Subject Name OPERATIONS RESEARCH Subject Code 10 CS

Subject Name: OPERATIONS RESEARCH Subject Code: 10 CS 661 Prepared By: Sindhuja K Department: CSE Date 12/1/2020

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. 12/1/2020

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. 12/1/2020

Text Books: 1. Frederick S. Hillier and Gerald J. Lieberman: Introduction to Operations Research, 8 th Edition, Tata Mc. Graw 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 Engineered for Tomorrow Algorithms, 4 th Edition, Thomson Course Technology, 2003. 2. Hamdy A Taha: Operations Research: An Introduction, 8 th Edition, Prentice Hall India, 2007. 12/1/2020

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 12/1/2020

Unit –I Introduction, Linear Programming – 1 12/1/2020

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 12/1/2020

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 12/1/2020

• 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. ) 12/1/2020

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. 12/1/2020

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 12/1/2020

AIMS OF OR Maxima of ; Profit Assembly line performance Crop yield Bandwidth Minima of; Loss Risks

• ORIGIN OF OPERATIONS RESEARCH The formal activities of Operations Research (OR) were initiated in England during World War II by Mc. Closky 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 12/1/2020

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) 12/1/2020

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: – – – 12/1/2020 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

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.

PHASES OF OR Defining the problem and gathering data Formulating of the model To obtain solution from model Testing of the model and the solution Implementation of the solution Operation Research 17

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. 12/1/2020

FORMULATING OF THE MODEL 12/1/2020

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”). 12/1/2020

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. 12/1/2020

PREPARING TO APPLY THE MODEL • Well documented installation • Computer based systems • Interactive decision support systems 12/1/2020

IMPLEMENTATİON • Implementation of the solution obtained from a validated model is a reliable solution to the real-life problems. • Implementation of the solution is the duty of operation research team. 12/1/2020

SOCIETIES • Operational Research Society, Turkey • The International Federation of Operational Research Societies(IFORS) • The Association of European Operational Research Societies(EURO) 12/1/2020

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. 12/1/2020

INTRODUCTION TO LINEAR PROGRAMMING (LP) Real-World Problem Implementation 12/1/2020 Recognition and Definition of the Problem Interpretation Validation and Sensitivity Analysis of the Model Formulation and Construction of the Mathematical Model Solution of the Model

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. 12/1/2020

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 nonnegativity • restrictions together form an LP model. 12/1/2020

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 4 x 6 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.

• 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 •

(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 12/1/2020

(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

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

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

(4) Graphical Solution The Wyndor Glass Co. example is used to illustrate the graphical solution. Shaded area shows values of (x 1, x 2) allowed by x 1 ≥ 0, x 2 ≥ 0, x 1 ≤ 4 Shaded area shows values of (x 1, x 2) , called feasible region

The value of (x 1, x 2) that maximize 3 x 1 + 5 x 2 is (2, 6)

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.

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. Feasible Region: The feasible region is the collection of all feasible solutions. 12/1/2020 A feasible solution is located in the feasible region. An infeasible solution is outside the feasible region.

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 3 x 1 + 5 x 2 ≤ 50 were added to the problem. In this case, there is no feasible region

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 = 3 x 1 + 2 x 2

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 x 1 ≤ 4, because x 2 then could be increased indefinitely in the feasible region without ever reaching the maximum value of Z = 3 x 1 + 2 x 2

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

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 prototype model has exactly one optimal solution, (x 1, x 2)=(2, 6), which is a CPF solution The modified problem has multiple optimal solution, two of these optimal solutions , (2, 6) and (4, 3), are CPF solutions.

Matrix Standard Form of an LP Model

Tabular Standard Form of an LP Model

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. 12/1/2020

• 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 5 eggs/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 5 kg and it contains 2 ingredients B 1 and B 2 costs Rs. 5 and Rs. 8/kg respectively. The brick contains not more than 4 kg of B 1 and a minimum of 2 kg of B 2. Since the demand for the profit is likely to be related to the price of brick. Formulate the problem as LPP. 12/1/2020

• Solve the following LPP by graphical method Minimize Z = 20 x 1+ 10 x 2 STC x 1+ 2 x 2≤ 40 3 x 1+ x 2≥ 30 4 x 1+ 3 x 2≥ 60 X 1, x 2 ≥ 0 maximize Z = 2 x 1+ 3 x 2 STC x 1+ 2 x 2≤ 4 x 1+ x 2=3 X 1, x 2 ≥ 0 maximize Z = 3 x 1+ 2 x 2 STC 5 x 1+ x 2 ≥ 10 x 1+ x 2≥ 6 x 1+ 4 x 2≥ 12 and X 1, x 2 ≥ 0 12/1/2020

maximize Z = 6 x 1+ 4 x 2 STC x 1+ 2 x 2 ≥ 4 x 1+ 2 x 2≤ 2 and X 1, x 2 ≥ 0 maximize Z = x 1+ x 2 /2 STC 3 x 1+ 2 x 2≤ 12 5 x 1≤ 10 x 1+ x 2≤ 18 -x 1+ x 2 ≥ 4 And X 1, x 2 ≥ 0 12/1/2020