Operations Research QM 350. Chapter 1 Introduction. Operations Research. University of Bahrain

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1 QM 350 Operations Research University of Bahrain INTRODUCTION TO MANAGEMENT SCIENCE, 12e Anderson, Sweeney, Williams, Martin Chapter 1 Introduction Introduction: Problem Solving and Decision Making Quantitative Analysis and Decision Making Quantitative Analysis Model Development Software Packages Operations Research The body of knowledge involving quantitative approaches to decision making is referred to as Management Science Operations Research Decision Science It had its early roots in World War II and is flourishing in business and industry due, in part, to: numerous methodological developments (e.g. simplex method for solving linear programming problems) a virtual explosion in computing power 1

2 Quantitative Analysis and Decision Making Decision-Making Process Structuring the Problem Analyzing the Problem Define the Problem Identify the Alternatives Determine the Criteria Identify the Alternatives Choose an Alternative Quantitative Analysis and Decision Making Analysis Phase of Decision-Making Process Qualitative Analysis based largely on the manager s judgment and experience includes the manager s intuitive feel for the problem is more of an art than a science Quantitative Analysis and Decision Making Analysis Phase of Decision-Making Process Quantitative Analysis analyst will concentrate on the quantitative facts or data associated with the problem analyst will develop mathematical expressions that describe the objectives, constraints, and other relationships that exist in the problem analyst will use one or more quantitative methods to make a recommendation 2

3 Quantitative Analysis and Decision Making Potential Reasons for a Quantitative Analysis Approach to Decision Making The problem is complex. The problem is very important. The problem is new. The problem is repetitive. Quantitative Analysis Quantitative Analysis Process Model Development Data Preparation Model Solution Report Generation Model Development Models are representations of real objects or situations Three forms of models are: Iconic models - physical replicas (scalar representations) of real objects Analog models - physical in form, but do not physically resemble the object being modeled Mathematical models - represent real world problems through a system of mathematical formulas and expressions based on key assumptions, estimates, or statistical analyses 3

4 Advantages of Models Generally, experimenting with models (compared to experimenting with the real situation): requires less time is less expensive involves less risk The more closely the model represents the real situation, the accurate the conclusions and predictions will be. Example: Production Problem (1) The total profit from the sale of a product can be determined by multiplying the profit per unit by the quantity sold. If we let x represent the number of units sold and P the total profit, then, with a profit of $10 per unit, the following mathematical model defines the total profit earned by selling x units The above mathematical expression describes the problem s objective and the firm will attempt to maximize profit. Example: Production Problem (2) A production capacity constraint would be necessary if, for instance, 5 hours are required to produce each unit and only 40 hours of production time are available per week. The production time constraint is given by The value of 5x is the total time required to produce x units; the symbol indicates that the production time required must be less than or equal to the 40 hours available 4

5 Example: Production Problem (3) The decision problem or question is the following: How many units of the product should be scheduled each week to maximize profit? A complete mathematical model for this simple production problem is This model is an example of a linear programming model. Mathematical Models Objective Function a mathematical expression that describes the problem s objective, such as maximizing profit or minimizing cost Constraints a set of restrictions or limitations, such as production capacities Uncontrollable Inputs environmental factors that are not under the control of the decision maker Decision Variables controllable inputs; decision alternatives specified by the decision maker, such as the number of units of Product X to produce Mathematical Models Deterministic Model if all uncontrollable inputs to the model are known and cannot vary Stochastic (or Probabilistic) Model if any uncontrollable are uncertain and subject to variation Stochastic models are often more difficult to analyze. Frequently a less complicated (and perhaps less precise) model is more appropriate than a more complex and accurate one due to cost and ease of solution considerations. 5

6 Transforming Model Inputs into Output Uncontrollable Inputs (Environmental Factors) Controllable Inputs (Decision Variables) Mathematical Model Output (Projected Results) Flowchart for the Production Model Exercise 8 p. 21 Suppose the firm in this example considers a second product that has a unit profit of $5 and requires 2 hours of production time for each unit produced. Use y as the number of units of product 2 produced. a. Show the mathematical model when both products are considered simultaneously. b. Identify the controllable and uncontrollable inputs for this model. c. Draw the flowchart of the input-output process for this model (see Figure 1.5). d. What are the optimal solution values of x and y? e. Is the model developed in part (a) a deterministic or a stochastic model? Explain 6

7 Exercise 8 p. 21 (1) b. Controllable inputs: x and y Uncontrollable inputs: profit (10,5), labor hours (5,2) and labor-hour availability (40) Exercise 8 p. 21 (2) Exercise 8 p. 21 (3) d. x = 0, y = 20 Profit = $100 (Solution by trial-and-error) e. Deterministic - all uncontrollable inputs are fixed and known. 7

8 QM 350: Course Content (1) Linear programming is a problem-solving approach developed for situations involving maximizing or minimizing a linear function subject to linear constraints that limit the degree to which the objective can be pursued. The transportation model is an example of a simple linear programming model. Simulation is a technique used to model the operation of a system. This technique employs a computer program to model the operation and perform simulation computations. QM 350: Course Content (2) Decision analysis can be used to determine optimal strategies in situations involving several decision alternatives and an uncertain or risk-filled pattern of events. Markov process models are useful in studying the evolution of certain systems over repeated trials. For example, Markov processes have been used to describe the probability that a machine, functioning in one period, will function or break down in another period. 8