Question

A small firm makes three products, which all follow the same three-step process, which consists of milling, inspection, and drilling. Product A requires 6 minutes of milling, 5 minutes of inspection, and 4 minutes of drilling; product B requires 2.5 minutes of milling, 2 minutes of inspection, and 2 minutes of drilling; and product C requires 5 minutes of milling, 4 minutes of inspection, and 8 minutes of drilling. The department has 20 hours available during the next period for milling, 15 hours for inspection, and 24 hours for drilling. Product A contributes $6.00 per unit to profit, product B contributes $4.00 per unit, and product C contributes $10.00 per unit. The following is the output maximizing profits for the next period: PROBLEM TITLE: LINEAR PROGRAMMING PROBLEM IS A MAX WITH 3 VARIABLES AND 3 CONSTRAINTS. Row X1 X2 X3 Cost 6.00 4.00 10.00 1 6.00 2.50 5.00 1,200.00 2 5.00 2.00 4.00 900.00 3 4.00 2.00 8.00 1,440.00 NUMBER OF ITERATIONS: 2 OPTIMAL SOLUTION: OBJECTIVE FUNCTION VALUE = 2,070 DECISION VARIABLE SECTION: Variable Status Value Reduced Cost X1 Non-basic 0 3.5 X2 Basic 180 0 X3 Basic 135 0 SLACK VARIABLES SECTION: Slack Status Value Shadow Price X4 Basic 75 0 X5 Non-basic 0 1.5 X6 Non-basic 0 .5

What would the objective function value be if 3 more units of each X4, X5, and X6 were added assuming they are within the range of optimality?

Answer 1

The status of the constraints or variables X4, X5, and X6 in the optimal solution is as follows:

Variables	Slack status	Value	Shadow Price
X4	Basic	75	0
X5	Non-basic	0	1.5
X6	Non-basic	0	0.5

The variable X4 is still basic variable, the amount of 75 hours is remaining after obtaining optimal solution. Thus, variable X4 is non-binding constraint, by adding 3 units of X4 will not change the objective function value.

The variable X5 is non-basic variable, all the units of X5 are consumed to obtain optimal solution. Thus, variable X4 is binding constraint, by adding extra unit of X5 will improve the objective function value by 1.5 (shadow price). By adding 3 units of X5 the objective function value will increase by 1.5 x 3 = $4.5.

Shadow price is of the constraint indicates how much a one-unit change in value of constraint would change the optimal value of objective function. The shadow price is valid until the change is with in range of optimality.

The variable X6 is non-basic variable, all the units of X6 are consumed to obtain optimal solution. Thus, variable X6 is binding constraint, by adding extra unit of X6 will improve the objective function value by 0.5 (shadow price). By adding 3 units of X6 the objective function value will increase by 0.5 x 3 = $1.5.

By adding 3 more units of each constraints – X4, X5 and X6 the objective function value will increase by $0, $4.5, and $1.5 respectively.