Inventory Models.

Inventory Models

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

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

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

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

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

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%.

Classificação ABC

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.

Classificação da criticidade dos itens
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.

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.

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.

Inventory Classifications
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 firm’s capital needs. Management of items with short shelf life and long shelf life is very different

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.

Type of Costs in Inventory Models
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

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 Ch = Annual holding cost per unit in inventory H = Annual holding cost rate C = Unit cost of an item Ch = H * C

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 Co = Order cost or setup cost

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: Cb = Fixed administrative costs of an out of stock item ($/stockout unit). Cs = Annualized cost of a customer awaiting an out of stock item ($/stockout unit per year).

Procurement/Manufacturing Cost Represents the unit purchase cost (including transportation) in case of a purchase. Unit production cost in case of in-house manufacturing. C = Unit purchase or manufacturing cost.

Demand in Inventory Models
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) D = Demand rate (usually per year)

Review Systems 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.

Economic Order Quantity Model - Assumptions
Demand occurs at a known and reasonably constant rate. 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.

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

Cost Equation for the EOQ Model
Total Annual Inventory Costs Total Annual Holding Costs Total Annual ordering Costs Total Annual procurement Costs = + + TC(Q) = (Q/2)Ch + (D/Q)Co + DC Q Ch The optimal order Size Q* =

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

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

Number of Orders per Year
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.

Cycle Time The cycle time, T, represents the time that elapses between the placement of orders. T = Q/D 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}.

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

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

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

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

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)Ch + (D/Q)Co + DC + ChSS Safety stock holding cost

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

Data Co = $12 ($8 for placing an order) + (20 min. to check)($12 per hr) Ch = $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:

Data The constant demand rate seems to be a good assumption. Annual demand = (120/week)(52weeks) = 6240 juicers.

AAC – Solution: EOQ and Total Variable Cost
Current ordering policy calls for Q = 600 juicers. TV( 600) = (600 / 2)($1.40) + (6240 / 600)($12) = $544.80 The EOQ policy calls for orders of size Savings of 16% Ö 2(6240)(12) 1.40 = 327 = Q* TV(327) = (327 / 2)($1.40) + (6240 / 327) ( $12) = $457.89

AAC – Solution: Reorder Point and Total Cost
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 TC(327) = ($10) + (13)($1.40) = $62,876.09 TV(327) + Procurement Safety stock cost holding cost

AAC – Solution: Sensitivity of the EOQ Results
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 Only 0.4% increase TV(327) = (327/2)($1.40) + (7500/327)($12) = $504.13

AAC – Solution: Cycle Time
For an order size of 327 juicers we have: T = (327/ 6240) = year. = (52)(5) = 14 days. This is useful information because: Shelf life may be a problem. Coordinating orders with other items might be desirable. working days per week

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

Service Levels and Safety Stocks

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 .

Two Types of Service Level
Comum Service levels can be viewed in two ways. 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. 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
Juicer Demand and Units on Backorder Cycle Number Demand # Units on backorder 1 585 2 610 3 628 15 4 572 5 605 Cycle service level = 4/5 = 80% Unit Service level = 1- 15/3000 = 99,5%

The Cycle Service Level Approach
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.

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 mL = demanda média no lead time sL= desvio padrão da demanda no lead time R = mL + zasL (1 –a) = Service level (use a Normal DistributionTable)

P(DL<R) = 1 – a m=192 P(DL>R) = a R P(DL> R) = P(Z > (R – mL)/sL) = a. Since P(Z > Za) = a, we have Za = (R – mL)/sL, which gives… R = mL + zasL

AAC - Cycle Service Level Approach
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 = juicers2.

AAC - Cycle Service Level Approach
To find mLand sL the parameters m (per week) and s (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 mL » (1.6)(120) = 192; s2L » (1.6)(83.33) =

AAC - Service Level for a given Reorder Point
Let us use the current reorder point of 205 juicers. 205 = 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 – Reorder Point for a given Service Level
Management wants to improve the cycle service level to 99%. The z value corresponding to 1% right hand tail is 2.33. R = (11.55) = 219 juicers.

AAC – Acceptable Number of Stockouts per Year
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
There will be an average of /327 = cycles (lead times) per year. The likelihood of stockouts = 1/19 = This translates into a service level of 94.76%

The Unit Service Level Approach
L(z)=partial expected value for the standard normal between some z and infinity 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) = aQ* / sL Solve for R using the equation R = mL + zsL

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

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

EOQ Models with Quantity Discounts
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.

All Units Discount Schedule
To determine the optimal order quantity, the total purchase cost must be included TC(Q) = (Q/2)Ch + (D/Q)Co + DCi + ChSS Ci represents the unit cost at the ith pricing level.

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

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

AAC – All units discount procedure
Step 1: Find the optimal order Qi* for each discount level “i”. Use the formula Step 2: For each discount level “i” modify Q i* as follows If Qi* is lower than the smallest quantity that qualifies for the i th discount, increase Qi* to that level. If Qi* is greater than the largest quantity that qualifies for the ith 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*) Ch=Ci.0,14

AAC – All units discount procedure
Step 1: Find the optimal order quantity Qi* for each discount level “i” based on the EOQ formula

TC(Q) = (Q/2)Ch + (D/Q)Co + DCi + ChSS

AAC – All Units Discount Procedure
Step 2 : Modify Q i * $10/unit $9.75/unit $9.50 Q1* Q3* 599 336 327 331 999 600 Q2*

Step 2 : Modify Q i * $10/unit Q3* Q3* Q3* Q3* Q3* $9.50 Q3* Q1* Q2* Q3* Q3* 336 327 331 999 600 Modified Q* and total Cost Qualified Price Modified Total Urder per Unit Q* Cost 1-299 10.00 300 **** 9.75 331 61,292.13 9.50 336 600 59,803.80 9.40 337 1000 59,388.88 5000 9.00 345 59,324.98 327

Step 3: Substitute Q I * in the total cost function Step 4 Modified Q* and total Cost Qualified Price Modified Total Urder per Unit Q* Q* Cost 1-299 10,00 327 327 62876,09 9,75 331 331 61.292,13 9,50 336 600 59.803,80 9,40 337 1000 59.388,88 5000 9,00 345 5000 59.324,98 AAC should order 5000 juicers

AAC – All Units Discount Excel Worksheet

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

Production Lot Size Model – Understanding the inventory profile
The production increases the inventory at a rate of P. Demand accumulation during production run = DT1 Production time T1 Maximum inventory Maximum inventory = (P – D)T1 = (P – D)(Q/P) = Q(1 – D/P) The inventory increases at a net rate of P - D Production Lot Size = Q = PT1 The demand decreases the inventory at a rate of D. Demand accumulation during production run

Production Lot Size Model – Total Variable Cost
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 (Co). In addition, we need to incorporate the annual production rate (P) in the model.

Production Lot Size Model – Total Variable Cost
TV(Q) = (Q/2)(1 - D/P)Ch + (D/Q)Co P is the annual production rate The average inventory Ch(1-D/P) The Optimal Order Size Q* = 2DCo

Production Lot Size Model – Useful relationships
Cycle time T = Q / D. Length of a production run T1 = Q / P. Time when machines are not busy producing the product T2 = T - T1 = Q(1/D - 1/P). Average inventory = (Q/2)(1-D/P).

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%.

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

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). T1 = (84,000 tubes per lot)/(8,760,000 tubes per year)= years (about 3.5 days). T2 = = years (about 46.5 days). TV(Q = 84,000) = (84,000/2) {1-(613,200/8,760,000)}(0.2) ,200/84,000)(150) = $8907.

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 Current cost = $8,907: savings = $3,057 or 34%

FARAH COSMETICS COMPANY – Production Lot Size Template (Excel)

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.

Planned Shortage Model – the Total Variable Cost Equation
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) - Cs. Reflects future reduction in profitability. Can be estimated from market surveys and focus groups. Backorder administrative cost per unit - Cb. Reflects additional work needed to take care of the backorder.

Planned Shortage Model – the Total Variable Cost Equation
Variáveis de controle: Q = Quantidade pedida, S = Quantidade em backorder quando chega o pedido The Annual holding cost = Ch[T1/T](Average inventory) = Ch[T1/T] (Q-S)/2 The Annual shortage cost = Cb(number of backorders per year) + Cs(T2/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, (T1/T) The proportion of time demand is backordered, (T2/T). Q-S Q T1 T2 S T

Finding T1/ T and T2/ T Q Average inventory = (Q - S) / 2
Proportion of time inventory exists = T1/T Q - S Q Q - S = (Q - S) / Q Q T2 T1 T1 Proportion of time shortage exists = T2/T T S T S = S / Q Average shortage = S / 2

Planned Shortage Model – The Total Variable Cost Equation
Annual holding cost: Ch[T1/T](Q-S)/2 = Ch[(Q-S) /Q](Q-S)/2 = Ch(Q-S)2/2Q Annual shortage cost: Cb(Units in short per year) + Cs[T2/T](Average number of backorders) = Cb(S)(D/Q) + CsS2/2Q

Planned Shortage Model – The Total Variable Cost Equation
The total annual variable cost equation The optimal solution to this problem is obtained under the following conditions Cs > 0 ; Cb < \/ 2CoCh / D (Q -S)2 D Q S2 2Q Ch + (Co + SCb) + TV(Q,S) = CS 2Q Holding costs Ordering costs Time independent backorder costs Time dependent backorder costs

Planned Shortage Model – The Optimal Inventory Policy
The Optimal Order Size - Q* = Ch + Cs Cs x 2DCo (DCb)2 ChCs Ch The Optimal Backorder level S*= Q* Ch - DCb Ch + Cs Reorder Point R = L D - S*

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.

SCANLON PLUMBING CORPORATION
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 – Solution
Input for the total variable cost function D = 780 saunas [(15)(52)] Co = $1,250 Ch = $525 Cs = $1,040 Cb = $10

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

SCANLON PLUMBING – Spreadsheet Solution

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.

Review Systems – Continuous Review
(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.

Continuous Review Systems
(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
(R,M) policies The R, M policy replenishes inventory up to a pre-determined level M. 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).

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 – –164 382 – 327 = 55 = nível de estoque abaixo de R = 219 quando foi colocado o novo pedido.

Periodic Review Systems
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.

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

(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.

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

AAC operates a (T, M) policy
Every three weeks AAC receives deliveries of different products from Citron. Lead time is eight days for ordering Citron’s 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 – Solution
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 260 days a year. (5)(52) = 260.

Review period demand = TD = ( 3/52)(6240) = 360 juicers, M = TD + SS = = 390 juicers, Q = M + LD – I = (6240) = 372 juicers.

Replenishment level Order Order M = maximum inventory Inventory position Inventory position SS SS SS L L Review point Review point T 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.

Single Period Inventory Model - Assumptions
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. Demand is stochastic with a known distribution. 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.

The Expected Profit Function
To find an optimal order quantity we need to balance the expected cost of over-ordering and under ordering. Expected Profit = S(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.

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 ³Q).

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 Optimal Solution 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*) ³ (p - c + g)/(p - s + g) For the continuous demand case find the Q* that solves F(Q*) = (p - c + g) /(p - s + g) Nível de serviço ótimo Probabilidade Acumulada Distribuição Acumulada Nível de serviço ótimo

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.

SENTINEL - Solution Input to the optimal order quantity formula
c = [ ] s = 0.01 g = 0.10 K = 1.20 p+ g - c p+ g - s The probability of the optimal service level = = 0.513 =

SENTINEL – Solution Finding the optimal order quantity Q*
1.0 P(D £ 39) = 0.50 P(D £ 40) = 0.55 0.513 0.55 0.50 Q* = 40 30 39 40 49

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

WENDELL’S BAKERY Management in Wendell’s 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.

WENDELL’S 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 = =

WENDELL’S BAKERY - Solution Finding the optimal order quantity
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* = m + zs For Wendell’s Q* = .8182 m=120 Q*

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

WENDELL’S BAKERY - Spreadsheet Solution
=(B5+B8-B6)/(B5+B8-B7) =NORMINV(E5,B10,B11) =(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

WENDELL’S – 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.

WENDELL’S – The commission strategy - Solution
The unit selling price changes to c = = $0.22 The optimal order: F(Q*) = ( ) / ( )= Z = .71 Q* = m + zs = (0.71)(20) » 134 donuts. .7616 m=120 Q*

Will the bakery’s expected profit increase? EP(134) = ( )(20) - ( )(134) - ( )x(20)L[( ) / 20] = $5.80 < 6.10 The bakery should not proceed with the alternative plan.

Comments The operator expected compensation will increase, but not as much as the bakery’s expected loss. An increase in the mean sales is probable when the commission compensation plan is implemented. This may change the analysis results.

Dimensionamento de Lotes (Lot Sizing)

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.

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).

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.

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.

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.

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.

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.

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.

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

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.

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

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.

WinQSB

Solve and Analyze

Wagner-Whitin

Silver-Meal

LOT for LOT

Comparação entre Métodos
Custo % acima do ótimo Wagner-Whitin $ 1368,00 - Silver-Meal EOQ $ 1521,00 11 POQ $ 1458,00 7 LUC LTC FPR $ 1472,00 8 PPB $ 1438,00 5 Lot for Lot Ótimo

Inventory Models.

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