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Lecture 15 - Integer

# Lecture 15 - Integer - 540: OPERATIONSRESEARCH...

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540:311 DETERMINISTIC MODELS IN OPERATIONS RESEARCH Lecture 15: Chapter 9.1‐9.4 Class MeeCng: Thu April 21 st 10:20‐11:40am Prof. W. Art Chaovalitwongse

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Integer Programming DefiniBon An Integer Programming problem (IP) is a Linear Programming (LP) in which some or all the variables are required to be nonnegaBve integers (non‐fracConal, whole numbers) . all variables are required to be integer is called a pure integer programming problem . some variables are restricted to be integer and some are not is called a mixed integer programming problem . The case where the integer variables are restricted to be 0 or 1 comes up surprisingly oKen. Such problems are called pure (mixed) 0‐1 programming problems or pure (mixed) binary integer programming problems .
Integer Programming 0 1 2 3 4 5 0 1 2 3 4 5 Question: What is the optimal integer solution? What is the optimal linear solution? Can one use linear programming to solve the integer program? maximize 3x + 4y subject to 5x + 8y 24 x, y 0 and integer What is the opBmal soluBon?

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A rounding technique that someBmes is useful, and someBmes not. 0 1 2 3 4 5 0 1 2 3 4 5 Solve LP (ignore integrality) get x=24/5, y=0 and z =14 2/5. Round, get x=5, y=0, infeasible! Truncate, get x=4, y=0, and z =12 Same solution value at x=0, y=3. Optimal is x=3, y=1 , and z =13
An All-Integer Programming Problem Boxcar Burger Restaurant \$2.7 million available expansion. Currently employs 19 managers to run the restaurants. Open at least 2 restaurants in downtown. Determine how many restaurants should be opened in suburban and downtown locations in order to maximize its total weekly net profit. Formulate the problem. Suburban Location Downtown Location Investment \$200,000 \$600,000 Net Weekly Profit \$1200 \$2000 Managers 3 1

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1 2 3 4 5 6 7 8 x 1 1 2 3 4 5 6 x 2 Solving Integer Programming Problem LP Optimal x 1 = 5 7 / 16 x 2 = 2 11 / 16 Max 1200 x 1 + 2000 x 2 ST: 2 x 1 + 6 x 2 27 x 2 2 3 x 1 + x 2 19 x 1 , x 2 0 and Integer LP relaxation, then round off ?
Solving Integer Programming Problem 1 2 3 4 5 6 7 8 x 1 1 2 3 4 5 6 x 2 LP Optimal x 1 = 5 7 / 16 x 2 = 2 11 / 16 Round up? x 1 = 6 x 2 = 3 Round off? x 1 = 5 x 2 = 3 Round down? x 1 = 5 x 2 = 2 Max 1200 x 1 + 2000 x 2 ST: 2 x 1 + 6 x 2 27 x 2 2 3 x 1 + x 2 19 x 1 , x 2 0 and Integer LP relaxation, then round off ?

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Solving Integer Programming Problem 1 2 3 4 5 6 7 8 x 1 1 2 3 4 5 6 x 2 LP relaxation gives an upper bound for Maximization Problem IP Optimal x 1 = 4 x 2 = 3
Types of Integer Programming Models Models Types of Decision Models All – integer (IP) All are integers Mixed-integer (MIP) Some, but not all, are integers Binary (BIP) All are either 0 or 1 The LP obtained by omitting all integer or 0-1 constraints on variables is called LP relaxation of the IP We also permit x j {0,1}. This is equivalent to 0 x j 1 and x j integer.

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Binary integer variables can be used to model yes/no decisions, such as whether to build a plant or buy a piece of equipment.
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