Question

the company ULW uses three special ship to deliver three different chemical-product X, Y and Z. The transportation time of a ship to its customers is one week. Each ship has got four special compartments. First compartment has a capacitiy of 13.000 tons. The second compartment has a capacitiy of 15.000 tons. The third one has got a capacity of 16.000 tons, and the last compartment has a capacity of 16.500 tons. The chemical-product X, Y and Z cannot be mixed with each other. The weekly market demand for the product X is about 1.250.000 tons; the weekly demand for the product Y is about 1.456.000 tons and the daily market demand for the product Z is about 241.000 tons. The market prices of the three chemical-product X, Y and Z are 6,67 TL; 7,53 TL and 7,84 TL (per tons). Develop the optimal weekly loading schedule for the maximum income.

Answer 1

Define the decision variables as follows:

Xjk = Tons of 'X' loaded in the j-th container of the k-th ship for j=1,2,3,4 and k=1,2,3
Yjk = Tons of 'Y' loaded in the j-th container of the k-th ship for j=1,2,3,4 and k=1,2,3
Zjk = Tons of 'Z' loaded in the j-th container of the k-th ship for j=1,2,3,4 and k=1,2,3

B1jk = binary integer such that B1jk =1 when Xjk > 0 for j=1,2,3,4 and k=1,2,3
B2jk = binary integer such that B2jk =1 when Yjk > 0 for j=1,2,3,4 and k=1,2,3
B3jk = binary integer such that B3jk =1 when Yjk > 0 for j=1,2,3,4 and k=1,2,3

Max Z = 6.67 * (X11 + X12 + X13 + X21 + X22 + X23 + X31 + X32 + X33 + X41 + X42 + X43) + 7.53 * (Y11 + Y12 + Y13 + Y21 + Y22 + Y23 + Y31 + Y32 + Y33 + Y41 + Y42 + Y43) + 7.84 * (Z11 + Z12 + Z13 + Z21 + Z22 + Z23 + Z31 + Z32 + Z33 + Z41 + Z42 + Z43)

Subject to,

X11 + X12 + X13 + X21 + X22 + X23 + X31 + X32 + X33 + X41 + X42 + X43 <= 1250000

Y11 + Y12 + Y13 + Y21 + Y22 + Y23 + Y31 + Y32 + Y33 + Y41 + Y42 + Y43 <= 1456000

Z11 + Z12 + Z13 + Z21 + Z22 + Z23 + Z31 + Z32 + Z33 + Z41 + Z42 + Z43 <= 241000*7

X1k - 13000 * B11k <= 0 for k=1,2,3
B11k - X1k <= 0 k=1,2,3

X2k - 15000 * B12k <= 0 for k=1,2,3
B12k - X2k <= 0 k=1,2,3

X3k - 16000 * B13k <= 0 for k=1,2,3
B13k - X3k <= 0 k=1,2,3

X4k - 16500 * B14k <= 0 for k=1,2,3
B14k - X4k <= 0 k=1,2,3

Y1k - 13000 * B21k <= 0 for k=1,2,3
B21k - Y1k <= 0 k=1,2,3

Y2k - 15000 * B22k <= 0 for k=1,2,3
B22k - Y2k <= 0 k=1,2,3

Y3k - 16000 * B23k <= 0 for k=1,2,3
B23k - Y3k <= 0 k=1,2,3

Y4k - 16500 * B24k <= 0 for k=1,2,3
B24k - Y4k <= 0 k=1,2,3

Z1k - 13000 * B31k <= 0 for k=1,2,3
B31k - Z1k <= 0 k=1,2,3

Z2k - 15000 * B32k <= 0 for k=1,2,3
B32k - Z2k <= 0 k=1,2,3

Z3k - 16000 * B33k <= 0 for k=1,2,3
B33k - Z3k <= 0 k=1,2,3

Z4k - 16500 * B34k <= 0 for k=1,2,3
B34k - Z4k <= 0 k=1,2,3

B1jk + B2jk + B3jk <= 1 for j=1,2,3,4 and k=1,2,3

Xjk, Yjk, Zjk >= 0 for j=1,2,3,4 and k=1,2,3

B1jk, B2jk, B3jk = {0, 1} for j=1,2,3,4 and k=1,2,3

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