Operation Research study material includes operation research notes , operation research books , operation research syllabus , operation research question paper, operation research case study, operation research questions and answers , operation research courses in operation research pdf form.
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Enter your email address to comment. Enter your website URL optional. Search this website Type then hit enter to search. Share via. Constraints: Restrictions placed on the firm by the operating environment stated in linear relationships of the decision variables.
Parameters: Numerical coefficients and constants used in the objective function and constraint equations. The formulation of the LPP as mathematical model involves the following key steps: Step 1.
Identify the decision variables to be determined and express them in terms of algebraic symbols as x1 , x2 ,. Step 2. Identify the objective which is to be optimized maximized or minimized and express it as a linear function of the above defined decision variables. Step 3. Identify all the constraints in the given problem and then express them as linear equations or inequalities in terms of above defined decision variables.
To produce one unit of A, 40kg iron, 30kg copper, 7kg zinc and 4kg manganese are needed. Similarly, to produce one unit of B, 70kg iron, 14kg copper and 9kg manganese are needed and for producing one unit of C, 50kg iron, 18kg copper and 8kg zinc are required. The total available quantities of metals are 1 metric ton iron, 5 quintals copper, 2 quintals of zinc and manganese each.
Formulate the problem mathematically. If the objective function z is a function of two variables then the problem can be solved by graphical method. The procedure is as follows. Step 1. First of all we consider the constraints as equalities or equations. Then we draw the lines in the plane corresponding to each equation obtained in step 1 and non-negative restrictions. Then we find the permissible region region which is common to all the equations for the values of the variables which is the region bounded by the lines drawn in step 2.
Step 4. Finally we find a point in the permissible region which gives the optimum value of the objective function. If there is no permissible region in a problem then we say that the problem has no solution using graphical method. If the maximum value of z appears at 2 points there exists infinite number of solutions to the LPP.
The Simplex method is an approach to solving linear programming models by hand using slack variables, tableaus, and pivot variables as a means to finding the optimal solution of an optimization problem. Such m variables are called basic variables and remaining n zero valued variables are called non basic variables.
Basic Feasible Solution are of two types,. Degenerate BFS: If one or more basic variables are zero. All the constraints with the exception of the non negativity restrictions on the variables are equations with non negative right hand side.
Step 1: The objective function of the given LPP is to be maximized if it is minimized convert into a maximisation problem by , Mimimum z. Step 2: All the bj should be non-negative. If some bj is negative multiply the corresponding equation by Step 3: The inequations of the constraint must be converted into equations by introducing slack or surplus variables in the constraints.
The cost of these variables are taken to be zero. Step 4: Find an initial basic feasible solution and formulate the simplex table as follows. CB1 s1 b1 a11 a CB2 s2 b2 a21 a CB3 s3 b3 a31 a Cj - Coefficient of objective function. CB - Coefficient of basic variables. YB - Basic variable. XB - Value of the basic variable. The entering column is know as key column or pivoting column. The leaving row is know as key row or pivoting row.
The intersection element of entering column and leaving row is called as key element or pivoting element or leading element. The vitamins available from a mango are 0. If the cost of a mango, an orange and an apple be Rs 0. Subject to 0. Type A vehicle can carry 7 tons solid and 3 tons liquid whereas B and C can carry 6 tons solid and 2 tons liquid and 3 tons solid and 4 tons liquid respectively. Find the minimum cost of transportation. How do we determined the solution of the LPP by?
Objective function B. Decision Variables C. Constraints D. Opportunity costs. In solving LPP by graphical method, the region of intersection of constraint functions is called as A. Infeasible region B. Unbounded region C. Infinite region D. Feasible region. Corner points of feasible region B. Both a and c C. A triangle B. A rectangle C. An unbounded region D. An empty region. The feasible region of a linear programming problem has four extreme points: A 0,0 , B 30,0 , C 24,8 , and D 0, How the entering variable column is denoted in the simplex algorithm?
How the leaving variable row is denoted in the simplex algorithm? What is the intersection element of pivoting column and pivoting row? The non? Slack variables B. Surplus variables C. A sequence is the order in which the jobs are processed. Sequence problems arise when we are concerned with situations where there is a choice in which a number of tasks can be performed. A sequencing problem could involve:. JOB : The jobs or items or customers or orders are the primary stimulus for sequencing.
There should be a certain number of jobs say n to be processed or sequenced. It may not be necessarily a mechanical device. Even human being assigned jobs may be taken as machines. There must be certain number of machines say k to be used for processing the jobs. If the time is certain then the determination of schedule is easy.
When the processing times are uncertain then the schedule is complex. Total Elapsed Time : It is the time between starting the first job and completing the last one.
Technological order : different jobs may have different technological order. It refers to the order in which various machines are required for completing the jobs e. Here the objective is to find out the optimum sequence of the jobs to be processed and starting and finishing time of various jobs through all the machines. The processing times on different machines are independent of the order in which they are processed.
The time involved in moving a job from one machine to another is negligibly small. Each job once started on a machine is to be performed up to completion on that machine. A job is processed as soon as possible but only in the order specified.
Step 1: All jobs are to be listed, and the processing time of each machine is to be listed. Ties in activity times can be broken arbitrarily. Step 4: Repeat steps2 and step3 to the remaining jobs, working towards the centre of the sequence.
Processing times are given in the following table:. Determine a sequence in which these jobs should be processed so as to minimize the total processing time. There are manuscripts of a number of different books.
Processing times for printing and binding are given in the following table:. Determine the sequence in which books should be processed on the machines so that the total time required is minimized. Let t11 , t12 , t13 ,. A mathematical model executive departments with a quantitative should include the following three important basic basis for decision regarding the operations factors under their control.
Phase III: Deriving the solutions from the model: This 3 OR is the application of scientific methods, phase is devoted to the computation of those values of techniques and tools to problems involving the decision variables that maximize or Minimize the operations of systems so as to provide these in objective function.
Such solution is called an optimal society in control of the operations with solution which is always in the best interest of the optimum solutions to the problem. The general techniques 4 OR is a scientific approach to problem solving for deriving the solution of OR model are discussed in for executive management. A good practitioner of OR and society. The model requires immediate be able to formulate the problem in the form of an modification as the controlled variables one or more appropriate model.
To do so, the following information changes significantly; otherwise the model goes out of will be required. As the conditions are constantly changing in 1 Who has to take the decision? Phase VI: Implementing the solution: Finally, the tested results of the model are implemented to work.
This phase is primarily executed with the cooperation of OR experts and those who are responsible for managing and operating the systems. We have to passed away from all above mentioned phases we get an appropriate model to solve the Linear Programming Problem.
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