Single Period Inventory Models

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Single Period Inventory Models
Yossi Sheffi Mass Inst of Tech Cambridge, MA
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Outline

• Single period inventory decisions
• Calculating the optimal order size
• ? Numerically
• Using spreadsheet
• Using simulation
• ? Analytically
• The profit function
• ? For specific distribution
• Level of Service
• Extensions
• ? Fixed costs
• ? Risks
• ? Initial inventory
• ? Elastic demand

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Single Period Ordering

• Seasonal items
• Perishable goods
• News print
• Fashion items
• Some high tech products
• Risky investments

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Selling Magazines

• Weekly demand

?? Total 4023 magazines ?? Average 77.4
Mag/week ?? Min 51 max 113 Mag/week
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Detailed Histogram
Frequency (Wks/Yr)
Cumm Freq. (Wks/Yr)
Demand (Mag/Wk)
Average77.4 Mag/wk
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Histogram
Cummulative Frequency
Cumm Freq. (Wks/Yr)
Frequency (Wks/Yr)
More
Demand (Mag/Wk)
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The Ordering Decision(Spreadsheet)

• Assume each magazine sells for 15
• Cost of each magazine 8

Order
d/wk
Prob
Exp.Profit
Newsboy Framework Panel 1 basic Scenario
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Expected Profits
Profit
Order Size
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Optimal Order (Analytical)
??

• The optimal order is Q
• At Q the probability of selling one more
  magazine
• is the probability that demand is
  greater than Q
• The expected profit from ordering the
  (Q1)st
• magazine is
• ?? If demand is high and we sell it
• ?? (REV-COST) x Pr( Demand is higher than Q)
• ?? If demand is low and we are stuck
• ?? (-COST) x Pr( Demand is lower or equal to Q)
• The optimum is where the total expected
  profit
• from ordering one more magazine is zero
• ? (REV-COST) x Pr( Demand gt Q) COST x
  Pr( Demand
• Q) 0

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Optimal Order
The critical ratio
Cummulative Frequency
Frequency (Wks/Yr
More
Demand (Mag/week)
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Salvage Value

• Salvage value 4/Mag.

Profit
Cummulative Frequency
Frequency (Wks/Yr)
Order Size
Demand (Mag/week)
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The Profit Function

• Revenue from sold items
• Revenue or costs associated with unsold items.
  These may include revenue from salvage or cost
  associated with disposal.
• Costs associated with not meeting
• customers demand. The lost sales cost can
  include lost of good will and actual
• penalties for low service.
• The cost of buying the merchandise in the first
  place.

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The Profit Function
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The Profit Function Simple Case
Optimal Order
and
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Level of Service
Cycle Service 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
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Level of Service
REV15 COST7
Service Level
Cycle Service Fill Rate
Order
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Normal Distribution of Demand
Expected Profit
Order Size
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Incorporating Fixed Costs
REV15 COST7

• With fixed costs of 300/order

Expected Profi
Order Size
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Risk of Loss
REV15 COST7
Probability of Loss
Order Quantity
300/(15-7)37.5. Below that there is no
possibility to make money even if we sell ALL the
order.
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Ordering with Initial Inventory
Given initial Inventory Q0, how to order? 1.
Calculate Q as before 2. If Q0 lt Q, order (Qlt
Q0 ) 3. If Q0 Q, order 0 With fixed costs,
order only if the expected profits from ordering
are more than the ordering costs 1. Set Qcr as
the smallest Q such that EProfits with
QcrgtEProfits with Q-F 2. If Q0 lt Qcr, order
(Qlt Q0 ) 3. If Q0 Qcr, order 0
Note the cost of Q0 is irrelevant
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Ordering with FixedCosts and Initial Inventory
REV15 COST7
Example F 150
Expected Profit
Initial Inventory
If initial inventory is LE 46, order up to
80 If initial inventory is GE 47, order nothing
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Elastic Demand
Expected Profit Function

• µ D(P) s f(µ)
• Procedure
• 1. Set P
• 2. Calculate µ
• 3. Calculate s
• 4.
• 5. Calculate optimal expected profits as a
• function of P.

Profit
Price
P 22 Q 65 Mag µ(p)56 Mag s 28 Exp.
Profit543
Rev 15 Cost 8 µ(p)165-5p s µ/2
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Any Questions?
Yossi Sheffi