Module A. The Simplex Solution Method
Module A. The Simplex Solution MethodThe simplex method , is a general mathematical solution technique for solving linear programming problems. In the simplex method, the model is put into the form of a table, and then a number of mathematical steps are performed on the table. These mathematical steps in effect replicate the process in graphical analysis of moving from one extreme point on the solution boundary to another. However, unlike the graphical method, in which we could simply search through all the solution points to find the best one, the simplex method moves from one better solution to another until the best one is found, and then it stops.
The manual solution of a linear programming model using the simplex method can be a lengthy and

Converting the Model into Standard FormThe first step in solving a linear programming model manually with the simplex method is to convert the model into standard form. At the Beaver Creek Pottery Company Native American artisans produce bowls ( x _{ 1 } ) and mugs ( x _{ 2 } ) from labor and clay. The linear programming model is formulated as
We convert this model into
standard form
by adding
The slack variables, s _{ 1 } and s _{ 2 } , represent the amount of unused labor and clay, respectively. For example, if no bowls and mugs are produced, and x _{ 1 } = 0 and x _{ 2 } = 0, then the solution to the problem is
and
Slack variables are added to constraints and represent unused resources. In other words, when we start the problem and nothing is being produced, all the resources are unused. Since unused resources contribute nothing to profit, the profit is zero.
It is at this point that we begin to apply the simplex method. The model is in the required form, with the inequality constraints converted to equations for solution with the simplex method. The Solution of Simultaneous Equations
Once both model constraints have been transformed into equations, the equations should be
For example, letting x _{ 1 } = 0 and s _{ 1 } = 0 results in the following set of equations.
and
First, solve for x _{ 2 } in the first equation:
Then, solve for s _{ 2 } in the second equation:
This solution corresponds with point
A
in Figure A1. The graph in Figure A1 shows that at point
A
,
x
_{
1
}
= 0,
x
_{
2
}
= 20,
s
_{
1
}
= 0, and
s
_{
2
}
= 60, the exact solution obtained by solving simultaneous equations. This solution is referred to as a
basic
Figure A1. Solutions at points A , B , and C
A basic feasible solution satisfies the model constraints and has the same number of variables with nonnegative values as there are constraints. Consider a second example where x _{ 2 } = 0 and s _{ 2 } = 0. These values result in the following set of equations.
and
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Then solve for s _{ 1 } :
This basic feasible solution corresponds to point C in Figure A1, where x _{ 1 } = 30, x _{ 2 } = 0, s _{ 1 } = 10, and s _{ 2 } = 0. Finally, consider an example where s _{ 1 } = 0 and s _{ 2 } = 0. These values result in the following set of equations.
and
These equations can be solved using row operations . In row operations, the equations can be multiplied by constant values and then added or subtracted from each other without changing the values of the decision variables. First, multiply the top equation by 4 to get 4 x _{ 1 } + 8 x _{ 2 } = 160 and then subtract the second equation:
Next, substitute this value of x _{ 2 } into either one of the constraints.
Row operations are used to solve simultaneous equations where equations are multiplied by constants and added or subtracted from each other. This solution corresponds to point B on the graph, where x _{ 1 } = 24, x _{ 2 } = 8, s _{ 1 } = 0, and s _{ 2 } = 0, which is the optimal solution point. All three of these example solutions meet our definition of basic feasible solutions . However, two specific questions are raised by the identification of these solutions.
The answers to both of these questions can be found by using the simplex method. The simplex method is a set of mathematical steps that determines at each step which variables should equal zero and when an optimal solution has been reached. 