Basics on Decision Making

Motivation Phases Implementation
Artiﬁcial Intelligence
Graph theory
G. Guérard
Department of Nouvelles Energies
Ecole Supérieure d’Ingénieurs Léonard de Vinci
Lecture 0
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OutlineOutlineOutlineOutlineOutlineOutlineOutlineOutlineOutlineOutlineOutlineOutlineOutlineOutlineOutlineOutlineOutline
1 Motivation
2 Phases
Information gathering phase
Design phase
Choice phase
3 Implementation
Deﬁnition
Implementation
Paradigm
Algorithm’s eﬃciency
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What is decision making ?What is decision making ?What is decision making ?What is decision making ?What is decision making ?What is decision making ?What is decision making ?What is decision making ?What is decision making ?What is decision making ?What is decision making ?What is decision making ?What is decision making ?What is decision making ?What is decision making ?What is decision making ?What is decision making ?
Motivation
Decision making is a process of choosing among two or more
alternate courses of action for the purpose of attaining a goal (or
goals).
How to choose between production’s strategies ?
How to ﬁnd a shortest path in a traﬃc ?
Etc.
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Decision Making PhasesDecision Making PhasesDecision Making PhasesDecision Making PhasesDecision Making PhasesDecision Making PhasesDecision Making PhasesDecision Making PhasesDecision Making PhasesDecision Making PhasesDecision Making PhasesDecision Making PhasesDecision Making PhasesDecision Making PhasesDecision Making PhasesDecision Making PhasesDecision Making Phases
Decision making involves three major phases followed by the
validation phase:
Information gathering,
Design,
Choice,
Implementation,
Validation.
To examine and to identify the problem (variables, functions, val-
ues, etc.).
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validation phase:
Design,
Choice,
Implementation,
Validation.
To construct a model that represents the system.
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validation phase:
Design,
Choice,
Implementation,
Validation.
To select a solution to the model.
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validation phase:
Design,
Choice,
Implementation,
Validation.
To implemente an informatic software to solve any instance of the
model.
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validation phase:
Design,
Choice,
Implementation,
Validation.
To validate the software if it shows reasonable computing time and
good solution.
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Information gathering phaseInformation gathering phaseInformation gathering phaseInformation gathering phaseInformation gathering phaseInformation gathering phaseInformation gathering phaseInformation gathering phaseInformation gathering phaseInformation gathering phaseInformation gathering phaseInformation gathering phaseInformation gathering phaseInformation gathering phaseInformation gathering phaseInformation gathering phaseInformation gathering phase
Information gathering includes several activities aimed at identifying
problem situations or opportunities :
1 Problem identification
2 Problem classification
3 Problem decomposition
Identification of organized goals and objectives related to an issue
of concern.
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To determine whether a problem exists, identify its symptoms, de-
termine its magnitude, and explicitly deﬁne it.
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Some issues may arise during data collection and estimation like
lack of data, inoccurate or inprecise data, bad estimation, information
overload, only simulated data, etc.
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Problem classiﬁcation is the placement of a problem in a deﬁnable
category. This leads to a standard solution approach.
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Many complex problems can be divided into sub-problems.
Some unstructured problems may have some highly structured sub-
problems.
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Design phaseDesign phaseDesign phaseDesign phaseDesign phaseDesign phaseDesign phaseDesign phaseDesign phaseDesign phaseDesign phaseDesign phaseDesign phaseDesign phaseDesign phaseDesign phaseDesign phase
The design phase involves ﬁnding and analyzing possible courses
of action :
1 To understand the problem
2 A model is constructed, tested, and validated
Modelling involves abstracting the problem to quantitative and/or
qualitative forms.
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of action :
For a mathematical model, the variables are identiﬁed and the rela-
tionships among them are established.
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of action :
All models are made up of three basic components: decision vari-
ables, uncontrollable variables, and result variables. Mathamtical
relationships link these components together.
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of action :
In a non-quantitative model, the relationships are symbolic or quali-
tative.
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All models are made up of three basic components :
1 Result variables
2 Decision variables
3 Uncontrollable variables
Result variables are outputs. The reﬂect the level of eﬀectiveness
of the system. These are dependent variables.
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1 Result variables
They describe alternative courses of action.
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1 Result variables
They are factors that aﬀect the result variables but are not under
control of the decision maker. Either these factors are ﬁxed or they
can vary.
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Structure of quantitative models
The components are linked together by mathematical
expressions.
Principle of Choice
A principle of choice is a criterion that describes the acceptability
of a solution approach
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Normative models
The chosen solution is demonstrably the best of all possible
alternatives. To find it, one should examine all alternatives and
prove that the one selected is indeed the best one. The process is
basically Optimization.
Descriptive models
They investigate alternate courses of action under different
configurations of inputs and processes. All the alternatives are not
checked, only a given set are.
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Choice phaseChoice phaseChoice phaseChoice phaseChoice phaseChoice phaseChoice phaseChoice phaseChoice phaseChoice phaseChoice phaseChoice phaseChoice phaseChoice phaseChoice phaseChoice phaseChoice phase
The choice phase includes:
Recommendation of a solution
Evaluation of a solution
When the problem is solved, a solution is chosen based on the
model. This one is recommended over the set of solutions.
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Evaluation is possible if the result variables give a quantitative so-
lution. A preference relation over the set of solution give an order:
f (a) f (b) means that f (b) is better or equal to f (a) where a, b
are variables and f (.) is a mathematical function based on result
variables.
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b is the best solution iﬀ: f (x) f (b) for any x in the set of solution.
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AlgorithmsAlgorithmsAlgorithmsAlgorithmsAlgorithmsAlgorithmsAlgorithmsAlgorithmsAlgorithmsAlgorithmsAlgorithmsAlgorithmsAlgorithmsAlgorithmsAlgorithmsAlgorithmsAlgorithms
An algorithm is a self-contained step-by-step set of operations to be
performed.
An algorithm is an effective method that can be expressed within
a finite amount of space and time and in a well-defined formal
language for calculating a function. Starting from an initial state
and initial input (perhaps empty), the instructions describe a
computation that, when executed, proceeds through a finite number
of well-defined successive states, eventually producing output and
terminating at a final ending state. The transition from one state
to the next is not necessarily deterministic; some algorithms, known
as randomized algorithms, incorporate random input.
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FlowchartFlowchartFlowchartFlowchartFlowchartFlowchartFlowchartFlowchartFlowchartFlowchartFlowchartFlowchartFlowchartFlowchartFlowchartFlowchartFlowchart
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PseudocodePseudocodePseudocodePseudocodePseudocodePseudocodePseudocodePseudocodePseudocodePseudocodePseudocodePseudocodePseudocodePseudocodePseudocodePseudocodePseudocode
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Classification by implementationClassification by implementationClassification by implementationClassification by implementationClassification by implementationClassification by implementationClassification by implementationClassification by implementationClassification by implementationClassification by implementationClassification by implementationClassification by implementationClassification by implementationClassification by implementationClassification by implementationClassification by implementationClassification by implementation
1 Classic, iterative implementation
2 Recursion
3 Logical
4 Deterministic or non-deterministic
5 Exact or approximate
6 Serial, parallel or distributed
Iterative algorithms use repetitive constructs like loops and some-
times additional data structures like stacks to solve the given prob-
lems.
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2 Recursion
3 Logical
A recursive algorithm is one that makes reference to itself repeat-
edly until a termination condition matches.
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2 Recursion
3 Logical
A logical algorithm is composed by logic component and control
component.
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2 Recursion
3 Logical
Deterministic algorithms solve the problem with exact decision at
every step of the algorithm whereas non-deterministic algorithms
solve problems via guessing. A deterministic algorithm is an algo-
rithm which, given a particular input, will always produce the same
output.
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2 Recursion
3 Logical
While many algorithms reach an exact solution, approximation algo-
rithms seek an approximation that is close to the true solution.
When it is not possible to ﬁnd a best solution in human time, one
can seek for an approximate solution by less complex algorithms.
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2 Recursion
3 Logical
Those implementations depend of computer architectures, i.e. num-
ber of processors that can work on a problem at the same time, and
how to decompose the whole process into independent processes.
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RecursionRecursionRecursionRecursionRecursionRecursionRecursionRecursionRecursionRecursionRecursionRecursionRecursionRecursionRecursionRecursionRecursion
A recursive algorithm is one that makes reference to itself
repeatedly until a termination condition matches.
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LogicalLogicalLogicalLogicalLogicalLogicalLogicalLogicalLogicalLogicalLogicalLogicalLogicalLogicalLogicalLogicalLogical
A logical algorithm is composed by logic component and
control component.
The logic component expresses the axioms that may be used in the
computation and the control component determines the way in
which deduction is applied to the axioms. This is the basis for the
logic programming paradigm. In pure logic programming languages
the control component is ﬁxed and algorithms are speciﬁed by
supplying only the logic component.
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Logical: prologLogical: prologLogical: prologLogical: prologLogical: prologLogical: prologLogical: prologLogical: prologLogical: prologLogical: prologLogical: prologLogical: prologLogical: prologLogical: prologLogical: prologLogical: prologLogical: prolog
Basics: man(adam). man(peter). man(paul). woman(marry).
woman(eve).
parent(adam,peter). parent(eve,peter). parent(adam,paul).
parent(marry,paul).
Controls: father(F,C):-man(F),parent(F,C).
mother(M,C):-woman(M),parent(M,C). is_father(F):-father(F,_).
is_mother(M):-mother(M,_).
Question: ?-is_father(adam).
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Classification by paradigmClassification by paradigmClassification by paradigmClassification by paradigmClassification by paradigmClassification by paradigmClassification by paradigmClassification by paradigmClassification by paradigmClassification by paradigmClassification by paradigmClassification by paradigmClassification by paradigmClassification by paradigmClassification by paradigmClassification by paradigmClassification by paradigm
1 Brute-force
2 Divide and conquer
3 Dynamic programming
4 Search
5 Randomized
6 Reduction
Naive algorithms try every possible solution to see which is best.
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1 Brute-force
4 Search
5 Randomized
6 Reduction
A divide and conquer algorithm repeatedly reduces an instance of
a problem to one or more smaller instances of the same problem until
the instances are small enough to solve easily.
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1 Brute-force
4 Search
5 Randomized
6 Reduction
A problem that can be solved optimally by breaking it into sub-
problems and then recursively ﬁnding the optimal solutions to the
sub-problems is said to have optimal substructure.
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1 Brute-force
4 Search
5 Randomized
6 Reduction
Many problems can be modeled as problems on graphs. A graph
exploration algorithm speciﬁes rules for moving around a graph and
is useful for such problems.
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1 Brute-force
4 Search
5 Randomized
6 Reduction
Such algorithms make some choices randomly. They can be very
useful in ﬁnding approximate solutions for problems where ﬁnding
exact solutions can be impractical.
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1 Brute-force
4 Search
5 Randomized
6 Reduction
This technique involves solving a diﬃcult problem by transforming
it into a better known problem for which we have optimal algo-
rithms. The goal is to ﬁnd a reducing algorithm whose complexity is
not dominated by the resulting reduced algorithm’s.
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Divide and conquerDivide and conquerDivide and conquerDivide and conquerDivide and conquerDivide and conquerDivide and conquerDivide and conquerDivide and conquerDivide and conquerDivide and conquerDivide and conquerDivide and conquerDivide and conquerDivide and conquerDivide and conquerDivide and conquer
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Dynamic programmingDynamic programmingDynamic programmingDynamic programmingDynamic programmingDynamic programmingDynamic programmingDynamic programmingDynamic programmingDynamic programmingDynamic programmingDynamic programmingDynamic programmingDynamic programmingDynamic programmingDynamic programmingDynamic programming
The longest common subsequence (LCS) problem is the problem of
ﬁnding the longest subsequence common to all sequences in a set
of sequences.
LCS (Xi ,Yj ) =



if i = 0 or j = 0
LCS (Xi−1,Yj−1) xi if xi = yj
longest(LCS (Xi ,Yj−1),LCS (Xi−1,Yj )) if xi = yj
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ReductionReductionReductionReductionReductionReductionReductionReductionReductionReductionReductionReductionReductionReductionReductionReductionReduction
We tranform a bipartite matching problem into a ﬂow problem :
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BasicsBasicsBasicsBasicsBasicsBasicsBasicsBasicsBasicsBasicsBasicsBasicsBasicsBasicsBasicsBasicsBasics
Termination
To determine whether the evaluation of a given program will
deﬁnitely terminate
Correctness
An input give a solution of the problem.
Completeness
For a feasible domain, the algorithm give a solution.
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ExampleExampleExampleExampleExampleExampleExampleExampleExampleExampleExampleExampleExampleExampleExampleExampleExample
Termination: in all condition, the algorithm returns a value.
Correctness: for an input n, the algorithm returns a factorial.
Completeness: n is an integer.
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ComplexityComplexityComplexityComplexityComplexityComplexityComplexityComplexityComplexityComplexityComplexityComplexityComplexityComplexityComplexityComplexityComplexity
The best, worst and average case complexity refer to three different ways of
measuring the time complexity (or any other complexity measure) of different inputs
of the same size. Since some inputs of size n may be faster to solve than others, we
define the following complexities:
Best-case complexity
This is the complexity of solving the problem for the best input of size n.
Worst-case complexity
This is the complexity of solving the problem for the worst input of size n.
Average-case complexity
This is the complexity of solving the problem on an average. This complexity is only
defined with respect to a probability distribution over the inputs. For instance, if all
inputs of the same size are assumed to be equally likely to appear, the average case
complexity can be defined with respect to the uniform distribution over all inputs of
size n.
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Complexity (time)Complexity (time)Complexity (time)Complexity (time)Complexity (time)Complexity (time)Complexity (time)Complexity (time)Complexity (time)Complexity (time)Complexity (time)Complexity (time)Complexity (time)Complexity (time)Complexity (time)Complexity (time)Complexity (time)
Notation Examples
O(1) Constant: condition, elementary computing
O(log(n)) Logarithm: tree, binary search
O(n) Linear: for, while, etc.
O(n ∗log(n)) Quasilinear: quicksort, mergesort, tree search
O(n²) Quadratic: multiplying two n-digit numbers, sort, matrix
O(np) Polynomial: grammar, matching, reduction
O(2n) Exponential: brute-force, search tree
O(pn) Exponential: dynamic programming, logical brute-force
O(n!) Factorial: naive combinatory algorithm
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ExampleExampleExampleExampleExampleExampleExampleExampleExampleExampleExampleExampleExampleExampleExampleExampleExample
Bubble sort: sort an array of n numbers by switching two
following indexes.
The best time complexity is when the algorithm do one loop:
O(n).
The worst time complexity is when the algorithm do n loop:
O(n2).
In average, the algorithm is quadratric: O(n2).
Data are stored in an array, data are reached directly by a
reference in memory. Data complexity is: O(1).
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Basics on Decision Making

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