Page 115 -

GRAPHICAL SENSITIVITY ANALYSIS 95
Can you calculate and Given that the value to the company of this extra hour is $4.375 then clearly manage-
interpret the dual price for
a constraint? Try ment should be prepared to offer a maximum bonus which is just less than this. In
Problem 4.
Copyright 2014 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has
deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
constraints: Sewing and inspection and packaging? Looking at equations (3.5) and
(3.6) we can see that increasing the time available to sewing and to inspection
and packaging will not affect these two equations (which relate, after all to cutting
and dyeing and to finishing). This implies that a marginal increase in sewing
time and a marginal increase in inspection and packaging time will have no effect
on the optimal solution and no effect on profit contribution.
What do we make of this? Recollect that these two constraints were non-binding
and had slack time associated with them. That is, at the current optimal solution
there is unused sewing time and also unused inspection and packaging time. So,
given we are not currently using all the existing available time for each of these
constraints, adding further time to capacity will have no effect on production and so
no effect on profit contribution.
The change in the value of the optimal solution per unit increase in the right-hand
side of a constraint is called the dual price. Here, the dual price for the cutting and
dyeing constraint is $4.375; in other words, if we increase the right-hand side of the
cutting and dyeing constraint by one unit, the value of the objective function will
improve by $4.375. Conversely, if the right-hand side of the cutting and dyeing
constraint were to decrease by one unit, the objective function would get worse by
$4.375. The dual price can generally be used to determine what will happen to the
value of the objective function when we make a one-unit change in the right-hand side
of a constraint. The dual price has a number of other names – dual value, shadow
price, opportunity cost are all used. Whichever name we use, it provides management
with important information. It tells management the change in the objective function
value for a marginal (one unit) change in the right-hand side value of a constraint. In
the case of the cutting and dying constraint, the dual price is $4.375. So, if we could
persuade our workforce to work one extra hour on cutting and dyeing, this will allow
additional production that will increase the total profit contribution by $4.375. The
dual price also allows management to place a value on such additional time. One of
our workers may come along and say they are prepared to put an extra hour’s work
into cutting and dyeing but they expect a bonus payment for doing so. How much of a
bonus should management be prepared to pay? The answer is given by the dual price.
other words, management can place a value on the acquisition of scarce resources.
We caution here that the value of the dual price may be applicable only for small
changes in the right-hand side. As more and more resources are obtained and the
right-hand side value continues to increase, other constraints will become binding and
limit the change in the value of the objective function. For example, in the problem for
GulfGolf we would eventually reach a point where more cutting and dyeing time would
be of no value; it would occur at the point where the cutting and dyeing constraint
becomes nonbinding. At this point, the dual price would equal zero. In the next section
we will show how to determine the range of values for a right-hand side over which the
dual price will accurately predict the improvement in the objective function.
To illustrate the correct interpretation of dual prices for a minimization problem,
suppose we had solved a problem involving the minimization of total cost and that
the value of the optimal solution was $100. Furthermore, suppose that the dual price
for a particular constraint was $10. The negative dual price tells us that the
objective function will not improve if the value of the right-hand side is increased
by one unit. Thus, if the right-hand side of this constraint is increased by one unit,
the value of the objective function will get worse by the amount of $10. Becoming
worse in a minimization problem means an increase in the total cost. In this case, the
value of the objective function will become $110 if the right-hand side is increased by
one unit. Conversely, a decrease in the right-hand side of one unit will decrease the
total cost by $10.