LINGO Output Is Given for the Following Linear Programming Problem

question 46

Essay

LINGO output is given for the following linear programming problem.
MIN 12 X1 + 10 X2 + 9 X3
SUBJECT TO
2)5 X1 + 8 X2 + 5 X3 >= 60
3)8 X1 + 10 X2 + 5 X3 >= 80
END
LP OPTIMUM FOUND AT STEP 1
OBJECTIVE FUNCTION VALUE
1)80.000000 $$\begin{array} { c c c } \text { VARIABLE } & \text { VALUE } & \text { REDUCED COST } \\\text { X1 } & .000000 & 4.000000 \\\text { X2 } & 8.000000 & .000000 \\\text { X3 } & .000000 & 4.000000\end{array}$$ $\begin{array}{l}\begin{array}{c}ROW&\text{SLACK OR SURPLUS}&\text{DUAL PRICE}\\\hline2)&4.000000 & .000000 \\3)&.000000 & -1.000000\end{array}\end{array}$ NO.ITERATIONS= 1
RANGES IN WHICH THE BASIS IS UNCHANGED: $\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\text { OBJ. COEFFICIENT RANGES }$
$\begin{array}{crrr}\underline{\text { VARIABLE }}&\frac{\text { CURRENT }}{\text { COEFFICIENT }} & \frac{\text { ALLOWABLE }}{\text { INCREASE }} & \frac{\text { ALLOWABLE }}{\text { DECREASE }}\\X 1 & 12.000000 & \text { INFINITY } & 4.000000 \\X 2 & 10.000000 & 5.000000 & 10.000000 \\X 3 & 9.000000 & \text { INFINITY } & 4.000000\end{array}$ $$\begin{array}{l}\begin{array} { c c c c } &&&\text { RIGHTHAND SIDE RANGES }\\& \text { CURRENT } & \text { ALLOWABLE } && \text { ALLOWABLE } \\\hline\text { ROW } & \text { RHS } & \text { INCREASE } && \text { DECREASE } \\2 & 60.000000 & 4.000000 && \text { INFINITY } \\3 & 80.000000 & \text { INFINITY } && 5.000000\end{array}\end{array}$$
a.What is the solution to the problem?
b.Which constraints are binding?
c.Interpret the reduced cost for x₁.
d.Interpret the dual price for constraint 2.
e.What would happen if the cost of x₁ dropped to 10 and the cost of x₂ increased to 12?

---

Definitions:

Unit Price

The cost assigned to a single unit of a product or service, facilitating price comparisons across different quantities.

Direct Labor Variance

The difference between the actual labor costs incurred and the standard labor costs expected for the actual production level, reflecting efficiency and cost control in production.

Per-unit Standards

Predetermined costs to produce a single unit of product, used in budgeting and cost control.

Direct Labor Hours

The total hours worked by employees directly involved in manufacturing products or providing services.