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Unformatted text preview: ENGRI 1101 Engineering Applications of QR Fall ’08 Prelim 2 SOLUTIONS Name: 1 Points Possible Ii You have 1 hour and 30 minutes to answer all the questions in the exam. The total number of points is 100.
For questions with multiple parts, the point value of each part is given at the start of that part. Be sure to give full explanations for all of your answers. In most cases, the correct reasoning alone
is worth more credit than the correct answer by itself. Your explanation need not be wordy, but it should be sufﬁcient to justify your answer. You receive much more credit for an incorrect answer if you recognize that something that you have computed
is not correct than if you merely pretend that it is correct. Not all the parts are in the same level of difﬁculty! Please make sure that you do not
spend too much time on one part. Good Luck! l l 1_ (30) Consider the following linear programming problem: .
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as the ent‘ermg l I KDerg‘ma 1‘ Cl 0 2. (25) During the next 2 months General Cars must meet (on time) the following demands for trucks and cars: month 1, 400. trucks and 800 cars; month 2, 300 trucks and 300
cars. During each month a total of at most 1000 vehicles can be produced. At the
beginning of month 1, 100 trucks and 200 cars are in inventory, available for immediate
delivery. ‘ Each truck uses 2 tons of steel, and each car uses 1 ton of steel. During month 1, steel
costs $400 per ton; during month 2, steel costs $600 per ton. At most 2500 tons of
steel can be purchased each month.‘ (Steel can be used only during the month in which
it is purchased.) ‘ There is an option of holding vehicles that remain on hand after satisfying the demand in month 1 in inventory, and using them to satisfy the demand in month 2. A holding
cost of $150 per each vehicle held in inventoryis assessed. Each car gets 20 mpg (miles per gallon), and each truck «gets 10 mpg. During each
month, the vehicles produced by the company must average at least 16 mpg. Formulate a linear programming model to determine how to meet the demand and
mileage requirements at minimum total cost (do not worry about integrality). Begin
by clearly deﬁning all the decision variables, and express the objective function and
constraints in terms of these variables. (/
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constrained diet problem (which is, as you should recall, a minimization problem). You
may assume that each objective function coefﬁcient is integer. At each node, you are
given the optimal value of the current version’s.LP relaxation, and the lowest indexed
variable (i.e. the variable with the smallest subscript) that has a fractional value in \
the LP optimal solution. (If no such variable exists, then it is indicated that this LP optimum is integer.) LP
infeasible (a) (8) Suppose that you are given only this information: what can you conclude about
the optimal value for this integer program? That is, give the best upper and lower .
bounds on the optimal value. ' ’ .13
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constraints should be imposed for each of these two branches. l (c) (8) Suppose that the LP values for these two wrenches are equal to 19.1 and ‘20.1
and the corresponding solutions are not integer. What can you conclude about the optimal value of the integer program now? Justify your answer.
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would your answer to part (c) be now? Justify your answer. 5 of {QAQROf‘ ~ was b
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show how to ﬁnd it, rather than just writing it down.) maximize 43:1 + 8372 .. 6x3 _ 83:4 subjectto
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