Sindbad Marine Co has two ships Barbarossa and Jebel Tarik based in Lattaquieh. Each ship has three sections: front, middle, and back. Sinbad Marine Co has received an offer to transport two commodities: flour and tea. It can accept all or part of them. The following table indicates data on both commodities.

The distribution of the tonnage according to the three sections for each ship is as follows (in t):

The volume for each ship expressed in m³.

In order to stabilize the ship, the weight in each section must be proportional to the tonnage. How must Sinbad Marine Co load both ships?

Answer:

The objective is to minimize the cost of shipping

X_ji= Quantity (in ton) of flour shipped in section i of ship j

Yji = Quantity (in ton) of tea shipped in section i of ship j

 i= 1 for the front section, i=2 for the middle section, i=3 for the back section

j= 1 for Barbarossa, j=2 for Jebel Tarik

The objective function Z = 400 (X₁₁+ X₁₂+ X₁₃+ X₂₁+ X₂₂+ X₂₃) + 1100 (Y₁₁+ Y₁₂ + Y₁₃ + Y₂₁ + Y₂₂ + Y₂₃)

Constraints:

• Flour (Barbarossa & Jebel Tarik): X₁₁ + X₁₂ + X₁₃ + X₂₁ + X₂₂ + X₂₃ ≤ 12000
• Tea (Barbarossa & Jebel Tarik): Y₁₁+ Y₁₂ + Y₁₃ + Y₂₁ + Y₂₂ + Y₂₃ ≤ 4000
• Barbarossa_Front_Qynt: X₁₁ + Y₁₁ ≤ 1500
• Barbarossa_Middle_Qynt:X₁₂ + Y₁₂ ≤ 4000
• Barbarossa_Back_Qynt:X₁₃ + Y₁₃ ≤ 3000
• JebelTarik_Front_Qynt:X₂₁ + Y₂₁ ≤ 1000
• JebelTarik_Middle_Qynt: X₂₂ + Y₂₂ ≤ 3000
• JebelTarik_Back_Qynt: X₂₃ + Y₂₃ ≤ 1500
• Barbarossa_Front_Vol: 1.5X₁₁ + 4Y₁₁ ≤ 2500
• Barbarossa_Middle_Vol: 1.5X₁₂ + 4 Y₁₂ ≤ 8000
• Barbarossa_Back_Vol: 1.5X₁₃ + 4 Y₁₃ ≤ 5000
• JebelTarik_Front_Vol: 1.5 X₂₁ + 4Y₂₁ ≤ 2000
• JebelTarik_Middle_Vol: 1.5 X₂₂ + 4 Y₂₂ ≤ 7000
• JebelTarik_Back_Vol: 1.5 X ₂₃ + 4 Y₂₃ ≤ 3500
• X_ji<= 0, Y_ji <=0