1、Inventory Basic Model,How can it be that mathematics, being after all a product of human thought which is independent of experience, is so admirably appropriate to the objects of reality? Albert Einstein,In our EOQ models, R and D are used interchangeable. D is demand, R is throughput. We assume R=D
2、 Everything produced is sold. A toy manufacturer uses 32000 silicon chips annually. The Chips are used at a steady rate during the 240 days a year that the plant operates. Holding cost is 60 cents per unit per year. Ordering cost is $24 per order. a) How much should we order each time to minimize ou
3、r total costs (total ordering and carrying costs)?,Problem 1: Optimal Policy,D = 32000, H = $0.6 per unit per year , S = $24 per order Ordering Quantity = Q # of orders = D/Q = 32000/Q Cost of each order = S = $24 OC = 24*32000/Q,Problem 1: EOQ,Problem 1,Cost of carrying one unit of inventory for on
4、e year = H Average Inventory At the start of cycle we have Q, at the end of the cycle we have 0. Average inventory = (Q+0)/2 = Q/2 Q/2 is also called cycle inventory.,In each cycle we have Q/2 inventory. In all cycles we have Q/2 inventory. Throughout the year we have Q/2 inventory. CC = HQ/2,Proble
5、m 1: Optimal Policy,Cost of carrying one unit of inventory for one year = HAt EOQ (Economic Order Quantity, OC=CC SD/Q = HQ/2 240(32000)/Q= 0.6Q/2 Q2= 2560000 Q = 1600 Q2 = 2DS/H,Problem 1,b) How many times should we order ? D = 32000 per year, EOQ = 1600 each time # of times that we order = D/EOQ D
6、/Q = 32000/1600 = 20 times.c) What is the length of an order cycle ? We order 20 times. Working days = 240/year 240/20 = 12 days. Alternatively 32000 is required for one year (240 days) Each day we need 32000/240 = 133.333 1600 is enough for how long? (1600/133.33) = 12 day,Problem 1,d) Compute the
7、average inventory At the start of cycle we have Q, at the end of the cycle we have 0. Average inventory = (Q+0)/2 = Q/2 Q/2 is also called cycle inventory.,In each cycle we have Q/2 inventory. In all cycles we have Q/2 inventory. Throughout the year we have Q/2 inventory.,Problem 1,d) Compute the av
8、erage inventory Average inventory = (Q+0)/2 = 1600/2 =800e) Compute the total carrying cost. We have Q/2 throughout the year Inventory carrying costs = average inventory (Q/2) multiplied by cost of carrying one unit of inventory for one year (H) Total Annual Carrying Cost = H(Q/2) = 0.6(1600/2) = $4
9、80f) Compute the total ordering cost and total cost. Ordering Cost = 24(32000/1600) = 24(20) = $480 Carrying Cost = H(Q/2) = 0.6(1600/2) = $480 Total Cost = Ordering cost + Carrying cost Total cost = $480+$480 = $960,Problem 1,Note that at EOQ total carrying costs is equal to total ordering costs. H
10、Q/2 = SD/Q HQ2=2DS If we solve this equation for Q we will have Q2=2DS/H,That is one way to compute EOQ and not to memorize it.,Problem 1,g) Compute the flow time ? Demand = 32000 per year Therefore throughput = 32000 per year Maximum inventory = EOQ = 1600 Average inventory = 1600/2 = 800 RT=I 3200
11、0T=800 T=800/32000=1/40 year Year = 240 days T=240(1/40)= 6 days Alternatively, the length of an order cycle is 12 days. The first item of an order when received spends 0 days, the last item spends 12 days. On average they spend (0+12)/2 = 6 days,Problem 1,h) Compute inventory turns. Inventory turn
12、= Demand divided by average inventory. Average inventory = I = Q/2 Inventory turns = D/(Q/2)= 32000/(1600/2) Inventory turns = 40 times per year.Notes: Cycle inventory is always defined as Max Inventory divided by 2. Cycle inventory = Q/2 If there is no safety stock Average inventory is the same as
13、Cycle inventory = Q/2. If there is safety stock- We will discuss it in ROP lectureAverage inventory = Cycle Inventory +Safety Stock = Q/2 +Is,Victor sells a line of upscale evening dresses in his boutique. He charges $300 per dress, and sells average 30 dresses per week. Currently, Vector orders 10
14、week supply at a time from the manufacturer. He pays $150 per dress, and it takes two weeks to receive each delivery. Victor estimates his administrative cost of placing each order at $225. His inventory carrying cost including cost of capital, storage, and obsolescence is 20% of the purchasing cost
15、. Assume 50 weeks per year.,Problem 2: Other Policies vs. Optimal Policy,a) Compute the total ordering cost and carrying cost under the current ordering policy? Number of orders/yr = D/Q = 1500/300 = 5 (D/Q) S = 5(225) = 1,125/yr. Average inventory = Q/2 = 300/2 = 150 H = 0.2(150) = 30,Flow unit = o
16、ne dress Flow rate D = 30 units/wk 50 weeks per year Ten weeks supply Q = 10(30) = 300 units. Demand 30(50)= 1500 /yr Fixed order cost S = $225 Unit Cost C = $150/unit H = 20% of unit cost. Lead time L = 2 wees,Problem 2,Annual holding cost = H(Q/2) = 30(150) = 4,500 /yr. Total annual costs = 1125+4
17、500 = 5625 b) Without any further computation, is EOQ larger than 300 or smaller? Why?,Problem 2,c) Compute the flow time. Average inventory = cycle inventory = I = Q/2 Average inventory = 300/2 = 150 Throughput? R? R= D, D= 30/week Current flow time RT= I 30T= 150 T= 5 weeks Did we really need this
18、 computations? Cycle is 10 weeks (each time we order demand of 10 weeks). The first item is there for 0 week. The last item is there for 10 weeks. On average (10+0)/2 = 5 weeks.,Problem 2,d) What is average inventory and inventory turns under this policy ? Inventory turn = Demand divided by average
19、inventory. I = Q/2 Inventory turns = D/(Q/2)= 1500/(300/2) = 10 timesInvTurn = R/I T=I/R InvTurn = 1/TWe already computed T T = 5 weeksTurn = 1/T= 1/5 ? Is InvTurn 10 or 1/5 Have we made a mistake? InvTurn = 1/5 per week, year = 50 weeks InvTurn =(1/5)(50) = 10=,Problem 2,e) Compute Victors total an
20、nual cost of inventory system (carrying plus ordering but excluding purchasing) under the optimal ordering policy?,Q* = EOQ =,= 150 units.,The total optimal annual cost will be,225(1500/150) + 30(150/2) = 2250 + 2250 = $4,500,Compared to 5,625, there is about 20% reduction in the total costs. Total
21、cost here is equal to carrying cost there.,Problem 2,f) When do you order (re-order point) ? An order for 150 units two weeks before he expects to run out. That is, whenever current inventory drops to 30 units/wk * 2 wks = 60 units. Which is the re-order point. When to order? When inventory on hand
22、is 60. How much to order? 150. R and Q Strategy.,Central Electric (CE) serves its European customers through a distribution network that consisted of four warehouses, in Poland, Italy, France, and Germany. The network of warehouses was built on the premise that it will allow CE to be close to the cu
23、stomer. Contrary to expectations, establishing the distribution network led to an inventory crisis. CE is considering to consolidate the regional warehouses into a single master warehouse in Austria. The following data is for the sake of analysis of this problem - not real world data. Currently, eac
24、h warehouse manages its ordering independently. Demand at each outlet averages 800 units per day. Assume a year is 250 days. Each unit of product costs $200, and CE has a holding cost of 20% per annum. The fixed cost of each order (administrative plus transportation) is $900 for the decentralized sy
25、stem and $2025 for the centralized system.,Problem 3; Centralization vs. Decentralization,Decentralized: Four warehouses in Poland, Italy, France, and Germany Centralized: One warehouse in Austria The holding cost will be the same in both decentralized and centralized ordering systems. H(decentraliz
26、ed) =20%(200) = $40 per unit per yr. H(centralized) = $40 per unit per yr. The ordering cost in the centralized ordering system is $2025. S(decentralized) = $900 per order. S(centralized) $900 = $2025 per unit per yr. The problem assumes this. It is also realistic, when we deliver centrally, S goes
27、up since the truck travel time in a route to 4 warehouses is longer than a trip to a single warehouse.,Problem 3; Centralization vs. Decentralization,Four outlets Each outlet demand D = 800(250) = 200,000 S= 900 C = 200 H = 0.2(200) = 40 If all warehouses merged into a single warehouse, then S= 2025
28、,Problem 3,=3000,With a cycle inventory of 1500 units for each warehouse.,b) Compute EOQ and cycle inventory in the centralized ordering In this problem, in the centralized system, S = $2025.,=9000,and a cycle inventory of 4500.,Compute EOQ and cycle inventory in decentralized ordering,The total cyc
29、le inventory across all four outlets equals 6000.,Problem 3,c) Compute the total annual holding cost + ordering cost (not including purchasing cost) for both policies TC = S(D/Q) + H(Q/2) Decentralized TC= 900(200000/3000) + 40(3000/2) TC = 60000+60000= 120000 Decentralized: TC for all 4 warehouses
30、= 4(120000)=480000 Centralized TC= 2025(800000/9000) + 40(9000/2) TC= 180000+180000 = 360000 480000 360000; about 25% improvement in the total costs,Problem 3,d) Compute the ordering interval in decentralized and centralized systems.,Decentralized = (3000/200000)(250) = 3.75 days Centralized = (9000
31、/800000)(250) = 2.821 dayse) Compute the average flow time3.75/2 = 1.875 days 2.821/2 = 1.41 days RT = I T= R/I 200000T= 1500 T = 1500/200000 year or 1.875 days 800000T= 4500 T = 4500/800000 year or 1.41 days The same computations,Problem 3: Inventory Turns,f) Compute inventory turns Inventory Turns
32、 = Demand /Average inventory = R/I = InvTurnDemand Average inventory Inventory Turns 200000 1500 200000/1500 = 133.33 800000 4500 800000/4500 = 177.78g) If the lead time is 2 days, when do you order? (re-order point)? Decentralized 2(800) = 1600 units Centralized = 2(4)(800) = 6400 units,Why We are
33、interested in reducing inventory.,Inventory adversely affects all competing edges (P/Q/V/T) Has cost Physical carrying costs Financial costs Has risk of obsolescence Due to market changes Due to technology changes Leads to poor quality Feedback loop is long Hides problems Unreliable suppliers, machi
34、ne breakdowns, long changeover times, too much scrap. Causes long flow time, not-uniform operations,How to Reduce EOQ,To reduce EOQ we may R, S, H Two ways to reduce average inventory - Reduce S - Postponement, Delayed Differentiation - CentralizeS does not increase in proportion of QEOQ increases a
35、s the square route of demand. - Commonality, modularization and standardization is another type of Centralization,Why not Always Centralized,If centralization reduces inventory, why doesnt everybody do it? Higher shipping costLonger response timeLess understanding of customer needsLess understanding
36、 of cultural, linguistics, and regulatory barriers These disadvantages my reduce the demand.,Formula Proof for Total Cost of EOQ,Total cost of any Q? TCQ = SR/Q + HQ/2 Total Cost of EOQ? The same as above, but can also be simplified = 2 + 2 2 = =2 2 2 = 2 = 2 2 = 2 = 2,Formula Proof for Flow Time Under EOQ,Flow time when we order of any Q? Throughput = R, average inventory I = Q/2 RT = Q/2 T = Q/2R Flow time when we order of EOQ? Total Cost of EOQ? The same as above, but can also be simplified I = EOQ/2 = 2 = 2 2 = 2 4 = 2 T = I/R T= = 2 = 22 = 2,
