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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 nonnegativesolutions(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


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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.


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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.,


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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 (notshown).

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.


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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.


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The conditions ofLP 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


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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.


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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. .


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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.


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Examples


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Example

A company makes cheap tables and chairsusing only wood and labor.

To make a chair requires 10 hoursof laborand20 board feet of wood.

To make a table requires 5 hours of laborand30 board feet of wood.

The profit per chair is $8 and $6 per table.

If it has 300 board feet of wood and 110 hoursof laboreach day, how many tables and chairsshould it make to maximize profits?

Objective

Constraints (Scarce Resources)


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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


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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


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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


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Inequalities

A resource may constrain a problem bybeing . . .

Equal-to =

Equal-to or greater-than => or >

Equal-to or less-than =< or

Less-than 0


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SURPLUS VARIABLES

If the labor constraint was greater than orequal to the 110 hours; expressed as

5 T + 10 C > 110 hours

Then a surplus variable would be needed tomake 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)


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Reformulation of the examplewith 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 theyare needed, they will appear in the program output. Below is how theprogram adds the slack variables.


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A company manufactures two products X and Y who se

prof i t co ntr ibut ions are Rs.10 and Rs. 20 respectively. Product X

requires 5 hou rs on machine I, 3 hours on machine II and2

hours on machine III. The requirement of product Y is 3 hou rs

on m achine I, 6 hours onmachine 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.


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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. Aunit 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


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Let the diet contain x units ofA and y units ofB.

Total cost = 2x + 4y

The LPP formulated for the given diet problem is

Minimize Z = 2x + 4y

subject to the constraints


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In the production of2 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.


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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


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Standard form ofLP problems

Standard form of LP problems must have following three

characteristics:

1. Objective function should be ofmaximization type

2. All the constraints should be ofequality type

3. All the decision variables should be nonnegative


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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


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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


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Other forms, such as minimization problems, problems withconstraints on alternative forms, as well as problems involving

negative variables can always be rewritten into an equivalent

problem in standard form.


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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


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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 suchcases, 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 = xj xj , where

It can be seen that xj will be negative, zero, or positive,

depending on whether xj is greater than, equal to, or less than

xj


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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 anonnegative slack variable xn+1 as follows:


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Similarly, if the constraint is in the form of a greater than or

equal totype 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 surp lus

variable.


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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, orslackvariables. The original set of

xi are non-basic variables.


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General form Vs Standard form

General form Violating points for standard

form ofLPP:

1.Objective function is of

minimization type.

2.Constraints are ofinequality

type.

3.Decision variable, x2, is

un restr icted, thu s, may take

negative values also.

How to transform a general form of a LPP to the standard form ?


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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.


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4.Third constraint 4.Third constraint

5. Constraints for decision

variables, x1and x2

5. Constraints for decision

variables, x1and x2


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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.


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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.


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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.


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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)


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Three pivotal operations were carried out to obtain the

canonical form of set of equations in last example having

three variables.


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Basic variable, Nonbasic variable,

Basic solution, Basic feasible solution


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Find all the basic solutions corresponding to the

system of equations


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Case 1

C 2


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Case 2

Case 3


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and x4 = 0 (nonbasic or independent variable). Since this

basic solution has all xj0 (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


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Flowchart for finding

the optimal solution

by the simplex

algorithm.


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Note that while comparing ( 3 4M) and ( 5 3M), it is decided

that ( 3 4M)


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Carry out one iteration of the Revised Simplex Method


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Where c1=1, c2=9 and c3=1 are the cost coefficients of the given

objective function and

b=9

15

14


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Solution:

Standard from:

Coefficients of

S1 S2 S3 in the


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S1,S2,S3 in the

objective function CB

Initially B-1 is Unit matrix

Standard form constraints are written in matrix form

and represented each column as P1, P2, P3 and P4


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P1 P2 P3 P4

1 2 3 1

3 2 2 1

2 3 1 1Where c1,c2,c3 are the cost

coefficients of the given objective

function


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XB =


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The Practical Importance of Duality

D lit i i li ( d li ) ti i ti d l i


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Duality arises in nonlinear (and linear) optimization models in a

wide variety of settings.

Models of electrical networks. The current flows are primal

variables and the voltage differences are the dualvariables

that arise in consideration of optimization (and equilibrium) in

electrical networks.

Models of economic markets. In these models, the primal

variables are production levels and consumption levels, and the

dual variables are prices of goods and services

Structural design. In these models, the tensions on the beams are

primal variables, and the nodal displacements are the dual

variables.

Nonlinear (and linear) duality is very useful. For example,

dual problems and their solutions are used in connection with:


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Identifying near-optimal solutions. A good dual solution can be

used to bound the values of primal solutions, and so can be used

to actually identify when a primal solution is near-optimal.

Proving optimality. Using a strong duality theorem, one can prove

optimality of a primal solution by constructing a dual solution with

the same objective function value.

Sensitivity analysis of the primal problem. The dual variable on a

constraint represents the incremental change in the optimal

solution value per unit increase in the RHS of the constraint

Convergence of improvement algorithms. The dual problem isoften used in the convergence analysis of algorithms.

Other uses, too . . . .


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Dual Problem. As a definition, the dual problem can be

formulated by transposing the rows and columns including

the right-hand side and the objective function, reversing the

inequalities and maximizing instead of minimizing. Thus bydenoting the dual variables as y1, y2, . . . , ym, the dual

problem becomes

Primal Problem


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Dual Problem


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Rules for PrimalDual Relations

General Rules for Constructing Dual


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1. The number of variables in the dual problem is equal to the

number of constraints in the original (primal) problem. The

number of constraints in the dual problem is equal to thenumber of variables in the original problem.

2. Coefficient of the objective function in the dual problem come

from the right-hand side of the original problem.

3. If the original problem is a maxmodel, the dual is a minmodel; if

the original problem is a minmodel, the dual problem is the max

problem.

4. The coefficient of the first constraint function for the dual

problem are the coefficients of the first variable in the

constraints for the original problem, and the similarly for other

constraints.

5. The right-hand sides of the dual constraints come from the

objective function coefficients in the original problem.

General Rules for Constructing Dual ( Continued)

6 The sense of the ith constraint in the dual is = if and only if the ith


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6. The sense of the ith constraint in the dual is if and only if the ith

variable in the original problem is unrestricted in sign.

7. If the original problem is man (min ) model, then after applying Rule 6,assign to the remaining constraints in the dual a sense the same as

(opposite to ) the corresponding variables in the original problem.

8. The ith variable in the dual is unrestricted in sigh if and only if the ith

constraint in the original problem is an equality.

9. If the original problem is max (min) model, then after applying Rule 8,

assign to the remaining variables in the dual a sense opposite to (the

same as) the corresponding constraints in the original problem.

Max model Min modelxj 0 jth constraint xj

0 jth constraint xj free jth constraint =

ith const yi 0ith const yi 0ith const = yi free


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Write the dual of the following linear programming problem:

Maximize f = 50x1 + 100x2


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1 2

subject to


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If a linear programming problem has an optimal solution,

then its dual has an optimal solution and the optimal

values of the two problems coincide


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Duality Theorems

Theorem 1 The dual of the dual is the primal


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Theorem 1 The dual of the dual is the primal.

Theorem 2 Any feasible solution of the primal gives an f value

greater than or at least equal to the value obtained by any

feasible solution of the dual.

Theorem 3 If both primal and dual problems have feasible

solutions, both have optimal solutions and minimum

f = maximum .

Theorem 4 If either the primal or the dual problem has an

unbounded solution, the other problem is infeasible.

Dual Simplex Method

Computationally, the dual simplex algorithm also involves a


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sequence of pivot operations, but with different rules

(compared to the regular simplex method) for choosing thepivot element


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