. Layout and Design Kapitel 4 / 9 (c) Prof. Richard F. Hartl Single product problems Simple assembly line balancing problem Basic model with alternative objectives

10. Layout and Design Kapitel 4 / 10 (c) Prof. Richard F. Hartl Single product problems Assumptions: 1 homogenuous product is produced by performing n operations given processing times t i 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. Layout and Design Kapitel 4 / 11 (c) Prof. Richard F. Hartl 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. Layout and Design Kapitel 4 / 12 (c) Prof. Richard F. Hartl Alternative 1 t(S k ) … workload of station k S k, k = 1,..., m Integer property Sum of inequalities and integer property of m   upper bound  t max + t(S k ) > c i.e.t(S k )  c + 1 - t max  k =1,...,m-1

13. Layout and Design Kapitel 4 / 13 (c) Prof. Richard F. Hartl Alternative 2 Minimization of cycle time (i.e. maximization of prodcution rate) lower bound for cycle time c: t max = max {t j  j = 1,..., n} … processing time of longest operation  c  t max Maximum production amount q max in time horizon T is given  Given number of stations m 

14. Layout and Design Kapitel 4 / 14 (c) Prof. Richard F. Hartl Alternative 2 lower bound for cycle time: upper bound for cycle time

15. Layout and Design Kapitel 4 / 15 (c) Prof. Richard F. Hartl 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. Layout and Design Kapitel 4 / 16 (c) Prof. Richard F. Hartl Alternative 3 Lower bound for cycle time: see Alternative 2 Upper bound for cycle time c max is given Lower bound for number of stations Upper bound for number of stations

17. Layout and Design Kapitel 4 / 17 (c) Prof. Richard F. Hartl ExampIe T = 7,5 hours Minimum production amount q min = 600 units seconds/unit

18. Layout and Design Kapitel 4 / 18 (c) Prof. Richard F. Hartl ExampIe Arbeitsgang jVorgängertjtj 1-6 2-9 314 415 524 632 73, 43 867 973 105, 91 118,110 12111 Summe 55  t j = 55  No maximum production amount  Minimum cycle time c min = t max = 10 seconds/unit

19. Layout and Design Kapitel 4 / 19 (c) Prof. Richard F. Hartl ExampIe Combinations of m and c leading to feasible solutions.

20. Layout and Design Kapitel 4 / 20 (c) Prof. Richard F. Hartl 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. Layout and Design Kapitel 4 / 21 (c) Prof. Richard F. Hartl # Stationen m theoretisch min Taktzeit minimale realisierbare Taktzeit c Bandwirkungsgrad 55/c  m 155 nicht möglich da c  45 - 228 0,982 319 0.965 414150,917 511120.917 610 0,917 Example Possible cycle times c for varying number of stations m 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. Layout and Design Kapitel 4 / 22 (c) Prof. Richard F. Hartl 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. Layout and Design Kapitel 4 / 23 (c) Prof. Richard F. Hartl 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. Layout and Design Kapitel 4 / 24 (c) Prof. Richard F. Hartl 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,..., m max

25. Layout and Design Kapitel 4 / 25 (c) Prof. Richard F. Hartl LP formulation for given cycle time Objective function: Constraints:  j = 1,..., n... j on exactly 1 station  k = 1,..., m max... Cycle time... Precedence cond. ... Binary variables  j and k

26. Layout and Design Kapitel 4 / 26 (c) Prof. Richard F. Hartl 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. Layout and Design Kapitel 4 / 27 (c) Prof. Richard F. Hartl LP formulation for given number of stations Replace m max by the given number of stations m c becomes an additional variable

28. Layout and Design Kapitel 4 / 28 (c) Prof. Richard F. Hartl 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. Layout and Design Kapitel 4 / 29 (c) Prof. Richard F. Hartl 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  c max c  c min

30. Layout and Design Kapitel 4 / 30 (c) Prof. Richard F. Hartl LP formulation for maximization of BG Derive a LP again  Weight cycle time and number of stations with factors w 1 and w 2 Objective function (linear): Minimize Z(x,c) = w 1  (  k  x nk ) + w 2  c  Large Lp-models!  Many binary variables!

31. Layout and Design Kapitel 4 / 31 (c) Prof. Richard F. Hartl Heuristic methods in case of given cycle time Many heuristic methods (mostly priorityrule methods) Shortened exact methods Enumerative methods

32. Layout and Design Kapitel 4 / 32 (c) Prof. Richard F. Hartl Priorityrule methods Determine a priortity value PV j 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. Layout and Design Kapitel 4 / 33 (c) Prof. Richard F. Hartl Priorityrule methods Requirements:  Cycle time c  Operations j=1,...,n with processing times t j  c  Precedence graph, defined by a sets of precedessors. Variables  knumber of current station  idle time of current station  L p set of already assigned operations  L s sorted list of n operations in respect to priority value

34. Layout and Design Kapitel 4 / 34 (c) Prof. Richard F. Hartl Priorityrule methods Operation j  L p can be assigned, if t j  and h  L p 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. Layout and Design Kapitel 4 / 35 (c) Prof. Richard F. Hartl Priorityrule methods Start: determine list L s by applying a prioritiy rule; k := 0; LP := <];... No operations assigned so far Iteration: repeat k := k+1; := c; whilethere 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)

36. Layout and Design Kapitel 4 / 36 (c) Prof. Richard F. Hartl Priorityrule methods Rule 1: Random choice of operations Rule 2: Choose operations due to monotonuously decreasing (or increasing) processing time: PV j : = t j Rule 3: Choose operations due to monotonuously decreasing (or increasing) number of direct followers: PV j : =  (j)  Rule 4: Choose operations due to monotonuously increasing depths of operations in G: PV j : = number of arcs in the longest way from a source of the graph to j

37. Layout and Design Kapitel 4 / 37 (c) Prof. Richard F. Hartl 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: