Transcript Introduction - Eastern Mediterranean University (EMU), Cyprus

Reduced cost
Slack or surplus
Dual price
Reduced cost
 In a LINGO solution report, you’ll find a reduced cost figure
for each variable. There are two valid, equivalent
interpretations of a reduced cost.
 First, you may interpret a variable’s reduced cost as the amount
that the objective coefficient of the variable would have to
improve before it would become profitable to give the variable
in question a positive value in the optimal solution. For
example, if a variable had a reduced cost of 10, the objective
coefficient of that variable would have to increase by 10 units
in a maximization problem and/or decrease by 10 units in a
minimization problem for the variable to become an attractive
alternative to enter into the solution
 Second, the reduced cost of a variable may be
interpreted as the amount of penalty you would have
to pay to introduce one unit of that variable into the
solution. Again, if you have a variable with a reduced
cost of 10, you would have to pay a penalty of 10
units to introduce the variable into the solution. In
other words, the objective value would fall by 10
units in a maximization model or increase by 10 units
in a minimization model.
Reduced Cost:
 If we increase one unit of a non-basic variable, how much the
objective function will degrade (decrease)? For example, if we want
to produce one unit of x2 then we have to produce 23 units of x1, so
the objective function will be: (5*1) + (3*23) = 74.
 Then the reduced cost will be: 75 – 74 = 1
Dual price
 The LINGO solution report also gives a dual price figure
for each constraint. You can interpret the dual price as the
amount that the objective would improve as the right-hand
side, or constant term, of the constraint is increased by one
unit
 Notice that "improve" is a relative term. In a maximization
problem, improve means the objective value would
increase. However, in a minimization problem, the
objective value would decrease if you were to increase the
right-hand side of a constraint with a positive
 Dual prices are sometimes called shadow prices, because
they tell you how much you should be willing to pay for
additional units of a resource.
Dual Price
 If for example we increase (or decrease) one unit to the 2nd
constraint (It is 25 now), then the objective function increase (or
decrease) 3 units.
 Note: The second constraint is satisfied equally, so it is a
Compulsory Constraint.
Slack or surplus
 The Slack or Surplus column in a LINGO solution
report tells you how close you are to satisfying a
constraint as an equality. This quantity, on less-thanor-equal-to (≤) constraints, is generally referred to as
slack. On greater-than-or-equal-to (≥) constraints, this
quantity is called a surplus.
 If a constraint is exactly satisfied as an equality, the
slack or surplus value will be zero
Slack or Surplus:
 Zero: if a constraint is completely satisfied equality
(third row or 2nd constraint).
 Positive: shows that how many more units of the
variable could be added to the optimal solution before
the constraint becomes an equality
 (2nd row: 60- (2*25)= 10)
 Negative: Constraint has been violated!
