Lot Sizing In MRP
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Transcript of 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
Combine periods 6 - 9
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 = setup cost + holding cost
Total cost = 3 x 200 + (50+10+10+30+55)x2
Total cost = 600+310
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
Combine periods 2 - 3
Combine periods 4 - 7
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
Combine periods 8 - 10
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 = setup cost + holding cost
Total cost = 3 x 200 + (40+70+30+55)x2
Total cost = 600+390
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 = setup cost + holding cost
Total cost = 3 x 200 + (80+60+60+110)
Total cost = 600+310
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
Do not include period 10 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
Do not include period 6 demand in the lot
Do not include period 10 demand in the lot
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
Queries…