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solutionexamjune8th

# solutionexamjune8th - This exam consists of three pages...

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This exam consists of three pages Faculty of Engineering and Natural Sciences Industrial Engineering Solution Final exam MS 301 june 8th 2011, 9.00-11.00, FENS L045, G035 and G032. Remark: keep your answers as short as possible but of course complete. Also read very carefully the questions. The total number of points of this exam is 100. At the beginning of each part of the exercise the total number of points are listed. Please write your student number at each page and write clearly. To avoid noise during the whole exam it is only allowed during the ﬁrst hour of the exam to ask questions. The results of the ﬁnal will be announced before the makeup on monday june 13th. Exercise 1. (30 points) Suppose that there are m factories that generate waste and n disposal sites. The amount of waste generated at factory i, 1 i m, is given by a i and the capacity of disposal site j is given by b j . The transport of waste should ﬁrst go from a factory to a transfer facility (these facilities need to be constructed) and then from this transfer facility to a disposal site. In order to minimize cost one now needs to determine at which of the K possible locations a transfer facility should be constructed. If a transfer facility is build on site k, 1 k K, it operates at a ﬁxed cost f k at that site, has a capacity of q k of taking waste material from factories and the unit processing cost per ton waste at site k is given by α k . Also let c ik be the unit shipping cost (cost per unit of waste) from factory i to site k and d kj the unit shipping cost from site k to disposal site j. The problem now is to choose at which sites to build a transfer facility so that the total ﬁxed and operating costs of the transfer facilities and the total transportation costs of the waste are minimized. 1. (15 points) Give a deﬁnition of the used decision variables. 2. ( 15 points) Construct the associated optimization problem and explain in detail how you obtained the objective function and the restrictions describing the feasible region. Solution. 1. Introduce for every 1 k K the binary decision variables y k = 1 if a transfer facility is build at site k 0 otherwise 1

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Also introduce for every 1 i m and 1 k K the decision variables x ik = units of waste transported from factory i to transfer site k and for every 1 k K and 1 j n z kj = units of waste transported from transfer site k to disposal site j 2. The restrictions are now given by the following: Amount of waste produced by factory 1 i m K k =1 x ik = a i . Units of waste arriving at transfer facility
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solutionexamjune8th - This exam consists of three pages...

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