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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 Don’t serve 1 and 7 by same truck x 2k + x 6k ≤ 1 ∀ k ∈ K Don’t serve 2 and 6 by same truck x 2k + x 9k ≤ 1 ∀ k ∈ K Don’t 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: Let’s assume that the number of bottles produced by machine k in time t is proportional to the number of labor days. (This wasn’t really clear in the problem specification, but most of you assumed it anyway). We’ll 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
 Ralphs
 Operations Research, Optimization, Jeff Linderoth, Prof Jeff Linderoth, Wolsey Problem, Problem Nemhauser

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