CEE 3000 HW 3 Solutions Fall 21

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Georgia Institute Of Technology *

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3000

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Civil Engineering

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Oct 30, 2023

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pdf

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11

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Georgia Institute of Technology | School of Civil & Environmental Engineering CEE 3000: CIVIL ENGINEERING SYSTEMS HOMEWORK 3: MATH TOOLS Fall 2021 Assigned:9/21/2021 | Due: 10/05/2021 Name: _____________________________________ Date: __________________________________ 1. Assume that you’ve been hired by a contractor specializing in the supply of storage sheds. Your employer has just obtained a contract from a Department of Transportation (DOT) in a northern state to manufacture sheds to hold sand or salts. These will used to keep their highways open to traffic in periods of heavy snow. The consultant will use a durable material recently developed to have an average useful life almost 15% longer than the average useful life of competing materials on the market, when properly maintained. Each shed must hold a volume of 14,500 m 3 . The rectangular base pad costs $75/m 2 , the materials on the sides and roof cost $45/m 2 and $50/m 2 , respectively. These special sheds are cuboid in shape. For stability, the DOT requires the width of the base to be half the height of the shed. The DOT desires the optimum design that results in the least total cost of materials. Your first task is to determine the optimum design dimensions for the sheds. Answer the following questions. (10 points) (a) What design dimensions will minimize the total material costs? (5 points)
2 Width = 18.31 meters Length = 21.63 meters Height = 36.62 meters (b) What is the optimum cost of each shed (to the nearest 100 thousand)? (3 points) Optimum cost = $200,000 (c) After doing some quick research to cross check the cost estimates you were given, you have just found out that the materials may cost up to 10% more than the original estimates given. The DOT would like to purchase 5 sheds. How much money should they budget for the sheds (in millions)? (2 points)
3 Budget: $0.995 million ~$995,000
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4 2. For the construction of a civil engineering facility, a contractor has found natural reserves of sand and gravel at Beaver Plains and Valley View for sale. The unit cost of material including delivery is $6 and $8 from Beaver Plains and Valley View, respectively. After the material is brought to the site, it is mixed thoroughly and uniformly. The contract specifications state that the mix should contain a minimum of 30% sand. A total volume of 100,000 m 3 of mixed material is needed for the project. The Beaver Plains Pit contains 25% sand and the Valley View Pit contains 50% sand. As the new construction engineer on the project, you have been asked to determine how much material should be taken from each pit in order to minimize the cost of the material. Answer the following questions. (10 points) (a) What are the decision variables? (1 point) X = Amount from Beaver Plains Y = Amount from Valley View (0.5pts each) (b) Write the constraints in mathematical form? (2 points) (0.5pts each) (c) How much material should the contractor take from each pit in order to minimize the overall cost of the material? Find the optimum solution. (6 points)
5 Optimal cost: $640,000 (d) If you had not been hired, the contractor would have used 60,000m 3 form Beaver Plains and 40,000m 3 from Valley View. How much did you save the company by giving them your advice? (1 point) $40,000 savings 3. Solve the following linear program using the graphical method (graph sheet on next page). Identify the decision variables. Compute the value of the decision variables and the objective function at optimality and indicate which statement best describes the solution. (10 points) Max Z = 9x + 6y
6 Subject to: 3x+2 y ≤ 30 -6x+ 3y ≤ 12 4x-2 y ≤ 20 y ≤ 7 x, y ≥ 0 (a) This linear program has a unique optimal solution (b) This linear program has alternate optima (c) This linear program is infeasible (d) This linear program is unbounded
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7
8 4. Officials are looking for sites that can fulfill future solid waste disposal needs for a city in an eastern state as the current landfill is filling up faster than expected due to a growing population. A landfill must not only be large enough to handle the weekly needs of the region; it also has to be as environmentally benign as possible. This means that certain types of materials that are placed in the landfill must not exceed certain threshold limits. Also, officials are primarily concerned with satisfying future waste disposal needs in the least costly manner possible. To simplify the problem of analyzing a particular site, city officials have divided up the region into three major areas and estimated the amount of waste (in tons) that can be transported from each part of the region on a weekly basis (due to public opposition to the number of trucks on the local roads). In addition, the amount of non- organic material per ton deposited in the landfill must be kept at a minimum in order for the landfill to provide the maximum capacity over its useful life. It is expected that the absolute limit of non-organic material allowed per week in the landfill will be a composite 1,000 pounds per ton. Answer the following questions. Additional pertinent data are in the table below. (10 points) Location 1 Location 2 Location 3 Cost ($/ton) 95 110 120 Supply limit per week (tons) 500 600 1,000 Non-organic (lbs/ton) 1,400 800 600 a) Formulate a linear program for the problem assuming that officials expect the landfill to handle 1,200 tons per week. (5 points)
9 b) If planners are expecting the landfill to handle 1,200 tons per week, what is the optimal distribution of waste delivery from the three locations in the region? (Hand in the results from Excel: the answer sheet and sensitivity report). How much will the city pay per week for the landfill service? (2 points) Optimal values: Location 1 = 450, Location 2 = 600, Location 3 = 150 Optimal cost: $126,750
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10 c) Suppose you want to do a sensitivity analysis of your results. In particular, you are interested in answering the following questions. How would the optimal cost change if location 2 was able to handle 650 tons per week instead of the current 600 tons? Show how you calculate this answer by referencing your sensitivity analysis form. (Hand in your sensitivity analysis report and supporting analysis from Excel templates). (1 point) New optimal cost = $126,562.50/week d) Suppose a trucking firm comes to you with a plan that could lower the cost per ton for wastes at location 2 from $110/ton to $108/ton, for a nominal fee. What effect will this change have on the following? (1 point) (i) The optimal values of the decision variables? -$2 is within the range of optimality, so the optimal values will not change. (0.5 pts) (ii) The optimal cost? The optimal cost will be reduced by $1200 to $125,550. (0.5 pts)
11 e) If the firm wants to charge the equivalent of a weekly flat fee of $1,000, should metro officials accept this offer? Why or why not? (1 point) $1,000/week < $1,200/week (from 4d(iii)), so city officials should accept this offer.

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