Inventory ModelsPLANNED SHORTAGE MODELPROPORTION OF TIME IN/OUT OF STOCKAverage Inventory Average Number of BackordersTOTAL ANNUAL COST EQUATIONOPTIMAL ORDER QUANTITY, Q* OPTIMAL # BACKORDERS, S*EXAMPLE SCANLON PLUMBINGRESULTSUsing the TemplateREORDER POINT ANALYSISExample What If Lead Time Were 4 Weeks?Using TemplateReviewInventory ModelsInventory ModelsPlanned Shortage ModelsPLANNED 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 backordersPROPORTION 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/QAverage 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/2QTOTAL ANNUAL COST EQUATIONTOTAL 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) + CDOPTIMAL ORDER QUANTITY, Q*OPTIMAL ORDER QUANTITY, Q*OPTIMAL # BACKORDERS, S*OPTIMAL # BACKORDERS, S* shbhshbsshhOCCDCQCSCCDCCCCCDCQ**)(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 Re201040525)10)(780()74)(525(*74)1040)(525()10*780(10401040525525)780)(1250(2*2SQUsing the TemplateUsing the TemplatePlanned ShortageWorksheetInputParametersOptimalValuesREORDER 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.•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.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.•Thus order should be placed when there are 60 - 20 = 40 saunas left in inventoryUsing TemplateUsing TemplateReorder Point = 40Enter Lead TimeReviewReview•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