Slack variables are added to … constraints and represent unused resources.

# Converting the Model into Standard Form

A-3

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

s 1 = 4 0 h r . o f l a b o r 0 + 2 ( 0 ) + s 1 = 4 0 x 1 + 2 x 2 + s 1 = 4 0

and

s 2 = 1 2 0 l b . o f c l a y 4 ( 0 ) + 3 ( 0 ) + s 2 = 1 2 0 4 x 1 + 3 x 2 + s 2 = 1 2 0

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:

Z = $40x_{1 }+ 50x_{2 }+ 0s_{1 }+ 0s_{2 }= 40(0) + 50(0) + 0(40) + 0(120) Z = $0

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 solved simultaneously to determine the values of the variables at every possible solu- tion point. However, notice that our example problem has two equations and four unknowns (i.e., two decision variables and two slack variables), a situation that makes direct simultaneous solution impossible. The simplex method alleviates this problem by assigning some of the variables a value of zero. The number of variables assigned values of zero is n - m, where n equals the number of variables and m equals the number of con- straints (excluding the nonnegativity constraints). For this model, n = 4 variables and m = 2 constraints; therefore, two of the variables are assigned a value of zero (i.e., 4 - 2 = 2).

# For example, letting x_{1 }= 0 and s_{1 }= 0 results in the following set of equations:

4 x 1 + 3 x 2 + s 2 = 1 2 0 x 1 + 2 x 2 + s 1 = 4 0

and

0 + 3 x 2 + s 2 = 1 2 0 0 + 2 x 2 + 0 = 4 0

# First, solve for x_{2 }in the first equation:

x 2 = 2 0 2 x 2 = 4 0