The shadow prices can be used to determine the effect of a new variable (like a new product in a production linear program). Suppose that, in formulation (8.1), a new variable w has coefficient 4 in the first constraint and 3 in the second. What objective coefficient must it have to be considered for adding to the basis?
If
we look at making w positive, then this is equivalent to
decreasing the right hand side of the first constraint by 4w
and the right hand side of the second constraint by
3w in the original formulation. We obtain the same effect
by making 
and 
.
The overall effect of this is to decrease the objective by
.
The objective value must be
sufficient to offset this, so the objective coefficient must be more than
10 (exactly 10 would lead to an alternative optimal solution with no
change in objective).
Answers:
(a) From the final tableau, we read that 
is basic and
are nonbasic. So 100 units of 
should be produced
and none of 
, 
and 
. The resuting profit is $ 600
and that is the maximum possible, given the constraints.
(b) The reduced cost for 
is 2 (found in Row 0 of the final
tableau). Thus, the effect on profit of producing units of
is 
. If 20 units of have been produced by mistake, then
the profit will be 
lower than the maximum
stated in (a).
(c) Let 
be the profit margin on . The reduced
cost remains nonnegative in the final tableau if 
.
That is 
. Therefore, as long as the profit margin
on is less than 4.5, the optimal basis remains unchanged.
(d) Let 
be the profit margin on . Since 
is
basic, we need to restore a correct basis. This is done by adding
times Row 1 to Row 0. This effects the reduced costs of
the nonbasic variables, namely 
, , 
and 
.
All these reduced costs must be nonnegative. This implies:
.
Combining all these inequalities, we get 
. So,
as long as the profit margin on is 6 or greater, the optimal
basis remains unchanged.
(e) The marginal value of increasing capacity in Workshop 1 is
.
(f) Let 
be the capacity of Workshop 1. The resulting RHS
in the final tableau will be:
in Row 1, and
in Row 2.
The optimal basis remains unchanged as long as these two quantities
are nonnegative. Namely, 
. So, the optimal
basis remains unchanged as long as the capacity of Workshop 1 is in
the range 0 to 800.
(g) The effect on the optimum profit of producing 
units
of 
would be 
. If the profit margin on is sufficient to offset this,
then should be produced. That is, we should produce if
its profit margin is at least 3.
