mit2 854f10 multipart

7/30/2019 MIT2 854F10 Multipart

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Single-stage, multiple-part-type s

Lecturer: Stanley B. Gershwi


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Setups

Setup: A setup change occurs when it coto make a Typej part after making a Type

than after making a Typej part.

Examples: Tool change (when making holes)

Die change (when making sheet metal p

Paint color change


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Costs

Setups

Setup costs can include Money costs, especially in labor. Also m

Time, in loss of capacity and delay.

Some machines create scrap while being during a setup change.

Setups motivate lots or batches: a set of are processed without interruption by setu


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Costs

Setups

Problem: Large lots lead to large inventories and lo

times.

Small lots lead to frequent setup changeReduction of setup time has been a very im

trend in modern manufacturing.


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Flexibility

Setups

Flexibility: a widely-used term whose meadiminishes as you look at it more closely. (

be the definition of abuzzword.)

Flexibility: the ability to make many differe ie, to operate on many different parts.Agility is also sometimes used.


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Flexibility

Setups

Which is more flexible?

A machine that can hold 6 different cutting tools, anchange from one to another with zero setup time.

A machine that can hold 25 different cutting tools, aa 30-second setup time.

Which is more flexible?

A final assembly line that can produce all variationsof cars, and can produce 100 cars per day.


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Setup Machines vs Batc

Setups

A machine has setups when there are costs or delassociated with changing part types. Machines thasetups may make one or many parts at a time.

A machine operates on batches of size n if it operan parts simultaneously each time it does an operatmachines may or may not operate on different part

they do, they may or may not require setup change

batches may or may not be homogeneous.)

E l O d h i l h b


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Loss of Capaci

Setups

Assume

there is one setup for every Q parts (Q=lothe setup time is S,the time to process a part is .Then the time to process Q parts is S+ Q

average time to process one part is + S/Q


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Loss of Capaci

Setups

If the demand rate is d parts per time unit, th

demand is feasible only if

Q 1+ S/Q < 1/d or d < =

S+ Q +

This is not satisfied if Sis too large or Q is t


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Deterministic Exa

Setups

Focus on a single part type (simplification!Short time scale (hours or days).Constant demand.Deterministic setup and operation times.Setup/production/(idleness) cycles.Policy: Produce at maximum rate until theis enough to last through the next setup tim


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Deterministic Exa

Setups

CumulativeProduction

Cycle: and DemandS= setup period for the

part type

P= period the machineis operating on the part

type

I= period the machineis making or setting up


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Deterministic Exa

Setups Objective in g

CumulativeProduction

and Demand

earliness

production P(t)

surplus/backlog x(t)

Objective the cumul

production

to the cum

d d li


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Deterministic Exa

Setups

Cycle:

Setup period. Duration: S. Production: 0.Sd. Net change of surplus, ie of PD i

S= Sd.Production period. Duration: t = Q. Pro

Q. Demand: td. Net change of PD is

P= Q td = QQd = Q(1 d)


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Deterministic Exa

Setups

Idleness period (for the part we focus on).I. Production: 0. Demand: Id. Net chang

PD is I= Id.

Total (desired) net change over a cycle: 0.

Therefore, net change of PD over whoS+P+I= Sd + Q(1 d)


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Deterministic Exa

Setups

Since I 0, Q(1 d) Sd 0.If I= 0, Q(1 d) = Sd.If d > 1, net change in PD will be ne


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80

S= 3, Q = 10, = 1, d = .5Production

DemandInventory/backlog

70

60

50

40

30

20

10

Deterministic Exa

Setups Production & invent

Production pduration = Q

Idle period dTotal cycle dMaximum inv

Q(1 d) =


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= 3, Q = 30, = 1, d = .50

ProductionDemand

Inventory/backlog0

0

0

0

0

0

0

Deterministic Exa

Setups Not frequent en

S8

Production p7

duration = Q6

Idle period d

5

27.4

Total cycle d32

Maximum inv1 Q(1 d) =


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Deterministic Exa

Setups Too frequen

S= 3, Q = 2, = 1, d = .5Batches to

demand n

Q(1

d

0.5

Backlog grToo much 20

spent on s

10

0

10

20

30

40

50Production



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0

= 3, Q = 3, = 1, d = .5Production


0

0

0

0

0

0

0

8

Deterministic Exa

Setups Just right

S

7

6 Small batc5

small inve4 Maximum 3

is Q(1 2

1


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70

Deterministic Exa

Setups Other parame

S= 3, Q = 10, = 1, d = .1Production


60

50

40

30

20

10


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Deterministic Exa

Setups Time in the sy

400

350

0 50 100 150 200

Time in system deterministic

Each batch300

Q+ Sti250

200

the system150 Q(1 d100

Optimal ba50

Q = Sd/(0


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Stochastic Exam

Setups

Batch sizes equal (Q); processing times ra Average time to process a batch is Q+

Random arrival times (exponential inter-ar Average time between arrivals of batches

Q/d = 1/.

Infinite buffer for waiting batches


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Stochastic Exam

Setups

400

350

0 50 100 150 200

Time in system deterministicTime in system exponential

Treat syste300 M/M/1 q250 batches.200

Average d150

batch is 1/100

50 Variability

0 delay.


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Batch size data

Setupsfrom a factory

April 97 Batch Sizes

0 20 40 60 80 100 120 140 1


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Batch size data


May 97 Batch Sizes

0 20 40 60 80 100 120 140 1


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Batch size data


January 98 Batch Sizes

Avg Lot Size

Std Dev=27

0 20 40 60 80 100 120 140 16


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Two Part Type

Setups

Assumptions: Cycle is produce Type 1, setup for Type 2

Type 2, setup for Type 1 .

Unit production times: 1, 2. Setup times: S1, S2.

Batch sizes: Q1, Q2.

Demand rates: d1, d2.

N idl


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Two Part Type

Setups

Cycle:

Type 1

Type 2

T

Demand

Cumulative

Production and


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Two Part Type

Setups

Let Tbe the length of a cycle. Then

S1 + 1Q1 + S2 + 2Q2 = T

To satisfy demand,

Q1 = d1T; Q2 = d2T

This implies

S1 + S2T=1 ( d + d )


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Two Part Type

Setups

idi is the fraction of time that is devoted tproducing part i.

1 (1d1 + 2d2) is the fraction of time thdevoted to production.

We must therefore have 1d1 + 2d2 < 1.feasibility condition.


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Multiple Part Typ

Setups

New issue: Setup sequence. In what order should we produce batche

different part types?

Sij is the setup time (or setup cost) for chafrom Type i production to Typej productioProblem: Select the setup sequence {i1, i2,...,in}


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Multiple Part Typ

Setups Cases

Sequence-independent setups: Sij = Sj. does not matter.

Sequence-dependent setups: traveling saproblem.


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Multiple Part Typ

Setups Cases

Paint shop: i indicates paint color numberSij is the time or cost of changing from Co

Colorj.

If i > j, i is darker thanj and Sij > Sji.


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Multiple Part Typ

Setups Cases

Hierarchical setups.Operations have several attributes.Setup changes between some attributes can be do

and easily.Setup changes between others are lengthy and ex


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Dynamic Lot Siz

Setups

Wagner-Whitin (1958) problemAssumptions: Discrete time periods (weeks, months, et

t = 1, 2,...,T. Known, but non-constant demand D1, D

Production, setup, and holding cost.

Infinite capacity.


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Dynamic Lot Siz

Setups Other notati

ct = production cost (dollars per unit) in peAt = setup or order cost (dollars) in periodht = holding cost; cost to hold one item in

from period t to period t + 1It = inventory at the end of period t the

variable

Qt = lot size in period t the decision var


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Dynamic Lot Siz

Setups Problem

minimize

t

T

=1

(At(Qt) + ctQt + htIt)

(where (Q) = 1 if Q > 0; (Q) = 0 if Q =

subject to

It+1 = It + Qt DtIt 0


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Dynamic Lot Siz

Setups Wagner-Whitin P

Characteristic of Solution:

I

Q


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Dynamic Lot Siz


Characteristic of Solution:

EitherIt = 0 orQt+1 = 0. That is, producwhen inventory is zero. Or,

kj,k,...,1+j=t,0>tI

dna,0>kQ,0>jQneth

.k,...,1+j=t,0=tQ

If we assume I = 0 and I = 0 (k > j


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Dynamic Lot Siz


nehT

,jDjQ=+1jI

...,+1jDjDjQ=+2jI

kD...+1jDjDjQ=0=kI

kD+...++1jD+jD=jQ,rO

tsactlyxaetohguoneceudorpsnaemichwh


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Dynamic Lot Siz


This is not enough to determine the solutiomeans that the search for the optimal is lim

It also gives a qualitative insight.


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Real-Time Schedu

Setups

Problem: How to decide on batch sizes (iechange times) in response to events.

Issue: Same as before. Changing too often causes capacity loss

too infrequently leads to excess inventor

time.


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ieypTfoteardnamed=id

yTfotearnctioudorpmuximam=i/1=i

etimptuse=S

tetimtaieypTfotearnctioudorp=)t(iu

yTfo)gklocabroyrtonevin(sluprsu=)t(ixixd

Real-Time Schedu

Setups One Machine, Two P

Model:


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Real-Time Schedu

Setups Heuristic: Corrido

Type 1 production1x

x235

30

25

20

15

10

5

0

35

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35

30

25

20

15

10

5

0

Setup 2

Setup 1

t j t

Draw two lines, laSetup 1 and Set

Keep the systemsetup i until x(t)

the Setupj line.

Change to setup


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Setups

Real-Time Schedu

Heuristic: Corrido

Setup

change

1x

x2t j t

35

30

25

20

15

10

5

0

35

30

25

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10

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0

Setup 2

Setup 1


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Real-Time Schedu


1x

x2

Type 2 production

t j t35

30

25

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10

5

0

35

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0

Setup 2

Setup 1


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Setups

Real-Time Schedu

Heuristic: Corrido

Setup

change

1x

x2t j t

35

30

25

20

15

10

5

0

35

30

25

20

15

10

5

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0

Setup 2

Setup 1


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Real-Time Schedu


1x

x2

Type 1 production

t j t35

30

25

20

15

10

5

0

35

30

25

20

15

10

5

0

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0

Setup 2

Setup 1


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Real-Time Schedu


1x

x2t j t

35

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0

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0

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0

Setup 2

Setup 1


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Real-Time Schedu


In this version, batch size is a function of tiAlso possible to pick parallel boundaries, w

upper limit. Then batch size is constant un

limit reached.


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Real-Time Schedu


Two possibilities (for two part types):

dnamediflynocycleitlimtosegrevnCo

.1


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Real-Time Schedu

Setups More Than Two Pa

Three possibilities (for more than two part ty

Limit cycle only if demand is within capaDivergence if demand is not within capacity, or

corridor boundaries are poorly chosen.

Chaos if demand is within capacity, and cboundaries chosen not well?


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2.854 / 2.853 Introduction to Manufacturing Systems

Fall 2010

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