Subject Name: OPERATIONS RESEARCH Subject Code: 10CS661 Prepared By: Sindhuja K

1. Subject Name: OPERATIONS RESEARCH Subject Code: 10CS661 Prepared By: Sindhuja K Department: CSE Date

2. OBJECTIVE With the growth of technology, the World has seen a remarkable changes in the size and complexity of organizations. An integral part of this had been the division of labour and segmentation of management responsibilities in these organizations. The results have been remarkable but with this, increasing specialization has created a new problem to meet out organizational challenges. The allocation of limited resources to various activities has gained significant importance in the competitive market. These types of problems need immediate attention which is made possible by the application of OR techniques. Operations Research (OR) is a science which deals with problem, formulation, solutions and finally appropriate decision making.

3. LEARNING OUTCOMES Identify and develop operational research models from the verbal description of the real world problems. Solve business problems and apply it's applications by using mathematical analysis. Develop the ideas of developing and analyzing mathematical models for decision problems, and their systematic solution. Understand the mathematical models that are needed to solve optimization problems.

4. Text Books: 1. Frederick S. Hillier and Gerald J. Lieberman: Introduction to Operations Research, 8th Edition, Tata McGraw Hill, 2005. (Chapters: 1, 2, 3.1 to 3.4, 4.1 to 4.8, 5, 6.1 to 6.7, 7.1 to 7.3, 8, 13, 14, 15.1 to 15.4) Reference Books: 1. Wayne L. Winston: Operations Research Applications and Algorithms, 4th Edition, Thomson Course Technology, 2003. 2. Hamdy A Taha: Operations Research: An Introduction, 8th Edition, Prentice Hall India, 2007. Engineered for Tomorrow

5. COURSE TOPICS UNIT 1 - Introduction, Linear Programming – 1 UNIT 2 - LP – 2, Simplex Method – 1 UNIT 3 - Simplex Method – 2 UNIT 4 - Simplex Method – 2, Duality Theory UNIT 5 - Duality Theory and Sensitivity Analysis, Other Algorithms for LP UNIT 6 - Transportation and Assignment Problems UNIT 7 - Game Theory, Decision Analysis UNIT 8 - Metaheuristics

6. Unit –I Introduction, Linear Programming – 1

7. Unit 1- TOPICS • Introduction: The origin, nature and impact of OR; • Phases of OR • Defining the problem and gathering data; • Formulating a mathematical model; • Deriving solutions from the model; • Testing the model; • Preparing to apply the model; • Implementation . • Introduction to Linear Programming: Prototype example; • The linear programming (LP) model

8. INTRODUCTION • Operations Research is a very important area of study, which tracks its roots to business applications. It combines the three broad disciplines of Mathematics, Computer Science, and Business Applications. • This course will formally develop the ideas of developing, analyzing, and validating mathematical models for decision problems, and their systematic solution. The course will involve programming and mathematical analysis

9. Operations research involves “research on operations.” Thus, operations research is applied to problems that concern how to conduct and coordinate the operations (i.e., the activities) within an organization . • The nature of the organization is essentially immaterial, and, in fact, OR has been applied extensively in such diverse areas as manufacturing, transportation, construction, telecommunications, financial planning, health care, the military, and public services, to name just a few. Therefore, the breadth of application is unusually wide. • The research part of the name means that operations research uses an approach that resembles the way research is conducted in established scientific fields. To a considerable extent, the scientific method is used to investigate the problem of concern. (In fact, the term management science sometimes is used as a synonym for operations research.)

10. WHAT IS OPERATIONS RESEARCH • Operations Research ( OR) Is the application of scientific methods in solving problems facing management and to help to taking decisions. • A rose by any other name… • Management Science • Systems Engineering • Industrial Engineering • Operations Management • Applied Mathematics • Operations research (also known as management science) is a collection of techniques based on mathematics and other scientific approaches that finds solutions to your problems.

11. DEFINITIONS • It is an Act of winning wars without actually fighting. • Aurther Clark • It is a Scientific Approach to problem solving for executive management. • H.M. Wagner • It is Art of giving bad answers to problem which otherwise have worse answers. • T.L. Saaty

12. AIMS OF OR Maxima of ; Profit Assemblylineperformance Cropyield Bandwidth Minima of; Loss Risks

13. ORIGIN OF OPERATIONS RESEARCH • The formal activities of Operations Research (OR) were initiated in England during World War II by McClosky and Trefthen in 1940. A team of British scientists set out to make decisions regarding the best utilization of war material. • Following the end of the war, the ideas advanced in military operations were adapted to improve efficiency and productivity in the civilian sector. • Today, OR is a dominant decision making tool, that seeks the optimum state in all conditions and thus provides optimum solution to organizational problems

14. THE NATURE OF OR • OR is applied to problems that concern how to conduct and coordinate the operation within a organization. • The steps followed are • Construct a model that attempts to abstract the essence of the real problem • Conduct suitable tests on the model, modify it as needed and verify some. • OR frequently attempts to find a best solution (optimal solution) for the problem under consideration(“Search for optimality”). • OR gives bad answers to the problems where worse could be given(it cannot give perfect answers to the problem. Thus OR improves only the quality of the solution)

15. IMPACT OF OR • Operations Research (OR) applies scientific method to the management of organized systems in business, industry, government and other enterprises. • OR can play a significant role in bringing a balance among different inter disciplinary people to managerial problem. • OR is regularly applied in areas such as: • supply chain management • marketing and revenue management systems • manufacturing plants • financial engineering • telecommunication networks • healthcare management • transportation networks • energy and the environment • service systems • web commerce • military defense

16. PHASES OF AN OR STUDY • As a decision-making tool, OR is both a science and an art. • The principal phases for implementing OR in practice includes: • Definition of the problem. • Construction of the model. • Solution of the model. • Validation of the model. • Implementation of the solution.

17. PHASES OF OR OperationResearch

18. DEFINING THE PROBLEM AND GATHERING DATA • In formulating a problem for O.R. study analysis must be made of the following major components: (i) The environment (ii) The objectives (iii) The decision maker (iv) The alternative courses of action and constraints. • Out of the above four components environment is most comprehensive as it provides a setting for the remaining three. • The operation researcher shall attend conferences, pay visits, send observation and perform research work thus succeeds in getting sufficient data to formulate the problems.

19. FORMULATING OF THE MODEL The operation researcher can now construct the model to show the relations and interrelations between a cause and effect or between an action and a reaction. The following steps are used in linear programming models • Specify solution objectives • Specify decision variables and stipulate constraints on the solution • Identify or construct an appropriate model for solution development • Determine and obtain required data • Develop solutions using analytical model

20. TO OBTAİN SOLUTİON FROM MODEL • A solution may be extracted form a model either by conducting experiments on it i.e. by simulation or by mathematical analysis. • It and even use standard algorithms.”Optimizing is the science of the ultimate; satisficing is the art of the feasible” • OR team uses Heuristic procedures and post optimality analysis(“what-if analysis”).

21. TESTİNG OF THE MODEL AND THE SOLUTİON • Before the application of the model solution, the validity of model and reliability of the solution should be tested. • Validity of the model can be decided by comparing its outputs with the results of past. • Retrospective test: If past behavior is repeated when provided similar inputs (a stable solution is being obtained), the model will be valid.

22. PREPARING TO APPLY THE MODEL • Well documented installation • Computer based systems • Interactive decision support systems

23. IMPLEMENTATİON • Implementation of the solution obtained from a validated model is a reliable solution to the the real-life problems. • Implementation of the solution is the duty of operation research team.

24. SOCIETIES • Operational Research Society,Turkey • The International Federation of Operational Research Societies(IFORS) • The Association of European Operational Research Societies(EURO)

25. OR SOFTWARE PACKAGES • A very popular approach now is to use today’s premier spreadsheet package, Microsoft Excel, to formulate small OR models in a spreadsheet format. The Excel Solver then is used to solve the models. • LINDO (and its companion modeling language LINGO) continues to be a dominant OR software package. Student versions of LINDO and LINGO now can be downloaded free from the Web. • CPLEX is a state-of-the-art software package that is widely used for solving large and challenging OR problems • MPL is a user-friendly modeling system that uses CPLEX as its main solver.

26. Recognition and Definition of the Problem Formulation and Construction of the Mathematical Model Real-WorldProblem Interpretation Validation and Sensitivity Analysis of the Model Solution of the Model Implementation INTRODUCTION TO LINEAR PROGRAMMING (LP)

27. LINEAR PROGRAMMING • In mathematics, linear programming (LP) is a technique for optimization of a linear objective function, subject to linear equality and linear inequality constraints. • Linear programming determines the way to achieve the best outcome (such as maximum profit or lowest cost) in a given mathematical model and given some list of requirements represented as linear equations.

28. MATHEMATICAL FORMULATION OF LINEAR PROGRAMMING MODEL: • Step 1 • - Study the given situation • - Find the key decision to be made • Identify the decision variables of the problem • Step 2 • Formulate the objective function to be optimized • Step 3 • Formulate the constraints of the problem • Step 4 • - Add non-negativity restrictions or constraints • The objective function , the set of constraints and the non-negativity • restrictions together form an LP model.

29. PROTOTYPE EXAMPLE • The Wyndor Glass Co. produces high-quality glass products, including windows and glass doors. It has three plants. • Plant 1 produces Aluminum frames • Plant 2 produces wood frames • Plant 3 produces the glass and assembles the products. • The company has decided to produce two new products. • Product 1: An 8-foot glass door with aluminum framing • Product 2: A 4x6 foot double-hung wood framed window • Each product will be produced in batches of 20. The production rate is defined as the number of batches produced per week. • The company wants to know what the production rate should be in order to maximize their total profit, subject to the restriction imposed by the limited production capacities available in the 3 plants.

30. To get the answer, we need to collect the following data. • (a) Number of hours of production time available per week in each plant for these two new products. (Most of the time in the 3 plants is already committed to current products, so the available capacity for the 2 new products is quite limited). • Number of hours of production time available per week in Plant 1 for the new products: 4 • Number of hours of production time available per week in Plant 2 for the new products: 12 • Number of hours of production time available per week in Plant 3 for the new products: 18

31. (b) Number of hours of production time used in each plant for each batch produced of each new product (Product 1 requires some of the production capacity in Plants 1 and 3, but none in Plant 2. Product 2 needs only Plants 2 and 3). Number of hours of production time used in Plant 1 for each batch produced of Product 1: 1 Number of hours of production time used in Plant 2 for each batch produced of Product 1: 0 Number of hours of production time used in Plant 3 for each batch produced of Product 1: 3 Number of hours of production time used in Plant 1 for each batch produced of Product 2: 0 Number of hours of production time used in Plant 2 for each batch produced of Product 2: 2 Number of hours of production time used in Plant 3 for each batch produced of Product 2: 2

32. (c) Profit per batch produced of each new product. Profit per batch produced of Product 1: $3,000 Profit per batch produced of Product 2: $5,000

33. Formulation as a Linear Programming Problem To formulate the LP model for this problem, let x1 = number of batches of product 1 produced per week x2 = number of batches of product 2 produced per week Z = total profit ( in thousands of dollars) from producing the two new products. Thus, x1 and x2 are the decision variables for the model. Using the data of Table , we obtain

34. HOW TO SOLVE LP PROBLEMS • Graphical Solution • Simplex Method for Standard form LP • Geometric Concepts • Setting up and Algebra • Algebraic solution of Simplex

35. (4) Graphical Solution The Wyndor Glass Co. example is used to illustrate the graphical solution. Shaded area shows values of (x1, x2) , called feasible region Shaded area shows values of (x1, x2) allowed by x1≥ 0, x2≥ 0, x1≤ 4

36. The value of (x1, x2) that maximize 3x1 + 5x2 is (2, 6)

37. Common Terminology for LP Model Objective Function: The function being maximized or minimized is called the objective function. Constraint: The restrictions of LP Model are referred to as constraints. The first m constraints in the previous model are sometimes called functional constraints. The restrictions xj>= 0 are called nonnegativity constraints.

38. Infeasible Solution: An infeasible solution is a solution for which at least one constraint is violated. Feasible Solution: A feasible solution is a solution for which all the constraints are satisfied. A feasible solution is located in the feasible region. An infeasible solution is outside the feasible region. Feasible Region: The feasible region is the collection of all feasible solutions.

39. Common Terminology for LP Model No Feasible Solutions: It is possible for a problem to have no feasible solutions. An Example he Wyndor Glass Co. problem would have no feasible solutions if the constraint 3x1 + 5x2 ≤ 50 were added to the problem. In this case, there is no feasible region

40. Common Terminology for LP Model Optimal Solution: An optimal solution is a feasible solution that has the maximum or minimum of the objective function. Multiple Optimal Solutions: It is possible to have more than one optimal solution. An Example The Wyndor Glass Co. problem would have multiple optimal solutions if the objective function were changed to Z = 3x1 + 2x2

41. Common Terminology for LP Model Unbounded Objective: If the constraints do not prevent improving the value of the objective function indefinitely in the favorable direction, the LP model is called having an unbounded objective. An Example The Wyndor Glass Co. problem would have no optimal solutions if the only functional constrait were x1 ≤ 4, because x2 then could be increased indefinitely in the feasible region without ever reaching the maximum value of Z = 3x1 + 2x2

42. Common Terminology for LP Model Corner-Point Feasible (CPF) Solution: A corner-point feasible (CPF) is a solution that lies at a corner of the feasible region. The five dots are the five CPF solutions for the Wyndor Glass Co. problem

43. Common Terminology for LP Model Relationship between optimal solutions and CPF solutions : Consider any linear programming problem with feasible solutions and a bounded feasible region. The problem must posses CPF solutions and at least one optimal solution. Furthermore, the best CPF solution must be an optimal solution. Therefore, if a problem has exactly one optimal solution, it must be a CPF solution. If the problem has multiple optimal solutions, at least two must be CPF solutions. (2,6) (4,3) The modified problem has multiple optimal solution, two of these optimal solutions , (2,6) and (4,3), are CPF solutions. The prototype model has exactly one optimal solution, (x1, x2)=(2,6), which is a CPF solution

44. Matrix Standard Form of an LP Model

45. Tabular Standard Form of an LP Model

46. ASSIGNMENT 1 • A Company manufactures FM radios and calculators.The radios contribute Rs 100 per unit and calculator Rs 150 per unit as a profit.Each radio requires 4 diodes,4 registers while each calculator requires 10 diodes and 2 registers.A radio takes 12 minutes and calculator takes 9.6 minutes on the company electronic testing machine and the product manager estimates that 160 hours of test time is available. The firm has 8000 diodes and 3000 resisters in the stock . Formulate the problem as LPP. • A computer company manufactures laptop and desktop that fetches a profit of Rs.700 and Rs.500 unit respectively . Eachunit of laptops takes 4 hours of assembly time and 2 hours of testing time while each unit desktop requires 3 hours of assembly time and 1 hour for testing. In a given month the total number of hours available for assembly is 210 hours and for inspection is 90 hours . Formulate the problem as LPP to maximize the profit.

47. Old hens can be bought at Rs.50 each but young one cost Rs.100 each . The old hens lay 3 eggs/week and young hens 5eggs/week . Each egg cost Rs.2 ,A hen cost Rs.5/week to feed. If a person has only Rs.2000 to spend for hens . Formulate the problem to decide how many each kind of hen should he buy. Assume he can not house more than 40 hens • A toy company manufactures two types of dolls A and B . Each doll of type B takes twice as long to produce as one of type A and the company would have time to make maximum of 2000 dolls/day. The supply of plastic is sufficient to produce 1500 dolls/day(both A and B combined). The doll B requires a fancy dress of which there a only 600/day available. If the company makes a profit of Rs.10 and Rs.18 /doll on doll A and B respectively. Formulate the problem as LPP to maximize the profit. • The standard of a special purpose brick is 5kg and it contains 2 ingredients B1 and B2 costs Rs.5 and Rs.8/kg respectively. The brick contains not more than 4kg of B1 and a minimum of 2kg of B2. Since the demand for the profit is likely to be related to the price of brick . Formulate the problem as LPP.

48. Solve the following LPP by graphical method Minimize Z = 20x 1+ 10x2 STC x 1+ 2x2≤40 3x 1+ x2≥30 4x 1+ 3x2≥60 X1, x2 ≥0 maximize Z = 2x 1+ 3x2 STC x 1+ 2x2≤4 x 1+ x2=3 X1, x2 ≥0 maximize Z = 3x 1+ 2x2 STC 5x 1+ x2 ≥10 x 1+ x2≥6 x 1+ 4x2≥12 and X1, x2 ≥0

49. maximize Z = 6x 1+ 4x2 STC x 1+ 2x2 ≥4 x 1+ 2x2≤2 and X1, x2 ≥0 maximize Z = x 1+ x2 /2  STC3 x 1+ 2x2≤12 5x 1≤10 x 1+ x2≤18 -x 1+ x2 ≥4 And X1, x2 ≥0