Assignment Or1

MEERA M NAIR (520845475)

ASSIGNMENT 1

Assignment 1 OPERATIONS RESEARCH MB0032 1. Describe in details the different scopes of application of Operations Research. The scope of Operations Research its application in various fields of everyday life are as follows: i) In Defence Operations: The application of modern warfare techniques in each of the components of military organizations requires expertise knowledge in respective fields. Further more, each component works to drive maximum gains from its operations and there is always a possibility that strategy beneficial to one component may have an adverse effect on the other. Thus in defence operations there is a necessity to co-ordinate the activities of various components which gives maximum benefit to the organization as a whole, having maximum use of the individual components. The final strategy is formulated by a team of scientists drawn from various disciplines who study the strategies of different components and after appropriate analysis of the various courses of actions, the best course of action, known as optimum strategy, is chosen. ii) In Industry: The system of modern industries are so complex that the optimum point of operation in its various components cannot be intuitively judged by an individual. The business environment is always changing and any decision useful at one time may not be so good some time later. There is always a need to check the validity of decisions continually, against the situations. The industrial revolution with increased division of labour and introduction of management responsibilities has made each component an independent unit having their own goals. Each department plan their own objectives and all these objectives of various department or components come to conflict with each other and may not conform to the overall objectives of the organization. The application of OR techniques helps in overcoming this difficulty by integrating the diversified activities of various components so as to serve the interest of the organization as a whole efficiently. OR methods in industry can be applied in the fields of production, inventory controls and marketing, purchasing, transportation and competitive strategies etc. iii) Planning: In modern times it has become necessary for every government to have careful planning, for economic development of the country. OR techniques can be fruitfully applied to maximize the per capita income, with minimum sacrifice and time. A government can thus use OR for framing future economic and social policies. iv)Agriculture: With increase in population there is a need to increase agriculture output. But this cannot be done arbitrarily. There are a number of restrictions under which agricultural production is to be studied. Therefore there is a need to determine a course of action, which serves the best under the given restrictions. The problem can be solved by the application of OR techniques. v) In Hospitals: The OR methods can be used to solve waiting problems in out-patient department of big hospitals. The administrative problems of hospital organization can also be solved by OR techniques. vi) In Transport: Different OR methods can be applied to regulate the arrival of trains and processing times, minimize the passengers waiting time and reduce congestion, formulate suitable transportation policy, reducing the costs and time of trans-shipment. vii) Research and Development: Control of R and D projects, product introduction planning etc. and many more applications.

MB0032

OPERATIONS RESEARCH

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MEERA M NAIR (520845475)

ASSIGNMENT 1

2. What do you understand by Linear Programming Problem? What are the requirements of L.P.P.? What are the basic assumptions of L.P.P.? The Linear Programming Problem (LPP) is a class of mathematical programming in which the functions representing the objectives and the constraints are linear. Here, by optimization, we mean either to maximize or minimize the objective functions. The general linear programming model is usually defined as follows: Maximize or Minimize Z = c1 x1 + c2 x 2 + - - - - + cn x n subject to the constraints,

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 the decision variables. Here ~ is either £ (less than), ³ (greater than) or = (equal). Note that, in terms of the above formulation the coefficient cj, aej, bj are interpreted physically as follows. If bi is the available amount of resources i, where aij is the amount of resource i, that must be allocated to each unit of activity j, the “worth” per unit of activity is equal to cj. Requirements of L.P.P. i. Decisions variables and their relationship ii. Well defined objective function iii. Existence of alternative courses of action iv. Non-negative conditions on decision variables. Basic assumptions of L.P.P 1. Liniarity: Both objective function and constraints must be expressed as linear inequalities. 2. Deterministic: All coefficient of decision variables in the objective and constraints expressions should be known and finite. 3. Additivity: The value of objective function for the given values of decision variables and the total sum of resources used, must be equal to sum of the contributions earned from each decision variable and the sum of resources used by decision variables respectively. 4. Divisibility: The solution of decision variables and resources can be any non-negative values including fractions. Q3. Describe the different steps needed to solve a problem by simplex method. To Solve problem by Simplex Method 1. Introduce stack variables (Si’s) for ” £” type of constraint. 2. Introduce surplus variables (Si’s) and Artificial Variables (Ai) for ” ³” type of constraint. 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 “M” Find Zj - Cj for each variable. 5. Slack and Artificial variables will form Basic variable for the first simplex table. Surplus variable will never become Basic Variable for the first simplex table.

MB0032

OPERATIONS RESEARCH

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MEERA M NAIR (520845475)

ASSIGNMENT 1

6. Zj = sum of [cost of variable x its coefficients in the constraints – Profit or cost 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 the next table. 8. Divide the quantities by the corresponding values of the key column to get ratios select the minimum ratio. This becomes the key row. The Basic variable corresponding to this row will be replaced by the variable found in step 6. 9. The element that lies both on key column and key row is called Pivotal element. 10. Ratios with negative and “a” value are not considered for determining key row. 11. Once an artificial variable is removed as basic variable, its column will be deleted from next iteration. 12. For maximisation problems decision variables coefficient will be same as in the objective function. For minimization problems decision variables coefficients will have opposite signs as compared to objective function. 13. Values of artificial variables will always is – M for both maximization and minimization problems. 14. The process is continued till all Zj - Cj ≥ 0. 4. Describe the economic 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 number of variables, the labour of computation can be considerably reduced by converting it into the dual problem and then solving it ii) The interpretation of the dual variable from the lost or economic point of view proves extremely useful in making future decisions in the activities being programmed. The linear programming problem can be thought of as a resource allocation model in which the objective is to maximize revenue or profit subject to limited resources. Looking at the problem from this point of view, the associated dual problem offers interesting economic interpretations of the L.P resource allocation model. 5. How can you use the Matrix Minimum method to find the initial basic feasible solution in the transportation problem. Step 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 ith row of the transportation table and decrease bj by ai go to step 3. if xij = bj cross off the ith column of the transportation table and decrease 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 transportation table until all the rim requirements are satisfied whenever the minimum cost is not unique make an arbitrary choice among the minima. 6. Describe the Integer Programming Problem. Describe the Gomory’s All-I.P.P. method for solving the I.P.P. problem.

MB0032

OPERATIONS RESEARCH

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MEERA M NAIR (520845475)

ASSIGNMENT 1

The Integer Programming Problem I P P is a special case of L P P where all or some variables are constrained to assume non-negative integer values. This type of problem has lot of applications in business and industry where quite often discrete nature of the variables is involved in many decision making situations. All And Mixed I P P An integer programming problem can be described as follows: Determine the value of unknowns x1, 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 and xj ≥ 0 j = 1, 2, … ,n where xj being an integral value for j = 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 of the variables are restricted to take integral value and rest (n – k) variables are free to take any non negative values, then the problem is known as mixed integer programming problem. Gomory’s All – IPP Method An optimum solution to an I. P. P. is first obtained by using simplex method ignoring the restriction of integral values. In the optimum solution if all the variables have integer values, the current solution will be the desired optimum integer solution. Otherwise the given IPP is modified by inserting a new constraint called Gomory’s or secondary constraint which represents necessary condition for integrability and eliminates some non integer solution without losing any integral solution. After adding the secondary constraint, the problem is then solved by dual simplex method to get an optimum integral solution. If all the values of the variables in this solution are integers, an optimum inter-solution is obtained, otherwise another new constrained is added to the modified L P P and the procedure is repeated. An optimum integer solution will be reached eventually after introducing enough new constraints to eliminate all the superior non integer solutions. The construction of additional constraints, called secondary or Gomory’s constraints, is so very important that it needs special attention.

MB0032

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