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*

sh

bh

sh

b

s

sh

h

O

CC

DCQCS

CC

DC

C

CC

C

DCQ

**

)(2*

2

•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

1040525

525

)780)(1250(2*

2

S

Q

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