2 Modelling_1

2 Modelling_1 - IEE 598 - Lecture 2 (I.1) Modelling with...

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IEE 598 - Lecture 2 (I.1) Modelling with Integer Variables Muhong Zhang DEPARTMENT OF INDUSTRIAL ENGINEERING Jan. 21, 2010
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Why do we need integer variables? Zhang IEE 376 Introduction to OR 2 / 17
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Why do we need integer variables? When the variables are associated with a physical identity indivisible . Example: a. Number of trucks to ship from point A to point B; b. Number of staff assigned to a shift; c. Number of products to produce; ... Zhang IEE 376 Introduction to OR 3 / 17
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Indicator of events (yes/no) Example: 0 - 1 Knapsack Problem We are given a set of items. Let the set be N . For any item i N , a value p i and weight w i are associated. Goal: Select a subset of items with the maximum value such that the total weight is less than a constant K . Decision variables: x i : = 1 if item i is in the knapsack; = 0 otherwise, i N . Objective function: max i N p i x i Constraints: i N w i x i K x i ∈ { 0 , 1 } , i N Zhang IEE 376 Introduction to OR 4 / 17
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Example: Matching/Assignment Problem There are n people and m jobs, where n m . Each job must be done by exactly one person. Each person can do, at most, one job. The cost of person j doing job i is c ij . Goal: Assign the people to the jobs to minimize the total cost. Decision variables: x ij : = 1 if people i is assigned to machine j ; = 0 otherwise, i = 1 , ..., n , j = 1 , ..., m . Objective function: min n i = 1 m j = 1 c ij x ij Constraints: n i = 1 x ij = 1, j = 1 , ..., m m j = 1 x ij 1, i = 1 , ..., n x ∈ { 0 , 1 } mn . Zhang
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This note was uploaded on 05/01/2011 for the course IEE 598 taught by Professor Hillary during the Spring '10 term at USC.

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2 Modelling_1 - IEE 598 - Lecture 2 (I.1) Modelling with...

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