Enterprising chemical engineering students have set up a still in a bathtub. The

Question

Enterprising chemical engineering students have set up a still in a bathtub. They can produce 225 bottles of pure alcohol each week. They bottle two products from alcohol: (i) wine, 20 proof, and (ii) whiskey, 80 proof. Recall that pure alcohol is 200 proof. They have an unlimited supply of water but can only obtain 800 empty bottles per week because of stiff competition from the local recycling center. The weekly supply of sugar is enough for either 600 bottles of wine or 1200 bottles of whiskey. They make $1.00 profit on each bottle of wine and $2.00 profit on each bottle of whiskey. They can sell whatever they produce. How many bottles of wine and whisky should they produce each week to maximize profit? Formulate the design optimization problem. You do not have to solve the problem; set it up as you did for the problems in Chapter 2 (using the 5-step method). Summarize the optimization problem as a cost function and a set of constraints. Express the constraints in standard form, i.e., equality constraints should be of the form () 0 i h x = and inequality constraints should be of the form () 0 i g x .

Explanation / Answer

Let
x1: bottles of wine produced in a week
x2: bottles of whiskey produced in a week

The objective function is to maximise profit.
Profit is gives as: profit = x1+2x2

there are below constarints.
Constraint 1: supply of bottles
x1 + x2 <= 800   
Constraint 2: supply of alcohol
0.1x1 + 0.4x2 <= 225
Constraint 3: sugar limitation
x1/600 + x2/1200 <=1
or 2x1 + x2 <=1200
Constraint 4 and 5:
x1 >= 0
x2 >= 0

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