Short answers

1. Define “Operation Research”.

Ans:- Operation Research is defined as the application of scientific method by inter-disciplinary teams to problems involving the control of organized systems so as to provide solutions which best serve the purpose of the organization as a whole.

Or

The application of scientific & mathematical concepts methods, tools & techniques to the operations of a systems in order to obtain optimum solution to problem.

2. Explain “Unrestricted Variable”

Ans: Usually in an LP it is assumed that all the variables xj (j = 1,2 ,….,n) should have non-negative values. In many practical situations however one or more of the variables can have either positive, negative or zero value. Variable which can assume positive, negative or zero value are called unrestricted Variables.

3. What is “Redundant Constraints?”

Ans: A redundant Constraints is one that does not affect feasible solution region (or space) and thus redundancy of any constraints does not cause any difficulty in solving an LP problem graphically. In other words constraints appears redundant when it may be more binding (restrictive ) than another.

4.Explain the principle of “Dominance in Game Theory”

Ans: The rule of dominance are used to reduce the size of the payoff matrix. These rules help in deleting certain rows and columns of the payoff matrix which are inferior (less attractive) to at least one of the remaining rows and columns (strategies ) in terms of payoffs to both the players. Rows and Columns once deleted will never be used for determining the optimum strategy for both the players.

5.What is the importance of “Mixed Strategy”

Ans: Courses of action that are to be selected on a particular occasion with some fixed proability are called mixed strategy. Thus, there is a probabilities situation and objective of the players is to maximize expected gains or to minimize expected losses by making choice among pure strategies with fixed probabilities.

6.Write the limitation of “Operation Research”

1. Dependence on electronic computer

2. non-quantifiable factor

3. Distance between manger and operation researcher

4. implementation

5.Money and Time costs

7.What do you mean by “Convex Set”

Ans: A convex set is a set of elements from a vector space such that all the points on the straight line line between any two points of the set are also contained in the set. If a and b are points in a vector space the points on the straight line between a and b are given by

x = λa + (1-λ)b for all λ from 0 to 1.

The above definition can be restated as: A set S is convex if for any two points a and b belonging to S there are no points on the line between a and b that are not members of S. Another restatement of the definition is: A set S is convex if there are no points a and b in S such that there is a point on the line between a and b that does not belong to S. The point of this restatement is to include the empty set within the definition of convexity. The definition also includes singleton sets where a and b have to be the same point and thus the line between a and b is the same point.

8. What is “Non-degenerate basic feasible solution

Ans: A basic feasible solution xB = B-1 b is said to be degenerate if at least one component of xB (Basic variable ) is zero. If all components of xB are non-zero (XB > 0) then it is called a non- degenerate basic feasible solution.

9. What is the structure of LPP

Ans: Under this step given data relating to a problem will be arranged first under the following three types of equations functions

Objective Functions

Constraints functions

Non-negative functions

10. What is looping in transportation problem

Ans: It consists of horizontal and vertical lines with an allocation at each corner which in turn is a joint of horizontal and vertical line. In other words it is an ordered set of four or more cells in which any two adjacent cells lie either in the same row or in the same column. A feasible solution to a transportation problem is basic, if and only if the corresponding cell in the transportation table don not contain loop .

11. What is Free Float

Ans: It is the spare time variable when all preceding activities occur at the earliest possible times and all successding activities occur at the earliest possible times. In other wordsit is the delay possible for an activity if all preceding activities start as early as possible whilst all subsequent activities start at their

earliest time. An equivcalent and easier definition is the delay possible in an activity if it starts at its earliest time and all succeeding activities start their earliest possible time

Free float = Total float - slack time of the head event

12. What are Artificial Variable

Ans: One type of variables introduced in a linear progr4am model in order to find an initial basic feasible solution and artificial variable is used for equality constraints and for greater than or equal inequality constraints.

13. What do you mean by “Objective Function”

Ans: The objective of a problem may be either to maximize or minimize some results. If it is a case of profit, income or output the objective must be maximization. But if it is a case of loss, cost or input, the objective will be minimization. For this the rate of profit or cost per variable in issue must be assessed first and then the number of each variable will represented in the function through some letters X,Y to be ascertained through the process of solution.

14. What is Dynamic Programming

Ans: Often decision –making process involves several decions to be taken at different time. For examples, problems of inventory control, evaluation of investment opportunities, long- term corporate planning and so on require sequential decision making. The mathematical technique of optimizing a sequence of interrelated decisions over a period of time is called dynamic programming.

15. What is Sensitivity Analysis

Ans: The objective of sensitivity analysis is to determines the effect of change in one or more parameters of a linear programming problem on its optimum solution. The data used for formulating a linear programming problem is based on estimates of per unit cost or profit, time taken in various production processed, maximum machine capacities, demand estimated, labor supply etc. There may be an error in the estimation of one or more of these parameters. Sensitivity analysis shows how and to what extents such errors affect the optimum solution to the problem. It can also determines the ranges within which the optimum solution is not affected by an errors in the parameters of the problem.

16. What is “VAM”

Ans: Vogel’s Approximation Method (VAM)

Hence penalty (reward) refers to the different between the two best costs (best revenue) in each r4ow and column of the table. This differences is considered as penalty (reward ) for making allocations in the second lowest cost (second highest revenue) entries instead of the lowest cost (highest reward) entries in each row or column.

17. What is “EOQ”

Ans: Economic order quantity (EOQ)

The amount of order that minimizes total variables costs requires to order and hold inventory. The optimal replenishment order size (or lot size) of inventory item or items which achieve the optimum total (or variable ) inventory cost during the given period of time.

18. Explain the terms Reneging, Jockying and Balking .

Ans: Banlking : Customer don not join the queue either by seeing the number of customer already in service system or estimating the excessive waiting time ofr desired service.

Jockying: Customer move from one queue to another hoping to receive service more quickly.

Reneging: Customer after joining the queue wait for sometime in the queue but leave before being served on account of certain reasons.

19. What is “Hurwitz Decision Criterion

Ans: This Criterion suggests that a rational decision marker should be neither completely optimistic nor pessimistic and therefore must display a mixture of both. Hurwitz who suggest this criterion introduced the idea of a coefficient of optimism (denoted by α) to measure ot decision maker degree of optimism. This coefficient lies between 0 and 1 where 0 represents a complete pessimistic attitude about the future and 1 a complete optimistic attitude about the future.

H (Criterion of realism) = α (Maximum in column) + (1 - α) (Minimum in column)

20. Write about “M/M/I Model”

21. Write short note on “Duality Theorem”

Ans: Linear programming duality implies that each linear programming problem can be analysed in tow different ways but having equivalent solutions. Each LP problem (both maximization and minimization ) stated in its original form has associated with another linear programming problem (called dual linear programming problem or in short dual) which is unique based on the same data. In general it is immaterial which of the two problems is called primal or dual since the dual of the dual is primal.

22. Explain “Feasible Region” with examples

Ans: Linear programming is just graphing a bunch of linear inequalities. when you graph inequalities, you need to shade the "good" region - pick a point that is not on the line, put it in the inequality, and the it the point makes the inequality true (like 0 <5) shade the side with that point. If it isn't true, (like 0 > 5) shade the other side.

Do this for all your inequalities, and the overlapping part is the feasible region - the region on the plane that is true for ALL the inequalities involved.

23. List some factors affecting the safety stock level

Ans:

24. What do you mean by “Zero-sum Game”

Ans: Situation or interaction in which one participant's gains result only from another's equivalent losses. In decision theory, situation where one or more participants' gain (loss) equals the loss (gain) of other participants. Thus, a gain (loss) for one must result in a loss (gain) for one or more others. Also called constant sum game. See also negative sum game and positive sum game.25. Write down the difference between “order” and “Reorder Point”

Ans:

26. What do you mean by “Saddle Point”

Ans: In a second-order linear difference equation system, if one root has absolute value greater than one, and the other root has absolute value less than one, then the steady state of the system is called a saddle point. In this case, the system is unstable for almost all initial conditions. The exception is the set of initial conditions that begin on the eigenvector associated with the stable eigenvalue

27. Explain “Strategy” and “Mixed Strategy” in Game Theory.

Ans: The strategy for a player is the list of all possible actions (moves or courses of action ) that he will take for every payoff (outcome ) that might arise. It is assumed that the rules govering the choices are know in advance to the players. The outcome 4esulting form a particular choices is also kjnwo to the players in advance and is expressed in terms of numerical values.

Pure Strategy: It is the decision rule which is always used by the player to select the particular strategy (course and action). Thus, each player knowns in advance of all strategies out of which he always selects only one a particular strategy regardless of the other player’s strategy and the objective of the player is to maximize gains or minimize losses.

Mixed Strategy: Courses of action that are to be selected on a particular occasion with some fixed proability are called mixed strategy. Thus, there is a probabilities situation and objective of the players is to maximize expected gains or to minimize expected losses by making choice among pure strategies with fixed probabilities.

28. Explain “Degeneracy” in LPP

Ans: An LP is degenerate if in a basic feasible solution, one of the basic variables takes on a zerovalue. Degeneracy is caused by redundant constraint(s) and could cost simplex method extraiterations.29. Explain “Slack and “Surplus Variables”

30. Explain “Queuing Theory” and list it various models

31. Differentiate between “Lonic Models and “Analogue Model”

Ans:

32. Define “Feasible Solution”

Ans: Term in Linear Programming (LP) used to designate a solution that occurs at the corner point of the feasible region in a graph. According to a theorem in LP, one or a linear combination of the basic feasible solutions will turn out to be an optimal solution

It refers to a feasible solution to a transportation problem with ‘m’ sources and ‘n’ destination in which the number of positive allocation is less than the sum of the rows and column minus one.

33. Explain the significant of “Duality” in linear programming

Ans: Linear programming duality implies that each linear programming problem can be analysed in tow different ways but having equivalent solutions. Each LP problem (both maximization and minimization ) stated in its original form has associated with another linear programming problem (called dual linear programming problem or in short dual) which is unique based on the same data. In general it is immaterial which of the two problems is called primal or dual since the dual of the dual is primal.

34. What is “Sensitivity Analysis” in LPP

Ans: Sensitivity analysis is the study of knowing the affect on optimal solution of the LP model due to variations in the input coefficients (also called parameter) one at a time, whereas parameters analysis is the study of measuring the affect on optimal solution of the LP model due to simultaneous changes in the inputs coefficients as a function of one parameters.

35. Explain the importance of “Replacement Models”

Ans: The models of replacement is felt when the job performing units such as men, machine3s, equipments, parts. Become less effective or useless due to either sudden or gradual deterioration in their efficiency, failure and breakdown. By replacing them with new ones at frequent intervals, maintenance and other overhead costs can be reduced. However, Such Replacements Would Increase the need of Capital cost for new ones.

36. Write the steps to find the “Initial Feasible Solution” of a “Transportation problem” by “MODI Method”

Ans: Modified Distribution Method (MODI)

This method evaluation is made of each of the empty cells (unoccupied cells ) rather than the closed paths relating to each of the empty cells that is done under the stepping stone method. Here, we are to trace out only one closed path in relation to the cell that shows the highest negative net change in the cost of improvement. Thus a considerable time is saved under this method.

Step 1. Find an initial basic feasible solution for the problem consisting of m+n-1 stones under any of the three methods.(NWCM, LCM or VAM).

Step 2. Determines the net change in the cost of each of the empty cells by the following equations

Net change = Cij - (R +C)

Step 3. Evaluate the net change in the improvement cost of each of the empty cells. If the net change in the improvement cost of all the empty cell is either positive or zero, the optimal solution is arrived at and accordingly the minimum cost will be determined as per the said optimum table. On the other hand if any of the empty cells shows any negative value, it will indicate that the optimum solution is yet to be arrived at and accordingly the process of solution will be repeated further for optimizing the result.

37. State the “Principle of Optimality” in “Dynamic programming”

use dynamic programming the problem must observe the principle of optimality that whatever the initial state is remaining decisions must be optimal with regard the state following from therst decision_Combinatorial problems may have this property butmay use too much memory_time to be e_cient_

38. Show that the “Assignment Problem” is a special case of “Transportation problem”

Ans: Since an assignment problem is a special case of the transportation problem it can also be solved by transportation methods. However every basic feasible solution of general assignment problem having square payoff matrix of order n should have m + n – 1= n + n -1 =2n-1 assignment. But due to the special structure of this problem, any solution cannot have more than n assignment. Thus the assignment problem is inherently degenerate. In order to remove degeneracy, (n-1) number of dummy allocation will be required in order to proceed with the algorithm solving a transportation problem. Thus, the problem of degeneracy at each solution makes the transportation method computationally inefficient for solving an assignment problems.

39. What is Transshipment problem.

Ans: In transportation problem shipment of commodity takes places among sources and destinations. But instead of direct shipments to destinations. The commodity can be transported to a particular destination through one or more intermediate or trans-shipment points. Each of these points in turn supply to other points. Thus, when the shipments pass from destination to destination and from source to source, we have a trans-shipment problem.

40. Explain & difference between PERT and CPM

Ans: PERT CPM

Probabilistic Model with uncertaintyin activity duration. The duration of eachactivity is normally computed frommultiple time estimates with a view totake into account time uncertainty. Theseestimates are ultimately used to arrive atthe probability of achieving any givenscheduled date of project completion.

A Deterministic Model with wellknown activity (single) times basedupon the past experience, ittherefore, does not deal withuncertainty in time.

It is said to be event oriented as theresults of analysis are expressed in termsIt is activity oriented as the results ofcalculations are considered in terms ofof events or distinct points in timeindicative of progress.

activities or operations of the project.The use of dummy activities is requiredfor representing the proper sequencing.The use of dummy activities is notnecessary.

It is used for repetitive jobs It is used for non-repetitive jobs.It is applied mainly for planning andscheduling research programmes.

It is used for construction and businessproblems.

PERT analysis does not usually considercosts.

CPM deals with costs of project schedulesand their minimization. The concept ofCrashing is used mainly in CPM models

41. Difference between Transportations problem and Assignment Problem

Ans: ...total supply must equal total demand in the transportation problem.

...the number of origins must equal the number of destinations in the transportation problem.

...each supply and demand value is 1 in the assignment problem.

...there are many differences between the transportation and assignment problems.

42. What is unbalanced assignment problem.

Ans: The number of columns and rows in the assignment matrix be equal. However when the given cost matrix is not a square matrix, the assignment problem is called an unbalanced problem. In such cases a dummy rows and column are added in the matrix (with zeros as the cost elements) to make it a square matrix. For example when the given cost matrix is of order 4*3 , a dummy column would be added with zero cost element in that column. After making the given cost matrix a square matrix, The Hungarian method may be used to solve the problem.

43. Differentiate between Linear Programming and Dynamic programming

Ans: Dynamic programming is different from linear programming on two counts.

1. There does not exist a standard mathematical formulation of dynamic programming problem. Accordingly there is no algorithm, like the simplex method, that may be programming problem. Dynamic programming is instead, a technique that permits us to dissect difficult problems into a sequence of sub-problems which are then evaluated by stages. This provides a generalized approach to problem solving.

2. While linear programming is a method which given single stage, that is one time-period solutions dynamic programming has the power to determines the optimal solution over, say, one year time horizon by breaking the problem into twelve smaller one month time horizon problems and to solve each of these optimally. Thus it uses a multistage approach to problem solving.

3. In DP there is no set procedure as in LP to solve any decision-problem. DP technique allows to break the given problem into a sequence of easier and smaller sub problems which are then solved in a sequential order.

4. LP approach provides one time period solution to a problem whereas DP approach is useful for decision making over time and solves each sub problems optimally.

44. Discuss travelling salesman problem

Ans: The Travelling Salesman Problem (TSP) is a problem in combinatorial optimization studied in operations research and theoretical computer science. Given a list of cities and their pair wise distances, the task is to find a shortest possible tour that visits each city exactly once.

Given n cities and the distances dij between any two of them, we wish to find the shortest tour going through all cities and back to the starting sity. Usually the TSP is given as a G = (V,D) where V = {1, 2, . . . , n} is the set of cities, and D is the adjacency distance matrix, with ∀i, j ∈ V, i 6= j, di,j > 0, the probem is to find the tour with minimal distance weight, that starting in 1 goes through all n citiesand returns to 1.The TSP is a well known and difficult problem, that can be solved inO(n!) ∼ O(nne−n) steps.