A brewery has the capability of producing a range of beers by blending existing stocks. Two beers (suds and premium) are currently available, with alcohol concentrations (by volume) of 3.0% for suds and 6.0% for premium. The manufacturing cost for suds is $0.30/gal, and for premium it is $0.40/gal.

Question

A brewery has the capability of producing a range of beers by blending existing stocks. Two beers (suds and premium) are currently available, with alcohol concentrations (by volume) of [latex]3.0 \%[/latex] for suds and [latex]6.0 \%[/latex] for premium. The manufacturing cost for suds is [latex]\$ 0.30 / gal[/latex], and for premium it is [latex]\$ 0.40 / gal[/latex]. In making blends, water can be added at no cost. An order for 10,000 gal of beer at [latex]5.0 \%[/latex] has been received for this week. There is a limited amount of suds available ( [latex]9000 gal[/latex] ), and, because of aging problems, the brewery must use at least 2,000 gal of suds this week. What amounts of suds, premium, and water must be blended to fill the order at minimum cost?

Accepted Answer

Let [latex]x_{1}[/latex] be gallons of suds blended [latex]x_{2}[/latex] be gallons of premium blended [latex]x_{3}[/latex] be gallons of water blended

Objective is to minimize cost

[latex]\min C=0.3 x_{1}+0.40 x_{2}[/latex] (1)

Subject to

[latex]\begin{aligned}&x_{1}+x_{2}+x_{3}=10,000 (2)\&0.03 x_{1}+0.060 x_{2}=0.050 imes 10,000 (3)\end{aligned}[/latex]

[latex]\begin{aligned}&x_{1} \geq 2000 (4) \&x_{1} \leq 9000 (5) \&x_{2} \geq 0 (6)\&x_{3} \geq 0 (7)\end{aligned}[/latex]

The problem given by Eqs. 1, 2, 3, 4, 5, 6, and 7 is optimized using Excel-Solver,

	x1	x2	x3
Initial values	1	0	0
Final values	2000	7333.333	666.6666667
Objective function	3533.333333


Constraints
x1+x2+x3	10000	=	1000
0.03x1+0.06x2	500	=	500
x1	2000	>=	2000
x1	2000	<=	9000
x2	7333.333333	>=	0
x3	666.6666667	>=	0

We obtain: suds = 2000 gallons; premium = 7333.3 gallons; water= 666.7 gallons, with the minimum cost of $3533.3.

Question 19.10: A brewery has the capability of producing a range of beers b...

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