MB0032 - P.Srinath - Completely Solved

8/9/2019 MB0032 - P.Srinath - Completely Solved

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MB0032: OperationsResearch[Assignment SET1 & SET2]

Name : P. SrinathSMDUE ID : 520923307Center : Mehbub College Campus, SecunderabadSubject Code : MB0032Subject : Operations Research


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ASSIGNMENT MBA SEM II Subject Code:MB0032 SET 1

1. Describe in details the different scopes of application of OperationsResearch?

Operations Research (OR) in the USA, South Africa and Australia,and Operational Research in Europe and Canada, is an interdisciplinarybranch of applied mathematics and formal science that uses methods such asmathematical modeling, statistics, and algorithms to arrive at optimal or nearoptimal solutions to complex problems. It is typically concerned withoptimizing the maxima (profit, assembly line performance, crop yield,bandwidth, etc) or minima (loss, risk, etc.) of some objective function.Operations research helps management achieve its goals using scientificmethods.

The terms operations research and management science are often

used synonymously. When a distinction is drawn, management sciencegenerally implies a closer relationship to the problems of businessmanagement. The field of operations research is closely related to Industrialengineering. Industrial engineers typically consider Operations Research (OR)techniques to be a major part of their toolset.

Some of the primary tools used by operations researchers arestatistics, optimization, probability theory, queuing theory, game theory,graph theory, decision analysis, and simulation. Because of thecomputational nature of these fields, OR also has ties to computer science,and operations researchers use custom-written and off-the-shelf software.

Operations research is distinguished by its frequent use to examine anentire management information system, rather than concentrating only onspecific elements (though this is often done as well). An operationsresearcher faced with a new problem is expected to determine whichtechniques are most appropriate given the nature of the system, the goals forimprovement, and constraints on time and computing power. For this andother reasons, the human element of OR is vital. Like any other tools, ORtechniques cannot solve problems by themselves.

Scope of operation Research

Examples of applications in which operations research is currently usedinclude:

Critical path analysis or project planning: identifying those processes ina complex project which affect the overall duration of the project

Designing the layout of a factory for efficient flow of materials constructing a telecommunications network at low cost while still

guaranteeing QoS (quality of service) or QoS (Quality of Experience) ifparticular connections become very busy or get damaged

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Road traffic management and 'one way' street allocations i.e. allocationproblems.

Determining the routes of school buses (or city buses) so that as fewbuses are needed as possible

designing the layout of a computer chip to reduce manufacturing time(therefore reducing cost)

Managing the flow of raw materials and products in a supply chain basedon uncertain demand for the finished products

Efficient messaging and customer response tactics Robotizing or automating human-driven operations processes Globalizing operations processes in order to take advantage of cheaper

materials, labor, land or other productivity inputs Managing freight transportation and delivery systems (Examples: LTL

Shipping, intermodal freight transport) Scheduling:

Personnel staffing Manufacturing steps Project tasks Network data traffic: these are known as queuing models or

queueing systems. sports events and their television coverage

blending of raw materials in oil refineries determining optimal prices, in many retail and B2B settings, within the

disciplines of pricing scienceOperations research is used extensively in government where evidence-

based policy is used.

2. What do you understand by Linear Programming Problem? What arethe requirement of L.P.P? What are the basic assumption of L.P.P?

Linear programming problem(LPP): The standard form of the linear programming problem is used to

develop the procedure for solving a general programming problem.

A general LPP is of the form

Max (or min) Z = c1x1 + c2x2 + +cnxnx1, x2, ....xn are called decision variable.

subject to the constraints

c1, c2,. Cn, a11, a12,. amn are all known constantsZ is called the "objective function" of the LPP of n variables which is to bemaximized orminimized.

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Requirements of L.P.P: There are mainly four steps in the mathematicalformulation of linear programming problem as a mathematical model. We willdiscuss formulation of those problems which involve only two variables.

Identify the decision variables and assign symbols x and y tothem. These decision variables are those quantities whose values wewish to determine. Identify the set of constraints and express them as linearequations/inequations in terms of the decision variables. Theseconstraints are the given conditions.

Identify the objective function and express it as a linear functionof decision variables. It might take the form of maximizing profit orproduction or minimizing cost. Add the non-negativity restrictions on the decision variables, asin the physical problems, negative values of decision variables have novalid interpretation.

There are many real life situations where an LPP may be formulated. Thefollowing examples will help to explain the mathematical formulation of anLPP.

Example 1: A diet is to contain at least 4000 units of carbohydrates,500 units of fat and 300 units of protein. Two foods A and B are available.Food A costs 2 dollars per unit and food B costs 4 dollars per unit. A unit offood A contains 10 units of carbohydrates, 20 units of fat and 15 units ofprotein. A unit of food B contains 25 units of carbohydrates, 10 units of fatand 20 units of protein. Formulate the problem as an LPP so as to find theminimum cost for a diet that consists of a mixture of these two foods and also

meets the minimum requirements.

Suggested answer:The above information can be represented as

Let the diet contain x units of A and y units of B.Total cost = 2x + 4y

The LPP formulated for the given diet problem isMinimize Z = 2x + 4ysubject to the constraints

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Basic Assumptions of L.P.P:Linear programming is applicable only to problems where the

constraints and objective function are linear i.e., where they can beexpressed as equations which represent straight lines. In real life situations,when constraints or objective functions are not linear, this technique cannotbe used.

Factors such as uncertainty, weather conditions etc. are not taken intoconsideration.

There may not be an integer as the solution, e.g., the number of menrequired may be a fraction and the nearest integer may not be theoptimal solution. i.e., Linear programming techqnique maygive practical valued answer which is not desirable.

Only one single objective is dealt with while in real life situations,

problems come with multi-objectives. Parameters are assumed to be constants but in reality they may not be

so.

3. Describe the different steps needed to solve a problem by Simplexmethod.

Simplex method

The simplex method is a method for solving problems in linearprogramming. This method, invented by George Dantzig in 1947, tests

adjacent vertices of the feasible set (which is a polytope) in sequence so thatat each new vertex the objective function improves or is unchanged. Thesimplex method is very efficient in practice, generally taking 2m to 3miterations at most (where m is the number of equality constraints), andconverging in expected polynomial time for certain distributions of randominputs (Nocedal and Wright 1999, Forsgren 2002). However, its worst-casecomplexity is exponential, as can be demonstrated with carefully constructedexamples (Klee and Minty 1972).

A different type of methods for linear programming problems areinterior point methods, whose complexity is polynomial for both average andworst case. These methods construct a sequence of strictly feasible points(i.e., lying in the interior of the polytope but never on its boundary) that

converges to the solution. Research on interior point methods was spurred bya paper from Karmarkar (1984). In practice, one of the best interior-pointmethods is the predictor-corrector method of Mehrotra (1992), which iscompetitive with the simplex method, particularly for large-scale problems.

Dantzig's simplex method should not be confused with the downhillsimplex method (Spendley 1962, Nelder and Mead 1965, Press et al. 1992). The latter method solves an unconstrained minimization problem in ndimensions by maintaining at each iteration n+1 points that define a simplex.

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At each iteration, this simplex is updated by applying certain transformationsto it so that it "rolls downhill" until it finds a minimum.

The Simplex Method is "a systematic procedure for generating andtesting candidate vertex solutions to a linear program." (Gill, Murray, andWright, p. 337) It begins at an arbitrary corner of the solution set. At eachiteration, the Simplex Method selects the variable that will produce the

largest change towards the minimum (or maximum) solution. That variablereplaces one of its compatriots that is most severely restricting it, thusmoving the Simplex Method to a different corner of the solution set andcloser to the final solution. In addition, the Simplex Method can determine ifno solution actually exists. Note that the algorithm is greedy since it selectsthe best choice at each iteration without needing information from previousor future iterations.

The Simplex Method solves a linear program of the form described in

Figure 3. Here, the coefficients represent the respective weights, or costs,

of the variables . The minimized statement is similarly called the cost ofthe solution. The coefficients of the system of equations are represented by

, and any constant values in the system of equations are combined on the

right-hand side of the inequality in the variables . Combined, thesestatements represent a linear program, to which we seek a solution ofminimum cost.

Figure 3

A Linear Program

Solving this linear program involves solutions of the set of equations. Ifno solution to the set of equations is yet known, slack variables

, adding no cost to the solution, are introduced. The initialbasic feasible solution (BFS) will be the solution of the linear program wherethe following holds:

Once a solution to the linear program has been found, successiveimprovements are made to the solution. In particular, one of the nonbasicvariables (with a value of zero) is chosen to be increased so that the value of

the cost function, , decreases. That variable is then increased,maintaining the equality of all the equations while keeping the other nonbasicvariables at zero, until one of the basic (nonzero) variables is reduced to zero

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and thus removed from the basis. At this point, a new solution has beendetermined at a different corner of the solution set.

The process is then repeated with a new variable becoming basic asanother becomes non-basic. Eventually, one of three things will happen. First,a solution may occur where no non-basic variable will decrease the cost, inwhich case the current solution is the optimal solution. Second, a non-basic

variable might increase to infinity without causing a basic-variable to becomezero, resulting in an unbounded solution. Finally, no solution may actuallyexist and the Simplex Method must abort. As is common for research in linearprogramming, the possibility that the Simplex Method might return to apreviously visited corner will not be considered here.The primary data structure used by the Simplex Method is "sometimes calleda dictionary, since the values of the basic variables may be computed(looked up) by choosing values for the non-basic variables." (Gill, Murray,and Wright, p. 337) Dictionaries contain a representation of the set ofequations appropriately adjusted to the current basis. The use of dictionariesprovide an intuitive understanding of why each variable enters and leavesthe basis. The drawback to dictionaries, however, is the necessary step of

updating them which can be time-consuming. Computer implementation ispossible, but a version of the Simplex Method has evolved with a moreefficient matrix-oriented approach to the same problem. This newimplementation became known as the Revised Simplex Method.

The steps of the Simplex Method also need to be expressed in thematrix format of the Revised Simplex Method. The basis matrix, B, consists ofthe column entries ofA corresponding to the coefficients of the variables

currently in the basis. That is if is the fourth entry of the basis, then

is the fourth column of B. (Note that B is therefore an

matrix.) The non-basic columns ofA constitute a similar though likelynot square, matrix referred to here as V.

Simplex Method Revised Simplex Method

Determine the current basis,d.

Choose to enter the basisbased on the greatest costcontribution.

If cannot decrease the cost,d is optimal solution.

If , d is optimal solution.

Determine that first exitsthe basis (becomes zero) as

increases.

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If can decrease withoutcausing another variable toleave the basis, the solution isunbounded.

If for all i, the solution isunbounded.

Update dictionary. Update .

Examples:

Use the two phase method to

Maximize z = 3x1 x2Subject to 2x1 + x2 2

x1 + 3x2 2x2 4,x1, x2 0

Rewriting in the standard form,Maximize z = 3x1 x2 + 0S1 MA1 + 0.S2 + 0.S3Subject to 2x1 + x2 S1 + A1 = 2

x1 + 3x2 + S2 = 2x2 + S3 = 4,x1, x2, S1, S2, S3, A1 0.

Phase I : Consider the new objective,Maximize Z* = A1Subject to 2x1 + x2 S1 + A1 = 2

x1 + 3x2 + S2 = 2x2 + S3 = 4,x1, x2, S1, S2, S3, A1 0.

Solving by Simplex method, the initial simplex table is given by

x1 x2 S1 A1 S2 S3

2 0 0 1 0

0 Ratio

A1 1 2* 1 1 1 0 0 2 2/2=1

S2 0 1 3 0 0 1 0 2 2/1=2

S3 0 0 1 0 0 0 1 4

2 1 1 0 0 0 2

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The maximum value of the objective function Z = 6 which is attained for x1 =2, x2 = 0.

4. Describe the economics importance of the Duality concept.

The Importance of Duality Concept Is Due To Two Main Reasons

i. If the primal contains a large number of constraints and a smaller numberof variables, the labour of computation can be considerably reduced byconverting it into the dual problem and then solving it.

ii. The interpretation of the dual variable from the loss or economic point ofview proves extremely useful in making future decisions in the activitiesbeing programmed.

Economic importance of duality concept

The linear programming problem can be thought of as a resource allocation

model in which the objective is to maximize revenue or profit subject tolimited resources. Looking at the problem from this point of view, theassociated dual problem offers interesting economic interpretations of the L.Presource allocation model. We consider here a representation of the generalprimal and dual problems in which the primal takes the role of a resourceallocation model.

Primal

Maximize

z =

=

n

j

jj xC1

.

Subject toj

n

jjij bxa=

1 , i = 1,2,., mxj 0, j = 1,2,., n

Dual

Minimize

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w =

=

m

i

ii yb1

.

Subject to

=

m

i

iiij cya1 , i = 1,2,., n

yj 0, i = 1,2,., m

From the above resource allocation model, the primal problem has neconomic activities and m resources. The coefficient cj in the primalrepresents the profit per unit of activity j. Resource i, whose maximum

availability is bi, is consumed at the rate aij units per unit of activity j.

Interpretation of Duel Variables

For any pair of feasible primal and dual solutions,

(Objective value in the maximization problem) (Objective value in theminimization problem)

At the optimum, the relationship holds as a strict equation. Note: Herethe sense of optimization is very important. Hence clearly for any two primaland dual feasible solutions, the values of the objective functions, when finite,

must satisfy the following inequality.

z =

==

=m

i

ii

n

j

jj wybxC11

The strict equality, z = w, holds when both the primal and dualsolutions are optimal.

Consider the optimal condition z = w first given that the primalproblem represents a resource allocation model, we can think of z asrepresenting profit in Rupees. Because bi represents the number of unitsavailable of resource i, the equation z = w can be expressed as profit (Rs) = (units of resource i) x (profit per unit of resource i)

This means that the dual variables yi, represent the worth per unit ofresource i [variables yi are also called as dual prices, shadow prices andsimplex multipliers].

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With the same logic, the inequality z < w associated with any twofeasible primal and dual solutions is interpreted as (profit) < (worth ofresources)

This relationship implies that as long as the total return from all theactivities is less than the worth of the resources, the corresponding primal

and dual solutions are not optimal. Optimality is reached only when theresources have been exploited completely, which can happen only when theinput equals the output (profit). Economically the system is said to remainunstable (non optimal) when the input (worth of the resources) exceeds theoutput (return). Stability occurs only when the two quantities are equal.

5. How can you use the Matrix Minimum method of finding the initialbasic feasible solution in the transportation problem?

The Initial basic Feasible solution using Matrix MinimumMethod

Let us consider a T.P involving m-origins and n-destinations. Since thesum of origin capacities equals the sum of destination requirements, afeasible solution always exists. Any feasible solution satisfying m + n 1 ofthe m + n constraints is a redundant one and hence can be deleted. This alsomeans that a feasible solution to a T.P can have at the most only m + n 1strictly positive component, otherwise the solution will degenerate.

It is always possible to assign an initial feasible solution to a T.P. insuch a manner that the rim requirements are satisfied. This can be achievedeither by inspection or by following some simple rules. We begin byimagining that the transportation table is blank i.e. initially all xij = 0. Thesimplest procedures for initial allocation discussed in the following section.

Matrix Minimum MethodStep 1:Determine the smallest cost in the cost matrix of the

transportation table. Let it be cij , Allocate xij = min ( ai, bj) in the cell ( i, j)

Step 2: If xij = ai cross off the i th row of the transportation table anddecrease bj by ai go to step 3.

if xij = bj cross off the ith column of the transportation table anddecrease ai by bj go to step 3.

if xij = ai= bj cross off either the ith row or the ith column but not both.

Step 3: Repeat steps 1 and 2 for the resulting reduced transportationtable until all the rim requirements are satisfied whenever the minimum costis not unique make an arbitrary choice among the minima.

6. Describe the Integer Programming Problem? Describe the GomorysAll-I.P.P. method for solving the I.P.P. problem.

Integer Programming Problem

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The Integer Programming Problem I P P is a special case of L P P whereall or some variables are constrained to assume nonnegative integer values.This type of problem has lot of applications in business and industry wherequite often discrete nature of the variables is involved in many decisionmaking situations. Eg. In manufacturing the production is frequentlyscheduled in terms of batches, lots or runs In distribution, a shipment must

involve a discrete number of trucks or aircrafts or freight cars

An integer programming problem can be described as follows:Determine the value of unknownsx1, x2, , xn

so as to optimize z = c1x1 +c2x2 + . . .+ cnxn

subject to the constraints

ai1 x1 + ai2 x2 + . . . + ain xn =bi , i = 1,2,,m

andxj 0 j = 1, 2, ,n

wherexj being an integral value forj = 1, 2, , k n.

If all the variables are constrained to take only integral value i.e. k = n,it is called an all(or pure) integer programming problem. In case only some ofthe variables are restricted to take integral value and rest (n k) variablesare free to take any non negative values, then the problem is known asmixed integer programming problem.

Gomorys All IPP MethodAn optimum solution to an I. P. P. is first obtained by using simplex

method ignoring the restriction of integral values. In the optimum solution ifall the variables have integer values, the current solution will be the desiredoptimum integer solution. Otherwise the given IPP is modified by inserting anew constraint called Gomorys or secondary constraint which representsnecessary condition for integrability and eliminates some non integer solutionwithout losing any integral solution. After adding the secondary constraint,the problem is then solved by dual simplex method to get an optimumintegral solution. If all the values of the variables in this solution are integers,an optimum inter-solution is obtained, otherwise another new constrained isadded to the modified L P P and the procedure is repeated. An optimuminteger solution will be reached eventually after introducing enough newconstraints to eliminate all the superior non integer solutions. Theconstruction of additional constraints, called secondary or Gomorysconstraints, is so very important that it needs special attention.

The iterative procedure for the solution of an all integer programmingproblem is as follows:Step 1: Convert the minimization I.P.P. into that of maximization, if it is inthe minimization form. The integrality condition should be ignored.

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Step 2: Introduce the slack or surplus variables, wherever necessary toconvert the inequations into equations and obtain the optimum solution ofthe given L.P.P. by using simplex algorithm.Step 3:Test the integrality of the optimum solutiona) If the optimum solution contains all integer values, an optimum basicfeasible integer solution has been obtained.

b) If the optimum solution does not include all integer values then proceedonto next step.Step 4: Examine the constraint equations corresponding to the currentoptimum solution. Let these equations be represented by

i

n

j

jij bxy ==

1

0 ( i 0 , 1, 2 , ........, m1 )

Where n denotes the number of variables and m the number of equations.

Choose the largest fraction ofb

i

s

ie to find {bi}i

Let it be [bk1]

or write is as kof

Step 5: Express each of the negative fractions if any, in the k th row of theoptimum simplex table as the sum of a negative integer and a nonnegativefraction.Step 6: Find the Gomorian constraint

ko

n

jjkj fxf =

1

0

and add the equation

Gsla ( 1 )=

=

+1

0

n

j

jkjko xff

to the current set of equation constraints.Step 7: Starting with this new set of equation constraints, find the newoptimum solution by dual simplex algorithm. (So that Gsla (1) is the initial

leaving basic variable).Step 8: If this new optimum solution for the modified L.P.P. is an integersolution. It is also feasible and optimum for the given I.P.P. otherwise returnto step 4 and repeat the process until an optimum feasible integer solution isobtained.

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ASSIGNMENT MBA SEM II Subject Code:MB0032 SET 2

1. What are the important features of Operations Research? Describein details the different phases of Operations Research.

Important features of OR are:i. It is System oriented: OR studies problem from over all point of view

of organizations or situations since optimum result of one part of the systemmay not be optimum for some other part.

ii. It imbibes Inter disciplinary team approach. Since no singleindividual can have a thorough knowledge of all fast developing scientificknowhow, personalities from different scientific and managerial cadre form ateam to solve the problem.

iii. It makes use of Scientific methods to solve problems.iv. OR increases the effectiveness of a management Decision making

ability.

v. It makes use of computer to solve large and complex problems.vi. It gives Quantitative solution.vii. It considers the human factors also.

Phases of Operations Research The scientific method in OR study generally involves the following

three phases:i)Judgment Phase: This phase consists of

a) Determination of the operation.b) Establishment of the objectives and values related to the operation.c) Determination of the suitable measures of effectiveness andd) Formulation of the problems relative to the objectives.

ii) Research Phase:This phase utilizes

a) Operations and data collection for a better understanding of theproblems.b) Formulation of hypothesis and model.c) Observation and experimentation to test the hypothesis on the basisof additional data.d) Analysis of the available information and verification of thehypothesis using pre established measure of effectiveness.e) Prediction of various results and consideration of alternativemethods.

iii) Action Phase: It consists of making recommendations for the decisionprocess by those who first posed the problem for consideration or by anyonein a position to make a decision, influencing the operation in which theproblem is occurred.

2. Describe a Linear Programming Problem in details in canonical form.Linear Programming

The Linear Programming Problem (LPP) is a class of mathematicalprogramming in which the functions representing the objectives and theconstraints are linear. Here, by optimization, we mean either to maximize or

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minimize the objective functions. The general linear programming model isusually defined as follows:

Maximize or MinimizeZ = c1 x1 + c2 x 2 +.. +cn x n

subject to the constraints,a11 x1 + a12 x2 + .+ a1n xn ~ b1

a21 x1 + a22 x2 +..+ a2n xn ~ b2am1x1 + am2 x2 +. +amn xn ~ bm

andx1 > 0, x2 > 0, xn > 0.Where cj, bi and aij (i = 1, 2, 3, .. m, j = 1, 2, 3.. n) are constants

determined from the technology of the problem and xj (j = 1, 2, 3 n) are thedecision variables. Here ~ is either < (less than), > (greater than) or =(equal). Note that, in terms of the above formulation the coefficient cj, aij, bjare interpreted physically as follows. Ifbi is the available amount of resourcesi, where aij is the amount of resource i, that must be allocated to each unit ofactivityj, the worth per unit of activity is equal to cj.

Canonical forms: The general Linear Programming Problem (LPP) defined above can

always be put in thefollowing form which is called as the canonical form:

MaximiseZ = c1 x1+c2 x2 + .+cn xnSubject to

a11 x1 + a12 x2 +.. +a1n xn < b1a21 x1 + a22 x2 +.. +a2n xn < b2am1x1+am2 x2 + + amn xn < bmx1, x2, x3, xn > 0.

The characteristics of this form are:1) all decision variables are nonnegative.2) all constraints are of < type.3) the objective function is of the maximization type.Any LPP can be put in the cannonical form by the use of five

elementary transformations:1. The minimization of a function is mathematically equivalent to themaximization of the negative expression of this function. That is, Minimize Z= c1 x1 + c2x2 + . + cn xn is equivalent to

Maximize Z = c1x1 c2x2 cnxn.2. Any inequality in one direction (< or>) may be changed to an inequality inthe opposite direction (> or 5 is equivalent to 2x13x2 < 5.3. An equation can be replaced by two inequalities in opposite direction. Forexample, 2x1+3x2= 5 can be written as 2x1+3x2 < 5 and 2x1+3x2 > 5 or 2x1+3x2 < 5 and 2x1 3x2 < 5.

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4. An inequality constraint with its left hand side in the absolute form can bechanged into two regular inequalities. For example: | 2x1+3x2 | < 5 isequivalent to 2x1+3x2 < 5 and 2x1+3x2 > 5 or 2x1 3x2 < 5.5. The variable which is unconstrained in sign (i.e., > 0, < 0 or zero) isequivalent to the difference between 2 nonnegative variables. For example, if

xis unconstrained in sign thenx= (x + x ) wherex + > 0,x < 0.

3. What are the different steps needed to solve a system of equationsby the simplex method?

To Solve problem by Simplex Method1. Introduce stack variables (Sis) for < type of constraint.2. Introduce surplus variables (Sis) and Artificial Variables (Ai) for > type ofconstraint.3. Introduce only Artificial variable for = type of constraint.4. Cost (Cj) of slack and surplus variables will be zero and that of Artificial

variable will be MFind Zj Cj for each variable.5. Slack and Artificial variables will form Basic variable for the first simplextable. Surplus variable will never become Basic Variable for the first simplextable.6. Zj = sum of [cost of variable x its coefficients in the constraints Profit orcost coefficient of the variable].7. Select the most negative value of Zj - Cj. That column is called key column.The variable corresponding to the column will become Basic variable for thenext table.8. Divide the quantities by the corresponding values of the key column to getratios select the minimum ratio. This becomes the key row. The Basicvariable corresponding to this row will be replaced by the variable found instep 6.9. The element that lies both on key column and key row is called Pivotalelement.10. Ratios with negative and a value are not considered for determiningkey row.11. Once an artificial variable is removed as basic variable, its column will bedeleted from next iteration.12. For maximisation problems decision variables coefficient will be same asin the objective function. For minimization problems decision variablescoefficients will have opposite signs as compared to objective function.13. Values of artificial variables will always is M for both maximisation andminimization problems.14. The process is continued till all Zj Cj > 0.

4. What do you understand by the transportation problem? What is thebasic assumption behind the transportation problem? Describe theMODI method of solving transportation problem.

This model studies the minimization of the cost of transporting acommodity from a number of sources to several destinations. The supply at

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each source and the demand at each destination are known. Thetransportation problem involves m sources, each of which has available ai (i= 1, 2, ..,m) units of homogeneous product and n destinations, each ofwhich requires bj (j = 1, 2., n) units of products. Here ai and bj are positiveintegers. The cost cij of transporting one unit of the product from the i thsource to the j th destination is given for each I and j. The objective is to

develop an integral transportation schedule that meets all demands from theinventory at a minimum total transportation cost. It is assumed that the totalsupply and the total demand are equal.

The condition (1) is guaranteed by creating either a fictitiousdestination with a demand equal to the surplus if total demand is less thanthe total supply or a (dummy) source with a supply equal to the shortage iftotal demand exceeds total supply. The cost of transportation from thefictitious destination to all sources and from all destinations to the fictitioussources are assumed to be zero so that total cost of transportation willremain the same.

The Transportation Algorithm (MODI Method)

The first approximation to (2) is always integral and therefore always afeasible solution. Rather than determining a first approximation by a directapplication of the simplex method it is more efficient to work with the tablegiven below called the transportation table. The transportation algorithm isthe simplex method specialized to the format of table it involves:

i) finding an integral basic feasible solutionii) testing the solution for optimalityiii) improving the solution, when it is not optimaliv) repeating steps (ii) and (iii) until the optimal solution is obtained.

The solution to T.P is obtained in two stages. In the first stage we find Basicfeasible solution by any one of the following methods a) Northwest cornerrale b) Matrix Minima Method or least cost method c) Vogels approximationmethod. In the second stage we test the B.Fs for its optimality either by MODImethod or by stepping stone method.

Modified Distribution Method / Modi Method / U V Method.Step 1: Under this method we construct penalties for rows and columns bysubtracting the least value of row / column from the next least value.Step 2: We select the highest penalty constructed for both row and column.Enter that row / column and select the minimum cost and allocate min (ai, bj)Step 3: Delete the row or column or both if the rim availability /requirements is met.Step 4: We repeat steps 1 to 2 to till all allocations are over.Step 5: For allocation all form equation ui + vj = cj set one of the dualvariable ui / vj to zero and solve for others.Step 6: Use these value to find ij = cij ui vj of all ij >, then it is theoptimal solution.Step 7: If any Dij 0, select the most negative cell and form loop. Startingpoint of the loop is +ve and alternatively the other corners of the loop are veand +ve. Examine the quantities allocated at ve places. Select theminimum. Add it at +ve places and subtract from ve place.Step 8: Form new table and repeat steps 5 to 7 till ij > 0

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5. Describe the North-West Corner rule for finding the initial basicfeasible solution in the transportation problem.

North West Corner RuleStep1: The first assignment is made in the cell occupying the upper

left hand (north west) corner of the transportation table. The maximumfeasible amount is allocated there, that is x11 = min (a1,b1). So that eitherthe capacity of origin O1 is used up or the requirement at destination D1 issatisfied or both. This value of x11 is entered in the upper left hand corner(small square) of cell (1, 1) in the transportation table

Step 2: If b1 > a1 the capacity of origin O, is exhausted but therequirement at destination D1 isstill not satisfied , so that at least one more other variable in the first columnwill have to take on a positive value. Move down vertically to the second rowand make the second allocation ofmagnitude x21 = min (a2, b1 x21) in the cell (2,1). This either exhausts thecapacity of origin O2 or satisfies the remaining demand at destination D1.

If a1 > b1 the requirement at destination D1 is satisfied but thecapacity of origin O1 is not completely exhausted. Move to the righthorizontally to the second column and make the second allocation ofmagnitude x12 = min (a1 x11, b2) in the cell (1, 2) . This either exhauststhe remaining capacity of origin O1 or satisfies the demand at destinationD2 .

If b1 = a1, the origin capacity of O1 is completely exhausted as well asthe requirement at destination is completely satisfied. There is a tie forsecond allocation, An arbitrary tie breaking choice is made. Make the secondallocation of magnitude x12 = min (a1 a1, b2) = 0 in the cell (1, 2) or x21 =min (a2, b1 b2) = 0 in the cell (2, 1).

Step 3: Start from the new north west corner of the transportationtable satisfying destination requirements and exhausting the origin capacitiesone at a time, move down towards the lower right corner of thetransportation table until all the rim requirements are satisfied.

6. Describe the Branch and Bound Technique to solve an I.P.P.problem.

The Branch And Bound TechniqueSometimes a few or all the variables of an IPP are constrained by their

upper or lower bounds or by both. The most general technique for thesolution of such constrained optimization problems is the branch and boundtechnique. The technique is applicable to both all IPP as well as mixed I.P.P.the technique for a maximization problem is discussed below:Let the I.P.P. be

Subject to the constraints

xj is integer valued ,j = 1, 2, .., r (< n) (3)xj > 0 .j = r + 1, .., n (4)

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Further let us suppose that for each integer valued xj, we can assign lowerand upper bounds for the optimum values of the variable by

Lj xj Uj j = 1, 2, . r (5)

The following idea is behind the branch and bound technique

Consider any variablexj, and let I be some integer value satisfying Lj I Uj 1. Then clearly an optimum solution to (1) through (5) shall also satisfyeither the linear constraint.

x j > I + 1 ( 6)Or the linear constraint xj I ...(7)

To explain how this partitioning helps, let us assume that there wereno integer restrictions (3), and suppose that this then yields an optimalsolution to L.P.P. (1), (2), (4) and (5). Indicatingx1= 1.66 (for example). Then we formulate and solve two L.P.Ps eachcontaining (1), (2) and (4). But (5) for j = 1 is modified to be 2 x1 U1 in

one problem and L1 x1 1 in the other.Further each of these problems process an optimal solution satisfying

integer constraints (3).Then the solution having the larger value for z isclearly optimum for the given I.P.P. However, it usually happens that one (orboth) of these problems has no optimal solution satisfying (3), and thus somemore computations are necessary. We now discuss step wise the algorithmthat specifies how to apply the partitioning (6) and (7) in a systematicmanner to finally arrive at an optimum solution.

We start with an initial lower bound for z, sayz (0) at the first iterationwhich is less than or equal to the optimal value z*, this lower bound may betaken as the starting Lj for somexj. In addition to the lower bound z (0) , wealso have a list of L.P.Ps (to be called master list) differing only in the bounds(5). To start with (the 0 th iteration) the master list contains a single L.P.P.consisting of (1), (2), (4) and (5). We now discuss below, the step by stepprocedure that specifies how the partitioning (6) and (7) can be appliedsystematically to eventually get an optimum integer valued solution.

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