Integer Programming

Integer Programming

An optimization method where some or all decision variables are constrained to integer values.

Total Integer Model

A model in which all decision variables must be integers, used when fractional values are not acceptable.

0

1 Integer Model

Mixed Integer Model

A model combining integer and continuous decision variables within the same formulation.

Decision Variable

A variable representing a controllable input in the model whose value is determined during optimization.

Objective Function

The formula that defines the goal of the model, such as maximizing profit or minimizing cost.

Constraint

A condition that decision variables must satisfy, reflecting limits like capacity or budget.

Feasible Solution

A solution that satisfies all constraints and integrality conditions of the model.

Optimal Solution

The best feasible solution according to the objective function.

Graphical Solution Method

A visual technique to solve two-variable linear models by plotting constraints and feasible regions.

Branch and Bound Method

A solution algorithm that breaks integer problems into subproblems and evaluates them systematically.

Binary Variable

A variable restricted to 0 or 1, used for inclusion/exclusion decisions.

Mutually Exclusive Constraint

A restriction ensuring only one variable in a group can be 1 at a time.

Conditional Constraint

A rule linking binary variables, such that one choice depends on another.

Corequisite Constraint

A condition that forces binary variables to be equal, ensuring joint inclusion or exclusion.

Slack

The unused portion of a ≤ constraint, representing spare capacity or resources.

Suboptimal Solution

A feasible but non-optimal solution that doesn't yield the best value of the objective function.

Set Covering Model

A binary model that selects the minimum number of options to cover all requirements.

Capital Budgeting Model

A 0

Fixed-Charge Model

A model with fixed setup costs plus variable costs, used in logistics and facility planning.

Rounding Down

A heuristic that floors fractional values, possibly resulting in feasible but suboptimal outcomes.

Rounding Up

A heuristic that ceilings fractional values, potentially causing infeasibility.

Enumeration Method

A brute-force method that checks all possible combinations of integer variables.

Solver (Excel)

A tool in Excel for solving optimization problems using linear and integer programming.

QM for Windows

Software used for modeling and solving various operations research problems, including integer programming.

Cutting Plane Method

An algorithm that adds linear constraints iteratively to eliminate non-integer solutions.

Constraint Violation

Occurs when a proposed solution does not satisfy one or more constraints.

Feasible Region

The set of all decision variable values that satisfy all constraints.

Operating Room Allocation Model

An integer programming application for assigning surgical time slots efficiently.

Scheduling Problem

An application where events or tasks are assigned to time slots or locations under constraints.

Multiple-Choice Constraint

A restriction requiring exactly one choice among a group of binary variables.

Investment Model

An optimization model that allocates capital among investment options under constraints.

Real-World Application

A practical use of integer programming in fields like logistics, finance, or healthcare.

Facility Location Problem

A model to find optimal facility sites minimizing fixed and transportation costs.

Set Covering Problem

A binary model aiming to cover all targets with the fewest selected sets or resources.

Real Variable

A decision variable that can assume any non-negative continuous value.

Budget Constraint

A restriction ensuring total costs do not exceed available funds.

Space Constraint

A limit on the amount of physical resource usage like area or volume.

Profit Maximization

An objective that seeks to maximize the difference between revenue and costs.

Constraint Coefficient

The multiplier of a decision variable in a constraint or objective function.

Integer Feasibility

A condition where all integer variables in a solution have valid integer values.

Infeasible Solution

A solution that violates one or more model constraints.

Enumeration Tree

A branching structure used in branch and bound to explore solution paths.

Optimality Gap

The difference between the best-known and best-possible solution values.

Constraint Binding

A constraint that holds as an equality and directly influences the optimal solution.