Prezentace aplikace PowerPoint

1
ProbabilisticInventory Models
_______________________________________________
____________________________ Operations
Research ? Jan Fábry
2
Inventory Models
Probabilistic Inventory Models

• When to order?

• How much to order?

• How much to store in safety stock?

_______________________________________________
____________________________ Operations
Research ? Jan Fábry
3
Probabilistic Inventory Models
Model with Continuous Demand
_______________________________________________
____________________________ Operations
Research ? Jan Fábry
4
Inventory Models
Probabilistic Model withContinuous Demand
Assumptions

• Single item

• Probabilistic distribution of demand(stationary
  demand)

• Deterministic lead time (constant)

• Continuous (but not uniform) depletion of the
  inventory

_______________________________________________
____________________________ Operations
Research ? Jan Fábry
5
Inventory Models
Probabilistic Model withContinuous Demand
Assumptions

• Purchasing cost is independent of the OQ

• Unit holding cost is independent of the OQ

• No additional cost in case of shortage

• Replenishment - exactly on the point when the
  shipment arrives

_______________________________________________
____________________________ Operations
Research ? Jan Fábry
6
Inventory Models
Probabilistic Model withContinuous Demand
Cycle I
Cycle II
_______________________________________________
____________________________ Operations
Research ? Jan Fábry
7
Inventory Models
Probabilistic Model withContinuous Demand
Probability distribution of demand
Mean of demand µQ
Standard deviation sQ
_______________________________________________
____________________________ Operations
Research ? Jan Fábry
8
Inventory Models
Probabilistic Model withContinuous Demand
Example Brewery

• Estimation of annual demand 120 000 cases

• Standard deviation of annual demand 12 000 cases

• Annual holding cost per case 20 CZK

• Ordering cost transportation 11 000 CZK per
  order other 1 000 CZK per order

• Lead time ½ of month

• Objective minimize total annual cost

_______________________________________________
____________________________ Operations
Research ? Jan Fábry
9
Inventory Models
Probabilistic Model withContinuous Demand
Example Brewery

• Mean of annual demand

µQ  120 000 cases

• Standard deviation of annual demand

sQ 12 000 cases

• Annual holding cost

c1 20 CZK per case

• Ordering cost

c2 12 000 CZK per order

• Lead time

d 1/2 of month 1/24 of year
_______________________________________________
____________________________ Operations
Research ? Jan Fábry
10
Inventory Models
Probabilistic Model withContinuous Demand
Example Brewery

• Optimum order quantity

_______________________________________________
____________________________ Operations
Research ? Jan Fábry
11
Inventory Models
Probabilistic Model withContinuous Demand
Example Brewery

• Mean of demand within the LEAD TIME
  Optimum reorder point

• Standard deviation of demand within the LEAD TIME

_______________________________________________
____________________________ Operations
Research ? Jan Fábry
12
Inventory Models
Probabilistic Model withContinuous Demand
Example Brewery
Standard deviation sd  500
Mean µd 5 000
_______________________________________________
____________________________ Operations
Research ? Jan Fábry
13
Inventory Models
Probabilistic Model withContinuous Demand
Example Brewery
Deterministic model planned shortages
Probabilistic model random occurance
of shortages
_______________________________________________
____________________________ Operations
Research ? Jan Fábry
14
Inventory Models
Probabilistic Model withContinuous Demand
Example Brewery
Service level - definition
1. Service Level is the PROBABILITY with which
DEMAND will be MET within the inventory cycle.
2. Service Level is the PROBABILITY with which
SHORTAGE WILL NOT OCCUR within the inventory
cycle.
3. Service Level is the PERCENTAGE of TIME that
all DEMAND is MET.
_______________________________________________
____________________________ Operations
Research ? Jan Fábry
15
Inventory Models
Probabilistic Model withContinuous Demand
Example Brewery

• Implemented reorder point (for the given service
  level p)

_______________________________________________
____________________________ Operations
Research ? Jan Fábry
16
Inventory Models
Probabilistic Model withContinuous Demand
_______________________________________________
____________________________ Operations
Research ? Jan Fábry
17
Inventory Models
Probabilistic Model withContinuous Demand
_______________________________________________
____________________________ Operations
Research ? Jan Fábry
18
Inventory Models
Probabilistic Model withContinuous Demand
Example Brewery

• Mean of total cost

• Objective find such SAFETY STOCK level w that
  satisfies the given SERVICE LEVEL
  p and minimizes MEAN of TOTAL
  COST ?TC

_______________________________________________
____________________________ Operations
Research ? Jan Fábry
19
Inventory Models
Probabilistic Model withContinuous Demand
Example Brewery
Determination of optimum SAFETY STOCK level
_______________________________________________
____________________________ Operations
Research ? Jan Fábry
20
Inventory Models
Probabilistic Model withContinuous Demand
Example Brewery
Determination of optimum SAFETY STOCK level
Real LEAD-TIME DEMAND Qd
Transformation
_______________________________________________
____________________________ Operations
Research ? Jan Fábry
21
Inventory Models
Probabilistic Model withContinuous Demand
Example Brewery
Determination of optimum SAFETY STOCK level
_______________________________________________
____________________________ Operations
Research ? Jan Fábry
22
Inventory Models
Probabilistic Model withContinuous Demand
Example Brewery
Determination of optimum SAFETY STOCK level
_______________________________________________
____________________________ Operations
Research ? Jan Fábry
23
Inventory Models
Probabilistic Model withContinuous Demand
Example Brewery

• Optimum SAFETY STOCK level

p  0.95
p  0.99
_______________________________________________
____________________________ Operations
Research ? Jan Fábry
24
Inventory Models
Probabilistic Model withContinuous Demand
Example Brewery

• Optimum mean of total annual cost

p  0.95
p  0.99
_______________________________________________
____________________________ Operations
Research ? Jan Fábry
25
Probabilistic Inventory Models
Single-Period Decision Model
_______________________________________________
____________________________ Operations
Research ? Jan Fábry
26
Inventory Models
Single-Period Decision Model
Assumptions

• Only one order in time period

• Probabilistic distribution of demand (continuous
  or discrete)

• End of time period

- surplus
- stockout
_______________________________________________
____________________________ Operations
Research ? Jan Fábry
27
Inventory Models
Single-Period Decision Model
Seasonal or perishable items

• Newspapers Newsboy problem

• Bread

• Flowers

• Fruits

• Seasonal clothing

• Christmas trees

• Halloween pumpkins

_______________________________________________
____________________________ Operations
Research ? Jan Fábry
28
Inventory Models
Single-Period Decision Model
Example Happyland

• Bakery department optimize everyday order of
  rolls

• Purchase price 1 CZK per roll

• Selling price 2 CZK per roll

• Remaining rolls are changed into crumbs

20 rolls in 1 sack of crumbs
Selling price of crumbs 12 CZK per sack
_______________________________________________
____________________________ Operations
Research ? Jan Fábry
29
Inventory Models
Single-Period Decision Model
Example Happyland

• Daily demand normal probabilistic distribution

? 10 000 rolls
? 500 rolls

• Objective determine optimum order quantity

_______________________________________________
____________________________ Operations
Research ? Jan Fábry
30
Inventory Models
Single-Period Decision Model
Example Happyland

• Real daily demand for rolls Q

• Daily quantity of ordered rolls - q

Evening
_______________________________________________
____________________________ Operations
Research ? Jan Fábry
31
Inventory Models
Single-Period Decision Model
Example Happyland
Q lt q

• Marginal loss per 1 roll

ML  purchase price  salvage value
_______________________________________________
____________________________ Operations
Research ? Jan Fábry
32
Inventory Models
Single-Period Decision Model
Example Happyland
Q gt q

• Marginal profit loss per 1 roll

MPL  selling price  purchase price
Q q

• No loss

_______________________________________________
____________________________ Operations
Research ? Jan Fábry
33
Inventory Models
Single-Period Decision Model
Example Happyland
No stockout
Expected ML  p(ML)
Stockout
Expected MPL  (1-p)MPL
_______________________________________________
____________________________ Operations
Research ? Jan Fábry
34
Inventory Models
Single-Period Decision Model
Example Happyland

• Optimum expected cost

• Probability with which no stockout occurs
  (optimum service level)

_______________________________________________
____________________________ Operations
Research ? Jan Fábry
35
Inventory Models
Single-Period Decision Model
Example Happyland
Determination of optimum order quantity
_______________________________________________
____________________________ Operations
Research ? Jan Fábry
36
Inventory Models
Single-Period Decision Model
Example Happyland
Determination of optimum order quantity
Real demand Q
Transformation
_______________________________________________
____________________________ Operations
Research ? Jan Fábry
37
Inventory Models
Single-Period Decision Model
Example Happyland
Determination of optimum order quantity
_______________________________________________
____________________________ Operations
Research ? Jan Fábry
38
Inventory Models
Single-Period Decision Model
Example Happyland

• Optimum order quantity

_______________________________________________
____________________________ Operations
Research ? Jan Fábry