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Scheduling
1. Scheduling of inventory releasing jobs to satisfy time
varying demand: an analysis of complexity
Stathis Grigoropoulos
Nils Boysen Stefan Bock Malte Fliedner
June 21, 2013
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3. Introduction
Motivation
Just-in-Time: (JIT) supply of some predetermined products, defining
specific demand events over time.
While all demand requests have to be fulfilled, the minimization of
resulting inventory is pursued, which is represented either by total or
maximum inventory levels.
Example: Mixed-model assembly line (cars)
Cars model and numbers are known by retailers
Predetermined deterministic and time varying demand for different
parts, depended on the car model
Single Line assembly of seat feeder scheduling s.t. minimum inventory
holdings
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4. Problem Specification
Set S of supplied products
Set J of jobs
Job j ∈ J with processing time pj
cj units of product type δj ∈ S
Set R of external requests, with
dr units of product ρr ∈ S
In stock at time τr
Request r ∈ R
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5. Problem Specification
Each job and demand request is associated with a supplied product
|S| disjoint subsets
Js = {j ∈ J : δj = s} and Rs = {r ∈ R : ρr = s}
Assume the machine as unique bottleneck in production
Non delay schedule
Determine: single job sequence σ
σ(l), job at position l(1 ≤ l ≤ n).
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6. Problem Specification
Define:
l
pσ(k) ∀l = 1..n
(1)
cj ∀s ∈ S; t ≥ 0
(2)
dr ∀s ∈ S; t ≥ 0
(3)
Inventory:Is (t) = As (t) − Ds (t), t ≥ 0.
(4)
Cσ(l) =
k=1
Total Supply:As (t) =
j∈Js |Cj ≤t
Total Demand:Ds (t) =
r ∈Rs |τr ≤t
Demand equals Supply for all products
Meet all demands,Is (t) ≥ 0, ∀s ∈ S
End of planning horizon: T = maxr ∈R {τr }.
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7. Problem Specification
Objectives:
Feasibility
I max = maxs∈S;t=1..T {Is (t)}.
Overall Efficiency
T
I =
Is (t).
s∈S t=1
Objective function values and all restrictions can be checked in polynomial
time and sequence σ constitutes a short certificate, the decision versions of
both scheduling problems are in N P.
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9. Simple Example
I max (σ 1 ) = 2, released by job 3 at time 2,
I max (σ 2 ) = 4, released by job 1 at time 5.
I (σ 1 ) = 16, 2 units of product 2 are on stock for 6 and one unit of
product 2 for another 4 time units,
I (σ 2 ) = 14.
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10. Feasibility
Formulation:
1|no − wait; pj = p; cj = cs |γ
Demand requests at t,with kp ≤ t ≤ (k + 1)p with k = 1...n
¯
ds,i = maxk=1..n {kp|(i − 1)cs ≥ Ds (kp − 1)}, ∀s ∈ S, i = 1..ns .
no material shortage for product s at any time before kp
The indexes are assumed to keep their order. We can swap jobs without
affecting feasibility
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June 21, 2013
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11. Feasibility
All integer deadlines are multiples of p, =>
Unit task time scheduling problem with integer deadlines, which
can be solved by the algorithm of Frederickson (1983), O(n) (EDD rule)
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12. Minimizing the sum of inventory
The formulation:
1|no − wait; pj = p; cj = cs |
I
The contribution of each job si in position k:
si
Ik =
T
min{cs ; max{0; i · cs − Ds (t)}}
t=kp
¯
∀s ∈ S; ∀i = 1, ..., ns ; ∀k = 1, ..., n|kp ≤ ds,i .
1
multiple jobs may compete for identical completion times,
2
coordination problem arises, which can be solved by a simple
assignment problem.
3
optimality in O(n3 ), e.g. by the Hungarian method.
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15. Minimizing maximum inventory
The formulation:
1|no − wait; pj = p; cj = cs |I max
We decompose the problem into a set of feasibility problems where a
maximum inventory level ¯ has to be obeyed
I
For any given ¯ the set Γsi of feasible times:
I
Γsi = {k = 1..n|(i − 1)cs ≥ Ds (kp − 1) ∧ i · cs − Ds (kp) ≤ ¯}
I
∀s ∈ S; ∀i = 1, ..., ns ;
1
we extend the constraints of the feasibility problem,
2
realise dates rds,i , deadlines dds,i .
3
optimality in O(n3 ), e.g. by the Hungarian method.
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16. Minimizing maximum inventory
If the difference is not higher than ¯, the release date goes just before the
I
deadline of the demand event(p multiple).
Else the realise date goes to a later point. Can be solved by the algorithm
of Frederickson (1983), O(n) (EDD rule)
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17. Minimizing maximum inventory
Solve the problem for interesting values of ¯
I
UB = maxs∈S,k=1..n {k · cs − Ds (kp)}
LB = max{0, maxs∈S {cs − ns − max{dr }}}
Binary search between UB − LB values in O(n log n)
Figure: Bipartite Graph for ¯ = 3, perfect matching
I
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18. Everything is NP-Hard
Varying processing times and identical supplies:NP-Hard.
Identical processing times and varying number of delivered product
units:NP-Hard.
Product dependent processing times and identical supplies:NP-Hard.
Varying processing times and varying number of delivered product
items:NP-Hard.
Strong NP-Hardness derived by reduction from 3-Partition.
However if we bound the number of product types, we can have a solution
using DP for Identical processing times.
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19. Dynamic Programming
Similar to the analysis in the restricted case:
Total inventory minimization
Maximum inventory minimization
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20. Table overview of complexity results
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21. Conclusion
Future work of authors: Heuristic methods for NP-Hard Problems.
Realistic case, the on-line version of the problems.
Trend: Tardiness problems based on this analysis.
Queuing Theory?
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