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Integer Programming $$\sum c_j\delta _j, \delta _j \in \ \{0,1\} \ \ \forall j$$ ∑ c j δ j , δ j ∈ { 0 , 1 } ∀ j

In: Combinatorial, Linear, Integer and Nonlinear Optimization Apps

Author

Listed:
  • J. MacGregor Smith

    (University of Massachusetts)

Abstract

Overview Integer Programming (IP) is one of the most useful optimization procedures because of its practical nature, however, it is also one of the most complex to solve because of the nonlinear, non-convex effects of the integer decision variables. Most of the IP problems for large scale instances are at least $$\mathscr{NP-}$$ NP - Complete for the decision problem, and many are $$\mathscr{NP-}$$ NP - Hard for the optimization part, so that getting optimal solutions is often out of the question. For example: Maximize Z= ( $$550*x_1 + 500*x_2$$ 550 ∗ x 1 + 500 ∗ x 2 ), subject to: $$4*x_1 + 5*x_2 \le 2000, 5*x_1 + 4*x_2 \le 2200, 2.5*x_1 + 7*x_2 \le 1750, x_1,x_2 \ge 0$$ 4 ∗ x 1 + 5 ∗ x 2 ≤ 2000 , 5 ∗ x 1 + 4 ∗ x 2 ≤ 2200 , 2.5 ∗ x 1 + 7 ∗ x 2 ≤ 1750 , x 1 , x 2 ≥ 0 and integer. Optimal solution, $$Z=249,800, x_1=336, x_2 = 130$$ Z = 249 , 800 , x 1 = 336 , x 2 = 130 (green dot in center). See Figure 4.1.

Suggested Citation

  • J. MacGregor Smith, 2021. "Integer Programming $$\sum c_j\delta _j, \delta _j \in \ \{0,1\} \ \ \forall j$$ ∑ c j δ j , δ j ∈ { 0 , 1 } ∀ j," Springer Optimization and Its Applications, in: Combinatorial, Linear, Integer and Nonlinear Optimization Apps, chapter 0, pages 133-178, Springer.
    • Handle: RePEc:spr:spochp:978-3-030-75801-1_4
    DOI: 10.1007/978-3-030-75801-1_4
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