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Inventory Models Planned Shortage Models 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 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 Average Inventory Average 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 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 OPTIMAL ORDER QUANTITY Q OPTIMAL 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 2CO D Ch Cs Q C h C s ChQ DCb S Ch Cs DCb 2 Ch C s EXAMPLE SCANLON PLUMBING Saunas cost 2400 each C 2400 Order cost 1250 CO 1250 Holding Cost 525 sauna yr Ch 525 Backorder Goodwill Cost 20 wk C S 1040 Backorder Admin Cost 10 order Cb 10 Demand 15 wk D 780 RESULTS 2 1250 780 525 1040 Q 525 1040 780 10 2 74 525 1040 525 74 780 10 S 20 525 1040 Re order 74 when ther e are 20 backorders Using the Template Input Parameters Optimal Values Planned Shortage Worksheet 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 Example 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 inventory Using Template Enter Lead Time Reorder Point 40 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