2Special Inventory Models Three common situations require relaxation of one or more of the assumptions on which the EOQ model is based.Noninstantaneous Replenishment occurs when production is not instantaneous and inventory is replenished gradually, rather than in lots.Quantity Discounts occur when the unit cost of purchased materials is reduced for larger order quantities.One-Period Decisions: Retailers and manufacturers of fashion goods often face situations in which demand is uncertain and occurs during just one period or season.
3Noninstantaneous Replenishment If an item is being produced internally rather than purchased, finished units may be used or sold as soon as they are completed, without waiting until a full lot is completed.Production rate, p, exceeds the demand rate, d.Cycle inventory accumulates faster than demand occursa buildup of p – d units occurs per time period, continuing until the lot size, Q, has been produced.
4Noninstantaneous Replenishment Production quantityDemand during production intervalMaximum inventoryProduction and demandDemand onlyTBOOn-hand inventoryQTimeImaxp – d
5Noninstantaneous Replenishment Cycle inventory is no longer Q/2, as it was with the basic EOQ method; instead, it is the maximum cycle inventory (Imax / 2)Imax = (p – d) = Q( )Qpp – dTotal annual cost (C) = Annual holding cost + annual ordering or setup costD = annual demand d = daily demand p = production rate S = setup costsQ = ELSC = ( ) (S)DQQ p – dp
6Economic Lot Size (ELS) Economic production lot size (ELS) is the optimal lot size in a situation in which replenishment is not instantaneous.D = annual demand d = daily demand p = production rate S = setup costsH = annual unit holding costELS =pp – d2DSH
7Finding the ELS Example D.1 The manager of a chemical plant must determine the following for a particular chemical:Determine the economic production lot size (ELS).Determine the total annual setup and inventory holding costs.Determine the TBO, or cycle length, for the ELS.Determine the production time per lot.What are the advantages of reducing the setup time by 10 percent?Demand = 30 barrels/day Setup cost = $200Production rate = 190 barrels/day Annual holding cost = $0.21/barrelAnnual demand = 10,500 barrels Plant operates 350 days/year
8Finding the ELS for the Example D.1 chemical D = annual demand d = daily demand p = production rate S = setup costsH = unit holding costQ = ELSELS =pp – d2DSHELS =190190 – 302(10,500)($200)$0.21ELS = barrelsDemand = 30 barrels/day Setup cost = $200Production rate = 190 barrels/day Annual holding cost = $0.21/barrelAnnual demand = 10,500 barrels Plant operates 350 days/year
9Finding the Total Annual Cost Example D.1D = annual demand d = daily demand p = production rate S = setup costsH = unit holding costQ = ELSC = ( )(H) (S)DQQ p – dpC = ( ) ($0.21) ($200)10,5004873.4– 30C = $ $430.91C = $861.82Demand = 30 barrels/day Setup cost = $200Production rate = 190 barrels/day Annual holding cost = $0.21/barrelAnnual demand = 10,500 barrels Plant operates 350 days/year
10Finding the TBO Example D.1 ELS TBOELS = (350 days/year) D 4873.4 D = annual demand d = daily demand p = production rate S = setup costsH = unit holding costQ = ELSTBOELS = (350 days/year)ELSDTBOELS = (350 days/year)4873.410,500TBOELS = , or 162 daysDemand = 30 barrels/day Setup cost = $200Production rate = 190 barrels/day Annual holding cost = $0.21/barrelAnnual demand = 10,500 barrels Plant operates 350 days/year
11Production Time per Lot Finding theProduction Time per LotExample D.1D = annual demand d = daily demand p = production rate S = setup costsH = unit holding costQ = ELSProduction time =ELSpProduction time =4873.4190Production time = 25.6, or 26 daysDemand = 30 barrels/day Setup cost = $200Production rate = 190 barrels/day Annual holding cost = $0.21/barrelAnnual demand = 10,500 barrels Plant operates 350 days/year
12Advantage of Reducing Setup Time OM Explorer Solver for the Economic Production Lot Size Showing the effect of a 10 Percent Reduction in setup cost. $180 vs original $200
15Quantity DiscountsQuantity discounts, which are price incentives to purchase large quantities, create pressure to maintain a large inventory.For any per-unit price level, P, the total cost is:Total annual cost = Annual holding cost + Annual ordering or setup cost + Annual cost of materialsD = annual demand S = setup costs P = per-unit price levelH = unit holding costQ = ELSC = (H) (S) + PDQ2D
16Total cost curves with purchased materials added Quantity DiscountsC for P = $4.00C for P = $3.50C for P = $3.00PD forP = $4.00P = $3.50P = $3.00EOQ 4.00EOQ 3.50EOQ 3.00First price breakSecond price breakTotal cost (dollars)Purchase quantity (Q)Total cost curves with purchased materials addedEOQs and price break quantities
17Finding Q with Quantity Discounts Step 1. Beginning with the lowest price, calculate the EOQ for each price level until a feasible EOQ is found.It is feasible if it lies in the range corresponding to its price.Step 2. If the first feasible EOQ found is for the lowest price level, this quantity is the best lot size.Otherwise, calculate the total cost for the first feasible EOQ and for the larger price break quantity at each lower price level. The quantity with the lowest total cost is optimal.
18Example D.2A supplier for St. LeRoy Hospital has introduced quantity discounts to encourage larger order quantities of a special catheter. The price schedule is:Order Quantity Price per Unit0 – 299 $60.00300 – 499 $58.80500 or more $57.00Annual demand (D) = 936 unitsOrdering cost (S) = $45Holding cost (H) = 25% of unit priceStep 1: Start with lowest price level:EOQ =2DSH2(936)(45)0.25(57.00)== 77 units
19Example D.2 continued Not feasible Not feasible Feasible EOQ 57.00 = 2DSH2(936)(45)0.25(57.00)== 77 unitsNot feasibleEOQ =2DSH2(936)(45)0.25(58.80)== 76 unitsNot feasibleEOQ =2DSH2(936)(45)0.25(60.00)== 75 unitsFeasibleThis quantity is feasible because it lies in the range corresponding to its price.Order Quantity Price per Unit0 – 299 $60.00300 – 499 $58.80500 or more $57.00Annual demand (D) = 936 unitsOrdering cost (S) = $45Holding cost (H) = 25% of unit price
20Example D.2 continuedStep 2: The first feasible EOQ of 75 does not correspond to the lowest price level. Hence, we must compare its total cost with the price break quantities (300 and 500 units) at the lower price levels ($58.80 and $57.00):C = (H) (S) + PDQ2DC75 = [(0.25)($60.00)] ($45) + $60.00(936)752936C75 = $57,284C300 = [(0.25)($58.80)] ($45) + $58.80(936)3002936= $57,382C500 = [(0.25)($57.00)] ($45) + $57.00(936)5002936= $56,999The best purchase quantity is 500 units, which qualifies for the deepest discount.
24One-Period DecisionsThis type of situation is often called the newsboy problem. If the newspaper seller does not buy enough newspapers to resell on the street corner, sales opportunities are lost. If the seller buys too many newspapers, the overage cannot be sold because nobody wants yesterday’s newspaper.List the different levels of demand that are possible, along with the estimated probability of each.Develop a payoff table that shows the profit for each purchase quantity, Q, at each assumed demand level.Calculate the expected payoff for each Q (or row in the payoff table) by using the expected value decision rule.Choose the order quantity Q with the highest expected payoff.
25Payoff = (Profit per unit)(Purchase quantity) = pQ One-Period DecisionsThe payoff for a given quantity-demand combination depends on whether all units are sold at the regular profit margin, which results in two possible cases.If demand is high enough (Q < D) then all of the cases are sold at the full profit margin, p, during the regular season.Payoff = (Profit per unit)(Purchase quantity) = pQIf the purchase quantity exceeds the eventual demand (Q > D), only D units are sold at the full profit margin, and the remaining units purchased must be disposed of at a loss, l, after the season.Payoff = (Profit per unit during season) (Demand) – (Loss per unit) (Amount disposed of after season) = pD – l(Q – D)
28Solved Problem 1For Peachy Keen, Inc., the average demand for mohair sweaters is 100 per week. The production facility has the capacity to sew 400 sweaters per week. Setup cost is $351. The value of finished goods inventory is $40 per sweater. The annual per-unit inventory holding cost is 20 percent of the item’s value.a. What is the economic production lot size (ELS)?b. What is the average time between orders (TBO)?c. What is the minimum total of the annual holding cost and setup cost?
30Solved Problem 3For Swell Productions, a concession stand will sell poodle skirts and other souvenirs of the 1950s a one-time event. Skirts are purchased for $40 each and are sold during for $75 each.Unsold skirts can be returned for a refund of $30 each. Sales depend on the weather, attendance, and other variables.The following table shows the probability of various sales quantities. How many skirts should be ordered?
31Solved Problem 3The highest expected payoff occurs when 400 skirts are ordered.Probabilities0.050.110.34