chapter 1 scm
TRANSCRIPT
CHAPTER 1
INTRODUCTION
1.1 BACKGROUND OF SUPPLY CHAIN MANAGEMENT
Competitiveness in today’s marketplace depends heavily on the
ability of a company to handle the several important challenges like reducing
total supply chain operating cost and reducing lead-times, increasing customer
service levels, and improving product quality.
In Figure 1.1 a typical non integral supply chain is shown, in which
the goods flow starts as raw materials at natural resources and ends with
products at final customers. Raw material winners keep raw materials on
stock and supply them to component manufacturers. Component
manufacturers have an inventory of materials at the start of the production
process and an inventory of components at the end. Product manufacturers
hold inventories of products and components, the latter being supplied by
component manufacturers. Wholesalers buy products from product
manufacturers and hold central and regional stock in central and regional
distribution centers respectively. Retailers get their supply from wholesalers
and have products in local stock for sales to final customers. In today’s
world, most supply chains still do not profit from supply chain management
from natural resources to final customers. Instead, the supply chains struggle
due to the increase of competition and dynamics in today’s markets,
organizations integral inventory management due to opposition between
organizations. Due to the increase of competition and dynamics in today’s
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markets, organizations are forced to further improve their business
performance. So far, very few supply chains have realized supply chain
management from natural resources to final customers, by one organization
dominating the other organizations in the supply chain. As presented in
Figure 1.2, a dominant organization can make use of hierarchical control to
achieve integral inventory management across organizational boundaries. The
subordinate organizations in the supply chain are forced to obey the
instructions of the dominant organization.
Figure 1.1 Non-integral inventory management by opposition between
organizations using central and procedural information
systems
To obtain the required flexibility, supply chains could introduce
supply chain management from natural resources to final customers, by co-
operation across a network of organizations. As is shown in Figure 1.3,
networked organizations apply lateral control to achieve integral inventory
management beyond their organizational boundaries.
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Figure 1.2 Integral inventory management by domination of one
organization over others, using a central and procedural
information system
When compared to non-integral inventory management in supply
chains due to opposition between organizations, the extension of integral
inventory management beyond organizational boundaries can increase the
productivity of supply chains by reducing inventory related costs and
improving inventory related quality (customer service). By focusing on the
inventory aspect, a network of supply chains can be considered as a network
of stock-points connected through transit processes.
Figure 1.3 Integral inventory management by co-operation across
networked organizations, using distributed and object-
oriented information systems
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Raw materials, components, assemblies and finished products all
represent inventory at the various stages of production and distribution.
Inventory occurs at all places in a supply chain, both in storage processes and
in production and in transportation processes. Inventory in a storage process is
called stock-point inventory, while inventory in production and transportation
processes is called transit inventory.
Inventory categories in supply chains are plotted for a particular
supply chain, from natural resources to final customers in Figure 1.4. For
every stage in the supply chain the inventory levels are indicated as per
category. Given the levels of the desired as well as the required inventory,
there would ideally be no extra inventory categories in a supply chain.
Integral inventory management concerns the coordinated planning, control
and monitoring of inventory levels in stock-points throughout supply chains,
in order to maximize overall supply chain performance. Integral inventory
management is a means for businesses by which they can raise their
productivity, including cost reduction and quality improvement.
Figure 1.4 Inventory categories in supply chains
Thus, integral inventory management and networked organization
management are supplementary directions for improvement of business
performance in supply chains. Ideally, both issues in supply chain
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management should be combined to satisfy customer requirements. The
combination of Integral Inventory Management (IIM) and Networked
Organization Management (NOM) is called Networked inventory
management (NIM) (Verwijmeren 1996a and Verwijmeren 1997). The
simultaneous achievement of integral inventory management and networked
organization management is depicted in Figure 1.5 .
Figure 1.5 Networked inventory management
1.2 SUPPLY CHAIN MANAGEMENT (SCM)
The challenges facing business organizations continue to grow
rapidly due to the globalization of commerce, rapid commoditization of
products, demand for customized products, global distribution of
manufacturing and warehousing facilities, etc. All these factors have driven
business organizations to concentrate on their supply chains so as to gain
competitive advantage in the market place. The ability to manage the
complete Supply Chain Network (SCN) and to optimize decisions is
increasingly being recognized as a crucial competitive factor in order to make
good decisions within a SCN. In today’s competitive world, the success of an
industry is contingent upon the management of its supply chain. The global
economy and the recent developments in information technology have
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significantly modified the business organization of enterprises and the way
that they do business. New forms of organizations such as extended
enterprises and virtual enterprises turn to appear and they are quickly adopted
by most leading enterprises. It is noticed that “competition in the future will
not be between individual organizations but between competing supply
chains” (Christopher, 1992). Thus, business opportunities are captured by
groups of enterprises in the same enterprise network. The main reason for this
change is the global competition that forces enterprises to focus on their core
competences. According to a visionary report of manufacturing challenges
2020 conducted in USA, this trend will continue and one of the six grand
challenges of this visionary report is the ability to reconfigure manufacturing
enterprises rapidly in response to changing needs and opportunities. Among
the techniques supporting a multi-decisional context, computer simulation
algorithms can undoubtedly play an important role in evaluating quantitative
and qualitative benefits in supply chain management environment. This
research focuses on the application of new heuristic optimization algorithms
in supply chain network architectures to meet above challenges effectively
and efficiently.
1.2.1 Definition of Supply Chain Management
In early 1990s, the phrase “supply chain management” came in to
use . In today’s global market, managing the entire supply chain becomes key
factor for the successful business. World-Class organizations now realize that
non-integrated manufacturing processes, non-integrated distribution processes
and poor relationship with suppliers and customers are inadequate for their
success. The challenges facing business organizations grow rapidly because
of the globalization of commerce, global distribution of manufacturing and
warehousing facilities, rapid commoditization of products, demand for
customized products, competitive pressures, rapid advances of information
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technology, etc. The task of getting the right product to the right customer at
the right place and at the right time is not easy, and this task leads to the
study on ‘supply chain’. Supply Chain Management (SCM) is a relatively
new term. It crystallizes concepts about integrated business planning that have
been espoused by logistics experts, strategists, and operations research
practitioners as far back as the 1950’s. Today, integrated planning is finally
possible due to advances in information technology, but most companies still
have much to learn about implementing new analytical tools needed to
achieve it.
Simchi-Levi et al (2000) defined SCM as a set of approaches
utilized to efficiently integrate suppliers, manufacturers, warehouses and
stores, so that merchandise is produced and distributed at the right quantities,
to the right location and at the right time in order to minimize system-wide
cost, while satisfying service level requirements. On the other hand,
Christopher (2000) defined SCM as the management of upstream and
downstream relationships with suppliers and customers to deliver superior
customer value at minimal cost in the supply chain as a whole. The common
thread in any definition is that supply chain management seeks to integrate
performance measures over multiple firms or processes, rather than taking the
perspective of a single firm or process. General supply chain network is
shown in Figure 1.6.
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Figure 1.6 Multi-echelon supply chain network
1.2.2 Functions and Tasks of Supply Chain Management
Using the concept of flow from raw material to consumer, Mabert
and Venkataramanan (1998) presented a general structure of the supply chain
and sample of elements (managerial functions and tasks) that configure it. The
chain contains five aggregate or major stages to represent important phases in
the flow.
Sourcing involves not only the supply of raw materials and
components through a network of vendors; it also includes product
development support through subassembly design and tooling production for
process changes.
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Inbound Logistics focuses on effective and efficient movement and
storage of required materials to meet production schedules.
Manufacturing uses provided inputs to produce a high quality and
price competitive product in a timely manner.
Outbound Logistics concentrates on movement of finished goods
through the distribution network to global markets for consumer use.
After-market Service recognizes the need to support the product
either through repair service, or customer service representatives, to answer
product-use questions.
1.2.3 Objectives of Supply Chain Management
The objective of supply chain management is to minimize total
supply chain cost to meet given demand in the market. This total cost may be
comprised of the following,
Raw material and other acquisition costs
In-bound transportation costs
Facility investment costs
Direct and indirect manufacturing costs
Direct and indirect distribution center costs
Inventory holding costs
Inter-facility transportation costs
Out-bound transportation costs
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1.3 SUPPLY CHAIN PLANNING OPTIMIZATION
FRAMEWORK
1.3.1 Importance of Performance Evaluation
A total supply chain cost analysis approach would look at all cost
consequences of supply chain structure and policy decisions. This approach
suggests that as a team participants in the supply chain should maximize the
overall profit in the chain, rather than optimizing their own portion of it. The
implication is that pricing of goods moving between participants can be
adjusted to reflect a fair sharing of the profit pie.
The supply chain is a complex network of facilities and
organizations, with different and conflicting objectives. Hence decision-
making is also a complex process. Hence, Planning processes are typically
subdivided in to multiple hierarchical-based planning levels. Each level has a
planning cycle that its processes follow. Currently, supply chain planning
decisions are usually done using three hierarchical planning levels:
Strategic or high-level planning is typically done yearly or on
an ad hoc basis
Tactical or mid-level planning is typically done quarterly or
monthly
Operational or low-level planning-largely involving
scheduling rescheduling, and execution is typically done
weekly, daily, or by shift
1.3.1.1 Strategic Level Planning Supply Chain Network Design
To support supply chain design, optimization determines the
location, size, and the number of plants, distribution centers, and suppliers.
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This level of planning includes sourcing and deployment plans for each plant,
each distribution center, and each customer. It also considers the flow of
goods through the supply chain network. Generally, supply chain network
design is done infrequently (i.e., every few years) as companies do not need to
add new plants or distribution centers on a routine basis.
1.3.1.2 Tactical Level Planning Supply Planning
Supply chain planning at a tactical level is called supply planning
and involves optimizing the flow of goods throughout a given supply chain
configuration over a time horizon. Similar to supply chain design, supply
planning develops sourcing, production, deployment, and distribution plans.
But there are some major differences:
The supply chain network configuration is already in place
with supply entities such as suppliers, plants, distribution
centers, and transportation lanes.
Supply plans are time-dependent using a "time buckets"
concept (generally, supply planning is done monthly or
weekly).
Supply plans may consider aggregate views of multistage
production processes by incorporating partial levels of a
plant's routings and a product's bill-of-materials.
Setup and changeover times may also be considered, but not
the sequencing of orders through a manufacturing facility.
1.3.1.3 Operational Level Planning-Production Scheduling
At an operational level, supply chain planning can be viewed as
supply scheduling. For a manufacturer, supply scheduling is essentially
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production scheduling done on a plant-by-plant basis. Production scheduling
develops a minute-to-minute or hour-to-hour schedule for all of a plant's
resource needs, including labor, equipment, and materials. Production
scheduling optimally sequences orders into the manufacturing process.
Generally, production scheduling is done frequently, potentially several times
a day to account for changes to orders, machine failures, material shortages,
and other plant disruptions. The process usually considers the lowest level of
detail on plant routings, product bills-of-material, and changeover and setup
times.
1.4 SUPPLY CHAIN NETWORK ARCHITECTURE
OPTIMIZATION
1.4.1 Introduction
Generally, optimization problems seek a solution where decisions
need to be made in a constrained or limited resource environment. Most
supply chain optimization problems require matching demand and supply
when one, the other, or both may be limited. By and large, the most important
limited resource is the time needed to procure, make, or deliver something.
Since the rate of procurement, production, distribution, and transportation
resources is limited, demand cannot be instantaneously satisfied. It always
takes some amount of time to satisfy demand, and this may not be quick
enough unless supply is developed well in advance of demand. In addition to
time, other resources, such as warehouse storage space or a truck's capacity,
may be constrained in meeting demand. An optimization problem comprises
of mainly following decision variables and constraints.
1) These SCN Decision Variables are within the planner's span of
control:
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When and how much of a raw material to order from a
supplier?
When to manufacture an order?
When and how much of the product to ship to a customer or
distribution center?
2) SCN Constraints are limitations placed upon the supply plan:
A supplier's capacity to produce raw materials or components
A production line that can only run for a specified number of
hours per day and a worker that must only work so much
overtime
A customer's or distribution center's capacity to handle and
process receipts
Constraints in an optimization problem are either hard or soft. Hard
constraints, such as the number of working hours in a shift or the maximum
capacity of a truck, must be adhered to or satisfied. Soft constraints can be
relaxed or violated. Examples of soft constraints include customer due dates
or warehouse space limitations. Customer due dates can be changed or a
product may be squeezed into a warehouse temporarily, making constraints
less stringent.
1.4.2 General Optimization Objectives of Supply Chain Network
SCN Objectives maximize, minimize, or satisfy something, such as
the following:
Maximizing profits or margins
Minimizing supply chain costs or cycle times
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Maximizing customer service
Minimizing lateness
Maximizing production throughput
Those unfamiliar with optimization are often confused about the
difference between a constraint and an objective. Fueling this confusion, some
factors can be formulated as either an objective or a constraint.
1.4.3 Optimization Models
Models describe the relationships among decisions, constraints, and
objectives. These are often expressed in the form of mathematical equations.
This is probably the most important but least understood part of an
optimization problem. Generally, the model must represent the "real world" to
the degree needed to capture the essence of the problem. It must represent the
important aspects of the supply chain in order to provide a useful solution.
Once an optimization problem is formulated, a algorithms / solver determines
the best course of action. A solver comprises a set of logical steps or
algorithms embodied in a computer program to search for a solution that
achieves the objective.
A solver can develop three types of solutions:
Feasible Solution-satisfies all the constraints of the problem.
Optimum Solution-the best feasible solution that achieves the
objective of the optimization problem. Although some problems
may yield more than one feasible solution, there is usually only
one optimum.
Optimized Solution-a solution that partially achieves the
objective of the optimization problem. It is not the optimum or
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best solution, but it is a satisfying or reasonable one. This is
usually one of the best feasible solutions. However, for
optimization problems that have no feasible solutions, it may be
one of the best infeasible solutions.
Figure 1.7 represents an optimization problem with a generalized set
of objectives to maximize. It depicts the different types of solutions that might
be developed by a solver. In some cases a solution may be a local optimized
one (see Figure 1.7).
Figure 1.7 Graphical representation of optimization solution types
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Optimized, Feasible Solution
Optimized, Infeasible Solution
Optimum, Feasible Solution
Decision Variables
Local Optimized,Infeasible Solution
Set of feasibleDecision Variables
Objective(S)
1.5 OPTIMIZATION
In recent years, optimization algorithms have received wide
attention by the research community as well as the industry. Many real world
issues are optimization problems and are combinatorial in nature. Generally,
combinatorial problems are NP-hard and they cannot be solved to optimality
within polynomial bounded computation time using exact methods like
branch and bound and dynamic programming.
To solve these combinatorial optimization problems, one often has
to use approximate methods which return near optimal solutions in a
relatively short time. Algorithms of these types are called as heuristics and
they often use some problem specific knowledge to either build or to improve
solutions. Recently, many researchers have focused their attention on a new
class of algorithms called meta-heuristics. It is a set of algorithmic concepts
that can be used to define heuristic methods applicable to wide set of varied
problems. The use of meta-heuristics has significantly produced good quality
solutions to hard combinatorial problems in a reasonable time
With rapidly advancing computer technology, computers are
becoming more powerful and correspondingly, the size and the complexity of
the problems being solved using optimization methods are also increasing.
1.5.1 Statement of an Optimization Problem
Optimization is the act of obtaining the best results for the given
problem under given circumstances. In design, manufacturing and
maintenance of any engineering system, engineers have to take many
technological and managerial decisions at several stages. The ultimate goal of
all such decisions is either to minimize the effort required or to maximize the
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desired benefit. The effort required or the benefit desired in any practical
situation is to find the conditions that give the maximum or minimum value of
a function. An optimization problem can be stated as follows:
Find which minimizes (1.1)
Subject to constraints
;
;
where ‘X’ is an ‘n’ dimensional vector called the design vector, f(X) is termed
as the objective function, and and are known as inequality and
equality constraints respectively. The number of variables (dimensions) ‘n’
and the number of constraints ‘m’ and/or ‘p’ need not be related in any way.
The problem stated in the equation (1.1) is called a constrained optimization
problem. Some optimization problems do not involve any constraints and are
called as unconstrained optimization problems (see equation 1.2). An
unconstrained optimization problem can be stated as follows:
Find which minimizes (1. 2)
Most of the practical problems are constrained in nature. A study of
unconstrained problems is also important for the following reasons:
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a) The constraints do not have the significant influence in certain
design and manufacturing problems.
b) Some of the powerful and robust methods of solving constrained
optimization problems require the use of unconstrained
minimization techniques.
c) The study of unconstrained minimization techniques provide the
basic understanding necessary for the study of constrained
minimization methods.
d) Most of the constrained optimization problems can be converted
into unconstrained problems using methods like penalty function
approach (Deb 2000) and are solved using unconstrained
algorithms.
1.5.2 Classification of Optimization Problems
Optimization problems can be broadly classified into continuous
and discrete problems. The design variables of an optimization
problem are permitted to take any real value and then the optimization
problem is called as continuous optimization problem. Continuous problems
are constrained and unconstrained in nature. Most of the real world
unconstrained and constrained problems are highly non linear, multimodal
and multi dimensional in nature. One of the major issues for solving
constrained optimization problems is the method of handling the constraints.
A simple approach is to convert the constrained optimization problem into an
unconstrained optimization problem by adding penalty for violation of
constraints. If some or all of the design variables of an optimization problem
are restricted to have only integer (discrete) values, the problem is known as
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discrete optimization problems. Examples of real world discrete problems
include travelling salesman problems, scheduling, quadratic assignment
problems etc.
There is no single method available for solving all optimization
problems efficiently. A number of optimization techniques have been
developed over the years for solving different types of optimization problems.
Optimization algorithms are broadly classified into exact methods (traditional
methods) and approximate methods (modern heuristics). Further, exact
methods are classified according to the solution construction procedures.
Approximate methods are also classified based on deterministic search
process or the stochastic search process. Finally the stochastic search methods
are classified based on the number of solutions obtained - a single solution or
population of solutions. Traditional methods such as linear programming,
branch and bound and dynamic programming methods give exact solutions to
the problem. A gradient based method initialized with a good starting point
gives the optimum solution for the unimodal function. The direct analytical
methods can be stopped at any time and return a solution because they
process complete solutions, whereas branch and bound and dynamic
programming methods construct solutions during the search process, and
therefore they cannot be stopped to return a complete solution before the
whole search is done. A diagrammatic representation of the above
classification is shown in Figure 1.8.
1.5.2.1 Exact Methods
Exact methods are often useful in optimization. They are
deterministic, fast and give exact solutions. One class of exact methods is
based on exhaustive search. Simple exhaustive search is even possible when
the number of solutions is small enough so that all of them can be checked
within an acceptable time-span. When linear programming is used, it is not
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Figure 1.8 Classification of Optimization methods
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Optimization methods
Exact methods (Traditional methods)
Approximate methods (Modern heuristics)
Direct analytical and work on
complete solutionsStochasticConstruct
SolutionsDeterministic
Examples:
LP, Local search, Gradient based
methods
Examples:
Branch and Bound, Dynamic
Programming
Example:
Tabu Search
Single solution method (point to point method)
Population based method
Example:
Simulated Annealing
Examples:Genetic algorithms
Evolutionary programmingEvolutionary strategyGenetic programming
Particle swarm optimization
necessary to check all solutions during the exhaustive search process. The
properties of the linear fitness function and convex search space makes it
feasible only to check a path on boundary of the search space. Optimization
methods such as branch and bound and dynamic programming methods work
on partial solutions, and can likewise cut off parts of the search space without
examining them. Algorithms that perform exhaustive search always find the
global optimum, but are often too time consuming or do not apply for solving
real-world problems. Either the search space of these problems is too large, or
the methods have to simplify the problem to be computationally efficient,
which is not possible for real world problems. When compared with
exhaustive search methods, local search methods and gradient based methods
are different.
In local search techniques, a new point is created within the
neighborhood of the current point, and if the neighborhood point is better than
the current point (better solution), it becomes the new current point. Gradient
based methods are analytical local search methods. These methods compute
the derivative information of the function and then move the search in the
direction of the derivative. These methods assume that the function is
differentiable with respect to the design variables and the derivatives are
continuous. Most of the real world industrial optimization problems are
discrete and combinatorial in nature, where objective functions are non
differentiable and discontinuous. In such cases gradient based methods are not
used to solve the problems.
Regarding problems with many optima (multi-modal), local search
strategies can only return a locally optimal solution. The neighborhood size
can of course be enlarged in an attempt to avoid ending up in a local
optimum, but this will increase the time complexity of the algorithm as the
size of the neighborhood is increased.
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1.5.2.2 Approximate Methods
It is not possible to change the time complexity from exponential to
polynomial for the exact search algorithms on NP hard problems.
Approximate methods (modern heuristics) focus on escaping local optima and
try to find the global optimum solution. The major advantage of approximate
algorithms is that their efficiency or applicability is not tied to any specific
problem domain. Approximate methods are generally implemented as meta-
heuristics to solve the hard combinatorial problems. Some of the important
approximate methods are discussed briefly in the next subsections.
1.5.2.3 Tabu search
Tabu Search (TS) is a recent addition to non-derivative optimization
algorithms developed by Glover (1989). Tabu Search is an adaptive heuristic
strategy that was primarily designed for combinatorial optimization. It has
been applied to a wide range of problems however, mostly of combinatorial in
nature. In the Tabu search method , flexible memory cycles (tabu lists) are
used to control the search. At each iteration we take the best move possible
that is not tabu, even if it means an increase in objective function value. The
idea is that when it reaches a local minimum, it is required to escape via
different path.
A short term memory is implemented as a list of previously visited
solutions that are classed as tabu. Whilst a solution is contained within the
tabu-list cannot be returned to. The tabu restrictions stop the search from
returning to the local optima and the new search trajectory ensures that new
regions of the search space are explored and the global optimum located.
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1.5.2.4 Simulated Annealing approach
The simulated annealing process is stochastic variant of the local
search method, but it can, in contrast, escape local optima. It simulates the
process of slow cooling of molten metal to achieve minimum value in a
minimization problem. The cooling phenomenon is simulated by controlling
the temperature like parameter introduced with the concept of Boltzmann
probability distribution. According to the Boltzmann probability distribution,
a system in thermal equilibrium at a temperature ‘T’ has its energy distributed
probabilistically according to , where k is the Boltzmann
constant. This expression suggests that a system at a high temperature has
almost uniform probability of being at any energy state, but at a low
temperature it has a small probability of being at a high energy state.
Metropolis et al (1953) suggested one way to implement Boltzmann
probability distribution in thermodynamic systems. The same can also be used
in the function minimization context. Let us say, at any instant the current
point and the function value at that point is . The
probability of the next point (from the current point s0) being at depends
on the difference in the function values at these two points or on
and is calculated using the Boltzmann probability
distribution:
(1.3)
If , this probability is one and the point is always accepted. In
the function minimization context, this makes sense because if the function
value at is better than that at , the point must be accepted. The
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interesting situation happens when , which implies that the function
value at is worse than that at . According to many traditional
algorithms, the point must not be chosen in this situation. But according
to the metropolis algorithm, there is some finite probability of selecting the
point s even though it is worse than the point . However, this probability is
not the same for all situations. This probability depends on relative magnitude
of and ‘T’ values. The pseudo code of simulated annealing is shown in
Figure 1.9.
Figure 1.9 Pseudo Code of Simulated Annealing Algorithm
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Step 1 : Choose an initial point , a termination criterion .
Set T a sufficiently high value, number of iterations to
be
performed at a particular temperature ‘n’, and set t = 0.
Step 2 : Calculate a neighborhood point = .
Usually a random point in the neighborhood is created.
Step 3 : If , set t = t + 1;
Else create a random number u, in the range (0, 1).
If set t = t + 1;
Else Go to Step 2.
Step 4 : If and T is small, Terminate;
Else if ( t mod n ) = 0 then lower ‘T’ according to a
cooling schedule.
Go to Step 2;
Else go to Step 2.
1.5.2.5 Evolutionary Algorithms (EA)
Evolutionary Algorithms (EA) are stochastic search methods
inspired from the metaphor of natural biological evolution. Genetic algorithm
(GA) is a Evolutionary Algorithm and was introduced by John Holland
(1975). These algorithms operate on a population of potential solutions
applying the principle of survival of the fittest to produce better
approximations to a solution.
Figure 1.10 Genetic algorithm flow chart
The initial population is randomly generated over the search space .
At each generation, operators borrowed from natural genetics such as
selection, recombination, mutation, migration, inversion, reinsertion, etc. are
applied to the individuals from the population. By applying genetic operators
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Yes
No
Generate initial population
Evaluate population
Select best individuals
Print the results
Selection
Recombination
Mutation
Generate a new population
Is the termination condition satisfied?
these individuals are evolved. Each individual from population is evaluated
by using a quality (fitness) function. Using this quality the best individuals are
selected at each generation. In this way new solutions are obtained. Some of
these new solutions can be better than the existing solutions. There are many
modalities to accept the new solutions (also called offspring) in population.
Some algorithms accept the new solution only if this solution is better than his
parent (or parents). The elitist algorithms accept the new obtained solution in
population. The general flow chart is depicted in Figure 1.10.
1.5.2.6 Ant Colony approach
The idea of imitating the behavior of ants for finding good solutions
to combinatorial optimization problems was initiated by Dorigo et al (1991).
Ant Colony Optimization (ACO) simulates the collective foraging habits of
ants, venturing out for food and bringing back to the nest. Real ants are
capable of finding the shortest path from a food source to their nest without
using visual cues as they have poor vision. They communicate information
concerning food sources via an aromatic essence. This chemical substance
deposited by ants as they travel is called a pheromone. A greater amount of
pheromone on the path gives an ant as a stronger stimulation and thus a higher
probability to follow it. They essentially move randomly, but when they
encounter a pheromone trail, they decide whether or not to follow it, and if
they do so, they deposit their own pheromone on the trail, which reinforces
the path. Since ants passing through a food source by a shorter path will have
higher traffic intensity and therefore will make the quantity of pheromone laid
down on the shorter path grow faster. However, there is always a small
probability that an ant will not follow a well-marked pheromone trail. This
small probability allows for exploration of other trails. The foraging behavior
of the real ant colonies can be used to solve combinatorial optimization
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problems. The detailed descriptions about the ACO algorithms and their
implementations are presented in the book “Ant Colony Optimization”
(Dorigo and Stutzle 2004).
1.5.2.7 Particle Swarm Optimization approach
The particle swarm optimization algorithm (PSO) is a relatively new
approach in modern heuristics for optimization. PSO is one of the
evolutionary computation methods. PSO algorithm was first proposed by
Kennedy and Eberhart (1995) for continuous function optimization. Since its
introduction, PSO has attracted a lot of researchers around the world.
PSO originated from the research of food hunting behaviors of
birds. Researchers found that in the course of flight flocks of birds would
always suddenly change direction, scatter and gather. Their behaviors are
unpredictable but always consistent as a whole, with individuals keeping the
most suitable distance. Through the research of the behaviors of similar
biological communities, it is found that there exists a social information
sharing mechanism in biological communities. This mechanism provides an
advantage for the evolution of biological communities, and provides the basis
for the formation of PSO. Every swarm of PSO is a solution in the solution
space. It adjusts its flight according to its own and its companion’s flying
experience. The best position in the course of flight of each swarm is the best
solution that is found by the swarm. The best position of the whole flock is
the best solution, which is found by the flock. The former is called pBest, and
the latter is called gBest. Every swarm continuously updates itself through the
above mentioned best solution. Thus a new generation of community comes
into being. In the practical operation, the fitness function, which is determined
by the optimization problem, assesses the extent to which the swarm is good
or bad.
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According to Angeline (1998), the two main distinctions between
PSO and evolutionary algorithms are:
i) Evolutionary algorithms rely on three mechanisms in their
processing: parent representation, selection of individuals
and the fine tuning of their parameters. In contrast, PSO only
relies on two mechanisms, since PSO does not adopt an
explicit selection function. The absence of a selection
mechanism in PSO is compensated by the use of leaders to
guide the search. However, there is no notion of offspring
generation in PSO as with evolutionary algorithms.
ii) A second difference between evolutionary algorithms and
PSO has to do with the way in which the individuals are
manipulated. PSO uses an operator that sets the velocity of a
particle to a particular direction. This can be seen as a
directional mutation operator in which the direction is
defined by both the particle’s personal best and the global
best (of the swarm). If the direction of the personal best is
similar to the direction of the global best, the angle of
potential directions will be small, whereas a larger angle will
provide a larger range of exploration. In contrast,
evolutionary algorithms use a mutation operator that can set
an individual in any direction (although the relative
probabilities for each direction may be different). In fact, the
limitations exhibited by the directional mutation of PSO has
led to the use of mutation operators similar to those adopted
in evolutionary algorithms.
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Following are the two key aspects by which PSO has become more
popular:
The main algorithm of PSO is relatively simple (since in its
original version, it only adopts one operator for creating new
solutions, unlike most evolutionary algorithms) and its
implementation is, therefore, straightforward.
PSO has been found to be very effective in a wide variety of
applications, being able to produce very good results at a very
low computational cost .
1.6 VARIANTS OF PARTICLE SWARM OPTIMIZATION
The variations introduced in the basic PSO algorithm by various
researchers are discussed in this section.
1.6.1 Basic Particle Swarm Optimization algorithm (B-PSO)
In order to establish a common terminology, in the following we
provide some definitions of several technical terms commonly used:
Swarm: Population of the algorithm.
Particle: Member (individual) of the swarm. Each particle represents
a potential solution to the problem being solved. The position of a particle is
determined by the solution it currently represents.
pbest (personal best): Personal best position of a given particle, so
far. That is, the position of the particle that has provided the greatest success
(measured in terms of a scalar value analogous to the fitness adopted in
evolutionary algorithms).
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lbest (local best): Position of the best particle member of the
neighborhood of a given particle.
gbest (global best): Position of the best particle of the entire swarm.
Leader: Particle that is used to guide another particle towards better
regions of the search space.
Velocity (vector): This vector drives the optimization process, that
is, it determines the direction in which a particle needs to “fly” (move), in
order to improve its current position.
Inertia weight: Denoted by w, the inertia weight is employed to
control the impact of the previous history of velocities on the current velocity
of a given particle.
Learning factor: Represents the attraction that a particle has toward
either its own success or that of its neighbors. Two are the learning factors
used: c1 and c2 , where c1 is the cognitive learning factor and represents the
attraction that a particle has toward its own success and c2 is the social
learning factor and represents the attraction that a particle has toward the
success of its neighbors. Both, c1 and c2, are usually defined as constants.
In PSO in each iteration t, each particle k keeps track of its
coordinates in the problem space, which are associated with the best solution
it has achieved so far. This value is called Pk. Another best value that is
tracked is the overall best value, and its location, obtained so far by any
particle in the population. This location is called G (Global Best). In the
original PSO concept proposed by Kennedy and Eberhart (1995), in every
iteration t, changing the velocity of each particle makes them to move toward
possibly new Pk and G locations. The kth particle in the multidimensional
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search space is represented by (i.e., (Xk1, kk2, …
XkD)). The best previous position (the position giving the best objective
function value) of the kth particle is recorded and represented by (i.e.,
(Pk1,Pk2, … PkD)), and the global best solution (obtained so far) is denoted by
( i.e., (G1, G2, …, GD). The rate of the position change (i.e., velocity) for
the particle is represented by (i.e., (vk1, vk2, … , vkD)).
The new velocity and new position of the particle is calculated by
using the following equations,
Velocity
for d= 1,2,..D. (1.3)
(1.4)
The equations (1.3) and (1.4) describe the flying trajectory of a population of
particles. Equation (1.3) describes how the velocity is dynamically updated
and equation (1.4) describes the position update of the flying particle. The
equation (1.3) consists of three parts. The first part is known as the
momentum part. The velocity cannot be changed abruptly. It is changed from
the current velocity. The second part is known as cognitive part which
represents private thinking of itself learning from its own flying experience.
The third part is the social part which represents the collaboration among
particles learning from group flying experience. In equation (1.3), if the sum
of the three parts on the right side exceeds a constant value specified by the
user, then the velocity on that dimension is assigned to be , that is,
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particles’ velocities on each dimension is clamped to a maximum velocity
, which is an important parameter. Originally is the only parameter
required to be adjusted by the users. Large leads the particles to fly past
the good solution areas. Small would lead the particles to be potentially
trapped into local minima, making them unable to fly into better solution
areas. Usually a fixed constant value is used as the , but a well designed
dynamically changing might improve the performance of PSO. The
velocity and position updates in Particle Swarm Optimization Depicted in
Figure 1.11(b).
PSO can be applied, like other algorithms in the field of
evolutionary computation, in the areas of solving discrete problems involving
system design, multi-objective optimization, classification, pattern
recognition, system modelling, scheduling, planning, robotic applications,
decision making, simulation and identification.
1.6.2 Linearly Decreasing Inertia Weight PSO algorithm
(LDIW-PSO)
In evolutionary programming, the balance between the global and
local search is adjusted through adapting the variance (strategy parameter) of
the Gaussian random function or step size, which can even be encoded in to
the chromosomes to undergo evolution itself.
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The generic PSO algorithm for a minimization problem is given below.
Figure 1.11 (a) The generic PSO algorithm for a minimization problem
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Step 1:Initialization of population: Generate K particles with random positions
or values and initial velocities in the multi (D) dimensional
problem space (with d = 1, 2, . . . , D).
Step 2:For each particle, evaluate the optimization function yielded by the
particle.
Step 3: Initialize particle best
for d = 1, 2, . . . , D, and
Global best , where , in
general, denotes the value of objective function yielded by particle k.
Step 4: Apply velocity, and move the particle according to Equation 1.3 and
Equation 1.4, respectively:
Velocity (vkd)new =
for d= 1,2,..D.
and hence obtain
In the
above,
c1, c2 are two positive constants, and r1 and r2 are two uniformly
distributed random numbers in the range (0, 1).
Step 5: Go back to step 4 until the termination criterion is met with.
Figure 1.11(b) Depiction of the velocity and position updates in
Particle Swarm Optimization
Velocity changes of a PSO consist of three parts, the “social” part,
the “cognitive” part, and the “momentum” part. The balance among these
parts determines the global and local search ability, and hence the
performance of a PSO. The first new parameter added into the original PSO
algorithm is the inertia weight ‘w’ (Shi and Eberhart 1998a, 1998b). The
inertia weight has characteristics that are reminiscent of the temperature in the
simulated annealing. A large inertia weight facilitates a global search, while a
small inertia weight facilitates local search. by linearly decreasing the inertia
weight from a relatively large value to a small value through the course of the
PSO run, the PSO tends to have more global search ability at the beginning of
the run while having more local search ability near the end of the run.
The dynamic equation of PSO with inertia weight is modified to be:
Velocity , =
for d = 1, 2, … , D. (1.5)
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The following weighting function is usually utilized in (equation 1.5):
(1.6)
where, : Maximum inertia weight
: Minimum inertia weight
: Maximum iteration number
: Current iteration number
and hence obtain new position of particle as follows,
(1.7)
Equation (1.5) is the same as equation (1.3), except for a new
parameter, inertia weight w. The inertia weight is introduced to balance
between the global and local search abilities. The large inertia weight
facilitates global search, while the small inertia weight facilitates local search.
The vmax can be simply set to the value of the range of each variable and the
PSO algorithm still performs well enough, if not better.
1.6.3 Clerc construction Factor Method PSO algorithm (CFM-PSO)
Another interesting variation of PSO has been reported by Clerc
and Kennedy (2002), they introduced the new concept of construction
coefficient which should control each of the velocity update relation to limit
the explosion of the particles beyond limits.
Another parameter called constriction coefficient is introduced with
the hope that it can insure a PSO to converge to the global minimum (see
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Clerc and Kennedy 2002). A simplified method of incorporating the
parameters is given in equation (1.8), where is a function of c1 and c 2.
Velocity, =
for d = 1, 2, … , D. (1.8)
and hence obtain
(1.9)
with , (1.10)
where = c1 + c2 and > 4 .
Mathematically, equations (1.5) and (1.8) are proved to be
equivalent by setting inertia weight w to be k, and c1 and c2 to meet the
condition = c1 + c2 and > 4. The PSO algorithm with the constriction
factor can be considered as a special case of the PSO algorithm with inertia
weight and the three parameters related through equation (1.10). A better
approach to use is to utilize the PSO with constriction factor while limiting
vmax to Xmax , the range of each variable on each dimension, or utilize the PSO
with inertia weight, w and c1 and c2 selected according to equation (1.8) (see
Eberhart and Shi 2000). When Clerc’s constriction method is used, is
commonly set to 4.1 and the constant multiplier k is approximately 0.729.
This is equivalent to the PSO with inertia weight w = 0.729 and
c1 = c2 =1.49445. Shi and Eberhart (1998a, 1999) introduced a linearly
decreasing inertia weight to the PSO and then further designed fuzzy systems
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to nonlinearly change the inertia weight (Shi and Eberhart 2001a, 2001b).
Recently, a new variation of PSO model introducing nonlinear variation of
inertia weight with dynamic adaptation was proposed by Chatterjee and Siarry
(2006). The search process of a PSO algorithm is nonlinear and complicated.
A PSO with well-selected parameter set can have good performance, but
much better performance could be obtained if a dynamically changing
parameter is well designed (Shi et al 2005).
1.6.4 Non-Linear Inertia Weight PSO algorithm (NLIW-PSO)
Although , the algorithms have shown some important advances by
providing high speed of convergence in specific problems it has also been
reported that the algorithms have tendency to get struck in near optimal
solution and may find it difficult to improve the solution accuracy by fine
tuning. This work uses the application of a new variation of PSO model
introducing a non-linear variations of inertia weight along with the particles
old velocity to improve the speed of convergence as well as fine tuning the
search in the multi dimensional space and this approach was proposed by
Chatterji and Siarry (2006).This method suggests setting a complete set of
free parameters for any given problem, saving the user from a tedious trial
and error based approach to determine them for each specific problem.
While introducing the concept of inertia weight (w), Shi and
Eberhart (2002) observed that a reasonable choice for ‘w’ should decrease
with a higher choice of vmax. In fact, when vmax is assigned the same value as
xmax, a reasonable w under this condition coincides with that value obtained
for ‘w’ , when vmax is independently chosen a high value. As a general remark,
Shi and Eberhart (2001a) opined that a better performance would be obtained
if the inertia weight were chosen a time varying, linearly decreasing quantity,
rather than being a constant value and supported their statement with a single
case study. A higher value of the inertia weight implied larger incremental
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changes in velocity per unit time step which meant exploration of new search
areas in pursuit of a better solution. However smaller inertia weight meant
less variation in velocity to provide slower updating for fine tuning a local
search. It was inferred that the system should start with a high inertia weight
for course global exploration and the inertia weight should linearly decrease
to facilitate final local search exploration. This should help the system to
approach the optimum of fitness function quickly. This method proposes a
new nonlinear function modulated inertia weight adaption with time for
improved performance of PSO algorithm.
This dynamic adaptation for PSO algorithm proposes to update the
velocity relation according to the following relation 1.11. The important
modification is the determination of the inertia weight witer as a nonlinear
function of the present iteration number (iter) at each time step. The proposed
adaption of witer is given by relation 1.12 .Where wintial is the initial inertia
weight at the start of a given run. wfinal the final inertia weight at the end of a
given run, when iter= (iter)max. (iter)max the maximum number of iterations in a
given run, iter the iteration number at the present time step, w iter the inertia
weight at the present time step and n the nonlinear modulation index.
The system stars with a high initial inertia weight (w initial) which
should allow it to explore new search areas aggressively and then decreases it
gradually according to relation 2, following different paths for different
values of n to reach wfinal at iter=(itermax).The proposed algorithm also
attempts to derive a reasonable set of choice for the free parameters of any
given system.ie { winitial,wfinal,n} on the basis of a fixed itermax .the objective is
to arrive at an attractive solution for any given problem with the known,
fixed free parameters applying our proposed PSO variation which should
require less computational burden and time compared to trial and error
approaches.
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Velocity , =
(1.11)
(1.12)
where (1.13a)
(1.13b)
iter The iteration number starting from 0. ;
itermax The maximum number of iterations that you are running the
PSO algorithm, For example if you are running 1000 iterations,
itermax = 999.
n Coefficient can be taken as 1.2
winitial Initial weight = 0.2
By choosing m = - 2.5 ×10-4, using the wfinal can be calculated.
c1 and c2 Two uniformly generated random numbers;; generated
from the seed.
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The above is the basic equation used to calculate the velocity and
hence obtain new position of particle as follows,
(1.14)
This algorithm found to be giving better solutions for the benchmark
functions considered in their paper comparing with the results obtained by
well- known PSO models.
1.7 THE MULTI-OBJECTIVE OPTIMIZATION PROBLEM
1.7.1 Basic concept of multi-objective programming
This section focuses on basic concept and up to date development of
Evolutionary Multi-objective Optimization (EMO) methods. Multi-objective
programming problems are formulated as follows:
Minimize or Maximize f(x) = ( f1(x), f2(x),……, fr(x) ) (1.15)
The constraint set X may be given by
gj (x) ≤ 0 , j = 1,………, m (1.16)
and/or a subset of Rn itself. For the problem (MOP), we define Pareto
solutions as follows:
Definition : A solution is said Pareto optimal, if there is no better
solution x Є X other than , namely, if
f(x) f( ) for any x ЄX (1.17)
In general, there may be many Pareto solutions. The final decision is
made among them taking the total balance over all criteria into account. This
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is a problem of value judgment of decision maker. The totally balancing over
criteria is usually called trade-off.
1.7.2 Constraint Handling
There are many algorithms underlying optimization problem is free
from any constraint. However; this is hardly the case, when it comes to
solving real-world optimization problems as most engineering problems are
constrained problems. General mathematical formulation of the constrained
optimization problem has been discussed in the above section 1.5.1.
Constraints divide the search space in to two divisions-feasible and infeasible
regions. Constraints can be two types: equality and inequality constraints.
Constraints can be hard or soft. A constraint is considered hard if it must be
satisfied in order to make a solution acceptable. A soft constraint, on the other
hand, can be relaxed to some extent in order to accept solution. Hard equality
constraints are difficult to satisfy, particularly if the constraints are nonlinear
in decision variables. Such hard equality constraints may possible to relax (or
made soft) by converting them in to an inequality constraint with some loss of
accuracy (Deb 1995). In all of the constraint handling strategies discussed
here, we assume greater-than-equal-to type inequality constraints only. It is
important to reiterate that this relaxation does not mean that the algorithms
cannot handle equality constraints. Instead, it suggests that equality
constraints should be handled by converting them in to relaxed inequality
constraints.
1.7.3 Transformation Methods
Transformation methods are the simplest and most popular
optimization methods of handling constraints. The constrained problem is
transformed in to a sequence of unconstrained problems by adding penalty
terms for each constraint violation, if a constraint is violated at any point, the
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objective function is penalized by an amount depending on the extent of
constraint violation. Penalty terms vary in the way the penalty is assigned.
Some penalty methods cannot deal with infeasible points at all and even
penalize feasible points that are close to the constraint boundary. These
methods are known as interior penalty methods. In these methods, every
sequence of unconstrained optimization method finds a feasible solution. The
solution found in one sequence is used as the starting solution for the next
sequence of the unconstrained optimization. In subsequent sequences, the
solution improves gradually and finally converges to the optimum solution.
The other kind of penalty methods penalizes infeasible points but do
not penalize feasible points. These methods are known as exterior penalty
methods. In these methods, every sequence of unconstrained optimization
finds an improved yet infeasible solution.
1.7.4 Penalty function approach
This penalty parameter approach is a popular constraint handling
strategy. Minimization of all objective functions is assumed here. However, a
maximization function can be handled by converting it in to a minimization
function by using the duality principle.
Before the constraint violation is calculated, all constraints are
normalized. Thus, the resulting constraint functions are g j (x(i)) ≥ 0 for
j = 1,2,……..,J. For each solution x(i), the constraint violation for each
constraint is calculated as follows:
ω j (x(1) ) =│g j (x(1)│ ,if g i (x(1)) < 0 ;
or ω j (x(1) ) = 0 ; otherwise (1.18)
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Thereafter, all constraint violation are added together to get the overall
constraint violation:
Ω ( x(i) ) (1.19)
This constraint violation is then multiplied with a penalty parameter Rm and
the product is added to each of the objective function values:
Fm (x(i)) = fm (x(i) ) + Rm Ω ( x(i) ) (1.20)
The function Fm takes into account the constraint violations. For a
feasible solution, the corresponding Ω term is zero and Fm becomes equal to
the original objective function fm. However, for an infeasible solution,
Fm > fm, thereby adding a penalty corresponding to total constraint violation.
The penalty parameter Rm is used to make both of the terms on the right side
of the above equation to have the same order of magnitude. Since the original
objective functions could be of different magnitudes, the penalty parameter
must also vary from one objective function to another. A number of static and
dynamic strategies to update the penalty parameter are suggested in the single
objective GA literature (Michalewicz 1992; Michalewicz and Schoenauer
1996 ; Homaifar et al 1994).any of these techniques can be used here as usual.
The change of penalty parameter Rm in successive sequences of the
penalty function method depends on whether an exterior or interior penalty
term is used. If the optimum point of the unconstrained objective problem, an
initial penalty parameter Rm = 0 (or another value of Rm) will solve the
constrained problem. In the exterior penalty method, a feasible or an
infeasible point can be used as the initial point of the first sequence, whereas
in the interior penalty method, usually an initial feasible point is desired. In
the case of exterior penalty term, a small initial value of R results in an
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optimum solution, close to the unconstrained optimum point. As Rm is
increased in successive sequences, the solution improves and finally
approaches the true constrained optimum point. With the interior penalty
term, however, a large initial value of Rm results in a feasible solution far
away from the constraint boundaries. As Rm is decreased, the solution is
improved and approaches the true optimum point. General algorithm of
constraint handling by penalty approach in shown below in Figure 1.12.
The mail advantage of this method is that any constraint (convex or
non convex) can be handled. The algorithm does not take in to account the
structure of the constraints, that is linear or nonlinear constraints can be
tackled with this algorithm
Figure 1.12 General algorithm of constraint handling by penalty
approach
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Step 1: Choose two termination parameters Є1 , Є2 , an initial
solution x(0), a penalty term, and an initial penalty
parameter R(0)m .choose parameter ‘c’ to update R such that
0 < c < 1 is used for interior penalty terms and c > 1 is
used for exterior penalty terms. Set t=0.
Step 2 : Form Fm (x(i)) = fm (x(i) ) + Rm Ω ( x(i) ) .
Step 3 : Starting with a solution x(i) , find x(i+1) such that Fm (x(i+1)) is
minimum for a fixed value of Rm . Use Є1 to terminate the
unconstrained search.
Step 4 : Is │ Fm (x(i+1)) - Fm (x(i)) │ ≤ Є2 ?
If yes, set xT =x(i+1) and Terminate ;
Else go to Step 5.
Step 5 : Choose Rm(i+1) = c *Rm
(i) ,
Set t =t+1 and go to Step 2.
1.8 OUTLINE OF THE PRESENT STUDY
In this thesis an attempt has been made to design, modeling and
analysis of supply chain network architectures using different variants of PSO
as solution methodologies. PSO algorithms are proposed for the performance
evaluation applications and are tested for their performance by solving the for
multi echelon supply chain networks. Novel solution procedures have been
presented for handling constrained multi echelon supply chain network
architecture for solving single and multi-objective analysis problems.
The outline of the thesis is as follows:
Chapter 1, presents an overview of supply chain management,
supply chain optimization and general classification of optimization problems
to solve optimization problems. A detailed literature review on single and
multi-objective analysis in integrated multi echelon supply chain network
design and models used are discussed in chapter 2. Development of
mathematical model for integrated tactical level three stage multi echelon
constrained supply chain networks configurations and proposes the
application of different PSO Algorithms for performance evaluation in
Chapter 3. Chapter 4, is the extension of chapter 3 , it presents mathematical
formulation and application of best PSO algorithm for the analysis and
performance evaluation of tactical level four-echelon constrained supply
chain network architecture. Chapter 5, is devoted to present the multi
objective analysis of the multi-stage multi echelon production-inventory-
distribution supply chain networks different sets of objectives. The
performance analysis is performed using weighted sum approach, and trade-
off solutions between the sets of objectives are proposed for managerial
decision making . The performance analysis and validation of the application
of proposed best PSO algorithm is done in chapter 6. Chapter 7, Summarizes
this research and concludes with a discussion on some possible research
extensions.
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