Linear Programming (LP) Linear Programming (LP)

8/14/2019 Linear Programming (LP) Linear Programming (LP)

1/25

Optimization in Engineering Design

1

Georgia Institute of Technology

Systems Realization Laboratory

Linear Programming (LP)

Linear Programming (LP)


2/25


2



Why Talk About Linear Programming?

Why Talk About Linear Programming?

LP is simpler than NLP, hence, good for a foundation

Linearity has some unique features for optimization

A lot of problems are or can be converted to a LP formulation

Some NLP algorithms are based upon LP simplex method


3/25


3



Bolted Joint Design

Bolted Joint Design

GivenAt - tensile strength area, function of d

Db - bolt circle diameter

Pt - total load

C - joint constant

Fi - preload (= 0.75 Sp At)Find

N - number of bolts, Sp - proof strength, d - diameter

Satisfy

3d Db / N good wrench rule Db / N 6d good seal ruleC Pt / N Sp At - Fi static loading constraint

Fi (1 - C) Pt / N joint separation constraint

Minimize Z= [ f1(N, d, Sp), f2(N, d, Sp), ..]

Question:

Is this a linear ornonlinear model?

Question:

Is this a linear or

nonlinear model?


4/25


4



Bolted Joint Design (2)

Bolted Joint Design (2)

Given

d - diameterAt - tensile strength area, function of d

Db - bolt circle diameter

Pt - total load

C - joint constant

Fi - preload (= 0.75 Sp At)

Find

N - number of bolts, Sp - proof strength

Satisfy

3d Db / N good wrench rule

Db / N 6d good seal rule

C Pt / N Sp At - Fi static loading constraint

Fi (1 - C) Pt / N joint separation constraint

Minimize Z= [ f1(N, Sp), f2(N, Sp), ..]

Question:

Is this a linear ornonlinear problem?

Question:

Is this a linear or

nonlinear problem?


5/25


5



Historical Perspective

Historical Perspective

1928 John von Neumann published related central theorem of gametheory

1944 Von Neumann and Morgenstern published Theory of Games andEconomic Behavior

1936 W.W. Leontief published "Quantitative Input and Output Relations

in the Economic Systems of the US" which was a linear model withoutobjective function.

1939 Kantoravich (Russia) actually formulated and solved a LPproblem

1941 Hitchcock poses transportation problem (special LP)

WWII Allied forces formulate and solve several LP problems related tomilitary

A breakthrough occurred in 1947...


6/25


6



SCOOP

SCOOP

US Air Force wanted to investigate the feasibility of applyingmathematical techniques to military budgeting and planning.

George Dantzig had proposed that interrelations between activities of alarge organization can be viewed as a LP model and that the optimalprogram (solution) can be obtained by minimizing a (single) linearobjective function.

Air Force initiated project SCOOP (Scientific Computing of OptimumPrograms)

NOTE:

SCOOP began in June 1947 and at the end of the same summer, Dantzig

and associates had developed:1) An initial mathematical model of the general linear programming

problem.

2) A general method of solution called the simplex method.


7/25


7



Simplex TodaySimplex Today

A large variety of Simplex-based algorithms exist to solveLP problems.

Other (polynomial time) algorithms have been developed forsolving LP problems:

Khachian algorithm (1979)

Kamarkar algorithm (AT&T Bell Labs, mid 80s)

See Section 4.10

BUT,

none of these algorithms have been able to beat Simplex inactual practical applications.

HENCE,

Simplex (in its various forms) is and will most likely remainthe most dominant LP algorithm for at least the near future


8/25


8



Fundamental TheoremFundamental Theorem

Extreme point (or Simplex filter) theorem:

If the maximum or minimum value of alinear function defined over a polygonalconvex region exists, then it is to be found

at the boundary of the region.

Extreme point (or Simplex filter) theorem:

If the maximum or minimum value of alinear function defined over a polygonalconvex region exists, then it is to be found

at the boundary of the region.

Convex set:

A set (or region) is convex if, for any two points (say, x1and x2) in that set, the line segment joining these points

lies entirely within the set.

A point is by definition convex.

Convex set:

A set (or region) is convex if, for any two points (say, x1and x2) in that set, the line segment joining these points

lies entirely within the set.A point is by definition convex.


9/25


9



What does the extreme point theorem imply?What does the extreme point theorem imply?

A finite number of extreme points implies a finite numberof solutions!

Hence, search is reduced to a finite set of points

However, a finite set can still be too large for practicalpurposes

Simplex method provides an efficient systematic searchguaranteed toconverge in a finite number of steps.


10/25


10



Basic Steps of SimplexBasic Steps of Simplex

1. Begin the search at an extreme point (i.e., a basic feasible

solution).

2. Determine if the movement to an adjacent extreme canimprove on the optimization of the objective function. If not,the current solution is optimal. If, however, improvement is

possible, then proceed to the next step.

3. Move to the adjacent extreme point which offers (or,perhaps, appears to offer) the most improvement in theobjective function.

4. Continue steps 2 and 3 until the optimal solution is found or itcan be shown that the problem is either unbounded orinfeasible.


11/25


11



Step 0 Obtain Canonical FormStep 0 Obtain Canonical Form

General Simplex LP model:

min (or max) z = ci xi

s.t.

A x = b

x 0In order to get and maintain this form, use

slack, if x b, then x + slack = b

surplus, if x b, then x - surplus = b

artificial variables (sometimes need to be added to ensure all

variables 0, see page 101)

Compare constraint conversion with goalconversions using deviation variables

Compare constraint conversion with goalconversions using deviation variables

IMPORTANT: Simplex only deals with equalities


12/25


12



Different "components" of a LP modelDifferent "components" of a LP model

LP model can always be split into a basic and a non-basicpart.

Transformed or reduced model is another good way

to show this.

This can be represented in mathematical terms as well asin a LP or simplex tableau.


13/25


13



Movement to Adjacent Extreme PointMovement to Adjacent Extreme Point

Given any basis we move to an adjacent extreme point(another basic feasible solution) of the solution space byexchanging one of the columns that is in the basis for acolumn that is not in the basis.

Two things to determine:

1) which (nonbasic) column of A should be brought intothe basis so that the solution improves?

2) which column can be removed from the basis such thatthe solution stays feasible?


14/25


14



Entering and Departing Vector (Variable) RulesEntering and Departing Vector (Variable) Rules

General rules:

The one non-basic variable to come in is the one whichprovides the highest reduction in the objective function.

The one basic variable to leave is the one which isexpected to go infeasible first.

NOTE: THESE ARE HEURISTICS!!

Variations on these rules exist, but are rare.


15/25


15



Simplex VariationsSimplex Variations

Various variations on the simplex method exist:

"regular" simplex (see Section 4.4)

two-phase method: Phase I for feasibility and Phase II

for optimality (see Section 4.5.1) condensed/reduced/revised method: only use the non-

basic columns to work with (see Section 4.6)

(revised) dual simplex (see Section 4.8), etc.


16/25


16



Computational ConsiderationsComputational Considerations

Unrestricted variables (unboundedness)

Redundancy (linear dependency, modeling errors)

Degeneracy (some basic variables = 0)

Round-off errors


17/25


17



Limitations of SimplexLimitations of Simplex

1. Inability to deal with multiple objectives

2. Inability to handle problems with integer variables

Problem 1 is solved using Multiplex

Problem 2 has resulted in:

Cutting plane algorithms (Gomory, 1958)

Branch and Bound (Land and Doig, 1960)

However,solution methods to LP problems with integer or Booleanvariables are still far less efficient than those which includecontinuous variables only


18/25


18



Example ProblemExample Problem

Maximize Z = 5x1 + 2x2 + x3

subject to

x1 + 3x2 - x3 6,

x2 + x3 4,

3x1 + x2 7,

x1, x2, x3 0.


19/25


19



Simplex and Example ProblemSimplex and Example Problem

Step 1. Convert to Standard Form

a11 x1 + a12 x2 + + a1n xn b1,

a21 x1 + a22 x2 + + a2n xn b2,

am1 x1 + am2 x2 + + amn xn bm,

a11 x1 + a12 x2 + + a1n xn + xxn+1n+1 = b1,

a21 x1 + a22 x2 + + a2n xn - xxn+2n+2 = b2,

am1 x1 + am2 x2 + + amn xn + xxn+kn+k = bm,

In our example problem:

x1 + 3x2 - x3 6,

x2 + x3 4,

3x1 + x2 7,

x1, x2, x3 0.

x1 + 3x2 - x3 + x4 = 6,

x2 + x3 + x5 = 4,

3x1 + x2 + x6 = 7,

x1, x2, x3, x4, x5, x6 0.


20/25


20



Simplex: Step 2Simplex: Step 2

Step 2. Start with an initial basic feasible solution (b.f.s.) and set up the

initial tableau.

In our example

Maximize Z = 5x1 + 2x2 + x3

x1 + 3x2 - x3 + x4 = 6,

x2 + x3 + x5 = 4,

3x1 + x2 + x6 = 7,

x1, x2, x3, x4, x5, x6 0.cj

5 2 1 0 0 0

cB Basis

x1 x2 x3 x4 x5 x6

Constants

0 x4 1 3 -1 1 0 0 6

0 x5 0 1 1 0 1 0 40 x6 3 1 0 0 0 1 7

c row 5 2 1 0 0 0 Z=0


21/25


21



Step 2: ExplanationStep 2: Explanation

Adjacent Basic Feasible Solution

If we bring a nonbasic variable xs into the basis, our system changes fromthe basis, xb, to the following (same notation as the book):

x1 + 1s xs=

xr + rsxr=

xm + ms xs=

1b

rb

sb

xi = for i =1, , misi ab

xs = 1

xj = 0 for j=m+1, ..., n and j s

The new value of the objective function becomes:

=

+=m

1i

sisii c)ab(cZ

Thus the change in the value of Z per unit increase in xs issc = new value of Z - old value of Z

= ==

+m

1i

ii

m

1i

sisii bcc)ab(c

=

m

1i

isis acc=

This is the Inner Product rule


22/25


22




Use the inner product rule to find the relative profit coefficients

cj5 2 1 0 0 0

cB Basis

x1 x2 x3 x4 x5 x6

Constants

0 x4 1 3 -1 1 0 0 6

0 x5 0 1 1 0 1 0 40 x6 3 1 0 0 0 1 7

c row 5 2 1 0 0 0 Z=0

jBjj Pccc =

c1 = 5 - 0(1) - 0(0) - 0(3) = 5 -> largest positivec2 = .

c3 = .

Step 4: Is this an optimal basic feasible solution?


23/25


23




Apply the minimum ratio rule to determine the basic variable to leave the basis.

The new values of the basis variables:

xi = for i = 1, ..., msisi xab

= >is

i0a

sa

bminxmaxis

In our example:

Row Basic Variable Ratio

1 x4 62 x5 -

3 x6 7/3


24/25


24




Perform the pivot operation to get the new tableau and the b.f.s.

jBjj Pccc=

c2 = 2 - (0) 8/3 - (0) 1 - (5) 1/3 = 1/3

c3 = 1 - (0) (-1) - (0) 1 - (5) 0 = 1

c6 = 0 - (0) 0 - (0) 0 - (5) 1/3 = -5/3

cB = (0 0 5)

New iteration:find enteringvariable:


25/25


25



Final TableauFinal Tableau

x3 enters basis,x5 leaves basis

Wrong value!4 should be 11/3

Date post:	30-May-2018
Category:	Documents
Upload:	dileep-pandey
View:	292 times
Download:	1 times

Linear Programming (LP) Linear Programming (LP)

Documents