Inventory Models

1
Inventory Models
2
Tipos de Demanda

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

3
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)

4
Tipos de estoques

• Matérias-primas
• Produtos em processo (WIP - Work In Process)
• Produtos acabados
• Em trânsito
• Em consignação

5
Importância dos Estoques

• Melhorar o serviço ao cliente
• Economia de escala
• Proteção contra mudanças de preço em épocas de
  inflação alta
• Proteção contra incertezas na demanda e no tempo
  de entrega
• Proteção contra contingências

6
Pressões para Manter Estoque Alto

• Estoque alto maior probabilidade de atender bem
  os clientes
• Mas
• Estoque alto certeza de alto custo em carregar
  estoques

7
Fontes de Elevação de Estoque

• Marketing
• Engenharia
• Controle de Qualidade
• Manufatura
• Suprimentos
• Gerentes

8
Segmentação de Estoques

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

9
Classificação ABC

•  

10
Segmentação de Estoques

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

11
Classificação XYZ

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

12
Segmentação

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

13
Classificação 123

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

14
Inventory Classifications

• Inventory can be classified in various ways

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

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

16
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

17
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

18
Type of Costs in Inventory Models

• 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
19
Type of Costs in Inventory Models

• 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).
20
Type of Costs in Inventory Models

• 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.
21
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)
22
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.

23
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.

24
The EOQ Model Inventory profile

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

Optimal EOQ policy consists of same-size orders.
This observation results in the following
inventory profile
25
Cost Equation for the EOQ Model
Total Annual ordering Costs
Total Annual Holding Costs
Total Annual procurement Costs
Total Annual Inventory Costs



TC(Q)
(Q/2)Ch
(D/Q)Co
DC
26
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 annual holding and ordering costs



o


Note at the optimal order size total holding
costs and ordering costs are equal
Q
27
Sensitivity Analysis in EOQ models
Deviations from the optimal order size cause
only small increase in the total cost.
The curve is reasonably flat around Q.
Q
28
Number of Orders per Year

• To find the number of orders per year

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.

29
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.

30
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
31
Lead Time and the Reorder Point Graphical
demonstration Short Lead Time
RReorder Point
Inventory position
L
R Inventory at hand at the beginning of lead
time
32
Lead Time and the Reorder Point Graphical
demonstration Long Lead Time
R inventory at hand at the beginning of lead
time one outstanding order demand during
lead time LD
Inventory at hand
33
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
34
Safety stock
Planned situation
Actual situation
Reorder Point
L
R LD
35
Safety stock
Actual situation
SSSafety stock
The safety stock prevents excessiveshortages.
SS
R LD
36
Inventory Costs Including safety stock
Total Annual Inventory Costs
Total Annual Holding Costs
Total Annual ordering Costs
Total Annual procurement Costs



TC(Q)
(Q/2)Ch
(D/Q)Co
DC ChSS
Safety stockholding cost
37
ALLEN APPLIANCE COMPANY (AAC)

• AAC wholesales small appliances.
• AAC currently orders 600 units of the Citron
  brand juicer each time inventory drops to 205
  units.
• Management wishes to determine an optimal
  ordering policy for the Citron brand juicer

38
ALLEN APPLIANCE COMPANY (AAC)

• 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

39
ALLEN APPLIANCE COMPANY (AAC)

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

40
AAC SolutionEOQ and Total Variable Cost

• Current ordering policy calls for Q 600
  juicers.
• TV(600)(600/2)(1.40)(6240/600)(12) 544.8
• The EOQ policy calls for orders of size

Savings of 16
TV(327) (327 / 2)(1.40) (6240 / 327) ( 12)
457.89
41
AAC SolutionReorder 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) 457.89 6240(10) (13)(1.40)
62,876.09
TV(327) Procurement Safety stock
cost
holding cost
42
AAC SolutionSensitivity 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.
  D7500 juicers.
• The new optimal order quantity is Q 359.
• The new variable total cost TV(359) 502
• If AAC still orders Q 327, its total variable
  costs becomes

TV(327) (327/2)(1.40) (7500/327)(12)
504.13
43
AAC SolutionCycle Time

• For an order size of 327 juicers we have
• T (327/ 6240) 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.

5 working days per week
44
AAC Excel Spreadsheet
45
Service Levels and Safety Stocks
46
Determining Safety Stock Levels

• Businesses incorporate safety stock requirements
  when determining reorder points.
• A possible approach to determining safety stock
  levels is by specifying desired service level .

47
Two Types of Service Level
Service levels can be viewed in two ways.
Comum

• 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.

• 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.

48
Two Types of Service Level

• Juicer Demand and Units on Backorder

Cycle 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
49
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.

50
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

mL demanda média no lead time e sL desvio
padrão da demanda no lead time
51
The Cycle Service Level Approach

• P(DLgt R) P(Z gt (R mL)/sL) a. SinceP(Z gt
  Za) a, we have Za (R mL)/sL, which gives

R mL zasL
52
AAC - Cycle Service Level Approach

• Assume that lead time demand is normally
  distributed.
• Estimation of normal distribution parameters
• Estimation of the mean weekly demand 10 weeks
  average demand 120 juicers/week.
• Estimation of the variance of the weekly demand
  Sample variance 83.33 juicers2.

53
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 demandmL (1.6)(120) 192 s2L
  (1.6)(83.33) 133.33

54
AAC - Service Level for a given Reorder Point

• 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.

55
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 192 2.33(11.55) 219 juicers.

56
AAC Acceptable Number of Stockouts per Year

• AAC is willing to run out of stock an average of
  at most one cycle/year with an order quantity of
  327 juicers.
• What is the equivalent service level for this
  strategy?

57
AAC Acceptable Number of Stockouts per Year

• There will be an average of 6240/327
  19.08 cycles (lead times) per year.
• The likelihood of stockouts 1/19 0.0524.
• This translates into a service level of 94.76

58
The Unit Service Level Approach

• 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

L(z) partial expected value for the standard
normal between some z and infinity
59
AAC Cycle Service Level (Excel spreadsheet)
60
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
  sellers holding costs.
• Quantity discounts reflect the savings inherent
  in large orders.

61
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.

62
All Units Discount Schedule
To determine 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.
63
AAC - All Units Quantity Discounts

• AAC is offering all units quantity discounts to
  its customers.
• Data

64
Should AAC increase its regular order of 327
juicers, to take advantage of the discount?
65
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 Qi value in
  the total cost formula TC(Qi ).
• Step 4 Select the Q i that minimizes TC(Q i)

ChCi.0,14
66
AAC All units discount procedure

• Step 1 Find the optimal order quantity Qi for
  each discount level i based on the EOQ formula

67
TC(Q) (Q/2)Ch (D/Q)Co DCi ChSS
68
AAC All Units Discount Procedure

• Step 2 Modify Q i

10/unit
Q1
Q3
336
999
1 299
327
331
600
Q2
69
AAC All Units Discount Procedure

• Step 2 Modify Q i

10/unit
Q3
Q3
Q3
Q3
Q3
Q2
Q3
Q3
Q1
Q3
336
999
1 299
327
331
600
70
AAC All Units Discount Procedure

• 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
62876,09
327
300-599
9,75
331
331
61.292,13
600-999
9,50
336
600
59.803,80
1000-4999
9,40
337
1000
59.388,88
5000
9,00
345
5000
59.324,98
AAC should order 5000 juicers
71
AAC All Units Discount Excel Worksheet
72
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.

• .

73
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

74
Production Lot Size Model Understanding the
inventory profile
Demand accumulation during production run DT1
Production Lot Size Q PT1
75
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.

76
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
77
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).

78
FARAH COSMETICS COMPANY

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

79
FARAH COSMETICS COMPANY Solution
Dozens

• Input for the total variable cost function
• D 613,200/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.

80
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) (84,000/2) 1-(613,200/8,760,000
  )(0.2) 613,200/84,000)(150)
  8907.

81
FARAH COSMETICS COMPANY Solution

• 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
Current cost 8,907 savings 3,057 or 34
82
FARAH COSMETICS COMPANY Production Lot Size
Template (Excel)
83
Planned Shortage Model

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

84
Planned Shortage Model 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.

85
Planned Shortage Model the Total Variable Cost
Equation
Variáveis de controle Q Quantidade pedida, S
Quantidade em backorder quando chega o pedido

• Annual holding cost ChT1/T(Average inventory)
  
• 
  ChT1/T (Q-S)/2
• 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
86
Finding T1/ T and T2/ T
Average inventory (Q - S) / 2
Proportion of time inventory exists
T1/T
Q - S
(Q - S) / Q
Q
T1
T2
Proportion of time shortage exists T2/T
T
S
S
S / Q
Average shortage S / 2
87
Planned Shortage Model The Total Variable Cost
Equation

• Annual holding costChT1/T(Q-S)/2 Ch(Q-S)
  /Q(Q-S)/2 Ch(Q-S)2/2Q
• Annual shortage costCb(Units in short per year)
  CsT2/T(Average number of backorders)
  Cb(S)(D/Q) CsS2/2Q

88
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 gt 0
• Cb lt \/ 2CoCh / D

(Q -S)2
D Q
S2 2Q
(Co SCb)
Ch
CS
TV(Q,S)
2Q
Holding costs
Time dependent backorder costs
Time independent backorder costs
Ordering costs
89
Planned Shortage Model The Optimal Inventory
Policy
The Optimal Order Size
2DCo
(DCb)2 ChCs
Ch
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 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

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
SCANLON PLUMBING Solution

• The optimal policy

-
R (4 / 52)(780) - 20 40
94
SCANLON PLUMBING Spreadsheet Solution
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
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.

97
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.

98
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).

99
Exemplo da Citron e AAC

• AAC usa política (R,M) com R219 e M 354 ( Q
  SS 327 27)
• Cliente pede 60 juicers quando I 224 (gt 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
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.

101
Periodic Review Systems

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

102
Periodic Review Systems

• (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.

103
Periodic Review Systems

• 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
104
AAC operates a (T, M) policy

• Every three weeks AAC receives deliveries of
  different products from Citron.
• Lead time is eight days for ordering Citrons
  juicers.
• AAC is now reviewing its juicer inventory and
  finds 210 in stock.
• How many juicers should AAC order for a safety
  stock of 30 juicers?

105
AAC operates a (T, M) policy Solution

• 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 260 days a year. (5)(52) 260.
106
AAC operates a (T, M) policy Solution

• 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.

107
AAC operates a (T, M) policy Solution
Replenishment level
Order
Order
M maximum inventory
Inventory position
Inventory position
SS
SS
SS
L
L
Reviewpoint
Reviewpoint
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.
108
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.

109
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 for DemandX)
  Prob(DemandX)
• The expected profit is a function of the order
  size, the random demand, and the various costs.

110
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 costs
• Q order quantity.
• EP(Q) Expected Profit if Q units are ordered.
• Scenarios
• Demand X is less than the order quantity (X lt Q).
• Demand X is greater than or equal to the order
  quantity (X ³Q).

111
The Expected Profit Function

• 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
112
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
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.

114
SENTINEL - Solution

• Input to the optimal order quantity formula
• p 0.30
• c 0.20 0.38-0.18
• s 0.01
• g 0.10
• K 1.20

The probability of the optimal service level
115
SENTINEL SolutionFinding 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
49
39
40
116
SENTINEL Spreadsheet Solution
117
WENDELLS BAKERY

• Management in Wendells wishes to determine the
  number of donuts to prepare for sale, on weekday
  evenings
• Data
• Unit cost is 0.15.
• Unit selling price is 0.35.
• Unsold donuts are donated to charity for a tax
  credit of 0.05 per donut.
• Customer goodwill cost is 0.25.
• Operating costs are 15 per evening.

Demand is normally distributed with a mean of
120, and a standard deviation of 20 donuts.
118
WENDELLS BAKERY - Solution

• Input to the optimal order quantity formula
• p 0.35
• c 0.15
• s 0.05
• g 0.25
• K 15.00

119
WENDELLS BAKERY - SolutionFinding the optimal
order quantity

• 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 m zs
• For Wendells Q 120 (0.3186)(20) _at_ 138

120
WENDELLS BAKERY - SolutionCalculating the
expected profit

• For the normal distribution
• L (Q - m ) /s is obtained from the partial
  expected value table.
• For Wendells
• 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

Ver slide 112
EP(Q) (p - s) m - (c - s)Q - (p g - s)
(s)L(Q - m ) /s - K
L(0.9) 0.1004
Apêndice B
121
WENDELLS BAKERY - Spreadsheet Solution
122
WENDELLS The commission strategy

• When commission replaces fixed wages
• Compare the maximum expected profit of two
  strategies
• 0.13 commission paid per donut sold,
• 15 fixed wage per evening (calculated before).
• Calculate first the optimal quantity for the
  alternative policy.
• Check the expected difference in pay for the
  operator.

123
WENDELLS 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 m zs 120 (0.71)(20) 134 donuts.

124
WENDELLS The commission strategy - Solution

• Will the bakerys 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 lt 6.10
• The bakery should not proceed with the
  alternative plan.

125
WENDELLS The commission strategy - Solution

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

126
Dimensionamento de Lotes (Lot 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 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.

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.

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
WinQSB

140
WinQSB
141
Solve and Analyze
142
Wagner-Whitin (Ótimo!)
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 11
POQ 1458,00 7
LUC 1458,00 7
LTC 1458,00 7
FPR 1472,00 8
PPB 1438,00 5
Lot for Lot 1472,00 8
Ótimo