operations reaserch assignment

8/2/2019 Operations Reaserch Assignment

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1) Structured approach to problems: A substantial amount of time andeffort can be saved in developing and solving OR models if one uses alogical and consistent approach. This implies that the decision makerhas to be careful while defining controllable and uncontrollable

variables, availability of resources, etc., functional relationshipsamong variables in the objective function and constraints.2) Critical Approach to problem solving: The decision maker will come to

understand various components of the problem and accordingly selecta mathematical model for solving the given problem. He will be awareof the explicit and implicit assumptions and inherent limitations of such models. Problem solutions are examined critically and the effectof any change and error in the problem data can be studied throughsensitivity analysis techniques.

Limitations:

1) Often solution to a problem is derived by either making it simplified orsimplifying assumptions and thus, such solutions have limitations.

2) Sometimes models do not present the realistic situations in whichdecisions must be made.

3) Often decision-maker is not fully aware of the limitations of themodels that he is using.

4) Many real world problems just cannot have an OR solution.

Question 3: It is said that the operation research increases the creativecapabilities of decision-maker. Do you agree with this view? Defendyour point of view with examples.

Answer: Yes, operations research increases the creative capabilities of adecision maker. I agree with this point because operations research is thesystematic application of quantitative methods, techniques and tools to theanalysis of problems involving the operation of systems. Operation researchoffers the decision-maker a method of evaluating every possible alternative

by using various techniques to know the potential outcomes. While solving areal-life problem, the decision maker must examine it both from quantitativeas well as qualitative perspective. Information about the problem from boththese perspective need to be brought together. Based on some mix of thesources of information, a decision is taken by the decision maker. The study of these methods and how decision-makers use them in decision process isthe essence of operations research process. As the term implies, operationsresearch involves research on military operations. This indicates theapproach as well as the area of its application. The operations researchapproach is particularly useful in balancing conflicting objectives wherethere are many alternative course of action available to the decision-makers.


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It is often called global optimum. The operations research approachattempts to find global optimum by analysing inter-relationships among thesystem components involved in the problem.

For example, consider the basic problem of maintaining stocks of finishedgoods. To the marketing manager, stocks of a large variety of products are ameans of supplying the companys customers with what they want andwhen they want it. Clearly, according to a marketing manager, a fully stocked warehouse is of prime importance to the company. But theproduction manager argues for long production runs preferably on a smallerproduct range, particularly if significant time is lost when production isswitched from one variety to another. The result would again be a tendency to increase the amount of stock carried but it is, of course, vital that theplant should be kept running. On the other hand, the finance managerseeks stocks in terms of capital tied up unproductively and argues strongly for their reduction.

In view of above examples, we can say that operations research increasesthe creative capabilities of the decision-maker irrespective of hisspecialisation.

Question 4: What is linear programming? What are its majorassumptions and limitations?

Answer: The word linear refers to linear relationship among variables in amodel. Thus, a given change in one variable will always cause a resultingproportional change in another variable. The word programming refers tomodelling and solving a problem mathematically that involves the economicallocation of limited resources by choosing a particular course of action orstrategy among various alternative strategies to achieve the desiredobjective.

Assumptions:

The following major assumptions of LP model are:

1) Certainty: In all LP models, it is assumed, that all parameters such asavailability of resources, profit (or cost) contribution of a unit of decision variable and consumption of resources by a unit of decisionvariable must be known and may be constant.

2) Additively: The value of the objective of function and the total amountof each resources used, must be equal to the sum of the respectiveindividual contributions by decision variables.


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3) Linearity: The amount of each resource used and its contribution tothe objective in the profit in objective function must be proportional tothe value of each decision variable.

4) Divisibility: The solution values of decision variables are allowed to

assume continuous values.Limitations:

1) LP treats all relationships among decision variable as linear. However,generally, neither the objective functions nor the constraints in reallife situations concerning business & individual problems are linearly related to the variables.

2) LP model does not take into consideration the effect of time anduncertainty. Thus LP model should be defined in such a way that any

change due to internal as well as external factors can be incorporated.3) Parameters appearing in the model are assumed to be constant but in

real life situations they are frequently neither known nor constant.4) It deals with only single objective, whereas in real life situation we

may come across conflicting multi objective problems.

Question 5: Linear programming is one the most frequently andsuccessfully applied operation research s technique to managerialdecisions. Elucidate this statement with some examples.

Answer: Linear programming helps in attaining the optimum use of productive resources. It also indicates how a decision maker can employ hisproductive factors effectively by selecting and distributing these resources.Linear programming technique improves the quality of decisions. Thedecision making approach of the use of this technique becomes moreobjective and less subjective.

Linear programming technique provides possible and practical solutionssince there might be other constraints operating outside the problem whichmust be taken into account.

Question 6: Explain the graphical method of solving an LP problem.

Answer

The graphical method is applicable to solve the LPP involving two decisionvariables x 1 , and x 2 , we usually take these decision variables as x, y insteadof x 1 , x 2 .

To solve an LPP, the graphical method includes two major steps.


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a) The determination of the solution space that defines the feasible solution(Note that the set of values of the variable x 1 , x 2 , x 3 ,....x n which satisfy all theconstraints and also the non-negative conditions is called the feasiblesolution of the LPP)

b) The determination of the optimal solution from the feasible region.

a) To determine the feasible solution of an LPP, we have the followingsteps.Step 1:Since the two decision variable x and y are non-negative, consider only thefirst quadrant of xy- plane

Draw the line ax + by = c ... (1)For each constraint, the line (1) divides the first quadrant in to two regionssay R 1 and R 2 , suppose (x 1 , 0) is a point in R 1 . If this point satisfies the inequation ax + by c or ( c), then shade the region R 1 . If (x 1 , 0) does notsatisfy the inequation, shade the region R 2 .

Step 3:Corresponding to each constant, we obtain a shaded region. The intersection

of all these shaded regions is the feasible region or feasible solution of theLPP.Let us find the feasible solution for the problem of a decorative item dealerwhose LPP is to maximise profit function.Z = 50x + 18y ... (1)

Subject to the constraints

Step 1:Since x 0, y 0, we consider only the first quadrant of the xy - plane

Step 2:We draw straight lines for the equation2x+ y = 100 ...(2)

x + y = 80 To determine two points on the straight line 2x + y = 100Put y = 0, 2x = 100


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x = 50(50, 0) is a point on the line (2)

put x = 0 in (2), y =100(0, 100) is the other point on the line (2)

Plotting these two points on the graph paper draw the line which representthe line 2x + y =100.

This line divides the 1 st quadrant into two regions, say R 1 and R 2 . Choose apoint say (1, 0) in R 1 . (1, 0) satisfy the inequation 2x + y 100. Therefore

R1 is the required region for the constraint 2x + y 100.

Similarly draw the straight line x + y = 80 by joining the point (0, 80) and(80, 0). Find the required region say R 1 ', for the constraint x + y 80.

The intersection of both the region R 1 and R 1 ' is the feasible solution of theLPP. Therefore every point in the shaded region OABC is a feasible solutionof the LPP, since this point satisfies all the constraints including the non-negative constraints.

b) There are two techniques to find the optimal solution of an LPP.

Corner Point Method The optimal solution to a LPP, if it exists, occurs at the corners of thefeasible region. The method includes the following steps

Step 1:Find the feasible region of the LLP.


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Step 2:Find the co-ordinates of each vertex of the feasible region. These co-ordinates can be obtained from the graph or by solving the equation of thelines.

Step 3:At each vertex (corner point) compute the value of the objective function.

Step 4:Identify the corner point at which the value of the objective function ismaximum (or minimum depending on the LPP). The co-ordinates of thisvertex is the optimal solution and the value of Z is the optimal value

Example:Find the optimal solution in the above problem of decorative item dealerwhose objective function is Z = 50x + 18y. In the graph, the corners of thefeasible region are

O (0, 0), A (0, 80), B (20, 60), C (50, 0)At (0, 0) Z = 0

At (0, 80) Z = 50 (0) + 18(80)= Rs. 1440

At (20, 60), Z = 50 (20) +18 (60)= 1000 + 1080 = Rs.2080

At (50, 0) Z = 50 (50) + 18 (0)= Rs. 2500.

Since our object is to maximise Z and Z has maximum at (50, 0) the optimalsolution is x = 50 and y = 0.

The optimal value is Rs. 2500.

If an LPP has many constraints, then it may be long and tedious to find allthe corners of the feasible region. There is another alternate and moregeneral method to find the optimal solution of an LPP, known as 'ISO profitor ISO cost method'

ISO- PROFIT (OR ISO-COST)

Method of Solving Linear Programming Problems

Suppose the LPP is to

Optimize Z = ax + by subject to the constraints


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Suggested answer:since x 0, y 0, consider only the first quadrant of the plane graph thefollowing straight lines on a graph paper

10x + 5y = 80 or 2x+y =166x + 6y = 66 or x+y =114x+ 8y = 24 or x+ 2y = 6

5x + 6y = 90Identify all the half planes of the constraints. The intersection of all thesehalf planes is the feasible region as shown in the figure.

Give a constant value 600 to Z in the objective function, and then we havean equation of the line120x + 100y = 600 ... (1)or 6x + 5y = 30 (Dividing both sides by 20)

P1Q 1 is the line corresponding to the equation 6x + 5y = 30. We give aconstant 1200 to Z, then the P 2 Q 2 represents the line.120x + 100y = 12006x + 5y = 60


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P2Q 2 is a line parallel to P 1Q 1 and has one point 'M' which belongs to feasibleregion and farthest from the origin. If we take any line P 3 Q 3 parallel toP2Q 2 away from the origin, it does not touch any point of the feasible region.

The co-ordinates of the point M can be obtained by solving the equation

2x + y = 16 x + y =11 which give x = 5 and y = 6

The optimal solution for the objective function is x = 5 and y = 6 The optimal value of Z120 (5) + 100 (6) = 600 + 600= 1200

Question 7: What is meant by the term feasible region? Why thismust be a well-defined boundary for maximization problem?

Answer: A feasible region is, in a constrained optimization problem, the setof solutions satisfying all equalities and/or inequalities. On the other hand alinear programming is a constrained optimization problem in which both theobjective function and the constraints are linear, therefore a feasible regionon a linear programming problem is the set of solutions of the a linearproblem. Many algorithms had been designed to successfully attainfeasibility at the same time as resolving the problem, e.g. reaching itsminimum. Perhaps one of the most famous and extensively utilized is theSimplex Method who travels from one extremal point to another, whichhappens to be the possible extrema given the convex nature of the problem,by maintaining a fixed number of components to zero, called basic variables.

Then, the algorithm arrives to a global minimum generally in polinomialtime even if its worst possible case has already been proved to beexponencial, see Klee-Minty's cube.


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GRAPHICAL ANALYSIS THE FEASIBLE REGION

THE SEARCH FOR AN OPTIMAL SOLUTION The figure shows how different constraints can be represented by straightlines to define a feasible region. There is an area outside the feasible regionthat is infeasible.

It may be seen that each of the constraints is a straight line. The constraintsintersect to form a point that represents the optimal solution. This is thepoint that results in maximum profit of 436,000 Rs. As shown in the slidebelow. The procedure is to start with a point that is the starting point say 200,000 Rs. Then move the line upwards till the last point on the feasibleregion is reached. This region is bounded by the lines representing theconstraints.


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Question 8: Define slack and surplus variables in a linear programmingproblem.

Answer: LP constraints are limited to only three possible relational

operators: = , , Strict inequalities ( < , > ) are not allowed in LP.

Graphically, an equality constraint traces a line. LP inequality constraints and consist of a line (the equality part of the relation) along with ahalf-plane (the inequality portion). Two-dimensional inequalities are easy towork with graphically because the regions they define are depicted quitenaturally by the Cartesian plane. Alas, that is not the case with analytical(algebraic) solution methods, which do not deal with geometric planes butwith purely abstract symbols. It's a bit harder to visualize relations withsymbols, but those representations can be very powerful. Algebra actually,math in general is famously powerful largely because of the logicalprecision afforded by symbolic equations. To say that something is equal tosomething else is, in math, a rigorously exact statement. On the other hand,to say that something is greater than or equal to some other quantity leavesopen an infinite (real) number of possibilities. Exactness goes down thedrain.

Consider the large block constraint: 2x + y 6. The economicinterpretation of this constraint is: the number of large blocks used toassemble tables plus the number of large blocks used to assemble chairscannot exceed the six units of large blocks in inventory. Suppose we buildone table and one chair. In order to determine how many blocks we have leftin stock, we must plug in those values into the variables of the constraint,compute the left hand side and subtract it from 6. We find we have threeblocks left in stock.

Accountants prefer another approach. Instead of computing how many blocks are left in stock every time a table or chair is assembled, they createa new account that keeps track of the blocks inventory continuously. Theaccount is initialized at 6 and decreased in real time (perpetual inventory) asblocks are used in production. The relation can now be expressed as: thenumber of large blocks used to assemble tables plus the number of largeblocks used to assemble chairs plus the number of large blocks left ininventory must equal the six units of large blocks originally stocked.Mathematically:

2x + y + s = 6
http://www.investorwords.com/3679/perpetual_inventory.htmlhttp://www.investorwords.com/3679/perpetual_inventory.html


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Where s is a new variable that keeps track of the large-blocks inventory.

The beauty of this arrangement is twofold. First, we can always tell at aglance how many blocks are idle (or slack, i.e., not being used in production)

by simply checking the slack variable. No need to recalculate balances.

Minutiae Aedificante

This slack-variable technique illustrates a general principle of computation:computational procedures can generally be reduced by increasing the size of the workspace (in our case, adding more variables). Actually, this goes backto a fundamental principle of physics: time and space are complementary concepts. That is the reason why relativity theory, and not classicalmechanics, provides the correct description of reality.

In second place, the constraint has been converted from an inequality to anequation. It is this conversion into an equation that makes analyticalsolution methods possible.

The variable s is called a slack variable because it measures the amount of slack (idle) resources still remaining in stock at any point in time during theproduction process. Since it is not possible to have negative slack,nonnegative constraints also apply to slack variables.

All constraints are automatically converted by LP software intotheir standard form (equations) by creating and adding slack variables.

The Lego problem has two constraints, so two slack variables arerequired:

2x + y + s 1 = 62x + 2y + s 2 = 8

Where s1 = number of large blocks remaining in stock, s2 = number of smallblocks remaining in stock, and s1, s 2 0.

To convert constraints to standard form, a surplus variable issubtracted on the left hand side of the constraint. For example, supposethat at least two items, either tables or chairs, are needed for demopurposes in the showroom. This requirement is captured by the followinginequality constraint:

x + y 2


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Since it is possible to exceed the minimum required production quantity, theonly way to obtain an equation when converting to standard form is tosubtract the excess production (the quantity produced in excess of the

minimum requirement):

x + y - s 3 = 2

Where s3 = number of items produced in excess of showroom requirements.Nonnegative constraints are also required for surplus variables. Thus, s3 0.

All constraints are automatically converted by LP software into theirstandard form by creating and subtracting surplus variables.

In practical terms, slack and surplus variables keep track of basically thesame thing: by how much is the left hand side of the constraint differentfrom the RHS constant. If there is a difference, it must be positive, by definition. If there is no difference, s = 0. The names slack and surplus servemerely to indicate the direction of the difference, that is, to remind us thatthe original (pre-standard) constraint was or .

Question 9: Explain various steps of the simplex method involved inthe computation of an optimum solution to a linear programmingproblem.

Answer: The steps for the computation of an optimum solution are asfollows:

Step-1 : Check whether the objective function of the given L.P.P is to bemaximized or minimized. If it is to be minimized then we convert it into aproblem of maximizing it by using the result Minimum Z = - Maximum (-z)

Step-2 : Check whether all right hand side values of the constrains are non-negative. If any one of values is negative then multiply the correspondinginequation of the constraints by -1, so as to get all values are non-negative.

Step-3 : Convert all the in equations of the constraints into equations by introducing slack/surplus variables in the constraints. Put the costs of these variables equal to zero.

Step-4 : Obtain an initial basic feasible solution to the problem and put it inthe first column of the simplex table.


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Step-5 : Compute the net evolutions j = Z j C j (j=1, 2n) by using therelation Z j C j = C B X j C j.

o Examine the signo

(i) If all net evolutions are non negative, then the initial basicfeasible solution is an optimum solution.o (ii) If at least one net evolution is negative, proceed on to the

next step.

Step-6 : If there is more than one negative net evolution, then choose themost negative of them. The corresponding column is called entering column.

o (i) If all values in this column are 0, then there is anunbounded solution to the given problem.

o (ii) If at least one value is > 0, then the corresponding variableenters the basis.

Step-7 : Compute the ratio {X B / Entering column} and choose theminimum of these ratios. The row which is corresponding to this minimumratio is called leaving row. The common element which is in both enteringcolumn and leaving row is known as the leading element or key element orpivotal element of the table.

Step-8 : Convert the key element to unity by dividing its row by the leadingelement itself and all other elements in its column to zeros by usingelementary row transformations.

Step-9 : Go to step-5 and repeat the computational procedure until either anoptimum solution is obtained or there is an indication of an unboundedsolution.

Question 10: What do you mean by an optimal basic feasible solution toa linear programming problem?

Answer: The basic feasible solution which optimizes (maximizes orminimizes) the objective function value of the given LP problem is called anoptimum basic feasible solution.

The terms basic solution and basic feasible solution are very importantparts of the standard vocabulary of linear programming, we now need toclarify their algebraic properties. For the augmented form of the example,notice that the system of functional constraints has 5 variables and 3

equations, so Number of variables _ number of equations _ 5 _ 3 _ 2.


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This fact gives us 2 degrees of freedom in solving the system, since any twovariables can be chosen to be set equal to any arbitrary value in order tosolve the three equations in terms of the remaining three variables.1 Thesimplex method uses zero for this arbitrary value. Thus, two of the variables

(called the nonbasic variables) are set equal to zero, and then thesimultaneous solution of the three equations for the other three variables(called the basic variables) is a basic solution. These properties aredescribed in the following general definitions.

A basic solution has the following properties:1. Each variable is designated as either a nonbasic variable or a basic

variable.2. The number of basic variables equals the number of functional

constraints (now equations). Therefore, the number of nonbasicvariables equals the total number of variables minus the number of functional constraints.

3. The nonbasic variables are set equal to zero.4. The values of the basic variables are obtained as the simultaneous

solution of the system of equations (functional constraints inaugmented form). (The set of basic variables is often referred to as thebasis. )

5. If the basic variables satisfy the nonnegative constraints, the basicsolution is a BF Solution.Xj _ 0, for j _ 1, 2. . . 5.

EXTREME POINTS AND OPTIMAL SOLUTIONS MULTIPLE OPTIMAL SOLUTIONS

There may be more than one optimal solutions. However, the condition isthat the objective function must be parallel to one of the constraints. If aweightage average of different optimal solutions is obtained, it is also anoptimal solution.


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Through the dual LP problem, production management attempts to achievea production plan that optimizes resource allocation so that each product isproduced at that quantity such that its marginal opportunity cost equals toits marginal return.

Thus the main focus of dual is to find out for each resource, its bestmarginal value (dual price or shadow price).

Essentially then, the shadow prices constitute an internal pricing systemfor the firms resource that:

1. Permits the rm to select which activity to pursue by considering only the marginal portability of its activities;

2. Allocate the contribution of the firm to its resources at the margin.

Functional properties of duality:

There are basically three functional properties of duality. These are:

1. Weak duality:

The first property is referred to as weak duality and provides a bound onthe optimal value of the Objective function of either the primal or the dual.Simply stated, the value of the objective function for any feasible solution tothe primal maximization problem is bounded from above by the value of the

objective function for any feasible solution to its dual. Similarly, the value of the objective function for its dual is bounded from below by the value of theobjective function of the primal.

The sequence of properties to be developed will lead us to the strongduality" property, which states that the optimal values of the primal anddual problems are in fact equal. Further, in developing this result, we showhow the solution of one of these problems is readily available from thesolution of the other.

2. Unboundedness Property: If the primal (dual) problem has anunbounded solution, then the dual (primal) problem is infeasible. We arenow in a position to give the main result of this section, the strong duality property. The importance of this property is that it indicates that we may infact solve the dual problem in place of origin conjunction with the primalproblem. The proof of this result depends merely on observing that theshadow prices determined by solving the primal problem by the simplemethod give a dual feasible solution.

3. Strong Duality Property: If the primal (dual) problem has a niteoptimal solution, then so does the dual (primal) problem and these twovalues are equal.


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Question 12 : Explain the primal dual relationship.

Answer: A summary of general relationship between primal and dual LPPis:-

If primal If dual(i) Objective is to maximize.(ii) jth primal value, xj(iii) ith primal constraint(iv) Primal variable xj

unrestricted in sign(v) Primal constraint i = type(vi) Primal constraints is less

than equal to type

(i) Objective is to minimize(ii) Jth dual constraint(iii) Its dual variable, yi(iv) Dual constraint j = type(v) Dual variable yi is

unrestricted in sign(vi) Dual constraints is greater

than equal to type

Question 13: What is duality? What is the significance of dual variablesin a LP model?

Answer: Duality implies that each LPP can analyse in two different ways butwould have equivalent solution. Any LPP can be stated in another equivalentform based on same data. The new LPP is called dual linear programming.

By using primal LPP, the production manager attempts to optimize resourceallocation by determining qualities of each product to be produced that willmaximize profit. Through a dual LPP approach, he attempts to achieve aproduction plan that optimizes resource allocation in a way that eachproduct is produced at that quantity so that its marginal opportunity costequals its marginal return. Thus, the main focus of a dual problem is to findfor each resource its best marginal value.

Question 14: Write a short note on sensitivity analysis.

Answer: A technique used to determine how different values of anindependent variable will impact a particular dependent variable under agiven set of assumptions. This technique is used within specific boundariesthat will depend on one or more input variables, such as the effect that

changes in interest rates will have on a bond's price. Sensitivity analysis is a


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way to predict the outcome of a decision if a situation turns out to bedifferent compared to the key prediction(s).

Sensitivity analysis is very useful when attempting to determine theimpact the actual outcome of a particular variable will have if it differs fromwhat was previously assumed. By creating a given set of scenarios, theanalyst can determine how changes in one variable(s) will impact the targetvariable.

For example, an analyst might create a financial model that will value acompany's equity (the dependent variable) given the amount of earnings pershare (an independent variable) the company reports at the end of the yearand the company's price-to-earnings multiple (another independent variable)at that time. The analyst can create a table of predicted price-to-earningsmultiples and a corresponding value of the company's equity based ondifferent values for each of the independent variables.

Duality allows us to identify those resources whose adjustment might bringincremental cost or profit improvement in the value of objective function.However duality variable values do not reveal possible magnitude suchchanges. But the study of sensitivity analysis shows us the magnitude of achange in the optimal solution of an LP model due to discrete variations inits parameters.

The degree of sensitivity of the optimal solution due to these variations canrange from no change at all to a substantial change in the optimal solutionof the given LP problem. Thus the sensitivity analysis, we determine therange (or limit) over which the LP model parameters can be change withoutaffecting the current optimal solution.

The sensitivity analysis is also referred to as post optimality analysisbecause it does not begin until the optimal solution to the given LP modelhas been obtained.

Question 15 : Discuss the role of sensitivity analysis in linearprogramming. Under what circumstances it is needed, and under whatconditions do you think it is not necessary?

Answer: The study of sensitivity analysis, however, does how the magnitudeof a change in the optimal solution of an LP model due to discrete variationsin its parameters. The degree of sensitivity of the solution due to thesevariations can change without affecting the current optimal solution. Forthis, instead of resolving the entire problem as a new problem with new

parameters. We may consider the optimal solution as an initial solution for


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the purpose of knowing the ranges, both lower and upper, within which aparameter may assume a value.

The circumstances when sensitivity analysis is needed

When there are changes in objective function coefficients (cj):

1. Change in the coefficient of a non-basic variable:If cj- zj 0 for all non basic variable in a maximization LPproblem, then the current optimal solution remains unchanged.

2. Change in the coefficient of a basic variable:In the maximization LP problem, the change in the coefficient,say ck, of a basic variable xk affects the cj-zj valuescorresponding to all non basic variables in the optimal Simplextable. It is because of coefficient ck is listed in the cB column of the Simplex table and affects the calculation of zj values.

3. Change in coefficient of a non basic variable in minimizationproblem

When there is a change in the availability of resources:

1. When slack variable is not in the solution mix

2. When a slack variable is in the basis (column B)

3. Changes in the right side when constrains are mixed type.

Question 16: Explain in brief the methods of initial feasible solution fortransportation problem.

Answer: There are several methods available to obtain an initial basicfeasible solution.

(a) NWCM North West Corner Method :- It is a simple and efficientmethod to obtain an initial solution. This method does not takeinto account the cost of transportation on any route of transportation.

(b) Least cost method:- The main objective is to minimize the totaltransportation cost, we must try to transport as much as possiblethrough those routes where the unit transportation cost is lowest.


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This method takes into account the minimum unit cost of transportation for obtaining the initial solution.

(c) VAM Vogels Approximation Method: - In this method, eachallocation is made on the basis of the opportunity cost that would

have been incurred if the allocation in certain cells with minimumunit transportation cost were missed. In this method allocationsare made so that the penalty cost is minimised. The advantage of this method is that it gives an initial solution or is the optimalsolution itself.

Question 17: State the transportation problem. Describe clearly thesteps involved in solving it.

Answer: The structure of transportation problem involves a large number of shipping routes from several supply origins to several demand destinations.

The objective is to determine the number of units of an item that should beshipped from an origin to a destination in order to satisfy the requiredquantity of goods or services at each destination centre. The transportationalgorithm discussed is applied to minimize the total cost of transporting ahomogeneous commodity from supply centres to demand centres.

The steps are as follows:-

1. Formulate the problem and arrange the data in matrix form:- theformulation of transportation problem is similar to the LPPformulation. Here objective function is the total transportation costand the constraints are the supply and demand available at eachsource and destination.

2. Obtain the initial basic feasible solution:- There are three methods :-a) North West Corner Methodb) Least cost methodc) Vogels a pproximation method

The initial solution obtained by any of three methods must satisfy following conditions:-i) The solution must be feasible, that is it must verify all the

supply and demand constraints.ii) The number of positive allocations must be equal to m+n-1

where m is the no. of rows and n is the no. of columns.Any solution that satisfies the above condition is called non-degenerate basic feasible solution, otherwise degenerate

solution.


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3. Test the initial solution for optimality:- The MODI method is discussedto test the optimality of the solution obtained in step 2. If the currentsolution is optimal, then stop. Otherwise, determine a new improvedsolution.

4.

Updating the solution:- Repeat the step 3 until the optimal solution isreached.

Question 19: What is an assignment problem? Give two applications.

Answer: An assignment problem is a particular case of a transportationproblem where the sources are assignment and the destinations are tasks.Every source has a supply of 1 and every destination has a demand of 1.Also, the objective is to minimize the total cost or to maximize the total profitof allocation.

Some of the problems where the assignment technique may be useful areassignment of: workers to machines, salesmen to different sales area, clerksto various checkout counters, classes to rooms, vehicles to routes, contractsto bidders, etc.

Applications of assignment problem

The linear programming model encompasses a wide variety of specific typesof problems. The general simplex method is a powerful algorithm that cansolve surprisingly large versions of any of these problems. However, some of these problem types have such simple formulations that they can be solvedmuch more efficiently by streamlined algorithms that exploit their specialstructure. These streamlined algorithms can cut down tremendously on thecomputer time required for large problems, and they sometimes make itcomputationally feasible to solve huge problems. This is particularly true forthe two types of linear programming problems studied in this chapter,namely, the transportation problem and the assignment problem. Bothtypes have a number of common applications, so it is important to recognizethem when they arise and to use the best available algorithms. Thesespecial-purpose algorithms are included in some linear programmingsoftware packages.

This problem has the interpretation of minimizing the cost for the flow of goods through a network. A streamlined version of the Simplex methodcalled the network simplex method is widely used for solving this type of problem, including its various special cases.


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A supplementary chapter on the books website describes various additionalspecial types of linear programming problems. One of these, called thetransshipment problem, is a generalization of the transportation problemwhich allows shipments from any source to any destination to first go

through intermediate transfer points. Since the transshipment problem alsois a special case of the minimum cost flow problem. Much researchcontinues to be devoted to developing streamlined algorithms for specialtypes of linear programming problems, including some not discussed here.At the same time, there is widespread interest in applying linearprogramming to optimize the operation of complicated large-scale systems.

The resulting formulations usually have special structures that can beexploited. Being able to recognize and exploit special structures is animportant factor in the successful application of linear programming.

Question 20: Give the mathematical formulation of an assignmentproblem. How does it differ from a transportation problem?

Answer: Given n resources and n activities and effectiveness of eachresource for each activity, the problem lies in assigning each resource to oneand only one activity so that given resource of activity is optimized.

Let there be n jobs which are to be assigned to n operators so that one job isassigned to only one operator.i = Index for job, i = 1, 2, n

j = Index for operators, j = 1, 2, n C ij = Unit cost for assigning job i to operator j

1 if job i is assigned to operator jXij = 0 Otherwise

The objective is to minimize the total cost of assignment. If job I is assignedto operator 1, the cost is (C 11 X11 ). Similarly, for job 1, operator 2 the cost is(C12 X12 ). The objective function is:

Minimize = n i=1 n j=1 C ij Xij (1)

Since one job (i) can be assigned to any one of the operators, we havefollowing constraint set:

n i=1 Xij = 1; for all j;j = 1, 2, ... n (2) Similarly for each operator, there may be only one assignment of job. Forthis, the constraint set is:

n i=1 Xij = 1; for all i;i = 1, 2, ... n (3) The non-negativity constraint is:

Xij > 0 (4) Minimize Z = n i=1 n j=1 C ij Xij

Subject to n i=1 Xij = 1; for all j;j = 1, 2, ... nn j=1 Xij = 1; for all i;i = 1, 2, ... n


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Xij > for all i and all j.

The assignment problem is nothing but a variation of the transportationproblem with two characteristics:-

i) The cost matrix is a square matrix.ii) The optimal solution for the problem would always be such that

there would be only one assignment in a given row or column of the cost matrix.

Question 21: Define

(1) Competitive game: The term game refers to a situation of conflictand competition in which two or more competitors are involved in

decision making process in anticipation of certain outcomes over aperiod of time. The competitors are referred to as players. A playermay be an individual, a group of individuals, or an organisation.

(2) Pay off matrix: The payoffs in terms of gains or losses, when playersselect their particular strategies, can be represented in the form of amatrix called payoff matrix.

(3) Pure strategies: This is the decision rule that is always used by theplayer to select the particular strategy. Thus, each player knows inadvance all strategies out of which he can always select only one

particular strategy, regardless of the other players strategy. Theobjective of the player is to maximize their gains or minimize theirlosses.

(4) Mixed strategies: Courses of action that are to be selected on aparticular occasion with some fixed probability are called mixedstrategies. Thus, there is a probabilistic situation and objective of theplayer is to maximize expected gains or to minimize expected losses by making the choice among pure strategies with fixed probabilities.

(5) Saddle point: Maxmin value equals the minmax value, then the game

is said to have a saddle point.(6) Optimal strategies: A course of action that puts any player in themost preferred position, irrespective of course of action his competitoradopt, is called as optimal strategy.

(7) Two-person zero-sum games: A game with only two players. Say player A and player B, is called a two-person zero-sum game, only if one players gain is equal to the los s of other player in a way that totalsum is zero.


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Question 22: Explain Minimax and Maxmin principle used in the theoryof game.

Answer: Maxmin principle: - For player A the minimum value in each rowrepresents the least gain (payoff) to him, if he chooses his particularstrategy. These are written in the matrix by row minima. He will then selectthe strategy that gives the largest gain among the row minimum values. Thechoice of player A is called the maxmin principle, and the correspondinggain is called the maxmin value of game.

Minimax principle: - For player B, who is assumed to be the looser, themaximum value in each column represents the maximum loss to him, if hechooses his particular strategy. These are written in the payoff matrix by column maximum. He will then select the strategy that gives the minimum

loss among the column maximum values. This choice of player B is calledthe minimax principle, and the corresponding loss is the minimax value of the game.

Question 23: Explain the following terms in PERT/CPM. (i) Earliesttime (ii) Latest time (iii) Total activity time (iv) Event slack (v) Criticalpath

Answer:

(i) It is the earliest time for an event to occur immediately after all thepreceding activities have been completed without delaying the entireproject.

(ii) It is the latest time at which an event can occur without causing adelay in the already determined projects completion time

(iii)Total activity time is the sum of the durations of all the activities.(iv) The float/slack or free time is the length of time to which a non-

critical activity and/or an event can be delayed or extended withoutdelaying the total project completion time.

(v) Certain activities in the network diagram of any project are calledcritical activities because delay in their execution will cause furtherdelay in the project completion time (activities with zero float).

The critical path is the sequence of critical activities that form a continuouspath between the start of the project and its end.

Question 24: What is float? What are the different types of float?

Answer: The float or free time is the length of the time in which a non-critical activity and/or an event can be delayed or extended without delaying

the total project completion time.


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The different types of floats are: -

Total float: - This is the length of time by which an activity can be delayedwhen all preceding activities are completed at their earliest possible timeand all successor activities can be delayed until their latest permissibletime.

The within which an activity must be scheduled is computed from LS andES values for each activitys start event and end event. That is for eachactivity (i, j) the total float is equal to the latest allowable time for the eventat the event at the end of the activity minus the earliest time for an event atthe beginning of the activity minus the activity duration. That is;

Total float (TFij) = (Lj Ei) tij = LSij Esij = LFij - EFij

Free float: - The free float of a non-critical activity is defined as the time by which the completion of an activity can be delayed without causing any delay in its immediate succeeding activities. Free float values for eachactivity (i, j) are compared as follows:

Free float (FFij) = (Ej EI) tij

= Min { ESij, for all immediate successors of activity (i, j)} Efij

Independent float: - This is the amount of acceptable delay in the completionof an activity so that it neither affects its predecessor nor the successoractivities. Thus, independent float is the amount of time available whenpreceding activities are completed at their latest permissible times and allthe following activities can still be completed at their earliest possible times.Independent float values for each activity are computed as follows: -

Independent float (IFij) = (Ej Li_ - tij = {ESij Lsij} tij

The negative value if independent float is considered to be zero.

Question 25: What is replacement? Describe some importantreplacement situation.

Answer: The problem of replacement is felt when the job performing unitssuch as men, machines, equipment, parts etc become less effective or


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useless due to either sudden or gradual deterioration in their efficiency,failure or breakdown. By replacing those with ones at frequent intervals,maintenance and other overhead costs can be reduced.

There are three types of replacement situation: -

1) Items such as machines, vehicles, tyres etc whose efficiency deteriorates with age due to constant use and which need increasedoperating and maintenance costs. In such cases the deterioration levelis predictable and is represented by a) Increased maintenance/operational costb) Its waste or scrap value and damage to item and safety risk.

2) Items such as light bulbs and tubes, electric motors, radio, televisionparts etc which do not give any indication of deterioration with time

but fail all of a sudden and are rendered useless. Such cases requirean anticipation of failure to specify the probability of any future timeperiod. With this probability distribution and the cost information, itis desired to formulate optimal replacement policy in order to balancethe wasted life of an item, replaced before failure against the costsincurred when the item fails in service.

3) The existing working staffs in an organisation gradually reduce due toretirement, death, retrenchment and other reasons.

operations reaserch assignment

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