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Unformatted text preview: 1 IE 220 Fall 2008 Professor P´olik October 2, 2008 MidTerm Exam 1 Solutions 1. ( 17 points ) A cargo plane has three compartments for storing cargo: front, center and back. These compartments have capacity limits on both the weight and space, as summarized below: Compartment Weight Capacity (Tons) Space Capacity (Cubic Feet) Front 12 7000 Center 18 9000 Back 10 5000 Furthermore, the weight of the cargo in the respective compartments must be the same propor tion of that compartment’s weight capacity to maintain balance of the airplane. This means that if the front is filled up to 50% of its weight capacity, then all the other compartments must be filled up to 50% of their weight capacity. The following four cargoes have been offered for shipment on an upcoming flight as space is available: Cargo Weight (Tons) Volume (Cubic Feet/Ton) Profit ($/Ton) 1 20 500 320 2 16 700 400 3 25 600 360 4 13 400 290 Any portion of these cargoes can be accepted, but for environmental reasons, the captain refuses to fly the plane if the cargo is less then 20 tons. The objective is to determine how much (if any) of each cargo should be accepted and how to distribute each among the compartments to maximize the total profit for the flight. (a) ( 6 points ) Formulate this problem as a linear programming model. Be sure to include all necessary constraints. You do not need to convert to standard form (maximization prob lem, only ≤ constraints, etc.). ( Hint: Your decision variables could be x 1 F ,x 2 F ,...,x 4 B .) The decision variables x 1 F ,x 2 F ,...,x 4 B will denote the weight of each cargo in each compartment. We maximize the total profit: max 320( x 1 F + x 1 C + x 1 B )+400( x 2 F + x 2 C + x 2 B )+360( x 3 F + x 3 C + x 3 B ))+290( x 4 F + x 4 C + x 4 B ) 2 The weight in each compartment must be less than the capacity: x 1 F + x 2 F + x 3 F + x 4 F ≤ 12 x 1 C + x 2 C + x 3 C + x 4 C ≤ 18 x 1 B + x 2 B + x 3 B + x 4 B ≤ 10 The volume in each compartment must be less than the capacity: 500 x 1 F + 700 x 2 F + 600 x 3 F + 400 x 4 F ≤ 7000 500 x 1 C + 700 x 2 C + 600 x 3 C + 400 x 4 C...
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 Spring '07
 Storer
 Linear Programming, Optimization, Compartment Front Center, x1F + x2F

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