1
SHADOW PRICES AND REDUCED COSTS
The following LP given in standard form which described the Primo Insurance Company’s venture into
special risk insurance and mortgages. We used x_1 to represent the monthly sales quota of SRIs and
x_2 to represent the sales quota of mortgages that can be recommended.
max
z = 5 x_1 + 2 x_2
Profit from sales
3 x_1 + 2 x_2 ≤ 2400
x_2 ≤ 800
2 x_1
≤ 1200
Subject to
Constraint on underwriting hours for a month
Constraint on administrative hours for a month
Constraint on other hours for a month
and the implicit constraints
x_1, x_2≥ 0.
The augmented form of the LP is the following linear system of equations:
3 x_1 + 2 x_2 + x_3
= 2400
x_2
+ x_4
= 800
2 x_1
+ x_5 = 1200
z - 5 x_1 - 2 x_2
=
0
with
x_1, x_2, x_3, x_4, x_5 ≥ 0.
The tableaux of the simplex method are all ways of expressing the system of 4 equations above. The
pivots are valid algebraic manipulations of the various systems of equations. For the simplex method,
we choose the pivots that we do by considering what the BFS would look like in that we don’t want
negative values of the variables, however, the system of equations that the tableaux represent are
correct reformulations of the system of equations above no matter what sign the variables that solve
them end up taking.
Below, Tableau 1 is the starting tableau and Tableau 2 is the optimal tableau. One can check that
(x_1, x_2, x_3, x_4, x_5, z) = (500, 500, -100, 300, 200, 3500) are solutions to both of the tableaux
below (though not the original LP because the non-negativity constraints on the variables are not all
held).
Tableau 1
Basic
Variables
z
x_3
x_4
x_5
Eqn
no.
(0)
(1)
(2)
(3)
z
x_1
x_2
x_3
x_4
x_5
RHS
1
0
0
0
-5
3
0
2
-2
2
1
0
0
1
0
0
0
0
1
0
0
0
0
1
0
2400
800
1200
2
Tableau 2
Basic
Variables
z
x_2
x_4
x_1
Eqn
no.
(0)
(1)
(2)
(3)
z
x_1
x_2
x_3
x_4
x_5
RHS
1
0
0
0
0
0
0
1
0
1
0
0
1
1/2
-1/2
0
0
0
1
0
1
-3/4
3/4
1/2
3600
300
500
600
The form of the objective function in this optimal tableau is z + x_3 + x_5 = 3600 or z = 3600 – x_3 –
x_5.
The optimal solution to the LP is (x_1, x_2, x_3) = (600, 300); that is, sell 600 SRIs and 300 mortgages
each month.
Now, what would happen if the number of underwriting hours available were to be increased by 1, say?
That is, there were now 2400 more hours available. Then, the underwriting constraint would be:
3 x_1 + 2 x_2 ≤ 2401.
Introduce a new slack variable w_3 ≥ 0 so that the augmented equation is
3 x_1 + 2 x_2 + w_3
= 2401
3
Now, if we define w_3 = x_3 + 1, that is to say, x_3 = w_3 – 1 then x_3 ≥ -1 and the equation now
reads as
3 x_1 + 2 x_2 + x_3
= 2400.
The only difference from what we worked with originally is the fact that we now allow x_3 ≥ -1 but,
as before all the other variables must be non-negative. Tableau 2 presents the Profit from Sales (z = 5
x_1 + 2 x_2 ) in the form z = 3600 – x_3 – x_5. One can see that now before when x_3 could not be
negative the maximum value of z was $3600 (with x_3=x_5=0) but our new problem allows x_3 to be -1
so the maximum value of z is $3601 (with x_3=-1 and x_5=0).
Thus, allowing there to be 1 more hour of underwriting would yield one more dollar of profit.
Basic
Variables
z
x_2
x_4
x_1
Eqn
no.
(0)
(1)
(2)
(3)
z
x_1
x_2
x_3
x_4
x_5
RHS
1
0
0
0
0
0
0
1
0
1
0
0
1
1/2
-1/2
0
0
0
1
0
1
-3/4
3/4
1/2
3600
300
500
600
4
For the sake of discussion suppose that the final tableau actually was the following instead:
Fake Tableau1
Basic
Variables
z
x_2
x_4
x_1
Eqn
no.
(0)
(1)
(2)
(3)
z
x_1
x_2
x_3
x_4
x_5
RHS
1
0
0
0
0
0
0
1
0
1
0
0
4
1/2
-1/2
0
0
0
1
0
2
-3/4
3/4
1/2
3600
300
500
600
Then the Profit from Sales would be z = 3600 – 4x_3 – 2x_5 and we would maximize z by using x_5=0
and x_3=-1 to give an optimal profit of $3604. Thus, by increasing the underwriting capacity by 1 hour
we would attain an increase in profit of $4. The coefficient of the associated slack variable in the z
equation of the optimal tableau gives the change in objective value per unit increase of the resource.
Definition: The shadow price associated with a particular constraint is the change in the optimal value
of the objective function per unit increase in the right hand side value for that constraint, all other
problem data remaining the same. Shadow prices are also referred to as dual variables or marginal
values for the constraints.
Definition: The reduced cost associated with the non-negativity constraint for each (original) variable is
the shadow price of that constraint (i.e. the corresponding change in the objective function per unit
increase in the lower bound of the variable).
For example, in Fake Tableau 2 below, the reduced cost associated with requiring that x_2 ≥ 1 instead
of x_2 ≥ 0 gives that the objective value now recorded as z = 3600 - 2 x_2 - 4x_3 has a maximum profit
of $3,598 instead of $3,600 so the profit decreases by $2. The reduced cost is $2. The coefficient of the
associated decision variable in the z equation of the optimal tableau gives the change in objective
value per unit increase in the lower bound of the decision variable.
Fake Tableau 2
Basic
Variables
z
x_5
x_4
x_1
Eqn
no.
(0)
(1)
(2)
(3)
z
x_1
x_2
x_3
x_4
x_5
RHS
1
0
0
0
0
0
0
1
2
2
-1
6
4
1/2
-1/2
0
0
0
1
0
0
1
0
0
3600
300
500
600
The shadow price associated with underwriting is $4/hour and the reduced cost associated with
mortgages is $2 per mortgage for this fake problem (not the one stated at the start of this document!)