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Optimal Inventory-Backorder Tradeoff in an Assemble-to-Order System with Random Leadtimes

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Title: Optimal Inventory-Backorder Tradeoff in an Assemble-to-Order System with Random Leadtimes


1
Optimal Inventory-Backorder Tradeoff in an
Assemble-to-Order System with Random Leadtimes
  • Yingdong Lu IBM T.J. Watson Research Center
  • Jing-Sheng Song University of California,
    Irvine
  • David Yao Columbia University

2
Outline
  • The Assemble-to-Order System
  • Model formulation
  • Properties of optimal solution
  • Solution techniques
  • Numerical results
  • Conclusion

3
Problem Background
  • Assemble-to-order
  • Mass customization Dell, Compaq, Ford
  • Only keep component inventory
  • Final product is assembled after an order is
    realized
  • Optimal tradeoff between inventory and service
  • Service measure
  • Average number of product backorders
  • EB Average of customer orders waiting
  • Proportion to average customer waiting time

4
The Assemble-to-Order System
  • Suppliers Components
    Products Backorders

  • (Items) (Customer demands)

L1
1
Q121
Q122
L2
l12
2
QK1
QK2
lK
QKm
Lm
m
5
The Demand Model(multivariate compound Poisson
process)
  • m different components
  • Overall demand Poisson process with rate l
  • Type-K demand requires only the components in K
  • K any subset of 1,, m
  • QKi required number of units of component i
    in K
  • qK probability a demand is of type-K
  • SqK 1
  • Aggregate demand of component i
  • Compound Poisson process with rate
  • li l Si in K qK

6
Other Modeling Assumptions
  • The leadtimes for each component are i.i.d.
    random variables
  • Li has distribution Gi
  • Base-stock policies (order-up-to policies)
  • si base-stock level for item i
  • FCFS
  • An order is backlogged if it is not yet
    completely filled.
  • Committed inventory
  • If we have some items in stock but not others
    that are requested by an order, we put aside
    those available items as committed inventory for
    that order.

7
The Optimization Problem
  • minimize EB(s1, , sm)
  • subject to c1s1cmsm lt C
  • where
  • B total number of customer backorders
  • For any demand type K, let
  • BK type-K backorders
  • number of type-K orders not yet
    completely satisfied
  • Then,
  • B SK BK

8
Solution Properties
  • Let si be the optimal base-stock level for
    component i.
  • If ci gt cj, li ELi lt lj ELj, and li lt lj ,
  • then si lt sj.
  • Example If i and j have the same cost, and are
    always requested together, then the one with
    longest leadtime has higher optimal base-stock
    level.

9
Solution Techniques
  • Surrogate the objective function by simple lower
    and upper bounds
  • Both the upper- and lower-bound problems share
    similar structures, which can be solved by
  • an exact network flow algorithm
  • (but the number of arcs grows exponentially in
    the number of components)
  • faster greedy heuristic algorithms
  • Numerical results show that the heuristic
    algorithm is effective.

10
The Supply Subsystem

Suppliers
X1
Xm
X2
QK2
Q121
Q122
QKm
QK1
l12
lK
Arrivals (Replenishment Orders)
11
The Lower Bound
  • Xi outstanding orders of component i in
    steady state,
  • has a Poisson distribution with mean
    liELi
  • Bi number of component i backorders Xi
    - si
  • BK,i number of type-K backorders that have
    component i backlogged
  • EBK,i lK EBi/li
  • BK number of type-K backorders maxi e K
    BK,i
  • EBK gt maxi e K EBK,i maxi e K lK
    EBi/li
  • EB average total number of backorders S K
    EBK
  • gt S K maxi e K lK EBi/li

?
?
12
Surrogate Problem I Lower Bound
  • Using the lower bound to approximate the
    objective function, we obtain the following
    surrogate problem
  • After a change of variables, the problem becomes

13
Solving Surrogate Problem I
  • Hierarchy structure
  • There exists a complete ordering of all the order
    types (the subsets). For any K and (i,j), such
    that i, j belong to K, but not any set lower than
    K, we have zizj
  • The hierarchy structure enables us to devise
  • an exact shortest-path algorithm (but still slow
    for large problems)
  • a much faster greedy-type heuristic

K1
K2
K3
14
Surrogate Problem II Upper Bound
  • Applying Lai-Robins inequality, we have the
    following surrogate problem

15
A Personal Computer Example
  • 6 major demand types
  • 2,5
  • 3,5
  • 1,2,5
  • 1,3,6
  • 1,3,4,5
  • 1,3,4,6
  • 6 differentiating items
  • 1. built-in zip drive
  • 2. standard hard drive
  • 3. high-profile hard drive
  • 4. DVD-Rom drive
  • 5. standard processor
  • 6. high-profile processor

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