Exact and Approximation Algorithms for Supply Chain Management Problems

1
(No Transcript)
2
Overview Examples of Supply Chain Management
Problems Some Computer Science
Preliminaries Exact Algorithms Structural
Properties of Local Search Algorithms for the
TSP Node Swapping Algorithms for the
TSP Conclusions
3
Examples of Supply Chain Management
Problems Production Mix. What mixture of crude
oils (from various countries) should be used in
an oil refining process to generate prescribed
volumes of fuels at the cheapest overall cost?
Capacity Planning. Given a set of
Bills-of-Material, what is the smoothest
production schedule that meets the requirements
of the Materials Requirement Plan?
Inventory-Production Planning. When and in what
quantities should goods be produced or bought to
meet inventory requirements while minimising
overall costs? Facility Layout. Where should
production and other facilities be located and
how should they be linked in a physical plant to
minimise operational costs? Vehicle Routing.
Specify the customer delivery sequence that a
fleet of lorries should follow to minimise
operational costs? Crew Rostering. What monthly
roster of work for individual employees of an
airline company will meet the (published) route
schedule while minimising costs? Facility
Location. Where should ambulance depots be
located in a country to meet emergency response
guarantees while minimising operational costs?
4
Some Computer Science Preliminaries I A decision
problem is one that admits a TRUE or FALSE
solution. We associate a natural size parameter n
(e.g. number of constraints) with each decision
problem. A problem belongs to the class P
(polynomially bounded -trivial) if it can be
solved on a computer in O (n) iterations. A
problem belongs to the class NP (non-polynomially
bounded) if a guessed solution can be verified on
a computer in O (n) iterations. A problem
belongs to the class NP-Complete (nontrivial) if
it is in NP and a (hypothetical) computer program
that solves it can be adapted to solve all other
problems in NP. An optimisation problem is one
where we seek the values of variables that will
minimise the value of an objective function,
subject to a set of constraints. We can associate
a decision problem with any optimisation problem,
by pre-specifying a bound B and asking if there
are variables that make the value of the
objective function less than B. An optimisation
problem is NP-Hard (intractable), if the
associated decision problem is NP-Complete.
All Supply Chain Management Problems of
practical interest are NP-Hard!
5
Computer Science Preliminaries II A Linear
Program (LP) is concerned with finding the
values of real variables that Minimise (Maximise)
the values of a linear objective function,
subject to linear inequalities. We know that LP
? P and we can solve problems with millions of
variables and constraints in a trivial amount of
time on a relatively small computer. However, if
some of the variables are integer valued (e.g. we
can not divide a pilot in two) the problem is a
Mixed Integer Program (MIP), if all the variables
are integers we get an Integer Program (IP) or if
all the variables are binary we get a Binary
Variable Program (BV). In all instances (apart
from very trivial cases), the problem is
NP-Hard. The computational difficulties are all
associated with the structure of the solution
space defined by the set of constraints. This is
the convex hull of the extreme points of the
MIP. The faces of an n-dimensional convex hull
consist of facets (faces of dimension n-1) and
lower dimension faces. The convex hull is plainly
the intersection of a set of linear inequalities
(cutting planes) of high dimension.
Find x1, x2, x3 ?? To Max x0 2x1
3x2 x3 x1 2x2 x3
? 4 4x1 x2 - 2x3 ? 5

6
Exact Algorithms Two practical difficulties
arise. Typically, we only have an approximate
(relaxed) mathematical formulation of the MIP.
We can think of the convex hull of the MIP as
being made of steel and contained within that of
the relaxed formulation, which consists of a
soft material. The second difficulty is that we
can never have a complete description of the
facial structure of the convex hull of an NP-Hard
problem. However, for many supply chain
management problem, we have partial knowledge of
the facial structure of the polyhedron associated
with the MIP researchers have identified
families of facet defining inequalities that can
be used to trim off the soft material of the
relaxed formulation and are tight with that of
the MIP. The accompanying notes show how this
approach is used on a set of supply chain
management problems. The resulting software
represents the state-of-the-art algorithms for
NP-Hard problems of practical interest.

7
Heuristic Algorithms Heuristic algorithms give
us approximate (hopefully, good) solutions to
NP-Hard problems, in cases where we do not have
the time or expertise to devise exact solution
algorithms. We shall motivate the discussion
with a case study. Travelling Salesperson
Problem (TSP). Given a network where the nodes
represent cities and the arcs are roads with
weights (distances), what is the shortest
Hamiltonian tour (cycle that visits each city
exactly once). The following network represents
towns/cities in Ireland. We shall assume that the
red tour is known and we wish to find one of
shorter distance. Lins r-swap is a local
search heuristic that removes r arcs from the
existing tour and replaces them with r new arcs,
in such a way as to generate a new tour (of
shorter distance). We can gain good insights
into the geometrical structure of the algorithm
by ignoring distances and concentrating on
enumeration issues.
8
TSP Example
DER
52
64
38
70
84
DON
ENN
OMA
ARM
BLF
40
32
44
44
68
42
46
38
40
38
18
24
44
48
SLG
MON
NEW
COS
CAV
38
40
34
36
68
62
30
25
40
50
60
40
18
62
CLB
ROS
LON
NAV
DUN
66
24
62
38
80
76
60
30
58
50
86
60
54
22
58
GAL
ATH
TUL
KLD
DUB
62
30
34
40
40
82
52
26
28
30
60
54
Sample Hamiltonian Tour
ENS
PTL
CAR
ARK
100
28
46
70
30
82
70
36
26
54
48
TRA
LIM
CLN
KIL
WEX
74
60
40
52
82
72
60
34
36
38
CRK
WTR
86
9
Tour Improvement - Lins r-Opt Swap

• Save Length
• Initial Tour -- 1373
• 2-Opt IN Lim-Ens Tra-Crk 30 1343
• OUT Lim-Crk Tra-Ens
• 3-Opt IN Kld-Car Kil-Ptl Wex-Wtr 14 1329
• OUT Ptl-Kld Kil-Wtr Wex-Car
• 3-Opt IN Mon-Arm Mon-New Cav-Nav 14 1315
• OUT Mon-Nav Mon-Cav Arm-New
• 3-Opt IN New-Blf Mon-Oma Enn-Der 6 1309
• OUT Mon-New Der-Blf Enn-Oma
• 2-Opt IN Don-Der Enn-Slg 6 1303
• OUT Don-Slg Enn-Der
• 3-Opt IN Enn-Oma Enn-Mon Slg-Don 12 1291
• OUT Enn-Slg Enn-Don Oma-Mon

10

• Structural Properties of Local Search Algorithms
  for the TSP
• Given a tour T in the complete graph
• Kn a 2-swap is a tour got by removing
• 2 arcs from T and replacing them with
• 2 new arcs (still generating a tour).
• A 3-swap is got by replacing 3 arcs in T.
• 4 Nonsingleton Cases Exist
• S1 S2 S0 S2S1 S0
  S2 S1 S0 S2 S1 S0
• 1 Singleton Case Arises
• S2 S1 S0

S1
T
S0
S1
S2
S0
11
Number of r-Swaps in Kn   r Singleton
Replications Occurrences Rotational r
Singleton Replications Occurrences
Rotational cases
adjustment cases
adjustment
2 ? 1 n n-31
/2 2 6 ?
2121 n n-115 /6 6 1
0 n
-- 1 840 n
n-104 --
1,2 265
n n-93 --
1,3 339 n n-93
-- 3 ?
4 n n-52 /3 3
1,4 339 n n-93
/2 2 1 1
n n-41 --
1,2,3 80 n n-82
-- 1,2 0
n -- 1,2,4
112 n n-82
-- 1,2,5 112
n n-82 -- 4 ?
25 n n-73 /4 4
1,3,5 138 n n-82 /3
3 1 8
n n-62 -- 1,2,3,4
27 n n-71 --
1,2 1 n
n-51 -- 1,2,3,5
35 n n-71 --
1,3 3 n n-51 /2
2 1,2,4,5 41 n
n-71 /2 2 1,2,3
0 n --
1,2,3,4,5 10 n
--
1,2,3,4,5,6 3 1
-- 5 ? 208
n n-94 /5 5 1
77 n n-83 --
1,2 20 n n-72
-- 1,3 30
n n-72 -- 1,2,3
5 n n-61 --
1,2,4 9 n n-61
-- 1,2,3,4 2
n -- 1,2,3,4,5
1 1 --
12
Structure of the Neighbour Search Space for
2-Swaps  2-swaps 01 05 12 23 34 45 02 13
24 35 04 15 03 14 25  012345(0) 1 1 1 1 1
1 . . . . . . . . .  015432(0) 1 .
. 1 1 1 1 . . . . 1 . . . 021345(0)
. 1 1 . 1 1 1 1 . . . . . .
. 013245(0) 1 1 . 1 . 1 . 1 1 . . .
. . . 012435(0) 1 1 1 . 1 . . . 1 1
. . . . . 012354(0) 1 . 1 1 . 1 . .
. 1 1 . . . . 043215(0) . 1 1 1 1 .
. . . . 1 1 . . .   012543(0) 1 . 1
. 1 1 . . . . . . 1 . 1 032145(0) .
1 1 1 . 1 . . . . . . 1 1
. 014325(0) 1 1 . 1 1 . . . . . . .
. 1 1 
13
Structure of the Search SpaceConsider a tour T
and an r-swap Hr T . Hr n-r n cos (?r
)       r-swaps occupy parallel hyperplanes in
a space of dimension ½ n (n-3)      
Hyperplanes are equally spaced and perpendicular
to T r-swaps partition the space of
tours in Kn  
14
Node Swapping Algorithms for the
TSP Arc-swapping algorithms give us a
decomposition of the local search
space. Node-swapping algorithms consist of
(repeatedly) removing a node from a tour and
inserting it between two other nodes (i.e.
visiting the nodes in a different order). High
order node-swaps duplicate many lower order
swaps. However, some node-swap algorithms
enable us to search an exponential number of
possibilities in a polynomially bounded number of
iterations. Node-swap algorithms form the basis
for Genetic Algorithms. The essential idea is
that we start with two (parent) tours p1 and p2
we then swap sequences of nodes between the tours
to generate two new (offspring) tours o1 and o2.
If the new tours are shorter than the parents, we
retain them and repeat the process. In the
accompanying notes, we summarise the algorithms
Ordinal Cross, Order (OX), Partial Map (MPX) and
Cycle (CX). The algorithms are easy to code and
quickly produce reasonable tours, but the quality
of the solutions is not competitive with exact
algorithms when implemented on large, practical
data sets.
15
Research Opportunities Concerted work by a
network of research centres over the past 25
years has identified families of facet defining
inequalities for many supply chain problems of
practical interest. These form the basis of
state-of-the-art exact solution algorithms found
in commercial software packages. However, new
facet defining classes are constantly being
discovered. A related problem is how to select
the inequalities that are likely to efficiently
trim off parts of the infeasible solution space,
in a particular instance. A good deal of work
needs to be done to identify the search space of
node-swap algorithms and to create a reference
framework for them. In this presentation, we
have not ad the opportunity to examine global
optimisation algorithms (e.g. Simulated
Annealing, Tabu Search, Ising Models), but much
research work needs to be undertaken to explore
their convergence properties. Research needs to
be undertaken to extend the range of applications
from supply chain management to bioinformatics,
drug design problems, financial mathematics and
so on.
16
(No Transcript)
17
(No Transcript)