02 or Optimization-01 Chapter-04

8/10/2019 02 or Optimization-01 Chapter-04

1/6

Dr. Vishnu Prasad PandeyMaster Degree Program, Purbanchal University

2014.08.07

LECTURE #02

Introduction to Optimization (1)

MEM124 Operations Research

Lecture#02: Operational Research

Contents of Chapter 4

4. Introduction to Optimization [9 hrs]

Linear and Multi-objective optimization models;

Modeling optimization problem in EXCEL; Building linear programing models;

Solving linear programing models;

Interpreting solver results and sensitivity analysis;

Solving multi-objective models;

Using premium solver for linear programming


Contents of this lecture

4.Introduction to Optimization

Introduction

Optimization modeling process

Solution of optimization models

Building linear programing models;

Solving linear programing models;

Graphical method

Simplex method


Optimization: Introduction

A mathematical Optimization model consists of

An Objective Function (OF), &

A set of constraints in a form of system of equations

inequalities

It seeks to answer the question What is the best?

Answer can be expressed as numerical value

Optimization models are used extensively in almost all

areas of decision-making, e.g. Engineering design

Physical, chemical & biological sciences

Economics

Management, financial portfolio selection


2/6


Optimization: Modeling Procedure

. Describe theproblem

. Prescribe a

solution, &

. Control the

problem byassessing/updating the

optimal solutioncontinuously,while changing

the parameters &structure of the

problem

Problemformulation


Optimization Model (OM): Basic contents

OMs are prescriptive or normative models seeking to findthe best possible strategy for decision-makers

Optimization problems are made up of following basicingredients:

Decision variables: the controllable inputs which affe

the value of the objective function

An objective funct ion: that we want to mini/maximize

Parameters: uncontrollable inputs which may be fixed

numbers

Constraints: relations between decision variables & thparameters


Optimization: Solutions

Feasible solution

A solution value for decision variables, where all the

constraints are satisfied

Most solution algorithms proceed by

first finding a feasible solution,

then seeking to improve upon it, and

finally changing the decision variables to move from one

feasible solution to another feasible solutionThe process is repeated until the OF has reached its

maximum or minimum

Optimal solution:

Gives the minimum or maximum result of the OF while

satisfying the constraints


Optimization: Solutions

Optimization problems are classified & solved according t

mathematical characteristics of the OF,

constraints, &

controlled decision variables

Linear Programming (LP) deals with: a case of optimizatio

problem, where both the OF & the constraints are lineariterms of the decision variables

Multi-objective Program (MP) or Goal Program, is: where a single objective characteristic of an optimizati

problem is replaced by several goals

in solving MP, one may represent some of the goals as

constraintsto be satisfied, while other objectives can b

weightedto make a composite single OF


3/6


Linear Programming (LP): Building LP

LP is a mathematical procedure for determining optimalallocation of scarce resources

LP deals with a class of programming problems where boththe OF & constraints are linear & all relations among the

variables corresponding to resources are linear

While building a LP model:

What are decision variables?

What are parameters?

What is the objective function?

What are the constraints?


Example#01 (same as in Lecture#1)

The Two Mines Company own two different mines that produceore which, after being crushed, is graded into three classes: higmedium and low-grade. The company has contracted to provide

smelting plant with 12 tons of high-grade, 8 tons of medium-graand 24 tons of low-grade ore per week. The two mines have

different operating characteristics as detailed below

Mine Cost/day ('000) Production (tons/day)

High Medium Low

A 180 6 3 4

B 160 1 1 6

If maximum operating hours/week of the mines will be 5, formulate

LP model to answer How many days per week should each mibe operated to fulfill the smelting plant contract?


Example#01: Formulation of LP

Decision variables:

x = days/week mine A is operated

y = days/week mine B is operated

Objective function (to minimize the cost):

min Z = 180x + 160y

Constraints:

High: 6x + 1y >= 12

Medium: 3x + 1y >= 8

Low: 4x + 6y >= 24

x


4/6


Solving LP: Extreme point solution method

Coordinates of all corners (or extreme) points of feasible regionare determined and values of OF at these points are computed &compared

Because: optimal solution to any LP problem always lie at one of

the corner points of feasible solution space

Steps/Procedure:

Plot each constraint on a graph paper, by considering all & are

=; and then plot the line

For each line, divide the region into 3 parts. Pick a point in either

side of the line & plug its coordinates into the constraint. If it

satisfies the condition, this side is feasible; otherwise the other side

is feasible. For equality constraints, only the points on the line are

feasible.

Throw away the sides that are NOT feasible

After all the constraints are graphed, you should have a non-empty

(convex) feasible region, unless the problem is infeasible


Solving LP: Example#01

Find graphical solution for a LP model formulated as below:

OF: Maximize profit function (7T + 5C) [T=Table, C=Chair]

S.T. Constraints: 3T + 4C 100; T, C > 0

SOLUTION:

Plot graphs for eachconstraints

Constraint line 1

(carpentry): 3T + 4C =

2400

Intercepts (for T = 0, C =600; for T = 800, C = 0) 0 T

600

C

0

Feasible

< 2400 hrs

Infeasible

> 2400 hrs



SOLUTION:

Constraint line 2(painting): 2T + 1C =

1000

Intercepts:

for T = 0, C = 1000;

for T = 500, C = 0

0 500 800 T

C

1000

600

0



SOLUTION:

Maximum cost for chair:C = 450.

Minimum cost for Table:

T = 100.

C

1000

600

450

0


5/6



OF line: 7T + 5C =

Profit.



If additionalconstraints were

available, such as,

need at least 75more chairs than

tables, then: C > T+ 75; or C T >

75.

New constraint

line would have

been added asshown in the

graph & Optimal

point would have

been shifted to anew point

T = 320

C = 360

No longe

feasible

New optimal point

T = 300, C = 375


Solving LP: Special cases of LP

Redundant Constraints do not affect the feasible region

Example: If two constraints are given as X 10 & X

12, the second constraint is redundant as it is less

restrictive

Infeasibility when no feasible solutions exist (there is nofeasible region)

Example: if X 10 & X 15



Al ternate optimalsolution : when there are

more than one optimal

solutions

Example:

Maximize (2T + 2C)

Subject to:

T + C < 10

T < 5

C < 6

T, C > 0

0 5 10

C

10

6

0

All points onRedsegment are

optimal


6/6



Unbounded solutions:when nothing prevents

the solution from

becoming infinitely large

Example:

Maximize (2T + 2C)

Subject to:

2T + 3C > 6

T, C > 0

C

2

1

0


Class Activities: Graphical methods

1. Solve the following LP problem using Graphical method:

Objective function: Max Z = 6x1 - 4x2

Subject to: 2x1+ 4x24;

4x1+ 8x216;

x1, x20

2. Solve the following LP problem using Graphical method:

Objective function: Max Z = 15x1 + 10x2

Subject to:

4x1+ 6x2360;

3x1180;

5x2200 & x1, x20


Thank you!

Thank You!!(For enquiries: [email protected])

02 or Optimization-01 Chapter-04

Documents

Transcript of 02 or Optimization-01 Chapter-04