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Capacity Planning

Sales and Operations Planning •Forecasting

•Capacity planning •Inventory optimization

•How much capacity assigned to each production unit? •Realistic capacity estimates

•Strategic level

•Moderately long time horizon •Larger time buckets

•Data not known exactly

1

Operations Planning

•Pochet-Wolsey’s model is too large, NP-hard •Decomposition→inaccurate model

•Medium-term planning horizon, strategic level •Larger time buckets

2

Some definitions

Work in Process inventory between the start and end

points of a product routing is called work in process (WIP)

WIP inventory work in process inventory

FGI inventory final goods inventory

Lead time of a given routing or line is the time allotted

for production of a part on that routing or line.

Littles Law

WIP=Throughput×cycle time

Master Production Scheduling Model (Pochet-Wolsey)

time horizon t=1, . . . ,n

•φi

tunit production cost •qtifixed production cost •πi

tinventory cost •Di

tdemand Decision variables

xtiproduction lot size in period t

ytibinary variable indicating production period tIi

t inventory at end of period t Moreover

Ck

t available capacity, resource k time t •ξikper unit resource consumption •βikfixed resource consumption

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Material Requirement Planning Model (Pochet-Wolsey)

Multi-item multi-level capacitated lot-sizing model •Optimize simultaneously production and purchase of

all items

•from raw materials to finished products •satisfy external demands

•satisfy internal demands •short-term horizon Bill of materials (BOM)

5

Material Requirement Planning Model (Pochet-Wolsey)

S(i)set of direct successors of i (items consuming i)ri jamount of item i required to make one item j •γilead-time to produce or deliver a lot of i

i.e. xitcan be delivered at time ti

Model min m

i=1 n

t=1 φi txti+qityititIti Subject to: Iti−1+x i t−γi= D i t+

jS(i) ri jxtj ! +Itii,t xitMtiyiti,t m

i=1 ξikxi t+ m

i=1 βikyi tCtkt xit∈R+,Iti∈R+,yti∈ {0,1}

Large, NP-hard model, difficult to solve

6

Capacity of Resources

•Gross capacity •Usable capacity •Productivity factor

Resource Gross Productivity Usable Description Capacity Factor Capacity

(hours/day) (hours/day)

Machine S100 8 0.95 7.6

Forklift 8 0.85 6.8

Machine ASS 16 0.85 13.6

Machine PPP 8 0.95 7.6

Capacity limit is soft (queuing theory) Capacity depends on - variation in arriving jobs - variation in lead times Lead time depends on load

Asmundsson, Single-stage multi-product planning

Simplified model min

i

t φitXititIit Subject to: Iit=Ii,t−1+Xi,tLDiti,t

i ξitXitCtt Xit,Iit∈R+ ∀i,t

•LP-model (easy to solve)

•Setup times are neglected (future work) •Constant lead time (production time) L

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Introduction

Accurate measurements of manufacturing capacity is hard to obtain (Elmaghraby 1991)

•Stochastic performance analysis models

Queuing models capture important aspects (Hopp, Spear-man 2000)

Stochastic model of whole production unrealistic •Deterministic techniques

Divide planning horizon in discrete buckets Assign capacity in each bucket

Solution satisfying aggregated constraint not feasible in practice

•Integrating approaches (Hung, Leachmand 1996) Solve LP model for production planning

Feed into simulator to estimate lead times Add cut if not feasible

nonlinear clearing function.

Figure 1 Relationships between throughput, WIP and lead time (Karmarkar 1989) Clearing Function WIP T h ro u g h p u t (C ap ac it y ) L ea d T im e Throughput Relationship with Lead Time

Theoretical Capacity T h eo re ti ca l C ap ac it y 9

Relation between WIP and system throughput

•Analytic congestion model for closed production sys-tems (Spearman 1991)

•Queuing models of manufacturing systems (Buzacott, Shanthikumar 1993)

Connection between WIP, variation, utilization (Medhi 1991) WIP= c

2ρ2

1−ρ+ρ

c variation service time and arrivals,ρutilization of server

ρ=

p

(WIP+1)2+4WIP(c21)(W IP+1)

2(c2−1)

Figure for different values of c

10

Clearing Function

Capacity limit depends on Work-in-process (W )

i

ξitXitft(

i

ξitWit)

If capacity depends on WIP we need to have cost associ-ated with WIP (internal inventory)

Asmundsson

Single Stage Multi-product Planning Model Using Clear-ing Functions

Xitnumber of units of product i produced time tRitnumber of units or product i released into the stage

at the beginning of time period t

Wit number of units of product i in WIP inventory at the end of period t

Iit number of units of product i in finished goods in-ventory at the end of period t

•ξitamount of resource (machine time) required to pro-duce one unit of item i time t

ft(W)clearing function representing resource in pe-riod t

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CF model (Clearing Function) min

t

i φitXititWititIititRit Subject to: Wit=Wi,t−1−Xit+Riti,t Iit=Ii,t−1+XitDiti,t

i ξitXitft(

i ξitWit)t Xit,Wit,Iit,Rit∈R+ ∀i,t where

•φitcost of producing unit i time t •ωitcost of WIP of unit i time t •πitcost of inventory unit i time t •ρitcost of releasing unit i time t

13

Problem with capacity constraint

XA+XBf(WA+WB)

for two products A,B

Solution exists XA>0, XB=0, WA=0, WB>0 •Maintain high WBif cheap

Create capacity for XA

No link between WIP available and production

14

Solution

Zit≥0 represent an allocation of expected throughput among different products XitZitft(

i ξitWit)i,t

i Zit=1 ∀t

Depends on total WIP.

Prefer WIP of specific product i.

AssumingξitWit=ZitiξitWitwe get (?) XitZitft( ?

i ξitWit Zit ) ∀i,t

i Zit=1 ∀t WIP T h ro u g h p u t (C ap ac it y ) Theoretical Capacity

ACF model (Allocated Clearing Function)

min

t

i φitXititWititIititRit Subject to: Wit=Wi,t−1−Xit+Riti,t Iit=Ii,t−1+XitDiti,t XitZitft( ?

i ξitWit Zit ) ∀t

i Zit=1 ∀t Xit,Wit,Iit,Rit∈R+ ∀i,t

(5)

Outer approximation of clearing function

Outer linear functionsαcWtcfor c=1, . . . ,C

i ξitWit≤αcWtcc,t then f(W) = min c=1,...,Cc Wtc} Assume that α1 >α2>α3> . . . >αc=0 Linear model min

t

i φitXititWititIititRit Subject to: Wit=Wi,t−1−Xit+Riti,t Iit=Ii,t−1+XitDiti,t

i ξitXit≤αcξitWit+Zitβci,t,c

i Zit=1 ∀t Zit,Xit,Wit,Iit,Rit∈R+ ∀i,t

Which is correct since

Zitf ξ itWit Zit = Zitmin c αcξitWit Zitc = min ccξ itWitcZit} 17

Asmundsson, experimental results

Single-stage system, 3 products

•Variation of arrival 0.5 variation of service 2 •Max throughput 10 items / period

•Minimize inventory holding costs(π1,π2,π3) = (1,2,3)

•Holding costs of WIP and FGI are same

Clearing function, linear approximation

18

Asmundsson, experimental results Production Plan 0 2 4 6 8 10 12 14 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 Time Period Demand WIP FGI TH

Figure 2 Production plan

Lead Time 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 Time Period P er io d s

Figure 3 Production lead time

5 Conclusions

Asmundsson, experimental results

Throughput Throughput 0 2 4 6 8 10 12 14 16 18 1 3 5 7 9 1113 151719212325 2729 3133353739 41434547 49 Period Throughput Demand FC PCF

Figure 5: Throughput and Demand of ACF and FC Models.

Demand (gray), throughput from ACF more smooth than FC

WIP and FGI

0 2 4 6 8 10 12 14 16 18 20 1 3 5 7 9 1113151719212325272931333537394143454749 Period Level PCF - WIP PCF - FGI FC - FGI

Figure 6:WIP and FGI levels, Cumulative across Items.

WIP high when throughput high, FC model does not capture WIP

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Asmundsson, experimental results Lead-Time 0.00 1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00 0 5 10 15 20 25 30 35 40 45 Period Lead-Time Item 1 Item 2 Item 3 Average Raw Processing

Figure 7:Production Lead-Time across all Items.

lead time greater than 0.1 (raw processing time), varies significantly

Marginal Cost of Capacity

0 5 10 15 20 25 1 3 5 7 9 1113151719212325 272931333537394143454749 Period MMC PCF FC

Figure 8:Marginal Cost of Capacity (MCC).

capacity constraint active in ACF, marginal cost positive

marginal cost is zero in FC for all periods where FGI is zero

21

Asmundsson, experimental results

Lead-Time vs. Throughput 0.00 0.20 0.40 0.60 0.80 1.00 1.20 0 1 2 3 4 5 6 7 8 9 10 Throughput Lead-Time

Figure 9:Nonlinear Relationship between throughput and lead time

Figure

Figure 1 Relationships between throughput, WIP and lead time (Karmarkar 1989)
Figure 2 Production plan

References

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