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GROUP MEMBERS



Roll Nos. Names:

25 Sumaiya Sakhani 26 Urmi Sampat

31 Sonia Verma

49 Shahan Engineer

53 Aseem Shah

50 Asees Ghura

ACKNOWLEDGEMENT:



We have taken efforts in this project. However, it would not have been possible without the

kind support and help of Prof. Nimalan. We would like to extend our sincere thanks to

him.

We are highly indebted to Prof. Nimalan for the guidance and constant supervision as well

as for providing necessary information regarding the project & also for the support in

completing the project.

We would like to express our gratitude towards our co-mates for their kind co-operation

and encouragement which helped us in completion of this project.

We would like to express our special gratitude and thanks to Nimalan Sir for giving us

such attention and time.

Our thanks and appreciations also go to our college, classmates and other group members

who have willingly helped each other out with their abilities.



Operations Research

Introduction:

Research applied to operations was practically prior to

the World War II. The first record of a modern operations

research effort is that of the Antiaircraft Command

Research Group organized in 1940 in the United Kingdom

to study problems arising from the interaction of radar

equipment and anti aircraft guns. Besides radar

operational improvements, studies included suchactivities as antisubmarine operations, aerial mining of

the sea, merchant convoy size determination, ship man

oeuvres under aerial attack, and statistical analysis of

bomb damage. Organizations of professional operations

researches were formed early in the post war period.

Operations research techniques were utilized in India

firstly in 1949, when an Operations Research unit wasestablished at the Regional Research Laboratory,

Hyderabad.

Characteristics of OR:

✔ It addresses decision making problems.

✔ It has inter disciplinary team approach.

✔Uses scientific approach for an optimum solution.

✔ Can handle maximization and minimization type

problems.

✔ Can handle problems with various constraints and

restrictions.



Techniques of OR :

1. Assignment of Jobs to Facilities.

2. Transportation Problems.3. Linear Programming Problem.

4. Competitive Strategies or Game Theory.

5. Decision Analysis.

6. Network analysis (PERT/CPM).

Applications of OR:

1. Financial Management:• To build cash management models

• Management of Portfolio investment.

• Forecasting cash flows and long term capital

needs.

• Optimum equipment replacement policies.

• Assigning audit teams effectively.

1. Production Management:• Balancing output level and market demand.

• Determination of Production schedule.

• Use in quality control.

• Machine scheduling problems.

• Determining landing and takeoff schedules.

1. Marketing Management:

•

Determining optimum product mix.• Determining advertising outlay between

different media.

• Choice of marketing strategy.

• Find the least cost shipment routes.



• Deciding sales man’s travel plans.

1. Human Resource Development:

• Scheduling training programs.

• Determining employment of permanent and

temporary workers.

• Deciding shift allocations and allocation of sales

force to various activities.

Limitations of OR :

a)It needs a clear statement of objectives and the

assumptions regarding constraints.b)The manager has to define precisely the relationship

between the different variables.

c) The solution depends on correctness of the model

build.

d)Requires lengthy calculations.

e)Many factors may affect decisions and some of these

may be difficult to be quantified.



Linear Programming Problem:

Linear programming (LP, or linear optimization) is amathematical method for determining a way to achievethe best outcome (such as maximum profit or lowestcost) in a given mathematical model for some list of requirements represented as linear relationships. Linearprogramming is a specific case of mathematicalprogramming (mathematical optimization).

Introduction :

More formally, linear programming is a technique for theoptimization of a linear objective function, subject tolinear equality and linear inequality constraints. It’sfeasible region is a convex polyhedron, which is a setdefined as the intersection of finitely many half spaces,each of which is defined by a linear inequality. Its

objective function is a real-valued affine function definedon this polyhedron. A linear programming algorithm findsa point in the polyhedron where this function has thesmallest (or largest) value if such point exists.

Linear programs are problems that can be expressed incanonical form:

http://en.wikipedia.org/wiki/Mathematical_model

http://en.wikipedia.org/wiki/Mathematical_optimization


http://en.wikipedia.org/wiki/Linear

http://en.wikipedia.org/wiki/Objective_function

http://en.wikipedia.org/wiki/Linear_equality

http://en.wikipedia.org/wiki/Linear_inequality

http://en.wikipedia.org/wiki/Constraint_(mathematics)

http://en.wikipedia.org/wiki/Feasible_region

http://en.wikipedia.org/wiki/Convex_polyhedron

http://en.wikipedia.org/wiki/Half_space

http://en.wikipedia.org/wiki/Real_number

http://en.wikipedia.org/wiki/Affine_function

http://en.wikipedia.org/wiki/Canonical_form



http://en.wikipedia.org/wiki/Linear

http://en.wikipedia.org/wiki/Objective_function

http://en.wikipedia.org/wiki/Linear_equality

http://en.wikipedia.org/wiki/Linear_inequality

http://en.wikipedia.org/wiki/Constraint_(mathematics)

http://en.wikipedia.org/wiki/Feasible_region

http://en.wikipedia.org/wiki/Convex_polyhedron

http://en.wikipedia.org/wiki/Half_space

http://en.wikipedia.org/wiki/Real_number

http://en.wikipedia.org/wiki/Affine_function

http://en.wikipedia.org/wiki/Canonical_form

http://en.wikipedia.org/wiki/Mathematical_model



where x represents the vector of variables (to bedetermined), c and b are vectors of (known) coefficientsand A is a (known) matrix of coefficients. The expressionto be maximized or minimized is called the objectivefunction (cTx in this case). The equations Ax ≤ b are theconstraints which specify a convex polytope over whichthe objective function is to be optimized. (In this context,

two vectors are comparable when every entry in one isless-than or equal-to the corresponding entry in theother. Otherwise, they are incomparable.)

Linear programming can be applied to various fields of study. It is used most extensively in business andeconomics, but can also be utilized for some engineeringproblems. Industries that use linear programming modelsinclude transportation, energy, telecommunications, and

manufacturing. It has proved useful in modeling diversetypes of problems in planning, routing, scheduling,assignment, and design.

Assumptions:

Linear programming is a mathematical technique forsolving constrained maximization and minimizationproblems when there are many constraints and theobjective function to be optimized, as well as theconstraints faced, are linear (i.e., can be represented bystraight lines). Linear programming was developed by theRussian mathematician L. V. Kantorovich in 1939 andextended by the American mathematician G. B. Danzig in1947. Its acceptance and usefulness have been greatly

http://en.wikipedia.org/wiki/Vector_space

http://en.wikipedia.org/wiki/Matrix_(mathematics)

http://en.wikipedia.org/wiki/Convex_polytope

http://en.wikipedia.org/wiki/Comparability

http://en.wikipedia.org/wiki/Assignment_problem

http://en.wikipedia.org/wiki/Vector_space

http://en.wikipedia.org/wiki/Matrix_(mathematics)

http://en.wikipedia.org/wiki/Convex_polytope

http://en.wikipedia.org/wiki/Comparability

http://en.wikipedia.org/wiki/Assignment_problem



enhanced by the advent of powerful computers, since thetechnique often requires vast calculations.

Firms and other organizations face many constraints inachieving their goals of profit maximization, costminimization, or other objectives. With only oneconstraint, the problem can easily be solved with thetraditional techniques presented in the previous twochapters. In order to maximize output subject to a givencost constraint (iso cost), the firm should produce at thepoint where the isoquant is tangent to the firm’s iso-cost.

Similarly, in order to minimize the cost of producing agiven level of output, the firm seeks the lowest iso-costthat is tangent to the given isoquant. In the real world,however, firms and other organizations often facenumerous constraints. For example, in the short run oroperational period, a firm may not be able to hire morelabor with some type of specialized skill, obtain morethan a specified quantity of some raw material, or

purchase some advanced equipment, and it may bebound by contractual agreements to supply a minimumquantity of certain products, to keep labor employed for aminimum number of hours, to abide by some pollutionregulations, and so on. To solve such constrainedoptimization problems, traditional methods break downand linear programming must be used.Linear programming is based on the assumption that theobjective function that the organization seeks to optimize(i.e., maximize or minimize), as well as the constraintsthat it faces, is linear and can be represented graphicallyby straight lines. This means that we assume that inputand output prices are constant, that we have constantreturns to scale, and that production can take place with



limited technologically fixed input combinations.Constant input prices and constant returns to scale meanthat average and marginal costs are constant and equal

(i.e., they are linear). With constant output prices, theprofit per unit is constant, and the profit function that thefirm may seek to maximize is linear. Similarly, the totalcost function that the firm may seek to minimize is alsolinear.

The limited technologically fixed input combinations thata firm can use to produce each commodity result inisoquants that are not smooth but will be made up of

straight line segments. Since firms and otherorganizations often face a number of constraints, and theobjective function that they seek to optimize as well asthe constraints that they face are often linear over therelevant range of operation, linear programming is veryuseful.

Examples of Application of LPP:

➢ Product Mix ProblemIt decides the combination of various types of

products which are supposed to be manufactured

with available resources keeping in mind to

maximize profit or to minimize cost.

➢ Advertising Problem

It decides the number of advertising units of

respective media which are supposed to be boughtto maximize the audience exposure.

➢ Staffing Problem



It decides a large restaurant, hospital, education

institute or a MNC to meet their manpower needs at

all hours with minimum number of employees.

➢ Investment Problem

To select specific areas for investments among the

alternative so as to maximize ROI & minimize risk.

➢ Transportation Problem

To decide how many units are to be transported from

a specific origin to a specific destination so that the

overall transportation cost is minimum.

➢ Assignment Problem To allocate different jobs to different entities to

achieve maximum profit or minimum cost.

Terminology Of Linear Programming:

A typical linear program has the following components:

1. An objective function.

2. Constraints or restrictions.

3. Non-negativity restriction.

And the following terms are commonly used to describe atypical L.P.P.

Decision variables: Decision variables are theunknowns whose values are to be determined from thesolution of the problem. E.g. decision variables in the

furniture manufacturing problem are say the tablesand chairs whose values or actual units of productionare to be found from the solution of the problem. Thesevariables should be inter-related in terms of consumption of resources. For example, both tablesand chairs require carpenter’s time and also wood and



other resources and any change in the quantityproduced of table affects the production level of chairs.Secondly the relationship among the variables should

be linear.Objective function: A firms objectives are expressedas a function of decision variables. It represents themathematical equation of the goals of the firm in termsof unknown values of the decision variables. Thus if theobjective is to maximize net profits in a furnituremanufacturing problem, then profits are expressed asfunction of (dependent on) the net per unit profits of

table and chair and the number of units produced (of tables and chairs).

Constraints: A constraint represents the limitationsimposed on the values of decision variables in thesolution. These limitations exist due to limitedavailability of resources as well as the requirements of these resources in the production of each unit of thedecision variable. For example manufacturer of a table

requires certain amount of time in a certaindepartment and the department works only for a givenperiod, (say 8 hours in a day for 5 days in a week). Theconstraints may represent some other type of limitations also. As in the production of a commoditythe market demand can put an upper limit on the valueof the decision variable in the optimal solution. Thus,the constraints define the limits within which a solution

to the problem must be found. These constraints mustbe capable of expression in mathematical form of anequality or inequality.

Linear relationships: Linear programming deals withproblems in which the objective function and theconstraints can be expressed as linear functions.



Hence, when the problem is solved graphically, in atwo variable case the constraints the objectivefunction, gives a straight line on a two dimensional

graph.Equations and inequalities: Equations arerepresented by = (equality) sign. They are specificstatements. But many business problems cannot beneatly expressed in equations (called strict equality).Instead of precise statements, we may have onlyminimum or maximum requirements or availability. Forexample we may state that available labour time is 40

hours per week, hence, labour time used in productionshould be less than or equal to 40 hours per week. Wethus need inequalities. Less than or equal torelationship is written as (≤) and (≥) sign indicatesgreater than or equal to relationship. Most of theconstraints in a LPP are expressed as inequalities. Theyindicate the upper or the lower limits of resource use orproduction level. They do not express exact levels.

Thus, they allow for many possibilities of the optimalvalues of the decision variable i.e. more than onecombination of the decision variables may give thesame optimal value of the objective function.

Non-negativity restrictions: The solution to theproblem implies finding values of the decisionvariables. These must be non-negative. As one cannotthink of manufacture of -4 tables or -6 chairs i.e.

negative production. Hence, decision variable shouldassume either zero or positive values. If we denote twodecision variables as x 1 and x 2 then the nonnegativity restriction is expressed as x 1 ≥ 0; x 2 ≥ 0.

-aseem



Advantages :

➢ Efficient use of factors of production.

➢ Scientific decision making.➢ Streamlined resource allocation.

Assumptions in LP:

➢ Available quantities of resources and consumption

per unit from resources is known exactly and with

certainty

➢ Production of finished products is possible in anyfractions, so is consumption of resources.

➢ All external factors are constant.

➢ The problem involves only one major objective.

Limitations of LP:

➢ Linearity is necessary in objective function and

constraints.

➢ If there are multiple objectives to be achieved, LP

cannot be used.

➢ All costs and benefits related to the problem may not

be quantifiable.➢ Economy of scale and learning curve effect cannot

be incorporated in LP.

Steps involved in Solving LPP :



The steps followed in solving a linear programmingproblem are:

1.Express the objective function of the problem as an

equation and the constraints as inequalities.2. Graph the inequality constraints, and define the

feasible region.3. Graph the objective function as a series of isoprofit

(i.e., equal profit) or iso-cost lines, one for each levelof profit or costs, respectively.

4. Find the optimal solution (i.e., the values of thedecision variables) at the extreme point or corner of

the feasible region that touches the highest isoprofitline or the lowest iso-cost line. This represents theoptimal solution to the problem subject to theconstraints faced.

Linear Programming: Profit Maximization

Formulation of the Profit Maximization LinearProgramming Problem.



Most firms produce more than one product, and acrucial question to which they seek an answer is howmuch of each product (the decision variables) the firmshould produce in order to maximize profits. Usually,firms also face many constraints on the availability of the inputs they use in their production activities. Theproblem is then to determine the output mix thatmaximizes the firm’s total profit subject to the inputconstraints it faces.

In order to show the solution of a profit maximizationproblem graphically, we assume that the firm producesonly two products: product X and product Y. Each unitof product X contributes $30 to profit and to coveringoverhead (fixed) costs, and each unit of product Ycontributes $40. Suppose also that in order to produceeach unit of product X and product Y, the firm requiresinputs A, B, and C in the proportions indicated in TableW-1. That is, each unit of product X requires 1 unit of input A, one-half unit of input B, and no input C, while 1

unit of product Y requires 1 A, 1 B, and 0.5 C. Table W-1 also shows that the firm has available only 7 units of input A, 5 units of input B, and 2 units of input C pertime period. The firm then wants to determine how touse the available inputs to produce the mix of productsX and Y that maximizes its total profits.



The first step in solving a linear programming problemis to express the objective function as an equation andthe constraints as inequalities. Since each unit of

product X contributes $30 to profit and overhead costsand each unit of product Y contributes $40, theobjective function that the firm seeks to maximize is

Max z = 30 Qx+ 40 Qy ---- Equation 1

where Max z is the total contribution to profit andoverhead costs faced by the firm (henceforth simplycalled the “profit function”), and X and Y refer,respectively, to the quantities of product X and product

Y that the firm produces. Thus, Equation W-1postulates that the total profit (contribution) function of the firm equals the per-unit profit contribution of product X times the quantity of product X producedplus the per-unit profit contribution of product Y timesthe quantity of product Y that the firm produces.

Let us now go on to express the constraints of theproblem as inequalities. From the first row of Table W-1, we know that 1 unit of input A is required to produceeach unit of product X and product Y and that only 7units of input A are available to the firm per period of time. Thus, the constraint imposed on the firm’sproduction by input A can be expressed as,

Qx + Qy ≤7 ------ Equation 2

That is, the 1 unit of input A required to produce each

unit of product X times the quantity of product Xproduced plus the 1 unit of input A required to produceeach unit of product Y times the quantity of product Yproduced must be equal to or smaller than the 7 unitsof input A available to the firm. The inequality signindicates that the firm can use up to, but no more than,



the 7 units of input A available to it to produceproducts X and Y. The firm can use less than 7 units of input A, but it cannot use more.

From the second row of Table W-1, we know that one-half unit of input B is required to produce each unit of product X and 1 unit of input B is required to produceeach unit of product Y, and only 5 units of input B areavailable to the firm per period of time. The quantity of input B required in the production of product X is then0.5 x, while the quantity of input B required in theproduction of product Y is 1 y and the sum of 0.5 x and

1 y can be equal to, but it cannot be more than, the 5units of input B available to the firm per time period.

Thus, the constraint associated with input B is

0.5 Qx + 1 Qy ≤5 ------- Equation 3

From the third row in Table W-1, we see that input C isnot used in the production of product X, one-half unit of input C is required to produce each unit of product Y,and only 2 units of input C are available to the firm pertime period. Thus, the constraint imposed onproduction by input C is

0.5 Qy ≤2 -------- Equation 4

In order for the solution to the linear programmingproblem to make economic sense, however, we mustalso impose non negativity constraints on the output of products X and Y. The reason for this is that the firm

can produce zero units of either product, but it cannotproduce a negative quantity of either product (or use anegative quantity of either input).

The requirement that x and y (as well as that thequantity used of each input) be nonnegative can beexpressed as



Qx, Qy ≥0

We can now summarize the linear programmingformulation of the above problem as follows:

Max z = 30 Qx+ 40 Qy (objective function)

Qx + Qy ≤7 (input A constraint)

0.5 Qx + 1 Qy ≤5 (input B constraint)

0.5 Qy ≤2 (input C constraint)

Qx, Qy ≥0 (non negativity constraint)

Thereby by solving the above algebraic equations

(mathematical model) we get the following graphs:



Shahan

Linear Programming: Cost Minimization :

Most firms usually use more than one input to produce aproduct or service, and a crucial choice they face is howmuch of each input (the decision variables) to use inorder to minimize the costs of production. Usually firmsalso face a number of constraints in the form of someminimum requirement that they or the product or servicethat they produce must meet. The problem is then todetermine the input mix that minimizes costs subject to

the constraints that the firm faces.

In order to show how a cost minimization linearprogramming problem is formulated and solved, assumethat the manager of a college dining hall is required toprepare meals that satisfy the minimum daily



requirements of protein (P), minerals (M), and vitamins(V). Suppose that the minimum daily requirements havebeen established at 14P, 10M,

and 6V. The manager can use two basic foods (say, meatand fish) in the preparation of meals. Meat (food X)contains 1P, 1M, and 1V per pound. Fish (food Y) contains2P, 1M, and 0.5V per pound. The price of X is $2 perpound, and the price of Y is $3 per pound.

This information is summarized in Table W-3. Themanager wants to provide meals that fulfill the minimumdaily requirements of protein, minerals, and vitamins atthe lowest possible cost per student.

The above linear programming problem can beformulated as follows:

Minimize C = 2Qx +3Qy (objective function)

Subject to, 1Qx + 2Qy = 14 (protein constraint)

1Qx + 1Qy = 10 (minerals constraint)

1Qx + 0.5Qy = 6 (vitamins constraint)

Qx, Qy = 0 (non negativity constraint)



By solving the problem algrebically we get the followinggraph:

The Corner point theorem :

If the feasible region is bounded, then the objectivefunction has both a maximum and a minimum value andeach occur at one or more corner points.

If the feasible region is unbounded ,the objective functionmay not have a maximum or a minimum. But if amaximum or minimum value exists, it will occur at oneor more corner points.

Example : Minimize 2x + 4y



subject to : x + 2y≥ 10 ; 3x + y ≥ 10

x ≥ 0 , y ≥ 0

Asees ghrrah



Special Cases in Linear Programming:

Infeasibility: Infeasible means not possible. Infeasiblesolution happens when the constraints havecontradictory nature. It is not possible to find a solutionwhich can satisfy all constraints.

In graphical method, infeasibility happens when wecannot find feasible region.

Example 14:

Max z = 4x + 3ySubject to,2x +3y ≤6

x + 4y≥10

x,y ≥0

Since there is no common feasible area the solution isinfeasible.Constraints are not favorable.

Unbounded LPP: Unbounded means infinite solution. Asolution which has infinity answer is called unboundedsolution.

In graphical solution, the direction with respect to originis as follows:



Max ZMin Z

Away from origin Towards the Origin

MaximizationMinimization

Now in a maximization problem, if we have following

Feasible region:there is no upper limit away from the origin, hence theanswer is Infinity.

y

Max Z.

x This is Called Unbounded solution.

urmi

R edundancy: a constraint is called redundant when itdoes not affect the solution. The feasible region does notdepend on that constraint.



Even if we remove the constraint from the solution, theanswer is not affected.

y

Max Z= 5x + 8ySubject to, (0,12)

3x + 2y ≤24

x ≤16

x + 3y ≤12redundant

x≤16

x,y ≥0 (0,4)A

B (0,0)oC(8,0) (12,0) (16,0) x

The feasible region for the above problem is OABC. The3rd constraint does not affect the feasible region.Hence, the constraint, (x ≤16) is a redundant constraint.

Alternate Optimal Solution: Multiple or alternateoptimal solutions mean a problem has more than onesolution which gives the optimal answer.

There are two or more sets of solution values which givemaximum profit or minimum cost.In graphical method, we come to know that there isoptimal solution which is Alternate when:

The iso–profit or iso–cost line is parallel to one of theboundaries of feasible region (they have the same slopevalue)

3x + 2y

X+3y≤1



y

x + y ≤2001

y≤1252

A c B

3x + 6y ≤900

0 D (200,0) (300,0)

x

Feasible Region: OABCD. There were two optimal solutions:At corner point B & CISO profit line: Max Z = 8x + 16yZ = 8x + 16y16y = -8x + Zy = -8x+Z16

Slope = -12

The iso-profit line is exactly parallel to boundary BC.Hence any point on line BC will give Optimal solution.

-sumaiya

0,20

0,15

0,12

Slope =



Case Study Problems: me

Q1 } Agashe and co. tries to reach target audiences

belonging to two different monthly income groups thefirst with incomes greater than rupees 15000 and the

second with income of less than rupees 15000. The total

advertising budget is rupees 200000. Advertising on TV

cost rupees 50000 for one program, whereas advertising

on radio cost rupees 20000 for one program. For contract

reason at least 3 programs must be given on TV and the

number of radio programs are limited to 5 only. One TVprograms covers 450000 audiences belonging to income

group having more than rupees 15000 monthly incomes

where as it reaches to 50000 audiences belonging to

below rupees 15000 monthly income group. Similarly one

radio programs reaches to 20000 and 80000 audiences

belonging to above rupees 15000 and below 15000

monthly income groups respectively. Formulate the linearprogramming problem and using graphical method

determines the media mix as o to maximize the total

number of target audience. Comment on the solution.

Solution:



Decision Variable:

Let, x= number of TV programs released.

y = number of Radio programs released.

Objective Function:

Max Z (target audience)

Max Z = 500000 x + 100000 y

Constraints:

Budget constraint : 50000 x + 20000 y ≤200000

Advertising constraint: x≥3

and y≤5

Non negative restrictio n : x, y ≥0

Answer Statement: 4 programs on TV should be

Released and 0 programs on radio to maximize total

target audience i.e. 200000.

Comments: y ≤5 is a redundant constraint. (Radio

programs)



Q 2} A television manufacturing firm is planning to

produce television sets of various designs and

specifications. The televisions are marked on the basis of

its overall quality appearance and warranty. the market

research survey and the firms past experience indicates

that all the 3 types flat screen ,black screen and normal

TV sets will all be sold whichever is produced. However

the firm plans to test the market response first by

manufacturing only 200 sets of all the 3 types all of which

1 will be definitely be sold because of reputation of the

firm. The manufacturing firm wants to decide how many

of flat screens and how many of black screen TV sets thefirm should produce whereas the number of TV sets of

normal type is automatically decided on the basis of first

two types. All the 3 types on TV sets differ significantly in

their quality tube costs and their other electronic



features. The following table summarizes the estimated

prices for the 3 types of TV sets and the corresponding

expenses for the firm. The manufacturing firm has hired a

high-tech plant to manufacture these TV sets at fixed

charges of rupees 200000 for a period of one month

Types of TV sets Prices Tube cost Labor

and other material charges

Flat screen 10000 3000

4750

Black screen 7000 2200

2500

Normal 6500 1900

2200

In planning the production the following considerations

must be taken into account:

1) The marketing management and manufacturing

conditions require that at least 120 TV sets be of flat and

black screen types.

2) At least 35% but not more than 70% must be of black

screen TV sets

3) At least 10% of the TV sets must be of flat screen

types

4) At least 30% of the total sets must be of normal type.

5) The maximum numbers of flat screen TV sets can be

manufactured at the plant. plant is restricted to 60 only.



a) The manufacturing firm wishes to determine the

number of TV sets to produce for each type, so as to

maximize to profit.

b) Formulate the following as LPP

c) Rewrite the above LPP in terms of two decision

variables taking advantages of the fact that all 200 TV

sets will be sold.

Find the optimal using graphical method for the restated

LPP, interpret you results.

Solution:

Decision variable;Let x1 be the total number of units of Flat Screen TVs.Let x2 be the total number of units of Black Screen TVs.Let x3 be the total number of units of normal Screen TVs.

Therefore, Max Z = 2250 x1 + 2300 x2 + 2400 x3

Subject to;x1 + x2 + x3 = 20 ; x1 + x2 ≥120 ; x2 ≥42

x2 ≤84 ; x1 ≥12 ; x3 ≥60 ; x1 ≤60.

Non negative Restriction;x1, x2 , x3 ≥0

(Rewrite in terms of 2 decision variables)x1 + x2 + x3 = 200

let, x3 = 200 –x1 –x2 (Substitute this equation in place of x3)Max Z = 2250x1+ 2300x2 + 2400 (200 –x1 –x2) – 200000

= 2250x1 + 2300x2 + 480000 –2400x1 –2400x2 –200000

Max Z = 280000 –150x1 –100x2



Subject to,

x1 + x2 ≥120 ; x2 ≥42

x2 ≤84 ; x1 ≥12

200 –x1 –x2 ≥60

x1+x2 ≤140

x1 ≤60

Non negativity constraint:

x1 + x2 + x3 ≥0

Answer Statement: 36 units of Flat Screen, 84 units of black Screen and 80 units of Normal TV should beproduced to maximize profit of Rs. 266200/-Comments: x1 ≥12 and x2 ≥42 are RedundantConstraints.



Q3} M/S Print well Pvt. Ltd are facing a tight financialsqueeze and hence are attempting cost saying, whereverpossible. The current is to print a book in hard cover andis paperback. The cost of hard cover type is Rs. 600/- per100 copies of paperback type. The company decides torun their 2 printing presses P1 and P2 for at least 80hours and 60 hours respectively every week. P1 canproduce 100 hard cover book in 1 hour and 100 paper

backs in hour. P2 can produce 100 hard cover books in !hour and 100 paper backs in 2 hours. Determine howmany books of each type should be produced to minimizecosts. Use graph to solve.

Decision variable:Let, x = no. of units produced of Hardcover books.y= no. of books produced of paperback books.

Objective function:Min Z: 6x + 5y

Constraints:P1 0.6 minutes x + 0.6 minutes y ≥4800minutes 80 x

60hours

P2 0.6 minutes x + 1.2 minutess y ≥3600minutes (60 x 60

hours)

Non negative restriction: The number of units produced of HC and PB can’t benegative.x, y ≥0



A nswer statement: There should be 0 hardcover booksproduced and 8000 paperback books produced so as tominimize the cost to Rs. 40000.

Comment: 0.6 x +1.2 y = 3600 is a redundantconstraint.

Bibliography:

http://www2.isye.gatech.edu/~wcook/papers/infeas.pdf

http://math.tutorvista.com/algebra/graphical-method.html

http://www.mathyards.com/lse/mm/LinearProgramming.pdf

http://www.phpsimplex.com/en/graphical_method_example.html

http://www.wikipedia.com









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