Integer Programming

Integer Programming - Lecture 3: Introduction to integer...

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Lecture 3: Introduction to integer linear programming Contents 1 Lumpy linear programs with fxed charges 1 2 Piece-wise linear costs 1 3 Knapsack and Capital Budgeting problems 2 4 Set packing, covering and partitioning models 3 5 Assignment and Matching problems 4 6 Facility location and network design problems 5 7 Traveling salesman models 7 8 Processor scheduling and sequencing problems 7 9 Solution methods 8 9 . 1 S o lv in gIP sa sLP s .......................................... 10 9.2 Rounding LP solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1 Lumpy linear programs with fxed charges In this lecture we will review optimization models where the constraints and the objective are linear but the domain D has a discrete component in the sense that some or all the variables are restricted to take only a discrete set of values. This restriction leads to interesting ramiFcations for modeling. We will breifly review the basic integer linear programming models in the lecture. Example 1 Suppose we want to invest in three mutual funds. Each of the three mutual funds speciFes a minimum amount l i that has to be invested in case one decides to invest in that fund. Each fund charges a front-end load consisting of a Fxed cost c i plus a proportional cost at the rate α i , i =1 ,..., 3. Suppose the investor has a capital W to invest. Describe the set of all feasible investments. Problem is crying out for decision variable y i , i , 2 , 3 deFned as follows. y i = ± 1 , invest non-zero amount in fund i, 0 , otherwise Let x i denote the amount invested in the mutual fund i 3. Using y i we want to ensure that if y i =0 then x i =0andif y i =1then l i x i W , i.e. l i y i x i Wy i The budget constraint is as follows m X i =1 ( c i y i +(1+ α i ) x i ) W. Thus, the set of all feasible investment decisions is given by X = ² ( x , y ): y i ∈{ 0 , 1 } ,l i y i x i i ,i , 2 , 3 , m X i =1 ( c i y i α i ) x i ) W ³
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Integer programming 2 Consider the following modiFcation. Suppose the mutual funds also stipulate that, after the minimum amount l i has been invested, all additional investments must be in multiples of d i , i =1 , 2 , 3. How does this change the set of feasible investments ? All this new constraint is saying is that x i = l i + d i z i for some z i 0 and integer. We will denote the set of integers by Z and the set of non-negative integers by Z + .Thu s , z i Z + . Incorporating this constraint we get X = ± ( x , z , y ): y i ∈{ 0 , 1 } ,z i Z + ,x = y i + d i z i , 0 d i z i ( W l i ) y i ,i , 2 , 3 , m i =1 ( c i y i +(1+ α i ) x i ) W ² 2 Piece-wise linear costs In the linear programming part of this course we discussed how to handle piecewise linear convex “ constraints and piecewise linear concave “ ” constraints. In this section we will discuss methods to handle the other two types of constaints and also piecewise linear functions with Fxed costs.
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Integer Programming - Lecture 3: Introduction to integer...

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