Question

Esiason Oil makes two blends of fuel by mixing oil from three wells, one each in Texas, Oklahoma, and California. The costs and daily availability of the oils are provided in the following table.

Source of Oil	Cost per Gallon	Daily Gallons Available
Texas well	0.30	12,000
Oklahoma well	0.40	20,000
California well	0.48	24,000

Because these three wells yield oils with different chemical compositions, Esiason’s two blends of fuel are composed of different proportions of oil from its three wells. Blend A must be composed of at least 35% of oil from the Texas well, no more than 50% of oil from the Oklahoma well, and at least 15% of oil from the California well. Blend B must be composed of at least 20% of oil from the Texas well, at least 30% of oil from the Oklahoma well, and no more than 40% of oil from the California well.

Each gallon of Blend A can be sold for $3.10 and each gallon of Blend B can be sold for $3.20. Long-term contracts require at least 20,000 gallons of each blend to be produced.

Let

x_i = number of gallons of oil from will i used in production of Blend A

y_i = number of gallons of oil from will i used in production of Blend B

i = 1 for the Texas well, 2 for the Oklahoma well, 3 for the California well

1. Develop the objective function, assuming that the client desires to maximize the total daily profit.

2. Show the mathematical expression for each of the following three constraints:

1. Total daily gallons of oil available from the Texas well is 12,000.
2. Total daily gallons of oil available from the Oklahoma well is 20,000.
3. Total daily gallons of oil available from the California well is 24,000.

3. Should this problem include any other constraints? If so, express them mathematically in terms of the decision variables.

Answer 1

a) Objective function is to maximize total daily profit

Maximize Z = (3.1-.3)X1 + (3.1-.4)X2 + (3.1-.48)X3 + (3.2-.3)Y1 + (3.2-.4)Y2 + (3.2-.48)Y3

or, Maximize Z = 2.8X1 + 2.7X2 + 2.62X3 + 2.9Y1 + 2.8Y2 + 2.72Y3

b) Mathematical expression for the constraints are following:

1. Total daily gallons of oil available from the Texas well is 12,000: X1 + Y1 <= 12000
2. Total daily gallons of oil available from the Oklahoma well is 20,000: X2 + Y2 <= 20000
3. Total daily gallons of oil available from the California well is 24,000: X3 + Y3 <= 24000

c) Yes, following other constraints are included:

X1 >= .35*(X1 + X2 + X3)

X2 <= .5*(X1 + X2 + X3)

X3 >= .15*(X1 + X2 + X3)

Y1 >= .2*(Y1 + Y2 + Y3)

Y2 >= .3*(Y1 + Y2 + Y3)

Y3 <= .4*(Y1 + Y2 + Y3)

X1 + X2 + X3 >= 20000

Y1 + Y2 + Y3 >= 20000

Xi, Yi >= 0