Op Tim Ization

Antoni Guasch 1

OPTIMIZATION OF

INDUSTRIAL PROCESSES

Antoni Guasch 2

Objective

Optimization is the art and science of allocating scarce resources to the best

possible effect. Optimization techniques are called into play every day in industrial

planning problems, industrial design, resource allocation, scheduling, decision-

making, etc. For example: how does an airliner know how to route its planes and

schedule its crews at minimum cost; while meeting constraints on airplane flight

hours between maintenance and maximum flight time for crews? Another example

could be how to schedule body cars into a painting line such as the planned

production can be achieved? The main goals of this course will be:

1) recognize problems that can be tackled using the tools of applied optimization,

2) formulate optimization problems correctly and appropriately,

3) solve optimization problems, primarily by selecting and applying the correct

solvers.

Antoni Guasch 3

Objective

This year the focus will be on Airline Operations and Scheduling. Many of

optimization problems in this area are common to other application areas:

Network flows Flight scheduling Fleet assignment Aircraft routing Crew scheduling Manpower planning Fuel management systems Gate assignment

Antoni Guasch 4

Bibliography & Software

Atenea documentation

Web links

Airline operations and schedulling, Massoud Bazargan, Ashgate, 2on Edition

AMPL software for mathematical programming

Antoni Guasch 5

Decision problems

Distinct Characteristics of Strategic, Tactical, and Operational Decisions

Antoni Guasch 6

Optimization

In mathematics, computer science, or management science, mathematical

optimization (alternatively, optimization or mathematical programming) is the

selection of a best element (with regard to some criteria) from some set of available

alternatives.

In the simplest case, an optimization problem consists of maximizing or minimizing

a real function by systematically choosing input values from within an allowed set

and computing the value of the function. The generalization of optimization theory

and techniques to other formulations comprises a large area of applied mathematics.

More generally, optimization includes finding "best available" values of some

objective function given a defined domain (or a set of constraints), including a

variety of different types of objective functions and different types of domains.

http://en.wikipedia.org/wiki/Optimization

Antoni Guasch 7

Optimum (global), optimal and locally optimum solutions

. (http://en.wikipedia.org/wiki/Local_optimum)

. (http://en.wikipedia.org/wiki/Global_optimum)

Antoni Guasch 8

Classification of Optimization models

Optimization

Continuous

Discrete

Discrete or Combinatorial Optimization

deals mainly with problems where we have to

choose an optimal solution from a finite (or

sometimes countable) number of

possibilities.

As opposed to discrete optimization, the

variables used in the objective function can

assume real values, e.g., values from

intervals of the real line

Antoni Guasch 9

Continuous optimization example

Antoni Guasch 10

Continuous optimization problem

http://en.wikipedia.org/wiki/Optimization_problem

Antoni Guasch 11

Continuous optimization models

Continuous

Optimization

Constrained

Unconstrained

Nonlinear equations Nonlinear least squares Global optimization Nondifferentiable optimization

Linear programming (LP) Nonlinear constrained Network programming

Jump to Introduction to AMPL

Antoni Guasch 12

AMPL

The AMPL book is available free for downloading from http://www.ampl.com/BOOK/download.html

AMPL download: Go to the Web page http://www.ampl.com/DOWNLOADS/index.html to download AMPL.

Follow the instructions of the page. We plan to use the solvers CPLEX and

MINOS with are included in the amplcml.zip zip archive. So, you do not

need to install additional solvers.

AMPL IDE download: The AMPL Integrated Development Environment, AMPL IDE, provides a simple and straightforward enhanced modeling

interface for AMPL users. Commands are typed at an AMPL prompt in the

usual way, and all installed solvers can be accessed.

Antoni Guasch 13

Software: AMPL

Lets try AMPL. Execute ampl.exe at the AMPL folder and try

model : reads the file into AMPL solve : to have AMPL translate your linear program, sends it to

a linear program solver, and then return the answer

MINOS 5.5 : indicates that AMPL uses version 5.5 of a solver called MINOS

Antoni Guasch 14

Software: AMPL ID

Lets try AMPL ID. Execute amplide.exe at the amplide folder

Antoni Guasch 15

Example linear model: prod0.mod

Optimization => Continuous => Constrained => Linear Programming

Linear objective function

Linear constraints

Antoni Guasch 16


Syntax:

Each variable is named in a var statement Each constraint by a statement that begins with subject to and a name like

X_limit or Time for the constraint

Multiplication requires an explicit * operator relation is written

Antoni Guasch 17


An steel company must decide how to allocate next weeks time on a rolling mill. The mill can produce bands and coils. The mills two products come off the rolling line at different rates.

Tons per hour: Bands 200; Coils 140

and they also have different profitabilities:

Profit per ton: Bands $25; Coils $30

To further complicate matters, the following weekly production amounts are the most that

can be justified in light of the currently booked orders:

Maximum tons: Bands 6,000; Coils 4,000

The question facing the company is as follows: If 40 hours of production time are available

this week, how many tons of bands and how many tons of coils should be produced

to bring in the greatest total profit?

Antoni Guasch 18


Time constraint

Antoni Guasch 19


25 + 30 = profit

Xc =

30

25

30

Antoni Guasch 20


Solving the linear program reduces to answering

the following question: Among all profit lines that

intersect the feasible region, which is highest and

furthest to the right? The answer is the middle line,

which just touches the region at one of the corners.

Antoni Guasch 21

Example linear model: prod.mod

Algebraic

notation

Antoni Guasch 22


Algebraic notation

Look at 04-tut1.pdf ampl

tutorial file from page 7 to 10

Antoni Guasch 23


Prod.mod

Prod.dat

Antoni Guasch 24


Antoni Guasch 25

Atenea milestone #1

Select a continuous optimization model, code it in AMPL and obtain the

solution.

Write a report with:

Group names. Maximum of three persons in each group Description of the problem Classification of the problem: linear or nonlinear, constrained or not

constrained, differentiable or not, . AMPL code with comments on the code Results and conclusions

Deadline: before the second class. A group may be chosen randomly to

present the problem in the second class.

Antoni Guasch 26

Maria wants to travel from Diamond to Einstein. The quickest route takes 31 minutes.

Highlight this route (http://www.oecd.org/pisa/test/)

Discrete optimization example

Antoni Guasch 27

Discrete optimization models

Discrete

Integer Programming (IP)

Antoni Guasch 28

Network flows and integer programming models

Basic elements of a network (graph)

http://en.wikipedia.org/wiki/Graph_theory

http://en.wikipedia.org/wiki/Flow_network

Nodes

Arcs, links or branches

Antoni Guasch 29

Network terminology

i j

i j

i j

= 100

Flow: The amount of goods, vehicles, flights, passengers and so on that move from one node to

another.

Directed flow: flow only allowed in one direction

Undirected flow: flow allowed in both directions

Arc capacity: the maximum amount of flow that can be sent through an arc.

Antoni Guasch 30

Network terminology

i Supply node: Node with a positive net flow. I.e. input flow > output flow.

Demand node: Node with a negative net flow

Transshipment node: node with 0 net flow

115

i

50

i

0

Antoni Guasch 31

Network terminology

Path: Sequence of different arcs that connect two nodes. In the first network there are to pahts that connect node 7 with node

9.

Source: First node of the path (7)

Destination: Last node of the path (9)

Cycle: A sequence of directed arcs that starts and finish at the same node. The second network has two cycles.

Connected network: here are two distinct notions of connectivity in a directed network. A directed network is weakly

connected if there is an undirected path between any pair of

vertices, and strongly connected if there is a directed path

between every pair of vertices. An undirected network is

connected if there is a path from any point to any other point in

the network

Antoni Guasch 32

Shortest path problem

Antoni Guasch 33


, = 1 ,

0

Antoni Guasch 34


# PINEAPPLE1.MOD

var X12 binary;

var X13 binary;

var X14 binary;

var X25 binary;

var X26 binary;

var X35 binary;

var X36 binary;

var X37 binary;

var X46 binary;

var X47 binary;

var X58 binary;

var X68 binary;

var X78 binary;

minimize Distance: 105*X12+75*X13+65*X14+45*X25+56*X26+71*X35+48*X36+63*X37...;

subject to Source: X12+X13+X14=1;

subject to Destination: X58+X68+X78=1;

subject to Tras2: X12-X25-X26=0;

subject to Tras3: X13-X35-X36-X37=0;

subject to Tras4: X14-X46-X47=0;

subject to Tras5: X25+X35-X58=0;

subject to Tras6: X26+X36+X46-X68=0;

subject to Tras7: X37+X47-X78=0;

Antoni Guasch 35


ampl: model pineapple1.mod;

ampl: solve;

MINOS 5.5: ignoring integrality of 13 variables

MINOS 5.5: optimal solution found.

5 iterations, objective 174

ampl: option solver cplex;

ampl: solve;

CPLEX 12.6.0.0: optimal integer solution; objective 174

0 MIP simplex iterations

0 branch-and-bound nodes

ampl: display Distance;

Distance = 174

ampl: display X14, X46, X68;

X14 = 1

X46 = 1

X68 = 1

Antoni Guasch 36

Shortest path problem (SPP) general mathematical model

M = set of nodes i,j,k = index of nodes , = t of flow along arc (i,j)

m = destination node

, = 1 ,

0

Minimize ,,

1, =1 j 1

, - , = 0 , 1

, =1 i

Sets

Index

Parameters

Decision variables

Objective function

Constraints

Antoni Guasch 37

Shortest path problem (SPP) general mathematical model

# PINEAPPLE2.MOD

set M; # intersections

param entr symbolic in M; # entrance to road network

param exit symbolic in M, entr; # exit from road network

set FLIGHTS within (M diff {exit}) cross (M diff {entr});

param time {FLIGHTS} >= 0; # times to travel roads

var Use {(i,j) in FLIGHTS} binary; # 1 iff (i,j) in shortest path

minimize Total_Time: sum {(i,j) in FLIGHTS} time[i,j] * Use[i,j];

subject to Start: sum {(entr,j) in FLIGHTS} Use[entr,j] = 1;

subject to End: sum {(i,exit) in FLIGHTS} Use[i,exit] = 1;

subject to Balance {k in M diff {entr,exit}}:

sum {(i,k) in FLIGHTS} Use[i,k] = sum {(k,j) in FLIGHTS} Use[k,j];

data;

set M := m1 m2 m3 m4 m5 m6 m7 m8;

param entr := m1 ;

param exit := m8 ;

param: FLIGHTS: time :=

m1 m2 105,m1 m3 75

m1 m4 75,m2 m5 45

m2 m6 56,m3 m5 45

m3 m6 48,m3 m7 63

m4 m6 44,m4 m7 57

m5 m8 88,m6 m8 65

m7 m8 76;

Antoni Guasch 38

Shortest path problem (SPP): general mathematical model

ampl: model pineapple2.mod;

ampl: option solver cplex;

CPLEX 12.6.0.0: optimal integer solution; objective 184

0 MIP simplex iterations

0 branch-and-bound nodes

ampl: display Use;

Use :=

m1 m2 0

m1 m3 0

m1 m4 1

m2 m5 0

m2 m6 0

m3 m5 0

m3 m6 0

m3 m7 0

m4 m6 1

m4 m7 0

m5 m8 0

m6 m8 1

m7 m8 0

;

Antoni Guasch 39

Shortest path problem (SPP): additional class work

9 80

30

Add the new node in the previous two models and obtain the solution

Antoni Guasch 40

Minimum cost flow problem

The minimum-cost flow problem is to

find the cheapest possible way of sending

a certain amount of flow through a flow

network. Solving this problem is useful for

real-life situations involving networks

with costs associated (e.g.

telecommunications networks), as well as

in other situations where the analogy is not

so obvious, such as where to locate

warehouses

http://en.wikipedia.org/wiki/Minimum-

cost_flow_problem

Antoni Guasch 41


1 3

2 4

5

6

7

75

75

50

60

40

5

8

7

4

1

5

8

3 4

5

Antoni Guasch 42


# variables definition

var X13 integer;

var X14 integer;

# more variables ...

minimize Cost: 5*X13+8*X14+7*X23...;

subject to N1constraint: X13+X14

Antoni Guasch 43

Minimum cost flow problem: general mathematical form

M = set of nodes i,j,k = index of nodes , = t of flow along arc (i,j)

= . = (, )

= (, )

, =

Minimize ,,

, - , =

, , ,

Sets

Index

Parameters

Decision variables

Objective function

Constraints

Antoni Guasch 44

Atenea milestone #2

Build a model for the minimum cost flow problem presented before using

general mathematical form

Write a report with:

Group names. Maximum of three persons in each group Description of the problem AMPL code with comments on the code Results and conclusions

Op Tim Ization

Documents

Transcript of Op Tim Ization