Flow shop production n Object-oriented n Assignment is

Flow shop production n Object-oriented n Assignment is derived from the item´s work plans. Uniform material flow: n 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 Kapitel 4 / 1

Flow shop production According to time-dependencies we distinguish between n Flow shop production without fixed time restriction for each workstation („Reihenfertigung“) n Flow shop prodcution with fixed time restriction for each workstation (Assemly line balancing, „Fließbandabgleich“) (c) Prof. Richard F. Hartl Layout and Design Kapitel 4 / 2

Flow shop production n 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 ¨ (c) Prof. Richard F. Hartl Layout and Design Kapitel 4 / 3

Flow shop production n 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. ¨ n Production lines, assembly lines ¨ automated system (simultaneous shifting) (c) Prof. Richard F. Hartl Layout and Design Kapitel 4 / 4

Assembly line balancing n 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. n Further possibilites: n n Line stops during processing time ¨ Intermittent transport: workpieces are transported between the stations. ¨ (c) Prof. Richard F. Hartl Layout and Design Kapitel 4 / 5

Assembly line balancing n n „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 Kapitel 4 / 6

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 (c) Prof. Richard F. Hartl Layout and Design Precedence graph Kapitel 4 / 7

Flow shop production n 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 ¨ n 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 ¨ (c) Prof. Richard F. Hartl Layout and Design Kapitel 4 / 8

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

Single product problems Assumptions: n n n n n 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 Kapitel 4 / 10

Alternative 1 Minimization of number of stations m (cycle time is given): Cycle time c: n lower bound for number of stations n upper bound for number of stations (c) Prof. Richard F. Hartl Layout and Design Kapitel 4 / 11

Alternative 1 t(Sk) … workload of station k Sk, k = 1, . . . , m Integer property tmax + t(Sk) > c i. e. t(Sk) c + 1 - tmax k =1, . . . , m-1 Sum of inequalities and integer property of m upper bound (c) Prof. Richard F. Hartl Layout and Design Kapitel 4 / 12

Alternative 2 Minimization of cycle time (i. e. maximization of prodcution rate) lower bound for cycle time c: n n n 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 Kapitel 4 / 13

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

Alternative 3 Maximization of efficiency („Bandwirkungsgrad“) n Determination of: Cycle time c ¨ Number of stations m ¨ Efficiency („BG“) n BG = 1 100% efficiency (no idle time) (c) Prof. Richard F. Hartl Layout and Design Kapitel 4 / 15

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

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

Examp. Ie 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 (c) Prof. Richard F. Hartl Layout and Design tj = 55 No maximum production amount Minimum cycle time cmin = tmax = 10 seconds/unit Kapitel 4 / 18

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

Examp. Ie n maximum BG = 1 (is reached only with invalid values m = 1 and c = 55) n 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 Kapitel 4 / 20

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. (c) Prof. Richard F. Hartl Layout and Design Kapitel 4 / 21

Further objectives Maximization of BG is equivalent to n Minimization of total processing time („Durchlaufzeit“): D=m c n Minimization of sum of idle times: n Minimization of ratio of idle time: n Minimization of total waiting time: (c) Prof. Richard F. Hartl Layout and Design LA = = 1 – BG Kapitel 4 / 22

LP formulation We distinguish between: n LP-Formulation for given cycle time n LP-Formulation for given number of stations n Mathematical formulation for maximization of efficiency (c) Prof. Richard F. Hartl Layout and Design Kapitel 4 / 23

LP formulation for given cycle time n Binary variables: j = 1, . . . , n k = 1, . . . , mmax n = number of station, where operation j is assigned to n Assumption: Graph G has only 1 sink, which is node n (c) Prof. Richard F. Hartl Layout and Design Kapitel 4 / 24

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

Notes Possible extensions: n Assignment restrictions (for utilities or positions) ¨ n 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 Kapitel 4 / 26

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

LP formulation for given number of stations Objective function: Minimize Z(x, c) = c Constraints: … cycle time c 0 integer j = 1, . . . , n . . . j on exactly 1 station k = 1, . . . , m . . . cycle time . . . precedence cond. j und k. . . binary variables (c) Prof. Richard F. Hartl Layout and Design Kapitel 4 / 28

LP formulation for maximization of BG n 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 Kapitel 4 / 29

LP formulation for maximization of BG n 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 xnk) + w 2 c Large Lp-models! Many binary variables! (c) Prof. Richard F. Hartl Layout and Design Kapitel 4 / 30

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

Priorityrule methods n Determine a priortity value PVj for each operations j n Prioritiy list n 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 Kapitel 4 / 32

Priorityrule methods n Requirements: Cycle time c ¨ Operations j=1, . . . , n with processing times tj c ¨ Precedence graph, defined by a sets of precedessors. ¨ n Variables k ¨ ¨ Lp ¨ Ls ¨ (c) Prof. Richard F. Hartl number of current station idle time of current station set of already assigned operations sorted list of n operations in respect to priority value Layout and Design Kapitel 4 / 33

Priorityrule methods n Operation j Lp can be assigned, if tj and h Lp is true for all h V(j) n Start with station 1 and fill one station after the other n From the list of operations ready to be assigned to the current station the highest prioritized is taken n Open a new station if the current station is filled to the maximum (c) Prof. Richard F. Hartl Layout and Design Kapitel 4 / 34

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 Kapitel 4 / 35

Priorityrule methods n Rule 1: Random choice of operations n Rule 2: Choose operations due to monotonuously decreasing (or increasing) processing time: PVj: = tj n Rule 3: Choose operations due to monotonuously decreasing (or increasing) number of direct followers: PVj : = (j) n 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 Kapitel 4 / 36

Priorityrule methods n Rule 5 Choose operations due to monotonuously decreasing positional weight („Positionswert“): n Rule 6: Choose operations due to monotonuously increasing upper bound for the minimum number of stations needed for j and all it´s predecessors: : n 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 Kapitel 4 / 37