The following linear programming problem $($Same problem as above$)$ has been solved by The Management Scientist and Excel.$$ Use the output from either one $($Management Scientist or Excel$)$ to answer the question, How much can the amount of resource $1$ increase before the dual price will change and the problem would need to be rerun?

LINEAR PROGRAMMING PROBLEM

MAX $25$X$1+30$X$2+15$X$3$

S$.$T$.$

$$

$1)4$X$1+5$X$2+8$X$3<1200$

$$

$2)9$X$1+15$X$2+3$X$3<1500$

OPTIMAL SOLUTION FROM MANAGEMENT SCIENTIST OUTPUT

Objective Function Value $=4700\mathrm{.}000$

Variable

Value

Reduced Cost

X$1$

$140\mathrm{.}000$

$0\mathrm{.}000$

X$2$

$0\mathrm{.}000$

$10\mathrm{.}000$

X$3$

$80\mathrm{.}000$

$0\mathrm{.}000$

$$

Constraint

Slack$/$Surplus

Dual Price

$1$

$0\mathrm{.}000$

$1\mathrm{.}000$

$2$

$0\mathrm{.}000$

$2\mathrm{.}333$

OBJECTIVE COEFFICIENT RANGES

Variable

Lower Limit

Current Value

Upper Limit

X$1$

$19\mathrm{.}286$

$25\mathrm{.}000$

$45\mathrm{.}000$

X$2$

No Lower Limit

$30\mathrm{.}000$

$40\mathrm{.}000$

X$3$

$8\mathrm{.}333$

$15\mathrm{.}000$

$50\mathrm{.}000$

RIGHT HAND SIDE RANGES

Constraint

Lower Limit

Current Value

Upper Limit

$1$

$666\mathrm{.}667$

$1200\mathrm{.}000$

$4000\mathrm{.}000$

$2$

$450\mathrm{.}000$

$1500\mathrm{.}000$

$2700\mathrm{.}000$

OPTIMAL SOLUTION FROM EXCEL OUTPUT

Profit$4700$X$1$X$2$X$3$Amount$140080$X$1$X$2$X$3$Limit$1458120012002915315001500$Profit$/$Cost Contribution$253015$

Sensitivity Report

Variable Cells$$FinalReducedObjectiveAllowableAllowableCellNameValueCostCoefficientIncreaseDecrease$B$$4$Amount X$1140025205\mathrm{.}714$$C$$4$Amount X$20-1030101$E$+30$$D$$4$Amount X$380015356\mathrm{.}667$Constraints$$FinalShadowConstraintAllowableAllowableCellNameValuePriceR$.$H$.$ SideIncreaseDecrease$F$$71200112002800533\mathrm{.}333$$F$$815002\mathrm{.}333150012001050$