6.1 Chapter 2 Galaxy LP Problem Updated

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Mjdah Al Shehri

Hamdy A. Taha, Operations Research: An

introduction, 8th Edition

Chapter 2:Modeling with LinearProgramming & sensitivityanalysis

1



Mute ur call



Hamdy A. Taha, Operations Research: An introduction, rentice Hall !

LINEAR PRORAMMIN !LP"

-In mathematics, linear programming (LP) is a technique foroptimization of a linear objective function, subject to linearequality and linear inequality constraints.

-Linear programming determines the way to achieve the bestoutcome (such as maimum profit or lowest cost! in a givenmathematical model and given some list of requirementsrepresented as linear equations.

!



Hamdy A. Taha, Operations Research: An introduction, rentice Hall "

Mathemati#al $orm%lation o$

Linear Programming model:"tep #

- "tudy the given situation

- $ind the %ey decision to be made- Identify the decision variables of the problem

"tep &- $ormulate the objective function to be optimized

"tep '- $ormulate the constraints of the problem

"tep

- )dd non-negativity restrictions or constraints*he objective function , the set of constraints and the non-negativity

restrictions together form an L+ model.

"



Hamdy A. Taha, Operations Research: An introduction, rentice Hall #

'O()ARIA*LE LP MO+EL

)+L/

0 THE GALAXY INDUSTRY PRODUCTION1

2 3alay manufactures two toy models/4 "pace 5ay.

4 6apper.

2 5esources are limited to

4 #&77 pounds of special plastic.

4 7 hours of production time per wee%.

#



Hamdy A. Taha, Operations Research: An introduction, rentice Hall $

2 ar%eting requirement

4 *otal production cannot eceed 877 dozens.

4 9umber of dozens of "pace 5ays cannot eceed number of dozens

of 6appers by more than :7.

2 *echnological input

4 "pace 5ays requires & pounds of plastic and

' minutes of labor per dozen.

4 6appers requires # pound of plastic and

minutes of labor per dozen.

$



Hamdy A. Taha, Operations Research: An introduction, rentice Hall %

2 ;urrent production plan calls for/

4 +roducing as much as possible of the more profitable product, "pace 5ay(<8 profit per dozen!.

4 =se resources left over to produce 6appers (<: profit

per dozen!.

2 *he current production plan consists of/

"pace 5ays > ::7 dozens

6apper > #77 dozens

+rofit > ?77 dollars per wee%

%



Management is8



Hamdy A. Taha, Operations Research: An introduction, rentice Hall &&

A Linear Programming ModelA Linear Programming Model

can provide an intelligentcan provide an intelligent

solution to this problemsolution to this problem



Hamdy A. Taha, Operations Research: An introduction, rentice Hall 1'

0OL1ION

2 @ecisions variables//

4 # > +roduction level of "pace 5ays (in dozens per wee%!.

4 & > +roduction level of 6appers (in dozens per wee%!.

2 Abjective $unction/

4 Bee%ly profit, to be maimized

1'



Hamdy A. Taha, Operations Research: An introduction, rentice Hall 11

The Linear Programming o!el

a 8# C :& (Bee%ly profit!

subject to&# C #& D > #&77 (+lastic!

'# C & D > &77 (+roduction *ime!

# C & D > 877 (*otal production! # - & D > :7 (i!

jE > 7, j > #,& (9onnegativity!

11



Hamdy A. Taha, Operations Research: An introduction, rentice Hall 1(

Feasible Solutions for Linear

Programs2 *he set of all points that satisfy all the constraints of the model is called

1(

!A"#$L! R!%#O&!A"#$L! R!%#O&



Hamdy A. Taha, Operations Research: An introduction, rentice Hall 1!

=sing a graphical presentation we can represent all the constraints,

the objective function, and the three types of feasible points.

1!



1"

1200

600

The Plastic constraint

easible

The plastic constraint:'()*('+)'--

X2

#neasible

Production

Time

/()*0('+'0--

Total production constraint:

()*('+1--600

800

Production mi2

constraint:

()3('+04-X1



1#

Solving Graphically for an

Optimal Solution



1$

R e c a l l t h e

. e a s i

b l e R e

g i o n

600

800

1200

400 600 800

X2

X1

)e no* demonstrate the search +or an optimal solution

Start at some aritrary pro+it, say pro+it - (,'''...

ro+it -

'''

(,

Then increase the pro+it, i+ possile...

!,",

...and continue until it ecomes in+eas

Proit 54-0-



1%

600

800

1200

400 600 800

X2

X1

/et0s tae a closer loo

at the optimal point

Feasible

regionFeasible

region

2n+easile



18

1200

600

The Plastic constraint

easible


X2

#neasible

Production

Time

/()*0('+'0--


()*('+1--600

800

Production mi2

constraint:

()3('+04-X1

A 6-,7--8

! 6-,-8

$ 601-,'0-8

9 644-,)--8

604-,-8



Hamdy A. Taha, Operations Research: An introduction, rentice Hall 1&

2 *o determine the value for # and & at the optimal

point, the two equations of the binding constraint

must be solved.

1&



Production mi2

constraint:

()3('+04-

('


Production

Time

/()*0('+'0--

(3143(-1(''

!314"3(-("''31- "8'

3(- ("'

(3143(-1(''

3153(-"#'31- ##'

3(- 1''



Hamdy A. Taha, Operations Research: An introduction, rentice Hall (1

Fy ;ompensation on /

a 8# C :&

*he maimum profit (:77! will be by producing/

"pace 5ays > 87 dozens, 6appers > &7 dozens

(1

(X"# X$) O%&e'ie *n

(7,7! 7

(:7,7! 'G77

(87,&7! :77

(::7,#77! ?77

(7,G77! '777



Hamdy A. Taha, Operations Research: An introduction, rentice Hall ((

ype o$ $easile points

2 Interior point/ satisfies all constraint but non with

equality.

2 Foundary points/ satisfies all constraints, at least one with equality

2 treme point/ satisfies all constraints, two with

equality.

((



(!

1200

600

The Plastic constraintThe plastic constraint:'()*('+)'--

X2

#neasibleProductionTime

/()*0('

+'0--


()*('+1--600

800

Production mi2

constraint:

()3('+04-X1

6'--, '--8

6

#nterior

point

6/--,-8

6

$oundary

point

644-,)--8

6

!2treme

point



Hamdy A. Taha, Operations Research: An introduction, rentice Hall ("

2 If a linear programming has an optimal solution , an

etreme point is optimal.

("



Hamdy A. Taha, Operations Research: An introduction, rentice Hall (#

0%mmery o$ graphi#al sol%tionpro#ed%re

#- graph constraint to find the feasible point

&- set objective function equal to an arbitrary value so that line

passes through the feasible region.'- move the objective function line parallel to itself until it

touches the last point of the feasible region .

- solve for # and & by solving the two equation that intersect

to determine this point

:- substitute these value into objective function to determine its

optimal solution. (#



Hamdy A. Taha, Operations Research: An introduction, rentice Hall ($

ORE EXAPLE

($



Hamdy A. Taha, Operations Research: An introduction, rentice Hall (%

E3ample 2/4(4!he Reddy Mi,,s Company"- 5eddy i%%s produces both interior and eterior paints from two raw materials #

and &

*ons of raw material per ton of

terior paint Interior paint aimum daily availability (tons!

5aw material # G &5aw material & # & GHHHHHHHH

+rofit per ton (<#777! :

-@aily demand for interior paint cannot eceed that of eterior paint by more

than # ton

-aimum daily demand of interior paint is & tons

-5eddy i%%s wants to determine the optimum product mi of interior and

eterior paints that maimizes the total daily profit

(%



Hamdy A. Taha, Operations Research: An introduction, rentice Hall (8

0ol%tion:

Let # > tons produced daily of eterior paint

& > tons produced daily of interior paint

Let z represent the total daily profit (in thousands of dollars!

O%&e'ie+

aimize z > : # C &

(=sage of a raw material by both paints! D (aimum raw material

availability!

=sage of raw material # per day > G# C & tons

=sage of raw material & per day > ## C && tons

- daily availability of raw material # is & tons

- daily availability of raw material & is G tons

(8



Hamdy A. Taha, Operations Research: An introduction, rentice Hall (&

Restri#tions:

G# C & D & (raw material #!

# C && D G (raw material &!

- @ifference between daily demand of interior (&! and eterior (#!

paints does not eceed # ton, so & - # D #- aimum daily demand of interior paint is & tons,

so & D &

- ariables # and & cannot assume negative values, so # E 7 , & E 7

(&



Hamdy A. Taha, Operations Research: An introduction, rentice Hall !'

Complete Reddy Mi,,s model:

aimize z > : # C & (total daily profit!

subject to

G# C & D & (raw material #!

# C && D G (raw material &! & - # D #

& D &# E 7

& E 7

- Abjective and the constraints are all linear functions in thiseample.

!'

Properie, o* he LP mo!el+



!1

Properie, o* he LP mo!el+ Linearity implies that the L+ must satisfy three basic properties/

#! +roportionality/

- contribution of each decision variable in both the objective

function and constraints to be directly proportional to the

value of the variable

&! )dditivity/

- total contribution of all the variables in the objective function

and in the constraints to be the direct sum of the individual

contributions of each variable

'! ;ertainty/

- )ll the objective and constraint coefficients of the L+ model are

deterministic (%nown constants!

- L+ coefficients are average-value approimations of the probabilistic

distributions - If standard deviations of these distributions are sufficiently small , then the

approimation is acceptable

- Large standard deviations can be accounted for directly by using stochastic L+

algorithms or indirectly by applying sensitivity analysis to the optimum solution



Hamdy A. Taha, Operations Research: An introduction, rentice Hall !(

E3ample 2/4(2

!Prolem Mi3 Model"- *wo machines and J

- is designed for :-ounce bottles

- J is designed for #7-ounce bottles

- can also produce #7-ounce bottles with some loss of

efficiency

- J can also produce :-ounce bottles with some loss of

efficiency

!(

achine : ounce bottles #7 ounce bottles



!!

achine :-ounce bottles #7-ounce bottles

87Kmin '7Kmin

J 7Kmin :7Kmin- and J machines can run 8 hours per day for : days a

wee%- +rofit on :-ounce bottle is &7 paise

- +rofit on #7-ounce bottle is '7 paise- Bee%ly production of the drin% cannot eceed :77,777

ounces- ar%et can utilize '7,777 (:-ounce! bottles and 8777 (#7-

ounce! bottles per wee%- *o maimize the profit



!"

Sol-ion+

Let # > number of :-ounce bottles to be produced per wee%

& > number of #7-ounce bottles to be produced per wee%

O%&e'ie+

aimize profit z > 5s (7.&7# C 7.'7&!

Con,rain,+

- *ime constraint on machine ,

(#K87! C (&K'7! D 8 G7 : > &77 minutes - *ime constraint on machine J,

(#K7! C (&K:7! D 8 G7 : > &77 minutes

- Bee%ly production of the drin% cannot eceed :77,777 ounces,

:# C #7& D :77,777 ounces

- ar%et demand per wee%,

#

E '7,777 (:-ounce bottles! & E 8,777 (#7-ounce bottles!

E.ample $ "01



!#

E.ample $/" 1

(Pro!-'ion Allo'aion o!el)

- *wo types of products ) and F

- +rofit of 5s. on type )- +rofit of 5s.: on type F- Foth ) and F are produced by and J machines

achine achine

+roducts J

) & minutes ' minutes

F & minutes & minutes

- achine is available for maimum : hours and '7 minutes during any

wor%ing day

- achine J is available for maimum 8 hours during any wor%ing day

- $ormulate the problem as a L+ problem.

Sol-ion+



!$

Let # > number of products of type )

& > number of products of type F

O%&e'ie+

- +rofit of 5s. on type ) , therefore # will be the profit on selling # units of type )- +rofit of 5s.: on type F, therefore :& will be the profit on selling & units of type F

*otal profit, z > # C :&

Con,rain,+

- *ime constraint on machine ,

&# C && D ''7 minutes

- *ime constraint on machine J,

'# C && D 87 minutes

- 9on-negativity restrictions are,

# E 7 and & E 7



!%

;omplete L+ model is, aimize z > # C :&

subject to

&# C && D ''7 minutes

'# C && D 87 minutes# E 7

& E 7

$ $ GRAPHICAL LP SOLUTION



!8

$/$ GRAPHICAL LP SOLUTION

*he graphical procedure includes two steps/#! @etermination of the feasible solution space.

&! @etermination of the optimum solution from

among all the feasible points in the solutionspace.

$ $ " Sol-ion o* a a.imi2aion mo!el



!&

$/$/" Sol-ion o* a a.imi2aion mo!el

E.ample $/$0" (Re!!3 i44, mo!el)"tep #/

#! @etermination of the feasible solution space/ - $ind the coordinates for all the G equations of the

restrictions (only ta%e the equality sign!

G# C & D &

# C && D G

& - # D #

& D &# E 7

& E 7

1

(

!

"

#

$

- ;hange all equations to equality signs



"'

- ;hange all equations to equality signs

G# C & > &

# C && > G

& - # > #

& > &# > 7

&

> 7

1

(

!

"

#

$

7 7



"1

- +lot graphs of # > 7 and & > 7- +lot graph of G# C & > & by using

the coordinates of the equation - +lot graph of # C && > G by using

the coordinates of the equation

- +lot graph of & - # > # by using


- +lot graph of & > & by using




"(

9ow include the inequality of all the G equations



"!

- 9ow include the inequality of all the G equations

- Inequality divides the (#, &! plane into two half spaces , one on

each side of the graphed line

- Anly one of these two halves satisfies the inequality

- *o determine the correct side , choose (7,7! as a reference point

- If (7,7! coordinate satisfies the inequality, then the side in which

(7,7! coordinate lies is the feasible half-space , otherwise the

other side is

- If the graph line happens to pass through the origin (7,7! , then

any other point can be used to find the feasible half-space

"tep &/



""

"tep &/

&! @etermination of the optimum solution from among

all the feasible points in the solution space/

- )fter finding out all the feasible half-spaces of allthe G equations, feasible space is obtained by the

line segments joining all the corner points ), F, ;,

@ , and $

- )ny point within or on the boundary of thesolution space )F;@$ is feasible as it satisfies all

the constraints

- $easible space )F;@$ consists of infinite number

of feasible points

- *o find optimum solution identify the direction in which the



"#

p y

maimum profit increases , that is z > :# C &

- )ssign random increasing values to z , z > #7 and z > #:

:# C & > #7

:# C & > #:

- +lot graphs of above two equations

- *hus in this way the optimum solution occurs at corner point ; which is the

point in the solution space

- )ny further increase in z that is beyond corner point ; will put points

outside the boundaries of )F;@$ feasible space

- alues of # and & associated with optimum corner point ; are

determined by solving the equations and

G# C & > &

# C && > G

- # > ' and & > #.: with z > : ' C #.: > &#

- "o daily product mi of ' tons of eterior paint and #.: tons of interior paint

produces the daily profit of <&#,777 .

1(

1 (



"$

- Important characteristic of the optimum L+ solution is that it is



"%

Important characteristic of the optimum L+ solution is that it isalways associated with a corner point of the solution space (wheretwo lines intersect!

- *his is even true if the objective function happens to beparallel to a constraint

- $or eample if the objective function is, z > G# C &

- *he above equation is parallel to constraint of equation

- "o optimum occurs at either corner point F or corner point

; when parallel

- )ctually any point on the line segment F; will be an

alternative optimum

- Line segment F; is totally defined by the corner points

F and ;

1

"i ti L+ l ti i l i t d ith i t f



"8

- "ince optimum L+ solution is always associated with a corner point of

the solution space, so optimum solution can be found by enumerating all

the corner points as below/-

HHHHHHHHHHHHHH;orner point (#,&! zHHHHHHHHHHHHHHHHH ) (7,7! 7

F (,7! &7

C (1#"/5) $" (opim-m ,ol-ion)

@ (&,&! #8

(#,&! #' $ (7,#!

- )s number of constraints and variables increases , the number of corner

points also increases

$/$/$ Sol-ion o* a inimi2aion mo!el



"&

$/$/$ Sol-ion o* a inimi2aion mo!el

E.ample $/$01

- $irm or industry has two bottling plants- Ane plant located at ;oimbatore and other plant located at

;hennai

- ach plant produces three types of drin%s ;oca-cola , $anta

and *humps-up

9umber of bottles produced per day



#'

by plant at

;oimbatore ;hennaiHHHHHHHHHHHHHHHHHHHHHH

;oca-cola #:,777 #:,777

$anta '7,777 #7,777

*humps-up &7,777 :7,777HHHHHHHHHHHHHHHHHHHHHHH

;ost per day G77 77

(in any unit!

- ar%et survey indicates that during the month of )pril there will be a demand of &77,777

bottles of ;oca-cola , 77,777 bottles of $anta , and 7,777 bottles of *humps-up

- $or how many days each plant be run in )pril so as to minimize the production cost ,

while still meeting the mar%et demand

Sol-ion+



#1

Let # > number of days to produce all the three types of bottles by plant

at ;oimbatore

& > number of days to produce all the three types of bottles by plant

at ;hennai

O%&e'ie+

inimize z > G77 # C 77 &

Con,rain+

#:,777 # C #:,777 & E &77,777

'7,777 # C #7,777 & E 77,777

&7,777 # C :7,777 & E 7,777

# E 7 & E 7

1(

!

"#



#(

;orner points ( ! z > G77 C 77 &



#!

;orner points (#,&! z > G77 # C 77 &

) (7, 7! #G777

F (#&,! 8877

; (&&,7! #'&77

- In #& days all the three types of bottles (;oca-cola, $anta, *humps-up!are produced by plant at ;oimbatore

- In days all the three types of bottles (;oca-cola, $anta, *humps-up!are produced by plant at ;hennai

- "o minimum production cost is 8877 units to meet the mar%et demand ofall the three types of bottles (;oca-cola, $anta, *humps-up! to beproduced in )pril



Hamdy A. Taha, Operations Research: An introduction, rentice Hall #"

"ensitivity Analysis

#"

he Role o$ 0ensitivity Analysis o$



Hamdy A. Taha, Operations Research: An introduction, rentice Hall ##

he Role o$ 0ensitivity Analysis o$the Optimal 0ol%tion

2 Is the optimal solution sensitive to changes in input

parameters

*he effective of this change is %nown as 0sensitivity1

##

0ensitivity Analysis o$ O5e#tive



Hamdy A. Taha, Operations Research: An introduction, rentice Hall #$

0ensitivity Analysis o$ O5e#tive6%n#tion Coe7i#ients/

2 5ange of Aptimality4 *he optimal solution will remain unchanged as long as

2 )n objective function coefficient lies within its range of optimality 2 *here are no changes in any other input parameters.

#$

The e++ects o+ chan7es in an ojectie +unction coe++icient

on the optimal solution



#%

600

800

1200

400 600 800

X2

X1


M a 2 1 2

) * 4 2 '

M a 9 "

9 1 4 # 9 ( M a 2 / . ; 4 2 ) * 4 2 ' M a 2 ' 2 ) * 4 2 '

The e++ects o+ chan7es in an ojectie +unction coe++icients




#8

600

800

1200

400 600 800

X2

X1


M a 2 1

2 ) * 4 2 '

M a 2 / . ; 4 2 ) * 4 2 '

M a 2 1 2 ) * 4 2 '

M a 2 / . ; 4 2 ) * 4

2 '

M a 2 ) - 2 ) * 4 2 '

/ . ; 4

) - Ran7e o+

optimalityM a

2 ) ) 2

) * 4 2 '



Hamdy A. Taha, Operations Research: An introduction, rentice Hall #&

2 It could be find the range of optimality for an

objectives function coefficient by determining the

range of values that gives a slope of the objective

function line between the slopes of the bindingconstraints.

#&



Hamdy A. Taha, Operations Research: An introduction, rentice Hall $'

2 *he binding constraints are/

&# C & > #&77

'# C & > &77

*he slopes are/ -&K#, and -'K respectively.

$'



Hamdy A. Taha, Operations Research: An introduction, rentice Hall $1

2 *o find range optimality for "pace 5ays, and

coefficient per dozen 6appers is ;&> :

*hus the slope of the objective function line can be

epressed as

4;#K:

$1



Hamdy A. Taha, Operations Research: An introduction, rentice Hall $(

2 5ange of optimality for ;# is found by sloving the

following for ;#/

-&K# M -;#K: M -'K

'.N: M ;#M #7

$(



Hamdy A. Taha, Operations Research: An introduction, rentice Hall $!

2 5ange optimality for 6apper, and coefficient per dozen space rays is ;#> 8

*hus the slope of the objective function line can be epressed as

48K;&

2

$!



Hamdy A. Taha, Operations Research: An introduction, rentice Hall $"

2 5ange of optimality for ;& is found by sloving the

following for ;&/

-&K# M -8K;& M -'K

M ;&M #7.GGN

$"



Hamdy A. Taha, Operations Research: An introduction, rentice Hall $#$#

<#&="$ #nput ata or<#&="$ #nput ata or

the %ala2y #ndustriesthe %ala2y #ndustriesProblemProblem



Hamdy A. Taha, Operations Research: An introduction, rentice Hall $$$$



Hamdy A. Taha, Operations Research: An introduction, rentice Hall $%$%



Hamdy A. Taha, Operations Research: An introduction, rentice Hall $8$8



Hamdy A. Taha, Operations Research: An introduction, rentice Hall $&$&



Hamdy A. Taha, Operations Research: An introduction, rentice Hall %'%'



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Hamdy A. Taha, Operations Research: An introduction, rentice Hall %!%!



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Documents

6.1 Chapter 2 Galaxy LP Problem Updated