MBA SEMESTER II MB0048 Operation Research- 4 Credits (Book ID: B1301) Assignment Set- 1 (60 Marks) Note: Each question carries 10 Marks. Answer all the questionsa .Define O.R and discuss its characteristics. [ 5 marks] Operations research, or Operational Research in British usage, is a discipline that deals with the application of advanced analytical methods to help make better decisions[1]. It is often considered to be a sub-field of Mathematics[2]. The terms management science and decision science are sometimes used as more modern-sounding synonyms[3]. Employing techniques from other mathematical sciences, such as mathematical modeling, statistical analysis, and mathematical optimization, operations research arrives at optimal or near-optimal solutions to complex decision-making problems. Because of its emphasis on human-technology interaction and because of its focus on practical applications, operations research has overlap with other disciplines, notably industrial engineering and operations management, and draws on psychology and organization science. Operations Research is often concerned with determining the maximum (of profit, performance, or yield) or minimum (of loss, risk, or cost) of some real-world objective. Originating in military efforts before World War II, its techniques have grown to concern problems in a variety of industries. Characteristics of Operations Research By an eHow Contributor Operations research, an interdisciplinary division of mathematics and science, uses statistics, algorithms and mathematical modeling techniques to solve complex problems for the best possible solutions. This science is basically concerned with optimizing maxima and minima of the objective functions involved. Examples of maxima could be profit, performance and yield. Minima could be loss and risk. The management of various companies has benefited immensely from operations research. Operations research is also known as OR. It has basic characteristics such as systems orientation, using interdisciplinary groups, applying scientific methodology, providing quantitative answers, revelation of newer problems and the consideration of human factors in relation to the state under which research is being conducted

b. Explain the nature of Operations Research and its limitations. Operations research has been used to solve only a fairly limited number of managerial problems. Its limitations should not be overlooked. In the first place, there is the sheer magnitude of the mathematical and computing aspects. The number of variables and interrelationships in many managerial problems, plus the complexities of human relationships and reactions, calls for a higher order of mathematics than nuclear physics does. The late mathematical genius John von Neumann found, in his development of the theory of games, that his mathematical abilities soon reached their limit in a relatively simple strategic problem. Managers are, however, a long way from fully using the mathematics now available.

In the second place, although probabilities and approximations are being substituted for unknown quantities and although scientific method can assign values to factors that could never be measured before, a major portion of important managerial decisions still involves qualitative factors. Until these can be measured, operations research will have limited usefulness in these areas, and decisions will continue to be based on non-quantitative judgments. Related to the fact that many management decisions involve un-measurable factors is the lack of information needed to make operations research useful in practice. In conceptualizing a problem area and constructing a mathematical model to represent it, people discover variables about which they need information that is not now available. To improve this situation, persons interested in the practical applications of operations research should place far more emphasis on developing this required information. Still another limitation is the gap between practicing managers and trained operation researchers. Managers in general lack a knowledge and appreciation of mathematics, just as mathematicians lack an understanding of managerial problems. This gap is being dealt with, to an increasing extent, by the business schools and, more often, by business firms that team up managers with operations research. But it is still the major reasons why firms are slow to use operations research. A final drawback of operations research at least in its application to complex problems is that analyses and programming are expensive and many problems are not important enough to justify this cost. However, in practice this has not really been a major limitation.

a. What are the essential characteristics of a linear programming model? marks]

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b. Explain the graphical method of solving a LPP involving two variables. [ 5 marks] Linear programming (LP, or linear optimization) is a mathematical method for determining a way to achieve the best outcome (such as maximum profit or lowest cost) in a given mathematical model for some list of requirements represented as linear relationships. Linear programming is a specific case of mathematical programming (mathematical optimization). More formally, linear programming is a technique for the optimization of a linear objective function, subject to linear equality and linear inequality constraints. Its feasible region is a convex polyhedron, which is a set defined 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 defined on this polyhedron. A linear programming algorithm finds a point in the polyhedron where this function has the smallest (or largest) value if such point exists. Linear programs are problems that can be expressed in canonical form: where x represents the vector of variables (to be determined), c and b are vectors of (known) coefficients and A is a (known) matrix of coefficients. The expression to be maximized or minimized is called the objective function (cTx in this case). The equations Ax b are the constraints which specify a convex polytope over which the objective function is to be optimized. (In this context, two vectors are comparable when every entry in one is less-than or equal-to the corresponding entry in the other. Otherwise, they are incomparable.) Linear programming can be applied to various fields of study. It is used in business and economics, but can also be utilized for some engineering problems. Industries that use linear programming models include transportation, energy, telecommunications, and manufacturing.

It has proved useful in modeling diverse types of problems in planning, routing, scheduling, assignment, and design. Linear Programming Graphical Method Learn about linear programming by graphical method. If the objective function Z is a function of two variables only, then the Linear Programming Problem can be solved effectively by the graphical method. If Z is a function of three variables, then also it can be solved by this method. But in this case the graphical solution becomes complicated enough. The linear programming problems are solved in applied mathematics models.

Method of Linear Programming Graph Draw the graph of the constraints. Determine the region which satisfies all the constraints and non-negative constraints (x > 0, y > 0). This region is called the feasible region. Determine the co-ordinates of the corners of the feasible region. Calculate the values of the objective function at each corner. Select the corner point which gives the optimum (maximum or minimum) value of the objective function. The co-ordinates of that point determine the optimal solution. Below you can see the problems linear programming by graphical method -

Problem 1:Use graphical method to solve the following linear programming problem. Maximize Z = 2x + 10 y Subject to the constraints 2 x + 5y < 16, x < 5, x > 0, y > 0. Solution:

Since x > 0 and y > 0 the solution set is restricted to the first quadrant.| i) 2x + 5y < 16 Draw the graph of 2x + 5y = 16 2x + 5y = 16 y= x 8 0

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0 3.2 2 y Determine the region represented by 2x + 5y < 16 ii) x < 5 Draw the graph of x = 5 Determine the region represented by x < 5. Shade the intersection of the two regions. The shaded region OABC is the feasible region B(5, 1.2) is the point of intersection of 2x + 5y = 16 and x = 5. The corner points of OABC are O(0,0), A(5,0), B(5,1.2) and C(0,3.2). Corners O(0,0) A(5,0) B(5,1.2) C(0,3.2) 0 Z = 2x + 10 y 10 22 32

Z is maximum at x = 0, y = 3.2 Maximum value of Z = 32. Problem 2: Use graphical method to solve the following linear programming problem. Maximize Z = 20 x + 15y Subject to 180x + 120y < 1500, x + y < 10, x > 0, y > 0 Solution:

Since x > 0 and y > 0, the solution set is restricted to the first quadrant. i) 180x + 120 y < 1500 180x + 120y < 1500 => 3x + 2y < 25. Draw the graph of 3x + 2y = 25 3x + 2y = 25 y= x 0 0 y Determine the region represented by 3x + 2y < 25. ii) x + y < 10 Draw the graph of x + y = 10 x + y = 10 y =10 - x x 0 10 5 10 0 y 5

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Determine the region represented by x + y < 10 Shade the intersection of the two regions. The shaded region OABC is the feasible region. B(5,5) is the point of intersection of 3x + 2y = 25 and x + y = 10. The corner points of OABC are O(0,0), A( Corners , 0), B (5,5) and C(0,10). O(0,0) A( ,0) 0 166.67

B(5,5) 175

C(0,10) 150

Z = 20x + 15 y

Z is maximum at x = 5 and y = 5. Maximum value of Z = 175. a. Explain the simplex procedure to solve a linear programming problem. [ 5 marks] b. Explain the use of artificial variables in L.P [ 5 marks] About Artificial Variables in Linear Programming (LP) Another frequently asked question by students is related to use of artificial variables while preparing the initial basic feasible solution table. The common flow of discussion forces the student to think in a logical way as he has been thinking about slack and surplus variables but the artificial variables can not be considered in the same logical category as the previous two. Just to recall, slack and surplus variables are used in LP to convert inequality constraints to that of equality. If the constraint is of