hw1sol - EGM6365 Homework #1 1. A beer can is supposed to...

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Unformatted text preview: EGM6365 Homework #1 1. A beer can is supposed to hold at least specific amount of beer and meet other design requirements. The can will be produced in billions, so it is desirable to minimize the cost of manufacturing them. Since the cost can be related directly to the surface area of the sheet metal used, it is reasonable to minimize the sheet metal required to fabricate the can. Fabrication, handling, aesthetic, and shipping considerations impose the following restrictions on the size of the can: 1. The diameter of the can should be no more that 8 cm. Also, it should not be less than 3.5 cm. 2. The height of the can should be no more than 18 cm and no less than 8 cm. The can is required to store at least 400 ml of fluid. Write a standard form of the optimization problem. D H Figure 1 Design of a beer can Minimize f ( D, H ) = π DH + 1− π D2 H 1600 ≤ 0, π 2 D2 , (cm 2 ) (cm3 ) Subject to 3.5 ≤ D ≤ 8 8 ≤ H ≤ 18 2. A rectangular underground storage tank is to be constructed and installed. Specifications require that the volume of the tank be 1000 m3 and that the ratio of the lengths of any two sides be no greater than two. The top surface of the tank is to be 3 m below the ground surface when installed. The cost of construction of the tank is $150/m2 based on the surface area. Installation cost in dollar is equal to 200 time the product of the area and the square of the depth of the hole to be excavated. For convenience, assume the tank and the hole cross sectional area are the same, i.e., no clearance required. Formulate the optimization problem to minimize the total project cost and write the result in standard form. Design variable: x = width of the tank y = breadth of the tank z = height of the tank Minimize Subject to 150[2 xy + 2 xz + 2 yz ] + 200[ xy ( z + 3) 2 ] xyz −1 = 0 1000 z −1 ≤ 0 2x x −1 ≤ 0 2y y −1 ≤ 0 2x y −1 ≤ 0 2z z −1 ≤ 0 2y x, y , z ≥ 0 3. A coal mining company has three coal mines and four coal cellars. The company is trucking coal from three mines to four cellars before shipping out of the customer. To make efficient usage of trucks, the total delivery distance from mines to cellars by the trucks needs to be minimized. The following tables show that distance between three mines (Mine number 1, 2, and 3) and four cellars (Cellar number A, B, C, and D). Each truck can carry two tons of coal at one time. The amount of coal produced at each coal mine and the amount of coal that can be stored at each cellar are given in the following table. Each delivery of the coal by truck means a round trip of the distance between the mine and cellar, since the truck will return empty to get more coal. Assume that the company has enough truck so each truck can move between a selected mine and cellar only. Formulate the problem in standard form. Do not normalize or attempt to solve. (Hint: There are 12 design variables) Mine number 1 2 3 Distance(meter) from Mine to Cellar Cellar A Cellar B Cellar C Cellar D 3500 2900 3450 1290 1670 4500 2390 4230 2340 1250 2880 3770 Coal Production at Mines (per day) 1 3500 ton 2 4000 ton 3 4500 ton Cellar Storage Capacity (per day) A 2000 ton B 3000 ton C 4500 ton D 2500 ton Let A1, B1, C1, D1 = Coal delivered (in ton) from mine 1 to cellars A, B, C, D, respectively. Then, design variables are u = [ A1 , A2 , A3 , B1 , , D3 ] . f = 3500 A1 + 2900 B1 + 3450C1 + 1290 D1 Minimize +1670 A2 + 4500 B2 + 2390C2 + 4230 D2 +2340 A3 + 1250 B3 + 2880C3 + 3770 D3 Subject to Coal mine production 1 [ A1 + B1 + C1 + D1 ] − 1 = 0 3500 1 [ A2 + B2 + C2 + D2 ] − 1 = 0 4000 1 [ A3 + B3 + C3 + D3 ] − 1 = 0 4500 Cellar storage 1 [ A1 + A2 + A3 ] − 1 = 0 2000 1 [ B1 + B2 + B3 ] − 1 = 0 3000 1 [C1 + C2 + C3 ] − 1 = 0 4500 1 [ D1 + D2 + D3 ] − 1 = 0 2500 Side constraints A1 , A2 , A3 , B1 , , D3 ≥ 0 ...
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