lot sizing in mrp

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To determine batch size for purchased or produced items.

It is deciding how much to order and when to order.

Lot sizing models determine the optimal timing and level of production.

DESCRIPTION

Lot-for-lot techniques order just what is required for production based on net requirements:

May not always be feasible.

If setup costs are high, costs may be high as well.

FEATURES

LOTSIZING TECHNIQUES

Economic Order Quantity (EOQ)Lot For Lot (L4L)Minimum Cost Per Period (Silver Method)Least Total Cost (LTC)Least Unit Cost (LUC)Part Period Balancing (PPB)McLaren’s Order Moment (MOM)Groff’s AlgorithmFreeland and Colley

Two types of demand :

Constant Demand A decision rule that orders the same quantity each time

an order is placed. EOQ is the example of this procedure.

Lumpy Demand The models consider the problem of determining

production lot sizes when demand varies with the time. Groff’s Algorithm, Least Unit Cost (LUC),Part Period

Balancing (PPB).

Heuristic rules: aim at achieving a low-cost solution that is not necessarily optimal . Least unit cost , Silver Method, Part Period Balancing.

Wagner- Whitin rules: is an optimization approach to lumpy demand.

RULES FOR LOT SIZING

TERMS USED

Gross RequirementActual demand in case of final product

Project On HandCurrent inventory at the end of period.

Planned Order ReleasedQuantity , when orders needs to be received.

1.ECONOMIC ORDER QUANTITY

ASSUMPTIONS The demand for the item is constant and known

with certainty. There are no upper or lower limits on the order

quantity (lotsize). There are no quantity discounts. Lead time and supply are known with certainty; lead

time is constant. Order quantities for individual items are made

independently.

EXAMPLE

0 1 2 3 4 5 6 7 8 9 10

GR 35 30 40 0 10 40 30 0 30 55

Holding cost = $2/week; Setup cost = $200; Average weekly gross requirements = 27 and Lead Time=1 week.

EOQ EXAMPLE

Demand should be dependent.

Q’= √(2ds/h) = √(2*27*200/2) = 74where,

D = Demand rate (in units per year) s = constant set -up to produce (purchase) a lot h = holding cost; Q’ = lot size (in units).

0 1 2 3 4 5 6 7 8 9 10

GR 35 30 40 0 10 40 30 0 30 55

OH 35 0 44 4 4 68 28 72 72 42 61

POR 74 74 74 74

Total cost = setup cost + holding cost

Total cost = 4 x 200 + (44+4+4+68+28+72+72+72+42+61)x2

Total cost = 600+790

Total cost = $1,590

AdvantagesYields minimum total setup/ordering plus holding costs. Assumes relatively constant demand. Limitations

Not valid for

- Lumpy demand

- Dependent demand. Variations

Least Total Cost , Least Unit Cost , Part Period

Balancing.

The L4L technique: The lots are put together by searching in a greedy

way for the combinations that reduce the costs most and ensure feasibility.

Order (or produce) exactly the quantity required in

each period to satisfy gross requirements and to

maintain safety stock at its required level. Simple to use, and agrees with Just-In-Time

philosophy of ordering/producing only when required.

2.LOT FOR LOT example

Lot size can be modified easily for purchase discounts or restrictions, scrap allowances, process

constraints etc. Lot (L4L) rule sets the lot size in such a way that no

inventory is carried from one period to the other.

No on-hand inventory is carried through the system total holding cost = $0

There are seven setups for this item in this plan

Total cost = setup cost + holding cost = 7 x $200 +0 = $1400

0 1 2 3 4 5 6 7 8 9 10

GR 35 30 40 0 10 40 30 0 30 55

OH 35 0 0 0 0 0 0 0 0 0 0

POR 30 40 0 10 40 30 0 30 55 0

Advantages Minimizes carrying costs. Is certainly the best method for

- highly discontinuous demand

- expensive purchased items

Limitations Minimizes on-hand inventory, but maximizes number

of orders placed (so can be expensive if setup/ordering costs are significant).

Assumption:Variable deterministic demand

Decision variable:

⇒ T – number of periods of demand that will becovered with a particular order

⇒ T is selected by minimizing the total inventorycosts in time interval T:

Inventory holding cost + Ordering cost

T

3.SILVER MEAL METHOD

PPB Example1 2 3 4 5 6 7 8 9 10

Gross requirements 35 30 40 0 10 40 30 0 30 55

Scheduled receipts

Projected on hand 35

Net requirements

Planned order receipts

Planned order releases

Holding cost = $1/week; Setup cost = $100;EPP = 100 units

2 30 200 200/1=2002, 3 70 280 = 40 x 2 x 1+200 280/2=1402, 3, 4 70 280 = 40 x 2 x 1+200 280/3=93.332, 3, 4, 5 80 340 = 40 x 2 x 1+10 x 2 x 3+200 340/4=852, 3, 4, 5, 6 120 660 = 40 x 2 x 1+10 x 2 x 3 + 40 x 2 x 4+200 660/5=132

Combine periods 2 - 5


6 40 200 200/1=2006, 7 70 260 = 30 x 2 x 1+200 260/2=1306, 7, 8 70 260 = 30 x 2 x 1+200 260/3=86.676, 7, 8, 9 100 440 = 30 x 2 x1+ 30 x 2 x 3 +200 440/4=110

9 30 200 200/1=20010 85 310=55 x 2 +200, 310/2=155

Total cost 600+ 310 910=

Trial Lot SizePeriods (cumulative net CPP

Combined requirements) Cumulative

0 1 2 3 4 5 6 7 8 9 10

GR 35 30 40 0 10 40 30 0 30 55

OH 35 0 50 10 10 0 30 0 0 55 0

POR 80 70 85


Total cost = 3 x 200 + (50+10+10+30+55)x2


Total cost = $910

Advantages It is for single level, capacitated lot sizing

problem. The largest decrease in average cost per unit

time (Silver Meal cost criterion) per unit of capacity absorbed.

Minimizes cost per period.

Limitations Not minimizes cost per unit.

Least Unit Cost is a heuristic similar to the Silver-

Meal method, except that instead of dividing the cost over j

periods by the number of periods, j, we divide it by the

total number of units demanded through period j, r1 + r2 +

… + rj.

4.LEAST UNIT COST

PPB Example1 2 3 4 5 6 7 8 9 10

Gross requirements 35 30 40 0 10 40 30 0 30 55

Scheduled receipts

Projected on hand 35

Net requirements

Planned order receipts

Planned order releases

Holding cost = $1/week; Setup cost = $100;EPP = 100 units

2 30 200 200/30=6.672, 3 70 280 = 40 x 2 x 1+200 280/70=4.002, 3, 4 70 280 = 40 x 2+200 280/70=4.002, 3, 4, 5 80 340 = 40 x 2 x 1+10 x 2 x 3 + 200 340/80=4.25



4,5 10 200 200/10=204,5,6 50 280 = 40 x 2 x 1+200 280/50=5.604,5,6, 7 80 400 = 40 x 2 x 1+ 30 x 2 x 2 + 200 400/80=5.004,5,6, 7, 8 80 Same 400/80=5.004,5,6, 7, 8, 9 110 640 = 400+ 30x2x4 640/110=5.82

8, 9 30 200 = 120 x 2 +200 200/30=6.678,9,10 85 310=55 x 2 +200 310/85=3.65

Trial Lot SizePeriods (cumulative net CPP

Combined requirements)Cumulative


0 1 2 3 4 5 6 7 8 9 10

GR 35 30 40 0 10 40 30 0 30 55

OH 35 0 40 0 0 70 30 0 0 55 0

POR 70 80 85


Total cost = 3 x 200 + (40+70+30+55)x2


Total cost = $990

Advantages Minimizes cost per unit.

Limitations The of both Silver-Meal and LUC approaches is

that they consider one lot at a time, and the cost per period (or unit) can vary widely from period to period.

Not significant when setup cost is high.

5.LEAST TOTAL TECHNIQUE

In the Least Total Cost, we tries to balance the total inventory cost with the order cost.

It is good where setup cost is significant.

Period Demand Periods Carried

Carrying Cost Cummulative CC

2 30 0 0 0

2,3 40 1 80 80

2,3,4 0 2 0 80

2,3,4,5 10 3 60 140

2,3,4,5,6 40 4 320 460

Since on adding 6 cummulative exceeds setup cost.

6 40 0 0 0

6,7 30 1 60 60

6,7,8 0 2 0 60

6,7,8,9 30 3 180 240

Since on adding 6 cummulative exceeds setup cost.

9 30 0 0 0

9,10 55 1 110 110

At end lot in period 9 is 85

5.LEAST TOTAL TECHNIQUE EXAMPLE

0 1 2 3 4 5 6 7 8 9 10

GR 35 30 40 0 10 40 30 0 30 55

OH 35 0 50 10 10 0 30 0 0 55 0

POR 80 70 85


Total cost = 3 x 200 + (80+60+60+110)


Total cost = $910

COMMONLY USED LOT SIZING TECHNIQUES

Sr. No

Lot Sizing Technique

Used/Suitable

1 Lot4Lot 1. Expensive/bulky items , perishable items or item that requires little or no ordering cost.

2. Items with highly discontinuous demand(Service parts).

3. Carrying cost is high.

2 EOQ , Luc 1. Items that are replenished in batches and whose usage rates are low compared to batch size.

6. Part Period Balancing

PPB approach is a variation of the LTC method.

The PPB procedure attempts to balance setup and holding

costs through the use of economic part periods (EPP).

EPP = (setup cost)/(holding cost)

Cost per setup = EPP * (holding cost/unit/period)

EPP=200/2

= 100

PPB Calculations

Periods combined

Requirement Cumulativerequirement

Periods carried Cumulative Part periods

2 30 30 0 0

2,3 40 70 1 40 = 40 * 1

2,3,4 0 70 2 40

2,3,4,5 10 80 3 70 = 40 *1 + 10 * 3

2,3,4,5,6 40 120 4 230=40*1+10*3+40*4

6 40 40 0 0

6,7 30 70 1 30

6,7,8 0 70 2 30

6,7,8,9 30 100 3 120 = 30 * 1 + 30 * 3

10 55 55 0 0

(combine periods 2 through 5)

(combine periods 6 through 9)

MRP Lot-Sizing Problem : PPB Approach

0 1 2 3 4 5 6 7 8 9 10

GR 35 30 40 0 10 40 30 0 30 55

OH 35 0 50 10 10 0 60 30 30 0 0

POR 80 100 55

Holding cost = 70 *2 + 120 * 2 = $380

Set up cost = 200 * 3 = $600

Total cost = 380 +600 = $980

7. McLaren’s Order Moment

MOM method is similar to PPB.

accumulate part periods until the target value is reached.

OMT = d(Σt=1,T-1 t + (TBO-T)T) OMT = order moment target

d = average requirements per period [270/10=27]

TBO = EOQ/d = time between orders [74/27=2.74]

T = largest integer less than (or equal) the TBO [2]

McLaren’s Order Moment

OMT = 27(Σt=1,2-1 t + (2.74-2)2)

= 67 When the accumulated parts period equal or exceed this

value, a second test is done that determines whether to include one more period in the lot:

h(k)Dt ≤ S h = holding cost per period [$2]

S = setup cost [$200]

k = number of periods the will be carried

Dt = current period requirement

2(k)Dt ≤ 200

MOM Calculations

Period Requirements Period carried Part periods Cumulative part periods

2 30 0 0 0

2,3 40 1 40 40

2,3,4 0 2 0 40

2,3,4,5 10 3 30 70

6 40 0 0 0

6,7 30 1 30 30

6,7,8 0 2 0 30

6,7,8,9 30 3 90 120

9 30 0 0 0

9,10 55 1 55 55

OMT = 67 , second test : Is 2(3)10 ≤ 200? :: yes, so include period 5

OMT = 67 , second test : Is 2(4)30 ≤ 200? :: no, so don’t include period 9

MRP Lot-Sizing Problem : MOM Approach

0 1 2 3 4 5 6 7 8 9 10

GR 35 30 40 0 10 40 30 0 30 55

OH 35 0 50 10 10 0 30 0 0 55 0

POR 80 70 85

Holding cost = 70 *2 + 30 * 2 + 55 * 2 = $310

Set up cost = 200 * 3 = $600

Total cost = 310 +600 = $910

8. groff’s algorithm

Similar to MOM in that it considers the addition of a

future demand in a lot.

If it satisfied:

n(n-1) Dn ≤ 2S/h

n = no. of periods carried

Dn = current period requirement

2S/h= 200

Groff calculationsPeriod Requirements n n(n-1) Dn ≤ 200 Carrying cost

2 30 0 0 Yes 0

2,3 40 1 0 Yes 40*2*1=80

2,3,4 0 2 0 Yes 0

2,3,4,5 10 3 60 Yes 10*2*3=60

2,3,4,5,6 40 4 480 No 40*2*4=320

6 40 0 0 Yes 0

6,7 30 1 0 Yes 30*2*1=60

6,7,8 0 2 0 Yes 0

6,7,8,9 30 3 180 Yes 30*2*3=180

6,7,8,9,10 55 4 660 No 55*2*4=440

10 55 0 0 Yes 0

Do not include period 6 demand in the lot


MRP Lot-Sizing Problem : Groff Approach

0 1 2 3 4 5 6 7 8 9 10

GR 35 30 40 0 10 40 30 0 30 55

OH 35 0 50 10 10 0 60 30 30 0 0

POR 80 100 55

Holding cost = (80 + 60) + (60 + 180) = $380

Set up cost = 200 * 3 = $600

Total cost = 380 +600 = $980

9. Freeland and colley method

This method also continues to add demands into a lot

until

h(t)Dt ≤ S

t = number of periods that inventory

carried

S= $200

FC calculationsPeriod Demand periods carried Carrying cost > 200?

2 30 0 0 No

2,3 40 1 80 No

2,3,4 0 2 0 No

2,3,4,5 10 3 60 No

2,3,4,5,6 40 4 320 Yes

6 40 0 0 No

6,7 30 1 60 No

6,7,8 0 2 0 No

6,7,8,9 30 3 180 No

6,7,8,9,10 55 4 440 Yes

10 55 0 0 No



MRP Lot-Sizing Problem : FC Approach

0 1 2 3 4 5 6 7 8 9 10

GR 35 30 40 0 10 40 30 0 30 55

OH 35 0 50 10 10 0 60 30 30 0 0

POR 80 100 55

Holding cost = (80 + 60) + (60 + 180) = $380

Set up cost = 200 * 3 = $600

Total cost = 380 +600 = $980

Comparison of lot sizing methods

Method Total cost

Minimum cost per period (Silver-Meal) $910

Least total cost $910

McLaren order moment (MOM) $910

Part period balancing (PPB) $980

Groff (GR) $980

Freeland and Colley (FC) $980

Period order quantity (POQ) $980

Least unit cost (LUC) $990

Lot-for-lot (L-4-L) $1400

Economic order quantity (EOQ) $1590

Evaluation of lot-sizing methods

Nydick and Wesis conducted a large number of

simulation experiments on many of the lot-sizing rules.

which results are:

L-4-L and EOQ rules performed very poorly.

PPB, GR and MCP were the best.

When the time period between orders is small, however,

almost all the rules tested provided the optimal solutions.

Use of lot sizing methods

In 1979 Wemmerlov interviewed thirteen MRP users in the mechanical and electronics industries

Technique Number of companies

Fixed period requirement 7

Lot-for-lot 6

Fixed order quantity 5

EOQ 4

Price breaks 3

Part period balancing 2

Planner decided lot sizes 2

Least total cost 1

result from survey

Dynamic lot sizing techniques such as LTC and PPB

were used by very few companies.

Companies avoid these techniques because changes in

top levels are transmitted down through lower stages,

producing system nervousness, or exaggerated

response at component level to small changes at parent

levels. At assembly and subassembly stages, the popular

lot-for-lot technique helped maintain stability and

minimized the amount of material tied up.

Conclusion from survey

Overall, the usages of more complex methods is very

limited, mainly because the more complex methods are

not even included in many MRP computer software

packages.

Some software companies will include them as custom

offerings, but there is a lack of interest in these

techniques.

referencesSecond Edition

PRODUCTION

PLANNING

AND

INVENTORY

CONTROL

Seetharama L. Narasimhan

University of Rode Island

Dennis W. McLeavey

University of Rhode Island

Peter J. Billington

University of southern Colorado

PHI Learning Private Limited

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lot sizing in mrp

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