The Bluegrass Distillery produces custom-blended whiskey. A particular blen consists of rye and bourbon whiskey. The company has received an order for a minimum of 400 gallons of the custon blend. The customer specified that the order must contain at least 40% rye and not more than 250 gallons of bourbon. The customer also specified that the blend should be mixed ion the ratio of two parts rye to one part bourbon. The distillery can produce 500 gallons per week, regardless of the blend. The production manager wants to complete the order in 1 week. The blend is sold for $5 per gallon. The distillery company's cost per gallon is $2 for rye and $q for bourbon. The company wants to determine the blend mix that will meet customer requirements and maximize profits.

a) Formulate a linear progrmming model for this problem.
b) Indicate the clack and surplus available at the optimal solution point and explain their meanings.
c) What increase in the objective function coefficients in this model would change the optimal solution point? Explain your answer.

What is that 'q' for the price of bourbon? A typo?

Cost per gallon price for bourbon is $1, not $q.

The distillery company's cost per gallon is $2 for rye and $1 for bourbon. The company wants to determine the blend mix that will meet customer requirements and maximize profits.

Yea it's a typo. Its $1 for bourbon

The distillery can handle 500 gallons and they got an order for a minimum of 400 gallons.

R=gallons of rye and B=gallns of bourbon.

Maximize Z=\underbrace{5(B+R)}_{\text{revenue}}-\overbrace{(2R+B)}^{\text{cost}}

subject to:

R\geq .4(B+R)... at least 40% of total produced is rye.

B\leq 250... no more than 250 gallons of bourbon

R+B\geq 400... at least 400 gallon of blend produced.

R+B\leq 500... max of 500 gallon od blend produced.

2B=R... twice as much rye as bourbon.

I ran this through the Excel Solver.
Do you have Excel? If so, do you know how to use the Solver?
There are the constraints. Can you take it from here?

Could you explain how you get the objective function, Z=5(B+R)-(2R+B)?

I really appreciate your help! And yeah, I know how to use the Solver. However, before I go for the Solver, I need to find out how to do (b) as well.. Could you also explain that for me? A similar problem will be on the exam so I really need to understand this problem.. Please help me out..!

Excel gives you the slack in the 'answer report'.

The difference between the right hand side and the left hand side of the (\leq) constraint yields the unused or SLACK amount of the resource.

The amount by which the left hand side exceeds the minimum limit represents the SURPLUS.

let s_{1} be the surplus variable, then the constraint can be converted to the equation:

The surplus is the difference between the 400 and 500. It is then 100.
Excel gives this as well. They wanted a minimum of 400 gallons, but no more than 500 gallons. We used the 500 gallons.

The slack for the bourbon is 83.333

Because 250-166.67=83.33

That is what is left over because the constraint said 'not more than' 250 gallons of bourbon. We only used 166.67 gallons.

The rye has a surplus of 133.33 gallons. We used 333.33, but 40% of 500 is 200 gallons. They asked for 'at least' 40% rye, but we used 66.67%. So, there is a surplus. 333.33-200=133.33.

See now what it is asking for?