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Unformatted text preview: IE 418 Integer Programming Problem Set #1 Answers 1 Formulation Practice 1.1 Problem Nemhauser and Wolsey Problem I.1.8.2 : The BST delivery company must make deliveries to 10 customers J = { 1,2,...,10 } whose demands are d j for j J . A company has four trucks K = { 1,2,3,4 } available each with capacity L k and daily operating cost c k for k K . A single truck cannot deliver to more than 5 customers and customer pairs { 1,7 } , { 2,6 } , { 2,9 } cannot be visited by the same truck. Formulate a model to determine which trucks to use so as to minimize the cost of delivering to all the customers. Answer: Define the indicator binary variables x jk = 1 if and only if truck k K was used to make a delivery to customer j J , and let z k ,k K be an indicator binary variable if truck k K was used. Define y jk be the amount of product delivered to customer j J by truck k K . There are many correct formulations. One mixed integer program that minimizes the daily cost of delivering to all the customers is the following: min X k K c k z k s.t. X k K y jk d j j J (Meet demand) X j J y jk L k k K Truck capacity (1) X j J x jk 5 k K 5 customers served by a truck x 1k + x 7k 1 k K Dont serve 1 and 7 by same truck x 2k + x 6k 1 k K Dont serve 2 and 6 by same truck x 2k + x 9k 1 k K Dont serve 2 and 9 by same truck y jk min { d j ,L k } x jk j J, k K Enforce y jk > 0 x jk = 1 x jk z k j J, k K Enforce x jk > 0 z k = 1 (2) A probably better formulation is to replace inequalities (1) with j J y jk L k z k k K , which removes the need to include inequalities (2). IE418 Problem Set #1 Prof Jeff Linderoth 1.2 Problem Nemhauser and Wolsey Problem I.1.8.3 1.3 Problem Nemhauser and Wolsey Problem I.1.8.4: The DuFour Bottling Company has two machines for its bottle production. The problem is to devise each year a maintenance schedule. Maintenance of each machine lasts 2 months. In addition, only half the workforce is available in July and August, so that only one machine can be used during that period. Monthly demands for bottles are d t ,t = 1,2,...,12 . Machine k,k = 1,2 , produces bottles at the rate of a k bottles per month but can produce less. There is also a labor constraint. Machine k requires l k labor days to produce a k , and the total available days per month are L t for t = 1,2,...12 . Formulate the problem of finding a feasible maintenance schedule in which all the demands are satisfied. Answer: Lets assume that the number of bottles produced by machine k in time t is proportional to the number of labor days. (This wasnt really clear in the problem specification, but most of you assumed it anyway). Well define the following variables: w kt Number of labor days used on machine k in month t x kt A binary indicator variable equal to 1 if and only if machine...
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 Spring '08
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