# Z94.1 - Analytical Techniques & Operations Research Terminology

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SAMPLE. A group of items, observations, test results, or portions of material, taken randomly from a larger collection of items, observations, tests results, or quantities of material, which serves to provide information that may be used as a basis for making a decision concerning the larger collection.

SAMPLE SIZE. Number of units in a sample. When a sample of 10 units is taken from a    population of 1,000 units, for example, we say that the sample size is 10.

SCREENING TEST. A test or combination of tests intended to remove unsatisfactory items or those likely to exhibit early failures. 

SECONDARY FAILURE. (See FAILURE, SECONDARY). 

SENSITIVITY. The responsiveness of a solution to changes in the numerical values of the problem parameters. 

SENSITIVITY ANALYSIS. An analysis of the effect on the solution of a mathematical problem, for example, as the parameters of the problem are varied. 

SENSITIVITY TESTING. A general technique which attempts to determine a relation between quantal response and some stimulus by using a destructive test.

SEPARABLE CONSTRAINT. Let g(x1,x2,...,xn) ≤ 0 be the ith constraint in a mathematical programming formulation. Then, if g(x1,x2,...,xn) = g1(x1) + g2(x2) + ... + gn(xn), the constraint is said to be separable. 

SEPARABLE OBJECTIVE FUNCTION. When the objective function of a mathematical programming problem in n variables is capable of being written as a sum of n functions each of which is a function of only a single variable, the objective function is said to be separable. More precisely Z = f(x1,x2...,xn) = f1(x1) + f2(X2) + ...+ fn(xn) is separable. 

SEPARABLE PROGRAMMING. A class of nonlinear programming problems in which each function appearing may be expressed as the sum of separate functions of single variables. When all the functions have been separated, it is possible to make a piecewise linear approximation to each one. For convex constraints, this is all that is required, since the LP algorithm will always choose the two nearest points surrounding the desired value, giving accuracy dependent only on the fineness of the approximation. For concave constraints, however, the ordinary LP algorithms will choose from the two most widely separated points, ignoring those in between and giving generally a most unsatisfactory accuracy. However, the separable programming algorithm assures that the points chosen are contiguous, and thus accurate. Of course, only a local optimum solution is assured.

SEQUENTIAL SAMPLING. Sampling technique in which a succession of samples, of a particular size each, are chosen at random.

SERVICE RATE DISTRIBUTION, CONSTANT. (See ARRIVAL RATE DISTRIBUTION, CONSTANT).

SERVICE TIME, CONSTANT. (See ARRIVAL RATE DISTRIBUTION, CONSTANT).

SERVICEABILITY. A measure of the degree to which servicing of an item will be accomplished within a given time under specified conditions. 

SERVICING. The replenishment of consumables needed to keep an item in operating condition, but not including any other preventive maintenance or any corrective maintenance. 

SERVICE RATE, MEAN. (See MEAN SERVICE RATE).

SET. Any well-defined collection of objects, things or symbols. To be well-defined, it must be possible to tell beyond doubt whether or not a given object, thing or symbol belongs to the collection being considered.

SET.  Any well-defined collection of objects, things or symbols.  To be well-defined, it must be possible to tell beyond doubt whether or not at given object, thing or symbol belongs to the collection being considered.

SHORTEST ROUTE PROBLEM. Given N nodes numbered i = 1, 2,..., N and a set of numbers (dij) where dij = the distance required travel from the ith node to the jth node. Find the minimum distance route from node 1 to N. 

SIGNAL FLOW GRAPH. A graphical model of a system in which nodes represent the values of system variables and directed branches between the nodes represent functional relationships between the system variables.

SIGNIFICANCE TESTING. Statistical appraisal of the outcomes of sampling to note whether or not, at a certain level of risk, the results represent real effects or chance fluctuations of sampling and measurement.

SIMPLEX. A simplex is an e-dimensional convex polyhedron having exactly n + 1 vertices. 

SIMPLEX ALGORITHM. (See SIMPLEX METHOD).

SIMPLEX METHOD. A computational procedure for solving a linear programming problem. The method embodies an algebraic algorithm, termed the simplex algorithm, which transforms by iterative steps a starting solution into an optimum solution (or determines that such a solution does not exist). 

SIMULATION. The design and operation of a model of a system. Commonly implies the use of a computer program designed to accept selected inputs; to treat those inputs in a manner analogous to the way the real system operates, and to read out measurements of the status and change results of the programmed operations affected by those inputs.

SINGULAR MATRIX. A square matrix whose rank is less than its dimension and whose determinant is zero. A singular matrix has no inverse. 

SINK. That single node in a capacitated network at which all flow is assumed to terminate.

SLACK. The difference between the larger and smaller members of an inequality. If u ≤ v, the slack is defined as s = v - u. 

SLACK VARIABLE. An auxiliary variable introduced to convert an inequality constraint to an equation. 

SLACK VECTOR. The column vector associated with a slack variable. It is a unit vector with + 1 or - 1 in the row in which the slack appears, and zeros elsewhere. 

SOLUTION. In linear programming, a set of values of the variables which satisfy all the given constraints. If all of these values are nonnegative, the solution is called feasible; if one or more are negative, it is called nonfeasible. Solutions are otherwise classified as basic (number of variables with nonzero values less than or equal to the number of constraints) and nonbasic (number of nonzero variables greater than the number of constraints). 

SOLUTION LEVEL. The set of values taken by the variables in a solution.

SOURCE. That single node in a capacitated network from which all flow is assumed to originate.

SPARSE VECTOR OR MATRIX. A vector or matrix whose elements are mostly zeros. 

STANDARD DEVIATION. The most usual measure of dispersion of observed values or results expressed as the positive square root of the variance.

STANDARD GAMBLE. A gamble where one gains XL with probability p or otherwise gains XH with probability (1-p).

STANDBY REDUNDANCY. (See REDUNDANCY, STANDBY). 

STARTING BASIS. The set of column vectors associated with the basic variables of a starting solution in linear programming. 

STARTING SOLUTION. The first solution used at the beginning of an iterative-type solution to an optimization problem.

STATE PROBABILITY IN QUEUING MODELS. The state of a system is specified by the number of units in the queue, waiting for service, the number of units in service, etc. The state probability is then the probability that the system is in a given state.

STATISTIC. A function of one or more random variables which does not depend upon any unknown parameter. Common statistics are the prior sample mean or variance of a random sample.

STATISTICAL QUALITY CONTROL. The application of statistical techniques to the control of quality.

STEADY STATE. A physical condition in which the variables (q.v.) affecting a system (q.v.) are either invariant or periodic functions of time.

STEPPINGSTONE METHOD. A name for a special form of the simplex method used for solving the transportation problem. Given a feasible solution, the algorithm tells how to improve this solution by introducing a new source- destination allocation and reallocating among existing allocations so that the constraints remain satisfied. Those existing allocations whose values are modified in this way are called steppingstones. If no such reallocation can improve the solution, the current solution is optimal. 

STEP STRESS TEST. A test consisting of several stress levels applied sequentially for periods of equal duration to a sample. During each period a state stress level is applied and the stress level is increased from one step to the next. 

STOCHASTIC. The adjective “stochastic” implies the presence of a random variable; e.g., stochastic variation is variation in which at least one of the elements is a variate and a stochastic process is one wherein the system incorporates an element of randomness as opposed to a deterministic system. [22:279, 4:6, 29:161]

STOCHASTIC PROGRAMMING. A generalization of linear programming in which any of the unit costs, the coefficients in the constraint equations, and the right hand sides may be random variables. The aim of such programming is to choose levels for the variables which will minimize some function of the cost. (See PERMANENTLY FEASIBLE SET). 

STORAGE LIFE (SHELF LIFE). The length of time an item can be stored under specified conditions and still meet specified requirements. 

STRATEGY. The decision rule used for making the choice from available courses of    action.

STRATEGY (MIXED). (See MINIMAX THEOREM).

STRATIFIED RANDOM SAMPLING. The process of selecting a simple random sample from each of the population strata. 

STRATUM. A group of units from a population, a subpopulation, usually defined by relevant population characteristics. 

STRICTLY CONCAVE FUNCTION. The negative of a strictly convex function. 

STRICTLY CONVEX FUNCTION. (See CONVEX FUNCTION).

STRICTLY INCREASING FUNCTION. A function f is strictly increasing on an interval (a,b) if f (y) > f (x) for any two numbers x and y (of this interval), for which x < y. 

SUBOPTIMAL. (1) Not yet optimal. (2) Optimal over a subregion of the feasible region. 

SUPPORTING HYPERPLANE. Let W be a boundary point of a convex set D. Then CTX = Z is called a supporting hyperplane at W if CTW = Z and if all of D lies in one closed half-space produced by the hyperplane. For every boundary point W of D, there exists at least one supporting hyperplane at W. 

SURPLUS VARIABLE. A slack variable when an inequality is of the “greater than or equal to” form.

SURVEILLANCE. (1) The continuing analysis and evaluation of records, methods, and procedures including the act of verification to assure conformance with technical requirements. (2) A system whereby supplies and equipment in storage are subjected to, but not limited to, cyclic, scheduled, and special inspection and continuous action to assure that material is maintained in ready for issue condition. 

SURVIVABILITY. The measure of the degree to which an item will withstand hostile man-made environment and not suffer abortive impairment of its ability to accomplish its designated mission. 

SYMMETRIC PARAMETRIC PROGRAMMING. The simultaneous parameterization of the right-hand side and the objective function. This is useful in economic studies when both costs and requirements change as a linear function of a single parameter - for example, time. Another use is as a sort of primal-dual algorithm to go from a pseudo right-hand side and objective function (for which the starting basis is optimal and feasible) to true values of the right-hand side and objective function. In some problems, this approach requires far fewer iterations than a more conventional algorithm. 

SYSTEM. A set of objects with relations between the objects and their attributes.

SYSTEM EFFECTIVENESS. A measure of the degree to which an item can be expected to achieve a set of specific mission requirements, and which may be expressed as a function of availability, dependability and capability. 

SYSTEM, LINEAR. A system is said to be linear if an input f1 (t) produces an output g1 (t) and an input f2 (t) produces an output g2 (t) and if then an input af1(t) + bf2(t) produces an output ag1(t) + bg2(t) for all f1(t) and f2(t) and all constants a and b.

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