Example $1$:

A paper mill converts pulpwood to low, medium and high grade newsprint. The pulpwood requirements for newsprint, availability of each pulpwood, and selling price $($per ton$)$ are shown below:

Resource Product Available $($Tons$)$

Low Grade Medium Grade High Grade

Virginia pine $221180$

White pine $123120$

Loblolly pine $112160$

Price $$900$ $$1000$ $$1200$

The associated linear program is

Max Revenue

Revenue $=900$ X$1+1000$ X$2+1200$ X$3$

S$.$T$.$

$2$ X$1+2$ X$2+$ X$3<=180$

X$1+2$ X$2+3$ X$3<=120$

X$1+$ X$2+2$ X$3<=160$

X$1,$ X$2,$ X$3>=0$

The sensitivity report for this problem is:

Adjustable Cells

Final Reduced Objective Allowable Allowable

Cell Name Value Cost Coefficient Increase Decrease

$B$$10$ Production Low Grade $8409001500250$

$C$$10$ Production Medium Grade $0-20010002001$E$+30$

$D$$10$ Production High Grade $12012001500499\mathrm{.}9999999$

Constraints

Final Shadow Constraint Allowable Allowable

Cell Name Value Price R$.$H$.$ Side Increase Decrease

$E$$5$ Virginia pine Used $18030018060140$

$E$$6$ White pine Used $12030012086\mathrm{.}6666666730$

$E$$7$ Loblolly pine Used $10801601$E$+3052$

Answer the following questions:

$1.$ What are the basic variables$/$non basic variable?

$2.$ What are the binding$/$non binding constraints?

$3.$ What is the optimal solution?

$4.$ In what range can the price of low grade paper vary without changing the optimal basis?

$5.$ What is the new optimal solution if the price of low grade paper changes to $$800?$

$6.$ If it is required to produce $2$ units from the medium grade paper, what is the effect on the solution?

$7.$ In what range can the availability of Virginia pine vary without changing the optimal basis?

$8.$ If $10$ additional tons of Virginia pine are obtained, by how much will the optimal revenue increase?

$9.$ If $10$ additional tons of Loblolly pine are obtained, by how much will the optimal revenue increase?

$10.$ If the pulpwood resources $($Virginia$,$ White, and Loblolly$)$ are increased by $10$ units, what is the new optimal solution $($i$.$e$.$ the optimal production levels of low, medium and high grade newsprint$)?$