Q.The degree of an objective function of a linear programming problem is 1
(A) 0
(B) 1
(C) 2
(D) Any natural number Assertion – Reason Based Questions Direction : Questions number 19 and 20 are Assertion and Reason based questions carrying 1 mark each. Two statements are given, one labelled Assertion (A) and other labelled Reason (R). Select the correct answer from the codes (A), (B), (C) and (D) as given below. (A) Both Assertion (A) and Reason (R) are true and the Reason (R) is the correct explanation of the Assertion (A). (B) Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion (A). (C) Assertion (A) is true and Reason (R) is false. (D) Assertion (A) is false and Reason (R) is true.

The question asks for the degree of the objective function in a linear programming problem. This is a concept check — it tests whether you understand what "linear" means in this context.

In linear programming, every constraint and the objective function must be linear. A linear expression in variables $x_1, x_2, \dots, x_n$ has the form $c_1 x_1 + c_2 x_2 + \dots + c_n x_n + d$, where each $c_i$ and $d$ are constants. No variable is squared, cubed, or multiplied by another variable — the highest power of any variable is exactly 1.

So the degree of the objective function — the highest exponent of any variable in it — is always 1. That is the defining property of linearity.

Now let's walk through the reasoning step by step.

1. Recall the definition of degree of a polynomial expression.

   The degree is the highest sum of exponents of variables in any term. For a single-variable term like $5x$, the degree is 1. For $3x^2$, it is 2. For a constant term like $7$, the degree is 0.

2. What form does an objective function take in LPP?

   It is always of the form $Z = c_1 x_1 + c_2 x_2 + \dots + c_n x_n$, possibly plus a constant. Every term is a constant times a single variable to the first power. No term has $x_i^2$, $x_i x_j$, $\sqrt{x_i}$, or any nonlinearity.

3. Therefore, the highest exponent on any variable is 1.

   Even if a constant term exists (degree 0), the overall degree of the expression is the maximum among its terms, which is 1.

4. Why can't it be 0, 2, or any natural number?

   • Degree 0 would mean a constant objective — no variables at all. That is a trivial case but not the general definition; the objective function in LPP always involves variables to optimize.
   • Degree 2 or higher would make it a quadratic or higher-degree programming problem, which is not linear programming.
   • "Any natural number" is too broad — only degree 1 is allowed.

Why degree matters in linear programming

The entire framework of linear programming rests on linearity. When we optimize something—maximize profit, minimize cost—the objective function must be a linear combination of the decision variables. This isn't just a preference; it's what makes the geometry work: linear objectives over linear constraints produce convex feasible regions where corner-point solutions exist.

The degree of a polynomial tells us the highest power to which any variable is raised. For an objective function like $Z = 3x + 5y - 7$, both $x$ and $y$ appear to the first power. That's degree $1$.

Let me show you why the other options don't fit.

Step-by-step reasoning

1. What is an objective function in LP? The objective function has the standard form:

where $c_i$ are constants (coefficients) and $x_i$ are decision variables. Sometimes a constant term is added, but that doesn't change the degree.

2. Check the degree of each term

   Every term is of the form $c_i x_i$, which is a variable raised to the power $1$. The degree of a polynomial is the highest degree among all its terms. Here, every term has degree $1$.

3. Why not degree $0$?

   A polynomial of degree $0$ is just a constant—no variables at all. An objective function with no variables would be meaningless: there's nothing to optimize. Option (A) is out.

4. Why not degree $2$ or higher?

   If we had terms like $x^2$, $xy$, or $x^3$, the function would be nonlinear. That's a different class of optimization problem (quadratic programming, nonlinear programming). The "linear" in linear programming explicitly rules this out. Options (C) and (D) describe nonlinear programs.

5. Confirming degree $1$

   The general form $Z = c_1 x_1 + c_2 x_2 + \cdots + c_n x_n$ is a first-degree polynomial in $n$ variables. This is the defining characteristic of LP.

Understanding the terminology

Linear programming is a method for finding the best outcome in a mathematical model whose requirements are represented by linear relationships. Every LP problem has three essential components, and understanding their names is crucial for setting up and solving these problems correctly.

The function you're trying to optimize — whether that's maximizing profit, minimizing cost, maximizing efficiency, or minimizing waste — is the heart of the problem. This is the objective function. It's called "objective" because it represents your objective or goal. For instance, if a factory wants to maximize profit $P = 50x + 40y$ where $x$ and $y$ are quantities of two products, then $P$ is the objective function.

Let me clarify what each term in the options actually means:

Objective function: The linear function $Z = c_1x_1 + c_2x_2 + \ldots + c_nx_n$ that you want to maximize or minimize. This is the target, the measure of success.

Feasible solution: Any set of values $(x_1, x_2, \ldots, x_n)$ that satisfies all the constraints. The collection of all feasible solutions forms the feasible region.

Optimal solution: The particular feasible solution that gives the best (maximum or minimum) value of the objective function. This is what we're searching for.

Constraints: The linear inequalities or equations like $a_1x_1 + a_2x_2 \leq b$ that restrict the values the variables can take. These represent limitations — available resources, capacity bounds, demand requirements, etc.

Notice that "feasible function" and "optimal function" aren't standard terms in linear programming. We have feasible solutions and optimal solutions, but the function itself — the expression we're optimizing — is always called the objective function.