1 / 37

Flow shop production

Flow shop production. Object-oriented Assignment is derived from the item´s work plans. Uniform material flow : Linear assignment (in most cases)

Télécharger la présentation

Flow shop production

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  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) 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 prodcution with fixed time restriction for each workstation (Assemly line balancing, „Fließbandabgleich“) 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 prodcuts • Some workstations may be skipped, but going back to a previous department is not possible • Processing times may differ between products 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) 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 begin of the station. • Further possibilites: • Line stops during processing time • Intermittent transport: workpieces are transported between the stations. 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 tjfor 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) Layout and Design

  7. Example Precedence graph Layout and Design

  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 Layout and Design

  9. Single product problems • Simple assembly line balancing problem • Basic model with alternative objectives 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 Layout and Design

  11. Alternative1 Minimization of number of stationsm (cycle time is given): Cycle time c: • lower bound for number of stations • upper bound for number of stations Layout and Design

  12. Alternative 1 t(Sk) … workload of station kSk, k = 1, ..., m Integer property Sum of inequalities and integer property of m  tmax + t(Sk) > c i.e. t(Sk) c + 1 - tmaxk =1,...,m-1   upper bound 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 qmaxin time horizon T is given  • Given number of stations m Layout and Design

  14. Alternative 2 • lower bound for cycle time: • upper bound for cycle time 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) 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 Layout and Design

  17. ExampIe • T = 7,5 hours • Minimum production amount qmin = 600 units • seconds/unit Layout and Design

  18. ExampIe tj = 55  No maximum production amount  Minimum cycle timecmin = tmax = 10 seconds/unit Layout and Design

  19. ExampIe Combinations of m and c leading to feasible solutions. 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 Layout and Design

  21. 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. 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: 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 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 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 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. 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 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 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 Layout and Design

  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! Layout and Design

  31. Heuristic methods in case of given cycle time • Many heuristic methods(mostly priorityrule methods) • Shortened exact methods • Enumerative methods Layout and Design

  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. Layout and Design

  33. Priorityrule methods • Requirements: • Cycle time c • Operations j=1,...,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 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 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; untilLs= <]; 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) 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 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: Layout and Design

More Related