Chapter 1 Management Science

Linear Programming Model

• The objective function is a linear mathematical relationship that describes the objective of the firm in terms of the decision variables.
• The objective function always consists of either maximizing or minimizing some value.

Model Constraints

• Model constraints are linear relationships of the decision variables that represent restrictions placed on the firm by the operating environment.
• Restrictions can be in the form of limited resources or restrictive guidelines.
• Parameters are actual numeric values in the objective function and constraints.

Maximization Model Example: Beaver Creek Pottery Company

• The company desires to maximize profit by deciding how many bowls and mugs to produce each day, given limited resources of pottery clay and skilled labor.
• The two products have different resource requirements for production and profit per item.

Decision Variables

• Decision variables are the unknowns that management needs to decide upon, such as how many bowls and mugs to produce.
• The solution occurs at an extreme point where constraint equation lines intersect with each other or with the axis.

Slack Variables

• Slack variables are added to each constraint to transform ≤ inequality constraints into equations.
• Slack variables represent the amount of unused resources.

Linear Programming Model in Standard Form

• The complete linear programming model can be written in standard form with slack variables.

Graphical Solution Steps

• Plot the model constraints as equations on the graph and indicate the feasible solution area.
• Plot the objective function and move it out from the origin to locate the optimal solution point.

Other Techniques

• Network Techniques: models represented as diagrams rather than mathematical relationships, offering a pictorial representation of the system under analysis.
• Analytical Hierarchy Process (AHP): a mathematical technique for helping the decision maker choose between several alternative decisions, given more than one objective.
• Simulation: a technique that replicates a real-world system under analysis, using a mathematical model constructed on a computer to solve problems in the simulated system.

Management Science Process

• The first step in the management science process is the identification of a problem that exists in the system (organization).
• The person who normally identifies a problem is the manager, but problems can also be identified by a management scientist.
• The problem must be clearly and concisely defined, and the goals (or objectives) of the organization must also be clearly defined.

Model Construction

• A management science model is an abstract representation of an existing problem situation.
• It can be in the form of a graph or chart, but most frequently a management science model consists of a set of mathematical relationships.
• These mathematical relationships are made up of numbers and symbols.

Components of a Break-Even Model

• Total variable cost = vcv, where cv = variable cost per unit and v = volume (number of units) sold.
• The total cost of an operation is computed by summing total fixed cost and total variable cost, as follows: TC = cf + vcv.
• Total revenue = vp, where p = price per unit.
• Profit is the difference between total revenue and total cost, and can be computed as: Z = $20x - 5x.

Sensitivity Analysis

• Sensitivity analysis is the study of how changes in parameters affect the model solution.
• It is used to see how sensitive the model is to changes in parameters such as fixed and variable costs, and price.

Management Science Modeling Techniques

• Two of the five steps of the management science process - model construction and solution - use management science techniques.
• Techniques can be loosely classified into four categories: mathematical programming, probabilistic techniques, network techniques, and other techniques.

Mathematical Programming Techniques

• Linear programming models help managers determine solutions for problems that will achieve some objective in which there are restrictions, such as limited resources or a recipe or perhaps production guidelines.

Probabilistic Techniques

• Probabilistic techniques are distinguished from mathematical programming techniques in that the results are probabilistic.
• An example of a probabilistic technique is decision analysis.

Network Techniques

• Networks consist of models that are represented as diagrams rather than as strictly mathematical relationships.
• These models offer a pictorial representation of the system under analysis.

Other Techniques

• The analytical hierarchy process (AHP) is a mathematical technique for helping the decision maker choose between several alternative decisions, given more than one objective.
• Simulation is a technique that can solve probabilistic and deterministic problems and is often the technique of last resort when no other management science technique will work.

Business Usage of Management Science Techniques

• Not all management science techniques are equally useful or equally used by business firms and other organizations.
• The most frequently used techniques are linear and integer programming, simulation, network analysis, inventory control, decision analysis, and queuing theory, as well as probability and statistics.

Management Science Models in Decision Support Systems

• A decision support system (DSS) is a computer-based system that helps decision makers address complex problems that cut across different parts of an organization and operations.
• A DSS is normally interactive, combining various databases and different management science models and solution techniques with a user interface.