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: –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

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

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

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

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.

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)

RESULTS

Using the Template Planned Shortage Worksheet Input Parameters Optimal Values

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 = years (for template) Want order to arrive when there are 20 backorders. Thus order should be placed when there are = 40 saunas left in inventory

Using Template Reorder Point = 40 Enter Lead Time

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