2008 ACADEMIC TOUR:
University of Paderborn, Germany
ADVANCED
METAHEURISTICS
Poznan University of
Technology
Prof. Jacek ZAK
Poznan University of Technology, Poland
ADVANCED METAHEURISTICS
CONTENTS
Poznan University of Technology
 INTRODUCTION TO METAHEURISTICS



BASIC NOTIONS, CONCEPTS AND FEATURES
REVIEW: LS, SA, GA, TS
SOLVING MULTIPLE OBJECTIVE OPTIMIZATION PROBLEMS
 SPECIALIZED SINGLE OBJECTIVE METAHEURISTICS




ANT COLONIES (SWARM – BASED METAHEURISTIC)
SPECIALIZED METAHEURISTICS FOR VEHICLE ROUTING PROBLEM
CASE STUDIES, COMPUTATIONAL RESULTS
COMMERCIAL SOFWARE (EVOLVER – GA; OPTQUEST – TS)
– PRESENTATION AND APPLICATION
 MULTIPLE OBJECTIVE METAHEURISTICS


PARETO SIMULATED ANNEALING (  Crew Assigmnent + Scheduling)
MULTIPLE OBJECTIVE GENETIC LOCAL SEARCH
– HYBRID GENETIC ALGORITHM
– SIGNLE OBJECTIVE GLS
– MULTIPLE OBJECTIVE GLS


PARETO MEMETIC ALGORITHM
CASE STUDIES, COMPUTATIONAL RESULTS
 CONLUSIONS
Slide 2
Vehicle Assignment Problem
ADVANCED METAHEURISTICS
INTRODUCTION TO
METAHEURISTICS
ADVANCED METAHEURISTICS
Poznan University of Technology
Motivation & Need for Metaheuristics
 Growing complexity of the real life problems;
mathematical sophistication of their discription;
 Many real life problems are NP-complete problems
(Traveling Salesman Problem, Set Covering Problem)


Computational time increases exponentially with the increase of the
size of instances;
Non-linear; non-proportional increase; NP – non polynomial
computational time
 Real need for efficient methods/ algorithms that would
be able to solve NP-complete problems

The algorithm is efficient if the cost (measured by the time of its
development and the size of used memory) of its application does
not grow too fast with the growing size of the problem.
Slide 4
ADVANCED METAHEURISTICS
Poznan University of Technology
Computational Complexity
 Computational complexity – theory of „how difficult” is to answer
a decision problem DP, where a DP is a question that has either a
„yes” or „no” answer
 The difficulty is measured by the number of operations an
algorithm needs to perform to find the correct answer to the DP
in the worst case
 A decision problem belongs to the class P of problems if there
exists a deterministic algorithm that answers the decision
problem and needs O(p(n)) operations; p is a polynomial in n; n
is the size of the instance
Slide 5
ADVANCED METAHEURISTICS
Poznan University of Technology
Computational Complexity
 A decision problem belongs to the class NP if there is a
nondeterministic polynomial time algorithm that solves the
decision problem
 For optimization problems it is possible to check
whether x belongs to X and f(x) < b; b is a constant in
polynomial time
 A decision problem DP is NP – complete if DP belongs to NP
and DP’ transforms into DP in a polynomial time for all DP’ that
belong to NP.
 NP-Complete is a subset of NP
Slide 6
ADVANCED METAHEURISTICS
COMPLETE PROBLEMS (NP – C)
Poznan University of Technology
Formal definiton of NP-C problem
A decision problem DP is NP- complete if:
1.
DP is in NP
2.
Each problem in NP is reducable/ transformable
to DP (in a polynomial time)
Slide 7
ADVANCED METAHEURISTICS
THE VIENNE DIAGRAM
Poznan University of Technology
 The Vienne Diagram of complexity clases shows
that P is not equal NP.
 It shows also existance of problems outside P
and NP-C.
NP
NP-C
P
Slide 8
ADVANCED METAHEURISTICS
COMPLETE PROBLEMS (NP – C)
Poznan University of Technology
Well konwn NP-complete problems
NP-C
Hamiltonian path problem
Traveling salesman problem
Knapsack problem
Vertex cover problem
Graph coloring problem
Boolean satisfiability problem
Slide 9
ADVANCED METAHEURISTICS
COMPLETE PROBLEMS (NP – C)
Poznan University of Technology
How to solve NP-Complete Problems?
Approximation
Randomization
Parametrisation
Restricion (in the sensce of restriction for
input)
Heuristics
Slide 10
ADVANCED METAHEURISTICS
COMPLETE PROBLEMS (NP – C)
Poznan University of Technology
Approximation
• Instead of searching for an optimal solution,
search for an "almost" optimal one.
• Many approximation algorithms emerge from
the linear programming relaxation of the
integer program
• It applies only to optimization problems and
not to "pure" decision problems like
satisfiability (although it's often possible to
conceive optimization versions of such
problems, such as the maximum satisfiability
problem).
Slide 11
ADVANCED METAHEURISTICS
COMPLETE PROBLEMS (NP – C)
Poznan University of Technology
Randomization
Randomized algorithm = probabilistic algorithm
• The algorithm typically uses the random bits as an
auxiliary input to guide its behavior, in the hope of
achieving good performance in the "average case"
• Use randomness to get a faster average running time, and
allow the algorithm to fail with some small probability.
Slide 12
ADVANCED METAHEURISTICS
COMPLETE PROBLEMS (NP – C)
Poznan University of Technology
Restricion (in the sensce of restriction for input)
By restricting the structure of the input (e.g. to graphs),
faster algorithms are usually possible.
Parametrisation
• The theory of parameterized complexity was developed in
the 1990s by Rod Downey and Michael Fellows
• Often there are fast algorithms if certain parameters of the
input are fixed
Slide 13
ADVANCED METAHEURISTICS
HEURISTICS
Poznan University of Technology
 Heuresis (gr.) = dicovering – the way of organizing the
learning process based on self-dependent search;
discovering the truth and solving the problem; Heuristic –
the way of learning without a well organized hypothesis;
„Trial-by error”
 Heurisko; Heuriskein (gr.) = find, discover, finding (they find)
– the art of discussing focused on discovering the truth, new
facts and relationships
 Heuristic – practical, experience – based, „intelligent” rule of
conduct and behavior
 Heuristic (algorithmic meaning) – „not fully valuable”
procedure that allows to find a „sufficiently good”,
approximate solution in the acceptable / reasonable time

Slide 14
Resigning from obtaining an optimal solution; trade – offs analysis;
searching for a satisfactory, high quality solution
ADVANCED METAHEURISTICS
HEURISTICS
Poznan University of Technology
 Heuristic algorithms should to be efficient to
generate „reasonalbe” solutions in a „resonable
time”
 Heuristics are typically used to solve complex
(large, nonlinear, nonconvex - containing many
local
minima)
multivariate
combinatorial
optimization problems that are difficult to solve to
optimality.
 Heuristics are good at dealing with local optima
without getting stuck in them while searching for
the global optimum.
Slide 15
ADVANCED METAHEURISTICS
METAEURISTICS
Poznan University of Technology
Greek: meta = megas = large, great, huge, universal
METAHEURISTICS = mega algorithms =
universal algorithms that help us to solve
independently in the approximate way a certain
decision problem
 METAHEURISTICS – Heuristic procedures; Provide general
schemes for solving similar categories of problems; need
customization
Slide 16
ADVANCED METAHEURISTICS
METAEURISTICS – general idea
Poznan University of Technology
 The goal of optimization is to find a discrete solution (vector of
bits, array or another structure)
 The solution optimizes (maximizes or minimizes) a function
created by the user (goal function)
 Solutions are called states and the whole set of all states
(candidate solutions) is called search space
 The nature of search space and states are different for particular
problems
 Metaheuristics are very often based on probabilistic procedures
Slide 17
ADVANCED METAHEURISTICS
METAEURISTICS - inspirations
Poznan University of Technology
Genetics
Metalurgy
Behaviour of
animals
Slide 18
ADVANCED METAHEURISTICS
TYPICAL IDEAS OF METAHEURISTICS
Poznan University of Technology
Metaheuristics are based often on
probabilistic procedures
Neihgbourhood relation
User specifies time budget (number
of iterations or time bounds)
Slide 19
ADVANCED METAHEURISTICS
METAEURISTICS - CLASSIFICATION
Poznan University of Technology
METAHEURISTICS
CLASSIC
HEURISTICS
LOCAL
SEARCH
BASED
LS
SA
TS
Slide 20
POPULATION
BASED
GA
ADVANCED METAHEURISTICS
METAEURISTICS - history
Poznan University of Technology
1965: I. Rechenberg - Evolution strategies
1975: J. Holland – Genetic Algorithms
1983: W.K. Hastings, S. Kirkpatrick, C.D.Gelatt
and M.P. Vecchi – Simulated Annealing
1986: F. Glover – Tabu Search (first mentioned the term
meta-heuristic)
1991: M. Dorigo –Ant Colonies Algorithms
Slide 21
ADVANCED METAHEURISTICS
METAEURISTICS - examples
Poznan University of Technology
Local search
Hill-climbing
Genetic algorithm
Tabu search
Memetic algorithm
Slide 22
ADVANCED METAHEURISTICS
Poznan University of Technology
REVIEW OF METAHEURISTICS
Slide 23
ADVANCED METAHEURISTICS
LOCAL SEARCH
Poznan University of Technology
LOCAL SEARCH
• One of the simplest and most popular
metaheuristics; often used as a basic algorithm
(component) of more advanced procedures
• Metaheuristic usually applied for solving hard
optimization problems
• General idea is moving from one solution to
another in the space of candidate solutions,
only in the neighbourhood of a current solution
Slide 24
ADVANCED METAHEURISTICS
LOCAL SEARCH
Poznan University of Technology
LOCAL SEARCH
• Major features of LS
 Iterative modification of the current solution a
 Defining the rule for generating the neighborhood V(a) of
the current solution a, which is a set of solutions similar to a
(LS is based on the neighbourhood relation)
 In each iteration one solution b from the neighborhood of
the current solution a is selected – usually b gives a better
value of the objective function)
 Solution b becomes a new current solution and a new
neighborhood V(b) is generated
 The cycle is repeated until the stop condition is reached
Slide 25
ADVANCED METAHEURISTICS
LOCAL SEARCH
Poznan University of Technology
LOCAL SEARCH
• Termination conditions:
– When the best solution is found
– Predefined time bound or number of steps
– Impossibility of improving the solution for a given
number of steps
• The family of LS is wide, e.g. „Hill climbing” algorithm
• Major versions
• Greedy – finishes search when any solution giving the
improvement of the objective function is found; next
the neighborhood of a new solution is analyzed
• Steepest descent – systematically reviews the
neighborhood and selects the solution that gives the
largest inprovement of the objective function
Slide 26
ADVANCED METAHEURISTICS
LOCAL SEARCH
Poznan University of Technology
PSEUDOCODE
N:=number of repetitions
~
s:=0;
for i=1 to N do
s:=initial solution;
while there is a neigbor of s with better quality do
s:=one arbitrary neighbor of s with better quality;
end while
~
if s is better than s then
~
s:=s;
end if
end for
~
return s;
Slide 27
ADVANCED METAHEURISTICS
LOCAL SEARCH
Poznan University of Technology
HILL CLIMBING
„Like climbing Everest in thick fog with amnesia”
At each step, move to a neighbor of higher value in hopes of getting to
an optimal solution (highest possible value)
Can easily modify this for problems where optimal means least
possible value
Slide 28
ADVANCED METAHEURISTICS
TABU SEARCH
Poznan University of Technology
TABU SEARCH
• Local (neighborhood) search based metaheuristic;
proved to be efficient and flexible optimization technique;
Some of the first TS algorithms did not yield impressive
results, but subsequent implementations were much more
successful (20 years of experience)
• The idea is using memory structures to remember
potential solutions to avoid cycling
 To improve efficiency of the exploration process one
needs to keep track not only of local information
(current value of the objective function) but also
information on the exploration process
• Inspiration - sociology
Slide 29
ADVANCED METAHEURISTICS
TABU SEARCH
Poznan University of Technology
INSPIRATION – TABU = TABOO
• TABU is a strong social prohibition against words, objects
or actions, that are considered undesirable or offensive by a
group, society or community
• Breaking TABU is usually considered objectionable
• Word TABU comes from Fijan and means „forbidden” or
„not allowed”
• Examples of TABU:
Gestures; subjects
Drags
Religion
Slide 30
ADVANCED METAHEURISTICS
TABU SEARCH
Poznan University of Technology
 In Tabu Search, sequences of solutions are examined and the
next move is made to the best neighbor of the current
solution a ; non – improving moves are acceptable (escaping
from local minima)
 To avoid cycling, solutions that were recently examined are
forbidden, or tabu, for a number of iterations; the use of
memory is helpful to forbid moves which might lead to
currently visited solutions
 The structure of the neighborhood V(a) depends upon the
itinerary and hence iteration k --- V (a, k)
 To alleviate time and memory requirements, it is customary to
record an attribute of tabu solutions rather than the solutions
themselves.
Slide 31
ADVANCED METAHEURISTICS
TABU SEARCH
Poznan University of Technology
TABU list:
X2
X3
X2
X1
X4
Slide 32
X3
ADVANCED METAHEURISTICS
TABU SEARCH
Poznan University of Technology
• Instead of recording solutions (impractical) – Tabu list of T solutions
we keep track of the last T moves
• For efficiency purposes several list Tr can be used at a time;
constituents are given a tabu status
• Relaxation of the tabu status – aspiration level conditions
• Short and Long Term Memory – changing the goal function;
intensification and diversification
• Intensification – giving high priority to the solutions which have
common features with the current solution
• Diversification – spreading the exploration over different regions of
the solution space
Slide 33
ADVANCED METAHEURISTICS
TABU SEARCH
Poznan University of Technology
PSEUDOCODE
Slide 34
ADVANCED METAHEURISTICS
SIMULATED ANNEALING
Poznan University of Technology
SIMULATED ANNEALING
• Statistical Mechanics:
 The behavior of systems with many degrees of freedom in
thermal equilibrium at a finite temperature.
• Combinatorial Optimization:

Finding the minimum of a given function depending on
many variables.
• Analogy:

If a liquid material cools and anneals too quickly, then the
material will solidify into a sub-optimal configuration. If the
liquid material cools slowly, the crystals within the material
will solidify optimally into a state of minimum energy (i.e.
ground state).
 This ground state corresponds to the minimum of the cost
function in an optimization problem.
Slide 35
ADVANCED METAHEURISTICS
SIMULATED ANNEALING
Poznan University of Technology
Fast cooling scheme
Slow cooling scheme
Example illustrating the effect of cooling scheme on the structure
of the material (cristalic structure of the metal)
Slide 36
ADVANCED METAHEURISTICS
SIMULATED ANNEALING
Poznan University of Technology
TERMINOLOGY
 X (or R or G) = Design Vector (i.e. Design,
Architecture, Configuration)
 E = System Energy (i.e. Objective Function
Value)
 T = System Temperature
 D = Difference in System Energy Between Two
Design Vectors
Slide 37
ADVANCED METAHEURISTICS
SIMULATED ANNEALING
Poznan University of Technology
THE SIMULATED ANNEALING ALGORITHM
1. Choose a random Xi, select the initial system temperature,
and specify the cooling (i.e. annealing) scheme
2. Evaluate E(Xi) using a simulation model
3. Perturb Xi to obtain a neighboring Design Vector (Xi+1)
4. Evaluate E(Xi+1) using a simulation model
5. If E(Xi+1)< E(Xi), Xi+1 is the new current solution
6. If E(Xi+1)> E(Xi), then accept Xi+1 as the new current solution
with a probability e(-D/T) where D = E(Xi+1) -E(Xi).
7. Reduce the system temperature according to the cooling
scheme
8. Terminate the algorithm.
Slide 38
ADVANCED METAHEURISTICS
SIMULATED ANNEALING
Poznan University of Technology
Scheme of SA
Slide 39
ADVANCED METAHEURISTICS
SIMULATED ANNEALING
Poznan University of Technology
value
value
Comparision of SA and LS
Local
Global
minimum
minimum
solution
Local and global extremes
in SA
Local
Global
minimum
minimum
solution
Local and global extremes
in LS
In SA algorithm not only the best solutions are evaluated,
so the algorithm may escape from local minimum region
Slide 40
ADVANCED METAHEURISTICS
SIMULATED ANNEALING
Poznan University of Technology
Pseudocode
Slide 41
ADVANCED METAHEURISTICS
GENETIC ALGORITHM
Poznan University of Technology
GENETIC ALGORITHMS
• PARTICULAR CLASS OF EVOLUTION - BASED
ALGORITHMS
• ALGORITHM INSPIARED BY EVOLUTIONERY
BIOLOGY
• TYPICAL FOR GA ARE:
• Crossover
• Selection
• Population
• Chromosome
• Goal Function called Fitness Function
Slide 42
ADVANCED METAHEURISTICS
GENETIC ALGORITHM
Poznan University of Technology
INSPIRATION IN EVOLUTION
POPULAION
POPULAION IN
ENVIRONMENT
INDIVIDUALS
STONES
Slide 43
PREDATORS
SURVIVING
POPULAION
AFTER SOME
TIME:
REPRODUCTION
ADVANCED METAHEURISTICS
GENETIC ALGORITHM
Poznan University of Technology
Initialization
SCHEME OF THE GENETIC ALGORITHMS
New population
Evolution
YES
Continue?
NO
Final polulation
Slide 44
Reproduction
ADVANCED METAHEURISTICS
GENETIC ALGORITHM
Poznan University of Technology
TYPICAL GENETIC ALGORITHM
Genetic algorithm
Genetic representation
of the solution domain
Fitness function
A typical representation of the solution is a vector or an
array of bits (but also of integers).
The fitness function measures the quality of the
solution and depends always on the problem.
Slide 45
ADVANCED METAHEURISTICS
GENETIC ALGORITHM
Poznan University of Technology
REPRESENTATION
• The representation of solution is called
chromosome
• Chromosome can be a vector or array of
bools, another data type or a tree data
structure
• Represetation has huge influence on efficiency
of algorithm
Slide 46
ADVANCED METAHEURISTICS
GENETIC ALGORITHM
Poznan University of Technology
REPRESENTATION
Chromosome as a
vector of bits
Chromosome as
an array of bits
Crossing-over
Slide 47
ADVANCED METAHEURISTICS
GENETIC ALGORITHM
Poznan University of Technology
FITNESS FUNCTION
• Every chromosome is ranked by fitness function
• Best chromosomes are allowed to crossover and
produce a new generation
• Fitness function should be very fast because of many
iterations of the algorithm
• The main problem is to create a proper fitness
function
Slide 48
ADVANCED METAHEURISTICS
GENETIC ALGORITHM
Poznan University of Technology
FITNESS FUNCTION
• Every chromosome is ranked by fitness function
• Best chromosomes are allowed to crossover and
produce a new generation
• Fitness function should be easily computed because
of many iterations of the algorithm
• The main problem is to create a proper fitness
function
Slide 49
ADVANCED METAHEURISTICS
GENETIC ALGORITHM
Poznan University of Technology
GENETIC PROCES
Slide 50
ADVANCED METAHEURISTICS
GENETIC ALGORITHM
Poznan University of Technology
PROCEDURE
t := 0;
Compute initial population B0;
WHILE stopping condition not fulfilled DO
BEGIN
select individuals for reproduction;
create offsprings by crossing individuals;
eventually mutate some individuals;
compute new generation
END
Slide 51
ADVANCED METAHEURISTICS
GENETIC ALGORITHM
Poznan University of Technology
APPLICATION
• VEHICLE ROUTING
• TRENING NEURAL NETWORK
• CONTEINER LOADING OPTIMIZATION
• AUTOMATIC DESIGN OF ELECTRICAL CIRCUITS
Slide 52
SOLVING MULTIPLE
OBJECTIVE
OPTIMIZATION
PROBLEMS –
INTRODUCTION TO
MULTIPLE OBJECTIVE
METAHEURISTICS
ADVANCED METAHEURISTICS
Poznan University of Technology
 MULTIPLE CRITERIA DECISION MAKING / AIDING
 MULTIPLE CRITERIA ANALYSIS (FRENCH)
 MULTIPLE CRITERIA DECISION MAKING (AMERICAN)
 MCDA IS A DYNAMICALLY DEVELOPING FIELD WHICH AIMS
AT GIVING THE DM SOME TOOLS IN ORDER TO ENABLE HIM/
HER TO SOLVE A COMPLEX DECISION PROBLEM WHERE
SEVERAL (CONTRADICTORY) POINTS OF VIEW MUST BE
TAKEN INTO ACCOUNT
 IN CONTRAST TO CLASSICAL OR TECHNIQUES MCDA/M
METHODS DO NOT YIELD “OBJECIVELY BEST SOLUTIONS”
BECAUSE IT IS IMPOSSIBLE TO GENERATE SUCH
SOLUTIONS WHICH ARE THE BEST SIMULTANEOUSLY, FROM
ALL POINTS OF VIEW
 MCDA/M CONCENTRATES ON SUGGESTING “COMPROMISE
SOLUTIONS” WHICH TAKE INTO ACCOUNT THE TRADE-OFFS
BETWEEN CRITERIA &THE DM’S PREFERENCES
Slide 54
ADVANCED METAHEURISTICS
CHARACTERISTICS OF MCD PROBLEMS
Poznan University of Technology
WHAT IS A MULTIPLE CRITERIA DECISION PROBLEM ?
 MULTIPLE CRITERIA DECISION PROBLEM IS A SITUATION IN
WICH, HAVING DEFINED A SET A OF ACTIONS AND A
CONSISTENT FAMILY OF CRITERIA F ONE WHISHES TO:

DETERMINE A SUBSET OF ACTIONS CONSIDERED TO BE
THE BEST WITH RESPECT TO F (CHOICE PROBLEM)

DIVIDE A INTO SUBSETS ACCORDING TO SOME NORMS
(SORTING PROBLEM)

RANK THE ACTIONS OF A FROM BEST TO WORST
(RANKING PROBLEM)
Slide 55
ADVANCED METAHEURISTICS
CHARACTERISTICS OF MCD PROBLEMS
Poznan University of Technology
 MULTIPLE OBJECTIVE MATHEMATICAL PROGRAM (MOMP) IS
A PROBLEM WHICH AIMS TO FIND A VECTOR x  RP
SATISFYING CONSTRAINTS OF THE TYPE
hi (x)  0, i = 1, 2, …, m
OBEYING EVENTUAL INTEGRALITY CONDITIONS AND
MAXIMIZING FUNCTIONS
MAX gj(x), j = 1, 2, …, n
 A MOMP IS THUS A MULTIPLE CRITERIA DECISION PROBLEM
IN WHICH:



Slide 56
A = { xi : (x) < 0, …i } …Rp
F = { g1 (x), …, gn (x)} IS A FAMILY OF TRUE CRITERIA
ONE AIMS TO FIND A BEST ACTION (CHOICE PROBLEM)
ADVANCED METAHEURISTICS
CHARACTERISTICS OF MCD PROBLEMS
Poznan University of Technology
 MULTIPLE CRITERIA DECISION PROBLEM IS DEFINED BY:


A SET A OF ACTIONS
A CONSISTENT FAMILY OF CRITERIA F
 A SET A IS A IS A COLLECTION OF OBJECTS, CANDIDADTES,
VARIANTS, DECISIONS THAT ARE TO BE ANALYZED AND EVALUTED
DURING THE DECISION PROCESS; A CAN BE DEFINED:


DIRECTLY – BY DENOMINATING ALL ITS ELEMENTS (FINITE SET,
RELATIVELY SMALL)
INDIRECTLY – BY DEFINING CERTAIN FEATURES OF ITS COMPONENTS
AND / OR CONSTRAINTS (INFINITE SET, FINITE SET BUT RELATIVELY
LARGE)
 A SET A CAN BE:


Slide 57
CONSTANT , A’ PRIORI DEFINED; NOT CHANGING DURING THE
DECISION PROCESS
EVOLVING, BEING MODIFIED IN THE DECISION PROCESS
ADVANCED METAHEURISTICS
CHARACTERISTICS OF MCD PROBLEMS
Poznan University of Technology
 A CONSISTENT FAMILY OF CRITERIA F IS A SET OF FUNCTIONS g –
CRITERIA THAT TOGETHER SHOULD GUARANTEE:



COMPREHENSIVE AND COMPLETE EVALUATION OF VARIANTS
(CONSIDERATION OF ALL ASPECTS OF THE DECISION PROBLEM)
CONSISTENCY OF THE EVALUATION (EACH CRITERION SHOULD
CORRESPOND TO THE DM’S GLOBAL PREFERENCES)
NON-REDUNDANCY OF CRITERIA (REPETITIONS SHOULD BE
ELIMINATED; MEANINGS AND SCOPES OF CRITERIA MUST BE CLEARLY
DEFINED)
 EACH CRITERION IN F IS A FUNCTION g – DEFINED ON A AND
REPRESENTING THE DM’S PREFERENCES TOWARDS A SPECIFIC
ASPECT (DIMENSION) OF THE DECISION PROBLEM. CATEGORIES
OF CRITERIA:



Slide 58
TRUE CRITERION („TRADITIONAL MODEL”)
SEMICRITERION („THRESHOLD MODEL”)
PSEUDOCRITERION („DOUBLE THRESHOLD MODEL ”)
ADVANCED METAHEURISTICS
CHARACTERISTICS OF MCD PROBLEMS
Poznan University of Technology
 DIFFICULTY OF MULTICRITERIA PROBLEMS

ILL – DEFINED MATHEMATICAL PROBLEMS – SEARCHING FOR A
SOLUTION x THAT MAXIMIZES MULTIPLE OBJECTIVE FUNCTION
F ( x)  max g1 ( x), g2 ( x),..., g J ( x)
subject to:
x A

THE CONCEPT OF A GLOBAL OPTIMAL SOLUTION DOES NOT MAKE ANY
SENSE IN A MULTICRITERIA CONTEXT; THERE IS NO SOLUTION THAT
WOULD BE THE BEST FROM ALL POINTS OF VIEW SIMULTANOUESLY;
INSTEAD THE NOTION OF A NON-DOMINATED OR EFFICIENT SOLUTION
IS INTRODUCED

SOLVING A MULTICRITERIA DECISION PROBLEM IS HELPING THE DM TO
MASTER THE DATA INVOLVED IN THE PROBLEM AND ADVANCE TOWARD
A “COMPROMISE SOLUTION”
Slide 59
ADVANCED METAHEURISTICS
BASIC DEFINITIONS
Poznan University of Technology
DOMINANCE RELATION - GIVEN TWO ELEMENTS a AND b OF A, a
DOMINANTES b (a D b) IFF
gj(a) ≥ gj (b) ; j = 1,2,…,n
WHERE AT LEAST ONE OF THE INEQUALITIES IS STRICT
EFFICIENT (PARETO – OPTIMAL) ACTION - ACTION a IS EFFICIENT IFF
NO ACTION OF A DOMINATES IT

Vilfredo Pareto (1906) – concept – cornerstone of traditional economic
theory;
A STATE OF THE WORLD A IS PREFERABLE TO A STATE OF THE WORLD B
IF AT LEAST ONE PERSON IS BETTER OFF IN A AND NOBODY IS WORSE
OFF
• EFFICIENT SET = PARETO OPTIMAL SET = SET OF
NONDOMINATED SOLUTIONS = NONINFERIOR SET
• FOR ALL NONDOMINATED SOLUTIONS THE
IMPROVEMENT ON ONE CRITERION IS COMPENSATED BY
DETERIORATION ON ANOTHER
Slide 60
ADVANCED METAHEURISTICS
BASIC DEFINITIONS
Poznan University of Technology
PARETO SET =
EFFICEINT
SOLUTIONS
CRITERIA
SOLUTI
ONS
I MAX
II MIN
III MAX
IV MAX
1
15
4,3
200
4
2
10
5,3
188
3
3
12
3,2
205
5
4
15
3,2
213
4
5
20
3,5
203
6
x1
N
X
0
Slide 61
x
x2
FIND
DOMINATED &
NONDOMINATED
ADVANCED METAHEURISTICS
BASIC DEFINITIONS
Poznan University of Technology
 If x belongs to X (set of feasible solutions) then x is
nondominated in X if there exists no other x1 in X such that x1 >
x and x1 ; x are different
 The main property of a set of nondominated solutions N is that
for every dominated solution (feasible solution not in N) we
can find a solution in N at which no vector components are
smaller and at least one is larger
 x in X is dominated by all points in N, indicating that the levels
of both components can be increased simultaneously; only for
points in N does this subregion of improvement extend
beyond the boundaries of X into the infeasible region
Slide 62
ADVANCED METAHEURISTICS
BASIC DEFINITIONS
Poznan University of Technology
 THE IMAGE OF A IN THE CRITERIA SPACE IS THE SET Za OF
POINTS IN Rn ONE OBTAINS WHEN EACH ACTION a IS
REPRESETED BY THE POINT WHOSE COORDINATES ARE:
{g1(a),...,gn(a)}
{g1(a), …,gn(a)}
a
c
b
Set of actions; decision space
Za
Zc
Zb
Set of evaluations; criteria space
 IN MULTIPLE OBJECTIVE DECISION PROBLEMS THE
CRITERIA SPACE IS VERY IMPORTANT FOR MAKING GOOD
CHOICES AND SELECTING APPROPRIATE – MOST RATIONAL
SOLUTIONS
Slide 63
ADVANCED METAHEURISTICS
BASIC DEFINITIONS
Poznan University of Technology
 PAY OFF MATRIX IS THE MATRIX G(nxn) DEFINED BY
Gkl = gk(âl) , k,l = 1,2,…,n
•
•
k
l
CRITERION 1
IT IS THUS THE MATRIX CONTAINING, FOR EACH ACTION âl, ITS
EVALUATIONS ACCORDING TO ALL THE CRITERIA
IN PARTICULAR Gll = Zl*
G = Z*
ll
SOLUTION 1
SOLUTION 2
l
SOLUTION 3
SOLUTION n
G11 = 250
G12 = 150
G13 = 125
G1n = 175
G21 = 0.60
G22 = 0.95
G23 = 0.80
G2n = 0.75
G31 = 67
G32 = 44
G33 = 29
G3n = 58
Gn1 = 0.12
Gn2 = 0.09
Gn3= 0.05
Gnn = 0.16
( Max)
CRITERION 2
(Max)
CRITERION 3
(Min)
CRITERION n
(Max)
Slide 64
ADVANCED METAHEURISTICS
BASIC DEFINITIONS
Poznan University of Technology
 IDEAL POINT IN Rn IS THE POINT WHOSE COORDINATES ARE
(Z1*,…, Zn*), WHERE
Zj* = Max gj(a) ; j = 1,2,…,n
A
ACTION âj IS BEST ACCORDING TO CRITERION j
gj(âj ) = Zj*
►
THE NADIR IS THE POINT WHOSE COORDINATES ARE
…, Zn) WHERE:
Zj = min Gjl ,
l
Slide 65
j=1,2,…,n
(Z1,
ADVANCED METAHEURISTICS
BASIC DEFINITIONS
Poznan University of Technology
x1
IDEAL POINT
x1max
A
x1min
THE NADIR
x2min
Slide 66
x2max x
2
ADVANCED METAHEURISTICS
SOLVING MOPs
Poznan University of Technology
COMPUTATIONAL PROCEDURE
GENERATING A GOOD
APPROXIMATION OF
THE PARETO SET
STEP
1
EXACT
APPROACHES
LARGE
SET
STEP 2
HEURISTIC
APPROACHES
METAHEURISTICS
REVIEW & EVALUATION OF
THE GENERATED SOLUTIONS
 PREFERENCES
 SEARCH PROCEDURE INTERACTIVE
METHODS
 TRADE- OFFS
ANALYSIS
STEP 3
Slide 67
COMPROMISE SOLUTION
ADVANCED METAHEURISTICS
APROXIMATION OF THE PARETO SET
Poznan University of Technology
 SOLVING MOPs IS UNDERSTOOD AS FINDING PARETO SETS =
SETS OF EFFICIENT/NONDOMINATED SOLUTIONS
 FOR A MAJORITY OF MOPs IT IS NOT EASY TO OBTAIN AN
EXACT DESCRIPTION OF THE PARETO SET


LARGE (INFINITE) NUMBER OF POINTS
POSSIBLE SITUATIONS
– computationally challenging & expensive  abandoned
– impossible – numerical complexity of mop
 EXACT SOLUTION SET IS NOT ATTAINABLE 
APPROXIMATED DESCRIPTION BECOMES AN APPEALING
ALTERNATIVE
 APPROXIMATING APPROACHES DEVELOPED TO:



Slide 68
REPRESENT THE PARETO SET WHEN THE SET IS NUMERICALLY
AVAILABLE (LINEAR OR CONVEX MOPS)
APPROXIMATE THE PARETO SET WHEN SOME BUT NOT ALL
PARETO POINTS ARE NUMERICALLY AVAILABLE (NONLINEAR
MOP’s)
APPROXIMATE THE PARETO SET WHEN PARETO POINTS ARE
NOT NUMERICALLT AVAILABLE (DISCRETE MOPS)
ADVANCED METAHEURISTICS
APROXIMATION OF THE PARETO SET
Poznan University of Technology
FOR ANY MOP APPROXIMATION



REQUIRES LESS EFFORT
USUALLY IS ACCURATE ENOUGH TO BE USED
AS A GENERATOR OF THE SOLUTION SET
REPRESENTS THE SOLUTION SET IN A
– SIMPLIFIED WAY
– STRUCTURED WAY
– UNDERSTADABLE WAY
APPROVIMATION – IMPORTANT RESEARCH
ASPECTS


Slide 69
QUALITY OF APPROXIMATION (Q of A)
MEASURING & EVALUATING Q of A
ADVANCED METAHEURISTICS
APROXIMATION OF THE PARETO SET
Poznan University of Technology
ITERATICE METHODS TO PRODUCE POINTS/OBJECTS
APPROXIMATING THE PARETO SET
EXACT APPROACHES
THEORETICAL PROOFS
FOR CORRECTNESS &
OPTIMALITY
HEURISTIC APPROACHES
THEORETICALLY
UNSUPPORTED
PARAMTERE
SPACE
INVESTIGATION
POINT-WISE
NONLINEAR
APPROXIMATION APPROXIMATION
PIECE-WISE LINEAR
APPROXIMATION
Slide 70
CLASSIC
HEURISTICS
POPULATION
BASED
METAHEURISTICS
LOCAL SEARCH
BASED
METAHEURISTICS
ADVANCED METAHEURISTICS
Ant colonies
ADVANCED METAHEURISTICS
Ant Colonies
Poznan University of Technology
 Ant Colony optimization algorithms are part of swarm intelligence
(SI)


SI – research field that studies algorithms inspired by the observation of the
behavior of swarms
SI algorithms are made up of simple individuals that cooperate through self
– organization (without central control)
 Ant Colony optimization was inspired by the observation of the
behavior of real ants; finding paths from a nest to food
 1940s – 1950s – Pierre – Paul Grasse (French entomologist) was
the first to investigate the social behavior of insects – termites


Insects are capable to react to „significant stimuli” – signals that activate
a genetically encoded reaction; those reactions can act as new significant
stimuli for both the insects that produced them and others in the colony
Stigmergy – type of indirect communication – „workers are stimulated
by the performance they have achieved”
Slide 72
ADVANCED METAHEURISTICS
Ant Colonies
Poznan University of Technology
Characteristics of Stigmergy
 The physical, nonsymbolic nature of the information
released by the communicating insects
– Modification of physical environmental states visited by the
insects
 Insects (ants) do not communicate using visual cues
 Local nature of the released information, which can
only be accessed by those insects that visit the place
where it was released (or its immediate neighborhood)
Slide 73
ADVANCED METAHEURISTICS
Ant Colonies
Poznan University of Technology
Behavior of ants
 Initially ants wander randomly to find food. While walking to and
from the food source ants deposit on the ground a chemical
substance called „pheromone”
 Other ants are able to smell the pheromone and its presence
influences on the choice of their path – they follow strong
pheromone concentrations
 After finding food ants return to the nest; the pheromone
deposited on the ground forms the pheromone trail
 Other ants follow the pheromone trail to find food
Path is not
very attractive
Slide 74
Pheromone
evaporates
Information to
other ants
ADVANCED METAHEURISTICS
Ant Colonies
Poznan University of Technology
Behavior of ants
 Ants select their paths randomly; however they prefer
in probability to follow a stronger pheromone trail; due
to random fluctuactions one path becomes more
acceptable until the colony of ants converges toward
one path only (Argentine ants; „binary bridge
experiment”) – J.-L. Deneubourg (1980s)
 Ants are capable of adapting to changes in their
environment – autocatalysis – exploitation of positive
feedback


Ants can find a new shortest path when the old one is not available
anymore
Ants can select the shortest path from available options – S. Goss
experiment – Argentine ants; two bridges of different lengths
(1980s)
Slide 75
ADVANCED METAHEURISTICS
Ant Colonies
Poznan University of Technology
Ants go from the nest to food using pheromone trail
FOOD
Slide 76
NEST/COLONY
ADVANCED METAHEURISTICS
Ant Colonies
Poznan University of Technology
An obstacle has interrupted the initial path – some ants
go right and some go left
FOOD
Slide 77
NEST/COLONY
obstacle
ADVANCED METAHEURISTICS
Ant Colonies
Poznan University of Technology
New shortest path around an obstacle was established
FOOD
NEST/COLONY
obstacle
Slide 78
ADVANCED METAHEURISTICS
Ant Colonies
Poznan University of Technology
Graph model of Ant Colonies
 Ant Colonies optimization focuses on finding good paths
through graphs
Before ants find
path to food
Slide 79
Many ants found
different
paths to food
The best path
to food
is established
ADVANCED METAHEURISTICS
Ant Colonies
Poznan University of Technology
Family of Ant
Colonies Algorithms
Marco Dorigo 1992 – Ant System
M. Dorigo, L. Gambardella, T. Stützle 1995 – Ant Colony System
T. Stützle, H. Hoos - 1995 – MAX-MIN Ant
System
M. Dorigo, L. Gambardella, T. Stützle
proposed also hybrid versions of AC and
LS
Slide 80
ADVANCED METAHEURISTICS
Ant Colonies
Poznan University of Technology
Principles of the AC Algorithm
 Calculation of probability how the real ants select paths;
probability is a function of the amount of the pheromone;
 Artificial ants may simulate pheromone depositing by
modifying appropriate pheromone variables associated with
problem states they visit while building solutions to the
optimization problem
 Stigmergy of the artificial ants (agents):


Associating state variables with different problem states
Giving the agents only local access to these variables
 Implicit evaluation of solutions – shorter paths are completed
earlier than longer ones; they receive pheromone
reinforcement quicker + autocatalysis can be very efficient; the
shorter the path the sooner the pheromone is deposited and
more ants use the shorter path
Slide 81
ADVANCED METAHEURISTICS
Ant Colonies
Poznan University of Technology
Principles of the Ant Colonies Algorithm
Stigmergy
Implicit Evaluation
Autocatalytic
Behavior
Similarities between real and artificial ants
 Population of individuals (independent agents) that work together to achieve
a certain goal (find food - good solution)
 Single ant is able to find a solution, but only cooperation enables ants to find a
good solution
 Ants deposit pheromone; real ants on the ground; artificial ants modify
numeric values (artificial pheromones) associated with different problem states;
a sequence of pheromone values is called the artificial pheromone trail
 Evaporation mechanism – allows artificial ants forget about history and focus
on new, promising search directins
 Step-wise, sequential process; real ants walk – pheromone concentration;
stochastic decision policy; artificial ants move through available problem states
and make stochastic decisions at each step
Slide 82
ADVANCED METAHEURISTICS
Ant Colonies
Poznan University of Technology
Differences between real and artificial ants
 Artificial ants live in a discrete world – they move sequentially through
a finite set of problem states
 The pheromone update (depositing and evaporation) is not
accomplished in exactly the same way by artificial ants as by real
ones. Sometimes done only by some of the artificial ants and often
only after a solution has been constructed
 Some implementations of artificial ants use additional mechanisms
that do not exist in the case of real ants; e.g. look-ahead, local search,
backtracking
Slide 83
ADVANCED METAHEURISTICS
Ant Colonies
Poznan University of Technology
Scheme of the Ant Colony Algorithm
 AC algorithm is based on probabilistic mechanism for solving
computational problems
 AC algorithm is a loop until termination condition is met
Set parameters, initialize pheromone trails
while termination conditions not met do
Construct Ant Solutions
Apply Local Search {optional}
Update Pheromones
end while
Slide 84
ADVANCED METAHEURISTICS
Ant Colonies
Poznan University of Technology
Ant Colonies Model
 A model P = (S, W, f) of a COP consists of:


a search space S defined over a finite set of discrete decision
variables and a set W of constraints among the variables
an objective function f: S
R+ to be minimized
 The search space S includes discrete decision variables Xi with
values vij; solution s in S that satisfies all constraints W is a
feasible solution
 A solution s* in S is called a global optimum if and only if
f(s*) < f(s) for each s in S
Slide 85
ADVANCED METAHEURISTICS
Ant Colonies
The Pheromone Model
Poznan University of Technology
 First Xi = vij (from its domain Di) is called a solution component – cij;
the set of all solution components is denoted by C
 A pheromone trail parameter Tij is associated with each component cij ;
the set of all pheromone parameters is denoted by T; the value of a
pheromone trail parameter Tij is denoted by tij (called pheromone
value, updated during the search); allows modeling the probability
distribution of different components of the solution
 Artificial ants build a solution by traversing the so-called construction
graph GC (V, E); V – vertices; E – edges; the set of components C can
be associated with V or E
 The ants move from vertex to vertex along the edges incrementally
building a partial solution; they deposit certain amount of pheromone
on the components (vertices or edges)
 The amount Dt of pheromone deposited may depend on the quality of
the solution found; subsequent ants utilize the pheromone information
as a guide toward more promising regions of the search space.
Slide 86
ADVANCED METAHEURISTICS
Ant Colonies
Poznan University of Technology
Choice of node in the graph
 Amount of pheromone on an arc
 Desirability of arc (a priori knowledge)
 Controlling influence of desirability and pheromone
Slide 87
ADVANCED METAHEURISTICS
Ant Colonies
Poznan University of Technology
Construct Ant Solution
Slide 88
ADVANCED METAHEURISTICS
Ant Colonies
Poznan University of Technology
Update pheromones
Slide 89
ADVANCED METAHEURISTICS
Ant Colonies
Poznan University of Technology
Example
Car accident – an
obstacle for
drivers(ants)
• Connection between two points is not available,
because there was an accident on the road
• By-pass required
Slide 90
ADVANCED METAHEURISTICS
Ant Colonies
Poznan University of Technology
Summary
 AC algorithm solves very well complex combinatorial
optimization problems, including the traveling salesman
problem – results are very close to optimum
 When graph can change dynamically AC is better than
other metaheuristics (SA,GA) and can operate in „real-time”
 AC is a brilliant idea for transportation, city logistics or
network routing
Slide 91
ADVANCED METAHEURISTICS
TYPES OF METACHEURISTIC FOR CVRP
Poznan University of Technology
Metaheuristics for the
Capacitated VRP
Slide 92
ADVANCED METAHEURISTICS
AGENDA
Introduction – CVRP
Types of Metaheuristics for CVRP






Simulated Annealing (SA)
Deterministic Annealing (DA)
Tabu Search (TS)
Genetic Algorithm (GA)
Ant Systems (AS)
Neural Networks (NN)
Conclusions
Slide 93
Poznan University of Technology
ADVANCED METAHEURISTICS
INTRODUCTION - CVRP
Poznan University of Technology
CVRP – CAPACITATED VEHICLE ROUTING PROBLEM
A fleet of vehicles supplies customers. Each vehicle
has a certain capacity and each customer has a
certain demand. There is a depot(s) and a distance
(length, cost, time) matrix between the customers.
We look for optimal vehicle routes (minimum
distance or number of vehicles).
The VRP is a NP complete problem.
The special cases of the VRP result in other popular
problems like the Travelling Salesman Problem (TSP) or even
Scheduling.
Slide 94
ADVANCED METAHEURISTICS
INTRODUCTION - CVRP
Poznan University of Technology
Given
• Complete graph G=(N,E)
• Set of nodes N={0,1,…,n}
• Set of edges (symmetric case) E={(i,j)|i,jN;i<j}
• Cost of traveling from node i to node j - cij
• Demand per node di(iN-{0})
• Vehicle capacity C
• Number of vehicles K
Find
• A set of at most K vehicle routes of total minimum
cost such that
– Every route starts and ends at the depot,
– Each customer is visited exactly once,
– The sum of the demands in each vehicle route does not
exceed the vehicle’s capacity
Slide 95
ADVANCED METAHEURISTICS
Mathematical formulation for CVRP:
INTRODUCTION - CVRP
Poznan University of Technology
.
r(S) = lower bound on the number of trucks required to service
If
Problem.
, then we have the Multiple Traveling Salesman
Alternatively, if the edge costs are all zero, then we have the
Bin
Slide
96Packing Problem
ADVANCED METAHEURISTICS
ITYPES OF METACHEURISTIC FOR CVRP
Poznan University of Technology
Four main types of metaheuristic that have
been applied to the VRP:
• Simulated Annealing (SA)
• Tabu Search (TS)
• Genetic Algorithm (GA)
• Ant Systems (AS)
Slide 97
ADVANCED METAHEURISTICS
Osman’s
Simulated Annealing Algorithms
TYPES OF METACHEURISTIC FOR CVRP
Poznan University of Technology
Features:
• Much more involved
• More successful
• Uses a better starting solution
• some parameters are adjusted in a trial
phase
• Richer solution neighborhoods are
explored
• Cooling schedule is more sophisticated
Slide 98
ADVANCED METAHEURISTICS
Osman’s Simulated Annealing Algorithms
TYPES OF METACHEURISTIC FOR CVRP
Poznan University of Technology
Algorithm:
Phase 1. Descent algorithm.
Step 1. (initial solution). Generate an initial solution by means of the Clarke
and Wright algorithm.
Step 2. (descent). Search the solution space using the -interchange
scheme. Implement an improvement as soon as it is identified. Stop
whenever an entire neighborhood exploration yields no impovement.
Phase 2. Simulated Annealing Search
Step 1.(initial solution). Use as a starting solution the incumbent obtained at
he end of Phase 1, or a solution produced by the Clarke and Wright
algorithm.
Preform a complete neighborhood search using -interchange generation
mechanism without, however, implementing any move.
Record Dmax and Dmin, the largest and the smallest absolute changes in the
objective function and compute , the number of feasible (potential
exchanges.
Slide 99
ADVANCED METAHEURISTICS
Osman’s Simulated Annealing Algorithms
TYPES OF METACHEURISTIC FOR CVRP
Poznan University of Technology
Algorithm :
Phase 2.
Step 2. (next solution). Explore the neighborhood of xt using
-interchange .
Step 3. (temperature update).
Occasional increment rule:
if =1, set t+1:=max {t/2, *}, :=0 and k:=k+1
Normal decrement rule:
if =0, set t+1+= t/[(n+nt)Dmax Dmin]. Set t:=t+1. If k=k3, stop.
Otherwise, go to step 2.
Slide 100
ADVANCED METAHEURISTICS
Tabu Search (TS)
TYPES OF METACHEURISTIC FOR CVRP
Poznan University of Technology
– Two Early Tabu Search Algorithms
– Osman’s Tabu Search Algorithms
– Taburoute
– Taillard’s Algorithm
– Xu and Kelly’s Algorithm
– Rego and Roucairol’s Algorithms
– Barbarosoglu and Ozgur’s Algorithm
– Adaptive Memory Procedure of Rochat and
Taillard
– Granular Tabu Search of Toth and Vigo
Slide 101
ADVANCED METAHEURISTICS
Tabu Search (TS)
TYPES OF METACHEURISTIC FOR CVRP
Poznan University of Technology
Taburoute – features:
 the neighbourhood structure is defined by all solutions that can
be reached from current solution by removing a vertex from its
current route and inserting it into another route containing on of
its p nearest neighbours using GENI (Generalized Insertion for
the TSP. This may result in eliminationing an existing route or in
creating new one
 Search process examines solutions that may be infeasible with
respect to the capacity or maximum route lengh constraints
 Does not use a tabu list but instead uses random tabu tags.
 Uses diversification strategy
Slide 102
Poznan University of Technology
Evolver nad PSP-OptQuest
Slide 103
Evolver
Genetic algorithm optimization for Microsoft Excel
Poznan University of Technology
1. The application of powerful genetic algorithmbased (GA) optimization techniques, can find
optimal solutions to problems which are
"unsolvable" for standard linear and nonlinear optimizers.
2. Add-in for Microsoft Excel.
3. Requires no knowledge of programming or GA
theory
4. By Palisade Corporation
Slide 104
Evolver
Genetic algorithm optimization for Microsoft Excel
Poznan University of Technology
Slide 105
Evolver
Adjustable Cells (options)
Poznan University of Technology
Solving Methods: grouping,
order, recipe, budget, project,
and schedule.
• The “Recipe” and “Order” solving
methods are the most popular and they
can be used together to solve complex
combinatorial problems
• The “Recipe” method treats each
variable as an ingredient in a recipe,
trying to find the “best mix” by changing
each variable’s value independently.
• In contrast, the “Order” solving method
swaps values between variables,
shuffling the original values to find the
“best order.”
Slide 106
Crassover and
Mutation Rate
Evolver
Optimization Operators (Genetic operators)
Poznan University of Technology
• Linear Operators – Designed to solve
problems where the optimal solution lies
on the boundary of the search space
defined by the constraints. This mutation
and crossover operator pair is well
suited for solving linear optimization
problems.
• Boundary Mutation – Designed to
Quickly optimize variables that affect
the result in a monotonic fashion and
can be set to the extremes of their
range without violating constraints.
Slide 107
Evolver
Optimization Operators (Genetic operators)
Poznan University of Technology
• Cauchy Mutation – Designed to
produce small changes in variables
most of the time, but can occasionally
generate large changes.
• Non-uniform Mutation – Produces
smaller and smaller mutations as
more trials are calculated. This allows
Evolver to “fine tune” answers.
• Arithmetic Crossover – Creates new offspring by arithmetically
combining the two parents (as opposed to swapping genes).
• Heuristic Crossover – Uses values produced by the parents to
determine how the offspring is produced. Searches in the most
promising direction and provides fine local tuning.
Slide 108
Evolver
Watcher
Poznan University of Technology
Evolver Watcher is
responsible for regulating and
reporting on all Evolver
activity. If you are running
applications other than Excel
that also use Evolver, such
as custom applications, the
populations they create will
also appear in Evolver
Watcher’s population list.
Slide 109
Premium Solver Platform (PSP) – OptQuest Engine
Tabu Search algorithm optimization for Microsoft Excel
Poznan University of Technology
1. The application of powerful tabu search
optimization techniques, can find optimal
solutions to problems which are "unsolvable"
for standard linear and non-linear optimizers.
2. Add-in for Microsoft Excel.
3. Requires no knowledge of programming or TS
theory
4. By Frontline Systems Inc.
Slide 110
Solver parametrs
Poznan University of Technology
Slide 111
Engine e.g.
OptQuest
Solver parametrs – OptQuest Engine
Poznan University of Technology
•
•
•
•
•
•
•
•
•
•
•
•
Max Time Solution
Iterations
Precision (Obj Fun)
Precision (Dec Var)
Population Size
Bounduary Freq
Use same sequence of
random numbers with
seed
Solve Without Integer
Constraints
Check for Duplicated
Solutions
Bypass Solver Raports
Assume Non-Negative
Show Iteration Results
Slide 112
OptQuest Engine vs. Evolver
Poznan University of Technology
Slide 113
Case study I– optimization by Evolver
Poznan University of Technology
1. Fleet management problem in the road
transportation company (4 old trucks; 16
months)
2. Mathematical model

Decision variables
1
xij  
0
Slide 114
truck i is used in the period j
otherwise
Case study I – optimization by Evolver
Poznan University of Technology
2. Mathematical model



Slide 115
The number of vehicles replaced per time period is
limited (e.g. 1 per quarter)
The vehicle withdrawn from utilization can not be
used again
The number of vehicles is constant in the time
horizon
Case study I – optimization by Evolver
Poznan University of Technology
3. Mathematical model

Criteria
– Total maintenance cost (PLN)
min FC   xij cij wij
i
j
Cost ratio
Decision
variables
Slide 116
Cost
Case study I – optimization by Evolver
Poznan University of Technology
4. Evolver Options







Slide 117
Solving method –
recipe
Crassover Rate – 0,5
Mutation Rate – 0,1
Population Size – 100
Random Number Seed
– Generated Randomly
Update the Display –
never
Valid Trails is Less
Than – 0,1%
Case study I – optimization by Evolver
Poznan University of Technology
5. Results
• Basic solution 339 800 PLN
Truck
Truck 1
Truck 2
Truck 3
Truck 4
New truck 1
New truck 2
New truck 3
New truck 4
1
1
1
1
1
0
0
0
0
2
1
1
1
1
0
0
0
0
3
1
1
1
1
0
0
0
0
4
1
1
1
1
0
0
0
0
5
1
1
1
1
0
0
0
0
6
1
1
1
1
0
0
0
0
7
1
1
1
1
0
0
0
0
Quarter
8 9
1 1
1 1
1 1
1 1
0 0
0 0
0 0
0 0
10
1
1
1
1
0
0
0
0
11
1
1
1
1
0
0
0
0
12
1
1
1
1
0
0
0
0
13
1
1
1
1
0
0
0
0
14
1
1
1
1
0
0
0
0
15
1
1
1
1
0
0
0
0
16
0
0
0
0
1
1
1
1
• Optimization by Evolver
• 116 436 PLN
Slide 118
Truck
Truck 1
Truck 2
Truck 3
Truck 4
New truck 1
New truck 2
New truck 3
New truck 4
1
1
1
1
1
0
0
0
0
2
1
1
1
0
0
0
1
0
3
1
1
1
0
0
0
1
0
4
1
1
1
0
0
0
1
0
5
1
1
1
0
0
0
1
0
6
1
0
1
0
0
1
1
0
Quarter
7 8 9 10
1 1 0 0
0 0 0 0
1 1 1 1
0 0 0 0
0 0 0 0
1 1 1 1
1 1 1 1
0 0 1 1
11
0
0
1
0
0
1
1
1
12
0
0
1
0
0
1
1
1
13
0
0
0
0
1
1
1
1
14
0
0
0
0
1
1
1
1
15
0
0
0
0
1
1
1
1
16
0
0
0
0
1
1
1
1
Case study II – optimization by OptQuest
Poznan University of Technology
1. Feet composition problem in the fuel
transportation/distribution company
2. Mathematical model

Decision variables
for j  1  asigning a vehicle to customer i ,

xij number of assigned vehicle


xij  1,P ,
xij  
for j  1 - assigning the vehicle chamber for transp orting fuel type j  2,3..., J to customer i,


xij  number of the assigned chamber i,

Slide 119
Case study II – optimization by OptQuest
Poznan University of Technology
2. Mathematical model
Constraints

–

p k
–
Capacity of the fuel chambers in each vehicle
 P such that
I
J
i 1 j  2
ij
xi1  p i xij  k   PK pk
Eliminating fuel mix in 1 fuel chamber
1  if xij  k , for a minimum 1 i  1, I ,
 
1
k
j  2 0  if otherwise
J
Slide 120
Case study II – optimization by OptQuest
Poznan University of Technology
2. Mathematical model

Constraints
–
Satisfying demand for fuel
Area A  1  xi1  P
i
–

Area B   1  xij  K
i j 1
Working time for vehicels/drivers

 LDśr  LRśr  LKl p  1  LPśr  LKrp Vep  Tmax
p
Slide 121
Case study II – optimization by OptQuest
Poznan University of Technology
2. Mathematical model
Criteria

–
Total distribution costs [PLN]


Min FC1   KZ p  LDśr  LRśr  LKl p  1  LPśr  KS p  [ PLN ]
P
1
Slide 122
Case study II – optimization by OptQuest
Poznan University of Technology
5. OptQuest Options






Slide 123
Max Time – 200 s
Iterations – 10 000
Precision (Obj Fun) – 0,0001
Precision (Dec Var) – 0,0001
Population Size – 75
Bounduary Freq – 0,25
Case study II – optimization by OptQuest
Poznan University of Technology
3. OptQuest Options
 Use same sequence of random
numbers with seed – inactive
 Solve Without Integer
Constraints – inactive
 Check for Duplicated Solutions –
active
 Bypass Solver Raports –
inactive
 Assume Non-Negative – active
 Show Iteration Results - inactive
Slide 124
Case study II – optimization by OptQuest
Poznan University of Technology
6. Results
Number of a vehicle
Basic solution
Optimal solution
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
Vehicles (4 – cells)
Vehicles (8 – cells)
Value of criterion [zł]
YES
YES
YES
YES
YES
YES
YES
YES
YES
YES
NO
NO
NO
NO
NO
NO
7
3
10 985
YES
YES
NO
NO
NO
NO
NO
NO
NO
NO
NO
NO
NO
NO
NO
YES
3
0
5 810
Slide 125
Case study III – optimization by Evolver
Poznan University of Technology
1. Traveling Salesman problem
table of decision variables
Find the best way to visit all 68 cities with the least amount of
traveling. The salesman must always return back to the first
city to form a complete loop.
Slide 126
table of distances
Case study III – optimization by Evolver
Poznan University of Technology
Traveling Salesman problem – NPcomplete Problem !!!
69 towns =1,82  10 solutions
94
Slide 127
Case study III – optimization by Evolver
Poznan University of Technology
2. Evolver Options
 Solving method – order
 Crassover Rate – 0,5
 Mutation Rate – 0,1
 Population Size – 100
 Random Number Seed –
Generated Randomly
 Update the Display – never
 Valid Trails is Less Than –
0,1%
Slide 128
Case study III – optimization by Evolver
Poznan University of Technology
RESULTS (Raport)
Valid Trials
Total Recalcs
Original Value
+ soft constraint penalties
= result
Best Value Found
+ soft constraint penalties
= result
Occurred on trial
Time to find this value
Stopped Because
Optimization Started At
Optimization Finished At
Total Optimization Time
Slide 129
182933
291505
25479
0
25479
7824
0
7824
169321
00:09:32
Halted by User
11:43:40
11:54:31
00:10:31
Basic Solution
1
2
3
4
5
…
13
14
15
16
17
…
29
30
…
65
66
67
68
1
Best Solution
1
58
21
23
13
…
30
28
61
18
17
…
48
35
…
65
8
36
27
1
MULTIPLE OBJECTIVE
METAHEURISTICS
ADVANCED METAHEURISTICS
Poznan University of Technology
MULTIOBJECTIVE APPROACH
LOCAL SEARCH BASED
POPULATION BASED
MOSA
PSA
TS - MOTS
VEGA
HYBRID
PMA
MOGLS
Slide 131
ADVANCED METAHEURISTICS
LOCAL SEARCH – BASED METAHEURISTICS
Poznan University of Technology
 MOSA





A PROTOTYPE OF A MULTIOBJECTIVE S.A. METHOD
FOR A SET OF WEIGHTING VECTORS  A S.A. PROCEDURE IS
PERFORMED ON THE PROBLEM SCALORIZED WITH THE
WEIGHTED SUM METHOD
STARTING SOLUTION x IS CHOSEN
A SOLUTION x’ IN SOME NEIGHBORHOOD IF x IS SELECTED AND
COMPARE WITH x
P
P
IF f(x’)f(x) OR

k 1
k
f k ( x' )   k f k ( x)
k 1
 YES: x’ IS ACCEPTED AS A BETTER SOLUTION
 NO: x’ IS ACCEPTED WITH SOME PROBABILITY


Slide 132
RESULT: SET OF POTETIALLY EFFICIENT SOLUTION IN
DIRECTION 
AFTER PROCEDURE FOR ALL   SETS OF POTENTIALLY
EFFICEINT SOLUTION ARE MERGED
ADVANCED METAHEURISTICS
LOCAL SEARCH – BASED METAHEURISTICS
Poznan University of Technology
 MOTS







Slide 133
BASED ON NEIGBORHOOD PRINCIPLES
STARTING POINT IS AN INITIAL SOLUTION x
NEW SOLUTION x’ IS SOME NEIGBORHOOD OF x IS
SELECTED, BUT IT IS VASED ON SELECTION USING A
WEIGTED DISTANCE FROM X POINT yu
IN ORDER TO OVERCOME LOCAL OPTIMA, SOME
SOLUTIONS IN THE NEIGHBOURHOOD ARE THE CLARED
AS ”TABU”
„”TABU” STATUS DEPENDS ON THE ITARATIONS PERFORM
SO FAR
FOR EACH WEIGHT SINGLE OBJECTIVE TS IS PERFORED
AT THE END OF THE ALGORITHM POTENTIALI EFFICENT
SETS OF SOLUTIONS ARE MERGED
ADVANCED METAHEURISTICS
LOCAL SEARCH – BASED METAHEURISTICS
Poznan University of Technology
POPULATION – BASED METAHEURISTICS



Slide 134
MAITAIN A WHOLE SET OF SOLUTIONS (THE
POPULATIONS)
TRY TO EVOLVE THE POPULATION TOWARDS TO
THE PARETO SET
MANY DIFFERENT TECHNIQUES (GA,
EVOLUTIONARY ALGORITHMS) TO EVALUATE
THE FITNESS OF IDIVIDUAL SOLUTIONS IN
MULTIOBJECTIVE CONTEXT
Pareto simulated
annealing
ADVANCED METAHEURISTICS
PARETO SIMULATED ANNEALING
Poznan University of Technology
Similarities to single objective simulated annealing &
genetic algorithms
PSA
SA
Slide 136
GA
New Concepts
ADVANCED METAHEURISTICS
PARETO SIMULATED ANNEALING
Poznan University of Technology
Similarities to single objective simulated annealing &
genetic algorithms
PSA
SA
GA
New Concepts
• The concept of the neighborhood
• Probabilistic acceptance of new neighbourhood solutions
(with a certain probability)
• Dependence of the acceptance probability on a parameter
(temperature)
• The scheme of the temperature changes
Slide 137
ADVANCED METAHEURISTICS
PARETO SIMULATED ANNEALING
Poznan University of Technology
Similarities to single objective simulated annealing &
genetic algorithms
PSA
SA
GA
New Concepts
• The use of a sample (population) of solutions; each of
them exploring the search space according to SA rules
• The solutions may be treated as independent agents,
exchanging information about their positions. A
separate weight vector is associated with each of the
generating solutions.
Slide 138
ADVANCED METAHEURISTICS
PARETO SIMULATED ANNEALING
Poznan University of Technology
Similarities to single objective simulated annealing &
genetic algorithms
PSA
SA
GA
New Concepts
• The use of scalarizing functions locally aggregating multiple
criteria functions and scalarizing functions based probabilities
for acceptance of new neighborhood solutions
• Automatic modifications of weights of particular objectives in
each iteration according to a certain rule. The rule for
updating the generating solutions’ weight vectors aims at
assuring dispersion of the solutions over all regions of the
nondominated set
Slide 139
ADVANCED METAHEURISTICS
PARETO SIMULATED ANNEALING
Poznan University of Technology
Multiple objective acceptance rules
 Single objective SA


New solution no worse than current solution – acceptance – P=1
Otherwise – acceptance P<1
 Each solution x can be modified (replaced) by accepting a
randomly generated solution from its neighbourhood. The
new solution is acceptable with some probability
 PSA uses the concept of multiple objective acceptance rules
(P. Serafini – 1994)
Slide 140
ADVANCED METAHEURISTICS
PARETO SIMULATED ANNEALING
Poznan University of Technology
MOA rules
In the multiple objective case one of the following exclusive
situations may occur

y dominates or is equal to x (new solution is not
worse than the current solution – P = 1)

y is dominated by x (new solution is worse than the
current solution – P < 1)

y is non dominated with respect to x (ambiguous
situation – P = ? )
y – new solution, x – current solution
Slide 141
ADVANCED METAHEURISTICS
PARETO SIMULATED ANNEALING
Poznan University of Technology
The probability of accepting solution y (compared with
x) based on MOA rules. The case of two maximized
objectives.
y
y
Criterion 2
(Max)
y
y
Criterion 1
(Max)
Slide 142
ADVANCED METAHEURISTICS
PARETO SIMULATED ANNEALING
Poznan University of Technology
y is non dominated with respect to x
 Rule could be interpreted as a local aggregation of all
objectives with the weighted Tchebycheff scalarizing
function with reference point at f(x)
P – probability of accepting the new solution y
T – temperature
Λ –weight vector
Slide 143
ADVANCED METAHEURISTICS
PARETO SIMULATED ANNEALING
Poznan University of Technology
y is non dominated with respect to x
Graphical illustration of the rule
Slide 144
ADVANCED METAHEURISTICS
PARETO SIMULATED ANNEALING
Poznan University of Technology
y is non dominated with respect to x
The rule may be also interpreted as a local aggregation of all
objectives with a weighted linear scalarizing function
P – probability of accepting the new solution y
T – temperature
Λ –weight vector
Slide 145
ADVANCED METAHEURISTICS
PARETO SIMULATED ANNEALING
Poznan University of Technology
y is non dominated with respect to x
Graphical illustration of the rule
Slide 146
ADVANCED METAHEURISTICS
PARETO SIMULATED ANNEALING
Poznan University of Technology
Management of the population of generating solutions
 The weights used in the acceptance rules allow to influence on the
direction of search in the objective space for particular generating
solutions
 The higher the weight associated with a given objective the higher the
influence of this objective on the probability of acceptance of new
solutions and the higher the pressure towards improvement of that
objective
 Controling the weight vectors the method may „push” generating
solutions into desired directions in the decision space
 PSA controls the weight vectors associated with particular generated
solution in order to achieve a form of repulsion between the solutions
 The weight vector associated with a given generating solution x is
modified in order to increase the probability of moving x away from its
closest neighbor x’ in the generating sample
 This is obtained by increasing the weights of the objectives on which x
is better than x’ and decreasing the weights of the objective on which x
is worse than x’
Slide 147
ADVANCED METAHEURISTICS
PARETO SIMULATED ANNEALING
Poznan University of Technology
Management of the population of generating solutions
 The Euclidean distance between solutions in the space of
normalized objectives is used
 The closest naighbor has to be non-dominated with respect to x.
If there is no generating solution that meets this requirement
each weight is either increased or decreased with probability =
0.5
 Repulsion mechanism never repulses the generating solutions
from the nondominated set
 During the computational process some generating solutions
may get stacked in regions far away from the nondominated set.


If a generating solution is dominated by at least one other generating
solution for a number od iterations it is considered not promising and
replaced by a solution from the set of potentially Pareto-optimal solutions,
having maximum distance to the closest generating solution
The idea is to move the generating solution to a poorly explored region of
the nondominated set
Slide 148
ADVANCED METAHEURISTICS
PARETO SIMULATED ANNEALING
Poznan University of Technology
Updating the set of potentially Pareto – optimal
solutions
 At the beginning of the computational procedure the set of
Pareto-opitmal solutions PP is empty
 PP is updated every time when a new solution is generated
Update:
• Add f(x) to PP if no point in PP dominates f(x)
• Remove from PP all points dominated by f(x)
Slide 149
ADVANCED METAHEURISTICS
PARETO SIMULATED ANNEALING
Poznan University of Technology
Updating the set of potentially Pareto – optimal
solutions
Updating could be very time consuming. The ways to avoid this
disadvantage are:
• A new solution y obtained from x should be used to update PP
set only when it is not dominated by x
• New solution y may be added to PP set only if they differ
enough from all solutions in PP set (threshold – minimum
Euclidean distance)
• Neglect updating PP in a number of starting iterations (solutions
added to PP in early iterations had good chances to be removed)
• Using the data structure called quad trees to accelerate the
process
of updating PP (4 and more objectives)
Slide 150
ADVANCED METAHEURISTICS
PARETO SIMULATED ANNEALING
Poznan University of Technology
Using partial preference information
 The information concerning the DM’s preferences may help
focusing on the interesting region of the nondominated set
(e.g. objective 1 is more important than objective 2, solutions
having value below a certain threshold on objective 3 are not
interesting)
 The most natural way of taking into account partial preference
information in PSA is to express it in the form of constraints
in the weight space
Slide 151
ADVANCED METAHEURISTICS
PARETO SIMULATED ANNEALING
Poznan University of Technology
Basic version
of PSA
algorithm
Slide 152
ADVANCED METAHEURISTICS
PARETO SIMULATED ANNEALING
Poznan University of Technology
APPLICATION OF PSA
MULTIOBJECTIVE BUS-DRIVER’S SCHEDULING
Slide 153
ADVANCED METAHEURISTICS
PARETO SIMULATED ANNEALING
Poznan University of Technology
Contents
 Introduction
 Problem definition and mathematical formulation
 Solution procedure
 Computational experiments
 Conclusions
Slide 154
ADVANCED METAHEURISTICS
PARETO SIMULATED ANNEALING
Poznan University of Technology
Combinatorial Problem
Specific Crew Scheduling
Problem
BUS DRIVER’S
SCHEDULING PROBLEM
Set of duties for bus drivers
Slide 155
ADVANCED METAHEURISTICS
PARETO SIMULATED ANNEALING
Poznan University of Technology
Multiple objective formulation
STAKEHOLDERS
Trasportation
company owner –
cost oriented
Slide 156
Bus driver –
convenience oriented
ADVANCED METAHEURISTICS
PARETO SIMULATED ANNEALING
Poznan University of Technology
Transportation company
 Publicly owned, inter-city passenger transportation company,
located in Poznan, Poland
 The company provides medium – haul transportation services
in Western Poland (Wielkopolska); operates 7 days a week
 Annual sales 35 mln zl = 10 mln Euro; 125 000 vkm/ week; 62% Night transportation jobs and 38% - Local transportation jobs
 3 categories of duties



Local transportation jobs – L (31) – 9040 km (avg. 292 km)
Night transportation jobs – N (20) – 5720 km (avg. 286 km)
Additional tasks – P (4)
 Fleet: 103 buses (Autosan and Jelcz) in different age and
technical condition
 Labor force – 98 employees, incl. 52 bus drivers
Slide 157
ADVANCED METAHEURISTICS
PARETO SIMULATED ANNEALING
Poznan University of Technology
Objectives
Balance the
workload
of drivers
Slide 158
Assure fair
assignment
of duties
ADVANCED METAHEURISTICS
PARETO SIMULATED ANNEALING
Poznan University of Technology
Optimization goals
Number of L, N, P
should be balanced
Slide 159
Days-off should
be grouped
Number of days-off
(including Sundays
and Saturdays)
sholud be grouped
ADVANCED METAHEURISTICS
PARETO SIMULATED ANNEALING
Poznan University of Technology
Decision variables

N if the i-th bus driver carries out the night
 transportation job on j-th day
xij   P ifadditional
the i-th bus driver carries out the
task on j-th day
 W if the i-th bus driver has a day-off on j-th
 day
L if the i-th bus driver carries out the local
transportation job on j-th day
i = 1, ..., I - BUS DRIVER INDEX
j = 1, ..., J - DAY INDEX
Slide 160
ADVANCED METAHEURISTICS
PARETO SIMULATED ANNEALING
Poznan University of Technology
Mathematical formulation
GOAL 1 - AVG. DEVIATION - LOCAL TRANSPORTATION JOBS
J
I

i 1
SL 
x
j 1
Min f1 
J
Lij
100%
I
where:
I
SL 
Slide 161
J
 x
i 1 j 1
IJ
Lij
100%
xLij
 1 if xij  L

 0 if otherwise
,
ADVANCED METAHEURISTICS
PARETO SIMULATED ANNEALING
Poznan University of Technology
Mathematical formulation
GOAL 2 - AVG. DEVIATION - NIGHT TRANSPORTATION JOBS
J
I

i 1
SN 
x
Min f 2 
j 1
J
I
Nij
100%
,
where:
I
SN 
Slide 162
J
 x
i 1 j 1
IJ
Nij
100%
xNij
 1 if xij  N

 0 if otherwise
ADVANCED METAHEURISTICS
PARETO SIMULATED ANNEALING
Poznan University of Technology
Mathematical formulation
GOAL 3 - AVG. DEVIATION – ADDITIONAL TASKS
J
I

i 1
SP 
x
j 1
Min f 3 
J
I
Pij
100%
,
where:
I
SP 
Slide 163
J
 x
i 1 j 1
IJ
Pij
100%
xPij
 1 if xij  P

 0 if otherwise
ADVANCED METAHEURISTICS
PARETO SIMULATED ANNEALING
Poznan University of Technology
Mathematical formulation
GOAL 4 – POINTS AWARDING THE AGGREGATION OF DAYS-OFF
I
J
Max f 4   xW ,
i 1 j 1
where:
0
1
xW   3
6

if
if
if
if
{x} > 0
{x} = 0
(xWij+xWij+1) = 0 and (xWij+1+xWij+2) = 0
(xWij+xWij+1) = 0 and (xWij+1+xWij+2) = 0 and (xWij+2+xWij+3) = 0
X = {(xWij+xWij+1) + (xWij+1+xWij+2) + ... + (xWin-1+xWin)} for n = J
 0 if xij  W
xW ij  
 1 if otherwise
Slide 164
ADVANCED METAHEURISTICS
PARETO SIMULATED ANNEALING
Poznan University of Technology
Mathematical formulation
GOAL 5 - AVG. DEVIATION – AGGREGATED DAYS-OFF
J
I
Min f 5 
 SW 
i 1
where:
I
SW 


xW  


Slide 165
x
j 1
J
I
W
100%
,
J
 x
i 1 j 1
IJ
W
100%
0`if {x} > 0
1if {x} = 0
3if (xWij+xWij+1) = 0 and (xWij+1+xWij+2) = 0
6if (xWij+xWij+1) = 0 and (xWij+1+xWij+2) = 0 and (xWij+2+xWij+3) = 0
X = {(xWij+xWij+1) + (xWij+1+xWij+2) + ... + (xWin-1+xWin)} for n = J
ADVANCED METAHEURISTICS
PARETO SIMULATED ANNEALING
Poznan University of Technology
Mathematical formulation
GOAL 6 - AVG. DEVIATION – SATURDAYS
J
I

i 1
SS 
Min f 6 
x
j 1
Sij
J
100%
,
I
where:
I
SS 
Slide 166
J
 x
i 1 j 1
IJ
Sij
100%
xSij
 1 if xij  W

 0 if otherwise
ADVANCED METAHEURISTICS
PARETO SIMULATED ANNEALING
Poznan University of Technology
Mathematical formulation
GOAL 7 - AVG. DEVIATION – SUNDAYS
J
I
S
i 1
Min f 7 
D

x
j 1
J
Dij
100%
,
I
where:
I
SD 
Slide 167
J
 x
i 1 j 1
IJ
Dij
100%
xDij
 1 if xij  W

 0 if otherwise
ADVANCED METAHEURISTICS
PARETO SIMULATED ANNEALING
Poznan University of Technology
Length of the
scheduling period
Types of
contracts (fulltime, part-time)
Number of drivers required
for each day
Constraints
Labour code
regulations
Expected absences
& preferences
Number of drivers
available on each day
Slide 168
ADVANCED METAHEURISTICS
PARETO SIMULATED ANNEALING
Solution procedure
Generation of solutions
PSA
Poznan University of Technology
Multiple objective metaheuristic
procedure
Generation of a sample of
schedules being a good
approximation of the whole set
of a non-dominated solution
Evaluation of schedules
according to DM’s preferences
LBS-D
Review of solution
Penetration of different regions
of the sample
Final selection of one solution
Slide 169
ADVANCED METAHEURISTICS
PARETO SIMULATED ANNEALING
Poznan University of Technology
Customization
Slide 170
ADVANCED METAHEURISTICS
PARETO SIMULATED ANNEALING
Poznan University of Technology
Results of the computational experiments
• 30000 feasible solutions, including 2062 PP
• Each solution – assignment metrix (40 drivers x 30 days);
allocation of jobs to the drivers
• Computational time - 4 minutes; PC Pentium 1GHz
• In the experiment 16 generating solutions have been
used and 1856250 steps (moves) of the procedure have
been preformed
• The exemplary solution - 4366
Slide 171
ADVANCED METAHEURISTICS
PARETO SIMULATED ANNEALING
Poznan University of Technology
Exemplary schedule
Slide 172
ADVANCED METAHEURISTICS
PARETO SIMULATED ANNEALING
Poznan University of Technology
LBS-D
 Interactive procedure for multiple objective mathematical
programming problem
 User – friendly interface
 Graphical facilities
 Phases of decision alternating with phases of computation
 Searches for a compromise solution in the neighborhood of
the selected solution (middle point)
 The search process is similar to projecting light onto the
solution set; it is based on the definition of the DM’s
preferences (aspiration levels; reference point)
Slide 173
Poznan University of Technology
General Scheme
LBS
Slide 174
ADVANCED METAHEURISTICS
PARETO SIMULATED ANNEALING
Poznan University of Technology
Decision phase
Fixing points z* and z*
Slide 175
ADVANCED METAHEURISTICS
PARETO SIMULATED ANNEALING
Poznan University of Technology
Decision phase
 DM’s preferences (q, p, v)
 Procedure finds starting middle points
Slide 176
ADVANCED METAHEURISTICS
PARETO SIMULATED ANNEALING
Poznan University of Technology
Decision phase
An outranking neighborhood is constructed
Slide 177
Acceptance of
worse values
Aspiration
Selection
ADVANCED METAHEURISTICS
PARETO SIMULATED ANNEALING
Poznan University of Technology
Graphical analysis
Slide 178
ADVANCED METAHEURISTICS
PARETO SIMULATED ANNEALING
Poznan University of Technology
Decision phase
Neighbor 3 selected as a new middle point and new neighbors
are generated
Final solution
Slide 179
ADVANCED METAHEURISTICS
PARETO SIMULATED ANNEALING
Poznan University of Technology
The most satisfactory schedule
Slide 180
ADVANCED METAHEURISTICS
PARETO SIMULATED ANNEALING
Poznan University of Technology
Comparison of solutions
Improvement:
3% to 24% on particular objectives
Slide 181
f2  24% ; f4  12%
ADVANCED METAHEURISTICS
PARETO SIMULATED ANNEALING
Poznan University of Technology
Conclusions
 Improvement of the real life solution
 Flexibility
 Good quality results
 Efficiency of work
Slide 182
Multiple objective genetic
local search
MOGLS
ADVANCED METAHEURISTICS
MULTIPLE OBJECTIVE GENETIC LOCAL SEARCH
Poznan University of Technology
Contents
General idea of hybrid algorithms
Single objective genetic local search
algorithm
Multiple objective genetic local
search algorithm
Slide 184
ADVANCED METAHEURISTICS
MULTIPLE OBJECTIVE GENETIC LOCAL SEARCH
Poznan University of Technology
General idea
Recombination
operators
Local search
MOGLS
MCDM
Slide 185
ADVANCED METAHEURISTICS
MULTIPLE OBJECTIVE GENETIC LOCAL SEARCH
Poznan University of Technology
General idea
 The algorithm hybridizes recombination
operations with local search (typical single
criterion algorithm)
 The idea of algorithm is more general, other local
huristic methods can be used
 MOGLS is a multiple objective version, contains
all aspects of MCDM
Slide 186
ADVANCED METAHEURISTICS
MULTIPLE OBJECTIVE GENETIC LOCAL SEARCH
Poznan University of Technology
Motivation for creating MOGLS
 Great success of other hybrid genetic meta-heuristics
HGAs
Memetic
algorithms
Slide 187
Genetic
Local Search
ADVANCED METAHEURISTICS
MULTIPLE OBJECTIVE GENETIC LOCAL SEARCH
Poznan University of Technology
Hybrid Genetic Algorithms (HGAs)
 Standard genetic/evolutionary algorithms working on a
reduced set of solutions
 Local heuristics = recombination operator (like crossover)
 The efficiency grows up, because the search space is
smaller
 Conclusion: the local optima can be achieved in very
efficient way
 HGAs may be also interpreted as a modification of multiple
start local search herusitcs with random initial solutions
Slide 188
ADVANCED METAHEURISTICS
MULTIPLE OBJECTIVE GENETIC LOCAL SEARCH
Poznan University of Technology
Single objective GLS algorithm
 Very good efficiency - > motivation to create
multiple objective version
 The algorithm stops if the current population was
not changed i K subsequent iterations (further
improvement is not possible)
Slide 189
ADVANCED METAHEURISTICS
MULTIPLE OBJECTIVE GENETIC LOCAL SEARCH
Poznan University of Technology
Single objective GLS algorithm
Slide 190
ADVANCED METAHEURISTICS
MULTIPLE OBJECTIVE GENETIC LOCAL SEARCH
Poznan University of Technology
Test of three single objective methods (for the same
instances)
Slide 191
ADVANCED METAHEURISTICS
MULTIPLE OBJECTIVE GENETIC LOCAL SEARCH
Poznan University of Technology
Test results on the graph
GLS generated best results
Slide 192
Poznan University of Technology
Application of MSLS
GA, GLS
Vehicle Routing Problem in a
Road Transportation
Company – case study
Slide 193
Vehicle Routing Problem in a Road
Transportation Company
Poznan University of Technology
 Road, freight transportation & logistic company, located in
Warsaw, Poland
 Activities: transportation and logistic services; forwarding;
customs clearence, national and international freight
transportation, maintenace / service of vehicles (MAN)
 Annual sales – 106 mln zl = 35 mlnEuro; 170 employees
 Fleet – 230 vehicles (tractors and trailors); capacity 20 – 30 T
 Transportation routes – 100 – 5000 km; 20 – 25 transportation
jobs / day
 Historical data – May – July 2005; 700 transportation jobs; 200
customers; 70% jobs generated by 20 customers (10%); 650
locations – 180 very important
 Analyzed case: 90 transportation jobs; 30 vehicles
Slide 194
Vehicle Routing Problem in a Road
Transportation Company
Poznan University of Technology
 The decision problem defined as a single objective
optimization problem


Correlation between criteria (time, cost, profit, distance)
Multiple vehicle, pick-up and delivery vehicle routing problem with time
windows (m-VPDVRPwTW)
 Not many reports about the solution procedures for this
problem

Slide 195
J. Desrosiers and others – small instance – application of Branch and
Bound
Mathematical Formulation of the Problem
Criterion
Poznan University of Technology
 Maximal profit in the time horizon [PLN]
R
Max Z   z r
r 1


W
R
B
 B wkm


Max z    i w  wr    cbr  lbr   cbprac  t bprac  
r 1  w 1
b 1
 b 1




B


 B wkm
dod
dod
prac
prac
prac
prac 
ir  c r   (cbr  lbr  cb  t b )   cb  t b 


b

1
b

1




B
B
    dod


dod
wkm
prac
prac
prac
prac 
i

c

(
c

l

c

t
)

c

t


r
r
br
br
b
b
b
b




b 1
b 1



(T  t r )
dod


tr
Slide 196
Mathematical Formulation of the Problem
Constarints
Poznan University of Technology
 Each order must be either completely fulfilled (by
1 or more vehicles) or rejected
 Each vehicle types and loads must match
 Capacity dimensions of the vehicle should
exceed weight/dimensions of the load
 Loading and unloading must be carried out in
concrete time windows
 Working time for drivers is defined by the labor
code
Slide 197
Mathematical experiment
Visualization of the optimal solution (GLS – 20 iterations)
Poznan University of Technology
Slide 198
The computational efficiency
MSLS vs. GLS vs. GA;Time of solution
Poznan University of Technology
PC Pentium IV/2,8 GHz; 30 vehicles and 90 orders
Slide 199
The computational efficiency
MSLS vs. GLS vs. GA;Time of solution
Poznan University of Technology
PC Pentium III 750 MHz; 30 vehicles and 90 orders
Slide 200
Intuitive vs. Computer planning
Poznan University of Technology
Parameter
Intuitive
planning
Forwarder Forwarder
1
2
Time of
planning
[min]
Profit
[PLN]
Slide 201
~ 230
~ 190
PC; program VR
MSLS
GA
GLS
~ 40
~ 40
~ 40
40000 36800 38500 38500 42200
Conclusion
Poznan University of Technology
 The real-life VRP is characterized by high
computational complexity
 GLS is the most efficient metaheuristic algorithm
(compared with MSLS & GA)
 For practical reasons it is advised to use
computers with high computational power to solve
VRP
 Practical results – computer system VR reduces
labor intensity by 80% and improves profits by
5.5%
Slide 202
ADVANCED METAHEURISTICS
MULTIPLE OBJECTIVE GENETIC LOCAL SEARCH
Poznan University of Technology
MOGLS
 Goal  generate good approximations of the nondominated
set
 Finding the whole nondominated set = finding the optima of
all weighted Tchebycheff and all weighted linear scalarizing
functions
 In fact the goal is a simultaneous optimalization of all
Tchebycheff and all weighted linear scalarizing functions
 „Optimization” is understood as a tendency of the
algorithm to improve values of all scalarizing functions
(with normalized weight vectors)
Slide 203
ADVANCED METAHEURISTICS
MULTIPLE OBJECTIVE GENETIC LOCAL SEARCH
Poznan University of Technology
MOGLS
 MOGLS implements the idea of simultanous optimization of
all weighted Tchebycheff, all weighted linear or all
composite scalarizing functions with normalized weight
vectors by random choice of the scalarizing function
optimized in each iteration
 MOGLS tries to improve the value of a randomly selected
scalarizing function in each iteration
 Single iteration consists of a single recombination of a pair
of solutions and application of a local heuristic that takes
into account the value of the current scalarizing function
Slide 204
ADVANCED METAHEURISTICS
MULTIPLE OBJECTIVE GENETIC LOCAL SEARCH
Poznan University of Technology
•
To draw at random the scalarizing funcion, a normalized weight vetor is
drawn at random by the algorithm
The algorithm assures that weight vectors are drawn with uniform
probability distribution p(Λ)
Rand() returns a value from <0,1>
Slide 205
ADVANCED METAHEURISTICS
MULTIPLE OBJECTIVE GENETIC LOCAL SEARCH
Poznan University of Technology
General scheme of the MOGLS
Slide 206
ADVANCED METAHEURISTICS
MULTIPLE OBJECTIVE GENETIC LOCAL SEARCH
Poznan University of Technology
Selection of solutions for recombination
 In single objective GLS the method combines features of
two good solutions
 MOGLS combines features of solutions that are already
good on the current scalarizing function
 In each iteration MOGLS algorithm constructs a temporary
elite population (TEP) composed of K different solutions
being the best on the current scalarizing function among all
known solutions.
 Two different solutions are drawn for recombination from
(TEP)
 The idea of recombining good solutions is motivated by
„global convexity”
– In single objective optimization this means – good solutions are similar
– In multiple objective optimization – good solutions on a given
scalarizing function being close in the objective space are similar
Slide 207
ADVANCED METAHEURISTICS
MULTIPLE OBJECTIVE GENETIC LOCAL SEARCH
Poznan University of Technology
Generating the initial set of solutions
 Construction by applying iteratively the local heuristic to
random starting solutions
 Local heuristic optimizes the scalarizing functions with
randomly generated weight vectors
 The number of the initial solutions S is the additional
parameter of the method
 The method allows to stop generating the initial solutions
when the avarage quality of K best solutions in the set of
initial solutions over all scalarizing functions is the same as
the avarage quality of solutions generated by the local
heuristic used for optimization of these functions
Slide 208
ADVANCED METAHEURISTICS
MULTIPLE OBJECTIVE GENETIC LOCAL SEARCH
Poznan University of Technology
Example of generated solutions for TSP
Slide 209
ADVANCED METAHEURISTICS
MULTIPLE OBJECTIVE GENETIC LOCAL SEARCH
Poznan University of Technology
Management of the current set of solutions
 The idea of storing all solutions in CS is very time and
memory consuming for MOGLS
 CS is organized as a queue of size KxS (K- number of best
solutions, S – number of initial solutions)
 In each iteration the newly generated solution is added to
the beginning of the queue (if the conditions are met); if it is
better than the worst solution in TEP and different form all
solutions in TEP
 If the size of queue is bigger than KxS the last solution is
removed
Slide 210
ADVANCED METAHEURISTICS
MULTIPLE OBJECTIVE GENETIC LOCAL SEARCH
Poznan University of Technology
Updating the reference point
 The reference point is an important parameter in case of
weighted Tchebycheff and composite scalarizing functions
 In MOGLS the ideal point (best known values of the
objective functions) is used as reference point
 The reference point changes in the run of the procedure
 The first approximation is obtained by applying local
heuristic to optimization of each objective individually.
Normalization of objectives, updating the set of PP,
using partial preference information
Analogy with PSA
Slide 211
Poznan University of Technology
MOGLS
pseudocode
Slide 212
Memetic algorithm and Pareto
memetic algorithm
ADVANCED METAHEURISTICS
MEMETIC ALGORITHM AND
PARETO MEMETIC ALGORITHM
Contents
 Memetic
 Memetic algorithm
 Pareto Memetic algorithm
Slide 214
Poznan University of Technology
ADVANCED METAHEURISTICS
MEMETIC ALGORITHM AND
PARETO MEMETIC ALGORITHM
Poznan University of Technology
Memetics - Genetics
Meme
transmission, or imitation”
“an element of culture that
may be considered to be
passed on by non-genetic
means”
Richard Dawkin , ethologist
English Oxford Dictionary
“the basic unit of cultural
Slide 215
ADVANCED METAHEURISTICS
MEMETIC ALGORITHM AND
PARETO MEMETIC ALGORITHM
Poznan University of Technology
Memetics - Genetics
 Mem is defined per analogy to gen
 Evolution is not only based on genetics
 Term Memetic algorithm was first used by Moscato in 1989
in the sense of population-based hybrid genetic algorithm
with some learing procedures
Slide 216
ADVANCED METAHEURISTICS
MEMETIC ALGORITHM AND
PARETO MEMETIC ALGORITHM
Poznan University of Technology
Memetic algorithm
Inspiration
Darwinian natural
evolution
Dawkins’ conception
of a meme
Techniques
Search algorithm
(LS)
Slide 217
Evolutionary algorithm
(GA)
ADVANCED METAHEURISTICS
MEMETIC ALGORITHM AND
PARETO MEMETIC ALGORITHM
Poznan University of Technology
General scheme of the memetic algorithm
 Initiation: generating an initial population
 Iteration (until termination conditions are reached)


Improvement of current solutions (by local optimalization methods)
Developing of new generation (solutions) by evolutionary algorithm
For improvements MA can use any local optimalization method
like local search, tabu search or another one
Slide 218
ADVANCED METAHEURISTICS
MEMETIC ALGORITHM AND
PARETO MEMETIC ALGORITHM
General scheme of the memetic algorithm
General initial population
Select individuals for nest generations
Crossover
Mutation
Local search
Population complete?
Enough generations found?
Slide 219
Poznan University of Technology
ADVANCED METAHEURISTICS
MEMETIC ALGORITHM AND
PARETO MEMETIC ALGORITHM
Pseudocode of the memetic algorithm
Slide 220
Poznan University of Technology
ADVANCED METAHEURISTICS
MEMETIC ALGORITHM AND
PARETO MEMETIC ALGORITHM
Poznan University of Technology
Pareto memetic algorithm
Author: A. Jaszkiewicz, Poznan University
of Technology
Modification of MOGLS (Multiple objective
genetic local search)
Slide 221
ADVANCED METAHEURISTICS
MEMETIC ALGORITHM AND
PARETO MEMETIC ALGORITHM
Poznan University of Technology
Two stages of the algorithm
Stage 1:
 Initiation
 Generation of the first approximation of the ideal
point
 Generation of the initial set of solutions
Stage 2
 Probabilistic choice of two solutions
 Recombination and improvement
Slide 222
ADVANCED METAHEURISTICS
MEMETIC ALGORITHM AND
PARETO MEMETIC ALGORITHM
Poznan University of Technology
Initiation

At the beginning a set of Pareto-optimal
solutions is empty.
PP:=Ø

The current set of solutions is empty,
too.
CS:=Ø
Slide 223
ADVANCED METAHEURISTICS
MEMETIC ALGORITHM AND
PARETO MEMETIC ALGORITHM
Poznan University of Technology
Generation of the first approximation of the ideal point
Random creation of a possible solution x
Optimalization x to x’ by local heuristic
algorithm
Adding x’ to CS
Updating set PP with x’
Slide 224
ADVANCED METAHEURISTICS
MEMETIC ALGORITHM AND
PARETO MEMETIC ALGORITHM
Poznan University of Technology
Generation of the initial set of solutions
• Randoming a weight vector Λ
 Random creation of a possible solution x
 Optimalization of the scalarizing function (z,.. Λ ) x to x’ by
local search
 Adding x’ to CS
• Updating set PP with x’
This phase is iterated until stopping condition is met
Slide 225
ADVANCED METAHEURISTICS
MEMETIC ALGORITHM AND
PARETO MEMETIC ALGORITHM
Probabilistic choice of two solutions
Randoming a weight vector Λ
Drawing randomly a sample of
solutions from CS
Slide 226
Poznan University of Technology
ADVANCED METAHEURISTICS
MEMETIC ALGORITHM AND
PARETO MEMETIC ALGORITHM
Poznan University of Technology
Recombination and improvement
Recombination of the best and second
best solution on s(z,..., Λ) – x1
Optimization of s(z,.. Λ ) x1 to x1’ by local
search
Adding x1’ to CS and updating PP if x1’ is
better than the second best solution in a
sample
Slide 227
Poznan University of Technology
Pseudo code of
PMA
Slide 228
Poznan University of Technology
Application of PMA
Vehicle Assignment
Problem in the Bus
Transportation Company
– case study
Slide 229
Introduction (I)
Poznan University of Technology
 The essence of the vehicle assignment problem (VAP) in a bus
transportation company


transportation companies utilise vehicles (buses) to transport passengers on
given routes according to a given timetable
general problem in such a situation is: How to assign particular buses to given
routes?
 Many formulations of the VAP are known, for example




Slide 230
linear programming formulations which can be solved with an application of
simplex method, network algorithms or assignment method (Cook 1985; Lotfi et
al. 1989)
linear, integer programming formulations (Löbel 1998, Rushmeier et al. 1997),
sometimes transformed into a non-linear, continuous form (Beaujon et al. 1991)
formulations based on the queuing theory (Green et al. 1995, Whitt 1992)
formulations considering the homogeneous (Beaujon et al. 1991) or a nonhomogeneous fleet (Ziarati et al. 1999)
Introduction (II)
Poznan University of Technology



formulations which combine the VAP with other fleet management
problems, such as: fleet sizing (Beaujon et al. 1991) or fleet scheduling
(Löbel 1998)
formulations referring to specific transportation environments, such as:
urban transportation (Löbel 1998), rail transportation (Ziarati et al.
1999) or air transportation (Rushmeier et al. 1997)
formulations with a single objective function (all mentioned above) or,
sometimes, with a multicriteria objective function (Zeleny 1982)
Proposed problem formulation



Slide 231
is expressed in terms of multicriteria, non-linear, integer, mathematical
programming
determines the optimal assignment of non-homogeneous fleet of buses
to a given set of routes in an international passenger transportation
company
one week time horizon is assumed for the problem analysis
Computational Experiment
Decision situation (I)
Poznan University of Technology
 A Polish, passenger transportation company operating on the
17 routes between 34 Polish and 47 European cities is
analysed
 All the routes are characterised by the following parameters:





length Si between 1818 and 4048 kilometres
average number of passengers travelling weekly on particular routes Pi
between 2 and 796
average income per one passenger (ticket price) ppas i between 188
and 721 PLN*
average load index wi between 0.25 and 0.46
fixed cost kij per route i and bus j between 3 530 and 14 809 PLN / ride
* PLN – Polish New – Polish currency. 1 PLN = 0.24 USD in December 2001
Slide 232
Computational Experiment
Decision situation (II)
Poznan University of Technology
Analysed company utilises a fleet of 30
buses (Hyundai, Neoplan, Scania, Volvo)
characterised by:



Slide 233
vehicle-kilometre cost kwkm ij between 1.49 and 2.01
PLN / kilometre
number of seats (capacity) cj between 31 and 57
comfort level fj between 3 and 9 points (comfort level
ranges from 1 to10 points)
Mathematical Formulation of the Problem
Input data – model parameters
Poznan University of Technology








Slide 234
Si – length of route i [kilometres]
Pi – average number of passengers travelling weekly on route i
[persons]
ppas i – average income per one passenger travelling on route i (ticket
price) [monetary units]
wi – average load index of a bus on route i [-], expressed as a quotient
of an average number of tickets sold for a particular ride on route i and
an average number of passengers in a bus during this ride, wi  {0,1}
kij – fixed cost per route i and bus j, including drivers’ salaries, highway
fares, tolls, insurance and licence fees etc. [monetary units / ride]
kwkm ij – variable (vehicle-kilometre) cost per bus j and route i, including
fuel and maintenance cost [monetary unit / kilometre]
cj – capacity of bus j – number of seats [-]
fj – travelling comfort level of bus j [-], expressed in points according to
the following characteristics of bus j: seats’ comfort (size, softness), air
conditioning, toilet, video etc., fj  {1, 2, 3, 4, ..., fmax = 10}
Mathematical Formulation of the Problem
Decision variables and Criteria
Poznan University of Technology
• The integer decision variable ij  {0, 1, 2, 3, ...}, denominates a
number of rides carried out weekly by a vehicle j on route i
Criterion
Unit
Dp
[monetary
units]
max
2. Capacity utilisation – WL
-
min
3. Total number of weekly lost
(rejected) customers
(passengers) – SK
-
min
[points]
max
1. Total weekly profit – Z
4. Comfort of travel for passengers –
WK
Slide 235
Consequence
The maximal number of
passengers should be
transported with minimal
costs
Average capacity utilisation
should be close to 80%
(assumed optimal level),
percentage of empty rides
should be minimal
Assures that all demand
will be satisfied,- high
customers’ satisfaction
High quality service
Mathematical Formulation of the Problem
Criteria
Poznan University of Technology
Criterion
Formula
J



 J


Max Z   Wi   S i    ij k wkm ij     ij  kij 
i 1 
j 1


 j 1

Wi  Pi  SKi   ppas i
I
1. Total weekly
profit – Z
2. Capacity
utilisation – WL
 I J  minPoi, cj

Min WL  WL opt   
  ij 
cj

 i 1 j 1 

Poi   Pi

J

j 1

I

J
  ij 

i 1 j 1
  ij   wi
I
3. Total number of
weekly lost
customers – SK
Min SK   SK i
i 1
J



 Poi cj 
i SKi  max 0, Pi   min  ,    ij  
 wi wi 
j 1 


I
J
4. Comfort of travel
Max WK   min Poi, cj  ij  fj
for passengers
i 1 j 1
– WK
Slide 236
I
J
 min P , c  
oi
i 1 j 1
j
ij
f
max
Mathematical Formulation of the Problem
Constraints
Poznan University of Technology
 The presented model takes into consideration the
following constraints:

real riding time by bus j on route i should be consistent with the
timetable

weekly working time of bus j should not be grater than its
maximal weekly working time, including maintenance (repair
and service) times
Slide 237
Mathematical Formulation of the Problem
Output data – the results
Poznan University of Technology
 As a result DM obtains the
most satisfactory solution of
the problem from the
company’s and its
customers’ point of view:

Slide 238
bus assignment
expected values of considered
criteria

ωijij =
Bus j

Route i
 1 0 ... 2 2 0 
 0 3 ... 1 0 0 


 ... ... ... ... ... ...


 1 0 ... 1 0 1 
 5 0 ... 0 1 0 


 0 0 ... 1 0 1 
 ... ... ... ... ... ...
 0 4 ... 0 0 0 


Computational Experiment
Stage one - results of PMA
Poznan University of Technology
 A sample of Pareto - optimal solutions generated after 60 000
iterations (recombination and local improvements) is
composed of 2 985 different solutions (possible assignments
of buses)
 The range of considered criteria:
Capacity
Number of lost
Profit - Z
utilisation - WL passengers – SK
[PLN]
Comfort of
travel - WK
[-]
[-]
[-]
Min
-2 669 000
0.11
0
0.85
Max
1 802 570
0.74
181
0.90
Slide 239
Computational Experiment
Stage two - settings of LBS method
Poznan University of Technology
Slide 240
Computational Experiment
Stage two - results of LBS method
Poznan University of Technology
 DM is interested in solution A2960 which outranks the present middle
point on criterion 1 (by 300 000 PLN) and is indifferent on the other
criteria
 Solution A2960 becomes a new middle point – its neighbourhood
consists of 38 solutions including solution A2959 which has been
Slide 241
selected by DM as the most satisfactory, compromise solution
Computational Experiment
Stage two – „the best” assignment of buses (solution
A2959)
Poznan University of Technology
Buses j
Routes i
1
1
0
2
0
3
0
4
0
5
0
6
0
7
0
8
0
10
11
12
13
14
15
3
1
2
1
1
0
1
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
1
0
1
0
0
0
0
0
0
0
0
1
0
0
1
0
0
0
0
0
0
0
0
0
0
0
9
0
…
0
0
1
0
0
0
10
0
11
0
12
0
13
0
14
0
15
0
16
0
17
0
1
0
0
0
0
0
1
0
0
0
0
0
1
1
0
1
1
0
0
0
1
0
0
0
1
0
0
0
0
0
0
0
0
1
0
0
0
1
0
0
1
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
1
0
1
0
...
20
21
22
23
24
25
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
...
30
Slide 242
1
0
0
0
1
1
1
0
0
Computational Experiment
Compromise solution vs. other Pareto - optimal
solutions
Other Pareto – optimal
solutions
Poznan University of Technology
Min
Max
The most
satisfactory
assignment
(solution A2959)
Profit – Z
- 2 669 000
1 802 670
1 752 140
Capacity utilisation - WL
Number of lost
passengers – SK
Comfort of travel - WK
0.11
0.74
0.15
0
181
22
0.85
0.90
0.87
Objective
Slide 243
Conclusions
Poznan University of Technology
 The presented methodology lets DM to define the most
satisfactory assignment of buses to particular routes
 The methodology can by applied in a long-distance passenger
transportation companies utilising a non-homogeneous fleet
of buses
 The methodology leads to the profitability analysis of
particular routes. Based on the analysis of criterion 1 certain,
non-profitable routes can be eliminated from the existing
portfolio of the transportation services. It also allows to define
the minimal ticket price for each route to assure its acceptable
profitability and maintain this service in the portfolio
 The methodology of solving VAP combined with an
appropriate database let us create the modern DSS for such a
problem in the future
Slide 244