BSAD030 Exam 2

Extreme point

Graphically speaking, extreme points are the feasible solution points occurring at the vertices, or "corners," of the feasible region. With two-variable problems, extreme points are determined by the intersection of the constraint lines

Reduced cost

If a variable is at its lower bound of zero, the reduced cost is equal to the shadow price of the non-negativity constraint for that variable.

alternative optimal solutions (from slides)

In the graphical method, if the objective function line is parallel to a boundary constraint in the direction of optimization, there are alternate optimal solutions, with all points on this line segment being optimal.

Blending Models

Involves mixing materials with individual properties and describing the properties of the blend with weighted averages.

linear function

Mathematical expressions in which the variables appear in separate terms and are raised to the first power.

infeasibility

No solution to the LP problem satisfies all the constraints, including the non-negativity conditions. Graphically, this means a feasible region does not exist. Causes include: -A formulation error has been made. -Management's expectations are too high. -Too many restrictions have been placed on the problem (i.e. the problem is over-constrained).

Right-hand-side allowable increase (decrease)

The allowable increase (decrease) of the right-hand side of a constraint is the amount the right-hand side may increase (decrease) without causing any change in the shadow price for that constraint. The allowable increase and decrease for the right-hand side can be used to calculate the range of feasibility for that constraint.

Alternative optimal solutions (from book)

The case in which more than one solution provides the optimal value for the objective function.

Shadow price

The change in the optimal objective function value per unit increase in the right-hand side of a constraint.

objective function

The expression that defines the quantity to be maximized or minimized in a linear programming model.

Feasible Region

The feasible region for a two-variable LP problem can be nonexistent, a single point, a line, a polygon, or an unbounded area Any linear program falls in one of four categories: -is infeasible -has a unique optimal solution -has alternative optimal solutions -has an objective function that can be increased without bound A feasible region may be unbounded and yet there may be optimal solutions. This is common in minimization problems and is possible in maximization problems.

problem formulation

The process of translating a verbal statement of a problem into a mathematical statement called the mathematical model.

Range of feasibility

The range of values over which the shadow price is applicable.

unbounded

The situation in which the value of the solution may be made infinitely large in a maximization linear programming problem or infinitely small in a minimization problem without violating any of the constraints. For real problems, this is the result of improper formulation. (Quite likely, a constraint has been inadvertently omitted.)

Sensitivity analysis

The study of how changes in the coefficients of a linear programming problem affect the optimal solution. is important to a manager who must operate in a dynamic environment with imprecise estimates of the coefficients. allows a manager to ask certain what-if questions about the problem.

Allocation Models

calls for a maximum objective, subject to less-than constraints (<=) on capacity

Covering Models

calls for a minimum objective function, subject to greater than constraints on required coverage.

Dual Value/Shadow Price

is determined by adding +1 to the right hand side value in question and then resolving for the optimal solution in terms of the same two binding constraints. equal to the difference in the values of the objective functions between the new and original problems.

range of optimality for objective function coefficients

only applicable for changes made to one coefficient at a time.

Slack and surplus variables

-A linear program in which all the variables are non-negative and all the constraints are equalities is said to be in standard form. -Standard form is attained by adding slack variables to "less than or equal to" constraints, and by subtracting surplus variables from "greater than or equal to" constraints. -Slack and surplus variables represent the difference between the left and right sides of the constraints. -Slack and surplus variables have objective function coefficients equal to 0.

Summary of the Graphical Solution Procedure for Minimization Problems

-Prepare a graph of the feasible solutions for each of the constraints. -Determine the feasible region that satisfies all the constraints simultaneously. -Draw an objective function line. -Move parallel objective function lines toward smaller objective function values without entirely leaving the feasible region. -Any feasible solution on the objective function line with the smallest value is an optimal solution.

Extreme Points and the Optimal Solution

-The corners or vertices of the feasible region --are referred to as the extreme points. -An optimal solution to an LP problem can be --found at an extreme point of the feasible region. -When looking for the optimal solution, you do not have to evaluate all feasible solution points. -You have to consider only the extreme points of the feasible region.

guidelines for problem formulation

-Understand the problem thoroughly. -Define the decision variables. -Describe the objective. -Describe each constraint. -Write the objective in terms of the decision -variables. -Write the constraints in terms of the decision variables.

redundant constraint

A constraint that does not affect the feasible region. If a constraint is redundant, it can be removed from the problem without affecting the feasible region.

decision variable

A controllable input for a linear programming model.

Relevant cost

A cost that depends upon the decision made. The amount of a relevant cost will vary depending on the values of the decision variables.

Sunk cost

A cost that is not affected by the decision made. It will be incurred no matter what values the decision variables assume.

standard form

A linear program in which all the constraints are written as equalities. The optimal solution of the standard form of a linear program is the same as the optimal solution of the original formulation of the linear program.

linear program

A mathematical model with a linear objective function, a set of linear constraints, and nonnegative variables.

mathmatical model

A representation of a problem where the objective and all constraint conditions are described by mathematical expressions.

non-negativity

A set of constraints that requires all variables to be nonnegative.

feasible solution

A solution that satisfies all the constraints simultaneously.

slack variable

A variable added to the left-hand side of a less-than-or-equal-to constraint to convert the constraint into an equality. The value of this variable can usually be interpreted as the amount of unused resource.

surplus variable

A variable subtracted from the left-hand side of a greater-than-or-equal-to constraint to convert the constraint into an equality. The value of this variable can usually be interpreted as the amount over and above some required minimum level.

Constraint

An equation or inequality that rules out certain combinations of decision variables as feasible solutions.

Network Models

Characterized by a set of supply sources, a set of demand locations, and the unit costs of transportation between supply-demand pairs Transportation models are the most common of all of these models.

Changes in Constraint Coefficients

Classical sensitivity analysis provides no information about changes resulting from a change in a coefficient of a variable in a constraint.

Non-intuitive Dual Values

Constraints with variables naturally on both the left-hand and right-hand sides often lead to dual values that have a non-intuitive explanation.