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Question 3 (23 points) Three refineries with daily capacities of 6, 5, and 8 million gallons, respectively, supply three distribution areas with daily demands of 4, 8, and 7 million gallons, respectively. Gasoline is transported to the three distribution areas through a network of pipelines. Each pipeline leaving refinery 1 has a daily capacity of 7 million gallons, each pipeline leaving refinery 2 has a daily capacity of 6 million gallons, and each pipeline leaving refinery 3 has a daily capacity of 9 million gallons. The transportation cost is 10 cents per 1000 gallons per pipeline mile. The table below gives the mileage between the refineries and the distribution areas. Refinery 1 is not connected to distribution area 3 so we use an arbitrarily high cost (M) along that arc. The LP for this problem is given with full problem data. The variables tell us how many thousands of gallons of oil to ship along each pipeline each day Min Z 12x1 18 x12 + Mx13 30xzi + 10xz2 8x23 20 x3i 25 x32 12 x33 St x11 +x12 x136000 x21x22+x235000 x3 x3 x338000 X11 321 x3 4000 x12 t X22 x328000 Xy3 + X23 + X33 7000 X11 7000 X12 S 7000 x13 7000 x21 S 6000 x22 6000 X23 6000 x39000 32 9000 33 9000 x11,12,X13,x21X22 ,x23,x31,X32,X33 20

Explanation / Answer

Answer:

By using,given data

Let refinery 1,2,3 be i
Let demand 1,2,3 be j
Xij are the arcs

Arc values were calculated by, miles (which are given) times cost to transport 1000 gallons (which is $0.10). The node values are x1000 for the actual number, as is the cost.

Min Z=12x11 + 18x12 + 30x21 + 10x22 + 8x23 + 20x31 + 25x32 + 12x33

s.t.

x11 + x12 <= 6,000
x21 + x22 + x23 <= 5,000
x31 + x32 + x33 <= 8,000
x11 + x21 + x31 = 4,000
x12 + x22 + x32 = 8,000
x23 + x33 = 7,000
xij >= 0 for i and j=1,2,3

Excel solver solution is as follows below:

x11 = 4,000
x12 = 2,000
x22 = 5,000
x32 = 1,000
x33 = 7,000
(answers times 1,000 for actual value).

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