The question is: What quantities does the manager control? What can the manager manipulate to influence

profit? It is incomplete simply to say that the manager controls the amount of each final product to make.

The manager controls, and must determine, how to make each final product and how much to make.

This can be expressed by letting xij $=$ number of barrels of raw gas $$i$=(1,2,3,4)$ used per day to make

final product $$j$=($R$,$ P$)$ be the decision variables.

Each barrel of raw gas i that is blended in final product j and then sold generates a profit equal to its selling

price minus its cost.

$1)$ Formulate the objective function for this problem.

Note that the coefficients for some variables are negative. For example, the company loses $$1\mathrm{.}00$ on each

barrel of raw gas $4$ that is blended into premium. Does this imply that the optimal value for these variables

must be zero and that they can be dropped from the problem? No$!$ In blending operations, it is common

for some low$-$cost materials to be combined with high$-$cost materials. Although it appears that we are

losing money on the high$-$cost materials, they make the low$-$cost materials more valuable, and often the

final product cannot be made without them. For example, tungsten steel combines low$-$cost iron ore or

scrap $($worth $$100/$ton$)$ with tungsten $($costing thousands of dollars per ton$)$ to make steel that might sell

for $$500$ per ton. The manufacturer loses money on the tungsten $($on a per ton basis$)$ but is more than

compensated by the enhanced value of the iron ore. Therefore, we do not omit variables from the problem

unless we can prove that their optimum value is zero.

The next step is to identify the constraints.

The availability constraint for each raw gasoline is barrels of raw gas i used per day $<=$ barrels of gas i

available per day.

The number of barrels of raw gas i used each day is the amount used to make regular gasoline xiR plus

the amount used each day to make premium gasoline xiP.

$2)$ Formulate the constraint $($both availability and demand$)$

If the model formulation is left at this stage, the optimal solution is to mix the lowest cost gasolines into

the final products, regardless of octane. Therefore, we need to include constraints that guarantee the

variables will take on values that produce final gasolines with at least the minimum specified octane

ratings. The octane rating of the regular gasoline that is produced will be a weighted average of the octanes

of the raw gasolines used.

Octane of regular $=[86*($barrels of raw gas $1$ used$/$day to make regular$)+88*($barrels of raw gas $2$

used$/$day to make regular$)+...+96*($barrels of raw gas $4$ used$/$day to make regular$)],$ divided by the

$[$total barrels of raw gases blended into regular gasoline$],$ which should be at least $89.$

$3)$ Can you formulate the constraint on Octane now?

$4)$ Enter the model in Excel and using its Solver, find the optimal mix.