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

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)



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“)



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

Inventory Station 1 Int. inventory Station 2 ... Station m Inventory



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)

Station 1 Station 2 Station 3 ...


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.


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)


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

t1=61

112

1011 3

9 37

78

26

43

54

..110

t2=92

45

Precedence graph



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


Single product problems

Simple assembly line balancing problem Basic model with alternative objectives


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


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

ctm j

n

j 1min :

11: max1

max

tctm j

n

j


Alternative 1

t(Sk) … workload of station k Sk, k = 1, ..., m

Integer property

Sum of inequalities

and integer property of m

max

1

111 tcmSt k

m

k

k

m

kj

n

jStt

1

11

upper bound

tmax + t(Sk) > c i.e. t(Sk) c + 1 - tmax k =1,...,m-1


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

maxqTc

mtc j

n

j 1


Alternative 2

lower bound for cycle time:

upper bound for cycle time

mtqTtcc j

n

j 1maxmaxmin ,,max:

minqTc


Alternative 3

Maximization of efficiency („Bandwirkungsgrad“)

Determination of: Cycle time c Number of stations m

Efficiency („BG“)

BG = 1 100% efficiency (no idle time)

j

n

jt

cmBG

1

1


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

max

1min : ctm j

n

j

11: maxmin1

max

tctm j

n

j


ExampIe

T = 7,5 hours Minimum production amount qmin = 600 units seconds/unit 45600/3600*5,7: minmax qTc

t1=6 1

1 12

10 11 3

9 3 7

7 8

2 6

4 3

5 4

..1 10

t2=9 2

4 5


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

m t cj

n

jmin max:

155 45 2


ExampIe

0

1

2

3

4

5

6

7

10 20 30 40 50 60

m BG = 1 BG = 0.982

c Combinations of m and c leading to feasible solutions.


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


# Stationen m

theoretisch min Taktzeit

minimale realisierbare Taktzeit c

Bandwirkungsgrad 55/cm

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

Example

Possible cycle times c for varying number of stations m

m55

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.


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:

j

n

jtcmL

1

mc

L

LtDW j

n

j

1


LP formulation

We distinguish between:

LP-Formulation for given cycle time

LP-Formulation for given number of stations

Mathematical formulation for maximization of efficiency


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

otherwise0

station toassigned is operation if1 k jx jk

j = 1, ..., n

k = 1, ..., mmax

max

1

m

kjkxk


LP formulation for given cycle time

Objective function:

Constraints:

nk

m

kxkxZMinimiere

max

1

1max

1

m

kjkx

ctx j

n

j=jk

1

maxmax

11

m

kjk

m

khk xkxk

10,x jk

j = 1, ... , n ... j on exactly 1 station

k = 1, ... , mmax ... Cycle time

... Precedence cond.

... Binary variables

Eh,j

j and k


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.

E(h,j)xkxkm

k

m

kjkhk with 1

1 1


LP formulation for given number of stations

Replace mmax by the given number of stations m

c becomes an additional variable


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

Eh,j

11

m

kjkx

ctx j

n

j=jk

1

m

kjk

m

khk xkxk

11

10,x jk

c 0 integer


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

nk

m

kxkccxZ

max

1, Minimize


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(kxnk) + w2c

Large Lp-models!

Many binary variables!


Heuristic methods in case of given cycle time

Many heuristic methods(mostly priorityrule methods)

Shortened exact methods

Enumerative methods


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.



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

c



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



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)



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



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:

mjNh

hj tt:PVj

cttmjVh

hjjj E:PV

cttmLmjNh

hjj 1:RW j

Documents

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