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Optimization is derived from the Latin word
“optimus”, the best.
Optimization characterizes the activities involved to
find “the best”.
What is Optimization?
Optimization is the mathematical discipline which is
concerned with finding the maxima and minima of
functions, possibly subject to constraints.
Optimization is the act of obtaining the best results
under given circumstances.
What is Optimization?
Objective function: This is the quantity (or quantities) that you are trying to optimize. It is sometimes referred to as a target.
Optimization variables: These are the variables you can change (sometimes called the changing variables) in order to achieve your optimum solution.
Maximize: In some optimization problems, you seek to make the objective function as large as possible. Such problems are maximization problems.
Minimize: In some optimization problems, you seek to make the objective function as small as possible. Such problems are minimization problems.
Explicit constraint: Explicit constraints describe those items that clearly
given to you as goals during your optimization process.
For example, limitations on resources (materials, labour) and
limitations on demand are often stated explicitly.
Implicit constraint: Implicit constraints refer to those quantities that you
must recognize are also constraints on your optimization process.
For example, in optimizing company profits by producing different
quantities of different goods, the number of units of each goods to produce
might need to be an integer. The quantity produced must also be non-
negative.
Optimality Criteria
In considering optimization problems, two questions
generally must be addressed:
1. Static Question. How can one determine whether
a given point x* is the optimal solution?
2. Dynamic Question. If x* is not the optimal point,
then how does one go about finding a solution
that is optimal?
General Ideas of Optimization
• There are two ways of examining optimization.
– Maximization (example: maximize profit)
• In this case you are looking for the highest
point on the function.
– Minimization (example: minimize cost)
• In this case you are looking for the lowest point
on the function.
6
Optimization theory finds ready application in all branches of
engineering in four primary areas:
1. Design of components or entire systems
2. Planning and analysis of existing operations
3. Engineering analysis and data reduction
4. Control of dynamic systems
We use optimization to obtain
Minimal Cost
Maximum Profit
Best Approximation
Optimal Design
Optimal Management or Control etc.,
Where would we use optimization?
• Design of civil engineering structures such as frames, foundations, bridges, towers, chimneys and dams for minimum cost.
• Optimal plastic design of frame structures (e.g., to determine the ultimate moment capacity for minimum weight of the frame).
• Design of water resources systems for obtaining maximum benefit.
• Design of optimum pipeline networks for process industry.
• Finding the optimal trajectories of space vehicles.
Optimum design of linkages, cams, gears, machine tools, and other mechanical components.
• Selection of machining conditions in metal-cutting processes for minimizing the product cost.
• Design of material handling equipment such as conveyors, trucks and cranes for minimizing cost.
• Design of pumps, turbines and heat transfer equipment for maximum efficiency.
Where would we use optimization? . .
Where would we use optimization? . .
•Optimum design of control systems.
• Optimum design of chemical processing equipments and plants.
• Selection of a site for an industry.
• Planning of maintenance and replacement of equipment to reduce operating costs.
• Allocation of resources or services among several activities to maximize the benefit.
• Controlling the waiting and idle times in production lines to reduce the cost of production.
• Planning the best strategy to obtain maximum profit in the presence of a competitor.
• Designing the shortest route to be taken by a salesperson to visit various cities in a single tour.
• Optimal production planning, controlling and scheduling.
• Analysis of statistical data and building empirical models to obtain the most accurate representation of the statistical phenomenon.
Where would we use optimization? . .
• Design of aircraft and aerospace structure for minimum weight
• Optimum design of electrical machinery such as motors, generators and transformers.
•Optimal location of telecommunication towers
Where would we use optimization? . .
What is a Function?
• Is a rule that assigns to every choice of x a unique value y =ƒ(x).
• Domain of a function is the set of all possible input values (usually x), which allows the function formula to work.
• Range is the set of all possible output values (usually y), which result from using the function formula.
• Unconstrained and constrained function
– Unconstrained: when domain is the entire set of real
numbers R
– Constrained: domain is a proper subset of R
• Continuous, discontinuous and discrete
What is a Function?
• Monotonic and unimodal functions– Monotonic:
– Unimodal:
What is a Function?
ƒ(x) is unimodal on the interval if and only if it is monotonic on either side of the single optimal point x* in the interval.
Unimodality is an extremely important functional property used in optimization.
A monotonic increasing function A monotonic decreasing function
An unimodal function
• An objective function is defined which needs to be
either maximized or minimized.
• The objective function may be technical or
economic.
Examples of economic objective are profits, costs of
production etc.. Technical objective may be the yield
from the reactor that needs to be maximized,
minimum size of an equipment etc..
The choice of the
Objective function
is governed by the nature of the problem
The geometric characteristics of the objective function plays an Important role in solution of the optimization.
Two different types of geometric Characteristics
ab
max min
A
B
C
D
E
ab
Uni-Model Function Multi -Model Function
A
Classification of the objective functions
Maximization f(x) is equivalent to minimization –f(x)
It can be seen from Fig.
that if a point x∗
corresponds to the
minimum value of function
f (x), the same point also
corresponds to the
maximum value of the
negative of the function, −f
(x).
The following operations on the objective function will not change the
optimum solution x∗
1. Multiplication (or division) of f (x) by a positive constant c.
2. Addition (or subtraction) of a positive constant c to (or
from) f (x).
STATEMENT OF A CONSTRAINED OPTIMIZATION PROBLEM
Problems where a set of optimal conditions needs to be find subject to a
set of additional constraints on the variables.
subject to the constraints
where X is an n-dimensional vector called the design vector, f (X) is termed
the objective function, and gj (X) and lj (X) are known as inequality and
equality constraints, respectively. The number of variables n and the
number of constraints m and/or p need not be related in any way
Optimization with constraints
2
2),(min
1,52
2),(min
0
2),(min
22
22
22
or
or
yx
yxyxf
yx
yxyxf
x
yxyxf
Problems where a set of optimal conditions needs to be find without any
additional constraints on the variables (or) Unconstrained Optimization is
concerned with the practical computational task of finding minima or
maxima of functions of one, several or even millions of variables
STATEMENT OF A UNCONSTRAINED OPTIMIZATION PROBLEM
f(X) is called the Objective Function
Where X is an n dimensional Vector Called Design Vector and the
variables are called the design or decision variables. The design variables
are collectively represented as a design vector X.
Unconstrained optimization
22 2),(min yxyxf
Essential features of optimization problem
An objective function is defined which needs to be
either maximized or minimized.
The objective function may be technical or
economic.
Examples of economic objective are profits,
costs of production etc..
Technical objective may be the yield from the
reactor that needs to be maximized, minimum
size of an equipment etc..
Underdetermined system:
If all the design variables are fixed.
There is no optimization.
Thus one or more variables is relaxed and the
system becomes an underdetermined system which
has at least in principle infinite number of solutions.
Essential features of optimization problem. .
Restrictions:
Usually the optimization is done keeping
certain restrictions or constraints. Thus, the
amount of row material may be fixed or there may
be other design restrictions.
Hence in most problems the absolute
minimum or maximum is not needed but a restricted
optimum i.e. the best possible in the given condition
Essential features of optimization problem. .
Constraint surfaces in a hypothetical two-dimensional design space
Depending on whether a particular design point belongs to the acceptable
or unacceptable region, it can be identified as one of the following four
types:
1. Free and acceptable
point
2. Free and
unacceptable point
3. Bound and
acceptable point
4. Bound and
unacceptable point
Design points that do not lie on any constraint surface are known as
free points
The set of values of X that satisfy the equation gj (X) = 0 forms a hyper-
surface in the design space and is called a constraint surface.
A design point that lies on one or more than one constraint surface is called
a bound point , and the associated constraint is called an active
constraint.
A contour line of a function of two variables is a curve along which the
function has a constant value.
A contour plot consists of contour lines where each contour line
indicates a specific value of the function
The locus of all points satisfying f (X) = C = constant forms a
hyper-surface in the design space, and each value of C corresponds to
a different member of a family of surfaces. These surfaces, called
objective function surfaces, are shown in a hypothetical two-
dimensional design space.
Once the objective function surfaces are drawn along with the
constraint surfaces, the optimum point can be determined without much
difficulty. But the main problem is that as the number of design variables
exceeds two or three, the constraint and objective function surfaces become
complex even for visualization and the problem has to be solved purely as a
mathematical problem
STEPS IN FORMULISATION OF AN OPTIMISATION PROBLEM
Phases of Solving Problems
“ There is no single method available for solving
all optimization problems efficiently”.
Hence a number of optimization methods have been
developed for solving different types of optimization problems
Classification of Optimization problems
Classification based on the existence of the constraints
Unconstrained
Constrained
Classification based on nature of the design variables
Parameter or Static optimization
( find the set of design parameters)
Trajectory or Dynamic optimization
(design variable is a function of one or more parameters)
Classification based on physical structure of the problem
optimal control (mathematical program problem involving no of stages )
non optimal control
Classification based on the nature of the equations involvedLinearNon linearGeometricQuadratic programming problems
Classification based on permissible values of the design variablesIntegerReal valued programming problems
(Design variables restricted to)Mixed
Classification based on no of objective functionsSingle Multi objective programming problems
Classification based on the deterministic of the variablesDeterministicStochastic programming
(in which some or all the parameters are probabilistic)
Classification based on Capability of the search algorithm
– search for a local minimum– global optimization; multiple objectives; etc.
Classification based on type of solution.
Analytical methods Search Methods Graphical methods Experimental methods Numerical methods
For an Unconstrained minimization problem
Function Characteristics
• Solution exists, smooth
• Complicated (multiple minima or maxima)
• Good starting points unknown/difficult to compute
Challenges
• Finding solution in reasonable amount of time
• Knowing when solution has been found
1. Descent method
2. Newton’s method
3. Conjugate direction method
4. Conjugate gradient algorithm
5. Quasi Newton’s method
SOLUTION METHODS FOR UNCONSTRAINED OPTIMIZATION
Unconstrained multi-parameter optimization techniques
Direct search (no information on derivatives used):
•Hooke-Jeeves’ pattern search•Nelder-Mead’s sequential simplex method•Powell's conjugate directions method•various evolutionary techniques
Unconstrained multi-parameter optimization techniques
Gradient-based methods (information on derivatives is
used):
•Steepest Descent
•Fletcher-Reeves' Conjugate Gradient method
Second order methods (information on the second
derivatives is used):
•Newton's Method
•Quasi-Newton Method (constructs an approximation
of the matrix of second derivatives)
Constrained Optimization
Constrained Optimization involves finding the
optimum to some decision problem in which the
decision-maker faces constraints.
Examples: constraints of money, time, capacity,
or energy.
Methods for Solving Constrained Optimization Problems
• Penalty Function Method
• Lagrange Multiplier
• Augmented Lagrange for Inequality Constraints
• Quadratic Programming
• Gradient Projection Method for Equality Constraints
• Gradient Projection Method for Inequality Constraints
Nonlinear Programming Optimization Methods:
• Sequential quadratic programming (SQP)
• Augmented Lagrangian method
• Generalized reduced gradient method
• Projected augmented Lagrangian
• Successive linear programming (SLP)
• Interior point methods etc.,
Convex programming studies the case when the objective function is
convex (minimization) or concave (maximization) and the constraint set is
convex. This can be viewed as a particular case of nonlinear programming
or as generalization of linear or convex quadratic programming.
Linear programming (LP), a type of convex programming, studies the case
in which the objective function f is linear and the set of constraints is
specified using only linear equalities and inequalities. Such a set is called a
polyhedron or a polytype if it is bounded.
Second order cone programming (SOCP) is a convex program, and
includes certain types of quadratic programs.
Methodologies in Optimization
Semidefinite programming (SDP) is a subfield of convex optimization
where the underlying variables are semidefinite matrices. It is
generalization of linear and convex quadratic programming.
Conic programming is a general form of convex programming. LP,
SOCP and SDP can all be viewed as conic programs with the
appropriate type of cone.
Geometric programming is a technique whereby objective and
inequality constraints expressed as polynomials and equality
constraints as monomials can be transformed into a convex program.
Integer programming studies linear programs in which some or all
variables are constrained to take on integer values. This is not convex,
and in general much more difficult than regular linear programming.
Quadratic programming allows the objective function to have quadratic
terms, while the feasible set must be specified with linear equalities and
inequalities. For specific forms of the quadratic term, this is a type of
convex programming.
Fractional programming studies optimization of ratios of two nonlinear
functions. The special class of concave fractional programs can be
transformed to a convex optimization problem.
Nonlinear programming studies the general case in which the objective
function or the constraints or both contain nonlinear parts. This may or may
not be a convex program. In general, whether the program is convex affects
the difficulty of solving it.
Stochastic programming studies the case in which some of the constraints
or parameters depend on random variables. Robust programming is, like
stochastic programming, an attempt to capture uncertainty in the data
underlying the optimization problem. This is not done through the use of
random variables, but instead, the problem is solved taking into account
inaccuracies in the input data. Combinatorial optimization is concerned with
problems where the set of feasible solutions is discrete or can be reduced to
a discrete one
Infinite-dimensional optimization studies the case when the set of feasible
solutions is a subset of an infinite-dimensional space, such as a space of
functions.
Heuristics and metaheuristics make few or no assumptions about the
problem being optimized. Usually, heuristics do not guarantee that any
optimal solution need be found. On the other hand, heuristics are used to
find approximate solutions for many complicated optimization problems.
Constraint satisfaction studies the case in which the objective function f is
constant (this is used in artificial intelligence, particularly in automated
reasoning).
Disjunctive programming is used where at least one constraint must be
satisfied but not all. It is of particular use in scheduling.
Calculus of variations seeks to optimize an objective defined over many
points in time, by considering how the objective function changes if there is
a small change in the choice path. Optimal control theory is a generalization
of the calculus of variations.
Dynamic programming studies the case in which the optimization strategy
is based on splitting the problem into smaller sub-problems. The equation
that describes the relationship between these sub-problems is called the
Bellman equation.
•Evolutionary Algorithms •Genetic Algorithm (GA)
•DE (Differential Evolution)
•Particle swarm optimization (PSO)
•Ant colony optimization
•Harmony search
•Gaussian adaptation etc.,
•Classical Optimization•Direct
•Snobfit.
•Hybrid approach etc.,
What are common for an optimization problem?
• There are multiple solutions to the problem; and the optimal solution is to
be identified.
• There exist one or more objectives to accomplish and a measure of how
well these objectives are accomplished(measurable performance).
• Constraints of different forms are imposed.
• There are several key influencing variables. The change of their values will
influence (either improve or worsen)the “measurable performance” and the
degree of violation of the “constraints.”
In any practical problems , the design variables cannot be chosen arbitrarily
rather they have to satisfy certain specified functional and other
requirements
The restrictions that must be satisfied to produce an acceptable design are
collectively called the Design constraints.
The constraints that represent limitations on the behavior or performance of
the system are termed Behavior or Functional constraints
The constraints that represents physical limitations on the design
variables such as, availability , etc are called Geometric or Side constraints
Properties of Practical Optimization Problems
• They are non-smooth problems having their objectives and constraints
are most likely to be non-differential and discontinuous
• Often, the decision variables are discrete making the search space
discrete as well
• The problems may have mixed types (real, discrete, Boolean,
permutation, etc.) of variables
• They may have highly non-linear objective and constraint functions due
to complicated relationships and equations which the decision
variables must form and satisfy. This makes the problems non-linear
optimization problems.
•There are uncertainties associated with decision variables, due to which the
true optimum solution may not of much importance to a practitioner.
•The objective and constraint functions may also non-deterministic.
•The evaluation of objective and constraint functions is computationally
expensive.
•The problems give rise to multiple optimal solutions, of which some are
globally best and many others are locally optimal.
•The problems involve multiple conflicting objectives, for which no one
solution is best with respect to all chosen objectives.
Properties of Practical Optimization Problems . .
Classical Optimization Techniques
The classical optimization techniques are useful in finding the
optimum solution or unconstrained maxima or minima of continuous
and differentiable functions.
These are analytical methods and make use of differential calculus in
locating the optimum solution.
The classical methods have limited scope in practical applications as
some of them involve objective functions which are not continuous
and /or differentiable.
Yet, the study of these classical techniques of optimization form a
basis for developing most of the numerical techniques that have
evolved into advanced techniques more suitable to today’s practical
problems
These methods assume that the function is differentiable twice
with respect to the design variables and the derivatives are
continuous.
Classical Optimization Techniques . .
optimization methods using calculus have several limitations
and thus not suitable for many practical applications.
Linear programming is Most widely used constrained form of
optimization method which deals with nonnegative
solutions(x1= 0 , x2= 1/2 x3= 5) to determine system of linear
equations with corresponding finite value of the objective
function.
Linear Programming is required that all the mathematical
functions in the model be linear functions.
Linear Programming
The term ‘linear’ implies that the objective function and
constraints are ‘linear’ functions of ‘nonnegative’ decision
variables (e.g., no squared terms, trigonometric functions, ratios
of variables)
Linear programming (LP) techniques consist of a sequence of
steps that will lead to an optimal solution to problems, in cases
where an optimum exists
The term ‘Linear’ is used to describe the proportionate
relationship of two or more variables in a model. The given
change in one variable will always cause a resulting proportional
change in another variable.
Applications of Linear Programming
The number of applications of linear programming has been so large,
some of them are:
Scheduling of flight times of aero planes
Distribution of resources
Selection of shares and stocks
Assignment of jobs to people and many other problems
Scheduling of production in many manufacturing units or industries.
Use of available resources in an organization
Engineering design problems
Shipping & transportation
Product mix
Marketing research
Food processing etc.,
Methods of Solving Linear Programming Problems
Trial and error: possible for very small problems; virtually
impossible for large problems.
Graphical or Geometrical approach : It is possible to solve a 2-
variable problem graphically to find the optimal solution (not
shown).
Simplex Method: This is a mathematical approach developed by
George Dantzig. Can solve small problems by hand.
Computer Software : Most optimization software actually uses
the Simplex Method to solve the problems.
Linearity: is a requirement of the model in both objective function
and constraints
Proportionality: Relationship between Outputs and inputs are
proportional
Additivity: Every function is the sum of individual contribution of
respective activities a1x1+a2x2
Divisibility: All decision variables are continuous (can take on any
non-negative value including fractional ones) x1=12, x2=3.8
Certainty or Deterministic: All the coefficients in the linear
programming models are assumed to be known exactly. a1=5, a2=2
Limitations of Linear Programming
The following will be the assumptions of linear programming problem that limit its applicability.
The conditions of LP problems are
1. Objective function must be a linear function of
decision variables.
2. Constraints should be linear function of decision
variables.
3. All the decision variables must be nonnegative.
For example
example shown above is in “general” form
There are mainly four steps in the mathematical formulation of
linear programming problem as a mathematical model.
Mathematical formulation of linear programming problem
Identify the decision variables and assign symbols x and y
to them. These decision variables are those quantities whose
values we wish to determine.
Identify the set of constraints and express them as linear
equations / in equations in terms of the decision variables.
These constraints are the given conditions.
Identify the objective function and express it as a linear
function of decision variables. It might take the form of
maximizing profit or production or minimizing cost.
Add the non-negativity restrictions on the decision
variables, as in the physical problems, negative values of
decision variables have no valid interpretation
Mathematical formulation of linear programming problem. .
There are many real life situations where an LPP may be
formulated. The following examples will help to explain
the mathematical formulation of an LPP.
Examples
Example
• A company makes cheap tables and chairs using only wood and labor.
• To make a chair requires 10 hours of labor and 20 board feet of wood.
• To make a table requires 5 hours of labor and 30 board feet of wood.
• The profit per chair is $8 and $6 per table.• If it has 300 board feet of wood and 110 hours
of labor each day, how many tables and chairs should it make to maximize profits?
Objective
Constraints (Scarce Resources)
Setting Up the Problem
• Profits: $6 per table and $8 per chair
Total Profits = 6T + 8 C• Constraints: 300 feet of wood per day and
110 hours labor per day• Wood Use: 30 feet per table
20 feet per chair• Labor Use: 5 hours per table
10 hours per chair
Writing the Equations
• Objective: Maximize Z = 6T + 8CMaximum Profits = ($6 x # of tables) + ($8 x # of chairs)
• Subject to:– 30T + 20C < 300 board feet (wood constraint)– 5T + 10C < 110 hours (labor constraint)
T,C > 0 (non-negativity)
Resources Requirements Tables Chairs
Amount Available
Unit profit $6 $8 Wood(board feet) Labor(hours)
30 20 5 10
300 board feet 110 hours
Writing the Equations
Maximize Z = 6 T + 8 C
Subject to: 30 T + 20 C < 300 (wood constraint)
5 T + 10 C < 110 (labor constraint)
T, C > 0 (non-negativity)
Resources Requirements Tables Chairs
Amount Available
Unit profit $6 $8 Wood(board feet) Labor(hours)
30 20 5 10
300 board feet 110 hours
Inequalities
• A resource may constrain a problem by being . . .– Equal-to… =– Equal-to or greater-than… => or >– Equal-to or less-than… =< or <– Greater-than… >– Less-than… <
. . .the amount of resource available.
Dealing with inequalities
• Converts “Less-than or Equal-to” variables, and “Less-than” variables to “Equal-to” variables by adding a slack variable.
30T + 20C < 300 (wood constraint)becomes
30T + 20C + Sw = 300• Sw represents the difference, if any, between the
amount of wood used and the amount available.• (It is unused resource)• Slack variables also cannot be negative so S > 0
SURPLUS VARIABLES
• If the labor constraint was greater than or equal to the 110 hours; expressed as…
–5 T + 10 C > 110 hours
• Then a surplus variable would be needed to make it an equality.
–5 T + 10 C - SL = 110 hours
• SL represents the excess labor need, if any, above 110 hrs.
– (Surplus variables cannot be negative so SL > 0)
Reformulation of the example with Slack Variables added
Maximize Z = 6T + 8C
Subject to: 30T + 20C < 300 board feet of wood
5T + 10C < 110 hours of labor
Maximize Z = 6T + 8C
Subject to: 30T + 20C + SW = 300 board feet of wood
5T + 10C + SL = 110 hours of labor
T, C, SW, SL > 0
The L.P. model adds any needed slack and surplus variables. But, if they are needed, they will appear in the program output. Below is how the program adds the slack variables.
A company manufactures two products X and Y whose
profit contributions are Rs.10 and Rs. 20 respectively. Product
X requires 5 hours on machine I, 3 hours on machine II and 2
hours on machine III. The requirement of product Y is 3 hours
on machine I, 6 hours on machine II and 5 hours on machine III.
The available capacities for the planning period for machine I, II
and III are 30, 36 and 20 hours respectively. Find the optimal
product mix.
A diet is to contain at least 4000 units of carbohydrates, 500
units of fat and 300 units of protein. Two foods A and B are available.
Food A costs 2 dollars per unit and food B costs 4 dollars per unit. A
unit of food A contains 10 units of carbohydrates, 20 units of fat and
15 units of protein. A unit of food B contains 25 units of
carbohydrates, 10 units of fat and 20 units of protein. Formulate the
problem as an LPP so as to find the minimum cost for a diet that
consists of a mixture of these two foods and also meets the
minimum requirements.
The above information can be represented as
Let the diet contain x units of A and y units of B.
Total cost = 2x + 4y
The LPP formulated for the given diet problem is
Minimize Z = 2x + 4y
subject to the constraints
In the production of 2 types of toys, a factory uses 3 machines A, B
and C. The time required to produce the first type of toy is 6 hours, 8 hours
and 12 hours in machines A, B and C respectively. The time required to
make the second type of toy is 8 hours, 4 hours and 4 hours in machines A,
B and C respectively. The maximum available time (in hours) for the
machines A, B, C are 380, 300 and 404 respectively. The profit on the first
type of toy is 5 dollars while that on the second type of toy is 3 dollars. Find
the number of toys of each type that should be produced to get maximum
profitThe data given in the problem can be represented in a table as follows.
.
Let x = number of toys of type-I to be produced
y = number of toys of the type - II to be produced
Total profit = 5x + 3y
The LPP formulated for the given problem is: Maximize Z = 5x + 3y
subject to the constraints
Standard form of LP problems
Standard form of LP problems must have following three
characteristics:
1. Objective function should be of maximization
type
2. All the constraints should be of equality type
3. All the decision variables should be nonnegative
Standard form Standard form is a basic way of describing a LP problem.
It consists of 3 parts:
A linear function to be maximized
maximize c1x1 + c2x2 + … + cnxn
Problem constraints
subject to a11x1 + a12x2 + … + a1nxn < b1 a21x1 + a22x2 + … + a2nxn < b2
… am1x1 + am2x2 + … + amnxn <
bm
Non-negative variables
e.g. x1, x2 > 0
• The problems is usually expressed in matrix form and then it
becomes:
maximize cTx
subject to ax < b, x > 0
where
X- Vector of decision variables
C- Objective function coefficients
a- Constraint coefficients
b- Right hand side of the constraint
Other forms, such as minimization problems, problems with
constraints on alternative forms, as well as problems involving
negative variables can always be rewritten into an equivalent
problem in standard form.
Any linear programming problem can be expressed in
standard form by using the following transformations.
1. The maximization of a function f (x1, x2, . . . , xn) is equivalent
to the minimization of the negative of the same function. For
example, the objective function
Consequently, the objective function can be stated in the
minimization form in any linear programming problem
2. The decision variables represent some physical dimensions,
and hence the variables xj will be nonnegative. However, a
variable may be unrestricted in sign in some problems. In such
cases, an unrestricted variable (which can take a positive,
negative, or zero value) can be written as the difference of two
nonnegative variables. Thus if xj is unrestricted in sign, it can be
written as
xj = x′ j − x′′ j , where
It can be seen that xj will be negative, zero, or positive,
depending on whether x′′ j is greater than, equal to, or less than
x′j
3. If a constraint appears in the form of a “less than or equal
to” type of inequality as
it can be converted into the equality form by adding a
nonnegative slack variable xn+1 as follows:
Similarly, if the constraint is in the form of a “greater than or
equal to” type of inequality as
it can be converted into the equality form by subtracting a
variable as
where xn+1 is a nonnegative variable known as a surplus
variable.
Converting linear program in standard form into linear
program in slack form:
N
Each constraint aijxj bi is represented
j=1
N
as xN+i= bi - aijxj and xN+i 0.
j=1
xN+i are basic variables, or slack variables. The original set of xi
are non-basic variables.
General form Vs Standard form
General form Violating points for standard
form of LPP:
1.Objective function is of
minimization type.
2.Constraints are of inequality
type.
3.Decision variable, x2, is
unrestricted, thus, may take
negative values also.
How to transform a general form of a LPP to the standard form ?
General form Transformation Standard form
General form
1.Objective function
Standard form
2. First constraint.
1.Objective function
3.Second constraint 3.Second constraint
2. First constraint.
4.Third constraint 4.Third constraint
5. Constraints for decision
variables, x1 and x2
5. Constraints for decision
variables, x1 and x2
Feasible solution. In a linear programming problem, any
solution that satisfies the constraints
is called a feasible solution
Basic solution. A basic solution is one in which n − m variables are set
equal to zero. A basic solution can be obtained by setting n − m variables to
zero and solving the constraint
simultaneously.
Basic Definitions
Basis. The collection of variables not set equal to zero to obtain the basic
solution is called the basis.
Basic feasible solution. This is a basic solution that satisfies
the nonnegativity conditions of Eq.
Non-degenerate basic feasible solution. This is a basic
feasible solution that has got exactly m positive xi .
Optimal solution. A feasible solution that optimizes the
objective function is called an optimal solution
Optimal basic solution. This is a basic feasible solution for
which the objective function is optimal.
Pivotal Operation
Operation at each step to eliminate one variable at a time, from
all equations except one, is known as pivotal operation.
Number of pivotal operations are same as the number of
variables in the set of equations.
Note: Pivotal equation is transformed first and using the
transformed pivotal equation other equations in the system
are transformed.
The set of equations (A3, B3and C3) is said to be in Canonical
form which is equivalent to the original set of equations (A0,
B0and C0)
Three pivotal operations were carried out to obtain the
canonical form of set of equations in last example having
three variables.
Basic variable, Nonbasic variable, Basic solution, Basic feasible solution
Find all the basic solutions corresponding to the
system of equations
Case 1
Case 2
Case 3
and x4 = 0 (nonbasic or independent variable). Since this
basic solution has all xj ≥0 (j = 1, 2, 3, 4), it is a basic
feasible solution
From case 3
The solution obtained by setting the independent variable
equal to zero is called a basic solution
Flowchart for finding the optimal solution by the simplex algorithm.
