In an Linear programming problem, the restrictions or limitations under which the objective function is to be optimised are called
Constraints
Understanding Linear Programming Problems
A Linear Programming Problem (LPP) is a mathematical method used to determine the best possible outcome (like maximum profit or minimum cost) in a mathematical model whose requirements are represented by linear relationships. The core components of an LPP are:
- Decision Variables
- Objective Function
- Constraints
Decision Variables in Linear Programming
Decision variables are the unknowns in the problem. They represent the quantities that we need to determine to achieve the objective. For example, if a company produces two products, the number of units of each product to produce would be the decision variables.
Objective Function in Linear Programming
The objective function is the mathematical expression that represents the goal of the problem. This goal is typically to maximize profit or minimize cost. The objective function is a linear equation involving the decision variables.
Constraints: The Limitations in Linear Programming
Constraints are the restrictions or limitations placed on the decision variables. These limitations usually arise from limited resources such as labor, materials, budget, or production capacity. Constraints are expressed as linear inequalities or equations involving the decision variables.
The question asks for the term used for the restrictions or limitations in a Linear programming problem. Based on the definitions:
- The Objective function is what you want to optimize (maximize or minimize).
- Decision variables are what you are trying to find the values for.
- Constraints are the conditions, restrictions, or limitations that must be satisfied.
Therefore, the restrictions or limitations under which the objective function is optimized are called Constraints.
Analyzing the Options
- Constraints: These are indeed the restrictions or limitations in an LPP.
- Objective function: This is the function to be optimized, not the restrictions.
- Decision variables: These are the quantities to be determined, not the restrictions themselves.
- None of the above: This is incorrect as Constraints fit the description.
| Term | Description |
|---|---|
| Decision Variables | Unknown quantities to be determined |
| Objective Function | Mathematical expression to optimize (maximize/minimize) |
| Constraints | Restrictions or limitations on decision variables (using resources, capacity, etc.) |
Revision Table: Key Linear Programming Terms
| Concept | Role in LPP | Example |
|---|---|---|
| Decision Variables | The quantities you control to solve the problem. | Number of items to produce, amount of resources to allocate. |
| Objective Function | The goal you want to achieve, expressed mathematically. | Maximize Profit $= 5x + 7y$ (where x, y are decision variables) |
| Constraints | The boundaries or limitations on your decisions due to limited resources or requirements. | Labor hours available $\le$ 100, Material used $\le$ 50 kg, Production capacity $\le$ 200 units. |
Additional Information on Linear Programming
Linear programming is a powerful tool used in various fields like business, economics, engineering, and operations research for making optimal decisions. The problems involve linear relationships, meaning the variables are only multiplied by constants and added together. The feasible region in an LPP is the set of all possible solutions that satisfy all the constraints. The optimal solution lies within this feasible region, usually at one of its corner points (vertices).
Understanding the distinct roles of decision variables, the objective function, and constraints is fundamental to formulating and solving Linear Programming problems.
Important Questions from Linear Programming
For the linear programming problem:
Maximum Z = 3X1 + 2X2
Subject to
-2X1 + 3X2 ≤ 9
X1 – 5X2 ≥ - 20
X1, X2 ≥ 0
The above problem has
The headquarters of the Eastern Railway Zone is located at _______.
Consider an LPP given as
Max Z = 2x 1 - x 2+ 2x 3
Subject to the constraints
2x 1+ x 2 ≤ 10
x 1+ 2x 2 - 2x 3 ≤ 20
x 1+ 2x 3 ≤ 5
x 1, x 2, x 3 ≥ 0
What shall be the solution of the LPP after applying first iteration of the Simplex Method?
Consider the following Linear programming problem (LPP):
Maximize z = x 1+ x 2
Subject to the constraints:
x 1+ 2x 2≤ 2000
x 1+ x 2≤ 1500
x 2≤ 600
and x 1, x 2≥ 0
The solution of the above LPP is:Consider the following statements with respect to duality in LPP
(a) The final simplex table giving optimal solution of the primal also contains optimal solution of its dual in itself.
(b) If either the primal or the dual problem has a finite optimal solution, then the other problem also has a finite optimal solution.
(c) If either problem has an unbounded optimum solution, then the other problem has no feasible solution at all
Which of the statements is (are) correct?
