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
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Mute ur call
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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.
!
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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.
"
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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%.
#
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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.
$
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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%
%
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Management is8
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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
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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 $unction/
4 Bee%ly profit, to be maimized
1'
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Hamdy A. Taha, Operations Research: An introduction, rentice Hall 11
The Linear Programming o!el
a 8# C :& (Bee%ly profit!
subject to&# C #& D > #&77 (+lastic!
'# C & D > &77 (+roduction *ime!
# C & D > 877 (*otal production! # - & D > :7 (i!
jE > 7, j > #,& (9onnegativity!
11
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Hamdy A. Taha, Operations Research: An introduction, rentice Hall 1(
Feasible Solutions for Linear
Programs2 *he set of all points that satisfy all the constraints of the model is called
1(
!A"#$L! R!%#O&!A"#$L! R!%#O&
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Hamdy A. Taha, Operations Research: An introduction, rentice Hall 1!
=sing a graphical presentation we can represent all the constraints,
the objective function, and the three types of feasible points.
1!
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1"
1200
600
The Plastic constraint
easible
The plastic constraint:'()*('+)'--
X2
#neasible
Production
Time
/()*0('+'0--
Total production constraint:
()*('+1--600
800
Production mi2
constraint:
()3('+04-X1
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1#
Solving Graphically for an
Optimal Solution
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1$
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-
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1%
600
800
1200
400 600 800
X2
X1
/et0s tae a closer loo
at the optimal point
Feasible
regionFeasible
region
2n+easile
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18
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
A 6-,7--8
! 6-,-8
$ 601-,'0-8
9 644-,)--8
604-,-8
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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&
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Production mi2
constraint:
()3('+04-
('
The plastic constraint:'()*('+)'--
Production
Time
/()*0('+'0--
(3143(-1(''
!314"3(-("''31- "8'
3(- ("'
(3143(-1(''
3153(-"#'31- ##'
3(- 1''
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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$) O%&e'ie *n
(7,7! 7
(:7,7! 'G77
(87,&7! :77
(::7,#77! ?77
(7,G77! '777
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Hamdy A. Taha, Operations Research: An introduction, rentice Hall ((
ype o$ $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.
((
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(!
1200
600
The Plastic constraintThe plastic constraint:'()*('+)'--
X2
#neasibleProductionTime
/()*0('
+'0--
Total production constraint:
()*('+1--600
800
Production mi2
constraint:
()3('+04-X1
6'--, '--8
6
#nterior
point
6/--,-8
6
$oundary
point
644-,)--8
6
!2treme
point
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Hamdy A. Taha, Operations Research: An introduction, rentice Hall ("
2 If a linear programming has an optimal solution , an
etreme point is optimal.
("
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Hamdy A. Taha, Operations Research: An introduction, rentice Hall (#
0%mmery o$ 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. (#
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Hamdy A. Taha, Operations Research: An introduction, rentice Hall ($
ORE EXAPLE
($
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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
(%
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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
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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
(&
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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+
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!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
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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
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!!
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
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!"
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
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!#
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
- $ormulate the problem as a L+ problem.
Sol-ion+
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!$
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
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!%
;omplete L+ model is, aimize z > # C :&
subject to
&# C && D ''7 minutes
'# C && D 87 minutes# E 7
& E 7
$ $ GRAPHICAL LP SOLUTION
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!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
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!&
$/$/" 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
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"'
- ;hange all equations to equality signs
G# C & > &
# C && > G
& - # > #
& > &# > 7
&
> 7
1
(
!
"
#
$
7 7
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"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
the coordinates of the equation
- +lot graph of & > & by using
the coordinates of the equation
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"(
9ow include the inequality of all the G equations
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"!
- 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 &/
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""
"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
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"#
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 (
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"$
- Important characteristic of the optimum L+ solution is that it is
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"%
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
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"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
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"&
$/$/$ 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
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#'
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+
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#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(
!
"#
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#(
;orner points ( ! z > G77 C 77 &
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#!
;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, $anta, *humps-up! to beproduced in )pril
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"ensitivity Analysis
#"
he Role o$ 0ensitivity Analysis o$
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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
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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
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#%
600
800
1200
400 600 800
X2
X1
on the optimal solution
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
on the optimal solution
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#8
600
800
1200
400 600 800
X2
X1
on the optimal solution
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 '
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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.
#&
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2 *he binding constraints are/
&# C & > #&77
'# C & > &77
*he slopes are/ -&K#, and -'K respectively.
$'
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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
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2 5ange of optimality for ;# is found by sloving the
following for ;#/
-&K# M -;#K: M -'K
'.N: M ;#M #7
$(
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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
$!
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2 5ange of optimality for ;& is found by sloving the
following for ;&/
-&K# M -8K;& M -'K
M ;&M #7.GGN
$"
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<#&="$ #nput ata or<#&="$ #nput ata or
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