There is a short notes on graph modelling.

1. JAHANGIRABAD INSTIUTE OF TECHNOLOGY
BARABANKI
Department of Mechanical Engineering
Graph Modeling
RAVI VISHWAKARMA
10/06/17 Ravi Vishwakarma ,Assistant Professor JIT 1

2. Graph Modeling
Linear graph system models provide a graphical representation of a
system model and the interconnection of its elements. A set of
differential model and the interconnection of its elements. A set of
differential and algebraic equations which completely define the system
may be derived directly from the linear graph model.
From the system linear graph, a procedure can be developed for
deriving a specific set of differential equations, known as the state
equations.
10/06/17 Ravi Vishwakarma ,Assistant Professor JIT 2

3. Network Flow Problems
The network flow problem is one that we all use in our daily lives:
what is the fastest route to take between two locations in the city during
the rush hour, what is the “most” scenic route to drive, or the cheapest
route to fly, between two cities in our vacation. The street intersections
or the cities are the nodes, and the street or highway segments in
between the intersections, or non stop fights between the cities are the
arcs. The objective is to find a series of arcs connected with correct
directions such that one can start at the origin and arrive at the
destination traversing the arcs and that the sum of the numbers on the
arcs are either minimum or maximum.
10/06/17 Ravi Vishwakarma ,Assistant Professor JIT 3

4. Optimization Concept
Finding an alternative with the most cost effective or highest
achievable performance under the given constraints, by maximizing
desired factors and minimizing undesired ones. E.g.
What is the size of the most profitable production run?
What is the least expensive shape of an oil can?
What is the stiffest beam we can cut from a 12 inch of diameter log?
10/06/17 Ravi Vishwakarma ,Assistant Professor JIT 4

5. Selection of goals & objectives-criteria
After determining the hierarchical system structure ,each level of the
system must be examined, and goals and objectives must be defined.
For each level of system, a specific purpose of function can be stated
that indicates the needs that must be met by the system. In essence
,these goals should reflect the stated purpose and function. Normally,
The purpose and function can be ascertained by defining the goal of the
system in relation to the next higher level system.
10/06/17 Ravi Vishwakarma ,Assistant Professor JIT 5

6. Method of optimization
All of us are optimizers. We all make decisions that maximize our
welfare in some way or another. Often the welfare we are maximizing
may come later in life. By optimizing, it reflects our evaluation of
future benefits versus current costs or benefits forgone. In economics,
the extent to which we value future benefits today is reflected by
what is called a discount rate. While economic criteria are only a part of
everything we consider when making decisions, they are often among
those deemed very important. Economic evaluation methods
involving discount rates can be used to consider and compare
alternatives characterized by various benefits and costs that occur
over time. This chapter begins with a quick and basic review of
what is often called applied engineering economics. Many of the
optimization methods used in practice incorporate concepts from
engineering economics.
10/06/17 Ravi Vishwakarma ,Assistant Professor JIT 6

7. Path Problem
The shortest path problem can be defined for graphs whether
undirected, directed, or mixed. It is defined here for undirected graphs;
for directed graphs the definition of path requires that consecutive
vertices be connected by an appropriate directed edge.
In real-life situations, the transportation network is usually stochastic
and time-dependent. In fact, a traveler traversing a link daily may
experiences different travel times on that link due not only to the
fluctuations in travel demand (origin-destination matrix) but also due to
such incidents as work zones, bad weather conditions, accidents and
vehicle breakdowns. As a result, a stochastic time-dependent (STD)
network is a more realistic representation of an actual road network
compared with the deterministic one.
10/06/17 Ravi Vishwakarma ,Assistant Professor JIT 7

8. Analytical Approach in Optimization
Differential calculus method of optimization is an analytical approach.
It is applicable to continuous, twice differentiable functions. In this
method, calculation of the numerical value of the objective function is
virtually the last step of the purpose. The optimum value of the
objective function is calculated after determining the optimum values
of the decision variables.
10/06/17 Ravi Vishwakarma ,Assistant Professor JIT 8

9. Subjective Optimization
The subjective optimization approach means that, instead of defining a
mathematical cost function, the human user directly user directly
evaluates the potential solutions and makes a decision that which
solutions are good and which are bad. In subjective optimization the
human user can directly evaluate the fitness of the solutions by
ordering the solutions or by the selection of the best ones.
Cont.……
10/06/17 Ravi Vishwakarma ,Assistant Professor JIT 9

10. 10/06/17 Ravi Vishwakarma ,Assistant Professor JIT 10
In real-life situations, the transportation network is usually stochastic and
time-dependent. In fact, a traveler traversing a link daily may experiences
different travel times on that link due not only to the fluctuations in travel
demand (origin-destination matrix) but also due to such incidents as work
zones, bad weather conditions, accidents and vehicle breakdowns. As a
result, a stochastic time-dependent (STD) network is a more realistic
representation of an actual road network compared with the deterministic
one.

11. Combinatorial Optimization
Combinatorial Optimization problems are connected with the efficient
allocations of limited resources to meet desired objectives when the
values of some or all of the variables are restricted to be integral. I
many practical problems, activities and resources, such as machines,
airplanes and people, are individuals.
Combinatorial Optimization is the process of finding one or more best
solutions in well defined discrete problem space. Such problems occur
in almost all fields of management as well as engineering disciplines.
10/06/17 Ravi Vishwakarma ,Assistant Professor JIT 11

12. Aluminum Extrusion
Aluminum Extrusion is a process in which an externally heated billet is
forced to go through a die with a predetermined orifice, herby
becoming a functional product. Inherently during this process heat is
also internally generated.
Extrusion is defined as the process of shaping material, such as
aluminum, by forcing it to flow through a shaped opening in a die.
Extruded material emerges as an elongated piece with the same profile
as the die opening.
10/06/17 Ravi Vishwakarma ,Assistant Professor JIT 12