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1 Inventory Models 2 Tipos de Demanda Demanda independente: são itens que dependem, em sua maioria, dos pedidos de clientes externos, como, por exemplo,

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Apresentação em tema: "1 Inventory Models 2 Tipos de Demanda Demanda independente: são itens que dependem, em sua maioria, dos pedidos de clientes externos, como, por exemplo,"— Transcrição da apresentação:

1

2 1 Inventory Models

3 2 Tipos de Demanda Demanda independente: são itens que dependem, em sua maioria, dos pedidos de clientes externos, como, por exemplo, produtos acabados em geral.

4 3 Demanda dependente é aquela de um item cuja quantidade a ser utilizada depende da demanda de um item de demanda independente. Exemplo: O item pneus em uma montadora é dependente do número de veículos demandados pelo público (5 pneus por carro) Tipos de Demanda

5 4 Tipos de estoques Matérias-primas Produtos em processo (WIP - work in process) Produtos acabados Em trânsito Em consignação

6 5 Importância dos Estoques –Melhorar o serviço ao cliente –Economia de escala –Proteção contra mudanças de preço em épocas de inflação alta –Proteção contra incertezas na demanda e no tempo de entrega –Proteção contra contingências

7 6 Pressões para Manter Estoque Alto Estoque alto = maior probabilidade de atender bem os clientes Mas Estoque alto = certeza de alto custo em carregar estoques

8 7 Fontes de Elevação de Estoque Marketing Engenharia Controle de Qualidade Manufatura Suprimentos Gerentes

9 8 Segmentação de Estoques Classificação ABC – é um processo de categorização de Pareto, baseado em algum critério relevante para a priorização dos esforços de gerenciamento. Na gestão de materiais, o critério usualmente mais utilizado consiste no consumo médio do item multiplicado pelo seu custo de reposição – conhecido como demanda valorizada. A partir do ranking destes itens, que podem ser separados em comprados e produzidos, estratifica-se três categorias através do corte considerando a percentagem acumulada em, por exemplo, 80%, 15% e 5%.

10 9 Classificação ABC

11 10 Segmentação de Estoques Classificação XYZ – Nessa classificação segmenta- se os itens baseando-se no critério de criticidade para facilitar as rotinas de planejamento, reposição e gerenciamento.

12 11 Classificação XYZ Classificação da criticidade dos itens Classe X Ordinário: Item de baixa criticidade, cuja falta naturalmente compromete o atendimento de um usuário interno (serviço ou produção) ou externos (clientes finais), mas não implica em maiores conseqüências. Classe Y Intercambiável: Apresenta razoável possibilidade de substituição com outros itens disponíveis em estoque sem comprometer os processos críticos, caso seja necessário e em detrimento dos custos envolvidos. Classe Z Vital: Item cuja falta acarreta conseqüências críticas, tais como interrupção dos processos da empresa, podendo comprometer a integridade de equipamentos e/ou segurança operacional.

13 12 Segmentação Classificação 123 – Essa classificação diz respeito a todo o processo de aquisição, incluindo tanto a identificação e qualificação dos fornecedores como o disparo e atendimento de requisições, em termos do grau de confiabilidade das especificações e prazos.

14 13 Classificação 123 Classificação da dificuldade na obtenção dos itens Classe 1 Complexa: São itens de obtenção muito difícil, pois envolvem diversos fatores complicadores combinados, tais como longos set-ups e lead- times (tempo de resposta, distâncias e variabilidades) e riscos quanto a pontualidade, qualidade, fontes alternativas e sazonalidades. Classe 2 Difícil: Envolve alguns poucos fatores complicadores relacionados acima, tornando o processo de obtenção relativamente difícil. Classe 3 Fácil: Fornecimento ágil, rápido e pontual e/ou o item é uma commodity, com amplas alternativas a disposição no mercado fornecedor.

15 14 Inventory can be classified in various ways: Used typically by accountants at manufacturing firms. Enables management to track the production process. Items are classified by their relative importance in terms of the firms capital needs. Management of items with short shelf life and long shelf life is very different Inventory Classifications

16 15 Overview of Inventory Issues Proper control of inventory is crucial to the success of an enterprise. Typical inventory problems include: –Basic inventory – Planned shortage –Quantity discount – Periodic review –Production lot size – Single period Inventory models are often used to develop an optimal inventory policy, consisting of: –An order quantity, denoted Q. –A reorder point, denoted R.

17 16 Inventory analyses can be thought of as cost-control techniques. Categories of costs in inventory models: –Holding (carrying costs) –Order/ Setup costs –Customer satisfaction costs –Procurement/Manufacturing costs Type of Costs in Inventory Models

18 17 Holding Costs (Carrying costs): These costs depend on the order size –Cost of capital –Storage space rental cost –Costs of utilities –Labor –Insurance –Security –Theft and breakage –Deterioration or Obsolescence C h = Annual holding cost per unit in inventory H = Annual holding cost rate C = Unit cost of an item C h = H * C Type of Costs in Inventory Models

19 18 Order/Setup Costs These costs are independent of the order size. –Order costs are incurred when purchasing a good from a supplier. They include costs such as Telephone Order checking Labor Transportation – Setup costs are incurred when producing goods for sale to others. They can include costs of Cleaning machines Calibrating equipment Training staff Type of Costs in Inventory Models C o = Order cost or setup cost

20 19 Customer Satisfaction Costs –Measure the degree to which a customer is satisfied. –Unsatisfied customers may: Switch to the competition (lost sales). Wait until an order is supplied. –When customers are willing to wait there are two types of costs incurred: Type of Costs in Inventory Models C b = Fixed administrative costs of an out of stock item ($/stockout unit). C s = Annualized cost of a customer awaiting an out of stock item ($/stockout unit per year).

21 20 Procurement/Manufacturing Cost –Represents the unit purchase cost (including transportation) in case of a purchase. –Unit production cost in case of in-house manufacturing. Type of Costs in Inventory Models C = Unit purchase or manufacturing cost.

22 21 Demand is a key component affecting an inventory policy. Projected demand patterns determine how an inventory problem is modeled. Typical demand patterns are: –Constant over time (deterministic inventory models) –Changing but known over time (dynamic models) –Variable (randomly) over time (probabilistic models) Demand in Inventory Models D = Demand rate (usually per year)

23 22 Two types of review systems are used: –Continuous review systems. The system is continuously monitored. A new order is placed when the inventory reaches a critical point. –Periodic review systems. The inventory position is investigated on a regular basis. An order is placed only at these times. Review Systems

24 23 The item has a sufficiently long shelf life. The item is monitored using a continuous review system. All the cost parameters remain constant forever (over an infinite time horizon). A complete order is received in one batch. Economic Order Quantity Model - Assumptions Demand occurs at a known and reasonably constant rate.

25 24 The constant environment described by the EOQ assumptions leads to the following observation: The optimal EOQ policy consists of same-size orders. QQQ The EOQ Model – Inventory profile This observation results in the following inventory profile :

26 25 QQQ Total Annual Inventory Costs = Total Annual Holding Costs Total Annual ordering Costs Total Annual procurement Costs ++ TC(Q) =(Q/2)C h +(D/Q)C o +DC ChCh ChCh The optimal order Size Q * = Cost Equation for the EOQ Model

27 26 Constructing the total annual variable cost curve Total Holding Costs Total ordering costs Add the two curves to one another Total annual holding and ordering costs Q TV(Q) Q*Q* The optimal order size o * * * * * TV(Q) = Total annual variable costs and Q * Note: at the optimal order size total holding costs and ordering costs are equal

28 27 The curve is reasonably flat around Q *. Q*Q* Deviations from the optimal order size cause only small increase in the total cost. Sensitivity Analysis in EOQ models

29 28 To find the number of orders per years : N = D/Q Example: The demand for a product is 1000 units per year. The order size is 250 units under an EOQ policy. How many orders are placed per year? N = 1000/250 = 4 orders. Number of Orders per Year

30 29 The cycle time, T, represents the time that elapses between the placement of orders. T = Q/D Cycle Time Example: The demand for a product is 1000 units per year. The order size is 250 units under an EOQ policy. How often orders need to be placed (what is the cycle time)? T = 250/1000 = ¼ years. {Note: the four orders are equally spaced}.

31 30 In reality lead time always exists, and must be accounted for when deciding when to place an order. The reorder point, R, is the inventory position when an order is placed. R is calculated by L and D must be expressed in the same time unit. R = L D Lead Time and the Reorder Point

32 31 Inventory position L Place the order now R=Reorder Point R = Inventory at hand at the beginning of lead time Lead Time and the Reorder Point – Graphical demonstration: Short Lead Time

33 32 Outstanding order Place the order now R = inventory at hand at the beginning of lead time + one outstanding order = demand during lead time = LD Inventory at hand L Lead Time and the Reorder Point – Graphical demonstration: Long Lead Time

34 33 Safety stocks act as buffers to handle: –Higher than average lead time demand. –Longer than expected lead time. With the inclusion of safety stock (SS), R is calculated by The size of the safety stock is based on having a desired service level. R = LD + SS Safety stock

35 34 L Place the order now Reorder Point R = LD Safety stock Planned situation Actual situation

36 35 L R = LD Safety stock Actual situation + SS New Reorder Point Place the order now SS=Safety stock The safety stock prevents excessive shortages. LD

37 36 Inventory Costs Including safety stock Total Annual Inventory Costs = Total Annual Holding Costs Total Annual ordering Costs Total Annual procurement Costs ++ TC(Q) =(Q/2)C h +(D/Q)C o +DC + C h SS Safety stock holding cost

38 37 ALLEN APPLIANCE COMPANY (AAC) AAC wholesales small appliances. AAC currently orders 600 units of the Citron brand juicer each time inventory drops to 205 units. Management wishes to determine an optimal ordering policy for the Citron brand juicer

39 38 Data –C o = $12 ($8 for placing an order) + (20 min. to check)($12 per hr) –C h = $1.40 [HC = (14%)($10)] –C = $10. –H = 14% (10% ann. interest rate) + (4% miscellaneous) –D = demand information of the last 10 weeks was collected: ALLEN APPLIANCE COMPANY (AAC)

40 39 Data –The constant demand rate seems to be a good assumption. –Annual demand = (120/week)(52weeks) = 6240 juicers. ALLEN APPLIANCE COMPANY (AAC)

41 40 Current ordering policy calls for Q = 600 juicers. TV( 600) = (600 / 2)($1.40) + (6240 / 600)($12) = $ The EOQ policy calls for orders of size AAC – Solution: EOQ and Total Variable Cost Savings of 16% 2(6240)(12) 1.40 = = Q*Q* TV(327) = (327 / 2)($1.40) + (6240 / 327) ( $12) = $457.89

42 41 TC(327) = ($10) + (13)($1.40) = $62, Under the current ordering policy AAC holds 13 units safety stock (how come? ): AAC is open 5 day a week. –The average daily demand = (120/week)/5 = 24 juicers/day. –Lead time is 8 days. Lead time demand is (8)(24) = 192 juicers. –Reorder point without Safety stock = LD = 192. –Current policy: R = 205. –Safety stock = 205 – 192 = 13. For safety stock of 13 juicers the total cost is TV(327) + Procurement + Safety stock cost holding cost AAC – Solution: Reorder Point and Total Cost

43 42 Changing the order size –Suppose juicers must be ordered in increments of 100 (order 300 or 400) –AAC will order Q = 300 juicers in each order. –There will be a total variable cost increase of $1.71. –This is less than 0.5% increase in variable costs. Changes in input parameters –Suppose there is a 20% increase in demand. D=7500 juicers. –The new optimal order quantity is Q * = 359. –The new variable total cost = TV(359) = $502 –If AAC still orders Q = 327, its total variable costs becomes TV(327) = (327 2)($1.40) + ( )($12) = $ Only 0.4% increase AAC – Solution: Sensitivity of the EOQ Results

44 43 For an order size of 327 juicers we have: –T = ( ) = year. = (52)(5) = 14 days. –This is useful information because: Shelf life may be a problem. Coordinating orders with other items might be desirable. AAC – Solution: Cycle Time working days per week

45 44 AAC – Excel Spreadsheet =SQRT(2*$B$10*$ B$14/$B$13) =1/E11 Copy to cell H12 =E10/B10 Copy to cell H11 =$B$10*$B$11+E14+$B$13*B16 Copy to Cell H15 =(E10/2)*$B$13+($B$10/E10)*$B$14 Copy to cell H14 =$B$15*$B$10+$B$16- INT(($B$15*$B$10+$B$16)/E10)*E10 Copy to cell H13

46 45 Service Levels and Safety Stocks Service Levels and Safety Stocks

47 46 Determining Safety Stock Levels Businesses incorporate safety stock requirements when determining reorder points. A possible approach to determining safety stock levels is by specifying desired service level.

48 47 The unit service level (fill rate) –The percentage of demands that are filled without incurring any delay. –Applied when the percentage of unsatisfied demand should be under control. Two Types of Service Level The cycle service level –The probability of not incurring a stockout during an inventory cycle. –Applied when the likelihood of a stockout, and not its magnitude, is important for the firm. Service levels can be viewed in two ways. Comum

49 48 Two Types of Service Level Juicer Demand and Units on Backorder Cycle NumberDemand# Units on backorder Cycle service level = 4/5 = 80% Unit Service level = 1- 15/3000 = 99,5%

50 49 In many cases short run demand is variable even though long run demand is assumed constant. Therefore, stockout events during lead time may occur unexpectedly in each cycle. Stockouts occur only if demand during lead time is greater than the reorder point. The Cycle Service Level Approach

51 50 To determine the reorder point we need to know: –The lead time demand distribution. –The required service level. In many cases lead time demand is approximately normally distributed. For the normal distribution case the reorder point is calculated by The Cycle Service Level Approach R = L + z L (1 – = Service level (use a Normal DistributionTable) L = demanda média no lead time L = desvio padrão da demanda no lead time

52 51 =192 P(D L > R) = P(Z > (R – L )/ L ) =. Since P(Z > Z ) =, we have Z = (R – L )/ L, which gives… The Cycle Service Level Approach P(D L >R) = Service level = P(D L

53 52 Assume that lead time demand is normally distributed. Estimation of the normal distribution parameters: –Estimation of the mean weekly demand = ten weeks average demand = 120 juicers per week. –Estimation of the variance of the weekly demand = Sample variance = juicers 2. AAC - Cycle Service Level Approach

54 53 To find L and L the parameters (per week) and (per week) must be adjusted since the lead time is longer than one week. –Lead time is 8 days =(8 / 5) weeks = 1.6 weeks. Estimates for the lead time mean demand and variance of demand L (1.6)(120) = 192; 2 L (1.6)(83.33) = AAC - Cycle Service Level Approach

55 54 Let us use the current reorder point of 205 juicers. 205 =192 + z (11.55 ) z = 1.13 From the normal distribution table we have that a reorder point of 205 juicers results in an 87% cycle service level. AAC - Service Level for a given Reorder Point

56 55 Management wants to improve the cycle service level to 99%. The z value corresponding to 1% right hand tail is R = (11.55) = 219 juicers. Management wants to improve the cycle service level to 99%. The z value corresponding to 1% right hand tail is R = (11.55) = 219 juicers. AAC – Reorder Point for a given Service Level

57 56 AAC is willing to run out of stock an average of at most one cycle per year with an order quantity of 327 juicers. What is the equivalent service level for this strategy? AAC is willing to run out of stock an average of at most one cycle per year with an order quantity of 327 juicers. What is the equivalent service level for this strategy? AAC – Acceptable Number of Stockouts per Year

58 57 AAC – Acceptable Number of Stockouts per Year There will be an average of = cycles (lead times) per year. The likelihood of stockouts = 1/19 = This translates into a service level of 94.76% There will be an average of = cycles (lead times) per year. The likelihood of stockouts = 1/19 = This translates into a service level of 94.76%

59 58 When lead time demand follows a normal distribution service level can be calculated as follows: –Determine the value of z that satisfy the equation L(z) = Q * L –Solve for R using the equation R = L + z L The Unit Service Level Approach L(z)=partial expected value for the standard normal between some z and infinity

60 59 =NORMDIST(B8,B5,B6,TRUE) AAC – Cycle Service Level (Excel spreadsheet) =NORMINV(B7,B5,B6)

61 60 Quantity Discounts are Common Practice in Business –By offering discounts buyers are encouraged to increase their order sizes, thus reducing the sellers holding costs. –Quantity discounts reflect the savings inherent in large orders. EOQ Models with Quantity Discounts

62 61 Quantity Discount Schedule –This is a list of per unit discounts and their corresponding purchase volumes. –Normally, the price per unit declines as the order quantity increases. –The order quantity at which the unit price changes is called a break point. –There are two main discount plans: All unit schedules - the price paid for all the units purchased is based on the total purchase (mais comum). Incremental schedules - The price discount is based only on the additional units ordered beyond each break point. EOQ Models with Quantity Discounts

63 62 To determine the optimal order quantity, the total purchase cost must be included TC(Q) = (Q 2)C h + (D Q)C o + DC i + C h SS C i represents the unit cost at the i th pricing level. All Units Discount Schedule

64 63 AAC - All Units Quantity Discounts AAC is offering all units quantity discounts to its customers. Data

65 64 Should AAC increase its regular order of 327 juicers, to take advantage of the discount? Should AAC increase its regular order of 327 juicers, to take advantage of the discount?

66 65 AAC – All units discount procedure – Step 1: Find the optimal order Q i * for each discount level i. Use the formula – Step 2: For each discount level i modify Q i * as follows If Q i * is lower than the smallest quantity that qualifies for the i th discount, increase Q i * to that level. If Q i * is greater than the largest quantity that qualifies for the i th discount, eliminate this level from further consideration. – Step 3: Substitute the modified Q * i value in the total cost formula TC( Q * i ). – Step 4: Select the Q i * that minimizes TC( Q i * ) C h =C i.0,14

67 66 Step 1: Find the optimal order quantity Q i * for each discount level i based on the EOQ formula AAC – All units discount procedure

68 67 TC(Q) = (Q 2)C h + (D Q)C o + DC i + C h SS

69 68 – Step 2 : Modify Q i * Q1*Q1* Q1*Q1* 327 $10/unit Q2*Q2* Q2*Q2* $9.75/unit Q3*Q3* Q3*Q3* 336 $9.50 AAC – All Units Discount Procedure

70 69 – Step 2 : Modify Q i * Q1*Q1* Q1*Q1* 327 $10/unit 331 Q2*Q2* Q2*Q2* Q3*Q3* Q3*Q3* 336 $9.50 AAC – All Units Discount Procedure Q3*Q3* Q3*Q3* Q3*Q3* Q3*Q3* Q3*Q3* Q3*Q3* Q3*Q3* Q3*Q3* Q3*Q3* Q3*Q3* Q3*Q3* Q3*Q3* Q3*Q3* Q3*Q3*

71 70 – Step 3: Substitute Q I * in the total cost function – Step 4 Modified Q* and total Cost QualifiedPriceModifiedTotal Urderper UnitQ* Cost , , , , , , , , , ,98 AAC should order 5000 juicers AAC – All Units Discount Procedure 327

72 71 AAC – All Units Discount Excel Worksheet

73 72 Demand rate is constant. Production rate is larger than demand rate. The production lot is not received instantaneously (at an infinite rate), because production rate is finite. There is only one product to be scheduled. The rest of the EOQ assumptions stay in place. Production Lot Size Model - Assumptions

74 73 The optimal production lot size policy orders the same amount each time. This observation results in the inventory profile below: Production Lot Size Model – Inventory profile

75 74 Production Lot Size = Q = PT 1 The inventory increases at a net rate of P - D The production increases the inventory at a rate of P. The demand decreases the inventory at a rate of D. Production time T1T1 Demand accumulation during production run Demand accumulation during production run = DT 1 Maximum inventory = (P – D)T 1 = (P – D)(Q/P) = Q(1 – D/P) Maximum inventory Production Lot Size Model – Understanding the inventory profile

76 75 The parameters of the total variable costs function are similar to those used in the EOQ model. Instead of ordering cost, we have here a fixed setup cost per production run (C o ). In addition, we need to incorporate the annual production rate (P) in the model. Production Lot Size Model – Total Variable Cost

77 76 TV(Q) = (Q 2)(1 - D P)C h +(D Q)C o P is the annual production rate C h (1-D/P) The Optimal Order Size Q * = 2DC o The average inventory Production Lot Size Model – Total Variable Cost

78 77 Cycle time T = Q / D. Length of a production run T 1 = Q / P. Time when machines are not busy producing the product T 2 = T - T 1 = Q(1 / D - 1 / P). Average inventory = (Q / 2)(1-D / P). Production Lot Size Model – Useful relationships

79 78 FARAH COSMETICS COMPANY Farah needs to determine optimal production lot size for its most popular shade of lipstick. Data The factory operates 7 days a week, 24 hours a day. Production rate is 1000 tubes per hour. It takes 30 minutes to prepare the machinery for production. It costs $150 to setup the line. Demand is 980 dozen tubes per week. Unit production cost is $.50 Annual holding cost rate is 40%.

80 79 Input for the total variable cost function D = 613,200 per year[(980 dozen/week (12) 7](365) C h = 0.4(0.5) = $0.20 per tube per year. C o = $150 P = (1000)(24)(365) = 8,760,000 per year. Dozens FARAH COSMETICS COMPANY – Solution

81 80 Current Policy Currently, Farah produces in lots of 84,000 tubes. T = (84,000 tubes per run) (613,200 tubes per year)= years (about 50 days). T 1 = (84,000 tubes per lot) (8,760,000 tubes per year)= years (about 3.5 days). T 2 = = years (about 46.5 days). TV(Q = 84,000) = (84,000 2) {1-(613,200 8,760,000)}(0.2) + 613,200 84,000)(150) = $8907. FARAH COSMETICS COMPANY – Solution

82 81 The Optimal Policy Using the input data we find TV(Q * = 31,499) = (31,499 / 2) [1-(613,200 / 8,760,000)](0.2) + ( 613,200 / 31,499)(150) = $5,850. The optimal order size (0.2)(1-613,200/8760,000) Q * = 2(613,200)(150) = 31,499 FARAH COSMETICS COMPANY – Solution Current cost = $8,907: savings = $3,057 or 34%

83 82 FARAH COSMETICS COMPANY – Production Lot Size Template (Excel)

84 83 Planned Shortage Model When an item is out of stock, customers may: –Go somewhere else (lost sales). –Place their order and wait (backordering). In this model we consider the backordering case. All the other EOQ assumptions are in place.

85 84 The parameters of the total variable costs function are similar to those used in the EOQ model. In addition, we need to incorporate the shortage costs in the model. –Backorder cost per unit per year (loss of goodwill cost) - C s. Reflects future reduction in profitability. Can be estimated from market surveys and focus groups. –Backorder administrative cost per unit - C b Reflects additional work needed to take care of the backorder. Planned Shortage Model – the Total Variable Cost Equation

86 85 Planned Shortage Model – the Total Variable Cost Equation The Annual holding cost = C h [T 1 T](Average inventory) = C h [T 1 T] (Q-S) 2 The Annual shortage cost = C b (number of backorders per year) + C s (T 2 T)(Average number of backorders). To calculate the annual holding cost and shortage cost we need to find –The proportion of time inventory is carried, (T 1 /T) –The proportion of time demand is backordered, (T 2 /T). T1T1 T2T2 T Variáveis de controle: Q = Quantidade pedida, S = Quantidade em backorder quando chega o pedido Q S Q-S

87 86 S Q - S Q T1T1 T2T2 S T Average inventory = (Q - S) / 2 Average shortage = S / 2 Proportion of time inventory exists = T 1 T T1T1 T Q - S Q Proportion of time shortage exists = T 2 T Finding T 1 / T and T 2 / T = (Q - S) / Q = S / Q

88 87 Annual holding cost: C h [T 1 T](Q-S) 2 = C h [(Q-S) Q](Q-S) 2 = C h (Q-S) 2 2Q Annual shortage cost: C b (Units in short per year) + C s [T 2 T](Average number of backorders) = C b (S)(D Q) + C s S 2 /2Q Planned Shortage Model – The Total Variable Cost Equation

89 88 –The total annual variable cost equation –The optimal solution to this problem is obtained under the following conditions C s > 0 ; C b < \ / 2C o C h / D TV(Q,S) = (Q -S) 2 2Q C h + DQDQ (C o + SC b S 2 2Q CSCS Holding costs Time dependent backorder costs Time independent backorder costs Ordering costs Planned Shortage Model – The Total Variable Cost Equation

90 89 The Optimal Backorder level S*=S*= Q * C h - DC b C h + C s Reorder Point R = L D - S * Planned Shortage Model – The Optimal Inventory Policy The Optimal Order Size ChCh (DC b ) 2 C h C s 2DC o Q * = C h + C s C s x

91 90 SCANLON PLUMBING CORPORATION Scanlon distributes a portable sauna from Sweden. Data –A sauna costs Scanlon $2400. –Annual holding cost per unit $525. –Fixed ordering cost $1250 (fairly high, due to costly transportation). –Lead time is 4 weeks. –Demand is 15 saunas per week on the average.

92 91 Scanlon estimates a $20 goodwill cost for each week a customer who orders a sauna has to wait for delivery. Administrative backorder cost is $10. Management wishes to know: –The optimal order quantity. –The optimal number of backorders. –Backorder costs SCANLON PLUMBING CORPORATION

93 92 SCANLON PLUMBING – Solution Input for the total variable cost function – D = 780 saunas [(15)(52)] – C o = $1,250 – C h = $525 – C s = $1,040 – C b = $10

94 93 x (780)(10) 2 (525)(1040) 525 2(780)(1250) Q * = 74 The optimal policy R = (4 / 52)(780) 20 = 40 _ S*=S*= (74)(525)(780)(10) SCANLON PLUMBING – Solution

95 94 SCANLON PLUMBING – Spreadsheet Solution

96 95 Review Systems – Continuous Review (R, Q) Policies –The EOQ, production lot size, and planned shortage models assume that inventory levels are continuously monitored Items are sold one at a time.

97 96 (R, Q) Policies –The above models call for order point (R) order quantity (Q) inventory policies. – Such policies can be implemented by A point-of-sale computerized system. The two-bin system. Review Systems – Continuous Review

98 97 (R, M) policies –When items are not necessarily sold one at a time, the reorder point might be missed, and out of stock situations might occur more frequently. –The order to level (R, M) policy may be implemented in this situation. Continuous Review Systems

99 98 (R,M) policies –The R, M policy replenishes inventory up to a pre-determined level M. Continuous Review Systems –Order Q = Q * + (R – I) = (M – SS) + (R – I) each time the inventory falls to the reorder point R or below. (Order size may vary from one cycle to another).

100 99 Exemplo da Citron e AAC AAC usa política (R,M) com R=219 e M = 354 (= Q + SS = ) Cliente pede 60 juicers quando I = 224 (> R) O novo pedido será feito quando estoque = 224 – 60 = 164 Novo pedido deverá ser = Q = Q * + (R – I) = (M – SS) + (R – I) = 382 = 354 – – – 327 = 55 = nível de estoque abaixo de R = 219 quando foi colocado o novo pedido.

101 100 It may be difficult or impossible to adopt a continuous review system, because of: –The high price of a computerized system. –Lack of space to adopt the two-bin system. –Operations inefficiency when ordering different items from the same vendor separately. The periodic review system may be found more suitable for these situations. Periodic Review Systems

102 101 Under this system the inventory position for each item is observed periodically. Orders for different items can be better coordinated periodically. Periodic Review Systems

103 102 – (T,M) Policies In a replenishment cycle policy (T, M), the inventory position is reviewed every T time units. An order is placed to bring the inventory level back up to a maximum inventory level M. M is determined by –Forecasting the number of units demanded during the review period T. –Adding the desired safety stock to the forecasted demand. Periodic Review Systems

104 103 T=Review period L = Lead time SS= Safety stock Q= Inventory position D= Annual demand I= Inventory position Periodic Review Systems Calculation of the replenishment level and order size Q = M + LD – I M = TD + SS

105 104 Every three weeks AAC receives deliveries of different products from Citron. Lead time is eight days for ordering Citrons juicers. AAC is now reviewing its juicer inventory and finds 210 in stock. How many juicers should AAC order for a safety stock of 30 juicers? AAC operates a (T, M) policy

106 105 Data –Review period T = 3 weeks = 3/52 = years, –Lead time = L = 8 days = 8/260 = years, –Demand D = 6240 juicers per year, –Safety stock SS = 30 juicers, –Inventory position I = 210 juicers AAC operates a (T, M) policy – Solution AAC operates 260 days a year. (5)(52) = 260.

107 106 Review period demand = TD = ( 3/52)(6240) = 360 juicers, M = TD + SS = = 390 juicers, Q = M + LD – I = (6240) = 372 juicers. AAC operates a (T, M) policy – Solution

108 107 Review point AAC operates a (T, M) policy – Solution T SS Inventory position Order Replenishment level Inventory position L Notice: I + Q is designed to satisfy the demand within an interval of T + L. To obtain the replenishment level add SS to I + Q. M = maximum inventory L

109 108 Demand is stochastic with a known distribution. Shelf life of the item is limited. Inventory is saleable only within a single time period. Inventory is delivered only once during a time period. Single Period Inventory Model - Assumptions At the end of each period, unsold inventory is disposed of for some salvage. The salvage value is less than the cost per item. Unsatisfied demand may result in shortage costs. At the end of each period, unsold inventory is disposed of for some salvage. The salvage value is less than the cost per item. Unsatisfied demand may result in shortage costs.

110 109 To find an optimal order quantity we need to balance the expected cost of over-ordering and under ordering. Expected Profit = (Profit when Demand=X)Prob(Demand=X) x The expected profit is a function of the order size, the random demand, and the various costs. The Expected Profit Function

111 110 –Developing an expression for EP(Q) Notation p = per unit selling price of the good. c = per unit cost of the good. s = per unit salvage value of unsold good. K = fixed purchasing costs Q = order quantity. EP(Q) = Expected Profit if Q units are ordered. Scenarios –Demand X is less than the order quantity (X < Q). –Demand X is greater than or equal to the order quantity (X Q). The Expected Profit Function

112 111 Scenario 1: Demand X is less than the units stocked, Q. Scenario 2: Demand X is greater than or equal to the units stocked. Profit = pX + s(Q - X) - cQ - K Profit = pQ - g(X - Q) - cQ - K EP(Q) = [pX+s(Q - X) - cQ - K]P(X) + [pQ - g(X - Q) - cQ - K]P(X) The Expected Profit Function

113 112 –To maximize the expected profit order Q * For the discrete demand case take the smallest value of Q * that satisfies the condition P(D Q * ) (p - c + g) (p - s + g) For the continuous demand case find the Q * that solves F(Q * ) = (p - c + g) (p - s + g) The Optimal Solution Distribuição Acumulada Probabilidade Acumulada Nível de serviço ótimo

114 113 THE SENTINEL NEWSPAPER Management at Sentinel wishes to know how many newspapers to put in a new vending machine. Data –Unit selling price is $0.30 –Unit production cost is $0.38. –Advertising revenue is $0.18 per newspaper. –Unsold newspaper can be recycled and net $0.01. –Unsatisfied demand costs $0.10 per newspaper. –Filling a vending machine costs $1.20. Demand distribution is discrete uniform between 30 and 49 newspapers.

115 114 SENTINEL - Solution Input to the optimal order quantity formula p = 0.30 c = 0.20 [ ] s = 0.01 g = 0.10 K = 1.20 The probability of the optimal service level = p+ g - c p+ g - s = =

116 P(D 39) = 0.50 P(D 40) = 0.55 Q * = 40 SENTINEL – Solution Finding the optimal order quantity Q *

117 116 =(B5+B8-B6)/(B5+B8-B7) =ROUNDUP(B10+E5*(B11- 10),0) =(E6-B10+1)/(B11-B10+1) SENTINEL – Spreadsheet Solution

118 117 WENDELLS BAKERY Management in Wendells wishes to determine the number of donuts to prepare for sale, on weekday evenings Data –Unit cost is $0.15. –Unit selling price is $0.35. –Unsold donuts are donated to charity for a tax credit of $0.05 per donut. –Customer goodwill cost is $0.25. –Operating costs are $15 per evening. Demand is normally distributed with a mean of 120, and a standard deviation of 20 donuts.

119 118 WENDELLS BAKERY - Solution Input to the optimal order quantity formula p = $0.35 c = $0.15 s = $0.05 g = $0.25 K = $15.00 The optimal service level = p+ g - c p+ g - s = =

120 =120 Q*Q* From the relationship F(Q * ) = we find the corresponding z value. From the standard normal table we have z = The optimal order quantity is calculated by Q * = + z For Wendells Q * = (0.3186)(20) 138 WENDELLS BAKERY - Solution Finding the optimal order quantity

121 120 EP(Q * ) = (p - s) - (c - s)Q * - (p + g - s) ( )L[(Q * - ) / - K For the normal distribution L [(Q * - ) / is obtained from the partial expected value table. For Wendells EP(138) = ( )(120) - ( )(138) - ( )x(20)L[( ) / 20] - 15 = $6.10 L(0.9) = WENDELLS BAKERY - Solution Calculating the expected profit Apêndice B Ver slide 112

122 121 =(B5-B7)*B10-(B6-B7)*E6-(B5+B8-B7)*B11*(EXP(- (((E6-B10)/B11)^2)/2)/((2*PI())^0.5)-((E6-B10)/B11)*(1- NORMSDIST((E6-B10)/B11)))-B9 =NORMINV(E5, B10,B11) =(B5+B8- B6)/(B5+B8-B7) WENDELLS BAKERY - Spreadsheet Solution

123 122 WENDELLS – The commission strategy When commission replaces fixed wages… –Compare the maximum expected profit of two strategies: $0.13 commission paid per donut sold, $15 fixed wage per evening (calculated before). –Calculate first the optimal quantity for the alternative policy. –Check the expected difference in pay for the operator.

124 123 WENDELLS – The commission strategy - Solution The unit selling price changes to c = = $0.22 The optimal order: F(Q * ) = ( ) / ( )= Z =.71 Q * = + z = (0.71)(20) 134 donuts =120 Q*Q*

125 124 Will the bakerys expected profit increase? EP(134) = ( )(20) - ( )(134) - ( )x(20)L[( ) / 20] = $5.80 < 6.10 The bakery should not proceed with the alternative plan. WENDELLS – The commission strategy - Solution

126 125 Comments –The operator expected compensation will increase, but not as much as the bakerys expected loss. –An increase in the mean sales is probable when the commission compensation plan is implemented. This may change the analysis results. WENDELLS – The commission strategy - Solution

127 126 Dimensionamento de Lotes ( Lot Sizing )

128 127 Introdução O problema de dimensionamento de lotes consiste em planejar a quantidade de itens a ser produzida em várias (ou única) máquinas, em cada período ao longo de um horizonte de tempo finito, de modo a atender uma certa demanda, podendo estar sujeito a algumas restrições.

129 128 Métodos Básicos de Dimensionamento de Lotes Lot for Lot (L4L); Silver-Meal Heuristic Procedure(SM ); Economic Order Quantity (EOQ) ; Periodic Order Quantity ( POQ); Least Unit Cost (LUC) ; Least Total Cost (LTC) ; Fixed Period Requirements (FPR); Part Period Balancing (PPB); Wagner-Whitin Algorithm(WW).

130 129 Lot for Lot Esta heurística consiste no método mais básico possível, onde a quantidade produzida visa atender somente o período em que o item será utilizado. Sendo assim, o estoque será sempre nulo e serão feitas preparações de máquina em todos os períodos com demanda positiva.

131 130 Silver-Meal Heuristic Procedure(SM ) Pode ser usado para achar um cronograma de produção perto do ótimo. A heurística do SM é baseado no fato de que a meta é minimizar o custo médio do período.

132 131 Economic Order Quantity (EOQ ) Consiste no principio de que sempre que seja necessário fazer uma encomenda, encomendar uma quantidade igual à EOQ. Assume-se que a demanda é constante, os itens são independentes, e nenhuma incerteza está envolvida no processo decisório. Esse método minimiza o custo total relevante do inventário.

133 132 Periodic Order Quantity ( POQ) Uma maneira de reduzir os altos custos de manter inventário associado com tamanhos de lotes fixos é usar a fórmula da EOQ para encontrar um período econômico de encomenda. Faz-se isso dividindo o EOQ pela taxa média de demanda.

134 133 Least Unit Cost (LUC) Este método tem como objetivo encontrar o tamanho da encomenda que se traduz no menor custo unitário do produto. O método segue os seguintes passos: 1. Calcular os lançamentos previstos acumulados até que o valor acumulado seja superior à quantidade de desconto. 2. Calcular se é vantajoso aceitar o desconto com base no menor custo unitário.

135 134 Least Total Cost (LTC) O tamanho da ordem cobrirá os próximos T períodos, onde T é o período onde o custo de transporte e o custo de preparação são muito próximos.

136 135 Fixed Period Requirements (FPR) Ordena-se uma quantidade suficiente para suprir a demanda de um número fixo de períodos consecutivos.

137 136 Part Period Balancing (PPB) Usa todas as informações providas pelo cronograma de pedidos, tentando igualar os custos totais de ordens feitas e do transporte de estoque.

138 137 Wagner-Whitin Algorithm (WW) Procedimento de programação dinâmica para obter o cronograma ótimo de dimensionamento de lotes no horizonte de planejamento.

139 138 Exemplo Certa firma que fabrica um determinado produto deseja fazer um planejamento da produção para um horizonte de quatro semanas. Sabe-se que a demanda para estas quatro semanas será de 104, 174, 46 e 112 unidades. Suponha que a firma faça no máximo uma preparação de máquina a cada semana e que não haja restrição de capacidade de produção.

140 139WinQSB

141 140 WinQSB

142 141 Solve and Analyze

143 142 Wagner-Whitin

144 143 Silver-Meal

145 144 EOQ

146 145 POQ

147 146 LUC

148 147 LTC

149 148 FPR

150 149 PPB

151 150 LOT for LOT

152 151 Comparação entre Métodos MétodoCusto % acima do ótimo Wagner-Whitin$ 1368,00 - Silver-Meal$ 1368,00- EOQ$ 1521,0011 POQ$ 1458,007 LUC$ 1458,007 LTC$ 1458,007 FPR$ 1472,008 PPB$ 1438,005 Lot for Lot$ 1472,008 Ótimo


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