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Inventory ModelsInventory Models
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Tipos de Demanda
• Demanda independente: são itens que dependem, em sua maioria, dos pedidos de clientes externos, como, por exemplo, produtos acabados em geral.
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• 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
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Tipos de estoques
• Matérias-primas• Produtos em processo (WIP - work in process)• Produtos acabados• Em trânsito• Em consignação
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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
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Pressões para Manter Estoque Alto
• Estoque alto = maior probabilidade de atender bem os clientes
Mas• Estoque alto = certeza de alto custo em carregar
estoques
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Fontes de Elevação de Estoque
• Marketing• Engenharia• Controle de Qualidade• Manufatura• Suprimentos• Gerentes
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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%.
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Classificação ABC
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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.
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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.
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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.
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Classificação 123Classificação 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.
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Inventory can be classified in various ways:
By Process By Importance By Shelf Life Raw materials Perishable Work in progress (A, B, C), (X,Y,Z), (1,2,3) Nonperishable Finished goods
Used typically by accountants at manufacturing firms.Enables management to track the production process.
Items are classified by their relative importancein terms of the firm’s capital needs.
Management of items with short shelf life and long shelf life is very different
Inventory Classifications
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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.
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• 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
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• 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 inventoryH = Annual holding cost rateC = Unit cost of an item
Ch = H * C
Type of Costs in Inventory Models
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• 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
Co = Order cost or setup cost
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• 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
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).
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• 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.
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• 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)
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• 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
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• 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.
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• The constant environment described by the EOQ assumptions leads to the following observation:
The optimal EOQ policy consists of same-size orders.
Q QQ
The EOQ Model – Inventory profile
This observation results in the following inventory profile :
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Q QQ
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
ChCh
The optimal order SizeThe optimal order Size
2DCo2DCoQ* = Q* =
Cost Equation for the EOQ Model
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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*
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
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The curve is reasonably flat around Q*.
Q*
Deviations from the optimal order size cause only small increase in the total cost.
Sensitivity Analysis in EOQ models
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• 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
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• 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}.
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• 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 DR = L D
Lead Time and the Reorder Point
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Inventory position
LPlace 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
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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
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• 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
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LPlace the order now
Reorder Point
R = LD
Safety stockPlanned situation
Actualsituation
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L
R = LD
Safety stock
Actualsituation
+ SS
New Reorder Point
Place the order now
SS=Safety stock
The safety stockprevents excessiveshortages.
LD
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Inventory Costs Including safety stock
Total Annual Inventory Costs
= Total Annual Holding Costs
Total Annual ordering Costs
Total Annual procurement Costs
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TC(Q) = (Q/2)Ch + (D/Q)Co + DC + ChSS
Safety stockholding cost
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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
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Sales of Juicers over the last 10 weeksWeek 1 2 3 4 5Sales 105 115 125 120 125Week 6 7 8 9 10Sales 120 135 115 110 130
• 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:
ALLEN APPLIANCE COMPANY (AAC)
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• Data– The constant demand rate seems to be a good
assumption.– Annual demand = (120/week)(52weeks) = 6240 juicers.
ALLEN APPLIANCE COMPANY (AAC)
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• 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
AAC – Solution:EOQ and Total Variable Cost
Savings of 16%
2(6240)(12)1.40 = 327.065 327 =Q*
TV(327) = (327 / 2)($1.40) + (6240 / 327) ( $12) = $457.89
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TC(327) = 457.89 + 6240($10) + (13)($1.40) = $62,876.09
• 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
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• 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) = (3272)($1.40) + (7500327)($12) = $504.13
Only 0.4% increase
AAC – Solution:Sensitivity of the EOQ Results
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• For an order size of 327 juicers we have:– T = (3276240) = 0.0524 year.
= 0.0524(52)(5) = 14 days.
– This is useful information because:
• Shelf life may be a problem.• Coordinating orders with other items might be desirable.
AAC – Solution:Cycle Time
working days per week
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AAC – Excel Spreadsheet
=SQRT(2*$B$10*$B$14/$B$13)
=1/E11Copy to cell H12
=E10/B10Copy to cell
H11
=$B$10*$B$11+E14+$B$13*B16Copy to Cell H15
=(E10/2)*$B$13+($B$10/E10)*$B$14Copy to cell H14
=$B$15*$B$10+$B$16-INT(($B$15*$B$10+$B$16)/E10)*E10
Copy to cell H13
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Service Levels and Safety Stocks
Service Levels and Safety Stocks
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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 .
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• 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
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Two Types of Service Level
Juicer Demand and Units on BackorderCycle Number Demand # Units on backorder
1 585 0
2 610 0
3 628 15
4 572 0
5 605 0
Cycle service level = 4/5 = 80%
Unit Service level = 1- 15/3000 = 99,5%
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• 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
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• 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 + zL (1 –= Service level(use a Normal
DistributionTable)
L = demanda média no
lead time
L= desvio padrão da demanda no lead time
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=192
P(DL> R) = P(Z > (R – L)/L) = . SinceP(Z > Z) = , we have Z = (R – L)/L,
which gives…
The Cycle Service Level Approach
P(DL>R) = Service level = P(DL<R) = 1 –
R
R = L + zL
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• 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 = 83.33 juicers2.
AAC - Cycle Service Level Approach
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• To find Land 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; 2L (1.6)(83.33) = 133.33
AAC - Cycle Service Level Approach
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• 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.
133 33.
AAC - Service Level for a given Reorder Point
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• Management wants to improve the cycle service level to 99%.
• The z value corresponding to 1% right hand tail is 2.33.
R = 192 + 2.33(11.55) = 219 juicers.
• Management wants to improve the cycle service level to 99%.
• The z value corresponding to 1% right hand tail is 2.33.
R = 192 + 2.33(11.55) = 219 juicers.
AAC – Reorder Point for a given Service Level
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• 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
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AAC – Acceptable Number of Stockouts per Year
• There will be an average of
6240327 = 19.08 cycles (lead times) per year.
• The likelihood of stockouts = 1/19 = 0.0524.
• This translates into a service level of 94.76%
• There will be an average of
6240327 = 19.08 cycles (lead times) per year.
• The likelihood of stockouts = 1/19 = 0.0524.
• This translates into a service level of 94.76%
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• 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 + zL
The Unit Service Level Approach
L(z)=partial expected value for the standard normal between
some z and infinity
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=NORMDIST(B8,B5,B6,TRUE)
AAC – Cycle Service Level (Excel spreadsheet)
=NORMINV(B7,B5,B6)
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• 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
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• 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
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• To determine the optimal order quantity, the total purchase cost must be included
TC(Q) = (Q2)Ch + (DQ)Co + DCi + ChSS
Ci represents the unit cost at the ith pricing level.
All Units Discount Schedule
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AAC - All Units Quantity Discounts
Quantity Discount Schedule
1-299 $10.00300-599 $9.75600-999 $9.40
1000-4999 $9.505000 $9.00
Quantity Discount Schedule
1-299 $10.00300-599 $9.75600-999 $9.40
1000-4999 $9.505000 $9.00
• AAC is offering all units quantity discounts to its customers.
• Data
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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?
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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*)
Q DC Co h* ( ) / 2 Ch=Ci.0,14
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Step 1: Find the optimal order quantity Qi* for each
discount level “i” based on the EOQ formula Lowest cost order size per discount level
Discount Qualifying Pricelevel order per unit Q*
0 1-299 10.00 3271 300-599 9.75 3312 600-999 9.50 3363 1000-4999 9.40 3374 5000 9.00 345
AAC – All units discount procedure
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TC(Q) = (Q2)Ch + (DQ)Co + DCi + ChSS
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– Step 2 : Modify Q i *
Modified Q* and total CostQualified Price Modified Total
Urder per Unit Q* Q* Cost1-299 10,00 327 **** ****
300-599 9,75 331 331 61.292,13600-999 9,50 336 600 59.803,80
1000-4999 9,40 337 1000 59.388,885000 9,00 345 5000 59.324,98
1 299
Q1*Q1*
327
$10/unit
599331Q2
*Q2*
$9.75/unit
999999600
Q3*Q3*
336
$9.50
AAC – All Units Discount Procedure
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– Step 2 : Modify Q i *
Modified Q* and total CostQualified Price Modified Total
Urder per Unit Q* Q* Cost1-299 10.00 300 **** ****
300-599 9.75 331 331 61,292.13600-999 9.50 336 600 59,803.80
1000-4999 9.40 337 1000 59,388.885000 9.00 345 5000 59,324.98
Modified Q* and total CostQualified Price Modified Total
Urder per Unit Q* Q* Cost1-299 10.00 327 **** ****
300-599 9.75 331 331 61,292.13600-999 9.50 336 600 59,803.80
1000-4999 9.40 337 1000 59,388.885000 9.00 345 5000 59,324.98
1 299
Q1*Q1*
327
$10/unit
331
Q2*Q2*
999999600
Q3*Q3*
336
$9.50
AAC – All Units Discount Procedure
Q3*Q3* Q3
*Q3* Q3
*Q3* Q3
*Q3* Q3
*Q3*Q3
*Q3*
Q3*Q3*
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– Step 3: Substitute Q I * in the total cost function
– Step 4
Modified Q* and total CostQualified Price Modified Total
Urder per Unit Q* Q* Cost1-299 10,00 327 62876,09
300-599 9,75 331 331 61.292,13600-999 9,50 336 600 59.803,80
1000-4999 9,40 337 1000 59.388,885000 9,00 345 5000 59.324,98
AAC should order 5000 juicersAAC should order 5000 juicers
AAC – All Units Discount Procedure
327
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Calculation of Optimal Inventory Policy Under All-Units Quantity Discounts
OPTIMALINPUTS Values OUTPUTS Values
Annual Demand, D = 6240.00 Order quantity, Q* = 5000Per Unit Cost, C = 10.00 Cycle Time (in years), T = 0.801282051Annual Holding Cost Rate, H = 0.14 # of Cycles Per Year, N = 1.248Annual Holding Cost Per Unit, Ch = 1.40 Reorder Point, R = 205.0000Order Cost, Co = 12.00 Total Annual Cost, TC(Q*) = 59341.36Lead Time (in years), L = 0.03077 Safety Stock, SS = 13.00
DISCOUNTSLevel Breakpoint Discount Price Q* TC(Q*) Modified Q*
0 1 10.00 327 62876.09 3271 300 9.75 331 61309.88 3312 600 9.50 336 59821.09 6003 1000 9.40 337 59405.99 10004 5000 9.00 345 59341.36 50005678
AAC – All Units Discount Excel Worksheet
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• 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
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• 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
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ProductionLot Size = Q = PT1
The inventory increasesat a net rate of P - D
The production increases theinventory at a rate of P.
The demand decreases theinventory at a rate of D.
Production time
T1
Demand accumulationduring production run
Demand accumulationduring production run = DT1
Maximum inventory = (P – D)T1
= (P – D)(Q/P) = Q(1 – D/P)
Maximum inventory
Production Lot Size Model – Understanding the inventory profile
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• 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
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TV(Q) = (Q2)(1 - DP)Ch + (DQ)Co
P is the annual production rate
Ch(1-D/P)
The Optimal Order Size
Q* = 2DCo
The average inventory
Production Lot Size Model –Total Variable Cost
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• 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).
Production Lot Size Model –Useful relationships
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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%.
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• 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.
Dozens
FARAH COSMETICS COMPANY – Solution
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• Current Policy
Currently, Farah produces in lots of 84,000 tubes.
T = (84,000 tubes per run)(613,200 tubes per year)= 0.137 years (about 50 days).
T1 = (84,000 tubes per lot)(8,760,000 tubes per year)= 0.0096 years (about 3.5 days).
T2 = 0.137 - 0.0096 = 0.1274 years (about 46.5 days).
TV(Q = 84,000) = (84,0002) {1-(613,2008,760,000)}(0.2)
+ 613,20084,000)(150) = $8907.
FARAH COSMETICS COMPANY – Solution
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• 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%
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FARAH COSMETICS COMPANY – Production Lot Size Template (Excel)
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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.
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• 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
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Planned Shortage Model –the Total Variable Cost Equation
• The Annual holding cost = Ch[T1T](Average inventory) =
Ch[T1T] (Q-S)2
• The Annual shortage cost = Cb(number of backorders per year) + Cs(T2T)(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).
T1 T2
T
Variáveis de controle:
Q = Quantidade pedida, S = Quantidade em backorder quando chega o pedido
Q
S
Q-S
86
S
Q - S
Q
T1 T2
S T
Average inventory = (Q - S) / 2
Average shortage = S / 2
Proportion of time inventory exists = T1T
T1
T
Q - S
Q
Proportion of time shortage exists
= T2T
Finding T1/ T and T2/ T
= (Q - S) / Q
= S / Q
87
• Annual holding cost:Ch[T1T](Q-S)2 = Ch[(Q-S) Q](Q-S)2 = Ch(Q-S)22Q
• Annual shortage cost:Cb(Units in short per year) + Cs[T2T](Average number of backorders) = Cb(S)(DQ) + CsS2/2Q
Planned Shortage Model –The Total Variable Cost Equation
88
– The total annual variable cost equation
– The optimal solution to this problem is obtained under the following conditions
• Cs > 0 ;
• Cb < \/ 2CoCh / D
TV(Q,S) = (Q -S)2
2Q Ch + DQ
(Co + SCb S2
2QCS
Holding costs
Time dependent backorder costs
Time independent backorder costs
Ordering costs
Planned Shortage Model –The Total Variable Cost Equation
89
The Optimal Backorder level
S*= Q* Ch - DCb
Ch + Cs Reorder Point R = L D - S*
Planned Shortage Model –The Optimal Inventory Policy
The Optimal Order Size
Ch
(DCb)2
ChCs
2DCo Q* = Ch + Cs
Cs
x
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.
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
92
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
93
x(780)(10)2
(525)(1040)5252(780)(1250) 525+1040
1040Q* = 74
• The optimal policy
R = (4 / 52)(780) 20 = 40
_S*= (74)(525) (780)(10)
525 + 104020
SCANLON PLUMBING – Solution
94
SCANLON PLUMBING – Spreadsheet Solution
Calculation of Optimal Inventory Policy for a Planned Shortage Model
OPTIMAL ASSIGNEDINPUTS Values OUTPUTS Values OUTPUTS Values
Annual Demand, D = 780.00 Order Quantity, Q* = 74.01 Q = 74.00Per Unit Cost, C = 2400.00 Backorder Level, S* = 19.84 S = 20.00Annual Holding Cost Rate, H = 0.22 Cycle Time (in years), T = 0.0949 T = 0.0949Annual Holding Cost Per Unit, Ch =525.00 # of Cycles Per Year, N = 10.5388 N = 10.5405Order Cost, Co = 1250.00 Reorder Point, R = 40.1531 R = 39.9976Annual Backorder Cost, Cs = 1040.00 Total Annual Variable Cost, TV(Q*) =28438.24 TV(Q) = 28438.51Fixed Admin. Backorder Cost, Cb =10.00 Total Annual Cost, TC(Q*) = 1900438.24 TC(Q) = 1900438.51Lead Time (in years), L = 0.07692 % of Customers Backordered = 26.81 % Back. = 27.03
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.
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
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
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).
99
Exemplo da Citron e AAC• AAC usa política (R,M) com R=219 e M = 354 (= Q + SS = 327 + 27)
• 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 – 27 + 219 –164
• 382 – 327 = 55 = nível de estoque abaixo de R = 219 quando foi colocado o novo pedido.
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
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
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
103
T =Review periodL = Lead timeSS= Safety stockQ = Inventory positionD = Annual demandI = Inventory position
Periodic Review Systems
• Calculation of the replenishment level and order size
104
• 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
105
• Data– Review period T = 3 weeks = 3/52 = .05769 years,– Lead time = L = 8 days = 8/260 = .03077 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.
106
• Review period demand = TD = ( 3/52)(6240) = 360 juicers,
• M = TD + SS = 360 + 30 = 390 juicers,
• Q = M + LD – I = 390 + .03077(6240) - 210 = 372 juicers.
AAC operates a (T, M) policy – Solution
107
Reviewpoint
Reviewpoint
AAC operates a (T, M) policy – Solution
T
SS SS SS
Inventory position
Order 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
108
• Demand is stochastic with a known distribution.
• 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.
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
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 costsQ = 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
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)X Q
X Q
The Expected Profit Function
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
Nível de serviço ótimo
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 isdiscrete uniform between 30 and 49 newspapers.
114
SENTINEL - Solution
• Input to the optimal order quantity formula
p = 0.30c = 0.20 [0.38-0.18]s = 0.01g = 0.10K = 1.20
The probability of the optimal service level = p+ g - cp+ g - s
0.30 + 0.10 - 0.200.30 + 0.10 - 0.01
= 0.513=
115
1.0
0.500.55
30 49
0.513
39 40
P(D 39) = 0.50P(D 40) = 0.55
Q* = 40
SENTINEL – SolutionFinding the optimal order quantity Q*
116
=(B5+B8-B6)/(B5+B8-B7)
=ROUNDUP(B10+E5*(B11-10),0)
=(E6-B10+1)/(B11-B10+1)
SENTINEL – Spreadsheet Solution
117
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 distributedwith a mean of 120, and a standard deviation of 20 donuts.
118
WENDELL’S BAKERY - Solution
• Input to the optimal order quantity formulap = $0.35c = $0.15s = $0.05g = $0.25K = $15.00
The optimal service level = p+ g - cp+ g - s
0.35+ 0.25 - 0.150.35+0.25 - 0.05
= 0.8182=
119
.8182
=120 Q*
• From the relationship F(Q*) = 0.8182 we find the corresponding z value.
• From the standard normal table we have z = 0.3186.• The optimal order quantity is calculated by
Q* = + z
• For Wendell’s Q* = 120 + (0.3186)(20) 138
WENDELL’S BAKERY - SolutionFinding the optimal order quantity
120
EP(Q*) = (p - s) - (c - s)Q* - (p + g - s) ()L[(Q* - ) / - KEP(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 Wendell’s
EP(138) = (0.35 - 0.05)(120) - (0.15 - 0.05)(138) - (0.35 + 0.25 - 0.05)x(20)L[(138 - 120) / 20] - 15 = $6.10
L(0.9) = 0.1004
WENDELL’S BAKERY - SolutionCalculating the expected profit
Apêndice B
Ver slide 112
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)
WENDELL’S BAKERY -
Spreadsheet Solution
122
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.
123
WENDELL’S – The commission strategy - Solution
• The unit selling price changes to
c = 0.35 - 0.13 = $0.22
• The optimal order:F(Q*) = (0.22 + 0.25 - 0.15) / (0.22 + 0.25 - 0.05)= 0.7616.
• Z = .71
• Q* = + z = 120 + (0.71)(20) 134 donuts.
.7616
=120 Q*
124
• Will the bakery’s expected profit increase?EP(134) = (0.22 - 0.05)(20) - (0.15 - 0.05)(134) - (0.22 + 0.25 - 0.05)x(20)L[(134 - 120) / 20] = $5.80 < 6.10
• The bakery should not proceed with the alternative plan.
WENDELL’S – The commission strategy - Solution
125
• 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.
WENDELL’S – The commission strategy - Solution
126
Dimensionamento de Lotes Dimensionamento de Lotes ((Lot SizingLot Sizing))
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.
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).
129
Lot for LotEsta 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.
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.
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.
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.
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.
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.
135
Fixed Period Requirements (FPR)
Ordena-se uma quantidade suficiente para suprir a demanda de um número fixo de períodos consecutivos.
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.
137
Wagner-Whitin Algorithm (WW)
Procedimento de programação dinâmica para obter o cronograma ótimo de dimensionamento de
lotes no horizonte de planejamento.
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.
139
WinQSBWinQSB
140
WinQSB
141
Solve and Analyze
142
Wagner-Whitin
143
Silver-Meal
144
EOQ
145
POQ
146
LUC
147
LTC
148
FPR
149
PPB
150
LOT for LOT
151
Comparação entre Métodos
Método Custo % acima do ótimo
Wagner-Whitin $ 1368,00 -Silver-Meal $ 1368,00 -
EOQ $ 1521,00 11POQ $ 1458,00 7LUC $ 1458,00 7LTC $ 1458,00 7FPR $ 1472,00 8PPB $ 1438,00 5
Lot for Lot $ 1472,00 8
Ótimo
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