1、1,CHAPTER 15: INVENTORY MODELS,OutlineDeterministic models The Economic Order Quantity (EOQ) model Sensitivity analysis A price-break Model Probabilistic Inventory models Single-period inventory models A fixed order quantity model A fixed time period model,2,Inventory Decision Issues,Demand of vario
2、us items Money tied up in the inventory Cost of storage space Insurance expense - risk of fire, theft, damage Order processing costs Loss of profit due to stock outs,3,How Much? When?,Inventory Decision Questions,4,THE EOQ MODEL,Demand rate,0,Time,Leadtime,Leadtime,Order Placed,Order Placed,Order Re
3、ceived,Order Received,Inventory Level,Reorder point, R,Order qty, Q,5,The EOQ Model Cost Curves,Slope = 0,Total Cost,Ordering Cost = SD/Q,Order Quantity, Q,Annual cost ($),Minimum total cost,Optimal order Q*,Holding Cost = HQ/2,6,EOQ Cost Model,D - annual demand Q - order quantity S - cost of placin
4、g order H - annual per-unit holding costOrdering cost = SD/Q Holding cost = HQ/2Total cost = SD/Q + HQ/2,7,Example 1: R & B beverage company has a soft drink product that has a constant annual demand rate of 3600 cases. A case of the soft drink costs R & B $3. Ordering costs are $20 per order and ho
5、lding costs are 25% of the value of the inventory. R & B has 250 working days per year, and the lead time is 5 days. Identify the following aspects of the inventory policy:a. Economic order quantity,8,b. Reorder pointc. Cycle time,9,d. Total annual cost,10,SENSITIVITY ANALYSIS,11,Some Important Char
6、acteristics of the EOQ Cost Function,At EOQ, the annual holding cost is the same as annual ordering cost.,12,The total cost curve is flat near EOQ So, the total cost does not change much with a slight change in the order quantity (see the total cost curve and the example on sensitivity),Some Importa
7、nt Characteristics of the EOQ Cost Function,13,EOQ WITH PRICE BREAKS,Assumptions Demand occurs at a constant rate of D items per year. Ordering Cost is $S per order. Holding Cost is $H = $CiI per item in inventory per year (note holding cost is based on the cost of the item, Ci). Purchase Cost is $C
8、1 per item if the quantity ordered is between 0 and x1, $C2 if the order quantity is between x1 and x2, etc. Delivery time (lead time) is constant.,14,EOQ with Price Breaks Formulae,FormulaeOptimal order quantity: the procedure for determining Q* will be demonstrated Number of orders per year: D/Q*
9、Time between orders (cycle time): Q*/D years Total annual cost: (1/2)Q*H + DS/Q* + DC(holding + ordering + purchase),15,EOQ with Price Breaks Procedure,Steps 1. Determine the largest (cheapest) feasible EOQ value: The most efficient way to do this is to compute the EOQ for the lowest price first, an
10、d continue with the next higher price. Stop when the first EOQ value is feasible (that is, within the correct interval). 2. Compare the costs: Compare the value of the average annual cost at the largest feasible EOQ and at all of the price breakpoints that are greater than the largest feasible EOQ.
11、The optimal Q is the point at which the average annual cost is a minimum.,16,Example 2: Nicks Camera Shop carries Zodiac instant print film. The film normally costs Nick $3.20 per roll, and he sells it for $5.25. Nicks average sales are 21 rolls per week. His annual inventory holding cost rate is 25
12、% and it costs Nick $20 to place an order with Zodiac. If Zodiac offers a 7% discount on orders of 400 rolls or more and a 10% discount for 900 rolls or more, determine Nicks optimal order quantity.,17,D = 21(52) = 1092; H = .25(Ci); S = 20Step 1: Determine the largest (cheapest) feasible EOQ,18,Ste
13、p 2: Compare the costsCompute the total cost for the most economical, feasible order quantity in each price category for which a was computed.,19,PROBABILISTIC MODELS,OutlineProbabilistic inventory models Single- and multi- period models A single-period model with uniform distribution of demand A si
14、ngle-period model with normal distribution of demand,20,Probabilistic Inventory Models,The demand is not known. Demand characteristics such as mean, standard deviation and the distribution of demand may be known. Stockout cost: The cost associated with a loss of sales when demand cannot be met. For
15、example, if an item is purchased at $1.50 and sold at $3.00, the loss of profit is $3.00-1.50 = $1.50 for each unit of demand not fulfilled.,21,Single- and Multi- Period Models,The classification applies to the probabilistic demand case In a single-period model, the items unsold at the end of the pe
16、riod is not carried over to the next period. The unsold items, however, may have some salvage values. In a multi-period model, all the items unsold at the end of one period are available in the next period. In the single-period model and in some of the multi-period models, there remains only one que
17、stion to answer: how much to order.,22,SINGLE-PERIOD MODEL,Computer that will be obsolete before the next order Perishable product Seasonal products such as bathing suits, winter coats, etc. Newspaper and magazine,23,Trade-offs in a Single-Period Models,Loss resulting from the items unsold ML= Purch
18、ase price - Salvage valueProfit resulting from the items sold MP= Selling price - Purchase priceTrade-off Given costs of overestimating/underestimating demand and the probabilities of various demand sizes how many units will be ordered?,24,Consider an order quantity Q Let P = probability of selling
19、all the Q units= probability (demandQ)Then, (1-P) = probability of not selling all the Q unitsWe continue to increase the order size so long as,25,Decision Rule:Order maximum quantity Q such that where P = probability (demandQ),26,Text Problem 21, Chapter 15: Demand for cookies:Demand Probability of
20、 Demand1,800 dozen 0.052,000 0.102,200 0.202,400 0.302,600 0.202,800 0.103,000 0,05 Selling price=$0.69, cost=$0.49, salvage value=$0.29 a. Construct a table showing the profits or losses for each possible quantity b. What is the optimal number of cookies to make? c. Solve the problem by marginal an
21、alysis.,27,Sample computation for order quantity = 2200: Expected number sold=1800(0.05)+2000(0.10)+2200(0.85) =2160 Revenue from sold items=2160(0.69)=$1490.4 Revenue from unsold items=(2200-2160)(0.29)=$11.6 Total revenue=1490.4+11.6=$1502 Cost=2200(0.49)=$1078 Profit=1502-1078=$424,28,29,Solution
22、 by marginal analysis: Order maximum quantity, Q such thatDemand, Q Probability(demand) Probability(demandQ), p,30,Demand Characteristics,Suppose that the historical sales data shows:Quantity No. Days sold Quantity No. Days sold14 1 21 1115 2 22 916 3 23 617 6 24 318 9 25 219 11 26 120 12,31,Demand
23、Characteristics,Mean = 20 Standard deviation = 2.49,32,Demand Characteristics,33,Example 3: The J&B Card Shop sells calendars. The once-a-year order for each years calendar arrives in September. The calendars cost $1.50 and J&B sells them for $3 each. At the end of July, J&B reduces the calendar pri
24、ce to $1 and can sell all the surplus calendars at this price. How many calendars should J&B order if the September-to-July demand can be approximated bya. uniform distribution between 150 and 850,34,Solution to Example 3:Loss resulting from the items unsold ML= Purchase price - Salvage value =Profi
25、t resulting from the items sold MP= Selling price - Purchase price =,35,P =Now, find the Q so that P(demandQ) =Q* =,36,Example 4: The J&B Card Shop sells calendars. The once-a-year order for each years calendar arrives in September. The calendars cost $1.50 and J&B sells them for $3 each. At the end
26、 of July, J&B reduces the calendar price to $1 and can sell all the surplus calendars at this price. How many calendars should J&B order if the September-to-July demand can be approximated byb. normal distribution with = 500 and =120.,37,Solution to Example 4: ML=$0.50, MP=$1.50 (see example 3)P =No
27、w, find the Q so that P =,38,We need z corresponding to area = From Appendix D, p. 780z = Hence, Q* = + z =,39,Example 5: A retail outlet sells a seasonal product for $10 per unit. The cost of the product is $8 per unit. All units not sold during the regular season are sold for half the retail price
28、 in an end-of-season clearance sale. Assume that the demand for the product is normally distributed with = 500 and = 100.a. What is the recommended order quantity?b. What is the probability of a stockout?c. To keep customers happy and returning to the store later, the owner feels that stockouts shou
29、ld be avoided if at all possible. What is your recommended quantity if the owner is willing to tolerate a 0.15 probability of stockout?d. Using your answer to part c, what is the goodwill cost you are assigning to a stockout?,40,Solution to Example 5: a. Selling price=$10, Purchase price=$8Salvage v
30、alue=10/2=$5MP =10 - 8 = $2, ML = 8-10/2 = $3Order maximum quantity, Q such that Now, find the Q so that P = 0.6 or, area (2)+area (3) = 0.6 or, area (2) = 0.6-0.5=0.10,41,Find z for area = 0.10 from the standard normal table given in Appendix D, p. 736 z = 0.25 for area = 0.0987, z = 0.26 for area
31、= 0.1025 So, z = 0.255 (take -ve, as P = 0.6 0.5) for area = 0.10 So, Q*=+z =500+(-0.255)(100)=474.5 units.b. P(stockout) = P(demandQ) = P = 0.6c. P(stockout)=Area(3)=0.15From Appendix D, find z for Area (2) = 0.5-0.15=0.35,42,z = 1.03 for area = 0.3485 z = 1.04 for area = 0.3508 So, z = 1.035 for a
32、rea = 0.35 So, Q*=+z =500+(1.035)(100)=603.5 units.d. P=P(demandQ)=P(stockout)=0.15For a goodwill cost of gMP =10 - 8+g = 2+g, ML = 8-10/2 = 3Now, solve g in p =Hence, g=$15.,43,MULTI-PERIOD MODELS,OutlineA fixed order quantity model A fixed time period model,44,A FIXED ORDER QUANTITY MODEL,Purchase
33、-order can be placed at any time On-hand inventory count is known always,Lead time for a high speed modem is two weeks and it has the following sales history in the last 25 weeks:Quantity/Week Frequency75-80 1 70-75 365-70 960-65 855-60 4,Will you order now if number of items on hand is:a. 200 b. 15
34、0 c. 100,45,A Fixed Order Quantity Model,The same quantity, Q is ordered when inventory on hand reaches a reorder point, R,46,A Fixed Order Quantity Model,An order quantity of EOQ works wellIf demand is constant, reorder point is the same as the demand during the lead time.If demand is uncertain, re
35、order point is usually set above the expected demand during the lead timeReorder point = Expected demand + Safety stock,47,Safety Stock,Lead Time,Time,Expected demand during lead time,Quantity,Safety stock,Reorder Point,48,Trade-Off with Safety Stock,Safety Stock - Stock held in excess of expected d
36、emand to protect against stockout during lead time.Safety stock Holding cost Stockouts Safety stock Holding cost Stockouts ,49,Acceptable Level of Stockout,Ask the manager!Acceptable level of stockout reflects managements toleranceA related term is service level. Example: if 20 orders are placed in
37、a year and management can tolerate 1 stockout in a year, acceptable level of stockout = 1/20 = 0.05 = 5% and the service level = 1- 0.05 = 0.95.,50,Computation of Safety Stock,51,Example 6: B&S Novelty and Craft Shop sells a variety of quality handmade items to tourists. B&S will sell 300 hand-made
38、carved miniature replicas of a Colonial soldier each year, but the demand pattern during the year is uncertain. The replicas sell for $20 each, and B&S uses a 15% annual inventory holding cost rate. Ordering costs are $5 per order, and demand during the lead time follows a normal probability distrib
39、ution with = 15 and = 6.a. What is the recommended order quantity?b. If B&S is willing to accept a stockout roughly twice a year, what reorder point would you recommend? What is the probability that B&S will have a stockout in any one order-cycle?c. What are the inventory holding and ordering costs?
40、,52,A FIXED TIME PERIOD MODEL,Purchase-order is issued at a fixed interval of time,A distributor of soft drinks prepares a purchase order for beverages once a week on every Monday. The beverages are received on Thursdays (the lead time is three days). Choose a method for finding order quantity for t
41、he distributor:a. Mean demand for 7 days + safety stockb. Mean demand for 10 days + safety stockc. Mean demand for 10 days + safety stock inventory on hand,53,Replenishment Level and Safety Stock,Replenishment level, M = Desired inventory to cover review period & lead time= Expected demand during re
42、view period & lead time + Safety stockOrder quantity, Q = M - HH = inventory on handTrade-off with safety stock Safety stock Holding cost Stockouts Safety stock Holding cost Stockouts ,54,The Fixed Time Period Model,55,Computation of Replenishment Level,56,Comparison Between P and Q models,57,Exampl
43、e 7: Statewide Auto parts uses a 4-week periodic-review system to reorder parts for its inventory stock. A 1-week lead time is required to fill the order. Demand for one particular part during the 5-week replenishment period is normally distributed with a mean of 18 units and a standard deviation of
44、 6 units.a. At a particular periodic review, 8 units are in inventory. The parts manager places an order for 16 units. What is the probability that this part will have a stockout before an order that is placed at the next 4-week review period arrives?B. Assume that the company is willing to tolerate
45、 a 2.5% chance of stockout associated with a replenishment decision. How many parts should the manager have ordered in part (a)? What is the replenishment level for the 4-week periodic review system?,58,Example 8: Rose Office Supplies, Inc., uses a 2-week periodic review for its store inventory. Mea
46、n and standard deviation of weekly sales are 16 and 5 respectively. The lead time is 3 days. The mean and standard deviation of lead-time demand are 8 and 3.5 respectively.A. What is the mean and standard deviation of demand during the review period plus the lead-time period?B. Assuming that the dem
47、and has a normal probability distribution, what is the replenishment level that will provide an expected stockout rate of one per year?C. If there are 18 notebooks in the inventory, how many notebooks should be ordered?,59,Example 9: Foster Drugs, Inc., handles a variety of health and beauty product
48、s. A particular hair conditioner product costs Foster Drugs $2.95 per unit. The annual holding cost rate is 20%. A fixed-quantity model recommends an order quantity of 300 units per order. a. Lead time is one week and the lead-time demand is normally distributed with a mean of 150 units and a standa
49、rd deviation of 40 units. What is the reorder point if the firm is willing to tolerate a 1% chance of stockout on any one cycle? b. What safety stock and annual safety stock cost are associated with your recommendation in part (a)? c. The fixed-quantity model requires a continuous-review system. Management is considering making a transition to a fixed-period system in an attempt to coordinate ordering,
