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Submodular Optimization Methods for Scheduling with Controllable Processing Times. Natalia Shakhlevich University of Leeds, U.K. Akiyoshi Shioura Tohoku University, Sendai, Japan Vitaly Strusevich University of Greenwich, London, U.K. This Talk. - PowerPoint PPT Presentation
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Submodular Optimization Methods for Scheduling with Controllable Processing
Times
Natalia ShakhlevichUniversity of Leeds, U.K.
Akiyoshi Shioura Tohoku University, Sendai, Japan
Vitaly StrusevichUniversity of Greenwich, London, U.K.
This Talk
• Illustrates the use of methods of Submodular Optimization for a bicriteria single machine scheduling problem to minimize the maximum processing cost and the total compression cost
• The problem is interpreted as a Make-or-Buy Production Planning Problem
Make-or-Buy Decision MakingIf the decision-maker (a production manager)
realizes that • the existing production capabilities are
insufficient to fulfill all orders internally or • if the cost of work-in-process of an order is too
high, the order can be partly subcontracted
Make-or-Buy Decision MakingSubcontracting incurs additional cost that can be • either compensated by quoting realistic
deadlines for all orders • or balanced by a reduction in internal
production expenses
Make-or-Buy Decision MakingThe make-or-buy decisions should be taken to
determine • which part of each order is manufactured
internally • and which is subcontracted
Closely related to the popular time-cost trade-off project management problems
Notation and ModelN = {1,…, n} set of orders (jobs) to be processed on a single
machine (internal manufacturing)
uj processing time of order j
pj actual processing time of order j (internal manufacturing)
lj lower bound on processing time of order j (a mandatory part for internal manufacturing)
Notation and Modelhj subcontracting time of order j
uj
lj
pj hj
uj = pj + hj
lj ≤ pj ≤ uj
subcontractedmanufatured internally
Notation and ModelA schedule can be given by the split-values pj and hj and by a sequence φ according to which the orders are processed by the machineThe completion time of order φ(k) sequenced in position k of permutation φ is
Cφ(k) = Cφ(k-1) + pφ(k),where for completeness Cφ (0)=0
The whole order φ(k) becomes available to the customer at time Cφ(k) (the
subcontractor is able to complete the required work hφ(k) by time Cφ(k))
Notation and ModelProducing an order j∈N incurs the following two costs:
•work-in-process cost at the main production facility fj(Cj)
•subcontracting cost αjhj, where all αj ≥0
Measures cost for completing j∈N at time Cj
Each fj is a non-decreasing piecewise linear function of mj pieces;
L – the total number of the linear pieces
Notation and ModelProducing an order j∈N incurs the following two costs: •work-in-process cost fj(Cj)•subcontracting cost αjhj
Functions to be minimized:•maximum work-in-process cost
F = max{fj(Cj)|j∈N}•total subcontracting cost
K= ∑j N ∈ αjhj
Notation and ModelFunctions to be minimized:• maximum work-in-process cost
F = max{fj(Cj)|j∈N} • total subcontracting cost
K= ∑j N ∈ αjhj
Bicriteria Model: find a set of Pareto optimal points with respect to the functions F and KSingle Criterion Model: minimized one of the functions, provided that the other is bounded from above
In This Talk1|pj =uj-hj |(F, K) Can be reformulated in terms of scheduling with controllable
processing timesHoogeveen & Woeginger (2002), O(L2(n4+logL))We reduce the problem to a polynomial number of parametric LP
problems over a submodular polyhedron intersected with a box
We show that such an LP problem can be solved in O(n2) time by establishing a link between its region and a base polyhedron with a special rank function
t
fj(t)
f1
0
fj(t)
f1f2
0 t
t
fj(t)
f1f2
f3
0
t
fj(t)
f1f2
f3
0
S1 consists of all break-points of all piecewise linear functions fj(t)
t
fj(t)
f1f2
f3
0
S1 consists of all break-points of all piecewise linear functions fj(t)S2 consists of intersection points of linear pieces
t
fj(t)
f1f2
f3
0
S1 consists of all break-points of all piecewise linear functions fj(t)S2 consists of intersection points of linear pieces
ju jl
S3 consists of intersection points with and jj utlt
)(),(, 32
21 nOSLOSLS
t
fj(t)
f1f2
f3
0 ju jl O(L2 ) stripescan be found in O(L2log L ) time
t
fj(t)
f1f2
f3
0 ju jl
y'
y''
Order 1 Order 2 Order 3
y'
y''
Order 1 Order 2 Order 3
j
j
j
j
ii
j
iij
jjj
QR
Qyp
pC
yRtQtfyyy
1
1
)('','Fix
Induces deadlines on Cj such that fj(Cj)≤ y
y'
y''
Order 1 Order 2 Order 3
,1where
'',' ,
,s.t.
max
LP parametric a solve toneed we'',' stripe in the points optimal Pareto find To
1
j
jj
jj
jjj
jj
j
ii
Njjj
QR
bQ
a
yyyNjupl
Njbyap
p
yy
Problem LP(y);
A solution is a piece-wise linear function of y
Solving for all stripesgives the efficient frontier
Submodular Systems For a set N={1,2,…,n}, let 2N denote the set of all subsets of NA vector x=(x1, x2,…, xn) ∈ X ⊆ ℝn is called maximal in X if there is no vector z=(z1, z2,…, zn)∈X such that
x ≤ z (componentwise)For a vector x=(x1, x2,…, xn)⊆ ℝn define
x(∅)=0 and
x(A)=∑j A ∈ xj for a non-empty set A 2∈ N
Submodular Systems A collection D of subsets of N is called a distributive lattice if for any two sets in D their union and their intersection are both in D, i.e.,
X∈ D and Y∈ D implies X∩Y∈ D and X∪Y∈ D
A set-function ψ: D →ℝ is called submodular if the inequality
ψ (AB)+ψ (AB) ≤ ψ(A)+ψ(B)
holds for all sets A,B D
Submodular Systems For a submodular function ψ defined on a distributive lattice D 2⊆ N such that ∅∈ D, N∈ D and ψ(∅)=0, the pair (D,ψ) is called a submodular system on N, while ψ is called the rank function of that system.
Submodular SystemsFor a submodular system (D,ψ) define two
polyhedraP(ψ) = {x ∈ ℝn ∣x(A)≤ψ(A), A∈D}
and B(ψ) = {x ∈ ℝn ∣x∈P(ψ), x(N)=ψ(N)}
B(ψ) represents the set of all maximal vectors
in P(ψ)
SubmodularPolyhedron
BasePolyhedron
Submodular SystemsA submodular polyhedron associated with the pair (2N,ψ) is called a polymatroid, provided that the rank function ψ is monotone, i.e., ψ satisfies ψ(A)≤ψ(B) for A⊆BShakhlevich & Strusevich (JoSch, 2005; Algorithmica, 2008) developed a unified approach to scheduling problems with controllable processing times based on reduction to LP problems over (generalized) polymatroids
Submodular Systems: 2D
x1
x2
x1 ≤ ψ({1})x2 ≤ ψ({2})x1 + x2 ≤ ψ({1,2})
Polymatroid
Base Polyhedron
LP over Base Polyhedra
Njj,...,ππ
n,...,π,πππ
x
NN j
n
Njjj
,1 ,
sets defineand
...such that21
npermutatio definefunction linear aFor
0
)((2)(1)
nix
B
x
NN iii
Njjj
,...,2,1),()(
issolution optimalan )(s.t.
max
problem LPFor
1*
)(
x
Base Polyhedron
,
0,0)(
Define
Njbya
jN
jjj
Problem LP(y) '',' fixed a Take yyy
system submodular a is ),(pair the0)(such that function setFor
lattice vedistributi a form sets These
,...,1,sets nested theDefine
nest
nest
0
D
D
any
jNN j
Njupl
Njbyap
jjj
jj
j
ii
,
,1
p(Nj)≤ψ(Nj, y),Submodular polyhedron
Submodular polyhedron
intersected with a box
Submodular Polydron with BoxFor a submodular system (D,ψ) and a submodular polyhedron
P(ψ) = {x ∈ ℝn ∣x(A)≤ψ(A), A∈D}introduce
P(ψ)lu = {x ∈ ℝn ∣x∈P(ψ),l≤x≤u}
We proveTheorem. Maximizing a linear function over P(ψ)l
u is equivalent to maximizing a linear function over a base polyhedron B(ψl
u) with the rank function
ψlu (A)=minDD {ψ(D)+u(A\D)- l (D\A)}
Application to Problem LP(y)Theorem. Problem LP(y) is equivalent to maximizing the same objective function over a base polyhedron B(ψl
u) with the rank functionψ'(A,y)=min1≤j≤n {ψ(Nj,y)+u(A\Nj)- l (Nj\A)}
slopes their oforder increasing-non in the taken lines 1 of enveloplower a
finding toreduces ),(' Computing
,...,2,1),,('),(')(
)LP( ProblemFor
1*
)(
n
y
niyyyp
y
NNN
i
iii
Van Hoesel et al. (1994), O(n)
= O(n2)
AlgorithmTo solve Problem 1|pj =uj-hj |(F, K) 1. Perform the pre-processing, i.e., find the stripes2. For the lowest stripe determine the linear piece of each function fj, j
= 1,...,n, related to that stripe. For each stripe based on the linear pieces of the functions in the previous stripe find the pieces in the current stripe.
3. For each stripe solve Problem LP(y).Step 1 of takes O(L² logL) time. Step 2 takes O(n logL) time for the lowest stripe, and O(L²n) all together. In Step 3, for each stripe Problem LP(y) can be solved in O(n²) time.
ConclusionOur algorithm for Problem 1|pj =uj-hj |(F, K) requiresO(L² (n2+logL) time, factor n² less than the algorithm
by Hogeveen and Woeginger (2002)The link between LP problem over a submodular
polyhedron intersected with a box and over a base polyhedron is a useful tool to handle various scheduling problems with controllable processing times