Simultaneous column-and-row generation for large-scale linear programs with column-dependent-rows

In this paper, we develop a simultaneous column-and-row generation algorithm that could be applied to a general class of large-scale linear programming problems. These problems typically arise in the context of linear programming formulations with exponentially many variables. The defining property for these formulations is a set of linking constraints, which are either too many to be included in the formulation directly, or the full set of linking constraints can only be identified, if all variables are generated explicitly. Due to this dependence between columns and rows, we refer to this class of linear programs as problems with column-dependent-rows. To solve these problems, we need to be able to generate both columns and rows on-the-fly within an efficient solution approach. We emphasize that the generated rows are structural constraints and distinguish our work from the branch-and-cut-and-price framework. We first characterize the underlying assumptions for the proposed column-and-row generation algorithm. These assumptions are general enough and cover all problems with column-dependent-rows studied in the literature up until now to the best of our knowledge. We then introduce in detail a set of pricing subproblems, which are used within the proposed column-and-row generation algorithm. This is followed by a formal discussion on the optimality of the algorithm. To illustrate the proposed approach, the paper is concluded by applying the proposed framework to the multi-stage cutting stock and the quadratic set covering problems

Simultaneous column-and-row generation for large-scale linear programs with column-dependent-rows

In this paper, we develop a simultaneous column-and-row generation algorithm for a general class of large-scale linear programming problems. These problems typically arise in the context of linear programming formulations with exponentially many variables. The defining property for these formulations is a set of linking constraints. These constraints are either too many to be included in the formulation directly, or the full set of linking constraints can only be identified, if all variables are generated explicitly. Due to this dependence between columns and rows, we refer to this class of linear programs as problems with column-dependent-rows. To solve these problems, we need to be able to generate both columns and rows on the fly within an efficient solution method. We emphasize that the generated rows are structural constraints and distinguish our work from the branch-and-cut-and-price framework. We first characterize the underlying assumptions for the proposed column-and-row generation algorithm and then introduce the associated set of pricing subproblems in detail. The proposed methodology is demonstrated on numerical examples for the multi-stage cutting stock and the quadratic set covering problems

High level rule modeling language for airline crew pairing

The crew pairing problem is an airline optimization problem where a set of least costly pairings (consecutive flights to be flown by a single crew) that covers every flight in a given flight network is sought. A pairing is defined by using a very complex set of feasibility rules imposed by international and national regulatory agencies, and also by the airline itself. The cost of a pairing is also defined by using complicated rules. When an optimization engine generates a sequence of flights from a given flight network, it has to check all these feasibility rules to ensure whether the sequence forms a valid pairing. Likewise, the engine needs to calculate the cost of the pairing by using certain rules. However, the rules used for checking the feasibility and calculating the costs are usually not static. Furthermore, the airline companies carry out what-if-type analyses through testing several alternate scenarios in each planning period. Therefore, embedding the implementation of feasibility checking and cost calculation rules into the source code of the optimization engine is not a practical approach. In this work, a high level language called ARUS is introduced for describing the feasibility and cost calculation rules. A compiler for ARUS is also implemented in this work to generate a dynamic link library to be used by crew pairing optimization engines

A note on "A LP-based heuristic for a time-constrained routing problem"

In their paper, Avella et al. (2006) investigate a time-constrained routing problem. The core of the proposed solution approach is a large-scale linear program that grows both row- and column-wise when new variables are introduced. Thus, a column-and-row generation algorithm is proposed to solve this linear program optimally, and an optimality condition is presented to terminate the column-and-row generation algorithm. We demonstrate by using Lagrangian duality that this optimality condition is incorrect and may lead to a suboptimal solution at termination

A note on "A LP-based heuristic for a time-constrained routing problem"

Avella et al. (2006) [Avella, P., D'Auria, B., Salerno, S. (2006). A LP-based heuristic for a time-constrained routing problem. European Journal of Operational Research 173:120-124] investigate a time-constrained routing (TCR) problem. The core of the proposed solution approach is a large-scale linear program (LP) that grows both row- and column-wise when new variables are introduced. Thus, a column-and-row generation algorithm is proposed to solve this LP optimally, and an optimality condition is presented to terminate the column-and-row generation algorithm. We demonstrate that this optimality condition is incorrect and may lead to a suboptimal solution at termination. We identify the source of this error and discuss how the generic column-and-row generation algorithm proposed by Muter et al. (2010) may be applied to this TCR problem in order to solve the proposed large-scale LP correctly

The set covering problem revisited: an empirical study of the value of dual information

This paper investigates the role of dual information on the performances of heuristics designed for solving the set covering problem. After solving the linear programming relaxation of the problem, the dual information is used to obtain the two main approaches proposed here: (i) The size of the original problem is reduced and then the resulting model is solved with exact methods. We demonstrate the effectiveness of this approach on a rich set of benchmark instances compiled from the literature. We conclude that set covering problems of various characteristics and sizes may reliably be solved to near optimality without resorting to custom solution methods. (ii) The dual information is embedded into an existing heuristic. This approach is demonstrated on a well-known local search based heuristic that was reported to obtain successful results on the set covering problem. Our results demonstrate that the use of dual information significantly improves the efficacy of the heuristic in terms of both solution time and accuracy

On EOQ cost models with arbitrary purchase and transportation costs

We analyze an economic order quantity cost model with unit out-of-pocket holding costs, unit opportunity costs of holding, fixed ordering costs, and general purchase-transportation costs. We identify the set of purchase-transportation cost functions for which this model is easy to solve and related to solving a one-dimensional convex minimization problem. For the remaining purchase-transportation cost functions, when this problem becomes a global optimization problem, we propose a Lipschitz optimization procedure. In particular, we give an easy procedure which determines an upper bound on the optimal cycle length. Then, using this bound, we apply a well-known technique from global optimization. Also for the class of transportation functions related to full truckload (FTL) and less-than-truckload (LTL) shipments and the well-known carload discount schedule, we specialize these results and give fast and easy algorithms to calculate the optimal lot size and the corresponding optimal order-up-to-level

Gold(i)/Zn(ii) catalyzed tandem hydroamination/annulation reaction of 4-yne-nitriles

The tandem hydroamination-annulation reaction of 4-pentyne-nitriles in the presence of amine nucleophiles and a cooperatively operating catalyst system, consisting of Ph3PAuCl and Zn(ClO4)2, provides an efficient route to 2-aminopyrroles. Two regioisomeric 2-aminopyrroles were formed in moderate to good yields. © 2010 The Royal Society of Chemistry.TÜBİTAK; TÜBA and MET

Effects of a saponin fraction extracted from Trigonella foenum-graecum L. and two commercially available saponins on sex ratio and gonad histology of Nile tilapa fry, Oreochromis niloticus (L.)

Over three million tonnes (t) of tilapia, mostly Nile tilapia (Oreochromis niloticus, L.), are produced annually making it the second most abundantly produced freshwater fish (FAO, 2010). Tilapia are mouthbreeders that often produce stunted populations under pond conditions; one means of prevention is to produce all-male fish with the additional advantage that males usually grow faster than females. All-male populations can be achieved by supplementing feed with androgens such as 17-α-Methyltestosterone (MT) during days 10–25 post-hatch (Pandian and Sheela, 1995). However, MT is considered to be carcinogenic (Velazquez and Alter, 2004), and Hulak et al. (2008) also showed that effluents of systems in which carp were fed diets containing MT caused masculinization of female fish. Furthermore, in aquaculture the application of hormones to fish destined for human consumption is prohibited in the European Union under directive 96/22/EC, article 5, which also prohibits import of animal products produced with hormones. Kwon et al. (2000) showed that Fadrozole, a non-steroidal compound, caused masculinization in tilapia by inhibiting aromatase, which is the enzyme responsible for the conversion of endogenous androgens to estrogens. Steinbronn et al. (2004) were able to show that a dose of 2000 ppm Quillaja saponins (Sigma S-2149) inhibited reproduction of tilapia after dietary application for 32 days to first-feeding fry, suggesting saponins as a possible alternative to MT. These secondary plant compounds consist of either a steroid or triterpenoid basic structure (aglycone or sapogenin) plus one or more sugar side chains (Francis et al., 2002a). In a previous experiment a saponin fraction from the soapbark tree (Quillaja saponaria M.) inhibited aromatase in vitro (Golan et al., 2008). The fenugreek plant (Trigonella foenum-graecum L), widely cultivated in the Middle East and Asia, also has a high saponin content. The experiment was therefore conducted to test whether saponin fractions from Q. saponaria and from T. foenum-graecum were able to influence the sex ratio and gonad histology of Nile tilapia