SCK3153 Lec 6 Single Order Quantities

8/9/2019 SCK3153 Lec 6 Single Order Quantities

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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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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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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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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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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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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.


9/19


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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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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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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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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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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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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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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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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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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STEP3: Substitute this ROL value into this Eq. to find ne Q

STEP4: Repeat Step 2 and 3

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SCK3153 Lec 6 Single Order Quantities