Notes Of Quantitative Techniques 3vn60

NOTES OF QUANTITATIVE TECHNIQUES

(1) Assignment is a type of linear programming problem. The assignment problem can be stated as a problem where different jobs are to be assigned to say different machines on the basis of the cost of doing these jobs. The objective is to minimize the total cost of doing all the jobs on different machines. The peculiarity of the assignment problem is only one job can be assigned to one machine i.e., it should be a one-to-one assignment. Assignment becomes a problem because each job requires different skills and the capacity or efficiency of each person with respect to these jobs can be different. (2) Complete Enumeration Method: This method can be used in case of assignment problems of small size. In such cases a complete enumeration and evaluation of all combinations of persons and jobs is possible. One can then select the optimal combination. The number of combinations increases manifold as the size of the problem increases. Hence the use of enumeration method is not feasible in real world cases. (3) Special problems in assignment : Some special type of assignment problems which are solved by using assignment methods are maximization case, unbalanced case, restricted assignments and travelling salesman problem. (4) Unbalanced assignment problem: There is a possibility that the number of rows may not be equal to the number of columns and vice versa. In such a situation, we introduce dummy row(s) or column(s) in the matrix. These rows or columns have a zero cost element. Thus we can balance the problem and then use Hungarian Method to find optimal assignment. (5) Restricted or constrained assignment problem : In practice , it is possible that a machine may not be in a position to perform a particular job as there may be some technical difficulties in using a certain machine for a certain job. In such cases the assignment is constrained by the given facts. To handle this problem of restriction on job assignment we assign a very high cost M to the constrained job-machine combination. 1

(6) Travelling salesman problem : Travelling salesman problem is a routine problem. It can be considered as a typical assignment problem with certain restrictions. Consider a salesman who is assigned the job of visiting n different cities. He knows the distances between all pairs of cities. He is asked to visit each of cities only once. The trip should be continuous and he should come back to the city from where he started using the shortest route. It does not matter from which city he starts. These restriction imply: (1) No assignment should be made along the diagonal. (2) No city should be included on the route more than once. (7) Maximization problem in assignment: But in some cases the assignment problem may include profits or revenue or sales. In such cases the assignments should be made in such a manner that these profits or sales are maximized. In order to solve a maximization case we find the regret values instead of opportunity cost. The problem can be solved in two ways: (1) Put a negative sign before the values in the assignment matrix. i.e. multiply each element of the matrix by -1 and then solve the problem as a minimization case using Hungarian Method. (2)Locate the largest value in the given matrix and subtract each element in the matrix from this value. Then solve the problem as a minimization case using this new modified matrix. INTRODUCTION OF OPERATION RESEARCH (1)CHARACTERISTICS OF OPERATIONS RESEARCH TECHNIQUE: (i) It has an inter-disciplinary team approach, to find the optimum solution to a problem. (ii) It uses scientific approach to arrive at the solution. (iii) It emphasizes on an overall approach to the system under consideration. (iv) It can handle maximization as well as minimization problems. (v) Most important of all it addresses itself directly to the decision making problem. (2) STEPS OF PROBLEM SOLVING APPROACH IN OPERTION RESEARH: The problem solving approach in operations research consist of following steps: (i) Recognise the existence of the problem. 2

(ii)

Formulate the problem after identifying the variables and their relationships. (iii) Determine appropriate quantitative tools to analyse the problem. (iv) Collect appropriate solution. (v) Obtain the solutions and interpret the results. (vi) Choose the appropriate solution. (vii) Implement the chosen solution. (3) APPLICATIONS OF OPERATIONS RESEARCH TECHNIQUES: Operations research approach gives an insight, into the problem areas , to the manager. He is then able to obtain a better solution to the problem under consideration. There is hardly any area in business which does not benefit from use of operations research techniques. Few areas where these techniques applies are: (i) Financial Management (ii) Production Management (iii) Marketing Management (iv) ing Management (v) Human Resource Development (4) ADVANTAGES OF OPERATIONS RESEARCH TECHNIQUES: (i) It needs a clear statement of the objectives, assumptions behind the model and their constraints. (ii) The manager, therefore, is forced to be precise in defining the relationships between the variables. (iii) It can point out gaps in the data. (iv) It helps in finding solution to a complex problem. (v) Use of computer in solving operations research problem has facilitated its application. It is also possible to revise a solution if necessary. (5) LIMITATIONS OF OPERATIONS RESEARCH : (i) They fail to represent the actual degree of complexity present in a given situation. (ii) Mathematical model representing the given situation fails to give enough weightage to human factors like a worker’s skill, customer’s satisfaction, ability of a manager. (iii) It is very expensive to build the model. A small firm cannot afford the expenses of running a model on a computer as computer time is also expensive. (iv) Persons using the model and building the model are different. Hence 3

It is very difficult sometimes to correctly interpret. (v) Sometime the manager may unknowingly use an incorrect value of the variable. (vi) Sometimes the model is not representative of the real situation and hence not workable. (vii) If the real world problems are very complex then it is difficult to build a truly representative model. (7)LIST OF OPERATIONS RESEARCH TECHNIQUES:These techniques are called Programming techniques which are a valuable guide to the decision maker. They are: (i) Decision Theory. (ii) Inventory Models. (iii) Linear Programming. (iv) Integer Programming. (v) Heuristic or rules of thumb. (vi) Simulation. (vii) Queing Theory. (viii) Networking Scheduling. LINEAR PROGRAMMING (1) DEFINITION: It is the analysis of problems in which a linear function of a number of variables is to be maximized or minimized, when those variables are subject to a number of restraints in the form of linear inequalities. (2) APPLICATION OF LP TECHNIQUE: A wide variety of problems can be efficiently solved using LP techniques. Few examples are: (i) A product mix problem. (ii) Allocation of bank funds. (iii) Manufacturing problem. (iv) Advertising application. (v) Portfolio selection (vi) Staffing problem. (vii) Trim loss problem. (3) COMPONENTS OF LP: A typical linear programming has the following components: (i) An objective function. 4

(ii) (iii)

Constrains or restrictions. Non-negativity restriction.

(4)DECISION VARIABLES :Decision variables are the unknowns whose values are to be determined from the solution of the problem. E.g. decision variables in the furniture manufacturing problem are say the tables and chairs whose values are actual units of production are to be from the solution of the problem. (5)OBJECTIVE FUNCTION: A firms objectives are expressed as function of decision variables. It represents the mathematical equation of the goals of the firm in of unknown values of the decision variables. Thus the objective is to maximize profit or minimise cost. (6)CONSTRAINTS : A constraint represents the limitations imposed on the values of decision variables in the solution. These limitations exist due to limited availability of resources as well as the requirements of the resources in the production of each unit of the decision variable. Thus the constraints define the limits within which a solution to the problem must be found. (7) NON-NEGATIVITY RESTRICTIONS :Linear Programming technique is used to obtain solution to real world problems. The solution to the problem implies finding values of the decision variables. These must be non – negative. As one cannot think of manufacturing or having negative production. (8) FORMULATION OF LINEAR PROGRAMMING PROBLEM (LPP): It can be done only if the following requirements are met: (i) There should be a clearly identifiable objective and it should be measurable in quantitative . (ii) The resources to be allocated in the problem should be identifiable and quantitatively measurable. (iii) The relationships representing the objective function and the constraints equations must be linear. (iv) There should be a series of feasible alternative courses of action available to the decision maker. (9) INFEASIBILITY : It is a case where there is no solution which satisfies all the constraints at the same time. This may occur if the problem is not correctly formulated. Graphically infeasibility is a case where there is no region which satisfies all constraints simultaneously.

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(10) UNBOUNDEDNESS :A LPP can fail to have an optimal solution if the objective can be made infinitely large without violating any of theconstraints. Unbounded solution results if in a maximisation problems all constraints are “ greater than or equal to type.”In such a situation there will be no upper limit on the feasible region. (11) REDUNDANCY : A constraint which does not affect the feasible region is called a redundant constraint. Such a constraint is not necessary for the solution of the problem. It can , therefore be omitted while formulating the problem. (12) ALTERNATIVE OPTIMA: This solution is given by an extreme point of the feasible region. The uniqueness implies that no other solution to the given problem shall give the same value of the objective function as given by the unique solution. SIMPLEX

(1)

SIMPLEX TABLEAU : A table used to keep track of the calculations made at each iteration of the simplex procedure and to provide basis for tableau revision

. (2)BASIC VARIABLES : The variables with non-zero values which make up the basis are called basic variables and the remaining variables are called non-basic variables. (3)ITERATION : A sequence of steps taken in moving from one basic feasible solution to another basic feasible solution. (4)KEY OR PIVOT COLUMN : The column with the largest positive and it indicates which variable will enter the next solution in a maximization case. (5)KEY OR PIVOT ROW: The row with the smallest positive value of the replacement ratio of the constant rows.

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(6)REPLACEMENT RATIO : The replacement ratio is obtained by dividing elements in the solution column by the corresponding elements in the key column. (7)KET 0R PIVOT ELEMENT: The element at the intersection of key row and key column. (8)SLACK VARIABLE : A variable used to convert a less than or equal constraint ( ≤ ) into equality constraint. It is added to the left hand side of the constraint. (9)SURPLUS VARIABLE : A variable used to convert a greater than or equal to type(≥) constraint. It is subtracted from the left hand side of the constraint. (10) ARTIFICIAL VARIABLE : It is a variable added to greater than or equal type(≥) constraint. This is in addition to surplus variables used. (11)DUALITY IN LP :Corresponding to every linear programming problem, there is another linear programming problem. The given problem is called the primal and the other its dual. When the primal problem is of the maximization type the dual is of the minimization type and vice versa. (12)SHADOW PRICES : The shadow prices are also called the marginal value products or marginal profitability of the resources. TRANSPORTATION (1)Transportation problem: It is concerned with transportation methods or selecting routes in a product distribution network among the manufacturing plants and distribution warehouses situated in different regions or local outlets. In applying the transportation method, management is searching for a distribution route, which can lead to minimization of transportation cost or maximization of profit. (2)Formulation of a transportation problem: It is based on the following assumptions: (i) Number of sources or origins is finite and given. (ii) Number of destinations is also finite and given. (iii) Supply at each source and demand at each destination is given for a given time. (iv) Total demand equals total supply, at the given point of time.

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(v)

Cost of transportation a unit from each source to each destination is given and constant.

(3)STEPS IN TRANSPORTATION PROBLEM SOLVING : (i) Set up a transportation tableau. (ii) Develop initial feasible solution. (iii) Test the solution for optimality or improvement. (iv) Develop the improved solution if the earlier solution is non optimal. (4) DEGENERACY: The initial basic feasible solution to a transportation problem should have a total number of occupied cell which is equal to the total number of rim requirements minus one i.e. m+ n – 1. When this rule is not met the solution is degenerate. PERT AND M (1) Critical Path method: M can be used in project planning when activity time can be estimated with a fair degree of certainty and costs also can be calculated accurately in advance. (2) PERT( PROGRAM EVALUATION REVIEW TECHNIQUE): It is useful for analyzing project scheduling problem when the completion time of different activities is not certain. It is used when time variation is inherent in the projects. (3)

USEFULNESS OF NETWORK MODELS : (i) Planning the flow of traffic to minimize congestion in big cities. (ii) Planning the shortest pickup and delivery routes for package handling companies. (iii) To find the best layout of , for, say water system in residential area. (iv) To plan the network of railways to connect important destinations in a huge country. (v) Ship or aircraft repair project. (vi) Satellite mission development and countdown programme. (vii) Deg and laying of new manufacturing plant.

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(4) ACTIVITIES : An activity is an task which has a definite beginning and a definite end . Activities start from left and end to the right. The tail or left hand side of the arrow represents the beginning of an activity and its head represent the completion of the activity. The length of the arrow is not related to the duration or time required for completion of the activity. It can be vertical or horizontal or even bent. (5) EVENTS OR NODES : The specific point of time at which an activity begins or ends is called a node. They are called tail nodes and head nodes respectively. (6) NETWORK : It is a diagrammatic representation of all the activities and nodes of a project’s operations. A project is made up of several activities. (7) DUMMY ACTIVITY :It is an activity which is used to establish a precedence relationship in a network. It does not require any time. It is denoted by broken lines. (8) STEPS IN THE USE OF PERT/M : (i) Identify activities (ii) Establish logical relationship between these activities. (iii) Construct a network with an arrow diagram. (iv) Number the events using Fulkerson’s rule. (v) Estimate time duration of each activity of the project. (vi) Find the critical path and critical activities. (vii) Perform time analysis and activity float analysis. (viii) Crash the project. (ix) Study resource allocation and resource leveling. (x) Prepare a time schedule of activities and monitor of the progress of the project. (9) CRITICAL PATH: It is the longest path in network. It determines the total time required for the completion of the project. It is shown in the network by a double or thick line or a red line. (10)THREE TIME ESTIMATES : (i) A small probability of reaching the “most optimistic time” represented by letter ‘a’. (ii) A small probability of reaching the “most pessimistic time” symbolized by letter ‘b’. (iii) One and only one” most likely time” denoted by ‘m’. (11) EXPECTED TIME : It is the weighted average of a, b, m and is given by te = (a + 4m + b) /6

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