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Mixed-integer linear programming approaches for nested $p$-center problems with absolute and relative regret objectives

Christof Brandstetter, Markus Sinnl

TL;DR

This work introduces the nested p$$ p $$ ‐center problem, a multi‐period variant of the well‐known p$$ p $$ ‐center problem, and develops branch‐and‐bound/branch‐and‐cut solution algorithms that include a preprocessing procedure that exploits the nesting property and starting heuristics and primal heuristics.

Abstract

We introduce the nested $p$-center problem, which is a multi-period variant of the well-known $p$-center problem. The use of the nesting concept allows to obtain solutions, which are consistent over the considered time horizon, i.e., facilities which are opened in a given time period stay open for subsequent time periods. This is important in real-life applications, as closing (and potential later re-opening) of facilities between time periods can be undesirable. We consider two different versions of our problem, with the difference being the objective function. The first version considers the sum of the absolute regrets (of nesting) over all time periods, and the second version considers minimizing the maximum relative regret over the time periods. We present three mixed-integer programming formulations for the version with absolute regret objective and two formulations for the version with relative regret objective. For all the formulations, we present valid inequalities. Based on the formulations and the valid inequalities, we develop branch-and-bound/branch-and-cut solution algorithms. These algorithms include a preprocessing procedure that exploits the nesting property and also begins heuristics and primal heuristics. We conducted a computational study on instances from the literature for the $p$-center problem, which we adapted to our problems. We also analyse the effect of nesting on the solution cost and the number of open facilities.

Mixed-integer linear programming approaches for nested $p$-center problems with absolute and relative regret objectives

TL;DR

This work introduces the nested p ‐center problem, a multi‐period variant of the well‐known p ‐center problem, and develops branch‐and‐bound/branch‐and‐cut solution algorithms that include a preprocessing procedure that exploits the nesting property and starting heuristics and primal heuristics.

Abstract

We introduce the nested -center problem, which is a multi-period variant of the well-known -center problem. The use of the nesting concept allows to obtain solutions, which are consistent over the considered time horizon, i.e., facilities which are opened in a given time period stay open for subsequent time periods. This is important in real-life applications, as closing (and potential later re-opening) of facilities between time periods can be undesirable. We consider two different versions of our problem, with the difference being the objective function. The first version considers the sum of the absolute regrets (of nesting) over all time periods, and the second version considers minimizing the maximum relative regret over the time periods. We present three mixed-integer programming formulations for the version with absolute regret objective and two formulations for the version with relative regret objective. For all the formulations, we present valid inequalities. Based on the formulations and the valid inequalities, we develop branch-and-bound/branch-and-cut solution algorithms. These algorithms include a preprocessing procedure that exploits the nesting property and also begins heuristics and primal heuristics. We conducted a computational study on instances from the literature for the -center problem, which we adapted to our problems. We also analyse the effect of nesting on the solution cost and the number of open facilities.
Paper Structure (26 sections, 11 theorems, 15 equations, 4 figures)

This paper contains 26 sections, 11 theorems, 15 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Literature review
  3. The p-center problem
  4. Nested and other multi-period facility location problems
  5. Mixed-Integer linear programming formulations for the nested p-center problem with the sum of absolute regrets objective
  6. First formulation
  7. Second formulation
  8. Third formulation
  9. Implementation details
  10. Preprocessing
  11. Starting heuristics
  12. Implementation details of formulation (nPCA2)
  13. Separation routine implemented in the LazyConstraintCallback
  14. Separation routine implemented in the UserCutCallback
  15. Primal heuristic
  16. ...and 11 more sections

Key Result

Proposition 7

For a given $h \in \mathcal{H}$, let $LB^h \geq 0$ be a lower bound on the value of decision variable $z^h$ of eq:nPC for any optimal solution. Then is a valid inequality for eq:nPC i.e. every feasible solution of eq:nPC fulfills eq:nPC-OPT.

Figures (4)

  • Figure 1: Instance eil51 with the optimal solutions for n-$p$CPA (rectangles) with $\mathcal{P} = \{4, 5, 6\}$ and for $p$CP (triangles) for $p=4,5,6$
  • Figure 2: Runtimes and optimality gaps for different settings and formulations
  • Figure 3: Comparison of the relative regrets of the optimal solution value and number of open facilities for $p$CP and n-$p$CP
  • Figure 4: Comparison of the different settings

Theorems & Definitions (26)

  • Definition 1
  • Definition 3
  • Definition 4
  • Proposition 7
  • proof
  • Proposition 8
  • proof
  • Proposition 9
  • proof
  • Proposition 10
  • ...and 16 more