Linear Programming Problems (LPPs) involve formulating an objective function and a set of constraints in order to optimize a linear objective subject to those constraints. Here’s a step-by-step guide to formulating an LPP:

1. Define Decision Variables:

• Identify the decision variables, which represent the quantities to be determined or optimized.
• Assign symbols to represent these variables, typically denoted by
  
  ${}_{}$

  x1​,x2​,…,xn​.

2. Formulate the Objective Function:

• Define the objective function, which represents the quantity to be maximized or minimized.
• Express the objective function as a linear combination of the decision variables.
• The objective function is typically of the form
  
  $$

  Maximize Z=c1​x1​+c2​x2​+…+cn​xn​ for maximization or
  $$

  Minimize Z=c1​x1​+c2​x2​+…+cn​xn​ for minimization, where
  $$

  c1​,c2​,…,cn​ are the coefficients of the decision variables.

3. Specify Constraints:

• Identify the constraints that limit the values of the decision variables.
• Express each constraint as a linear inequality or equality involving the decision variables.
• Constraints are typically represented in the form
  
  ${}_{}$

  a11​x1​+a12​x2​+…+a1n​xn​≤b1​,
  ${}_{}$

  a21​x1​+a22​x2​+…+a2n​xn​≤b2​, and so on, where
  ${\mathrm{<br>}}_{}$

   

  aij​ are the coefficients of the decision variables, and bi​ are the constants.

4. Identify the Feasible Region:

• Determine the feasible region, which is the set of all feasible solutions that satisfy all the constraints.
• Graphically represent the feasible region in
  
  $\mathrm{<br>}$

  n-dimensional space for visualization purposes, where
  $\mathrm{<br>}$

  n is the number of decision variables

• The feasible region is typically bounded by the constraints and can be represented as a polygon, polyhedron, or higher-dimensional shape.

5. Formulate the Complete LPP:

• Combine the objective function and the constraints to formulate the complete Linear Programming Problem.
• Write the LPP in standard form, which involves rewriting all constraints as equalities by introducing slack variables for inequalities.
• The standard form of an LPP is:
  
  

  $$\text{<br>}$$

  Maximize Z=c1​x1​+c2​x2​+…+cn​xn​

  subject to:

  $$\begin{array}{rl}{\displaystyle {\mathrm{<br>}}_{}}& \end{array}$$
  
  
  
  a11​x1​+a12​x2​+…+a1n​xn​a21​x1​+a22​x2​+…+a2n​xn​am1​x1​+am2​x2​+…+amn​xn​x1​,x2​,…,xn​​=b1​=b2​⋮=bm​≥0​

  Where

  $$

  m is the number of constraints, and
  ${\mathrm{<br>}}_{}$

  x1​,x2​,…,xn​ are the decision variables.

6. Interpretation:

• Interpret the solution obtained from solving the LPP in the context of the problem.
• The optimal solution provides the values of the decision variables that maximize or minimize the objective function while satisfying all the constraints.
• Analyze sensitivity to changes in coefficients or constraints to understand the robustness of the solution.

By following these steps, one can effectively formulate a Linear Programming Problem to address various optimization challenges in fields such as operations research, economics, engineering, and management.