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Lecture 1

Pages In Book: 1, 16, 23-39

Operations Research = Management Science

The scientific approach to management decision making.

The science of better decision making.

1. Decisions

2. Constraints

3. Objective(s)

EMIS 8360

Section 1.6

Deterministic Models – all data are known

Stochastic Models – quantities are known only by probability distributions

Stochastic models are much more difficult to solve!

(Example: point-to-point demand for service in a telecommunication data network)

EMIS 8360

Chapter 2

Deterministic Optimization Models equal Mathematical Programs

Composed Of The Following:

1. Subscripts

2. Constants and Sets

3. Decision Variables

4. Constraints

5. Objective Function

EMIS 8360

Example 2.1 – Page 24

Subscripts (we only have one for this example)

j – denotes the source of crude petroleum

j = 1 implies Saudi Arabia

j = 2 implies Venezuela

EMIS 8360

Supply and Cost

EMIS 8360

Decision Variables & Constraints

X1 – denotes the number of barrels of Saudi crude processed each day

X2 – denotes the number of barrels of Venezuela crude processed each day

(Demand On Gasoline)

0.3X1 + 0.4X2> 2000

EMIS 8360

Constraints Continued

(Demand On Jet Fuel)

0.4X1 + 0.2X2 > 1500

(Demand On Lubricants)

0.2X1 + 0.3X2> 500

(Max Supply From Saudi Arabia)

X1< 9000

EMIS 8360

The Model

Minimize 20X1 + 15X2

Subject To

0.3X1 + 0.4X2> 2000

0.4X1 + 0.2X2> 1500

0.2X1 + 0.3X2> 6000

0 < X1< 9000

0 < X2< 6000

EMIS 8360

The Graph

X2 (000s)

0.3X1 + 0.4X2 = 2000

(6666.7,0)

(0,5000)

0.3X1 + 0.4X2> 2000

8

6

4

2

X1 (000s)

2

4

6

8

EMIS 8360

The Graph - Continued

X2 (000s)

0.4X1 + 0.2X2 = 1500

(3750,0)

(0,7500)

0.4X1 + 0.2X2> 1500

8

6

4

2

X1 (000s)

2

4

6

8

EMIS 8360

The Graph - Continued

X2 (000s)

0.2X1 + 0.3X2 = 500

(2500,0)

(0,1666.7)

0.2X1 + 0.3X2> 500

Redundant!

8

6

4

2

X1 (000s)

2

4

6

8

EMIS 8360

The Graph - Continued

X2 (000s)

Optimum

(2000, 3500)

8

6

0.3X1+0.4X2 = 2000

0.4X1+0.2X2 = 1500

Cost = 92,500

4

2

X1 (000s)

2

4

6

8

EMIS 8360

AMPL Software

http://www.ampl.com

Try AMPL

download the Student Edition

2. (for Windows Users)

amplcml.zip

place this file in some directory – such as 8360/AMPL/

EMIS 8360

AMPL Software - Continued

extract all

to run double click sw

Note: My model is in a:ex1.txt

type

sw: ampl

ampl: model a:ex1.txt;

EMIS 8360

Example in a:ex1.txt

# AMPL Model Ex2.1 – a:ex1.txt

option solver cplex;

printf"AMPL Model Ex2.1 - run on PC\n\n";

var x1 >= 0, <= 9000;

var x2 >= 0, <= 6000;

minimize cost: 20*x1 + 15*x2;

subject to Gas: 0.3*x1 + 0.4*x2 >= 2000;

subject to Jet_Fuel: 0.4*x1 + 0.2*x2 >= 1500;

subject to Lubricants: 0.2*x1 + 0.3*x2 >= 500;

expand cost; expand Gas; expand Jet_Fuel;

expand Lubricants;

solve;

display x1; display x2;

EMIS 8360

Solution Obtained

ampl: model a:ex1.txt;

AMPL Model Ex2.1 - run on PC

minimize cost:

20*x1 + 15*x2;

subject to Gas:

0.3*x1 + 0.4*x2 >= 2000;

subject to Jet_Fuel:

0.4*x1 + 0.2*x2 >= 1500;

subject to Lubricants:

0.2*x1 + 0.3*x2 >= 500;

CPLEX 8.0.0: optimal solution; objective 92500

2 dual simplex iterations (0 in phase I)

x1 = 2000

x2 = 3500

EMIS 8360

To Save Solution

While still in sw:

use edit to select all

use edit to copy

then paste to Word or notepad document

then you can turn in your output

EMIS 8360

Generic Model

The generic model does not have the data with the model.

The data is in a separate file.

Hence we can solve any problem of this type.

We simply provide the proper data in a data file.

EMIS 8360

# Ex2.1 AMPL Model - Generic - a:ex2.txt

option solver cplex;

set S; # sourses - sa = Saudi Arabia, v = Venezuela

set P; # products - g = gas, j = jet fuel, l = lubs

param a {S}; # availability of crude from sources

param c {S}; # cost/barrel

param d {P}; # demand for products

param cf {S,P}; # conversion factor - crude to product

data a:data.txt;

var x {s in S} >= 0, <= a[s];

minimize cost: sum {s in S} c[s]*x[s];

subject to Demand {p in P}:

sum {s in S} cf[s,p]*x[s] >= d[p];

solve;

display x;

expand cost; expand Demand;

EMIS 8360

# Ex2.1 AMPL Model - Generic - a:data.txt

set S := sa v;

set P := g j l;

param a := sa 9000 v 6000;

param c := sa 20 v 15;

param d := g 2000 j 1500 l 500;

param cf :=

sa g 0.3

v g 0.4

sa j 0.4

v j 0.2

sa l 0.2

v l 0.3;

EMIS 8360

ampl: model a:ex2.txt;

CPLEX 8.0.0: optimal solution; objective 92500

2 dual simplex iterations (0 in phase I)

x [*] :=

sa 2000

v 3500;

minimize cost:

20*x[\'sa\'] + 15*x[\'v\'];

subject to Demand[\'g\']:

0.3*x[\'sa\'] + 0.4*x[\'v\'] >= 2000;

subject to Demand[\'j\']:

0.4*x[\'sa\'] + 0.2*x[\'v\'] >= 1500;

subject to Demand[\'l\']:

0.2*x[\'sa\'] + 0.3*x[\'v\'] >= 500;

EMIS 8360

Optimal Solutions For Linear Programs (LPs)

There will be either a unique optimum or an infinite number of optimal solutions.

There cannot be exactly 2 optimal solutions to a LP.

EMIS 8360

Infinite Number Of Solutions

All points are optimal

Max 2X + 2Y

Subject To

X + Y < 5

0 < X < 3

0 < Y < 4

EMIS 8360

Output From AMPL For An Infinite Number Of Solutions

# Example With An Infinite Number Of Solutions

# runL1c.txt

option solver cplex;

var X >= 0, <= 3; var Y >= 0, <= 4;

maximize profit: 2*X + 2*Y;

subject to Constraint: X + Y <= 5;

expand profit; expand Constraint;

solve;

display X, Y;

EMIS 8360

Output Is A Single Solution

sw: ampl

ampl: model runL1c.txt;

maximize profit:

2*X + 2*Y;

s.t. Constraint:

X + Y <= 5;

CPLEX 7.1.0: optimal solution; objective 10

0 simplex iterations (0 in phase I)

X = 3

Y = 2

ampl:

Other Optimal Solutions:

(1,4), (2,3),(1.5,3.5),

Etc.

EMIS 8360

Example Of A Problem With No Feasible Solutions

max X

subject To:

X + Y < 2

X + Y > 4

X > 0, Y > 0

EMIS 8360

Example Of An Unbounded Problem

Max X

Subject To:

X + Y > 10

X > 0

0 < Y < 10

Try (100,5), (1000, 7) (10000, 10)

EMIS 8360

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