IENG313 Operation Research I

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-than-or-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!