Unformatted text preview: d weight. Or one wants to select a subset of potential products in which to invest. Or,
one has a set of different integers, and one wants to select a subset that sums to a value K. In these
cases, it is typical for the integer variables to be as follows: ⎧1
⎪
xi = ⎨
⎪0
⎩ if element i is selected
otherwise. Example: knapsack/capital budgeting. In this example, there are six items to select from.
Item
Cost
Value 1
5
16 2
7
22 3
4
12 4
3
8 5
4
11 6
6
19 Problem: choose items whose cost sums to at most 14 so as to maximize the utility. maximize
Formulation: 16 x1 + 22 x2 + 12 x3 + 8 x4 + 11x5 + 19 x6
5x1 + 7 x2 + 4 x3 + 3x4 + 4 x5 + 6 x6 ≤ 14 subject to xi ∈{0,1}
In general: for i = 1 to 6. Maximize the value of the selected items such that the weight is at most b.
Ci = value of item i for i = 1 to n.
ai = weight of item i for i = 1 to n.
b = bound on total weight.
n maximize ∑c x
i =1 ii n subject to ∑a x
i =1 ii ≤ b xi ∈{0,1} for i = 1 to n. Covering and packing problems. In some selection problems, each item is associated with a subset of a lar...
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This note was uploaded on 03/18/2014 for the course MGMT 15.053 taught by Professor Jamesorli during the Spring '07 term at MIT.
 Spring '07
 JamesOrli
 Management

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