LP Quantitative Techniques

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Quantitative technique forManagerial DecisionCourse code: MAS 501

By

Nagendra Bahadur Aatya

!ead of De"artent

LinearProgramming


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Introduction

Linear programming is a widely used mathematical modeling

technique to determine the optimum allocation of scarce resourcesamong competing demands. Resources typically include rawmaterials, manpower, machinery, time, money and space.

The technique is very powerful and found especially useful becauseof its application to many dierent types of real business problems

in areas lie !nance, production, sales and distribution, personnel,mareting and many more areas of management.

"s its name implies, the linear programming model consists of linearob#ectives and linear constraints, which means that the variables ina model have a proportionate relationship. $or e%ample, an increase

in manpower resource will result in an increase in wor output.


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#SS#N$%A&S '( &%N#A) *)'+)AMM%N+ M'D#&

(or a given "ro,le situation- there are certain essential

conditions that need to ,e solved ,y using linear "rograing.

&. Limited resources ' limited number of labour, materialequipment and

!nance

(. )b#ective ' refers to the aim to optimi*e +ma%imi*e thepro!ts or minimi*e the costs.

-. Linearity ' increase in labour input will have aproportionate

increase in output.

. /omogeneity ' the products, worers0 e1ciency, andmachines are

assumed to be identical.

2. 3ivisibility ' it is assumed that resources and products can

be divided into fractions. +in case the fractions are notpossible, lie production of one4third of a


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*)'*#)$%#S '( &%N#A)*)'+)AMM%N+ M'D#&

The following properties form the linearprogramming model'

&. Relationship among decision variables mustbe linear in nature.

(. " model must have an ob#ective function.

-. Resource constraints are essential.

. " model must have a non4negativity

constraint.


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*)'+)AMM%N+

$ormulation of linear programming is the representation of problem

situation in a mathematical form. It involves well de!ned decision

variables, with an ob#ective function and set of constraints.

',ective function:

The ob#ective of the problem is identi!ed and converted into asuitable ob#ective function. The ob#ective function represents the aim

or goal of the system +i.e., decision variables which has to bedetermined from the problem. 5enerally, the ob#ective in most caseswill be either to ma%imi*e resources or pro!ts or, to minimi*e thecost or time.Constraints:

6hen the availability of resources are in surplus, there will be no problemin maing

decisions. 7ut in real life, organi*ations normally have scarce resourceswithin which the #ob has to be performed in the most eective way. Therefore, problem situations are within con!ned limits in which theoptimal solution to the problem must be found.

Nonnegativity constraint

8egative values of physical quantities are impossible, lie producing

negative number of chairs, tables, etc., so it is necessary to include theelement of non4negativity as a constraint


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+#N#)A& &%N#A) *)'+)AMM%N+ M'D#&

" general representation of LP model is given as follows'

9a%imi*e or 9inimi*e, : ; p1 %& < p( %( ======pn %n

>ub#ect to constraints,

w&& %& < w&( %( < ======w&n %n ? or ; or @ w& =====+i

w(& %& < w(( %( ======w(n %n ? or ; or @ w( ===== +ii. . . .

. . . .

. . . .

wm& %& < wm( %(


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#2a"le" company manufactures two types of bo%es, corrugated and ordinary

cartons.

The bo%es undergo two ma#or processes' cutting and pinning operations.

The pro!ts per unit are Rs. A and Rs. respectively.

Bach corrugated bo% requires ( minutes for cutting and - minutes forpinning operation, whereas each carton bo% requires ( minutes forcutting and & minute for pinning.

The available operating time is &(C minutes and AC minutes for cuttingand pinning machines.

The manager has to determine the optimum quantities to be

manufacture the two bo%es to ma%imi*e the pro!ts.


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SolutionDecision varia,les completely describe the decisions to be made +in thiscase, by 9anager. 9anager must decide how many corrugated and ordinarycartons should be manufactured each wee. 6ith this in mind, he has to

de!ne'

%l be the number of corrugated bo%es to be manufactured.%( be the number of carton bo%es to be manufactured

',ective function is the function of the decision variables that the

decision maer wants to ma%imi*e +revenue or pro!t or minimi*e +costs.9anager can concentrate on ma%imi*ing the total weely pro!t +*./ere pro!t equals to 4ee6ly revenues7 8 4ra aterial "urchase cost7 8 4other

varia,le costs7./ence 9anagerDs ob#ective function is'

9a%imi*e

* ; AE

( < %( Constraints show the restrictions on the values of the decision variables.6ithoutconstraints manager could mae a large pro!t by choosing decision

variables to be verylarge. /ere there are three constraints'

"vailable machine4hours for each machine Time consumed b each roduct


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"ll these characteristics e%plored above give thefollowing &inear *rograing +LP problem

ma% * ; A%& < %( +The )b#ective function

s.t. (%& < -%( ? &(C +cutting timeconstraint

(%& < %( ? AC +pinning constraint

%&, %( @ C +>ign restrictions

" value of +%&,%( is in the feasi,le region if it

satis!es all the constraints and sign restrictions.

This type of linear programming can be solve bytwo methods&5raphical method(>imple% algorithm method


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5raphic 9ethod

Step 1: Convert the inequality constraint as equations and fnd co-ordinates o the line.

Step 2: Plot the lines on the graph.

+Note: If the constraint is @ type, then the solution *one lies away from the centre.

If the constraint is ? type, then solution *one is towards the centre.

Step 3: Obtain the easible zone.

Step 4: Find the co-ordinates o the objectives unction (proft line and plot it on the graphrepresenting it with a dotted line.

Step 5: !ocate the solution point.

+"ote# $ the given proble% is %a&i%ization' z %a& then locate the solution point at the far mostpoint of the feasible *one from the origin and if minimi*ation, :min then locate the solution at theshortest point of the solution *one from the origin.

Step 6: Solution typei. If the solution point is a single point on the line, tae the corresponding values of %& and %(.

ii. If the solution point lies at the intersection of two equations, then solve for %& and %( using the

two equations.

iii. If the solution appears as a small line, then a multiple solution e%ists.

iv. If the solution has no con!ned boundary, the solution is said to be an unbound solution.


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G)8T=

The inequality constraint of the first line is

(less than or equal to) ? type which meansthe

feasible solution zone lies towards the origin.

("ote# $ the constraint type is thenthe solution zone area lies a)ay ro%the origin in the opposite direction."o) the second constraints line isdra)n.

6hen the second constraint is drawn, you may notice that aportion of feasible area is cut. This indicates that while

considering both the constraints, the feasible region getsreduced further. 8ow any point in the shaded portion willsatisfy the constraint equations.

the ob#ective is to ma%imi*e the pro!t. The point that liesat the furthermost point of the feasible area will give thema%imum pro!t. To locate the point, we need to plot theob#ective function +pro!t line.


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Gont..

',ective function line 4*ro9t &ine7

Bquate the ob#ective function for any speci!c pro!t value :,

Gonsider a :4value of AC, i.e.,A%& < %( ; AC

>ubstituting %& ; C, we get %( ; &2 and

if %( ; C, then %& ; &C

Therefore, the co4ordinates for the ob#ective function line are +C,&2,

+&C,C as indicated ob#ective function line. The ob#ective function linecontains all possible combinations of values of %l and %(.

Therefore, we conclude that to ma%imi*e pro!t, &2 numbers ofcorrugated bo%es and -C numbers of carton bo%es should be producedto get a ma%imum pro!t. >ubstituting

%& ; &2 and %(; -C in ob#ective function, we get:ma% ; A%& < %(

; A+&2 < +-C

9a%imum pro!t ' Rs. (&C.CC


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5raphic 9ethod on Tora

>teps for shoving linear programming bygraphic method using Tora shoftware

>tep & >tart ⇒ Tora⇒select linear programming ⇒


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>imple% 9ethod

In practice, most problems contain more than twovariables and are consequently too large to be tacledby conventional means. Therefore, an algebraictechnique is used to solve large problems using>imple% 9ethod. This method is carried out through

iterative process systematically step by step, and!nally the ma%imum or minimum values of theob#ective function are attained.

The simple% method solves the linear programmingproblem in iterations to improve the value of theob#ective function. The simple% approach not only

yields the optimal solution but also other valuableinformation to perform economic and 0what if0 analysis.


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ADD%$%'NA& A)%ABS /S#D %NS'&%N+ &**

Three types of additional variables are used in simple% method suchas,

+a >lac variables +>&, >(, >-..=>n' >lac variables refer to the amount

of unused resources lie raw materials, labour and money.

+b >urplus variables +4>&, 4>(, 4>-..=4>n' >urplus variable is the amount

of resources by which the left hand side of the equation e%ceeds theminimum limit.

+c "rti!cial Hariables +a&, a(, a-.. =an' "rti!cial variables are temporary

slac variables which are used for purposes of calculation, and areremoved later.

The above variables are used to convert the inequalities into equalityequations, as given in the 5iven Table below.


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Gont=


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*rocedure of si"le2 Method

Step 1: For%ulate the !P proble%.

Step 2: $ntroduce slac* +au&iliary variables.

i constraint type is , introduce

i constraint type is introduce / a and

i constraint type is 0 introduce a

Step 3: Find the initial basic solution.

Step 4: 1stablish a si%ple& table and enter all variable coe2cients. $ theobjective unction is %a&i%ization' enter the opposite sign co-e2cient and i%ini%ization' enter )ithout changing the sign.

Step 5: Tae the most negative coe1cient in the ob#ective function, :# toidentify the ey column +the corresponding variable is the entering variableof the ne%t iteration table.

Step 6: $ind the ratio between the solution value and the coe1cient of theey column. Bnter the values in the minimum ratio column.


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Gont=.

Step 7: 3a*e the %ini%u% positive value available in the %ini%u% ratio

colu%n to identiy the *ey ro). (3he corresponding variable is theleaving variable o the table.

Step 8: 3he intersection ele%ent o the *ey colu%n and *ey ro) is the pivotal ele%ent.

Step 9: Gonstruct the ne%t iteration table by eliminating the leavin variableand introducing the entering variable.

Step 10: Convert the pivotal ele%ent as 4 in the ne&t iteration table andco%pute the other ele%ents in that ro) accordingly. 3his is the pivotalequation ro) (not *ey ro).

Step 11: Other ele%ents in the *ey colu%n %ust be %ade zero. Forsi%plicity' or% the equations as ollo)s# Change the sign o the *eycolu%n ele%ent' %ultiply )ith pivotal equation ele%ent and add thecorresponding variable.


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Gont=.

Step 12: Chec* the values o objective unction. $ there are

negative values' the solution is not an opti%al one5 go tostep 6. 1lse' i all the values are positive' opti%ality isreached. "on-negativity or objective unction value is notconsidered. 7rite do)n the values o &4' &8'99..&i andcalculate the objective unction or %a&i%ization or

%ini%ization.Note:

(i $ there are no &4' &8 variables in the fnal iteration table'the values o &4 and &8 are zero.

(ii "eglect the sign or objective unction value in the fnaliteration table.


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B%ample

Previous the pacaging product mi% problem is solved usingsimple% method.

9a%imi*e : ; A%& < %(>ub#ect to constraints,

(%&- which represents the unused resources.Introducing the slac variable, we have the equation (%&< -%(

< >- ; &(C

>imilarly for pinning machine, the equation is

(%&< %( < > ; AC


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B%ample cont=.

If variables %& and %( are equated to *ero,i.e., %& ; C and %( ; C, then>- ; &(C

> ; AC

This is the basic solution of the system, andvariables >- and > are nown as 7asic Hariables,

>7 while %& and %( nown as 8on47asic

Hariables. If all the variables are non negative,

a basic feasible solution of a linearprogramming problem is called a 7asic $easible>olution.


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Gont=.

Rewriting the constraints with slac variables gives us,

:ma% ; A%& < %( < C>- < C>

>ub#ect to constraints,(%& < -%( < >- ; &(C ....................+i

(%& < %( < > ; AC ....................+ii

where %&, %( @ C

6hich can shown in following simple% table form


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Gont=

If the ob#ective of the given problem is a ma%imi*ation one, enter the co4e1cient of the ob#ective function :# with opposite sign as shown in table.

Tae the most negative coe1cient of the ob#ective function and that is theey column c. In this case, it is 4A.

$ind the ratio between the solution value and the ey column coe1cient

and enter it in the minimum ratio column.

The intersecting coe1cients of the ey column and ey row are called thepivotal element i.e. (.

The variable corresponding to the ey column is the entering element ofthe ne%t iteration table and the corresponding variable of the ey row is

the leaving element of the ne%t iteration table +$n other )ords' &4 replaces: in the ne&t iteration table. ;iven indicates the *ey colu%n' *ey ro)and the pivotal ele%ent.


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Gont..

In the next iteration, enter the basic variables by eliminating the leaving variable (i.e., ey row) and

introducing the entering variable (i.e., ey column).

9ae the pivotal element as & and enter the values of other elements in that rowaccordingly.

In this case, convert the pivotal element value ( as & in the ne%t iteration table.

$or this, divide the pivotal element by (. >imilarly divide the other elements inthat row by (. The equation is >J(.

This row is called as Pivotal Bquation Row Pe.

The other co4e1cients of the ey column in iteration Table 2. must be made as*ero in the iteration Table 2.2.


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Gont=

>olver, K ; >7 < + c M Pe

The equations for the variables in the iteration number & oftable N are,

(or S; Q < SB = 48 > c ? *e7

; >- < +(% Pe

; >- (Pe ==========+i

$or : , K ; >7 < + c M Pe

; : < ++ A M Pe

; : < APe ==========+iiOsing the equations +i and +ii the values of >- and : for

the values of Table & are found as shown in Table 2.


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Gont=.

!sing these equations, enter the values of basic variables "#

and ob$ective function %. If

all the values in the ob$ective function are non&negative, the solution is optimal.

/ere, we have one negative value &. Repeat the steps to !nd the ey rowand pivotal equation values for the iteration ( and chec for optimality.

6e get 8ew Table as below'


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Gont=.

The solution is,x

' ' corrugated boxes are to be produced and

x* + carton boxes are to be produced to yield a

-rofit, %max

s. *'.


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Gont=.

It can be solve in dierent inds of softwareas Tora, BEGBL, Lindo, etc.

In Tora "s follows,

>teps for shoving linear programming bygraphic method using Tora shoftware

>tep & >tart ⇒ Tora⇒select linear programming ⇒


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LP Quantitative Techniques

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