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Linear programming is one of the most widely used techniques in operations research for optimizing the allocation of limited resources to achieve a particular objective, such as minimizing cost or maximizing profit. The simplex method, developed by George Dantzig in 1947, is the most popular algorithm for solving linear programming problems (LPPs). It is an iterative procedure that moves from one feasible solution to another while improving the value of the objective function at each step until an optimal solution is obtained. The method is based on the idea that if an optimal solution exists, it lies at a vertex, or corner point, of the feasible region formed by the constraints of the linear program. The general form of a linear programming problem can be expressed as: maximize or minimize $Z = c_1x_1 + c_2x_2 + \cdots + c_nx_n$, subject to a set of linear constraints such as $a_{11}x_1 + a_{12}x_2 + \cdots + a_{1n}x_n \leq b_1$, and so on, with non-negativity restrictions $x_i \geq 0$. To apply the simplex method, the problem is usually transformed into standard form. In this form, the problem is typically set as a maximization problem with all constraints expressed as equalities by introducing slack, surplus, or artificial variables, and all decision variables required to be non-negative. The computational procedure of the simplex method begins with setting up the initial simplex tableau. The first step is to convert inequalities into equalities by adding slack variables in the case of “less than or equal to” constraints, subtracting surplus variables for “greater than or equal to” constraints, and adding artificial variables if necessary. Once the problem is converted into standard form, an initial basic feasible solution is obtained, usually by setting the decision variables to zero and solving for the slack variables, which ______ ____ ___ _______ ___ ___.
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