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Operations Research Models and Methods
Models Section


Decision variables describe the quantities that the decision makers would like to determine. They are the unknowns of a mathematical programming model. Typically we will determine their optimum values with an optimization method. In a general model, decision variables are given algebraic designations such as . The number of decision variables is n, and is the name of the jth variable. In a specific situation, it is often convenient to use other names such as or or . In computer models we use names such as FLOW1 or AB_5 to represent specific problem-related quantities. An assignment of values to all variables in a problem is called a solution.

The objective function evaluates some quantitative criterion of immediate importance such as cost, profit, utility, or yield. The general linear objective function can be written as

Here is the coefficient of the jth decision variable. The criterion selected can be either maximized or minimized.


A constraint is an inequality or equality defining limitations on decisions. Constraints arise from a variety of sources such as limited resources, contractual obligations, or physical laws. In general, an LP is said to have m linear constraints that can be stated as

One of the three relations shown in the large brackets must be chosen for each constraint. The number is called a "technological coefficient," and the number is called the "right-hand side" value of the ith constraint. Strict inequalities (< and >) are not permitted. When formulating a model, it is good practice to give a name to each constraint that reflects its purpose.


Associated with each variable, , may be a specified quantity, , that limits its value from above;

When a simple upper is not specified for a variable, the variable is said to be unbounded from above.


In most practical problems the variables are required to be nonnegative;

This special kind of constraint is called a nonnegativity restriction. Sometimes variables are required to be nonpositive or, in fact, may be unrestricted (allowing any real value).


Combining the aforementioned components into a single statement gives:

The constraints, including nonnegativity and simple upper bounds, define the feasible region of a problem.


The collection of coefficients for all values of the indices i and j are called the parameters of the model. For the model to be completely determined all parameter values must be known.

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Operations Research Models and Methods
by Paul A. Jensen
Copyright 2004 - All rights reserved