Inventory Models
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Transcript of Inventory Models
Inventory ModelsInventory Models
Planned Shortage Models
PLANNED SHORTAGE MODELPLANNED 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
PROPORTION OF TIME PROPORTION OF TIME IN/OUT OF STOCKIN/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
Average InventoryAverage InventoryAverage Number of BackordersAverage Number of Backorders
• Average Inventory =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)(Q-S)22/2Q/2Q
• Average Backorders =Average Backorders = (Average B/O When Out of Stock)(Proportion of time out of stock)
= (S/2)(S/Q) = SS22/2Q/2Q
TOTAL ANNUAL COST EQUATIONTOTAL ANNUAL COST EQUATION
• TC(Q,S) = CO(Number of 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
OPTIMAL ORDER QUANTITY, Q*OPTIMAL ORDER QUANTITY, Q*OPTIMAL # BACKORDERS, S*OPTIMAL # BACKORDERS, S*
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•Take partial derivatives with respect to Q and S and set = 0. •Solve the two equations for the two unknowns Q and S.
EXAMPLEEXAMPLESCANLON PLUMBINGSCANLON 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)
RESULTSRESULTS
backorders20 are e when ther74order Re
201040525
)10)(780()74)(525(*
74)1040)(525(
)10*780(
1040
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525
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S
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Using the TemplateUsing the Template
Planned ShortageWorksheet
InputParameters
OptimalValues
REORDER POINT ANALYSISREORDER 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.
• R = LD – S*– If R is negative, an order is placed when there
are S* - LD backorders.– If R is positive, an order is placed when there
are LD - S* items left inventory.– If R = 0, an order is placed when there is no item
left and no backorder
ExampleExampleWhat If Lead Time Were 1 Week?What If Lead Time Were 1 Week?
• Demand over 1 week = 15
• Want order to arrive when there are 20 backorders. (S* = 20)
• R = LD – S* = 15 – 20 = -5
• Thus order should be placed when there are 5 backorders
ExampleExampleWhat If Lead Time Were 4 Weeks?What 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. (S* = 20)
• R = LD – S* = 60 - 20
• Thus order should be placed when there are 60 - 20 = 40 saunas left in inventory
Using TemplateUsing Template
Reorder Point = 40Enter Lead Time
ReviewReview• 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