# Z94.1 - Analytical Techniques & Operations Research Terminology

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IDENTITY MATRIX. A square matrix with all main diagonal elements unity and the remaining elements zero.

IMPLICIT ENUMERATION. A method (used in branch and bound algorithms) of checking all possible realizations of a binary vector without explicitly enumerating all possible cases.

IMPLICIT PRICES. Same as marginal values, shadow prices, dual variable levels, i.e., numbers giving the incremental worth of a relaxation of one unit in the right-hand side of a constraint. 

INDEPENDENT EQUATIONS. A set of equations none of which can be expressed as a linear combination of the others. With linear equations, the condition for independence is that the matrix (coefficient columns) shall be nonsingular or, equivalently, have rank equal to the number of equations. 

INDEPENDENT TRIALS. The successive trials of an event are said to be independent if the probability of outcome of any trial is independent of the outcome of the others. The expression is usually confined to cases where the probability is the same for all trials. In the sampling of attributes, such a series of trials is often referred to as “Bernoullian Trials.” It includes all the classical cases of drawing colored balls from urns with replacement after each draw, coin tossing, dice rolling, and the events associated with other games of chance. [22:136, 13:118, 25:95, 29:44]

INDEPENDENT VARIABLE. This term is regularly used in contradistinction to “dependent variable” in regression analysis. When a variate y is expressed as a function of variables xl, x2, etc., plus a stochastic term the x’s are known as “independent variables.” The terminology is rather unfortunate since the concept has no connection with either mathematical or statistical dependence. The usage is so convenient, however, and so common that strongly coordinated action would be necessary to change it. 

INEQUALITY. A proposition which relates the magnitudes of two mathematical expressions or functions A and B. Inequalities are of four types: A is greater than B (A > B); A is less than B (A < B); A is greater than or equal to B       (A ≥ B); A is less than or equal to B (A ≤ B). The first two types are called “strict” and the last two “relaxed” or “weak.” 

INEQUALITY RELATION. A constraint which is an inequality. 

INFEASIBLE BASIS. A basis corresponding to an infeasible solution. Postmultiplying  the inverse of such a basis by the constant vector, one obtains a solution in which at least one variable value is negative. 

INITIAL BASIS. The set of column vectors associated with the basic variables for which a starting solution will be obtained in linear programming. It is often an identity matrix. 

INITIAL FEASIBLE BASIS. The set of column vectors associated with the basic variables of the first feasible solution to a linear programming problem. 

INITIAL SOLUTION. The solution used at the beginning of an iterative solution to a mathematical problem.

INPUT-OUTPUT COEFFICIENT. A coefficient of an activity vector of the activity analysis problem. The (i,j) coefficient represents the amount of resource i required to produce one unit of activity j. 

INSPECTION. Activities, such as measuring, examining, testing, gauging one or more characteristics of a product or service and comparing these with specified requirements to determine conformity.

INTEGER FORM. A cutting plan used to solve integer linear programs. 

INTEGER LINEAR PROGRAMMING. A linear programming problem with the added restriction that some or all of the variables are constrained to be integers. 

INTERARRIVAL TIME. The time between two successive arrivals. (Mean interarrival time: the arithmetic average of the actual interarrival times or the expected value of the interarrival time distribution; reciprocal of interarrival rate).

INTERINDUSTRY (INPUT-OUTPUT) ANALYSIS. An interpretation of the economy in terms of the interactions of industrial organizations. Also termed “Leontief Model.” 

INVENTORY, ACTIVE. That group of items in a storage assigned an operational status. 

INVENTORY, INACTIVE. That group of items in storage being held in reserve for possible future commitment to the operational inventory. 

INVERSE OF A MATRIX. An inverse of the square matrix A is another matrix B of the same dimension such that AB = BA = I, where I is the identity matrix. 

INVERSION. An operation on a matrix yielding the matrix’s inverse. 

ITEM, INTERCHANGEABLE. One which (1) possesses such functional and physical    characteristics as to be equivalent in performance, reliability, and maintainability, to another item of similar or identical purpose; and (2) is capable of being exchanged for the other item a) without selection for fit or performance, and b) without alteration of the items themselves or of adjoining items, except for adjustment. 

ITEM, REPLACEABLE. One which is interchangeable with another item, but which differs physically from the original item in that the installation of the replaceable item requires operations such as drilling, reaming, cutting, filing, shimming, etc., in addition to the normal application and methods of attachment. 

ITEM, SUBSTITUTE. One which possesses such functional and physical characteristics    as to be capable of being exchanged for another only under specified conditions or for particular applications and without alteration of the items themselves or of adjoining items. 

ITEM, UNIT. An object or quantity of material and/or service on which a set of observations can be made.

ITERATION. A single cycle of operations in a solution algorithm made up of a number of such cycles. 

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