P1254325987eVcdE

1
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.
2
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

3
Make-or-Buy Decision Making

• If 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

4
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

5
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

6
Notation and Model

• N 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)

7
Notation and Model

• hj subcontracting time of order j

pj
hj
uj
subcontracted
manufatured internally
lj
uj pj hj lj pj uj
8
Notation and Model

• A schedule can be given by the split-values pj
  and hj and by a sequence f according to which the
  orders are processed by the machine
• The completion time of order f(k) sequenced in
  position k of permutation f is
• Cf(k) Cf(k-1) pf(k),
• where for completeness Cf (0)0
• The whole order f(k) becomes available to the
  customer at time Cf(k) (the subcontractor is able
  to complete the required work hf(k) by time
  Cf(k))

9
Notation and Model

• Producing an order j?N incurs the following two
  costs
• work-in-process cost at the main production
  facility fj(Cj)
• subcontracting cost ajhj, where all aj 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
10
Notation and Model

• Producing an order j?N incurs the following two
  costs
• work-in-process cost fj(Cj)
• subcontracting cost ajhj
• Functions to be minimized
• maximum work-in-process cost
• F maxfj(Cj)j?N
• total subcontracting cost
• K ?j?N ajhj

11
Notation and Model

• Functions to be minimized
• maximum work-in-process cost
• F maxfj(Cj)j?N
• total subcontracting cost
• K ?j?N ajhj
• 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

12
In This Talk

• 1pj uj-hj (F, K)
• Can be reformulated in terms of scheduling with
  controllable processing times
• Hoogeveen Woeginger (2002), O(L2(n4logL))
• 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

13
fj(t)
f1
t
14
fj(t)
f1
f2
t
15
fj(t)
f3
f1
f2
t
16
S1 consists of all break-points of all piecewise
linear functions fj(t)
fj(t)
f3
f1
f2
t
17
S1 consists of all break-points of all piecewise
linear functions fj(t)
fj(t)
f3
S2 consists of intersection points of linear
pieces
f1
f2
t
18
S1 consists of all break-points of all piecewise
linear functions fj(t)
fj(t)
f3
S2 consists of intersection points of linear
pieces
f1
f2
S3 consists of intersection points with
t
19
fj(t)
f3
f1
f2
O(L2 ) stripes can be found in O(L2log L ) time
t
20
fj(t)
f3
f1
f2
y''
Order 1
Order 2
Order 3
y'
t
21
Induces deadlines on Cj such that fj(Cj) y
22
Problem LP(y) A solution is a piece-wise linear
function of y Solving for all stripes gives the
efficient frontier
23
Submodular Systems

• For a set N1,2,,n, let 2N denote the set of
  all subsets of N
• A 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(Ø)0
• and
• x(A)?j?A xj
• for a non-empty set A?2N

24
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 XnY? D and X?Y? D
• A set-function ? D ?R is called submodular if
  the inequality
• ? (A?B)? (A?B) ?(A)?(B)
• holds for all sets A,B ? D

25
Submodular Systems

• For a submodular function ? defined on a
  distributive lattice D? 2N such that
• Ø? D, N? D and ?(Ø)0,
• the pair (D,?) is called a submodular system on
  N, while ? is called the rank function of that
  system.

26
Submodular Systems

• For a submodular system (D,?) define two
  polyhedra
• P(?) x? Rn x(A)?(A), A?D
• and
• B(?) x? Rn x?P(?), x(N)?(N)
• B(?) represents the set of all maximal vectors in
  P(?)

Submodular Polyhedron
Base Polyhedron
27
Submodular Systems

• A submodular polyhedron associated with the pair
  (2N,?) is called a polymatroid, provided that the
  rank function ? is monotone, i.e., ? satisfies
  ?(A)?(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

28
Submodular Systems 2D

• x1 ?(1)
• x2 ?(2)
• x1 x2 ?(1,2)

Polymatroid
Base Polyhedron
29
LP over Base Polyhedra
Base Polyhedron
30
Problem LP(y)
p(Nj)?(Nj, y), Submodular polyhedron
Submodular polyhedron intersected with a box
31
Submodular Polydron with Box

• For a submodular system (D,?) and a submodular
  polyhedron
• P(?) x? Rn x(A)?(A), A?D
• introduce
• P(?)lu x? Rn x?P(?),lxu
• We prove
• Theorem. Maximizing a linear function over P(?)lu
  is equivalent to maximizing a linear function
  over a base polyhedron B(?lu) with the rank
  function
• ?lu (A)minD?D ?(D)u(A\D)- l (D\A)

32
Application to Problem LP(y)

• Theorem. Problem LP(y) is equivalent to
  maximizing the same objective function over a
  base polyhedron B(?lu) with the rank function
• ?'(A,y)min1jn ?(Nj,y)u(A\Nj)- l (Nj\A)

Van Hoesel et al. (1994), O(n)
33
Algorithm

• To solve Problem 1pj uj-hj (F, K)
• Perform the pre-processing, i.e., find the
  stripes
• 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.
• 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.

34
Conclusion

• Our algorithm for Problem 1pj uj-hj (F, K)
  requires
• O(L² (n2logL) 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