Linear Programming MCQs

A constraint is considered binding if it is fully utilized in the optimal solution, meaning all available resources are used.

The shadow price of a constraint represents the change in the objective function (usually profit) for each additional unit of the constrained resource.

In a transportation problem, a demand constraint for a warehouse ensures that the total shipments arriving at that warehouse equal its required demand.

The objective function in production planning aims to minimize the total cost of production, including costs associated with manufacturing, holding inventory, and changes in production levels.

The cost of holding inventory (storage cost) is usually a component of the objective function, reflecting the expenses associated with storing finished goods.

The cost for changing production levels (increasing or decreasing production) is often included in the objective function. It reflects the expenses associated with adjusting production volume.

The production cost per unit is the cost of manufacturing a single unit of a product. It usually includes raw materials, labor, and manufacturing overhead.

Production planning involves determining the optimal production levels for different products to meet demand while minimizing costs.

Ensures that the total amount of labor used for producing two products (P1 and P2) does not exceed the available labor hours (600 hours).

Represents the total amount of units shipped from both factories (Factory 1 and Factory 2) to the warehouse. It ensures that at least 400 units are shipped to the warehouse.

This constraint guarantees a balance in the flow of products through a warehouse. It ensures that the total amount of products entering the warehouse (from various sources) equals the total amount leaving (going to different destinations).

Ensures that the proportion of Material A in a blend is at least 30%. This constraint is used in blending problems where certain materials must be present in a minimum proportion to meet quality standards.

The total cost of production (including costs related to both products A and B) plus the cost of holding inventory and the cost of changes in production. The formula calculates the total cost and can be used for optimizing production decisions.

A solution to the constraint (3x1 + 5x2 ≤ 15) is feasible if it satisfies the constraint. In this case, only a certain combination of x1 and x2 values will satisfy the inequality.

Represents a constraint that ensures the production quantity of each product (x1 and x2) does not exceed the available resources. In this case, the constraint ensures that the combined labor hours required for producing both products remains within the total available labor hours (600 hours).

A mathematical expression that describes the relationship between variables and how they affect the total cost (Z). This formula accounts for production costs, inventory holding costs, and changes in production.

Decision variables represent the quantities that need to be figured out to solve the problem. For example, how many units should be produced, shipped, or allocated. These variables are always measurable and directly linked to the problem's requirements and the goal of the optimization.

The objective function in linear programming tells you what you're trying to achieve, either minimizing costs or maximizing profits. It's expressed as a linear equation, like Z = 3x1 + 5x2.

Constraints in linear programming define the restrictions or limits within which the solution must operate. Think of them as boundaries or rules to follow. For example, limited machine capacity or the maximum number of units you can produce.

The feasible region in linear programming is the area on a graph where all the constraints are satisfied simultaneously. It's the area where your solutions are allowed to exist based on your limitations.

The shadow price tells you how much the objective function (like profit or cost) would change if you increased one unit of a constrained resource. This only applies to constraints that are already fully used (binding constraints).

An unbounded problem in linear programming happens when the feasible region (the solution space) can continue indefinitely in the direction you're trying to optimize (min/max). This means there's no clear best solution.

An infeasible problem in linear programming means there's no solution that can satisfy all of the constraints at the same time. Your problem is set up with impossible requirements.

Slack variables are used in linear programming to turn inequality constraints (≤ or ≥) into equality constraints (=). This allows you to work with the problem in a more standardized form. These variables represent unused resources.

The area where all constraints overlap, representing all possible solution combinations that satisfy all the defined limitations.

A constraint is considered non-binding if it is not fully utilized in the optimal solution, meaning there is unused capacity or resources. This means increasing the resource would not lead to a change in the objective function (e.g., profit).

The objective function in production planning aims to minimize the total cost of production, including costs associated with manufacturing, holding inventory, and changes in production levels.

In a linear programming model, infeasibility occurs when there is no solution that satisfies all constraints. It means the constraints are so stringent that an overlapping feasible region does not exist.

A transshipment model is a type of network flow problem where goods can be shipped through intermediate nodes (transshipment points) before reaching their final destination.

Slack variables are introduced in linear programming when an inequality constraint is transformed into an equality constraint. They represent the difference between the left and right sides of the inequality.

The constraint xin+xjn=xnk+xnlx_{in} + x_{jn} = x_{nk} + x_{nl} in a transshipment model ensures that for each node, the total flow entering the node equals the total flow leaving the node.