INVENTORY MANAGEMENT WITH INDEPENDENT DEMAND Cases

Transcription

1 INVENTORY MANAGEMENT WITH INDEPENDENT DEMAND Cases

2 CASE 1. ABC ANALISYS AND INVENTORY MANAGEMENT SYSTEM Ref. Annual Demand in Euro Cumulated in Euro Demand % total demand Class A B C Hypothesis: Regular demand with uniform monthly demand Management classes: 1ª. Same management system for all articles. Buying Lot Size = 2 months of demand. 2ª. Differentiated management according to the article class: Class Buying Lot Size in Months of Demand A 1 B 2 C 4 2

3 Calculate: Number of Order Releases and value of average inventory for each Management system and compare the results. 3

4 SOLUTION Management System 1: For all articles we adopt a Lot Size of two months of demand. Annual number of Releases Average Inventory Management System 2: Differentiated management: Class Rule Annual number of Releases Average Inventory A Each month B Each 2 months C Each 4 months The fact of having a differentiated management using ABC analysis, using smaller Lot Sizes for the highest rotation items, allows as to reduce the average inventory from euro to euro. Total number of releases drops from 60 to 57. The impact of this reduction on the Order Cost ( C ) should be evaluated for each product family. l In case the Order Cost is the same for all items, Management using ABC criteria will reduce the total ordering cost (57 in front of 60 orders) 4

5 CASE 2: WILSON S.A Wilson S.A. company is considering applying rational inventory models for the management of its warehouses. Considering the stability of its management parameters, they opt to use simple management models. The demand is constant through the year with a total value of units per year. The Holding Cost is estimated at 0.02 /day, and the Order Cost is estimated at 50 /order. The company wants to calculate the economic order quantity, the optimal reordering period and the minimal management cost associated, in order to verify its own calculations. We assume 1 year = 360 days. 5

6 SOLUTION Problem data are as follows: D = units / year h = 0,02 / day x unit (Holding Cost) S = 50 / order (Order Cost) = 1 year = 360 days First we will calculate EOQ (Economic Order Quantity) by determining the total cost function for each period T (time between receptions): 1 T S QhT 2 The Total Cost Function for the management period will be obtained using the relation between the Period T and the Management Period D Q T So we will obtain the annual Total Cost Function by multiplying the Total Cost Function per period by Q N (The number of periods T per year). D 1 S Qh T Q 2 T N 1 S Qh Q 2 Once obtained the Total Cost Function for the management period, we will find the optimum value of Q (minimum cost) by finding the partial derivative with respect to Q and equaling it to 0: 2 0 ; 2 Q Q 0 Making the derivative and rearranging we obtain: Q 2SD h 2*50* ,02* units aprox. The Reordering period will be: T Q 1.666*360 3 days D

7 And the management cost: * *1.666 * 0,02 * To compare with the formulas on the slides, remember H (Holding cost per unit per year is equal to 7

8 CASE 3 USED OIL The owner of a small mechanical shop located in a seaside village has recently been fined for making uncontrolled spills of used oil to the sewers. They haven t called before to the public waste retrieval services because they were charging 100 for each trip from his shop to the village dump. He also had to keep the used oil in 8 liter drums and store the drums in his small facility full of cars to be repaired causing an important detriment, especially if the number of drums was growing. A holiday-maker taking its car to be repaired in the shop has decided to help him reconcile ecology and business and to do so has asked him for some data: - Holiday-Maker: How many oil changes you do each week? - Mechanic: An average of three a day, so by the end of the week I may have generated about 60 liters of used oil. - Holiday-Maker: How much it cost each drum to keep the used oil? - Mechanic: The City Hall provides the drums for free, they come stacked inside each other and they aren t a problem while empty. I could keep those over a tool cabinet as its weight is very small. - Holiday-Maker: How many full drums can you keep on your shop? - Mechanic: A maximum of 25 drums, otherwise I wouldn t be able to move by there. - Holiday-Maker: In how much you would rate the weekly cost of keeping a drum full of used oil in your shop? - Mechanic: In about 5. - Holiday-Maker: As today it s raining and I can t go to the beach, I will be going to my hotel to craft some numbers and this afternoon I will give you an answer. 8

9 Answer: You should keep your used oil until you have, then you call the municipality services to pick it up and take it to the dump. This mean you will have to call them approximately each weeks. Anyway I have to tell you something, even going against all my principles, if the fine is smaller than, what really would be the best for you is to pay the fine again next year. 9

10 SOLUTION The proposal is to use a Stock management model with constant demand. Weekly demand constant d 60 liters / week or 7,5 drums / week Ordering Cost S 100 Holding Cost h 0, 625 / liter x week or 5 / drum x week Management period 52 weeks d 1 S Qh Q 2 2Sd Q 138,564 liters. Approximately 17 drums. h Q Interval between pick-ups 2, 3 d weeks approximately approximately. Answer You should keep your used oil until you have 17 drums, then you call the municipality services to pick it up and take it to the dump. This mean you will have to call them approximately each 2,3 weeks. Anyway I have to tell you something, even going against all my principles, if the fine is smaller than 4.503, what really would be the best for you is to pay the fine again next year. 10

11 CASE 4 SISCU BROTHERS The Siscu Brothers company is thinking in adopting a fixed quantity stock system (Q system). The demand for its product is distributed according to a Normal Law, with a mean of 200 units per week and a typical deviation of 25 units per week. Making an order costs 160 /lot and takes 2 weeks to receive it from the supplier. Keeping a unit of the product in the warehouse for a week has been calculated it represents a cost of 0,1. The buying cost of the product is 2 /unit. They want to manage the stocks for a period of 50 weeks. The company doesn t want to have a stockout risk bigger than 5% as they compete in a market of commodity products. 11

12 SOLUTION 0) Problem data: Demand: d ~ N (200, 25 2 ) (weekly distribution) Lead Time: Ordering Cost: Buying Cost: Holding Cost: Management Period: l = 2 weeks S = 160 /lot C b = 2 /unit h = 0,1 /unit * week = 50 week Stock Management Quality: = 0,05 1) Costs Function: D 1 c b D S Q SS h Q D h S Q Q 2 0 Q * 2SD h u where the average demand for the management period (50 weeks) is: D u. 2) Reorder Point: The demand distribution during the uncertainty period (l ), would be: D ~ N ( L l, l 2 ) N ( 400, 1250 ) This operation we perform to calculate the demand distribution during the uncertainty period (both for Q systems and P systems) is known as demand Convolution. The Reorder Point will be the inventory level for which the following property is accomplished: Pr( ROP D ) 1 l If we express this property in terms of a Standard Normal Distribution, we will have: 12

13 ROP l Pr where ~ N (0,1) l We will have stockouts with a maximum probability equal to when the demand during the Lead Time is bigger than the Reorder Point (the probability of this happening is equal to ), so from here the previous expression. In the case of this problem, we will have to find at the Standard Normal Distribution N(0,1) tables the value for which the Cumulative Probability for a higher value is equal to 5% (So the area under the curve for the values - to z is equal to 95%). This value is 1,65. To find the Reorder Point we have to transform this value from the N(0,1) distribution to the Convoluted Demand Distribution (unstandardize the value): 400 1,65 ROP 1250 And so we find: ROP = 458 units. 3) Safety Stock: Let s remember the Safety Stock (SS) is the quantity we kkep in inventory to prevent the demand growing above the mean value during the uncertainty period. So, if once we have released an order to the supplier (when we reach the Reorder Point) we have an average demand during all the lead time, just before receiving the ordered quantity (Q) our inventory level will be exactly our Safety Stock (Safety Stock would remain unused). As a consequence, we will calculate the Safety Stock as: SS ROP l u. (Remember, if the Demand is Variable with a Normal Distribution N(µ,σ) and the Lead Time is Constant with a value l, then ) SS = 58 units 4) Management Cost for the policy designed: *10000 *80058 *0.1*50160* euro 13

14 CASE 5 SAINT SEGISMOND RIDING CENTRE Mr. Mariano is the responsible of the stables at the Saint Segismond Riding Centre, mainly dedicated to the care and rent of horses. The number of horses they have at the stables in a certain day is variable, as some new horses may arrive to the stables and others may be retrieved by their owners, or be ceded to other Riding Centers to be rented. Anyway, Mr. Mariano says the number of horses in the stables follows a Normal Law with a mean of 50 horses per day and a standard deviation of 10 horses per day; and under no circumstances they can be deprived of their fodder ration. (We assume the maximum demand is equivalent to a 3,56 value of Z in the Standard Normal Distribution tables) Each horse eats 10 Kg of special fodder from a close village cooperative at a cost of 0,20 /Kg. The costs associated with making an order of fodder, which will take 4 days to arrive, are 110. The fodder, once arrived to the Riding Center, is kept in a warehouse which generates a cost of 0,02 /kg * day and the payment of a yearly salary of /year to a watchman who takes care also of the placement and retrieval of the fodder on the warehouse. With all these data, we want to know the quantity to order, reorder point, safety stock and minimum management cost, considering we want to optimize the fodder stock management for a year. 14

15 SOLUTION 1) Problem data: Number of horses per day ~ N (50,100) Daily consumption per horse: 10 Kg Stock management quality: = 0 Ordering Cost: Lead Time: Holding Cost: Wage: Buying Cost: Maximum demand 360days 1year S = 110 /lot l = 4 days h = 0,02 /Kg * day W = / year C b 0,2euro / Kg z=3,56 We will use a Q management system, as we are asked for a reorder point (P systems have no reorder point). The average consumption of fodder per day will be: d Kg. So the average consumption of fodder per year will be: D d Kg. 2) Cost Function: C b D 1 DS Q SS h W Q 2 Optimization: D S Q Q h 0 Q 2SD h 2*110* ,21Kg. 0.02*360 3) Reorder Point As we don t accept any stockouts, the Reorder Point will equal the maximum demand during the uncertainty period (L). Convolution of the number of horses during 4 days: 15

16 4*50,4* ,400 D L ~ N N We unestandardize the maximum number of horses: max hor hor. max As each horse will eat 10 Kg of fodder per day, this means a Reorder Point of Kg of fodder. 4) Safety stock SS ROP Dl Kg. 5) Total Cost per year ,2* *2345,21 712*0,02* , ,9 euro 16

17 CASE 6 XIPLAND The Xipland S.A. company, importer of computer chips wants to have a correct management of its stocks. This company imports all the material from Taiwan being the cost of each chip equal to 5. In concept of customs they pay a 1% of the purchasing price for each element. Once the order has been made (with a cost of 500 per order), the material will take 10 days to be received. When the product arrives, the company proceeds to perform a thorough quality control, made by an employee with a /year wage. The company doesn t has its own warehouse, using instead a rented warehouse with a cost of 1200 /month, paid whereas the warehouse is used or not. The cost is allocated to Inventory Management. Finally, Xipland has an insurance policy over the material on the warehouse, with a cost of 1% of the value in inventory per month (see note 1). The demand follows a Normal Law with a Mean of 50 units per day and a Variance of 100 units at the square power per day at the square power. The probability of a stockout can t be greater than 2%. We want to know: Which has to be the value of the Order Quantity, Reorder Point, Safety Stock and minimum management cost to manage these stocks in an optimal way for a period of one year? Note 1: The value of a unit in storage is calculated as the purchasing price plus the storage and quality control costs. Note 2: The z value for a cumulative probability of 98% equals to 2,05 Note 3: 1 year = 360 days 17

18 SOLUTION 0) Problem data: C b 5euro / unit. S 500euro l = 10 days Quality Control cost = w = /year Warehouse rent = r = /month = /year α = 12% of the year inventory value β = 2% d ~ N(50,100) daily = 1 year = 360 days 1) Cost Function and EOQ: D 1 Cb DS Q SS Cb Q 2 Optimization: D S Q Q 1 w a Cb 2 2D 2 w r D 0 (1% per month) wr Q 2SD 2 2*500*18000 C D w a 0.12* 5,05* b 2 Q chips 2) Reorder Point and Safety Stock 2% t Demand convolution for 10 days: D l ~N 500,1000 Unestandardizing the value 2.05 we will determine the Reorder Point: ROP ROP chips 1000 SS = = chips 18

19 4) Annual Cost ,05* ,05 *0, ,46 19

20 Case 7 Rebolex Imports, SA A Bolex watches import company wants to establish a sound management of their stocks. It has carried out a study of the expected demand for next year, with the result that the daily demand follows a Normal Law with a mean of 350 and a standard deviation of 80. Bolex watches are manufactured in South Korea and are charged at 6. Moreover, the cost of transporting each order is 17,827. For shipping features once an interval between orders has been set, it must remain always constant, being the Lead Time equal to 24 days. Society values a financial cost of 1% per day for the capital invested in their stocks, regardless of the cost of transport. Rebolex Imports, SA wants to determine the full parameters for the management of their stocks over the next year (1 year = 360 days). The quality of inventory management they want to offer allows for a maximum of 2% of stockouts. 20

21 SOLUTION Problem data are: d ~ N (350, 80) D = d * = 350 * 360 = Units/year Buying cost c b = 6 /unit Order (Transport) Cost S = /order Lead Time l = 24 days Financial cost c 1% daily 0.01 / day (We don t include transport cost) f Inventory management quality (β) = 2% = = Total cost for the management period Q* = Economic Order Quantity 1) Calculation of the average order size. First we have to consider this problem implies the need to keep the Interval between orders constant, so we will need to apply a fixed period model (P). In these models, the order size will be different each time, but the interval between orders will remain constant. The average order will be equal to the Economic Order Quantity. To find the EOQ we will determine the Total Cost function and minimize it: D 1 cbd S Q SS cbc f Q 2 D cc b f - -S 0 2 Q Q 2 2SD 2*17827* Q units cc 6*0.01*360 b f 2) Interval between orders (P) Q P 41,2 days d 350 Where d is the average daily demand. 3) Safety Stock In Fixed-Period (P) systems, the uncertainty period is equal to P + L, so is 65.2 days. The demand convolution for this 65.2 days is: Demand P+L ~ N ; N22,820 ; The maximum demand covered (or Target), (DC) will be the demand level for which: P(Demand P+L > DC) = 1-β =

22 At the Standard Normal Distribution N (0,1) the value which leaves only 2% of the area under the Normal Curve at its right is Unestandardizing this value we find the demand covered: Dc D c = u D c The Safety Stock is equal to the Demand Covered minus the average demand for the uncertainty period: SS = (P+L) = = 1331 units 4) Management Cost * * = * *6*0.01* Average Lot Size u. Interval between orders 41,2 days Safety Stock 1331 uds. Order Quantity OHi Management Cost