hw3_sol.pdf - CO 370 Homework assignment 3 Fall 17 Page 1...

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CO 370 - Homework assignment 3 Fall ’17 Page 1 CO 370 - Fall ’17 Homework assignment #3: (Due on Friday, Nov 3, at 11:59pm) Instructions: You will be graded not only on correctness, but in clarity of exposition. Allowed sources of help: CO370 course notes and your CO370 classmates. Please write down the name of the people you collaborate with. Any source of help other than the ones stated above are considered cheating. In the models, please define clearly what are your decision variables. Also, state clearly what your constraints represent. Your models should be as general as possible. All answers must be justified unless otherwise stated. You are responsible to make sure your submission on crowdmark is legible by the grader and that it is received on time. Submissions that are poorly scanned may be penalized depending on how bad the scan is. Question 1 (20 points) A company is considering opening factories to produce a given product in a set of cities F = { 1 , . . . , f } . Each factory i F has a nominal capacity to produce p i units per week of a given product. The weekly fixed cost of keeping each factory open is c i , for all i F . The country is divided into a set of regions R = { 1 , . . . , r } and each region j R requires q j units of the product per week. The costs of shipping one unit from a factory to a region are π ij for all i F , j R . In addition to the nominal capacity of each factory, the company can invest some money in each factory to increase its capacity (up to a maximum of D dollars per factory i F ). The capacity increase g ( x ) as a function of the investment x is given as the following function: g ( x ) = a 1 x if x [0 , d 1 ] a 1 d 1 + a 2 ( x - d 1 ) if x ( d 1 , d 2 ] a 1 d 1 + a 2 ( d 2 - d 1 ) + a 3 ( x - d 2 ) if x ( d 2 , D ] (you cannot assume anything about the convexity/concavity of g ) You may assume that any fraction of a product can be shipped. In addition we must satisfy the following: 1. Factories have a minimum operating threshold of shipping. That is, for instance, the company does not want to ship 0.0000001 unit of a product. Thus, if factory i ships to customer j , then it must ship at least l ij units. 2. If factories 3 and 4 are opened, then either factory 5 or 6 must remain closed. 3. If factories 7 or 8 are opened AND factory f remains closed, then factories 10 and 11 must be opened OR factory 12 must remain closed. (here “or” means either one condition, or the other, or both) Write an IP to minimize the costs subject to the above constraints.
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CO 370 - Homework assignment 3 Fall ’17 Page 2 Solution: Decision variables: Let the binary variable y i ∈ { 0 , 1 } , i F denote whether or not factory i is open. Also, let variable x ij represent the amount to ship from factory i to region j , for all i F, j R .
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