DRIVE SPRING 2015PROGRAM- MBADS/ MBAFLEX/ MBAHCSN3/ MBAN2/ PGDBAN2SEMESTER- 2SUBJECT CODE & NAME- MB0048 Operations ResearchBK ID- B1631

OUESTION 1. Discuss the methodology of Operations Research. Explain in brief the phases of Operations Research.Ans: Operations Research means the use of scientific methods to provide criteria for decisions regarding man, machine, and systems involving repetitive operations and with application of scientific methods, techniques and tools to the operation of a system is to get optimum solutions to the problems. Operations Research MethodologyOperations Research methodology consists of five steps. a) Defining the problemThe first and the most important step in the Operations Research approach of problem solving is to define the problem. One needs to ensure that the problem is identified properly because this problem statement will indicate the following three major aspects: Description of the goal or the objective of the study Identification of the decision alternative to the system Recognition of the limitations, restrictions, and requirements of the systemb) Model ConstructingBased on the problem definition, you need to identify and select the most appropriate model to represent the system. While selecting a model, you need to ensure that the model specifies quantitative expressions for the objective and the constraints of the problem in terms of its decision variables. A model gives a perspective picture of the whole problem and helps in tackling it in a well-organized manner. c) Model SolutionA solution to a model implies determination of a specific set of decision variables that would yield an optimum solution. An optimum solution is one which maximizes or minimizes the performance of any measure in a model subject to the conditions and constraints imposed on the model.d) Model ValidationA model is a good representation of a system. However, the optimal solution must work towards improving the systems performance. You can test the validity of a model by comparing its performance with some past data available from the actual system. e) Result ImplementationYou need to apply the optimal solution obtained from the model to the system and note the improvement in the performance of the system. You need to validate this performance check under changing conditions. Phases of Operations Research Judgment phaseThis phase includes the following activities: Determination of the operations Establishment of objectives and values related to the operations Determination of suitable measures of effectiveness Formulation of problems relative to the objectives Research phaseThis phase utilizes the following methodologies: Operation and data collection for a better understanding of the problems Formulation of hypothesis and model Observation and experimentation to test the hypothesis on the basis of additional data Analysis of the available information and verification of the hypothesis using pre-established measure of effectiveness Prediction of various results and consideration of alternative methods Action phaseThis phase involves making recommendations for the decision process. The recommendations can be made by those who identify and present the problem or by anyone who influences the operation in which the problem has occurred.QUESTION 2: a. Explain the graphical method of solving Linear Programming Problem.b. A paper mill produces two grades of paper viz., X and Y. Because of raw material restrictions, it cannot produce more than 400 tons of grade X paper and 300 tons of grade Y paper in a week. There are 160 production hours in a week. It requires 0.20 and 0.40 hours to produce a ton of grade X and Y papers. The mill earns a profit of Rs. 200 and Rs. 500 per ton of grade X and Y paper respectively. Formulate this as a Linear Programming Problem.

ANS:

Linear programming is a mathematical technique designed to help managers in their planning and decision making. It is usually used in an organization that is trying to make the most effective use of its resources. Resources typically include machinery, manpower, money, time, warehouse space and raw materials.While obtaining the optimal solution to an LPP by the graphical method, the statement of the following theorems of linear programming is used: Step 1) Formulate the problem in terms of a series of mathematical equations representing objective function and constraints of LPP.Step 2) Plot each of the constraints equation graphically. Replace the inequality constraint equation to form a linear equation. Plot the equations on the planar graph with each axis representing respectable variables.Step 3) Identify the convex polygon region relevant to the problem. The area which satisfies all the constraints simultaneously will be the feasible region. Step 4) Determine the vertices of the polygon and find the values of the given objective function Z at each of these vertices. Identify the greatest and least of these values. These are respectively the maximum and minimum value of Z.Step 5) Identify the values of (X1, X2) which correspond to the desired extreme value of Z. This is an optimal solution of the problem.Objective functions and constraints :Objective functions and constraints are formulated from information extracted from the problem statement. The managements dilemma is to optimise the output or the objective function subject to the set of constraints. Optimisation of resources in which both the objective function and constraints are represented in a linear form is called Linear programming Problem.Formulation 1) Study the given situation to find the key decisions to be made2) Identify the variables involved and designate them by symbols Xj(j=1,2....)3) State the feasible alternatives which generally are Xj 0 for all j4) Identify the constraints in the problem and express them as linear inequalities or equations, LHS of which are linear functions of the decision variables.5) Identify the objective function and express it as a linear function of the decision variables.

b) Formulation of LPP (Objective function & Constraints)

Let x1 and x2 be the numbers of units of two grades of X and YSince the profit for the two grades of paper X and Y are given, the objective function is to maximize the profit MAX (Z)=200X1+500X2There are 2 constrains one w.r.t. to raw materials and the other w.r.t to production hours.The complete LPP isMax(Z)=200X1=500x2Sunject to X1= 0, then it is the optimal solution.Step 7 if any ij= 0.

QUESTION 4: a. Explain the steps involved in Hungarian method of solving Assignment problems.ANS:Hungarian method algorithm is based on the concept of opportunity cost and is more efficient in solving assignment problems. The following steps are adopted to solve an AP using the Hungarian method algorithm. Step 1: Prepare row ruled matrix by selecting the minimum values for each row and subtract it from the other elements of the row. Step 2: Prepare column-reduced matrix by subtracting minimum value of the column from the other values of that column. Step 3: Assign zero row-wise if there is only one zero in the row and cross (X) or cancel other zeros in that column.Step 4: Assign column wise if there is only one zero in that column and cross other zeros in that row.Step 5: Repeat steps 3 and 4 till all zeros are either assigned or crossed. If the number of assignments is equal to number of rows present, you have arrived at an optimal solution, if not, proceed to step 6. Step 6: Mark () the unassigned rows. Look for crossed zero in that row. Mark the column containing the crossed zero. Look for assigned zero in that column. Mark the row containing assigned zero. Repeat this process till all the makings are done. Step 7: Draw a straight line through unmarked rows and marked column. The number of straight line drawn will be equal to the number of assignments made. Step 8: Examine the uncovered elements. Select the minimum. Subtract it from the uncovered elements. Add it at the point of intersection of lines. Leave the rest as is. Prepare a new table.Step 9: Repeat steps 3 to 7 till optimum assignment is obtained. Step 10: Repeat steps 5 to 7 till number of allocations = number of rows.

b. Find an optimal solution to an assignment problem with the following cost matrix:J1J2J3J4

M110978

M25877

M35465

M42345

ANS:First, the minimum element in each row is subtracted from all the elements in that row.This gives the following reduced-cost matrix.J1J2J3J4

M13201

M20322

M31021

M40123

Since both the machines M2 and M4 have a zero cost corresponding to job J1 only, a feasible assignment using only cells with zero costs is not possible. To get additional zeros subtract the minimum element in the fourth column from all the elements in that column. The result is shown in the table below.

J1J2J3J4

M13200

M20321

M31020

M40122

Only three jobs can be assigned using the zero cells, so a feasible assignment is still not possible. In such cases, the procedure draws a minimum number of lines through some selected rows and columns in such a way that all the cells with zero costs are covered by these lines. The minimum number of lines needed is equal to the maximum number of jobs that can be assigned using the zero cells.

J1J2J3J4

M14200

M20210

M32020

M40011

A feasible assignment is now possible and an optimal solution is assignedM1 to J3M2 to J1M3 to J4M4 to J2The total cost is given by: 7 +5+ 5+ 3 = 20An alternate optimal solution is:M1 to J3M2 to J4M3 to J2M4 to J1QUESTION 5: a. Explain Monte Carlo Simulation.MONTE CARLO SIMULATIONMonte Carlo simulation is useful when same elements of a system, such as arrival of parts to a machine, etc., exhibit a chance factor in their behavior. Experimentation on probability distribution for these elements is done through random sampling. Following five steps are followed in the Monte Carlo simulation:Procedure of Monte Carlo Simulation:1. Decide the probability distribution of important variables for the stochastic process.2. Calculate the cumulative probability distributing for each variable in Step 13. Decide an interval of random numbers for each variable.4. Generate random numbers.5. Simulate a series of trials and determine simulated value of the actual random variables.

b. A Company produces 150 cars. But the production rate varies with the distribution.Production Rate147148149150151152153

Probability0.050.100.150.200.300.150.05

At present the track will hold 150 cars. Using the following random numbers determine the average number of cars waiting for shipment in the company and average number of empty space in the truck. Random Numbers 82, 54, 50, 96, 85, 34, 30, 02, 64, 47.ANS:Production RateProbabilityCumulative probabilityRN Range

1470.050.0500 04

1480.100.1505 14

1490.150.3015-29

1500.200.5030 49

1510.300.8050 79

1520.150.9580 94

1530.051.0095 99

Simulation for 10 days using the given random numbersDaysRNProduction RateCar Waiting Space in the truck

1821522

254150-

350150-

4961533

5851522

634150-

730150-

802147-3

9641511

10471502

Total 8 3Therefore, Avg number of cars waiting =8/10= 0.8 /dayAvg number of empty space =3/10= 0.3/day

QUESTION 6:a. Explain the dominance principle in game theory.b. Describe the Constituents of a Queuing System.c. Differentiate between PERT and CPM

ANS: Dominance principle in game theory.In game theory, strategic dominance (commonly called simply dominance) occurs when one strategy is better than another strategy for one player, no matter how that player's opponents may play. Many simple games can be solved using dominance. The opposite, intransitivity, occurs in games where one strategy may be better or worse than another strategy for one player, depending on how the player's opponents may play.In a rectangular game, the pay-off matrix of player A is pay-off in one specific row (rth row ) exceeding the corresponding pay-off in another specific row (sth row ). This means thatwhatever course of action is adopted by player B, for A, the course of action Ar yields greater gains than the course of action As . Therefore, Ar is a better strategy than As irrespective of Bs strategy. Hence, you can say that Ar dominates As .Alternatively, if each pay-off in a specific column ( pth column) is less than the corresponding pay-off in another specific column ( qth coloum) , it means strategy Bp offers minor loss than strategy Bq irrespective of As strategy. Hence, you can say that Bp dominates Bq . Therefore, you can say that: a) In the pay-off matrix, if each pay-off in rth row is greater than (or equal to) the corresponding pay-off in the sth row , Ar dominates As .b) In the pay-off matrix, if each pay-off in pth coloum is less than (or equal to) the corresponding pay-off in the qth coloum ,Bp dominates Bq .At times, a convex combination of two or more courses of action may dominate another course of action. Whenever a course of action (say As or Bq ) is dominated by others, then that course of action ( As or Bq ) can be deleted from the pay-off matrix. Such a deletion will not affect the choice of the solution, but it reduces the order of the pay-off matrix. Successive reduction of the order using dominance property helps in solving games.

The Constituents of a Queuing System.The constituents of a queuing system include arrival pattern, service facility and queue discipline. Arrival pattern: It is the average rate at which the customers arrive. Service facility: Examining the number of customers served at a time and the statistical pattern of time taken for service at the service facility. Queue discipline: The common method of choosing a customer for service amongst those waiting for service is First Come First Serve.

Differentiate between PERT and CPMPERTCPM

1PERT was developed in connection with an Research and Development (R&D) work. Therefore, it had to cope with the uncertainties that are associated with R&D activities. In PERT, the total project duration is regarded as a random variable.CPM was developed in connection with a construction project, which consisted of routine tasks whose resource requirements and duration were known with certainty. Therefore, it is basically deterministic.

2It is an event-oriented network as in the analysis of a network, emphasis is given on the important stages of completion of a task rather than the activities required to be performed to reach a particular event or task.CPM is suitable for establishing a trade-off for optimum balancing between schedule time and cost of the project.

3PERT is normally used for projects involving activities of non-repetitive nature in which time estimates are uncertain.CPM is used for projects involving activities of repetitive nature.