Inventory Models

1
Inventory Models

• Planned Shortage Models

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PLANNED SHORTAGE MODEL

• Assumes no customers will be lost because of
  stockouts
  
• Instantaneous reordering
• This can be modified later using standard reorder
  point analyses
  
• Stockout costs
• Cb -- fixed administrative cost/stockout
• Cs -- annualized cost per unit short
• Acts like a holding cost in reverse
• Reorder when there are S backorders

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PROPORTION OF TIME IN/OUT OF STOCK

• T1 time of a cycle with inventory
• T2 time of a cycle out of stock
• T T1 T2 time of a cycle
• IMAX Q-S total demand while in stock.
• T1/T Proportion of time in stock.
• Multiplying by D/D gives T1D/TD
• (Demand while in stock)/(Demand for cycle)
  (Q-S)/Q
  
• T2/T Proportion of time out of stock
• Multiplying by D/D gives T2D/TD
• (Demand while out of stock)/(Demand for cycle)
  S/Q

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Average InventoryAverage Number of Backorders

• Average Inventory
• (Avg. Inv. When In Stock)(Proportion of time in
  stock)
  
• (IMAX/2)((Q-S)/Q) ((Q-S)/2)((Q-S)/Q)
  (Q-S)2/2Q
  
• Average Backorders
• (Average B/O When Out of Stock)(Proportion of
  time out of stock)
  
• (S/2)(S/Q) S2/2Q

5
TOTAL ANNUAL COST EQUATION

• TC(Q,S) CO(Avg. Cycles Per Year)
  CH(Average
  Inv.) Cs (Average
  Backorders) Cb (Number B/Os Per
  Cycle) (Avg. Cycles Per Year) CD
• CO(D/Q)
  Ch((Q-S)2/2Q)
  Cs(S2/2Q) CbS(D/Q) CD
  

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OPTIMAL ORDER QUANTITY, QOPTIMAL BACKORDERS,
S

• Take partial derivatives with respect to Q and S
  and set 0.
  
• Solve the two equations for the two unknowns Q
  and S.

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EXAMPLESCANLON PLUMBING

• Saunas cost 2400 each (C 2400)
• Order cost 1250 (CO 1250)
• Holding Cost 525/sauna/yr. (Ch 525)
• Backorder Goodwill Cost 20/wk (CS 1040)
• Backorder Admin. Cost 10/order (Cb 10)
• Demand 15/wk (D 780)

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RESULTS

9
Using the Template
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REORDER POINT ANALYSIS

• Reorder point can be affected by lead time.
• If lead time is fixed at L years, order is placed
  accounting for the fact that LD items would be
  demanded during lead time.
  
• Thus order is places when there are S-LD
  backorders.
  
• If this is a negative number, this implies an
  order is placed when there are LD - S items left
  in ventory.

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ExampleWhat If Lead Time Were 4 Weeks?

• Demand over 4 weeks 4(15) 60
• 4 weeks .07692 years (for template)
• Want order to arrive when there are 20
  backorders.
  
• Thus order should be placed when there are 60 -
  20 40 saunas left in inventory

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Using Template
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Review

• In planned shortage models there can be both
  time-dependent and time-independent shortage
  costs
  
• There are 2 unknowns which are found by taking
  partial derivatives of the total cost equation
  
• Q -- the amount to order
• S -- the number of backorders when order is
  placed
  
• The actual reorder point may be adjusted for lead
  time.
  
• Use of template