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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: –C b -- fixed administrative cost/stockout –C s -- 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 T 1 = time of a cycle with inventory T 2 = time of a cycle out of stock T = T 1 + T 2 = time of a cycle I MAX = Q-S = total demand while in stock. T 1 /T = Proportion of time in stock. Multiplying by D/D gives T 1 D/TD = (Demand while in stock)/(Demand for cycle) = (Q-S)/Q T 2 /T = Proportion of time out of stock Multiplying by D/D gives T 2 D/TD = (Demand while out of stock)/(Demand for cycle) = S/Q

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Average Inventory Average Number of Backorders Average Inventory =Average Inventory = (Avg. Inv. When In Stock)(Proportion of time in stock) (Q-S) 2 /2Q =(I MAX /2)((Q-S)/Q) = ((Q-S)/2)((Q-S)/Q) = (Q-S) 2 /2Q Average Backorders =Average Backorders = (Average B/O When Out of Stock)(Proportion of time out of stock) S 2 /2Q = (S/2)(S/Q) = S 2 /2Q

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TOTAL ANNUAL COST EQUATION TC(Q,S) = C O (Avg. Cycles Per Year) + C H (Average Inv.) + C s (Average Backorders) + C b (Number B/Os Per Cycle) (Avg. Cycles Per Year) + CD = C O (D/Q) + C h ((Q-S) 2 /2Q) + C s (S 2 /2Q) + C b S(D/Q) + CD

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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.

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EXAMPLE SCANLON PLUMBING Saunas cost $2400 each (C = 2400) Order cost = $1250(C O = 1250) Holding Cost = $525/sauna/yr.(C h = 525) Backorder Goodwill Cost $20/wk (C S =1040) Backorder Admin. Cost = $10/order (C b = 10) Demand = 15/wk(D = 780)

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RESULTS

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Using the Template Planned Shortage Worksheet Input Parameters Optimal Values

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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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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

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Using Template Reorder Point = 40 Enter Lead Time

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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

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