Question

Interpreting an LP output after solving the problem using the software.

              

The following linear programming problem has been solved using the software. Use the output to answer the questions below.

LINEAR PROGRAMMING PROBLEM:

MAX 25X1+30X2+15X3

S.T.

  1) 4X1+5X2+8X3<1200

2) 9X1+15X2+3X3<1500

OPTIMAL SOLUTION:

Objective Function Value =         4700.000

  Variable Value Reduced Costs

X1 140.000 0.000

X2 0.000 10.000

X3 80.000 0.000

Constraint Slack/Surplus Dual Prices

1 0.000 1.000

2 0.000 2.333

OBJECTIVE COEFFICIENT RANGES:

Variable Lower Limit Current Value Upper Limit

X1 19.286 25.000 45.000

X2 no lower limit 30.000 40.000

X3 8.333 15.000 50.000

RIGHT HAND SIDE RANGES

Constraint Lower Limit Current Value Upper Limit

1 666.667 1200.000 4000.000

2 450.000 1500.000 2700.000

a.   Give the complete optimal solution.

b.   Give the reduced costs? Explain fully its meaning.

c.   What is the dual price for the second constraint? What interpretation does this have?

d.  Over what range can the objective function coefficient of x₂ vary before a new solution point becomes optimal?

e.   By how much can the amount of resource 2 decrease before the dual price will change?

f. Which constraint(s) have a surplus ?Explain

f. Which constraint(s) have a slack ? Explain

Each value is less than the right side of the less than or equal to sign

g. Can this problem be solved using the graphical method?

h. What does the objective function value represent? (Give the breakdown to show how x1, x2 and x3 contribute to the objective function value).

Answer 1

SOLUTION

a) The optimal solution is following:

X1 = 140

X2 = 0

X3 = 80

Objective value = 4700

b) Slack/surplus for both constraints is 0. So both constraints are binding.

c) Dual price for second constraint is 2.33. It means a unit increase in RHS of this constraint will increase the objective value by 2.33 units

d) Range of optimality for objective coefficnt of x2 is no lower limit to 40 (upper limit)

e) lower limit for resource 2 is 450. Therefore, resource 2 can be decreased by 1500-450 = 1050.