The quantity column values are the solution values for the vari- ables in the basic feasible solution.

The initial simplex tableau always begins with the solution at the origin, where x_{1 }and x_{2 }equal zero. Thus, the basic variables at the origin are the slack variables, s_{1 }and s_{2}. Since the quantity values in the initial solution always appear as the right-hand-side values of the constraint equations, they can be read directly from the original constraint equations.

The number of rows in a tableau is equal to the number of constraints plus four.

The top two rows and bottom two rows are standard for all tableaus; however, the num- ber of middle rows is equivalent to the number of constraints in the model. For example, this problem has two constraints; therefore, it has two middle rows corresponding to s_{1 }and s_{2}. (Recall that n variables minus m constraints equals the number of variables in the prob- lem with values of zero. This also means that the number of basic variables with values other than zero will be equal to m constraints.)

The number of columns in a tableau is equal to the number of variables (including slacks, etc.)

Similarly, the three columns on the left side of the tableau are standard, and the remain- ing columns are equivalent to the number of variables. Since there are four variables in this model, there are four columns on the right of the tableau, corresponding to x_{1}, x_{2}, s_{1}, and s_{2}.

plus three.

The c values are the contribution to profit (or cost) for each variable.

j

The next step is to fill in the c values, which are the objective function coefficients, rep- resenting the contribution to profit (or cost) for each variable x or s_{j }in the objective func- tion. Across the top row the c values 40, 50, 0, and 0 are inserted for each variable in the model, as shown in Table A-3.

j

j

j

The Simplex Method

A-7

# Table A-3

The Simplex Tableau with c j V a l u e s

c j

Basic

40

50

0

0

Variables

Quantity

x_{1 }

x_{2 }

s_{1 }

s_{2 }

0 0

s_{1 }s 2

40 120

____________________________

z j c j - z j

____________________________

T h e v a l u e s f o r o n t h e l e f t s i d e o f t h e t a b l e a u a r e t h e c o n t r i b u t i o n s t o p r o f i t o f o n l y r those variables in the basic feasible solution, in this case s_{1 }and s_{2}. These values are inserted at this location in the tableau so that they can be used later to compute the values in the o w . z j c j

The columns under each variable (i.e., x_{1}, x_{2}, s_{1}, and s_{2}) are filled in with the coefficients of the decision variables and slack variables in the model constraint equations. The s_{1 }row represents the first model constraint; thus, the coefficient for x_{1 }is 1, the coefficient for x_{2 }is 2, the coefficient for s_{1 }is 1, and the coefficient for s_{2 }is 0. The values in the s_{2 }row are the second constraint equation coefficients, 4, 3, 0, and 1, as shown in Table A-4.

# Table A-4

The Simplex Tableau with Model Constraint Coefficients

c j

0 0

Basic Variables

s s_{2 }1

z j c j - z j

40

50

0

0

Quantity

x_{1 }

x_{2 }

s_{1 }

s_{2 }

40

1

2

1

0

120

4

3

0

1

____________________________