Prsentation PowerPoint

1
A performance evaluation approach of an
inventory system with batch orders   Jie LI,
Alexandru SAVA, Xiaolan XIE INRIA/LGIPM Metz
France
2
PLAN

• Motivation
• Model of a two stages production-distribution
  system
• Performance evaluation
• Numerical results
• Conclusion

3
MOTIVATION

• Realistic production-distribution systems are
  subject to
• - production capacity constraints
• - transport constraints
• - random demand
• - batch order
• The goal of our research is to develop efficient
  methods for design of production-distribution
  systems by taking into account transportation
  constraints and optimizing both performances and
  costs.

4
A TWO-STAGE PRODUCTION-DISTRIBUTION NETWORK

• A warehouse supplied by a manufacturing plant
  with
• random customer demand
• finite production capacity
• Batch orders and batch transportation
• transport time from the plant to the warehouse

5
A TWO-STAGE PRODUCTION-DISTRIBUTION NETWORK
Customer arrival (compound Poisson) - according
to a Poisson process of rate l of orders -
random order size X
6
A TWO-STAGE PRODUCTION-DISTRIBUTION NETWORK
Inventory control policy base stock policy with
base stock level R
Inventory level
Order quantity x
R
Replenish order to the plant
Fill order to the customer
time
Customer arrival
Replenishment order process Compound Poisson
process (l, X)
7
A TWO-STAGE PRODUCTION-DISTRIBUTION NETWORK
Priority at manufacturing plant - First Come
First Served Transportation - Order by order
with constant transportation time Lt.
8
A TWO-STAGE PRODUCTION-DISTRIBUTION NETWORK

• Performance indicators
• - Order to delivery time lead time in plant
  transportation time
• Inventory cost at the warehouse
• Only two first moment of each random variable be
  used

9
PERFORMANCE MEASURE
Inventory Cost
Holding cost (No Stockout) ChEI
backlog cost (Stockout) CbEB
Inventory level
Inventory level

N Inventory on order
time
B backlogged quantities
I(R-N)
B(N-R)
?
Distribution of Inventory On Order (N)
10
PERFORMANCE EVALUATION

• Derive first the distribution of inventory on
  order N
• Derive the performance indicators

Mx/ G/1 queue
Mx/ D/8 queue
11
PLANT MODEL (1st moment)

• Mean number of orders in the plant
• ENB queue length of M/G/1 with l arrival
  rate, mTB service rate of an order, (sTB)2
  variance of order service time
• average time a batch spends at the plant
• ELB ENB / l
• average time a unit spends at the plant
• ELs ? ELB
• average number of units at the plant
• ENs ELs.EX. l

12
PLANT MODEL (2nd moment)

• Assuming mutual independence of order queue
  length and size of the order in service leads to
  
• Based on observation of numerical experiences,
  the following approximation is used

13
THE TRANSPORT SYSTEM MODEL

• Approximate the departure process of orders from
  the plant by a compound Poisson process (l, X)
• Estimate the first and second moment of total
  number of units in transportation to the warehouse

14
INVENTORY ON ORDER

• mean and variance of inventory on order N

• Assuming Ns and Nt to be mutually independent
  implies

15
PERFORMANCE MEASURE

• By definition of the base stock policy,
• R I B N
• R - base stock level I -
  inventory on hand
• B - backlogged quantities N - total
  inventory on order.
• As a result,
• I (R-N)
• B (N-R)
• where ( x ) maxx, 0.

16
COST FUNCTION

• An approximation of inventory on order by a
  log-normal distribution with mean equal to EN
  and variance equal to Var(N).

C(R) ChEI Cb EB
where z is the solution of equation
17
NUMERICAL RESULTS

• arrival rate l 1 - 1.6.
• Order size X UNIF 3, 9.
• Service time of a product EXP(0.1)
• Transportation time Lt 3
• Base stock level is set at R 50
• Inventory holding cost Ch 1
• Backlogging cost Cb 1

18
NUMERICAL RESULTS
19
CONCLUSION

• Performance evaluation of lead time and
  inventory cost
• Only the two first moments of each random
  variable be used
• customer order size, finite production capacity
  of the plant, and transport time be taken into
  account.
• Numerical examples show that the approximate
  analytical approach provides quite accurate
  performance/cost estimations.