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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 x1 and x2 equal zero. Thus, the basic variables at the origin are the slack variables, s1 and s2. 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 s1 and s2. (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 x1, x2, s1, and s2.

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 sj 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

x1

x2

s1

s2

0 0

s1 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 s1 and s2. 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., x1, x2, s1, and s2) are filled in with the coefficients of the decision variables and slack variables in the model constraint equations. The s1 row represents the first model constraint; thus, the coefficient for x1 is 1, the coefficient for x2 is 2, the coefficient for s1 is 1, and the coefficient for s2 is 0. The values in the s2 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 s2 1

z j c j - z j

40

50

0

0

Quantity

x1

x2

s1

s2

40

1

2

1

0

120

4

3

0

1

____________________________

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