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SINGLE ORDER QUANTITIY MODELS
SCI 3133Inventory Control and MRP
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2009 by Dr Mohammad Ishak Desa FSKSM, UTM
For it wit onlyone repleni ent.
Items reor ered for specific timeperiod to
satisfydemandduring t at period. e itemsusually aveveryshort lifeandhighly
perishablebut highlydemanded.
Eg:Fresh fish, newspaper, Christmas tree, flowers
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2009 by Dr Mohammad Ishak Desa FSKSM, UTM
To determine the best val e of Q, quantity to order,
for a single cycle. g ne spaper.
The objective is to maximize expected profit and
minimized expected loss.
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2009 by Dr Mohammad Ishak Desa FSKSM, UTM
The expected profit on the Qth unit is
= Probability selling the unit x Profit made from selling it.
= Prob (D Q) x (SP-U )
The expected loss on the Qth unit is
= Probability not selling the unit x loss incurred for not
selling.= Prob (D < Q) x (U -SV)
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2009 by Dr Mohammad Ishak Desa FSKSM, UTM
Only buy (Order) Q units henthe expected profit > the expected loss
Prob (D Q) x (SP-U ) Prob (D < Q) x (U -SV)
(1 Prob (D Q)) x (U -SV)
Or (Prob (D Q) x SP) (Prob (D Q) x U ) (U -SV) ((Prob (D Q)xU ) - (Prob (D Q) x SV))
Rearrange:U -SV
Prob (D Q) ------SP-SV
{The best Q is the largest Q that is still fulfill the above relationship and is obtainedinteractively start ith small value of Q}
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2009 by Dr Mohammad Ishak Desa FSKSM, UTM
xample: (Special Heater for Winter)
Demand 1 2 3 4 5 Prop. of the month 0.2 0.3 0.3 0.1 0.1
Q Prob(D=Q) Prob(D=> Q)
1 0.2 1.0
2 0.3 0.8
3 0.3 0.54 0.1 0.2
5 0.1 0.1
(UC(UC--SV)/(SPSV)/(SP--SV) = 0.33SV) = 0.33
Prob(D=>3) = 0.5 >0.33Prob(D=>3) = 0.5 >0.33Prob(D=>4) = 0.2 < 0.33Prob(D=>4) = 0.2 < 0.33Q* = 3Q* = 3
UC=RM1000UC=RM1000SP=RM2000SP=RM2000
SV=RM500SV=RM500
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2009 by Dr Mohammad Ishak Desa FSKSM, UTM
Ne sboy problem is a typical example ofsingleperiod problemofstock with uncertaindemand.
A ne sboy selling papers on a street corner has to
decidehow many paperstobuyfromsuppliers. Buying too many papers, left the risk of unsold
papers hich have no values at the end of the day. Buying too fe , loss opportunity to get higher profit
due to unsatisfied demand.
Formalize the marginal analysis technique.
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2009 by Dr Mohammad Ishak Desa FSKSM, UTM
Thenewsboybuys Q papers: (suppose D = demand)
If D > Q then Profit = Q x (SP- U ).
If D < Q then Profit = (D x SP) + (Q-D) x SV (Q x U )
= (D x SP) (Q x U ) if SV=0
The optimal Q* maximizes these expected profits (totalexpected profit ( P(Q)))
P(Q) = Sumof(profit x probability) .Case when SV=0
Q
= [DxSP-QxU ] x Prob(D) + Q x (SP-U ) x Prob(D)
D=0 D= Q+1
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2009 by Dr Mohammad Ishak Desa FSKSM, UTM
P(Q) = sum of (profit x probability)
Q
= SP x[ D x Prob(D) + Q x Prob(D] - (Q x U )D=0 D= Q+1
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2009 by Dr Mohammad Ishak Desa FSKSM, UTM
EP(Q*) EP(Q* 1) > 0 > EP(Q* + 1) EP(Q*)
SP x [ Prob(D) UC/SP ] > 0 > SP x [ Prob(D) UC/SP]D=Q* D=Q*1
oror
givinggiving
Prob(D Q*) > UC/SP> Prob(D(Q*+1)
Prob(D Q*) > (UC-SV)/(SP-V) > Prob(D(Q*+1) inthecaseinthecase
SV positiveSV positive
Find the best Q* in such a way that
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2009 by Dr Mohammad Ishak Desa FSKSM, UTM
Units 1 2 3 4 5 6 7 8 Prob. 0.05 0.1 0.15 0.2 0.2 0.15 0.1 0.05
Q Prob(DQ)Prob(D)
1 1.0 0.05
2 0.95 0.1
3 0.85 0.15
4 0.70 0.25 0.50 0.2
6 0.30 0.15
7 0.15 0.1
8 0.05 0.05
Choose Q*Choose Q* such thatsuch that
Prob(D Q*) > UC/SP > Prob(D(Q*+1)
Prob (D>Prob (D> 4) > 0.67 >4) > 0.67 > Prob (D>4+1)Prob (D>4+1)0.7 > 0.67 > 0.50.7 > 0.67 > 0.5
Q* = 4Q* = 4 and EP(4) = $100and EP(4) = $100
UC= 80UC= 80SP=120SP=120UC/SP= 0.67UC/SP= 0.67
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2009 by Dr Mohammad Ishak Desa FSKSM, UTM
Extend the ne sboy problem by looking at modelsfor discrete demand over several periods andallo shortages.
Assumptions: Demand uncertain follo s a kno n PDF eg Poisson.
Small demands and lo stock level
Replace a unit of the item every time one is used
The objective is to find the optimal number of units tostock (the maximum stock levelAo)
Eg: Stock of spare parts for production equipments,vehicle ..
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2009 by Dr Mohammad Ishak Desa FSKSM, UTM
When # of stock A > D, then a (A-D) x H holdingcost per unit time is incurred.
When D >A
, there is ashortage cost (D-A) x S per unit time.
The TotalExpected ost (TEC) for all values ofexpected demand :
TEC(Q) = SUM (prob. of no shortage x holdingcost for unused units) + SUM (prob. of a shortagex shortage cost for unmet demand)
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2009 by Dr Mohammad Ishak Desa FSKSM, UTM
A
TEC(A) = HC x Prob(D) x (A-D) + SC x Prob(D) x (D-A)
D=0 D= A+1
Find the best Ao (Stock Level) such that
TEC(Ao) TEC(Ao-1) < 0 (SC/(HC + SC) > Prob(D Ao-1))
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Example:
Demand 0 1 2 3 4 5 Prop. of the month 0.8 0.1 0.05 0.03 0.015 0.005
A rob( A) rob( A)
. .
. .
. .
. .
. .
. .
SC/(HC + SC) = 0.952SC/(HC + SC) = 0.952
Prob(D 0.952 > 0.95=Prob(D 0.95=Prob(D
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2009 by Dr Mohammad Ishak Desa FSKSM, UTM
STEP 1:Calculate EOQ=Sqrt((2x40x40/4)) = 28.28
Set Q = 28.28
STEP 2: Substitute this Q value into
(HCxQ)/(SCxD) = Prob(D), solve this tofind ROL ROL
STEP3: Subtitute this ROL value into this Eq. to find ne Q
STEP: Repeat Step 2 and 3
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2009 by Dr Mohammad Ishak Desa FSKSM, UTM
STEP 1:EOQ=Sqrt((2xRCxD/HC)) as an initial estimate of Q =
28.28
STEP 2: Substitute this Q value into
(HC
xQ)/(SC
xD) = Prob(D), solve this to findROL ROL
The Prob (D>ROL) =0.014
Since Demand follow N(40,4), the corresponding ROL = D + Z = 48.8 (where Z=2.2)
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2009 by Dr Mohammad Ishak Desa FSKSM, UTM
STEP3: Substitute this ROL value into this Eq. to find ne Q
STEP4: Repeat Step 2 and 3