submodular optimization methods for scheduling with controllable processing times natalia...

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 Making

Subcontracting 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 Making

The 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 Model

hj 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 machine

The 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 K

Single 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 times

Hoogeveen & 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

Q

R

Q

yp

pC

yRtQtf

yyy

1

1

)(

'','Fix

Induces deadlines on Cj such that fj(Cj)≤ y

y'

y''

Order 1 Order 2 Order 3

,1

where

'','

,

,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

Q

Rb

Qa

yyy

Njupl

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 Systems

For 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⊆B

Shakhlevich & 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 the

0)(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 stripes

2. 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.

Conclusion

Our algorithm for Problem 1|pj =uj-hj |(F, K) requires

O(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