OR topics in MRP-II - · PDF fileOverview Why bother ? Push and Pull systems Manufacturing...

OR topics in MRP-II

Mads Jepsens

OR topics in MRP-II – p.1/25

OverviewWhy bother ?



Push and Pull systems




Manufacturing Resource Planning (MRP-II)





Rough-Cut Capacity Planning(RCCP)






Work center scheduling


Why bother?

MRP-II is still used in many production planningsystems including Dynamics AX


Why bother?


Room for improvements, many constraints aretightened to obtain feasible solutions.


Why bother?



Academically challenging. Problems are typicallyNP-hard in the strong sense.


Why bother?



Academically challenging. Problems are typicallyNP-hard in the strong sense.

Closely related to other interesting problems


Production Planning Problem

A set of resources (Machines, personal etc).




Each resource have different types of constraints.





Each product (item) is defined by its bill of material(BOM) and a routing






The BOM describes how many and which componentsshould be used to produce the item







The routing describes which work processes areneeded. Each Bom may have several different routings.







The routing describes which work processes areneeded. Each Bom may have several different routings.

For a finite time period and a finite set of orders theProduction Planning Problem is to determine when andwhere to produce a given item


Solution Models

EOQ inventory model. Limited to single product withconstant lead time.


Solution Models


Material Requirement Planning(MRP), does notconsider capacity on resources.


Solution Models


Material Requirement Planning(MRP), does notconsider capacity on resources.

Manufacturing Resource Planning (MRP-II). Considerscapacity on critical resources.


Solution Models continued

Kanban/lean manufacturing. Can require huge changesin factory layout, in the philosophy at the company andhas some issues handling less frequent ordered items.




Enterprise Resource Planning. Is a comprehensivesystem and much more than just planning. Introducessupply chain optimization.




Enterprise Resource Planning. Is a comprehensivesystem and much more than just planning. Introducessupply chain optimization.

Advanced Planning Systems. Specialized plug-in basedsystems to handle specific groups of companies.



A example of a pull systems is the kanban system




In a push system items are pushed towards thecustomer. MRP-II is a example of a push system.





In a Pull/Kanban system the amount of wip is reduced.






In a Pull system bottlenecks are visible since machinesafter the bottleneck have no kanban cards.







Bottlenecks are less visible in a push system sinceitems pile up several places.







Bottlenecks are less visible in a push system sinceitems pile up several places.

Less frequent ordered items is easier handled in a pushsystem.


Material Requirement Planning(MRP)



Originally system was based on reorder point.




Reorder point is suited for independent demand.





But not for dependent demand.






MRP works backwards from independent demand toderive a schedule.






MRP works backwards from independent demand toderive a schedule.

MRP is called a push system since it pushes items inthe production chain.


Schematic of MRP


MRP Inputs and Outputs



Master Production Schedule:



Master Production Schedule: Item, Quantity and duedates.




Erp Database:




Erp Database: BOM, Routing, lot-sizing rule(LSR),lead time(PLT) and On-Hand Inventory.





Scheduled Receipts: Out standing orders and Jobs.Work in process.





Scheduled Receipts: Out standing orders and Jobs.Work in process.

MRP outputs: Planned order release, Change noticesand Exception reports.


MRP Procedure continued



Lot sizing:



Lot sizing:

Wagner Whitin



Lot sizing:

Wagner Whitin

lot for lot



Lot sizing:

Wagner Whitin

lot for lot

fixed order period



Lot sizing:

Wagner Whitin

lot for lot

fixed order period

Fixed order Quantity and EOQ.



Lot sizing:

Wagner Whitin

lot for lot

fixed order period


Part-Period Balancing. Balancing inventory cost andSetup Cost.



Lot sizing:

Wagner Whitin

lot for lot

fixed order period



Time fasing. All lead times are considered for items, notfor status on floor



Lot sizing:

Wagner Whitin

lot for lot

fixed order period



Time fasing. All lead times are considered for items, notfor status on floor

Bom Explosion. Netting and lot sizing is done for eachsub item.


Issues with MRP


Issues with MRP

Assume constant lead times. To take care of variationSafety Stock and Safety Lead time is used.


Issues with MRP


Capacity Infeasibility, there is no capacity check.


Issues with MRP



Long Planned Lead time due to variation in deliverytime.


Issues with MRP



Long Planned Lead time due to variation in deliverytime.

System Nervousness. Plans that are feasible canbecome infeasible.


MRP-II

Resourceplanning

Aggregate

planning

production Firmorders

Short−termforecast

planning

Rough−cut

capacity

scheduling

Master

production

materialBills of

I/O control

planning

Materialrequirement

On−hand &

scheduledreciepts

Jobpool

Capacity

planning

Jobrelease

Routingdata

Jobdispatching

Long−term

forecast

managementDemand

requirement

Long−range

planning

Intermediate−range

planning

Short−term

control





Considers aggregated work




Allocates work in time buckets





Determines resources in order to reach due dates






Both regular and nonregular (outsourcing over timeetc.) is considered







Time driven RCCP is when project dates must be meet.








In resource-driven RCCP only regular capacity can beused








In resource-driven RCCP only regular capacity can beused

This session will consider the time driven


Some notation


Some notation

n jobs J1, J2, · · · , Jn


Some notation

n jobs J1, J2, · · · , Jn

k resources R1, · · · , Rk


Some notation

n jobs J1, J2, · · · , Jn


T time buckets


Some notation

n jobs J1, J2, · · · , Jn


T time buckets

Qkt is the regular capacity for resource k in period t


Some notation

n jobs J1, J2, · · · , Jn


T time buckets


Job Jj requires qjk units of resource k


Some notation

n jobs J1, J2, · · · , Jn


T time buckets



xkt denotes the fraction of job Jj performed in period t


Some notation

n jobs J1, J2, · · · , Jn


T time buckets




Jj must be performed in time window [rj , dj ]


Some notation

n jobs J1, J2, · · · , Jn


T time buckets





pj is the minimum number of periods job Jj can use.


Some notation

n jobs J1, J2, · · · , Jn


T time buckets





pj is the minimum number of periods job Jj can use.

1

pjis the maximum fraction of a job that can be

completed in single time bucket.


Precedence constraints

If job Ji must finish before Jj there is a precedence relation.For a period τ this can be modelled as:




xjτ > 0 →τ−1∑

t=dt

xit = 1




xjτ > 0 →τ−1∑

t=dt

xit = 1

There will be dj − rj constraints per precedence relation.




xjτ > 0 →τ−1∑

t=dt

xit = 1

There will be dj − rj constraints per precedence relation.

Time windows can in some cases be tightened, due to

precedence constraints.


Nonregular capacity

Let Qkt denote the nonregular capacity for resource k inperiod t. Then for each resource and time period:


Nonregular capacity


Ukt = max{0,n

∑

j=1

qjkxjt − Qkt}


Nonregular capacity


Ukt = max{0,n

∑

j=1

qjkxjt − Qkt}

The cost of using nonregular capacity for resource k in time

period t is ckt


Nonregular capacity


Ukt = max{0,n

∑

j=1

qjkxjt − Qkt}

The cost of using nonregular capacity for resource k in time

period t is ckt It is assumed that there is no limit on the

nonregular resources.


Mathematical Model

min∑T

t=1

∑Kk=1

cktUkt

subject to∑dj

t=rjxjt = 1 1 ≤ j ≤ n

xjt ≤ 1

pj1 ≤ j ≤ n, 1 ≤ t ≤ T

∑nj=1

qjkxjt − Ukt ≤ 0 1 ≤ j ≤ n, 1 ≤ t ≤ T

Precedencerelations

xjt, Ukt ≥ 0 1 ≤ j ≤ n, 1 ≤ t ≤ T


Controlling feasibility

Allowed To Work window for job Jj is defined as [Sj , Cj ].




Job Jj cannot start before Sj or after Cj





A ATW for job Jj is feasible if:






1. Sj ≥ rj and Cj ≤ dj

2. Cj − Sj ≥ pj − 1







2. Cj − Sj ≥ pj − 1

A set S of ATW windows is feasible if:







2. Cj − Sj ≥ pj − 1

A set S of ATW windows is feasible if:

1. Every ATW window is feasible

2. Sj > Cj if Ji → Jj


Mathematical Model ATW windows

sjt =

{

1 Sj ≤ t ≤ Cj

0 otherwise

(PS) min∑T

t=1

∑Kk=1

cktUkt

subjectto∑dj

t=rjxjt = 1 1 ≤ j ≤ n

xjt ≤ sjt

pj1 ≤ j ≤ n, 1 ≤ t ≤ T

∑nj=1

qjkxjt − Ukt ≤ 0 1 ≤ j ≤ n, 1 ≤ t ≤ T

xjt, Ukt ≥ 0 1 ≤ j ≤ n, 1 ≤ t ≤ T


Constructive heuristics (HBASIC)

Construct a feasible set S of ATW windows.




Solve problem PS




Solve problem PS

To obtain a feasible set of ATW windows construct them asfollows:




Solve problem PS


Set Sj = rj




Solve problem PS


Set Sj = rj

Set Cj = min{dj ,min{k|Ji →Jk} rk − 1}


Constructive heuristics (HCPM)

Define the slack of job Jj as Lj = Cj − (Sj + pj)




{Ji1 , Ji2 , ·, Jik} is a ordered set if:





Ji1 → Ji2, Ji3 → Ji4 , ·, Jik−1→ Jik





Ji1 → Ji2, Ji3 → Ji4 , ·, Jik−1→ Jik

A critical path is a path where:





Ji1 → Ji2, Ji3 → Ji4 , ·, Jik−1→ Jik


Lj1 = Lj2 = · · · = min1≤j≤n Lj





Ji1 → Ji2, Ji3 → Ji4 , ·, Jik−1→ Jik


Lj1 = Lj2 = · · · = min1≤j≤n Lj

Sji= Sji−1

+ pi−1 for 2 ≤ i ≤ R





Ji1 → Ji2, Ji3 → Ji4 , ·, Jik−1→ Jik


Lj1 = Lj2 = · · · = min1≤j≤n Lj

Sji= Sji−1

+ pi−1 for 2 ≤ i ≤ R

Cji= Cji+1

− pji+1for1 ≤ i ≤ R





Ji1 → Ji2, Ji3 → Ji4 , ·, Jik−1→ Jik


Lj1 = Lj2 = · · · = min1≤j≤n Lj

Sji= Sji−1

+ pi−1 for 2 ≤ i ≤ R

Cji= Cji+1


A critical path is maximal if for all Jl:





Ji1 → Ji2, Ji3 → Ji4 , ·, Jik−1→ Jik


Lj1 = Lj2 = · · · = min1≤j≤n Lj

Sji= Sji−1

+ pi−1 for 2 ≤ i ≤ R

Cji= Cji+1


A critical path is maximal if for all Jl:

{Ji1 , Ji2 , · · · , Jik , Jil} is critical

{Jil, Ji1 , Ji2 , · · · , Jik} is critical


(HCPM) continued

Initialize Sj = rj and Cj = dj for all Jj


(HCPM) continued


Compute the slack for all jobs


(HCPM) continued



Find a maximal critical path {Ji1 , Ji2 , ·, Jik}


(HCPM) continued




Compute the total slack L = CjR− (Sj1 +

∑Ri=1

pji)


(HCPM) continued





∑Ri=1

pji)

Set Sji= Sj1 +

∑i−1

k=1pik + |L

∑i−1

k=1pik/

∑Rk=1

pik |


(HCPM) continued





∑Ri=1

pji)

Set Sji= Sj1 +

∑i−1

k=1pik + |L

∑i−1

k=1pik/

∑Rk=1

pik |

Set Cik = Sik+1− 1


HCPM example

1 2 3 4

S 0 5 15 20

C 10 15 25 30

p 5 5 10 5


HCPM example

1 2 3 4

S 0 5 15 20

C 10 15 25 30

p 5 5 10 5

L = C4 − (0 + 20) = 5


HCPM example

1 2 3 4

S 0 5 15 20

C 10 15 25 30

p 5 5 10 5

L = C4 − (0 + 20) = 5

Recall Sji= Sji

+∑i−1

k=1pjk

+ |L∑i−1

k=1/∑R

k=1pjk|


HCPM example

1 2 3 4

S 0 5 15 20

C 10 15 25 30

p 5 5 10 5

L = C4 − (0 + 20) = 5

Recall Sji= Sji

+∑i−1

k=1pjk

+ |L∑i−1

k=1/∑R

k=1pjk|

S2 = 0 + 5 + 5∗525

= 6


HCPM example

1 2 3 4

S 0 5 15 20

C 10 15 25 30

p 5 5 10 5

L = C4 − (0 + 20) = 5

Recall Sji= Sji

+∑i−1

k=1pjk

+ |L∑i−1

k=1/∑R

k=1pjk|

S2 = 0 + 5 + 5∗525

= 6

S3 = 0 + 10 + 10∗525

= 12


HCPM example

1 2 3 4

S 0 5 15 20

C 10 15 25 30

p 5 5 10 5

L = C4 − (0 + 20) = 5

Recall Sji= Sji

+∑i−1

k=1pjk

+ |L∑i−1

k=1/∑R

k=1pjk|

S2 = 0 + 5 + 5∗525

= 6

S3 = 0 + 10 + 10∗525

= 12

S4 = 0 + 20 + 20∗525

= 24


HCPM example

1 2 3 4

S 0 5 15 20

C 10 15 25 30

p 5 5 10 5

L = C4 − (0 + 20) = 5

Recall Sji= Sji

+∑i−1

k=1pjk

+ |L∑i−1

k=1/∑R

k=1pjk|

S2 = 0 + 5 + 5∗525

= 6

S3 = 0 + 10 + 10∗525

= 12

S4 = 0 + 20 + 20∗525

= 24

C1 = S2 − 1 = 5, C2 = 11, C3 = 23


Questions?


Questions?

Thank you!


Date post:	04-Feb-2018
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OR topics in MRP-II - · PDF fileOverview Why bother ? Push and Pull systems Manufacturing...

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