A stochastic integer programming approach to solving a synchronous optical network ring design problem

Abstract

We develop stochastic integer programming techniques tailored toward solving a Synchronous Optical Network (SONET) ring design problem with uncertain demands.Our approach is based on an L-shaped algorithm, whose (integer) master program prescribes a candidate network design, and whose (continuous) subproblems relay information regarding potential shortage penalty costs to the ring design decisions. This naive implementation performs very poorly due to two major problems: (1) the weakness of the master problem relaxations, and (2) the limited information passed to the master problem by the optimality cuts. Accordingly, we enforce certain necessary conditions regarding shortage penalty contributions to the objective function within the master problem, along with a corresponding set of valid inequalities that improve the solvability of the master problem. We also detail how a nonlinear reformulation of the model can be used to capture an exponential number of optimality cuts generated by the linear model. We augment these techniques with a powerful upper-bounding heuristic to further accelerate the convergence of the algorithm, and demonstrate the effectiveness of our methodologies on a test bed of randomly generated stochastic SONET instances.

Locating Hybrid Fuel Cell–Turbine Power Generation Units under Uncertainty

“…However, the single cut algorithm has the fastest running time for three of the instances, and was within 3% of the multicut algorithm for the other two. This provides further evidence for the hypothesis put forth by Smith, Schaefer, and Yen (2002) that the single cut method is preferable for stochastic integer programs with continuous second stage even when the number of scenarios is small.…”

“…Numerical experiments for stochastic linear programs indicate that the multi-cut version is preferred when the number of realizations r is not significantly larger than the number of first-stage constraints Louveaux, 1988, 1997;Gassmann, 1990). However, Smith, Schaefer, and Yen (2002) found that when the first-stage problem is an integer program, the single cut version was preferable even with many fewer scenarios relative to the number of first-stage constraints.…”

Stochastic programming for nurse assignment

“…(On the first iteration, let η ← −∞, and let X be a feasible assignment) for all n ∈ N ,ξ ∈ do bound on the optimal solution. On each iteration, we consider a subset of dual extreme points ⊆ , and let constraint set (27 ) be the subset of (27) over . We solve a restricted master problem (2,5,26) and (27 ) to find an assignment X and an anticipated objective value η.…”

Two-stage quadratic integer programs with stochastic right-hand sides

“…Stochastic programs have many applications, including supply chain network design [53], telecommunications [34,52], server location [43] and dynamic capacity acquisition [8]. Imposing integrality restrictions on the second-stage variables increases the problem complexity significantly as the expected recourse function becomes nonconvex and discontinuous in general [55].…”