Single Period Inventory Models Yossi Sheffi Mass Inst of Tech Cambridge, MA.
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Transcript of Single Period Inventory Models Yossi Sheffi Mass Inst of Tech Cambridge, MA.
Single Period Inventory Models
Yossi Sheffi
Mass Inst of Tech
Cambridge MA
Outlinebull Single period inventory decisionsbull Calculating the optimal order sizebull Numericallybull Using spreadsheetbull Using simulationbull Analyticallybull The profit functionbull For specific distributionbull Level of Servicebull Extensionsbull Fixed costsbull Risksbull Initial inventorybull Elastic demand
Single Period Ordering
bull Seasonal itemsbull Perishable goodsbull News printbull Fashion itemsbull Some high tech productsbull Risky investments
Selling Magazines
bull Weekly demand
1048708 Total 4023 magazines
1048708 Average 774 Magweek
1048708 Min 51 max 113 Magweek
Detailed Histogram
Average=774 Magwk
Fre
qu
en
cy
(WksY
r)
Demand (MagWk)
Cu
mm
Fre
q (W
ksY
r)
Histogram
Demand (MagWk)
Fre
qu
en
cy (W
ksY
r)
Cu
mm
Fre
q (W
ksY
r)Cu
mm
ula
tive F
req
uen
cy
More
The Ordering Decision(Spreadsheet)bull Assume each magazine sells for $15bull Cost of each magazine $8
Order
ExpProfit
dwk Prob
Newsboy Framework ndash Panel 1 ldquobasic Scenariordquo
Expected ProfitsP
rofi
t
Order Size
Optimal Order (Analytical)
bull The optimal order is Qbull At Q the probability of selling one more magazine is the probability that demand is greater than Qbull The expected profit from ordering the (Q+1)st magazine isbull 1048708 If demand is high and we sell itbull 1048708 (REV-COST) x Pr( Demand is higher than Q)bull 1048708 If demand is low and we are stuckbull 1048708 (-COST) x Pr( Demand is lower or equal to Q)
bull The optimum is where the total expected profit from ordering one more magazine is zerobull (REV-COST) x Pr( Demand gt Q) ndash COST x Pr( Demandbull le Q) = 0
bull
Optimal OrderThe ldquocritical ratiordquo
Fre
qu
en
cy (
WksY
r
Cu
mm
ula
tive F
req
uen
cy
Demand (Magweek) More
Salvage Valuebull Salvage value = $4Mag
Demand (Magweek)
Order Size
Cu
mm
ula
tive
Fre
qu
en
cy
Fre
qu
en
cy
(WksY
r)
Pro
fit
The Profit Functionbull Revenue from sold itemsbull Revenue or costs associated with unsold
items These may include revenue from salvage or cost associated with disposal
bull Costs associated with not meeting customersrsquo demand The lost sales cost
can include lost of good will and actual penalties for low servicebull The cost of buying the merchandise in the
first place
The Profit Function
The Profit Function ndash Simple Case
Optimal Order
and
Level of ServiceCycle Service ndash The probability that
there will be a stock-out during a
cycle
Cycle Service = F(Q)
Fill Rate - The probability that a
specific customer will encounter a
stock-out
Level of ServiceS
erv
ice L
evel
Order
Cycle Service
Fill Rate
REV=$15
COST=$7
Normal Distribution of Demand
Order Size
Exp
ecte
d P
rofi
t
Incorporating Fixed Costsbull With fixed costs of $300order
REV=$15
COST=$7
Exp
ecte
d
Pro
fi
Order Size
Risk of Loss
300(15-7)=375 Below that there is no possibility to make money even if we sell ALL the order
REV=$15
COST=$7
Order Quantity
Pro
bab
ilit
y o
f Loss
Ordering with Initial Inventory
Note the cost of Q0 is irrelevant
Given initial Inventory Q0 how to order1 Calculate Q as before2 If Q0 lt Q order (Qlt Q0 )3 If Q0 ge Q order 0With fixed costs order only if the expectedprofits from ordering are more than theordering costs1 Set Qcr as the smallest Q such that E[Profits with Qcr]gtE[Profits with Q]-F2 If Q0 lt Qcr order (Qlt Q0 )3 If Q0 ge Qcr order 0
Ordering with FixedCosts and Initial Inventory
bullIf initial inventory is LE 46 order up to 80
bullIf initial inventory is GE 47 order nothing
REV=$15
COST=$7
Example F = $150Exp
ecte
d P
rofi
t
Initial Inventory
bull μ =D(P) σ = f(μ)bull Procedurebull 1 Set Pbull 2 Calculate μbull 3 Calculate σbull 4 bull 5 Calculate optimal expected profits as a function of P
Elastic DemandExpected Profit Function
Pro
fit
Price
Rev = $15Cost= $8μ(p)=165-5pσ= μ2
P = $22Q= 65 Magμ(p)=56 Magσ= 28Exp Profit=$543
Any Questions
Yossi Sheffi
- Slide 1
- Slide 2
- Slide 3
- Slide 4
- Slide 5
- Slide 6
- Slide 7
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- Slide 22
- Slide 23
-
Outlinebull Single period inventory decisionsbull Calculating the optimal order sizebull Numericallybull Using spreadsheetbull Using simulationbull Analyticallybull The profit functionbull For specific distributionbull Level of Servicebull Extensionsbull Fixed costsbull Risksbull Initial inventorybull Elastic demand
Single Period Ordering
bull Seasonal itemsbull Perishable goodsbull News printbull Fashion itemsbull Some high tech productsbull Risky investments
Selling Magazines
bull Weekly demand
1048708 Total 4023 magazines
1048708 Average 774 Magweek
1048708 Min 51 max 113 Magweek
Detailed Histogram
Average=774 Magwk
Fre
qu
en
cy
(WksY
r)
Demand (MagWk)
Cu
mm
Fre
q (W
ksY
r)
Histogram
Demand (MagWk)
Fre
qu
en
cy (W
ksY
r)
Cu
mm
Fre
q (W
ksY
r)Cu
mm
ula
tive F
req
uen
cy
More
The Ordering Decision(Spreadsheet)bull Assume each magazine sells for $15bull Cost of each magazine $8
Order
ExpProfit
dwk Prob
Newsboy Framework ndash Panel 1 ldquobasic Scenariordquo
Expected ProfitsP
rofi
t
Order Size
Optimal Order (Analytical)
bull The optimal order is Qbull At Q the probability of selling one more magazine is the probability that demand is greater than Qbull The expected profit from ordering the (Q+1)st magazine isbull 1048708 If demand is high and we sell itbull 1048708 (REV-COST) x Pr( Demand is higher than Q)bull 1048708 If demand is low and we are stuckbull 1048708 (-COST) x Pr( Demand is lower or equal to Q)
bull The optimum is where the total expected profit from ordering one more magazine is zerobull (REV-COST) x Pr( Demand gt Q) ndash COST x Pr( Demandbull le Q) = 0
bull
Optimal OrderThe ldquocritical ratiordquo
Fre
qu
en
cy (
WksY
r
Cu
mm
ula
tive F
req
uen
cy
Demand (Magweek) More
Salvage Valuebull Salvage value = $4Mag
Demand (Magweek)
Order Size
Cu
mm
ula
tive
Fre
qu
en
cy
Fre
qu
en
cy
(WksY
r)
Pro
fit
The Profit Functionbull Revenue from sold itemsbull Revenue or costs associated with unsold
items These may include revenue from salvage or cost associated with disposal
bull Costs associated with not meeting customersrsquo demand The lost sales cost
can include lost of good will and actual penalties for low servicebull The cost of buying the merchandise in the
first place
The Profit Function
The Profit Function ndash Simple Case
Optimal Order
and
Level of ServiceCycle Service ndash The probability that
there will be a stock-out during a
cycle
Cycle Service = F(Q)
Fill Rate - The probability that a
specific customer will encounter a
stock-out
Level of ServiceS
erv
ice L
evel
Order
Cycle Service
Fill Rate
REV=$15
COST=$7
Normal Distribution of Demand
Order Size
Exp
ecte
d P
rofi
t
Incorporating Fixed Costsbull With fixed costs of $300order
REV=$15
COST=$7
Exp
ecte
d
Pro
fi
Order Size
Risk of Loss
300(15-7)=375 Below that there is no possibility to make money even if we sell ALL the order
REV=$15
COST=$7
Order Quantity
Pro
bab
ilit
y o
f Loss
Ordering with Initial Inventory
Note the cost of Q0 is irrelevant
Given initial Inventory Q0 how to order1 Calculate Q as before2 If Q0 lt Q order (Qlt Q0 )3 If Q0 ge Q order 0With fixed costs order only if the expectedprofits from ordering are more than theordering costs1 Set Qcr as the smallest Q such that E[Profits with Qcr]gtE[Profits with Q]-F2 If Q0 lt Qcr order (Qlt Q0 )3 If Q0 ge Qcr order 0
Ordering with FixedCosts and Initial Inventory
bullIf initial inventory is LE 46 order up to 80
bullIf initial inventory is GE 47 order nothing
REV=$15
COST=$7
Example F = $150Exp
ecte
d P
rofi
t
Initial Inventory
bull μ =D(P) σ = f(μ)bull Procedurebull 1 Set Pbull 2 Calculate μbull 3 Calculate σbull 4 bull 5 Calculate optimal expected profits as a function of P
Elastic DemandExpected Profit Function
Pro
fit
Price
Rev = $15Cost= $8μ(p)=165-5pσ= μ2
P = $22Q= 65 Magμ(p)=56 Magσ= 28Exp Profit=$543
Any Questions
Yossi Sheffi
- Slide 1
- Slide 2
- Slide 3
- Slide 4
- Slide 5
- Slide 6
- Slide 7
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- Slide 22
- Slide 23
-
Single Period Ordering
bull Seasonal itemsbull Perishable goodsbull News printbull Fashion itemsbull Some high tech productsbull Risky investments
Selling Magazines
bull Weekly demand
1048708 Total 4023 magazines
1048708 Average 774 Magweek
1048708 Min 51 max 113 Magweek
Detailed Histogram
Average=774 Magwk
Fre
qu
en
cy
(WksY
r)
Demand (MagWk)
Cu
mm
Fre
q (W
ksY
r)
Histogram
Demand (MagWk)
Fre
qu
en
cy (W
ksY
r)
Cu
mm
Fre
q (W
ksY
r)Cu
mm
ula
tive F
req
uen
cy
More
The Ordering Decision(Spreadsheet)bull Assume each magazine sells for $15bull Cost of each magazine $8
Order
ExpProfit
dwk Prob
Newsboy Framework ndash Panel 1 ldquobasic Scenariordquo
Expected ProfitsP
rofi
t
Order Size
Optimal Order (Analytical)
bull The optimal order is Qbull At Q the probability of selling one more magazine is the probability that demand is greater than Qbull The expected profit from ordering the (Q+1)st magazine isbull 1048708 If demand is high and we sell itbull 1048708 (REV-COST) x Pr( Demand is higher than Q)bull 1048708 If demand is low and we are stuckbull 1048708 (-COST) x Pr( Demand is lower or equal to Q)
bull The optimum is where the total expected profit from ordering one more magazine is zerobull (REV-COST) x Pr( Demand gt Q) ndash COST x Pr( Demandbull le Q) = 0
bull
Optimal OrderThe ldquocritical ratiordquo
Fre
qu
en
cy (
WksY
r
Cu
mm
ula
tive F
req
uen
cy
Demand (Magweek) More
Salvage Valuebull Salvage value = $4Mag
Demand (Magweek)
Order Size
Cu
mm
ula
tive
Fre
qu
en
cy
Fre
qu
en
cy
(WksY
r)
Pro
fit
The Profit Functionbull Revenue from sold itemsbull Revenue or costs associated with unsold
items These may include revenue from salvage or cost associated with disposal
bull Costs associated with not meeting customersrsquo demand The lost sales cost
can include lost of good will and actual penalties for low servicebull The cost of buying the merchandise in the
first place
The Profit Function
The Profit Function ndash Simple Case
Optimal Order
and
Level of ServiceCycle Service ndash The probability that
there will be a stock-out during a
cycle
Cycle Service = F(Q)
Fill Rate - The probability that a
specific customer will encounter a
stock-out
Level of ServiceS
erv
ice L
evel
Order
Cycle Service
Fill Rate
REV=$15
COST=$7
Normal Distribution of Demand
Order Size
Exp
ecte
d P
rofi
t
Incorporating Fixed Costsbull With fixed costs of $300order
REV=$15
COST=$7
Exp
ecte
d
Pro
fi
Order Size
Risk of Loss
300(15-7)=375 Below that there is no possibility to make money even if we sell ALL the order
REV=$15
COST=$7
Order Quantity
Pro
bab
ilit
y o
f Loss
Ordering with Initial Inventory
Note the cost of Q0 is irrelevant
Given initial Inventory Q0 how to order1 Calculate Q as before2 If Q0 lt Q order (Qlt Q0 )3 If Q0 ge Q order 0With fixed costs order only if the expectedprofits from ordering are more than theordering costs1 Set Qcr as the smallest Q such that E[Profits with Qcr]gtE[Profits with Q]-F2 If Q0 lt Qcr order (Qlt Q0 )3 If Q0 ge Qcr order 0
Ordering with FixedCosts and Initial Inventory
bullIf initial inventory is LE 46 order up to 80
bullIf initial inventory is GE 47 order nothing
REV=$15
COST=$7
Example F = $150Exp
ecte
d P
rofi
t
Initial Inventory
bull μ =D(P) σ = f(μ)bull Procedurebull 1 Set Pbull 2 Calculate μbull 3 Calculate σbull 4 bull 5 Calculate optimal expected profits as a function of P
Elastic DemandExpected Profit Function
Pro
fit
Price
Rev = $15Cost= $8μ(p)=165-5pσ= μ2
P = $22Q= 65 Magμ(p)=56 Magσ= 28Exp Profit=$543
Any Questions
Yossi Sheffi
- Slide 1
- Slide 2
- Slide 3
- Slide 4
- Slide 5
- Slide 6
- Slide 7
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- Slide 22
- Slide 23
-
Selling Magazines
bull Weekly demand
1048708 Total 4023 magazines
1048708 Average 774 Magweek
1048708 Min 51 max 113 Magweek
Detailed Histogram
Average=774 Magwk
Fre
qu
en
cy
(WksY
r)
Demand (MagWk)
Cu
mm
Fre
q (W
ksY
r)
Histogram
Demand (MagWk)
Fre
qu
en
cy (W
ksY
r)
Cu
mm
Fre
q (W
ksY
r)Cu
mm
ula
tive F
req
uen
cy
More
The Ordering Decision(Spreadsheet)bull Assume each magazine sells for $15bull Cost of each magazine $8
Order
ExpProfit
dwk Prob
Newsboy Framework ndash Panel 1 ldquobasic Scenariordquo
Expected ProfitsP
rofi
t
Order Size
Optimal Order (Analytical)
bull The optimal order is Qbull At Q the probability of selling one more magazine is the probability that demand is greater than Qbull The expected profit from ordering the (Q+1)st magazine isbull 1048708 If demand is high and we sell itbull 1048708 (REV-COST) x Pr( Demand is higher than Q)bull 1048708 If demand is low and we are stuckbull 1048708 (-COST) x Pr( Demand is lower or equal to Q)
bull The optimum is where the total expected profit from ordering one more magazine is zerobull (REV-COST) x Pr( Demand gt Q) ndash COST x Pr( Demandbull le Q) = 0
bull
Optimal OrderThe ldquocritical ratiordquo
Fre
qu
en
cy (
WksY
r
Cu
mm
ula
tive F
req
uen
cy
Demand (Magweek) More
Salvage Valuebull Salvage value = $4Mag
Demand (Magweek)
Order Size
Cu
mm
ula
tive
Fre
qu
en
cy
Fre
qu
en
cy
(WksY
r)
Pro
fit
The Profit Functionbull Revenue from sold itemsbull Revenue or costs associated with unsold
items These may include revenue from salvage or cost associated with disposal
bull Costs associated with not meeting customersrsquo demand The lost sales cost
can include lost of good will and actual penalties for low servicebull The cost of buying the merchandise in the
first place
The Profit Function
The Profit Function ndash Simple Case
Optimal Order
and
Level of ServiceCycle Service ndash The probability that
there will be a stock-out during a
cycle
Cycle Service = F(Q)
Fill Rate - The probability that a
specific customer will encounter a
stock-out
Level of ServiceS
erv
ice L
evel
Order
Cycle Service
Fill Rate
REV=$15
COST=$7
Normal Distribution of Demand
Order Size
Exp
ecte
d P
rofi
t
Incorporating Fixed Costsbull With fixed costs of $300order
REV=$15
COST=$7
Exp
ecte
d
Pro
fi
Order Size
Risk of Loss
300(15-7)=375 Below that there is no possibility to make money even if we sell ALL the order
REV=$15
COST=$7
Order Quantity
Pro
bab
ilit
y o
f Loss
Ordering with Initial Inventory
Note the cost of Q0 is irrelevant
Given initial Inventory Q0 how to order1 Calculate Q as before2 If Q0 lt Q order (Qlt Q0 )3 If Q0 ge Q order 0With fixed costs order only if the expectedprofits from ordering are more than theordering costs1 Set Qcr as the smallest Q such that E[Profits with Qcr]gtE[Profits with Q]-F2 If Q0 lt Qcr order (Qlt Q0 )3 If Q0 ge Qcr order 0
Ordering with FixedCosts and Initial Inventory
bullIf initial inventory is LE 46 order up to 80
bullIf initial inventory is GE 47 order nothing
REV=$15
COST=$7
Example F = $150Exp
ecte
d P
rofi
t
Initial Inventory
bull μ =D(P) σ = f(μ)bull Procedurebull 1 Set Pbull 2 Calculate μbull 3 Calculate σbull 4 bull 5 Calculate optimal expected profits as a function of P
Elastic DemandExpected Profit Function
Pro
fit
Price
Rev = $15Cost= $8μ(p)=165-5pσ= μ2
P = $22Q= 65 Magμ(p)=56 Magσ= 28Exp Profit=$543
Any Questions
Yossi Sheffi
- Slide 1
- Slide 2
- Slide 3
- Slide 4
- Slide 5
- Slide 6
- Slide 7
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- Slide 22
- Slide 23
-
Detailed Histogram
Average=774 Magwk
Fre
qu
en
cy
(WksY
r)
Demand (MagWk)
Cu
mm
Fre
q (W
ksY
r)
Histogram
Demand (MagWk)
Fre
qu
en
cy (W
ksY
r)
Cu
mm
Fre
q (W
ksY
r)Cu
mm
ula
tive F
req
uen
cy
More
The Ordering Decision(Spreadsheet)bull Assume each magazine sells for $15bull Cost of each magazine $8
Order
ExpProfit
dwk Prob
Newsboy Framework ndash Panel 1 ldquobasic Scenariordquo
Expected ProfitsP
rofi
t
Order Size
Optimal Order (Analytical)
bull The optimal order is Qbull At Q the probability of selling one more magazine is the probability that demand is greater than Qbull The expected profit from ordering the (Q+1)st magazine isbull 1048708 If demand is high and we sell itbull 1048708 (REV-COST) x Pr( Demand is higher than Q)bull 1048708 If demand is low and we are stuckbull 1048708 (-COST) x Pr( Demand is lower or equal to Q)
bull The optimum is where the total expected profit from ordering one more magazine is zerobull (REV-COST) x Pr( Demand gt Q) ndash COST x Pr( Demandbull le Q) = 0
bull
Optimal OrderThe ldquocritical ratiordquo
Fre
qu
en
cy (
WksY
r
Cu
mm
ula
tive F
req
uen
cy
Demand (Magweek) More
Salvage Valuebull Salvage value = $4Mag
Demand (Magweek)
Order Size
Cu
mm
ula
tive
Fre
qu
en
cy
Fre
qu
en
cy
(WksY
r)
Pro
fit
The Profit Functionbull Revenue from sold itemsbull Revenue or costs associated with unsold
items These may include revenue from salvage or cost associated with disposal
bull Costs associated with not meeting customersrsquo demand The lost sales cost
can include lost of good will and actual penalties for low servicebull The cost of buying the merchandise in the
first place
The Profit Function
The Profit Function ndash Simple Case
Optimal Order
and
Level of ServiceCycle Service ndash The probability that
there will be a stock-out during a
cycle
Cycle Service = F(Q)
Fill Rate - The probability that a
specific customer will encounter a
stock-out
Level of ServiceS
erv
ice L
evel
Order
Cycle Service
Fill Rate
REV=$15
COST=$7
Normal Distribution of Demand
Order Size
Exp
ecte
d P
rofi
t
Incorporating Fixed Costsbull With fixed costs of $300order
REV=$15
COST=$7
Exp
ecte
d
Pro
fi
Order Size
Risk of Loss
300(15-7)=375 Below that there is no possibility to make money even if we sell ALL the order
REV=$15
COST=$7
Order Quantity
Pro
bab
ilit
y o
f Loss
Ordering with Initial Inventory
Note the cost of Q0 is irrelevant
Given initial Inventory Q0 how to order1 Calculate Q as before2 If Q0 lt Q order (Qlt Q0 )3 If Q0 ge Q order 0With fixed costs order only if the expectedprofits from ordering are more than theordering costs1 Set Qcr as the smallest Q such that E[Profits with Qcr]gtE[Profits with Q]-F2 If Q0 lt Qcr order (Qlt Q0 )3 If Q0 ge Qcr order 0
Ordering with FixedCosts and Initial Inventory
bullIf initial inventory is LE 46 order up to 80
bullIf initial inventory is GE 47 order nothing
REV=$15
COST=$7
Example F = $150Exp
ecte
d P
rofi
t
Initial Inventory
bull μ =D(P) σ = f(μ)bull Procedurebull 1 Set Pbull 2 Calculate μbull 3 Calculate σbull 4 bull 5 Calculate optimal expected profits as a function of P
Elastic DemandExpected Profit Function
Pro
fit
Price
Rev = $15Cost= $8μ(p)=165-5pσ= μ2
P = $22Q= 65 Magμ(p)=56 Magσ= 28Exp Profit=$543
Any Questions
Yossi Sheffi
- Slide 1
- Slide 2
- Slide 3
- Slide 4
- Slide 5
- Slide 6
- Slide 7
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- Slide 22
- Slide 23
-
Histogram
Demand (MagWk)
Fre
qu
en
cy (W
ksY
r)
Cu
mm
Fre
q (W
ksY
r)Cu
mm
ula
tive F
req
uen
cy
More
The Ordering Decision(Spreadsheet)bull Assume each magazine sells for $15bull Cost of each magazine $8
Order
ExpProfit
dwk Prob
Newsboy Framework ndash Panel 1 ldquobasic Scenariordquo
Expected ProfitsP
rofi
t
Order Size
Optimal Order (Analytical)
bull The optimal order is Qbull At Q the probability of selling one more magazine is the probability that demand is greater than Qbull The expected profit from ordering the (Q+1)st magazine isbull 1048708 If demand is high and we sell itbull 1048708 (REV-COST) x Pr( Demand is higher than Q)bull 1048708 If demand is low and we are stuckbull 1048708 (-COST) x Pr( Demand is lower or equal to Q)
bull The optimum is where the total expected profit from ordering one more magazine is zerobull (REV-COST) x Pr( Demand gt Q) ndash COST x Pr( Demandbull le Q) = 0
bull
Optimal OrderThe ldquocritical ratiordquo
Fre
qu
en
cy (
WksY
r
Cu
mm
ula
tive F
req
uen
cy
Demand (Magweek) More
Salvage Valuebull Salvage value = $4Mag
Demand (Magweek)
Order Size
Cu
mm
ula
tive
Fre
qu
en
cy
Fre
qu
en
cy
(WksY
r)
Pro
fit
The Profit Functionbull Revenue from sold itemsbull Revenue or costs associated with unsold
items These may include revenue from salvage or cost associated with disposal
bull Costs associated with not meeting customersrsquo demand The lost sales cost
can include lost of good will and actual penalties for low servicebull The cost of buying the merchandise in the
first place
The Profit Function
The Profit Function ndash Simple Case
Optimal Order
and
Level of ServiceCycle Service ndash The probability that
there will be a stock-out during a
cycle
Cycle Service = F(Q)
Fill Rate - The probability that a
specific customer will encounter a
stock-out
Level of ServiceS
erv
ice L
evel
Order
Cycle Service
Fill Rate
REV=$15
COST=$7
Normal Distribution of Demand
Order Size
Exp
ecte
d P
rofi
t
Incorporating Fixed Costsbull With fixed costs of $300order
REV=$15
COST=$7
Exp
ecte
d
Pro
fi
Order Size
Risk of Loss
300(15-7)=375 Below that there is no possibility to make money even if we sell ALL the order
REV=$15
COST=$7
Order Quantity
Pro
bab
ilit
y o
f Loss
Ordering with Initial Inventory
Note the cost of Q0 is irrelevant
Given initial Inventory Q0 how to order1 Calculate Q as before2 If Q0 lt Q order (Qlt Q0 )3 If Q0 ge Q order 0With fixed costs order only if the expectedprofits from ordering are more than theordering costs1 Set Qcr as the smallest Q such that E[Profits with Qcr]gtE[Profits with Q]-F2 If Q0 lt Qcr order (Qlt Q0 )3 If Q0 ge Qcr order 0
Ordering with FixedCosts and Initial Inventory
bullIf initial inventory is LE 46 order up to 80
bullIf initial inventory is GE 47 order nothing
REV=$15
COST=$7
Example F = $150Exp
ecte
d P
rofi
t
Initial Inventory
bull μ =D(P) σ = f(μ)bull Procedurebull 1 Set Pbull 2 Calculate μbull 3 Calculate σbull 4 bull 5 Calculate optimal expected profits as a function of P
Elastic DemandExpected Profit Function
Pro
fit
Price
Rev = $15Cost= $8μ(p)=165-5pσ= μ2
P = $22Q= 65 Magμ(p)=56 Magσ= 28Exp Profit=$543
Any Questions
Yossi Sheffi
- Slide 1
- Slide 2
- Slide 3
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- Slide 5
- Slide 6
- Slide 7
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- Slide 23
-
The Ordering Decision(Spreadsheet)bull Assume each magazine sells for $15bull Cost of each magazine $8
Order
ExpProfit
dwk Prob
Newsboy Framework ndash Panel 1 ldquobasic Scenariordquo
Expected ProfitsP
rofi
t
Order Size
Optimal Order (Analytical)
bull The optimal order is Qbull At Q the probability of selling one more magazine is the probability that demand is greater than Qbull The expected profit from ordering the (Q+1)st magazine isbull 1048708 If demand is high and we sell itbull 1048708 (REV-COST) x Pr( Demand is higher than Q)bull 1048708 If demand is low and we are stuckbull 1048708 (-COST) x Pr( Demand is lower or equal to Q)
bull The optimum is where the total expected profit from ordering one more magazine is zerobull (REV-COST) x Pr( Demand gt Q) ndash COST x Pr( Demandbull le Q) = 0
bull
Optimal OrderThe ldquocritical ratiordquo
Fre
qu
en
cy (
WksY
r
Cu
mm
ula
tive F
req
uen
cy
Demand (Magweek) More
Salvage Valuebull Salvage value = $4Mag
Demand (Magweek)
Order Size
Cu
mm
ula
tive
Fre
qu
en
cy
Fre
qu
en
cy
(WksY
r)
Pro
fit
The Profit Functionbull Revenue from sold itemsbull Revenue or costs associated with unsold
items These may include revenue from salvage or cost associated with disposal
bull Costs associated with not meeting customersrsquo demand The lost sales cost
can include lost of good will and actual penalties for low servicebull The cost of buying the merchandise in the
first place
The Profit Function
The Profit Function ndash Simple Case
Optimal Order
and
Level of ServiceCycle Service ndash The probability that
there will be a stock-out during a
cycle
Cycle Service = F(Q)
Fill Rate - The probability that a
specific customer will encounter a
stock-out
Level of ServiceS
erv
ice L
evel
Order
Cycle Service
Fill Rate
REV=$15
COST=$7
Normal Distribution of Demand
Order Size
Exp
ecte
d P
rofi
t
Incorporating Fixed Costsbull With fixed costs of $300order
REV=$15
COST=$7
Exp
ecte
d
Pro
fi
Order Size
Risk of Loss
300(15-7)=375 Below that there is no possibility to make money even if we sell ALL the order
REV=$15
COST=$7
Order Quantity
Pro
bab
ilit
y o
f Loss
Ordering with Initial Inventory
Note the cost of Q0 is irrelevant
Given initial Inventory Q0 how to order1 Calculate Q as before2 If Q0 lt Q order (Qlt Q0 )3 If Q0 ge Q order 0With fixed costs order only if the expectedprofits from ordering are more than theordering costs1 Set Qcr as the smallest Q such that E[Profits with Qcr]gtE[Profits with Q]-F2 If Q0 lt Qcr order (Qlt Q0 )3 If Q0 ge Qcr order 0
Ordering with FixedCosts and Initial Inventory
bullIf initial inventory is LE 46 order up to 80
bullIf initial inventory is GE 47 order nothing
REV=$15
COST=$7
Example F = $150Exp
ecte
d P
rofi
t
Initial Inventory
bull μ =D(P) σ = f(μ)bull Procedurebull 1 Set Pbull 2 Calculate μbull 3 Calculate σbull 4 bull 5 Calculate optimal expected profits as a function of P
Elastic DemandExpected Profit Function
Pro
fit
Price
Rev = $15Cost= $8μ(p)=165-5pσ= μ2
P = $22Q= 65 Magμ(p)=56 Magσ= 28Exp Profit=$543
Any Questions
Yossi Sheffi
- Slide 1
- Slide 2
- Slide 3
- Slide 4
- Slide 5
- Slide 6
- Slide 7
- Slide 8
- Slide 9
- Slide 10
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- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- Slide 22
- Slide 23
-
Expected ProfitsP
rofi
t
Order Size
Optimal Order (Analytical)
bull The optimal order is Qbull At Q the probability of selling one more magazine is the probability that demand is greater than Qbull The expected profit from ordering the (Q+1)st magazine isbull 1048708 If demand is high and we sell itbull 1048708 (REV-COST) x Pr( Demand is higher than Q)bull 1048708 If demand is low and we are stuckbull 1048708 (-COST) x Pr( Demand is lower or equal to Q)
bull The optimum is where the total expected profit from ordering one more magazine is zerobull (REV-COST) x Pr( Demand gt Q) ndash COST x Pr( Demandbull le Q) = 0
bull
Optimal OrderThe ldquocritical ratiordquo
Fre
qu
en
cy (
WksY
r
Cu
mm
ula
tive F
req
uen
cy
Demand (Magweek) More
Salvage Valuebull Salvage value = $4Mag
Demand (Magweek)
Order Size
Cu
mm
ula
tive
Fre
qu
en
cy
Fre
qu
en
cy
(WksY
r)
Pro
fit
The Profit Functionbull Revenue from sold itemsbull Revenue or costs associated with unsold
items These may include revenue from salvage or cost associated with disposal
bull Costs associated with not meeting customersrsquo demand The lost sales cost
can include lost of good will and actual penalties for low servicebull The cost of buying the merchandise in the
first place
The Profit Function
The Profit Function ndash Simple Case
Optimal Order
and
Level of ServiceCycle Service ndash The probability that
there will be a stock-out during a
cycle
Cycle Service = F(Q)
Fill Rate - The probability that a
specific customer will encounter a
stock-out
Level of ServiceS
erv
ice L
evel
Order
Cycle Service
Fill Rate
REV=$15
COST=$7
Normal Distribution of Demand
Order Size
Exp
ecte
d P
rofi
t
Incorporating Fixed Costsbull With fixed costs of $300order
REV=$15
COST=$7
Exp
ecte
d
Pro
fi
Order Size
Risk of Loss
300(15-7)=375 Below that there is no possibility to make money even if we sell ALL the order
REV=$15
COST=$7
Order Quantity
Pro
bab
ilit
y o
f Loss
Ordering with Initial Inventory
Note the cost of Q0 is irrelevant
Given initial Inventory Q0 how to order1 Calculate Q as before2 If Q0 lt Q order (Qlt Q0 )3 If Q0 ge Q order 0With fixed costs order only if the expectedprofits from ordering are more than theordering costs1 Set Qcr as the smallest Q such that E[Profits with Qcr]gtE[Profits with Q]-F2 If Q0 lt Qcr order (Qlt Q0 )3 If Q0 ge Qcr order 0
Ordering with FixedCosts and Initial Inventory
bullIf initial inventory is LE 46 order up to 80
bullIf initial inventory is GE 47 order nothing
REV=$15
COST=$7
Example F = $150Exp
ecte
d P
rofi
t
Initial Inventory
bull μ =D(P) σ = f(μ)bull Procedurebull 1 Set Pbull 2 Calculate μbull 3 Calculate σbull 4 bull 5 Calculate optimal expected profits as a function of P
Elastic DemandExpected Profit Function
Pro
fit
Price
Rev = $15Cost= $8μ(p)=165-5pσ= μ2
P = $22Q= 65 Magμ(p)=56 Magσ= 28Exp Profit=$543
Any Questions
Yossi Sheffi
- Slide 1
- Slide 2
- Slide 3
- Slide 4
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- Slide 6
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- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- Slide 22
- Slide 23
-
Optimal Order (Analytical)
bull The optimal order is Qbull At Q the probability of selling one more magazine is the probability that demand is greater than Qbull The expected profit from ordering the (Q+1)st magazine isbull 1048708 If demand is high and we sell itbull 1048708 (REV-COST) x Pr( Demand is higher than Q)bull 1048708 If demand is low and we are stuckbull 1048708 (-COST) x Pr( Demand is lower or equal to Q)
bull The optimum is where the total expected profit from ordering one more magazine is zerobull (REV-COST) x Pr( Demand gt Q) ndash COST x Pr( Demandbull le Q) = 0
bull
Optimal OrderThe ldquocritical ratiordquo
Fre
qu
en
cy (
WksY
r
Cu
mm
ula
tive F
req
uen
cy
Demand (Magweek) More
Salvage Valuebull Salvage value = $4Mag
Demand (Magweek)
Order Size
Cu
mm
ula
tive
Fre
qu
en
cy
Fre
qu
en
cy
(WksY
r)
Pro
fit
The Profit Functionbull Revenue from sold itemsbull Revenue or costs associated with unsold
items These may include revenue from salvage or cost associated with disposal
bull Costs associated with not meeting customersrsquo demand The lost sales cost
can include lost of good will and actual penalties for low servicebull The cost of buying the merchandise in the
first place
The Profit Function
The Profit Function ndash Simple Case
Optimal Order
and
Level of ServiceCycle Service ndash The probability that
there will be a stock-out during a
cycle
Cycle Service = F(Q)
Fill Rate - The probability that a
specific customer will encounter a
stock-out
Level of ServiceS
erv
ice L
evel
Order
Cycle Service
Fill Rate
REV=$15
COST=$7
Normal Distribution of Demand
Order Size
Exp
ecte
d P
rofi
t
Incorporating Fixed Costsbull With fixed costs of $300order
REV=$15
COST=$7
Exp
ecte
d
Pro
fi
Order Size
Risk of Loss
300(15-7)=375 Below that there is no possibility to make money even if we sell ALL the order
REV=$15
COST=$7
Order Quantity
Pro
bab
ilit
y o
f Loss
Ordering with Initial Inventory
Note the cost of Q0 is irrelevant
Given initial Inventory Q0 how to order1 Calculate Q as before2 If Q0 lt Q order (Qlt Q0 )3 If Q0 ge Q order 0With fixed costs order only if the expectedprofits from ordering are more than theordering costs1 Set Qcr as the smallest Q such that E[Profits with Qcr]gtE[Profits with Q]-F2 If Q0 lt Qcr order (Qlt Q0 )3 If Q0 ge Qcr order 0
Ordering with FixedCosts and Initial Inventory
bullIf initial inventory is LE 46 order up to 80
bullIf initial inventory is GE 47 order nothing
REV=$15
COST=$7
Example F = $150Exp
ecte
d P
rofi
t
Initial Inventory
bull μ =D(P) σ = f(μ)bull Procedurebull 1 Set Pbull 2 Calculate μbull 3 Calculate σbull 4 bull 5 Calculate optimal expected profits as a function of P
Elastic DemandExpected Profit Function
Pro
fit
Price
Rev = $15Cost= $8μ(p)=165-5pσ= μ2
P = $22Q= 65 Magμ(p)=56 Magσ= 28Exp Profit=$543
Any Questions
Yossi Sheffi
- Slide 1
- Slide 2
- Slide 3
- Slide 4
- Slide 5
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- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- Slide 22
- Slide 23
-
Optimal OrderThe ldquocritical ratiordquo
Fre
qu
en
cy (
WksY
r
Cu
mm
ula
tive F
req
uen
cy
Demand (Magweek) More
Salvage Valuebull Salvage value = $4Mag
Demand (Magweek)
Order Size
Cu
mm
ula
tive
Fre
qu
en
cy
Fre
qu
en
cy
(WksY
r)
Pro
fit
The Profit Functionbull Revenue from sold itemsbull Revenue or costs associated with unsold
items These may include revenue from salvage or cost associated with disposal
bull Costs associated with not meeting customersrsquo demand The lost sales cost
can include lost of good will and actual penalties for low servicebull The cost of buying the merchandise in the
first place
The Profit Function
The Profit Function ndash Simple Case
Optimal Order
and
Level of ServiceCycle Service ndash The probability that
there will be a stock-out during a
cycle
Cycle Service = F(Q)
Fill Rate - The probability that a
specific customer will encounter a
stock-out
Level of ServiceS
erv
ice L
evel
Order
Cycle Service
Fill Rate
REV=$15
COST=$7
Normal Distribution of Demand
Order Size
Exp
ecte
d P
rofi
t
Incorporating Fixed Costsbull With fixed costs of $300order
REV=$15
COST=$7
Exp
ecte
d
Pro
fi
Order Size
Risk of Loss
300(15-7)=375 Below that there is no possibility to make money even if we sell ALL the order
REV=$15
COST=$7
Order Quantity
Pro
bab
ilit
y o
f Loss
Ordering with Initial Inventory
Note the cost of Q0 is irrelevant
Given initial Inventory Q0 how to order1 Calculate Q as before2 If Q0 lt Q order (Qlt Q0 )3 If Q0 ge Q order 0With fixed costs order only if the expectedprofits from ordering are more than theordering costs1 Set Qcr as the smallest Q such that E[Profits with Qcr]gtE[Profits with Q]-F2 If Q0 lt Qcr order (Qlt Q0 )3 If Q0 ge Qcr order 0
Ordering with FixedCosts and Initial Inventory
bullIf initial inventory is LE 46 order up to 80
bullIf initial inventory is GE 47 order nothing
REV=$15
COST=$7
Example F = $150Exp
ecte
d P
rofi
t
Initial Inventory
bull μ =D(P) σ = f(μ)bull Procedurebull 1 Set Pbull 2 Calculate μbull 3 Calculate σbull 4 bull 5 Calculate optimal expected profits as a function of P
Elastic DemandExpected Profit Function
Pro
fit
Price
Rev = $15Cost= $8μ(p)=165-5pσ= μ2
P = $22Q= 65 Magμ(p)=56 Magσ= 28Exp Profit=$543
Any Questions
Yossi Sheffi
- Slide 1
- Slide 2
- Slide 3
- Slide 4
- Slide 5
- Slide 6
- Slide 7
- Slide 8
- Slide 9
- Slide 10
- Slide 11
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- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- Slide 22
- Slide 23
-
Salvage Valuebull Salvage value = $4Mag
Demand (Magweek)
Order Size
Cu
mm
ula
tive
Fre
qu
en
cy
Fre
qu
en
cy
(WksY
r)
Pro
fit
The Profit Functionbull Revenue from sold itemsbull Revenue or costs associated with unsold
items These may include revenue from salvage or cost associated with disposal
bull Costs associated with not meeting customersrsquo demand The lost sales cost
can include lost of good will and actual penalties for low servicebull The cost of buying the merchandise in the
first place
The Profit Function
The Profit Function ndash Simple Case
Optimal Order
and
Level of ServiceCycle Service ndash The probability that
there will be a stock-out during a
cycle
Cycle Service = F(Q)
Fill Rate - The probability that a
specific customer will encounter a
stock-out
Level of ServiceS
erv
ice L
evel
Order
Cycle Service
Fill Rate
REV=$15
COST=$7
Normal Distribution of Demand
Order Size
Exp
ecte
d P
rofi
t
Incorporating Fixed Costsbull With fixed costs of $300order
REV=$15
COST=$7
Exp
ecte
d
Pro
fi
Order Size
Risk of Loss
300(15-7)=375 Below that there is no possibility to make money even if we sell ALL the order
REV=$15
COST=$7
Order Quantity
Pro
bab
ilit
y o
f Loss
Ordering with Initial Inventory
Note the cost of Q0 is irrelevant
Given initial Inventory Q0 how to order1 Calculate Q as before2 If Q0 lt Q order (Qlt Q0 )3 If Q0 ge Q order 0With fixed costs order only if the expectedprofits from ordering are more than theordering costs1 Set Qcr as the smallest Q such that E[Profits with Qcr]gtE[Profits with Q]-F2 If Q0 lt Qcr order (Qlt Q0 )3 If Q0 ge Qcr order 0
Ordering with FixedCosts and Initial Inventory
bullIf initial inventory is LE 46 order up to 80
bullIf initial inventory is GE 47 order nothing
REV=$15
COST=$7
Example F = $150Exp
ecte
d P
rofi
t
Initial Inventory
bull μ =D(P) σ = f(μ)bull Procedurebull 1 Set Pbull 2 Calculate μbull 3 Calculate σbull 4 bull 5 Calculate optimal expected profits as a function of P
Elastic DemandExpected Profit Function
Pro
fit
Price
Rev = $15Cost= $8μ(p)=165-5pσ= μ2
P = $22Q= 65 Magμ(p)=56 Magσ= 28Exp Profit=$543
Any Questions
Yossi Sheffi
- Slide 1
- Slide 2
- Slide 3
- Slide 4
- Slide 5
- Slide 6
- Slide 7
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- Slide 22
- Slide 23
-
The Profit Functionbull Revenue from sold itemsbull Revenue or costs associated with unsold
items These may include revenue from salvage or cost associated with disposal
bull Costs associated with not meeting customersrsquo demand The lost sales cost
can include lost of good will and actual penalties for low servicebull The cost of buying the merchandise in the
first place
The Profit Function
The Profit Function ndash Simple Case
Optimal Order
and
Level of ServiceCycle Service ndash The probability that
there will be a stock-out during a
cycle
Cycle Service = F(Q)
Fill Rate - The probability that a
specific customer will encounter a
stock-out
Level of ServiceS
erv
ice L
evel
Order
Cycle Service
Fill Rate
REV=$15
COST=$7
Normal Distribution of Demand
Order Size
Exp
ecte
d P
rofi
t
Incorporating Fixed Costsbull With fixed costs of $300order
REV=$15
COST=$7
Exp
ecte
d
Pro
fi
Order Size
Risk of Loss
300(15-7)=375 Below that there is no possibility to make money even if we sell ALL the order
REV=$15
COST=$7
Order Quantity
Pro
bab
ilit
y o
f Loss
Ordering with Initial Inventory
Note the cost of Q0 is irrelevant
Given initial Inventory Q0 how to order1 Calculate Q as before2 If Q0 lt Q order (Qlt Q0 )3 If Q0 ge Q order 0With fixed costs order only if the expectedprofits from ordering are more than theordering costs1 Set Qcr as the smallest Q such that E[Profits with Qcr]gtE[Profits with Q]-F2 If Q0 lt Qcr order (Qlt Q0 )3 If Q0 ge Qcr order 0
Ordering with FixedCosts and Initial Inventory
bullIf initial inventory is LE 46 order up to 80
bullIf initial inventory is GE 47 order nothing
REV=$15
COST=$7
Example F = $150Exp
ecte
d P
rofi
t
Initial Inventory
bull μ =D(P) σ = f(μ)bull Procedurebull 1 Set Pbull 2 Calculate μbull 3 Calculate σbull 4 bull 5 Calculate optimal expected profits as a function of P
Elastic DemandExpected Profit Function
Pro
fit
Price
Rev = $15Cost= $8μ(p)=165-5pσ= μ2
P = $22Q= 65 Magμ(p)=56 Magσ= 28Exp Profit=$543
Any Questions
Yossi Sheffi
- Slide 1
- Slide 2
- Slide 3
- Slide 4
- Slide 5
- Slide 6
- Slide 7
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- Slide 22
- Slide 23
-
The Profit Function
The Profit Function ndash Simple Case
Optimal Order
and
Level of ServiceCycle Service ndash The probability that
there will be a stock-out during a
cycle
Cycle Service = F(Q)
Fill Rate - The probability that a
specific customer will encounter a
stock-out
Level of ServiceS
erv
ice L
evel
Order
Cycle Service
Fill Rate
REV=$15
COST=$7
Normal Distribution of Demand
Order Size
Exp
ecte
d P
rofi
t
Incorporating Fixed Costsbull With fixed costs of $300order
REV=$15
COST=$7
Exp
ecte
d
Pro
fi
Order Size
Risk of Loss
300(15-7)=375 Below that there is no possibility to make money even if we sell ALL the order
REV=$15
COST=$7
Order Quantity
Pro
bab
ilit
y o
f Loss
Ordering with Initial Inventory
Note the cost of Q0 is irrelevant
Given initial Inventory Q0 how to order1 Calculate Q as before2 If Q0 lt Q order (Qlt Q0 )3 If Q0 ge Q order 0With fixed costs order only if the expectedprofits from ordering are more than theordering costs1 Set Qcr as the smallest Q such that E[Profits with Qcr]gtE[Profits with Q]-F2 If Q0 lt Qcr order (Qlt Q0 )3 If Q0 ge Qcr order 0
Ordering with FixedCosts and Initial Inventory
bullIf initial inventory is LE 46 order up to 80
bullIf initial inventory is GE 47 order nothing
REV=$15
COST=$7
Example F = $150Exp
ecte
d P
rofi
t
Initial Inventory
bull μ =D(P) σ = f(μ)bull Procedurebull 1 Set Pbull 2 Calculate μbull 3 Calculate σbull 4 bull 5 Calculate optimal expected profits as a function of P
Elastic DemandExpected Profit Function
Pro
fit
Price
Rev = $15Cost= $8μ(p)=165-5pσ= μ2
P = $22Q= 65 Magμ(p)=56 Magσ= 28Exp Profit=$543
Any Questions
Yossi Sheffi
- Slide 1
- Slide 2
- Slide 3
- Slide 4
- Slide 5
- Slide 6
- Slide 7
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- Slide 22
- Slide 23
-
The Profit Function ndash Simple Case
Optimal Order
and
Level of ServiceCycle Service ndash The probability that
there will be a stock-out during a
cycle
Cycle Service = F(Q)
Fill Rate - The probability that a
specific customer will encounter a
stock-out
Level of ServiceS
erv
ice L
evel
Order
Cycle Service
Fill Rate
REV=$15
COST=$7
Normal Distribution of Demand
Order Size
Exp
ecte
d P
rofi
t
Incorporating Fixed Costsbull With fixed costs of $300order
REV=$15
COST=$7
Exp
ecte
d
Pro
fi
Order Size
Risk of Loss
300(15-7)=375 Below that there is no possibility to make money even if we sell ALL the order
REV=$15
COST=$7
Order Quantity
Pro
bab
ilit
y o
f Loss
Ordering with Initial Inventory
Note the cost of Q0 is irrelevant
Given initial Inventory Q0 how to order1 Calculate Q as before2 If Q0 lt Q order (Qlt Q0 )3 If Q0 ge Q order 0With fixed costs order only if the expectedprofits from ordering are more than theordering costs1 Set Qcr as the smallest Q such that E[Profits with Qcr]gtE[Profits with Q]-F2 If Q0 lt Qcr order (Qlt Q0 )3 If Q0 ge Qcr order 0
Ordering with FixedCosts and Initial Inventory
bullIf initial inventory is LE 46 order up to 80
bullIf initial inventory is GE 47 order nothing
REV=$15
COST=$7
Example F = $150Exp
ecte
d P
rofi
t
Initial Inventory
bull μ =D(P) σ = f(μ)bull Procedurebull 1 Set Pbull 2 Calculate μbull 3 Calculate σbull 4 bull 5 Calculate optimal expected profits as a function of P
Elastic DemandExpected Profit Function
Pro
fit
Price
Rev = $15Cost= $8μ(p)=165-5pσ= μ2
P = $22Q= 65 Magμ(p)=56 Magσ= 28Exp Profit=$543
Any Questions
Yossi Sheffi
- Slide 1
- Slide 2
- Slide 3
- Slide 4
- Slide 5
- Slide 6
- Slide 7
- Slide 8
- Slide 9
- Slide 10
- Slide 11
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- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- Slide 22
- Slide 23
-
Level of ServiceCycle Service ndash The probability that
there will be a stock-out during a
cycle
Cycle Service = F(Q)
Fill Rate - The probability that a
specific customer will encounter a
stock-out
Level of ServiceS
erv
ice L
evel
Order
Cycle Service
Fill Rate
REV=$15
COST=$7
Normal Distribution of Demand
Order Size
Exp
ecte
d P
rofi
t
Incorporating Fixed Costsbull With fixed costs of $300order
REV=$15
COST=$7
Exp
ecte
d
Pro
fi
Order Size
Risk of Loss
300(15-7)=375 Below that there is no possibility to make money even if we sell ALL the order
REV=$15
COST=$7
Order Quantity
Pro
bab
ilit
y o
f Loss
Ordering with Initial Inventory
Note the cost of Q0 is irrelevant
Given initial Inventory Q0 how to order1 Calculate Q as before2 If Q0 lt Q order (Qlt Q0 )3 If Q0 ge Q order 0With fixed costs order only if the expectedprofits from ordering are more than theordering costs1 Set Qcr as the smallest Q such that E[Profits with Qcr]gtE[Profits with Q]-F2 If Q0 lt Qcr order (Qlt Q0 )3 If Q0 ge Qcr order 0
Ordering with FixedCosts and Initial Inventory
bullIf initial inventory is LE 46 order up to 80
bullIf initial inventory is GE 47 order nothing
REV=$15
COST=$7
Example F = $150Exp
ecte
d P
rofi
t
Initial Inventory
bull μ =D(P) σ = f(μ)bull Procedurebull 1 Set Pbull 2 Calculate μbull 3 Calculate σbull 4 bull 5 Calculate optimal expected profits as a function of P
Elastic DemandExpected Profit Function
Pro
fit
Price
Rev = $15Cost= $8μ(p)=165-5pσ= μ2
P = $22Q= 65 Magμ(p)=56 Magσ= 28Exp Profit=$543
Any Questions
Yossi Sheffi
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Level of ServiceS
erv
ice L
evel
Order
Cycle Service
Fill Rate
REV=$15
COST=$7
Normal Distribution of Demand
Order Size
Exp
ecte
d P
rofi
t
Incorporating Fixed Costsbull With fixed costs of $300order
REV=$15
COST=$7
Exp
ecte
d
Pro
fi
Order Size
Risk of Loss
300(15-7)=375 Below that there is no possibility to make money even if we sell ALL the order
REV=$15
COST=$7
Order Quantity
Pro
bab
ilit
y o
f Loss
Ordering with Initial Inventory
Note the cost of Q0 is irrelevant
Given initial Inventory Q0 how to order1 Calculate Q as before2 If Q0 lt Q order (Qlt Q0 )3 If Q0 ge Q order 0With fixed costs order only if the expectedprofits from ordering are more than theordering costs1 Set Qcr as the smallest Q such that E[Profits with Qcr]gtE[Profits with Q]-F2 If Q0 lt Qcr order (Qlt Q0 )3 If Q0 ge Qcr order 0
Ordering with FixedCosts and Initial Inventory
bullIf initial inventory is LE 46 order up to 80
bullIf initial inventory is GE 47 order nothing
REV=$15
COST=$7
Example F = $150Exp
ecte
d P
rofi
t
Initial Inventory
bull μ =D(P) σ = f(μ)bull Procedurebull 1 Set Pbull 2 Calculate μbull 3 Calculate σbull 4 bull 5 Calculate optimal expected profits as a function of P
Elastic DemandExpected Profit Function
Pro
fit
Price
Rev = $15Cost= $8μ(p)=165-5pσ= μ2
P = $22Q= 65 Magμ(p)=56 Magσ= 28Exp Profit=$543
Any Questions
Yossi Sheffi
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Normal Distribution of Demand
Order Size
Exp
ecte
d P
rofi
t
Incorporating Fixed Costsbull With fixed costs of $300order
REV=$15
COST=$7
Exp
ecte
d
Pro
fi
Order Size
Risk of Loss
300(15-7)=375 Below that there is no possibility to make money even if we sell ALL the order
REV=$15
COST=$7
Order Quantity
Pro
bab
ilit
y o
f Loss
Ordering with Initial Inventory
Note the cost of Q0 is irrelevant
Given initial Inventory Q0 how to order1 Calculate Q as before2 If Q0 lt Q order (Qlt Q0 )3 If Q0 ge Q order 0With fixed costs order only if the expectedprofits from ordering are more than theordering costs1 Set Qcr as the smallest Q such that E[Profits with Qcr]gtE[Profits with Q]-F2 If Q0 lt Qcr order (Qlt Q0 )3 If Q0 ge Qcr order 0
Ordering with FixedCosts and Initial Inventory
bullIf initial inventory is LE 46 order up to 80
bullIf initial inventory is GE 47 order nothing
REV=$15
COST=$7
Example F = $150Exp
ecte
d P
rofi
t
Initial Inventory
bull μ =D(P) σ = f(μ)bull Procedurebull 1 Set Pbull 2 Calculate μbull 3 Calculate σbull 4 bull 5 Calculate optimal expected profits as a function of P
Elastic DemandExpected Profit Function
Pro
fit
Price
Rev = $15Cost= $8μ(p)=165-5pσ= μ2
P = $22Q= 65 Magμ(p)=56 Magσ= 28Exp Profit=$543
Any Questions
Yossi Sheffi
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Incorporating Fixed Costsbull With fixed costs of $300order
REV=$15
COST=$7
Exp
ecte
d
Pro
fi
Order Size
Risk of Loss
300(15-7)=375 Below that there is no possibility to make money even if we sell ALL the order
REV=$15
COST=$7
Order Quantity
Pro
bab
ilit
y o
f Loss
Ordering with Initial Inventory
Note the cost of Q0 is irrelevant
Given initial Inventory Q0 how to order1 Calculate Q as before2 If Q0 lt Q order (Qlt Q0 )3 If Q0 ge Q order 0With fixed costs order only if the expectedprofits from ordering are more than theordering costs1 Set Qcr as the smallest Q such that E[Profits with Qcr]gtE[Profits with Q]-F2 If Q0 lt Qcr order (Qlt Q0 )3 If Q0 ge Qcr order 0
Ordering with FixedCosts and Initial Inventory
bullIf initial inventory is LE 46 order up to 80
bullIf initial inventory is GE 47 order nothing
REV=$15
COST=$7
Example F = $150Exp
ecte
d P
rofi
t
Initial Inventory
bull μ =D(P) σ = f(μ)bull Procedurebull 1 Set Pbull 2 Calculate μbull 3 Calculate σbull 4 bull 5 Calculate optimal expected profits as a function of P
Elastic DemandExpected Profit Function
Pro
fit
Price
Rev = $15Cost= $8μ(p)=165-5pσ= μ2
P = $22Q= 65 Magμ(p)=56 Magσ= 28Exp Profit=$543
Any Questions
Yossi Sheffi
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Risk of Loss
300(15-7)=375 Below that there is no possibility to make money even if we sell ALL the order
REV=$15
COST=$7
Order Quantity
Pro
bab
ilit
y o
f Loss
Ordering with Initial Inventory
Note the cost of Q0 is irrelevant
Given initial Inventory Q0 how to order1 Calculate Q as before2 If Q0 lt Q order (Qlt Q0 )3 If Q0 ge Q order 0With fixed costs order only if the expectedprofits from ordering are more than theordering costs1 Set Qcr as the smallest Q such that E[Profits with Qcr]gtE[Profits with Q]-F2 If Q0 lt Qcr order (Qlt Q0 )3 If Q0 ge Qcr order 0
Ordering with FixedCosts and Initial Inventory
bullIf initial inventory is LE 46 order up to 80
bullIf initial inventory is GE 47 order nothing
REV=$15
COST=$7
Example F = $150Exp
ecte
d P
rofi
t
Initial Inventory
bull μ =D(P) σ = f(μ)bull Procedurebull 1 Set Pbull 2 Calculate μbull 3 Calculate σbull 4 bull 5 Calculate optimal expected profits as a function of P
Elastic DemandExpected Profit Function
Pro
fit
Price
Rev = $15Cost= $8μ(p)=165-5pσ= μ2
P = $22Q= 65 Magμ(p)=56 Magσ= 28Exp Profit=$543
Any Questions
Yossi Sheffi
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Ordering with Initial Inventory
Note the cost of Q0 is irrelevant
Given initial Inventory Q0 how to order1 Calculate Q as before2 If Q0 lt Q order (Qlt Q0 )3 If Q0 ge Q order 0With fixed costs order only if the expectedprofits from ordering are more than theordering costs1 Set Qcr as the smallest Q such that E[Profits with Qcr]gtE[Profits with Q]-F2 If Q0 lt Qcr order (Qlt Q0 )3 If Q0 ge Qcr order 0
Ordering with FixedCosts and Initial Inventory
bullIf initial inventory is LE 46 order up to 80
bullIf initial inventory is GE 47 order nothing
REV=$15
COST=$7
Example F = $150Exp
ecte
d P
rofi
t
Initial Inventory
bull μ =D(P) σ = f(μ)bull Procedurebull 1 Set Pbull 2 Calculate μbull 3 Calculate σbull 4 bull 5 Calculate optimal expected profits as a function of P
Elastic DemandExpected Profit Function
Pro
fit
Price
Rev = $15Cost= $8μ(p)=165-5pσ= μ2
P = $22Q= 65 Magμ(p)=56 Magσ= 28Exp Profit=$543
Any Questions
Yossi Sheffi
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Ordering with FixedCosts and Initial Inventory
bullIf initial inventory is LE 46 order up to 80
bullIf initial inventory is GE 47 order nothing
REV=$15
COST=$7
Example F = $150Exp
ecte
d P
rofi
t
Initial Inventory
bull μ =D(P) σ = f(μ)bull Procedurebull 1 Set Pbull 2 Calculate μbull 3 Calculate σbull 4 bull 5 Calculate optimal expected profits as a function of P
Elastic DemandExpected Profit Function
Pro
fit
Price
Rev = $15Cost= $8μ(p)=165-5pσ= μ2
P = $22Q= 65 Magμ(p)=56 Magσ= 28Exp Profit=$543
Any Questions
Yossi Sheffi
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bull μ =D(P) σ = f(μ)bull Procedurebull 1 Set Pbull 2 Calculate μbull 3 Calculate σbull 4 bull 5 Calculate optimal expected profits as a function of P
Elastic DemandExpected Profit Function
Pro
fit
Price
Rev = $15Cost= $8μ(p)=165-5pσ= μ2
P = $22Q= 65 Magμ(p)=56 Magσ= 28Exp Profit=$543
Any Questions
Yossi Sheffi
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Any Questions
Yossi Sheffi
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