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# Operations Research

Operations research, or management science, is the application of advanced analytical methods in mathematics to complex business decision-making problems. By using mathematical methods, analysts can model business decisions and solve optimization problems (such as profit maximization or cost minimization problems) using linear and nonlinear programming and statistical analysis. These problems are simplified with the use of computer programs such as Excel Solver.

Linear Programming (LP)

Linear programming is concerned with the optimization (minimization or maximization) of a linear function while satisfying a set of linear equalities and/or inequalities (collectively known as constraints). Linear programming in operations research can be applied to a variety of problems.

Resource allocation: The most typical problems in business are resource allocation problems. For example, a common type of resource allocation problem is the problem of product mix, which looks at the optimal mix of products that will maximize profit subject to resource constraints such as limited machine time or a limited quantity of a necessary input. A solution to this problem will also provide other valuable information that may allow the student to identify bottlenecks, find the range over which the unit profit can change without affecting the product mix, identify the marginal benefit of adding additional units of the limited resource, and find the range of quantity over which the availability of the resource can change without affecting the identity of the bottleneck in the system.

Blending: Blending problems are problems that involve finding the optimal blend or mix of ingredients in order to minimize costs and satisfy some additional requirement. For example, crude oil blending allows refineries to mix different crude oil products to achieve some optimal and consistent product. Similarly, ingredients in animal feed stock may be combined in different amounts to achieve some optimal blend that meets minimum nutritional requirements.

Queuing: Queuing models look at resource decisions such as determining optimal inventory levels to minimize costs in the face of fixed ordering costs (such as delivery costs), carrying costs for units on hand, seasonal price fluctuations, and fluctuations in demand (and costs associated with stock outs).

Scheduling: Scheduling problems include problems such as personnel staffing, project tasks, and determining the optimal coverage of sporting events.

Network flow and network optimization: Network flow problems are a special case in linear programming that look at things like transportation or routing problems, critical path analysis (project management), the assignment problem, and other flow problems.

Facility location: Facility location problems look to minimize things like transportation costs, or address other criteria such as taking advantage of price-differentials in the case of low-cost labour in other jurisdictions.  Facility location problems also include problems such as floor planning.

Investment decisions: Linear programming can also be used to solve investment problems such as designing the optimal portfolio to minimize, or hedge, risk.

Nonlinear Programming

Nonlinear programming problems are similar to linear programming problems in that they involve the optimization of an objective function while satisfying a set of equalities and/or inequalities. However, unlike linear programs, nonlinear programs exist where the objective function or some of the constraints are nonlinear. For example, linear problems may become nonlinear problems if one of the cost functions exhibits economies of scale.

Integer Programming

Integer programming refers to linear or nonlinear optimization problems where some of the variables are required to take on discrete values. They can be modeled with binomial variables that take on values of either zero or one. Many business decision problems require integer programming because their variable units must be whole numbers. For example, it would not make sense for a optimal solution to require a company to build half a factory in one location, to produce only one-third of a widget, to keep 20 and 1/8 units of inventory on hand, to hire less than one person, or to route half a bus one way and the other half another. Thus program designers are faced with a finite set of alternatives and face discrete decision problems.

Dynamic Programming

Dynamic programming involves problems that require a sequence of decisions over time, where one decision must follow logically from the preceding outcome. Many investment decisions are modeled using dynamic programming, because sell, hold, or buy decisions can be programmed in the model based on fluctuations in the prices of securities.

Monte Carlo Simulation

Monte Carlo methods involve repeated random sampling in order to obtain variables or outcomes. Therefore, by running the simulation again and again, a computer program can provide results in the form of statistical probabilities. The name derives from the fact that the simulation can be likened to gambling at a casino, recording your winnings, and determining from this information the probability of winning your next hand.

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