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Given a set of duties (sequences of activities to be accomplished by a bus driver on a specific day) to be operated over a cyclic one-week horizon and groups of drivers with similar characteristics...

The integral simplex using decomposition (ISUD) is a primal algorithm dedicated to solve set partitioning problems (SPP). Given an integer solution, the integral simplex using decomposition (ISUD) seeks a descent direction that leads to an improved adjacent integer solution. It uses a horizontal decomposition (of a linear transformation of the constraint matrix). We propose the integral simplex using double decomposition (ISU2D) which is a parallel version of ISUD. It uses an innovative disjoint vertical decomposition to find in parallel orthogonal descent directions leading to an integer solution with a larger improvement. Each descent direction identifies a set of variables that will leave the current solution and a set of entering variables with better costs. To find these directions, we develop a dynamic decomposition approach that splits the original problem into subproblems that are then solved in parallel by ISUD. Our main innovation is the use of the current solution as a foundation for the construction of the set of subproblems; the set changes during the optimization process as the current solution changes. In addition, we use bounding and pricing strategies and implement parallel processing techniques. We show that ISU2D is 3 to 4 times faster than ISUD on large instances.

Integral simplex using decomposition (ISUD) is a method that efficiently solves set partitioning problems. It is an iterative method that starts from a known integer solution and moves through a sequence of integer solutions, decreasing the cost at each iteration. At each iteration, the method decomposes the original problem into a reduced problem (RP) and a complementary problem (CP). Given an integer solution to RP (that is also solution to the original problem), CP finds a descent direction having the minimum ratio between its cost and the number of its positive variables. We loop on until an optimal or near-optimal solution to the original problem is reached. In this paper, we introduce a modified model for CP. The new model finds a descent direction that minimizes the ratio between the cost of the direction and an overestimation of the number of variables taking one in the next solution. The new CP presents higher chances of finding improved integer solutions without branching. We present results for the same large instances (with up to 570,000 columns) as the ones previously used to test ISUD. For all the instances, optimality is always reached with a speedup factor of at least five.

In this paper, we introduce a general framework for vector space decompositions that decompose the set partitioning problem into a reduced problem, defined in the vector subspace generated by the columns corresponding to nonzero variables in the current integer solution, and a complementary problem, defined in the complementary vector subspace. We show that the integral simplex using decomposition algorithm (ISUD) developed in Zaghrouti et al. (Oper Res 62:435–449, 2014. https://doi.org/10.1287/opre.2013.1247) uses a particular decomposition, in which integrality is handled mainly in the complementary problem, to find a sequence of integer solutions with decreasing objective values leading to an optimal solution. We introduce a new algorithm using a new dynamic decomposition where integrality is handled only in the reduced problem, and the complementary problem is only used to provide descent directions, needed to update the decomposition. The new algorithm improves, at each iteration, the current integer solution by solving a reduced problem very small compared the original problem, that we define by zooming around the descent direction (provided by the complementary problem). This zooming algorithm is superior than ISUD on set partitioning instances from the transportation industry. It rapidly reaches optimal or near-optimal solutions for all instances including those considered very difficult for both ISUD and CPLEX.

The problem of buffer sizing and inspection stations positioning in unreliable production lines is a complex mixed integer nonlinear optimization problem. In this problem, we have a production line with n machines and n fixed-size (storage) buffers in series. The machines produce parts that are either conforming or nonconforming, and the line includes inspection stations that reject the nonconforming pats. The goal is to find the optimal buffer sizes, the number and positions of the inspection stations, and satisfy the customer demand on conforming parts while minimizing the total cost. We present in this paper an exact method to solve this complex manufacturing problem. We also present new theoretical results on buffer-size bounds, stationarity, and cost function convexity permitting to significantly reduce the problem complexity. These theoretical and algorithmic developments allow solving to optimality instances with up to 30 machines tools developed previously cannot solve.

Starting from the improved primal simplex (IPS) decomposition, introduced by Elhallaoui et al. (2011) to tackle degeneracy in general linear programs, we introduce and discuss the mathematical foundations of improved column-generation decompositions (ICG) that are better than the standard column-generation decomposition when degeneracy is an issue. We also present an improved dynamic constraint aggregation (IDCA), which is a specialization of ICG to efficiently solve set partitioning problems. We show that IDCA improves the dynamic constraint aggregation (DCA) and the multiphase dynamic constraint aggregation (MPDCA) algorithms (Elhallaoui et al., 2005, 2010) that not only reduce degeneracy but also profit from it to efficiently solve set partitioning problems. IDCA solves at each iteration a complementary problem (CP) to obtain a group of variables that can be aggregated or pivoted into the basis to bypass degenerate pivots and decrease the objective value and to obtain more “central” dual solutions to generate new columns. Our numerical results show reduction factors in the solution times exceeding 10 for IDCA compared to a state-of-the-art standard column-generation solver.

Designing competitive manufacturing systems with high levels of productivity and quality at a reasonable cost is a complex task. Decision makers must face numerous decision variables which involve multiple and iterative analysis of the estimated cost, quality and productivity of each design alternative. This paper adresses this issue by providing a fast algorithm for solving the buffer sizing and inspection positioning problem of large production lines by combining heuristic and exact algorithms. We develop a multilevel hybrid search method combining a genetic algorithm and tabu search to identify promising locations for the inspection stations and an exact method that optimizes rapidly (in polynomial time) the buffers' sizes for each location. Our method gives valuable insights into the problem, and its solution time is a small fraction of that required by the exact method on production lines with 10-30 machines.