1 inventory models 2 tipos de demanda demanda independente: são itens que dependem, em sua maioria,...

Inventory ModelsInventory 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.

• 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

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 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 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.

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

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.

• 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

• 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

• 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 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)

• 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

• 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.

• The constant environment described by the EOQ assumptions leads to the following observation:

The optimal EOQ policy consists of same-size orders.

The EOQ Model – Inventory profile

This observation results in the following inventory profile :

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

The optimal order SizeThe optimal order Size

2DCo2DCoQ* = Q* =

Cost Equation for the EOQ Model

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

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

The curve is reasonably flat around Q*.

Deviations from the optimal order size cause only small increase in the total cost.

Sensitivity Analysis in EOQ models

• 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

• 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}.

• 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

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

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

Lead Time and the Reorder Point –Graphical demonstration: Long Lead Time

• 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

LPlace the order now

Reorder Point

R = LD

Safety stockPlanned situation

Actualsituation

R = LD

Safety stock

Actualsituation

New Reorder Point

Place the order now

SS=Safety stock

The safety stockprevents excessiveshortages.

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 stockholding 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

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:

• Data– The constant demand rate seems to be a good

assumption.– Annual demand = (120/week)(52weeks) = 6240 juicers.

• 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

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

• 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

• 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

AAC – Excel Spreadsheet

=SQRT(2*$B$10*$B$14/$B$13)

=1/E11Copy to cell H12

=E10/B10Copy to cell

=$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

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 .

• 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.

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%

• 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

• 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

R = L + zL (1 –= Service level(use a Normal

DistributionTable)

L = demanda média no

lead time

L= desvio padrão da demanda no lead time

P(DL> R) = P(Z > (R – L)/L) = . SinceP(Z > Z) = , we have Z = (R – L)/L,

which gives…

P(DL>R) = Service level = P(DL<R) = 1 –

R = L + zL

• 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

• 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

• 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

• 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

• 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

• 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%

• 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

=NORMDIST(B8,B5,B6,TRUE)

AAC – Cycle Service Level (Excel spreadsheet)

=NORMINV(B7,B5,B6)

• 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.

EOQ Models with Quantity Discounts

• 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

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

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?

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

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

TC(Q) = (Q2)Ch + (DQ)Co + DCi + ChSS

– 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

Q1*Q1*

$10/unit

599331Q2

$9.75/unit

999999600

Q3*Q3*

AAC – All Units Discount Procedure

– Step 2 : Modify Q i *

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

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

Q1*Q1*

$10/unit

Q2*Q2*

999999600

Q3*Q3*

Q3*Q3* Q3

*Q3* Q3

*Q3*Q3

Q3*Q3*

– Step 3: Substitute Q I * in the total cost function

– Step 4

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

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

• 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

• 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

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

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

• 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) = (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

• 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

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%.

• 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

• 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.

• 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.

• 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

• 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).

Variáveis de controle:

Q = Quantidade pedida, S = Quantidade em backorder quando chega o pedido

Average inventory = (Q - S) / 2

Average shortage = S / 2

Proportion of time inventory exists = T1T

Proportion of time shortage exists

Finding T1/ T and T2/ T

= (Q - S) / Q

= S / Q

• 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

– 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

Holding costs

Time dependent backorder costs

Time independent backorder costs

Ordering costs

Planned Shortage Model –The Total Variable Cost Equation

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

(DCb)2

2DCo Q* = Ch + Cs

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 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

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

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

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

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.

• (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

• (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.

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).

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.

• 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

• 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.

T =Review periodL = Lead timeSS= Safety stockQ = Inventory positionD = Annual demandI = Inventory position

• Calculation of the replenishment level and order size

• 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

• 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.

• 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.

Reviewpoint

SS SS SS

Inventory position

Order Order

Replenishment level

Inventory position

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

• 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.

• 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

– 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).

• 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

– 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

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.

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=

0.500.55

P(D 39) = 0.50P(D 40) = 0.55

Q* = 40

SENTINEL – SolutionFinding the optimal order quantity Q*

=(B5+B8-B6)/(B5+B8-B7)

=ROUNDUP(B10+E5*(B11-10),0)

=(E6-B10+1)/(B11-B10+1)

SENTINEL – Spreadsheet Solution

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.

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=

=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

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

=(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

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 = 0.35 - 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.

=120 Q*

• 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.

• 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 Dimensionamento de Lotes ((Lot SizingLot 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 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.

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.

WinQSBWinQSB

WinQSB

Solve and Analyze

Wagner-Whitin

Silver-Meal

LOT for LOT

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

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