1.1.Origin of the operations research Advent of the industrial revolution The artisans’ small shops of an earlier era have evolved into the billion-dollar corporations of today. The computer revolution
1.2 THE NATURE OF OPERATIONS RESEARCH operations research involves “research on operations The process begins by carefully observing and formulating the problem, including gathering all relevant data. The next step is to construct a scientific (typically mathematical) model that attempts to abstract the essence of the real problem
suitable experiments are conducted to test this hypothesis it attempts to resolve the conflicts of interest among the components of the organization OR frequently attempts to find a best solution
1.3 THE IMPACT OF OPERATIONS RESEARCH improving the efficiency of numerous organizations around the world Has a significant contribution to increasing the productivity of the economies of various countries
CHAPTER.2 Overview of the Operations Research Modeling Approach
Major phase of OR study; 1. Define the problem of interest and gather relevant data. 2. Formulate a mathematical model to represent the problem. 3. Develop a computer-based procedure for deriving solutions to the problem from the model. 4. Test the model and refine it as needed. 5. Prepare for the ongoing application of the model as prescribed by management. 6. Implement
2.1 DEFINING THE PROBLEM AND GATHERING DATA to study the relevant system and develop a well-defined statement of the problem to be considered. It is difficult to extract a “right” answer from the “wrong” problem! Ascertain the appropriate objectives
Example! An OR study done for the San Francisco Police Department1 resulted in the development of a computerized system for optimally scheduling and deploying police patrol officers. The new system provided annual savings of $11 million, an annual $3 million increase in traffic citation revenues, and a 20 percent improvement in response times. In assessing the appropriate objectives for this study, three fundamental objectives were identified: 1. Maintain a high level of citizen safety. 2. Maintain a high level of officer morale. 3. Minimize the cost of operations
2.2 FORMULATING A MATHEMATICAL MODEL A mathematical model is a description of a system using mathematical concepts and language. systemmathematical The process of developing a mathematical model is termed mathematical modeling.
Mathematical models are used not only in the natural sciences (such as physics, biology, earth science, meteorology) and engineering disciplines (e.g. computer science, artificial intelligence), but also in the social sciences (such as economics, psychology, sociology and political science);
Therefore, when formulating MM, following is considered decision variables objective function example, x1 +3x1x2 + 2x2 < 10 Such mathematical expressions for the restrictions often are called constraints. parameters of the model( objectives and constraints) Sensitivity analysis Linear programming model
EXAMPLE OF MATHEMATICAL MODEL The potential field is given by a function V : R 3 → R and the trajectory is a solution of the differential equation Note this model assumes the particle is a point mass, which is certainly known to be false in many cases in which we use this model; for example, as a model of planetary motion
Advantages describes the problem more concisely reveal important cause and effect relationships it facilitates dealing with the problem in its entity and considering all its interrelationships simultaneously forms a bridge to the use of high powered mathematical techniques and computers to analyze the problem
Disadvantage Abstract idealization of the problem
2.3 DERIVING SOLUTIONS FROM THE MODEL Heuristic procedures Post-optimality analysis This analysis also is sometimes referred to as what-if analysis sensitivity analysis
2.4 TESTING THE MODEL Model validation; process of testing and improving a model to increase its validity retrospective test test involves using historical data to reconstruct the past
2.5 PREPARING TO APPLY THE MODEL Model procedure Decision support system; In other cases, an interactive computer-based system
2.6 IMPLEMENTATION The implementation phase involves; OR team gives operating management a careful explanation of the new system to be adopted and how it relates to operating realities.
Definitions of linear programming Linear programming (LP, or linear optimization) is a mathematical method for determining a way to achieve the best outcome (such as maximum profit or lowest cost) in a given mathematical model for some list of requirements represented as linear relationships.mathematical model More formally, linear programming is a technique for the optimization of a linear objective function, subject to linear equality and linear inequality constraintsoptimizationlinear objective functionlinear equalitylinear inequalityconstraints
Its feasible region is a convex polyhedron, which is a set defined as the intersection of finitely many half spaces, each of which is defined by a linear inequality.feasible regionconvex polyhedronhalf spaces Its objective function is a real-valued affine function defined on this polyhedron. A linear programming algorithm finds a point in the polyhedron where this function has the smallest (or largest) value if such point existsrealaffine functionalgorithm
Linear programs are problems that can be expressed in canonical form canonical form Max cTx Subject to: Ax ≤ b And x≥ 0 In fact, any problem whose mathematical model fits the very general format for the linear programming model is a linear programming problem. Furthermore, a remarkably efficient solution procedure, called the simplex method, is available for solving linear programming problems of even enormous size.
3.1 PROTOTYPE EXAMPLE The WYNDOR GLASS CO. produces high-quality glass products, including windows and glass doors. It has three plants. Aluminum frames and hardware are made in Plant 1, wood frames are made in Plant 2, and Plant 3 produces the glass and assembles the products. Because of declining earnings, top management has decided to revamp the company’s product line. Unprofitable products are being discontinued, releasing production capacity to launch two new products having large sales potential: Product 1: An 8-foot glass door with aluminum framing Product 2: A 4 _ 6 foot double-hung wood-framed window 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. The marketing division has concluded that the company could sell as much of either product as could be produced by these plants. However, because both products would be competing for the same production capacity in Plant 3, it is not clear which mix of the two products would be most profitable. Therefore, an OR team has been formed to study this question
TABLE 3.1 Data for the Wyndor Glass Co. problem Production Time per Batch, Hours Production Time Plant 1 2 Available per Week, Hours products Plant1 2 11 04 20 212 33 218 Profit per batch$3000 $5000
Formulation as a Linear Programming Problem To formulate the mathematical (linear programming) 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 per week (in thousands of dollars) from producing these two products Thus, x1 and x2 are the decision variables for the model. Using the bottom row of Table 3.1, we obtain Z + 3x1 +5x2.
The objective is to choose the values of x1 and x2 so as to maximize Z =3x1 + 5x2, subject to the restrictions imposed on their values by the limited production capacities available in the three plants. Table 3.1 indicates that each batch of product 1 produced per week uses 1 hour of production time per week in Plant 1, whereas only 4 hours per week are available. This restriction is expressed mathematically by the inequality x1 ≤ 4. Similarly, Plant 2 imposes the restriction that 2x2 ≤ 12. The number of hours of production rates would be 3x1 +2x2. Therefore, the mathematical statement of the Plant 3 restriction is 3x1 +2x2 ≤18. Finally, since production rates cannot be negative, it is necessary to restrict the decision variables to be nonnegative: x1 ≥ 0 and x2 ≥0.
To summarize, in the mathematical language of linear programming, the problem is to choose values of x1 and x2 so as to Maximize Z = 3x1 + 5x2, subject to the restrictions 3x1 + 2x2 ≤ 4 3x1 +2x2 ≤12 3x1 + 2x2 ≤18 and x1 ≥ 0, x2 ≥ 0.