Shadow Prices:
Definition and Explanation of Shadow Price:
The determination of the optimum mix to
maximize the
contribution margin or to minimize cost assumes a defined set
of constraints.
It is useful to consider the sensitivity of the solution if
a constraint is relaxed. This effect is often referred to as shadow
price and simply shows the change in
contribution margin (in a
contribution margin maximization problem) or the change in cost (in a cost
minimization problem) resulting from relaxing a constraint.
To present the idea of shadow price, the value
of additional grinding or polishing hours (from the
contribution margin maximization example) can be considered, i.e., the
worth of additional grinding and polishing hours. If the machine shop had
more grinding or polishing hours, the contribution margin could be increased
by using more of each. The index row of
the third (optimum solution) simplex tableau (see
contribution margin maximization example) shows the shadow prices
in the slack variable columns, which is the location for both ≤ and ≥
constraints, while the artificial variable column is used for the =
constraint, with the m value ignored. In this illustration, only ≤
constraints are encountered. The
coefficients under the s1 (grinding) and s2 (polishing) slack variable
columns give the tradeoff in terms of product mix as the constraints are
increased or decreased. Thus, one more hour of grinding time will increase
the contribution margin by $0.625, computed as follows: as one more grinding
hour is made available, 0.25 units of y (deluxe models) with a unit
contribution margin of $4 (0.25 ×
$4 = $1) will replace 0.125 units of x (standard models) with a $3 unit
contribution margin (0.125 ×
$3 = $0.375), for a net contribution margin of $0.625 ($1  $0.375 =
$0.625). If additional grinding time
could be obtained at no increase in unit variable cost, an increase of
$0.625 contribution margin per grinding hour would result; and as much as
$0.625 more than the present unit variable cost of grinding time could be
incurred before reaching a point at which a zero per unit contribution
margin would occur. Thus, overtime hours might be considered. The $0.4375
has the same meaning for each hour of polishing time, and in each case the
observations assume that the sales price per unit remains unchanged.
The range of hours over which the shadow prices
of $0.625 and $0.4375 for grinding and polishing hours are valid can be
found as follows: 1. For the lower limit
of range, divide each unit in the solution mix by the coefficient under the
slack variable column, i.e., the s1 and s2 column. The smallest positive
number that result in a column is the maximum decrease for that constraint:
Product 
(1) Units 
(2) Grinding 
(3) Polishing 
(4)
s1
Grinding
1/2 
(5)
s2
Polishing
1/3 
y 
20 
0.250 
0.1250 
80 
160 
x 
10 
0.125 
0.3125 
80 
32 
For the grinding constraint, the decrease is
80 hours. Since the original number of hours available was 120, the lower
limit is 40 hours. For the polishing constraint, the decrease is 32 hours.
Since 80 hours were originally available, the lower limit is 48 hours.
2. For the upper limit of the range,
multiply each coefficient by 1 and repeat the step 1 process. The smallest
positive number that result in a column is the maximum increase for that
constraint:

(1) 
(2) 
(3) 
(4)
s1 
(5)
s2 
Product 
Units 
Grinding 
Polishing 
Grinding
1/2 
Polishing
1/3 
y 
20 
0.250 
0.1250 
80 
160 
x 
10 
0.125 
0.3125 
80 
32 
For the grinding constraint, the maximum
increase is 80; and since the original number of hours available was 120,
the upper limit is 200 hours. For the polishing constraint, the increase is
160, and the upper limit is 240 hours (160 plus the original constraint of
80 polishing hours). The limits occur
for a constraint because increases and decreases beyond the limits will
change the shadow price. In summary, the lower and upper limits for this
example are:

Lower
limit 
Upper
limit 
Grinding hours 
40 
200 
Polishing hours 
48 
240 
Both the constraints in this example are of
the ≤ type. The same method for finding the lower and upper constraint range
limits is used for the = type of constraint except that the coefficient
under artificial variable column for the = constraint are used in the
computations. For a ≥ constraint, the method for finding the lower and upper
constraint range limits differs in that the signs of the coefficients under
the slack variable column for the ≥ constraints are changed in step 1. With
this exception, the procedure is the same.
When there is a zero shadow price (not the case
in the above example), there is no defined upper limit for the ≥ type
constraint because there is already more of this constraint used than is
required. The lower and upper limit
computations apply, assuming that only one constraint is to be relaxed, and
provided there is a unique solution to the linear programming problem. The
limits for cases in which two or more constraints are relaxed simultaneously
can be computed following a methodology that is beyond the scope of this
discussion. The described method is
equally applicable to cost minimization problems. It should also be observed
that the shadow price indicates the opportunity cost of using a resource
(such as grinding or polishing hours) for some other purpose. For example,
if an hour of grinding time could instead be used to produce some other
product at a
contribution margin greater than $0.625 per hour, then use of the
grinding hour resource in producing the alternate product would be
preferable. 