OD Linear Programming 2010

7/24/2019 OD Linear Programming 2010

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1

LINEAR PROGRAMMING

Introduction

Development of linear programming was among the

most important scientific advances of mid-20th cent.

Most common type of applications: allocate limited

resourcesto compet ing act ivit iesin an optimal way.

Very large and wide range of applications from local to

very global ones.

Linear programming uses a mathematical model.

Linear because it requires linear functions.

Programming as synonymous ofplanning.

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Introduction

Besides allocate resources to activities, anyproblem

described by a mathematical model can be solved.

Very efficient solution procedure: simplex method.

This method will be described next as well as the

interior-point algorithmfor larger problems.

Besides these algorithms, we will also discuss

sensit ivit y analysis.

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Prototype example

Wyndor Glass Co. produces glass products, including

windows and glass doors.

Plant 1 produces aluminum frames, Plant 2 produces

wood frames and Plant 3 produces glass and

assembles.

Two new products:

Glass door with aluminum framing (Plant 1 and 3)

New wood-framed glass window (Plant 2 and 3)

As the products compete for Plant 3, the most

profitable mixof the two products is needed.

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Data for Wyndor Glass Co.

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Wyndor Glass Co. Produ ct-Mix Probl em

Doors Windows

Profit Per Batch $3.000 $5.000

(Batches of 20)

Hours available

per week

Plant 1 1 0 4

Plant 2 0 2 12

Plant 3 3 2 18

Hours Used Per Batch Produced

Formulation of the LP problem

x1 = number of batches of product 1 produced per week

x2 = number of batches of product 2 produced per week

Z= total profit per week (in thousands)

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1 2maximize 3 5Z x x

1

2

1 2

subject to 4

2 12

3 2 18

x

x

x x

1 2and 0, 0.x x


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Graphical solution

Trial-and-error: Z = 10, 20, 36

Slope of the line is 3/5:

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1

2

1 2

subject to 42 12

3 2 18

xx

x x

1 2and 0, 0.x x

2 1

3 1

5 5x x Z

Feasible region:

satisfiesall

constraints

Solution (using Excel)

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Wyndo r Glass Co. Produ ct-Mix Problem

Doors Windows Range Name Cel ls

Profit Per Batch $3.000 $5.000 BatchesProduced C12:D12

Hours Hours HoursAvailable G7:G9

Used Available HoursUsed E7:E9

Plant 1 1 0 2


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Standard form

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1 1 2 2maximize n nZ c x c x c x

11 1 12 2 1 1

21 1 22 2 2 2

1 1 2 2

subject to

n n

n n

m m mn n m

a x a x a x b

a x a x a x b

a x a x a x b

1 2and 0, 0, , 0.

nx x x

Other possible forms

Minimizing rather then maximizing

Some functional constraints are greater-than-or-equal

Some functional constraints in equation form

Deleting the non-negativity constraints for some vars

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1 1 2 2minimize n nZ c x c x c x

1 1 2 2 , for some values ofi i in n ia x a x a x b i

1 1 2 2, for some values of

i i in n ia x a x a x b i

unestricted in sign for some values ofj

x i


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Allocation of resources to activities

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Resource

Resource Usage per Unit of Activity

Amount of

Resource

available

Activity

1 2 n

1 a11 a12 a1n b1

2 a21 a22 a2n b2

m am1 am2 amn bm

Contribution

toZper unitof activity

c1 c2 cn

Terminology for solutions

Solution: any specification of values for the decision

variables (x1,x2,,xn).

Feasible solution: solution for which allthe constraints

are satisfied.

Infeasible solution: solution for which at least one

constraint is violated.

Feasible region

No feasible solution: no solution satisfies all

constraints.

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Example: no feasible solution

If in the Wyndor Glass Co. problem the profit should

be above $50000 per week:

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Terminology for solutions

Optimal solution: solution with the most favorable

value.

Most favorable value: largest valueifmaximizingor

smallest value ifminimizing.

No optimal solutions: (1) no feasible solutions or,

(2) unbounded Z(see next slide).

More than one solution: see next slide.

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Examples of solutions

More than one solution Unbounded cost function

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Corner-point feasible solution

Lies in a corner. Corners must have at least one

optimal solution.

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Assumptions of linear programming

Proportionality: value of objective function Z is

proportional to level of activity xj.

Violations: 1) Setup costs

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Violations of proportionality

2) Increase of marginal returndue to economy of

scale (longer production runs, quantity discounts,

learning-curve effect, etc.)

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Violations of proportionality

3) Decrease of marginal returndue to e.g. marketing

costs: to sell more the advertisement grows

exponentially.

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Assumptions of linear programming

Additivity: everyfunction in a linear programming

model is the sum of the individual contributions.

Divisibility: decision variables can have any values.

If values are limited to integer values, the problem

needs an integer programmingmodel.

Certainty: parameters of the model cj,aij andbi are

assumed to be known constant s.

In real applications parameters have always some

degree of uncertainty, and sensitivity analysis must be

conducted.

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Conclusions

Linear programming models can be solved using:

Spreadsheet as Excel Solver for relatively simple

problems

Problems with thousands (or even millions!) of decision

variables and/or functional constraints must use

modeling languages such as: CPLEX/MDL or

LINGO/LINDO

Matlab Optimization Toolbox

Linear programming solves many problems.

However, when one or more assumptions are violated

seriously, other programming techniques are needed.

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SIMPLEX METHOD


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Simplex method

Algebraic procedure with underlying geometric

concepts.

Definitions:

Constraint boundary: line of boundary of the

constraint.

Corner-point solution

Corner-point feasible solution (CFP solution)

CFP solutions are adjacent if they share n 1 constraint

boundaries.

The line segment between two CFP solutions is an edge.

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Example revisited

Wyndor Glass Co.

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Steps of simplex algorithm

Sequence of CFP solutions:

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Setting up the simplex method

Original form Augmented form

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1

2

1 2

subject to 4

2 12

3 2 18

x

x

x x

1 2and 0, 0.x x


3

4

5

1

2

1 2

subject to

(1) 4

(2) 2 12

(3) 3 2 18

x

x

x

xx

x

x

and 0, 1,2,3,4,5.j

x j

Convert functional inequality constraintsto equivalent

equality constraintsusing slack variables.


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Properties of a basic solution

Each variable is basic or nonbasic.

Number of basic variables is equal to number of

functional constraints (equations).

The nonbasic variables are set to zero.

Values ofbasic variables are solving system of

equations (functional constraints in augmented

form).

Set of basic variables are called the basis. If basic variables satisfy the nonnegativity constraints

basic solution is a BF solution.

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Final form

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maximize Z

1 2

1 3

2 4

1 2 5

subject to

(0) 3 5 0

(1) 4

(2) 2 12

(3) 3 2 18

Z x x

x x

x x

x x x

and 0, 1, ,5.j

x j


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Tabular form (simplex tableau)

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a) Algebraic form b) Tabular Form

Basic

variable

Eq.

Coefficient of:

Right

side

x

1

x

2

x

3

x

4

x

5

(0) 3x1 5x2 = 0 Z (0) 1 -3 -5 0 0 0 0

(1) x1 +x3 = 4 x3 (1) 0 1 0 1 0 0 4

(2) 2x

2 +x

4 =12 x

4 (2) 0 0 2 0 1 0 12

(3) 3x1+ 2x2 +x5 =18 x5 (3) 0 3 2 0 0 1 18

Simplex method in tabular form

Initialization. Introduce slack variables. Select decision

variables to be initial nonbasic variables and slack

variables to be initial basic variables.

Initial solution of example: (0, 0, 4, 12, 18)

Optimality test. Current BF solution is optimal if every

coefficient in row zero is nonnegative. If yes, stop.

Example: coefficients ofx1 andx2 are 3 and 5.

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Complete set of simplex tableau

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Iteration Basic variable Eq.

Coefficient of:

Right side

x

1

x

2

x

3

x

4

x

5

Z (0) 1 -3 -5 0 0 0 0

0 x3 (1) 0 1 0 1 0 0 4

x4 (2) 0 0 2 0 1 0 12

x5 (3) 0 3 2 0 0 1 18

Z (0) 1 -3 0 0 2.5 0 30

1 x3 (1) 0 1 0 1 0 0 4

x2 (2) 0 0 1 0 0.5 0 6

x5 (3) 0 3 0 0 -1 1 6Z (0) 1 0 0 0 1.5 1 36

2 x3 (1) 0 0 0 1 1/3 -1/3 2

x2 (2) 0 0 1 0 0.5 0 6

x1 (3) 0 1 0 0 -1/3 1/3 2

Other model forms

Equality constraints

In this case, artificial variables are used. The Big M

method is used.

Functional constraints in form

Surplus variables are used.

Minimization

Maximize Z.

Big M method can have two phases: 1) all artificial

variables driven to zero; 2) compute BF solutions.

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Postoptimality analysis

Reoptimization: when the problem changes slightly,

new solution is derived from thefinalsimplex tableau.

Shadow price for resource i(denoted yi*) is the

marginal value ofi.

i.e. the rate at whichZcan be increased by slightly

increasing the amount of this resource bi.

Example: recall bivalues ofWyndor Glass problem.

The final tableau gives:

y1* = 0, y2

* = 1.5, y3* = 1.

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Graphical check

Increasing b2 in 1 unit, the new profit is 37.5

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Shadow prices

Increasing b2 in 1 unit increases the profit in $1500.

Should this be done? Depends on the marginal

profitability of other products using that hour time.

Shadow prices are part of the sensit ivit y analysis.

Constraint on resource 1 is not bindingthe optimal

solution. Resources as this are free goods.

Constraints on resources 2 and 3 are binding

constraints. Resources are scarce goods.

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Sensitivity analysis

Identify sensitive parameters (those that cannot be

changed without changing optimal solution).

Parameters bican be analyzed using shadow prices.

Parameters ci for Wyndor example (see figure in next

slide).

Parameters aijcan also be analyzed graphically.

For problems with a large number of variables usually

only biand ciare analyzed, because aijare determined

by the technology being used.

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Sensitivity analysis of parameters ci

Graphical analysis of

Wyndor example:

c1 = 3 can have values in

[0, 7.5] without changing

optimal

c2 = 5 can have any value

greater than 2 without

changing optimal

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Interior-point approach

Discovery made in 1984 by Narendra Karmarkar.

Used in hugelinear programming problems, beyond

the scope of the simplex method.

Starts by identifying a feasible t rial solution, moving a

better one until it reaches the optimal trial solution.

Trial solutions are interior points.

Interior-point or barrier algorithms (each constraint

boundary is treated as a barrier).

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Augmented form

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1 1 1 1maximize n n n n n m n mZ c x c x c x c x

11 1 12 2 1 1 1

21 1 22 2 2 2 2

1 1 2 2

subject to

n n n

n n n

m m mn n n m m

a x a x a x x b

a x a x a x x b

a x a x a x x b

and 0, 1,2, , .j

x j m

Matrix form (standard form)

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maximize Zcx

subject to

and Ax b x 0

1 2[ , , , ],nc c cc

1

2,

n

x

x

x

x

1

2,

m

b

b

b

b

0

0,

0

0

11 12 1

21 22 2

1 2

.

n

n

m m mn

a a a

a a a

a a a

A


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Revised simplex method

1. Initialization: (as original) Introduce slack variables xs.

Decision variables are initial nonbasic variables.

2. Iteration:

Step 1: (as original) Determine entering basic variable

(nonbasic variable with the largest negative

coefficient).

Step 2: Determine leaving basic variable. As original,

but computations are simplified.

Step 3: Determine new BF solution: set xB = B1b.

3. Optimality test: solution is optimal if every

coefficient inZis 0. Computations are simplified.

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DUALITY THEORY ANDSENSITIVITY ANALYSIS


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Duality theory

Every linear programming problem (primal) has a dual

problem.

Most problem parameters are estimates, others (as

resource amounts) represent managerial decisions.

The choice of parameter values is made based on a

sensit ivit y analysis.

Interpretation and implementation of sensitivity

analysis are key uses of duality theory.

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Primal and dual problems

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1

maximizen

j j

j

Z c x

1

subject to

, for 1,2, ,n

ij j i

j

a x b i m

and 0 for 1,2, , .j

x j n

Primal problem

1

minimizem

i i

i

W b y

1

subject to

, for 1,2, ,m

ij i j

j

a y c j n

and 0 for 1,2, , .i

y i m

Dual problem


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Duality theory

Relations between parameters:

The coefficients of the objective functions of the primal

are the right-hand sides of the functional constraints in

the dual.

The right-hand side of the functional constraints in the

primal are coefficients of the objective functions in the

dual.

The coefficients of a variable in the functional

constraints of the primal are the coefficients in thefunctional constraints of the dual.

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Primal-dual table

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Primal problem

Coefficient of: Right

side

x

1

x

2

xn

C

c

e

o

y

1

a11 a12 a1n b1C

c

e

s

o

o

e

v

u

o

m

n

m

z

e

D

p

o

e

m

y

2

a21 a22 a2n b2

y

an1 an2 amn bmR

g

s

d

c1 c2 cn

Coefficients of objective function(maximize)


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Example: Wyndor Glass Co.

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1

2

3 5 ,x

Maximize Zx

1

2

subject to

1 0 4

0 2 12

3 2 18

x

x

1

2

0and .

0

x

x

Primal problem Dual problem

1 2 3

4

Minimize 12 ,

18

Z y y y

1 2 3

subject to

1 0

0 2 3 5

3 2

y y y

1 2 3and 0 0 0 .y y y

Example: Wyndor Glass Co.

Tableau representation

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x

1

x

2

y1 1 0 4

y2 0 2 12

y3 3 2 18

3 5


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Primal-dual relationships

Weak duality property: Ifx is a feasible solution for

the primal problem andy is a feasible solution for the

dual problem, then

cxyb

Strong duality property: Ifx* is an optimal solution for

the primal problem andy* is an optimal solution for

the dual problem, then

cx

*

=y*

b

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Complementary solutions property: At each iteration,

the simplex method identifies a CFP solutionx for the

primal problem and a complementary solution for the

dual problem, where

cx =yb

Ifx is not optimalfor the primal problem, theny is not

feasiblefor the dual problem.

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Complementary optimal solutions property: At the

final iteration, the simplex method identifies an

optimal solutionx* for the primal problem and a

complementary optimal solutiony* for the dual

problem, where

cx* =y*b

Theyi* are the shadow prices for the primal problem.

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Symmetry property: For anyprimal problem and its

dual problem, all relationships between them must be

symmetricbecause the dual of this dual problem is

this primal problem.

Duality theorem: The following are the only possible

relationships between the primal and dual problems:

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Applications of duality theory

If the number of functional constraints m is bigger than

the number of variablesn, applying simplex directly to

the dual problem will achieve a substantial reduction

in computational effort.

Evaluating a proposed solution for the primal

problem. i) ifcx =yb,x andy must be optimal even

without applying simplex; ii) ifcx


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Duality theory in sensitivity analysis

Sensitivity analysis investigate the effect of changing

parameters aij,bi,cj in the optimal solution.

Can be easier to study these effects in the dual

problem

Optimal solution of the dual problem are the shadow

prices of the primal problem.

Changes in coefficients of a nonbasic variab le. Only

changes one constraint in the dual problem. If this

constraint is satisfied the solution is still optimal.

Introducing a new variable in the object ive function.

Only introduces a new const raintin the dual problem. If

the dual problem is still feasible, solution is still optimal.

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Essence of sensitivity analysis

LP assumes that all the parameters of the model (aij,biandcj) are known constants.

Actually:

The parameters values used are just estimates based on

a prediction of future conditions;

The data may represent deliberate overestimates or

underestimates to protect the interests of the

estimators.

An optimal solution is optimal only with respect to

the specific model being used to represent the realproblem sensit ivit y analysisis crucial!

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Simplex tableau with parameter changes

What changes in the simplex tableau if changes aremade in the model parameters, namely b ,c ,

A ?

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A

b c

Coefficientof:

Right side

q. Z Original variables Slack variables

New initialtableau

(0) 1 0 0

(1,2,,m) 0 I

Revisedfinal tableau

(0) 1 y*

(1,2,,m) 0 S*

c

A b

* *Z y b

* *b S b

* * z c y A c

* *A S A

Applying sensitivity analysis

Changes inb:

Allowable range: range of values for which the current

optimal BF solution remains feasible (find range ofbisuch that , assuming this is the only change

in the model). The shadow price for bi remains valid ifbi

stays within this interval. The 100% rule for simult aneous changes in right-hand

side: the shadow prices remain valid as long as the

changes are not too large. If the sum of the % changes

does not exceed 100%, the shadow prices will still be

valid.

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* * b S b 0


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Applying sensitivity analysis

Changes in coefficients of nonbasic variable:

Allowable range to stay optimal: range of values over

which the current optimal solution remains optimal

(find , assuming this is the only change in the

model).

The 100% rule for simult aneous changes in object ive

funct ion coeff icients: if the sum of the % changes does

notexceed 100%, the original optimal solution will still

be valid. Introduction of a new variable:

Same as above.

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*j j

c y A

Other simplex algorithms

Dual simplex method

Modification useful for sensitivity analysis

Parametric linear programming

Extension for systematic sensitivity analysis

Upper bound technique A simplex version for dealing with decision variables

having upper bounds:xj uj, where uj is the maximumfeasible value ofxj.

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OD Linear Programming 2010

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