Question

A beer brewery currently manufactures 5 different beers. Each beer uses ingredients in different quantities. Not all of the beers use the same ingredients. The beers that the brewery offers are Willy P's Porter, Willy P's Lager, Willy P's Amber Lager, Willy P's Pale Ale, and a special blend of Willy P's Special Academic Edition. The brewery is able to realize a profit of p dollars per 5 gallons of beer i sold. The ingredients used to manufacture the beers are lager yeast, ale yeast, specialty malts, base malts, low-bitter hops, and high-bitter hops. Water is also an ingredient, but it is not taken into account for this problem. The current inventory levels for these are shown in the table below. A single unit of production is a 5-gallon batch. The brewery would like to know how many batches of 5-gallon beers they should manufacture. Use this information to answer questions 1 - 10 Amount of Ingredient Needed for One Batch of 5-Gallon Beer Beer Porter Lager Amber Lager Pale Ale Special Academic Ingredient Current Inventory Lager Yeast Ale Yeast Specialty Malt Base Malt Low-Bitter Hops High-Bitter Hops 0 d0,1 0 d0,2 d04 a1,4 a24 a3,4 a4,4 a5,A P4 bi a2,0 a2,2 a3,2 a3,1 44,1 45,2 Profit per 5-Gallons Po

Now suppose that if the Porter is produced, a fixed production cost of K0 is incurred. To include this requirement, we introduce a binary variable z0, which is set equal to 0 if we do not produce any of the porter, and is set equal to 1 if we do produce the porter. Which changes to the objective function and the constraints need to be made? (In all equations, M represents a very very very big number, and it is assumed that any possible decision xi will always be less than this number).

(a) Objective Function:
f (x0, x1, x2, x3, x4) = p0x0 + p1x1 + p2x2 + p3x3 + p4x4 − K0z0

Constraint:

x0 ≥z0M

(b) Objective Function:
f (x0, x1, x2, x3, x4) = p0x0 + p1x1 + p2x2 + p3x3 + p4x4 + K0z0

Constraint:

x0 =z0M+(1−z0)

(c) Objective Function:
f (x0, x1, x2, x3, x4) = p0x0 + p1x1 + p2x2 + p3x3 + p4x4 − K0z0

Constraint:

x0 ≤z0M

(d) Objective Function:
f (x0, x1, x2, x3, x4) = p0x0 + p1x1 + p2x2 + p3x3 + p4x4 + K0z0

Constraint:

x0 ≤z0M

6. Now suppose that the total amount of yeast remaining after production (both lager and ale yeast) should be larger than 5% of the amount of base malt remaining. Which constraint will reflect this requirement?

   1. (a) (a0,1x1+a0,2x2+a0,4x4+a1,0x0+a1,3x3+a1,4x4)≥0.05(a3,0x0+a3,1x1+a3,2x2+a3,3x3+ a3,4x4)

   2. (b) b0 + b1 − (a0,1x1 + a0,2x2 + a0,4x4 + a1,0x0 + a1,3x3 + a1,4x4) ≥ 0.05(b3 − a3,0x0 + a3,1x1 + a3,2x2 + a3,3x3 + a3,4x4)

   3. (c) b0 + b1 ≥ 0.05b3

   4. (d) b0 + b1 − (a0,1x1 + a0,2x2 + a0,4x4 + a1,0x0 + a1,3x3 + a1,4x4) ≤ 0.05(b3 − a3,0x0 + a3,1x1 +

      a3,2x2 + a3,3x3 + a3,4x4)

7. Now suppose that distributors always purchase twice as much pale ale than amber lager, and hence, the amount of pale ale produced must at least be as large as double the amount of amber lager produced. Which constraint would reflect this?

   (a) x0 ≤ x1 (b) 2x2 ≤ x3 (c) x3 ≤ 2x1 (d) 2x3 ≤ x2

Answer 1

1. (c) Objective Function:

f (x0, x1, x2, x3, x4) = p0x0 + p1x1 + p2x2 + p3x3 + p4x4 − K0z0

Constraint:

x0 ≤z0M

2. b) b0 + b1 − (a0,1x1 + a0,2x2 + a0,4x4 + a1,0x0 + a1,3x3 + a1,4x4) ≥ 0.05(b3 − a3,0x0 + a3,1x1 + a3,2x2 + a3,3x3 + a3,4x4)

3. (b) 2x2 ≤ x3