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How Do You Write Out a Standard Minimization Problem?
in Mathematics by Platinum (109k points) | 38 views

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A standard minimization problem in the context of optimization, particularly linear programming, is typically presented in a structured format that clearly specifies the objective function to be minimized, subject to a set of constraints. Here's how you can write out a standard minimization problem:

Objective Function:

First, you define the objective function, which is the function you want to minimize. It is usually a linear function of several variables. The general form is:\[ \text{Minimize} \, Z = c_1x_1 + c_2x_2 + \cdots + c_nx_n \]- \(Z\) is the objective function value you want to minimize.
- \(c_1, c_2, \ldots, c_n\) are the coefficients of the variables in the objective function.
- \(x_1, x_2, \ldots, x_n\) are the decision variables.

Constraints:

Next, you list the constraints that the solution must satisfy. These constraints are typically inequalities (either ≤ or ≥) or equations (=), and they represent limitations or requirements in the problem. The general form of constraints for a minimization problem is:\[ a_{11}x_1 + a_{12}x_2 + \cdots + a_{1n}x_n \leq b_1 \]
\[ a_{21}x_1 + a_{22}x_2 + \cdots + a_{2n}x_n \leq b_2 \]
\[ \vdots \]
\[ a_{m1}x_1 + a_{m2}x_2 + \cdots + a_{mn}x_n \leq b_m \]- \(a_{ij}\) are the coefficients of the variables in the constraints.
- \(b_i\) are the right-hand side values of the constraints.
- \(m\) is the number of constraints.

Non-negativity Constraints:

Finally, you need to specify the non-negativity constraints for the decision variables, which ensure that the solution does not include negative values for variables that cannot be negative by their nature (e.g., quantities of products, time, etc.).

by Diamond (64.5k points)
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