Introduction to Operations Research and Linear Programming

Introduction to Operations Research

1. Definition, Origin and Modelling Approach - AN INTRODUCTION TO OPERATIONS RESEARCH -

2. OPERATIONS RESEARCH Often referred as MANAGEMENT SCIENCE . It is simply a scientific approach to decision making that seeks to best design and operate a system, usually under conditions requiring allocation of scarce resources. Operations research (OR) is the discipline of using models, either quantitative or qualitative, to aid decision-making in complex implementation problems.

3. OPERATIONS RESEARCH In a sense, every effort to apply science to management of organized systems, and to their understanding, was a predecessor of operations research.  in 1937 in Britain as a result of the initiative of A.P. Rowe led British scientists to teach military leaders how to use the then newly developed radar to locate enemy aircraft. By the end of World War II there were 26 operations research groups in the Air Force. At the end of World War II a number of British operations research workers moved to government and industry. Albert Percival Rowe Wurzburg Radar British Iron and Steel Research Association

4. MODELLING APPROACH Operations research basically uses MODELS to analyze different situations. A MODEL is an idealized representation of reality.

5. MODELLING APPROACH The 7-Step Model-Building Process STEP 1: FORMULATE THE PROBLEM It deals with defining the organization’s problem. Defining the problem includes specifying the organization’s objectives and the parts of the organization that must be studied before the problem can be solved. STEP 2: OBSERVE THE SYSTEM It deals with collecting data to estimate the value of the parameters that affects the organization. STEP 3: FORMULATE A MATHEMATICAL MODEL OF THE PROBLEM Using collected data, mathematical model for the problem is formed.

6. MODELLING APPROACH The 7-Step Model-Building Process STEP 4: VERIFY THE MODEL AND USE THE MODEL FOR PREDICTION In this step, the mathematical model developed will be determined if its an accurate representation of the problem. STEP 5: SELECT A SUITABLE ALTERNATIVE Given a model and set of alternatives, one will select the alternative that best meets the organization’s objectives. STEP 6: PRESENT THE RESULTS AND CONCLUSION STEP 7: IMPLEMENT AND EVALUATE RECOMMENDATIONS

7. MODELLING APPROACH Most of the models used for analyzing an organizations situation are PRESCRIPTIVE or OPTMIZATION models. A prescriptive model “prescribes” behavior for an organization that will enable it to best meet it goal. Optimization models seek to find values of the decision variables that optimize (maximize or minimize) an objective function that satisfy given constraints.

8. MODELLING APPROACH Three Components of an Optimization Model DECISION VARIABLES These are values that are under the decision maker’s control and influence the performance of a system. For example, the decision variable X can represent the number of pounds of product A that a company will produce. OBJECTIVE FUNCTION it defines the criterion of evaluating the solution. It is a mathematical function that converts the solution into mathematical evaluation of a solution. Also, it indicates the direction of optimization, either to MAXIMIZE or MINIMIZE.

9. MODELLING APPROACH Three Components of an Optimization Model CONSTRAINTS These are the restrictions to the values of the decision variables. Objective Function These are the set of constraints. These are decision variables.

10. MODELLING APPROACH PLANT PRODUCT AVAILABLE TIME PER WEEK 1 2 1 1 4 2 2 12 3 3 2 18 Profit per Batch $3,000 $5,000 These values will form the objective function. Products 1 and 2 will the decision variables represented as X and Y, respectively. These values will be part of the constraints.

11. MODELLING APPROACH Developing a Mathematical Model 1. Determine the decision variables. PLANT PRODUCT AVAILABLE TIME PER WEEK 1 2 1 1 4 2 2 12 3 3 2 18 Profit per Batch $3,000 $5,000 As per definition, decision variables are variables that a decision maker can control. In this case, one can control the number of batches of product 1 and 2 that can be produced. For representation, let the number of batches of Product 1 = X the number of batches of Product 2 = Y

12. MODELLING APPROACH Developing a Mathematical Model 2 . Determine the objective function (z). PLANT PRODUCT AVAILABLE TIME PER WEEK 1 2 1 1 4 2 2 12 3 3 2 18 Profit per Batch $3,000 $5,000 Since the problem is dealing with PROFIT , the optimization model will incline to MAXIMIZATION . For the objective function, Maximize z = 3,000X + 5,000Y For every batch of Product 1 and 2 there is a corresponding profit.

13. MODELLING APPROACH Developing a Mathematical Model 3. Establish the constraints. PLANT PRODUCT AVAILABLE TIME PER WEEK 1 2 1 1 4 2 2 12 3 3 2 18 Profit per Batch $3,000 $5,000 Constraint 1 Constraint 2 Constraint 3 NOTE: Constraints denote that there are limited resources one must properly allocate. Subject to X ≤ 4 2Y ≤ 12 3X + 2Y ≤ 18 X, Y ≥ 0 Constraint 1 Constraint 2 Constraint 3 Non-negativity Constraint

14. MODELLING APPROACH Developing a Mathematical Model PLANT PRODUCT AVAILABLE TIME PER WEEK 1 2 1 1 4 2 2 12 3 3 2 18 Profit per Batch $3,000 $5,000 Maximize z = 3,000X + 5,000Y Subject to X ≤ 4 2Y ≤ 12 3X + 2Y ≤ 18 X, Y ≥ 0