OPERATIONS REASERCH

1. OPERATIONS RESEARCH Unit – I Introduction OPERATIONS REASERCH UNIT -1

2. Origin of OR • Complexity and specialization increases ,it becomes difficult to allocate available resource to various activity. • These type of problems can be solved by emergence of operation technique. • In the Second world war operation research come to world of complexity. • The calculation of OR done by computer.

3. Various other names • Operational Analysis • Operations Evaluation • Systems Analysis • Systems Evaluation • Systems Research • Management Science

4. IndianIndustries making use of OR.. • Indian railways • Indian Airlines • Defense Organizations • Hindustan Lever • Tata Steel……….

5. Definition of OR • Scientific method of providing executive departments with a quantitative basis for decision regarding the operations under their control. • It is the art of giving bad answers to problems to which otherwise worse answers are given. • OR is a scientific approach to problem solving for executive management.

6. Optimization means… • A decision, which taking into account all the present circumstances can be considered the best one, is called an optimal decision.

7. Nature of OR • Operation research means research in operation. • Operation means any activity in the environment. • OR is used to solve problem in transportation ,manufacturing ,health ,construction ,telecommunication ,financial planning ,military and public services

8. Process begins with careful observation. • Gather data • Construct a mathematical model that represents your problem. • Find the solution • If not satisfied change the assumptions • OR must provide quantitative results. • Mainly concentrate on the objectives of an organization. • Always try to get Best solution from many solution.

9. Phases of OR • Definition of the problem • Formulating a mathematical model • Solution of the model • Testing/Validation of the model • Preparing to apply the model • Implementation of the solution

10. Defining a problem • This is the first step in OR process. • Study the system and prepare well defined statement of problem. • Includes Objectives ,constraints , relationship between area to be studied and other area. • Is a crucial step in OR

11. Team will presents assumption and alternatives to management • Management also takes decision. • Defining a problem means setting up objectives. • Objectives are formulated for entire organization and not for individual.

12. Formulating Mathematical model • Is representation of the problem in terms of some equations that is convenient for analysis. • If there is ‘n’ related quantifiable decision to be made they are represented as “decision variables” are represented as x1,x2,…..xn. who’s values to be determined

13. The appropriate measure of performance (eg. Profit) is expressed as a mathematical function of these decision variables(P=3x1+4x2…….Nxn) this function is called “Objective function” • Any restrictions to value assigned to the decision variable are called “Constraints” • They are represented as inequalities. • Constant in the objective function and constraints are called “parameter”. f(x1,x2)=3x1+4x2-----objective function x1+x2<=100,2x1+x2>=50----constraints

14. Deriving solution from the model • Develop procedure to for solution to the problem. • Theme of OR is optimal solution. • Solve by Linear programming ,Graphical method , simplex methods etc. • OR occasionally uses Heuristic procedure. • Post optimal Analysis to be done which includes sensitive analysis.

15. Testing the solution • The solution is been tested for bugs and get a valid result. • Some of the parameters are not been estimated properly every time . • This type of improvement in the model known as validation. • Retro positive test-uses historical data

16. Preparing to apply model • Once we test the model now it is ready to install in the organization. • Includes model , solution procedure and operating procedure for implementation. • This system is actualy computer based. • Includes data base too.

17. Implementation • The last phase of OR • Translating real problem into computer system.

18. Characteristics of OR • Existence of problem • Intension to solve problem • Application of system concept and system analysis to the problem. • Scientific approach to the problem where research methods are used • Formation of group consist of different specialist. • Solution must be optimal • Solution must be more appropriate. • Solution must be in measurable terms • Solution must meet the objective

19. Objectives of OR • It is decision making and improve quality. • Identification of optimum solution • Integrating the system as whole • Improve the objectivity of analysis and classify of thoughts. • Minimize the cost and maximize the profit. • Improves the productivity and efficiency. • Success in the competition and market leadership.

20. Functions of OR • Provides scientific basis to decision making. • Reduce the complex problem to a number of problem. • Provide system integration. • Optimization of resource • Maximize the work time • Minimize the cost • maximize the .profit • Select the best alternatives • Solution of specific problem.

21. Application Of OR • Agriculture • Finance • Industry • Marketing • Personnel Management • Production Management • public Administration • Town planning • Education training • Business Management • Research and development

22. Limitation of OR • Information Gap • Qualification technique • finite variable • Limited number of Constraints • single objective • Many real world problem cannot have OR • Often solution is obtained by assumption so solution will have limitation.

23. Linear programmingUNIT -2 • Linear programming is very important tool of operation research. • Developed during the world war II • Trial and error method used for solution • No satisfied solutions • LPP gives some certainty .

24. Every business has limited resources (men,meterial, labour ,money etc.). • But wants to achieve its main objective of getting the maximum profit. • There may be a number of alternative. • LPP gives best possible solution. • LPP deals with the optimization(minimization and maximization) of function of variable know as objective function.

25. It is subjected to set of linear equalities and/or inequalities knows as constraints. • LPP has two basic part : objective function which describe the primary purpose (max and min) • second part is constraint set: restrictions under which optimization to be made. • Definition: It is method of of planning where by some objective function is minimized or maximized while at the same time satisfying the various restriction placed on the potential solutions

26. Definition 2: Linear programming is a planning technique that permits some objective functions to be minimized or maximized with in the frame work of given situational restrictions. • Formulation of LP problems: • Objective functions: An Objective function is some sort of mathematical relationship between variable under consideration. This relationship is linear in LPP.

27. Decision variable: Are variables that seek to determine. • Represented as X1,X2,……,Xn. • They are activities. • In other way that decided your problem • That are need to be measured.

28. Constraints on the variable : Objective functions are optimized under certain restrictions imposed on the variable or some combination of few of all the variables occurring in the function. Feasible solutions: A set of values of x1,x2,,…..xn which satisfies the constraints and non negativity restriction is called “Feasible solution” A feasible solution that optimizes the objective function is known as “optimal solution”

29. Assumption of LPP • In all mathematical model we need to have some assumptions to reduce complexity of problems. • Certainty • Additively • proportionality • divisibility • and mutual exclusive ,finiteness ,non negative constraints

30. Certainty: • It is assumed that all model parameters availability resource ,profit ,contribution of decision variable .consumption of resource must be known and it is constant. • If a random variable represented by distribution may tend to change then it is solved by stichistic LP model or parametric programming. • “The value assigned to each parameter of a linear programming model is assumed to be a known constant.”

31. Divisibility • Assumption: Decision variables in linear programming model are allowed to have any value including non integer that satisfy functional and non negativity constraints. • Variables can have fraction value. • For instance 3.567 gallons of oil produced divisible but 2.5 machine manufactured?.....

32. Additively • Assumption :” Every function in a linear programming model is the sum of the individual contributions of the respective activities”. • Eg. Total profit earned by selling product A and B is must equal to profit by product A + profit by product B.

33. Proportionality • Assumption : “Any changes in the input must change the output linearly”. • Contributions from the decision variables Xi to objective function Ci is Xi*Ci. • The amount of each resource used and its contribution to the profit in objective function must be proportional to the value of the each decision variable. • One unit requires 5 hr machine for 3 unit = 5*3=15 this is proportional.

34. Mutualexclusive • All decision parameters and variables are assumed to be mutually exclusive. Non negativity: All the variables are positive like resource, profit etc. Finiteness: Presence of finite number of activities and constraints is helpful for obtaining best solution.

35. Advantages of using LPP • LP helps in obtaining the optimum use of productive resources ,It also indicate how a decision makers by selecting and distributing these resource. • LP technique improves the quality of decision . The decision making approach of the user of this technique more objective and less subjective.

36. LP technique provides and practical solution since there might be other constraints operating outside the problem which must be taken in account. • Highlighting of bottleneck in the production is the most significant advantage of this technique. • LP also helps in revalution of a basic plan for changing condition.

37. Limitation of LP • LP treats all the relationships between the variable is linear. • No guarantee that solution is integer. • Does not take consideration of time and uncertainty. • Large problems can be solved using computer instead of LPP . • Parameters are assumed to be constant . • Deals with single objective.

38. Mathematical model of LPP • Linear programming with ‘n’ decision variable and ‘m’ constraints can be stated as follows, • find the decision variables x1,x2,x3,…..,xn optimize(max/min) Z=c1x1+c2x2+….+cnxn subjected to, a11x1+a12x2+a13x3+…..+a1nxn(<=,=,>=) b1 a21x1+a22x2+a23x3+…..+a2nxn (<=,=,>=) b2 . . am1x1+am2x2+am3x3+…..+amnxn (<=,=,>=) bm where x1,x2,x3,…xn>=0;

39. Table of data to solve LP

40. Application of LPP • Military • Agriculture • Environmental protection • Facilities Location • Product Mix • Production • Mixing or Blending • Transportation • Portfolio selection • Profit planning • Travelling salesmen problem

41. Problems 1) A farmer owns 200 pigs that consumes minimum 90 kg of special feed daily. The feed is prepared as a mixture of corn and soybean meal with following. requirements are as follows, 1)At most 1% calcium 2) At least 30% protein 3) At most 5% fiber formulate LPP for Minimize cost.

42. 2.The manager of an oil refinery has to decide upon the optimum mix of two possible blending processes of which the input and the output per production run are as follows.

43. The amount of available of crude A and B are 200 and 150 units respectively . Market requirements show that at least 100 units of Gasoline X and 80 units of gasoline Y must be produced. The profit per production must run from process 1 and process 2 are Rs 3 and Rs 4 resp. Formulate the LP model.

44. 3) A marketing manager wishes to allocate his annual advertising budgets of Rs 2000 in two media A and B .The unit cost of message in media A is Rs 100 and in B 150 .media A is monthly magazine and not more than one insertion is desired in one issue. at least 5 messages should appear in media B .The expected effective audience for unit message for media A is 4000 and for B is 5000.Formulate LPP.

45. 4) A manufacturer produce two types of model M1 and M2 .Each model of the type M1 requires 4hrs grinding and 2 hrs of polishing; where as each model of type M2 requires 2hrs grinding and 5 hrs of polishing. The manufacturer have 2 grinder and 3 polishers each grinder work 40 hours a week and each polisher works for 60 hrs a week. Profit on M1 model is rs.3 and on model rs.4.Whatever is produces in week is sold in the market. How should the manufacturer allocate his production capacity to the two types of module so that he may make the maximization profit in a week?

46. 5) A company manufactures two products A and B. These products are processed in the same machine. It takes 10 min to process one unit of A and 2 min for B. Machine operates for maximum 35 hrs in a week. Product A is requires 1kg and B requires 0.5 kg of raw material per unit the supply of which is 600kg per week. Market constraints on product B is Known to be 800 unit every week. Product A costs rs 5 and sold at rs 10 , product B costs rs 6 and sold at 8. Determine the no of unit of A and B per week to maximize the profit.

47. 6) A person requires 10 ,12 and 12 units of chemical A,B and C respectively for his garden. A liquid product contain 5 ,2 and 1 units of A,B,C respectively. A dry product contains 1 ,2 and 4 units of A ,B and C per carton. If the liquid product sells Rs 3 per jar and dry product sells at Rs 2 how many of each should be purchased in order to minimize the cost and meet the requirement?

48. An oil refinery can blended three grades of crude oil to produce quality R and S petrol. Two possible blending processes are available. For each production run the older process uses 5 unit of crude A ,7 unit of crude B and 2 unit of crude C to produce 9 units of R and 7 units of S. The newer process uses 3 units of crude A ,9 units of crude B and 4 units of crude C to produce 5 units of R and units of S petrol. Because of prior contract commitments the refinery must produce at least 500 units of crude R and at least 300 units of S for the next month. It has available 1500 units of crude A. 1900 units of crude B and 1000 units of crude C.For each unit of R

49. refinery receives Rs 60 and for each unit of S Rs 90.Find the linear programming formulation of the problem so as to maximize the revenue.

50. 8)A high quality furniture two products, tables and chairs. Both the product have to be processes through two machines M1 and M2 the total machine hours available are 200 hrs of M1 and 400 hrs of M2 respectively. Time in hours required for producing a chair and table on the both the machine is as follows. • Machine Table Chair M1 7 4 M2 5 5 from table profit is Rs 40 and from chair Rs 30 Maximize the profit.