Mean-risk stochastic programs model uncertainty by including risk measures in the objective function. This allows for modeling risk averseness for many problems in science and engineering. This dissertation addresses gaps in the literature on stochastic programs with mean-risk objectives. This includes a need for a computational study of the few available algorithms for this class of problems. The study was aimed at implementing and performing an empirical investigation of decomposition algorithms for stochastic linear programs with absolute semideviation (ASD) and quantile deviation (QDEV) as mean-risk measures. Specifically, the goals of the study were to analyze for specific instances how algorithms perform across different levels of risk, investigate the effect of using ASD and QDEV as risk measures, and understand when it is appropriate to use the risk-averse approach over the risk-neutral one. We derive two new subgradient based algorithms for the ASD and QDEV models, respectively. These algorithms are based on decomposing the stochastic program stage-wise and using a single (aggregated) cut in the master program to approximate the mean and deviation terms of the mean-risk objective function. We also consider a variant of each of the algorithms from the literature in which the mean-risk objective function is approximated by separate optimality cuts, one for the mean and one for the deviation term. These algorithms are implemented and applied to standard stochastic programming test instances to study their comparative performance. Both the aggregated cut and separate cut algorithms have comparable computational performance for ASD, while the separate cut algorithm outperforms its aggregate counterpart for QDEV. The computational study also reveals several insights on mean-risk stochastic linear programs. For example, the results show that for most standard test instances the risk-neutral approach is still appropriate. We show that this is the case due to the test instances having random variables with uniform marginal distributions. In contrast, when these distributions are changed to be non-uniform, the risk-averse approach is preferred. The results also show that the QDEV mean-risk measure has broader flexibility than ASD in modeling risk.

In this thesis, we present variants of Shor and Zhurbenko's r-algorithm, motivated by the memoryless and limited memory updates for differentiable quasi-Newton methods. This well known r-algorithm, which employs a space dilation strategy in the direction of the difference between two successive subgradients, is recognized as being one of the most effective procedures for solving nondifferentiable optimization problems. However, the method needs to store the space dilation matrix and update it at every iteration, resulting in a substantial computational burden for large-sized problems. To circumvent this difficulty, we first develop a memoryless update scheme. In the space transformation sense, the new update scheme can be viewed as a combination of space dilation and reduction operations. We prove convergence of this new algorithm, and demonstrate how it can be used in conjunction with a variable target value method that allows a practical, convergent implementation of the method. For performance comparisons we examine other memoryless and limited memory variants, and also prove a modification of a related algorithm due to Polyak that employs a projection on a pair of Kelley's cutting planes. These variants are tested along with Shor's r-algorithm on a set of standard test problems from the literature as well as on randomly generated dual transportation and assignment problems. Our computational experiments reveal that the proposed memoryless space dilation and reduction algorithm (VT-MSDR) and the proposed modification of the Polyak-Kelly cutting plane method (VT-PKC) provide an overall competitive performance relative to the other methods tested with respect to solution quality and computational effort. The r-Algorithm becomes increasingly more expensive with an increase in problem size, while not providing any gain in solution quality. The fixed dilation (with no reduction) strategy (VT-MSD) provides a comparable, though second-choice, alternative to VT-MSDR. Employing a two-step limited memory extension over VT-MSD sometimes helps in improving the solution quality, although it adds to computational effort, and is not as robust a procedure. / Master of Science

This dissertation addresses the development of enhanced formulations for minimax and mixed-integer programming models for certain industrial and logistical systems, along with the design and implementation of efficient algorithmic strategies. We first examine the general class of minimax mixed-integer 0-1 problems of the type that frequently arise in decomposition approaches and in a variety of location and scheduling problems. We conduct an extensive polyhedral analysis of this problem in order to tighten its representation using the Reformulation-Linearization/Convexification Technique (RLT), and demonstrate the benefits of the resulting lifted formulations for several classes of problems. Specifically, we investigate RLT-enhanced Lagrangian dual formulations for the class of minimax mixed-integer 0-1 problems in concert with deflected/conjugate subgradient algorithms. In addition, we propose two general purpose lifting mechanisms for tightening the mathematical programming formulations associated with such minimax optimization problems. Next, we explore novel continuous nonconvex as well as lifted discrete formulations for the notoriously challenging class of job-shop scheduling problems with the objective of minimizing the maximum completion time (i.e., minimizing the makespan). In particular, we develop an RLT-enhanced continuous nonconvex model for the job-shop problem based on a quadratic formulation of the job sequencing constraints on machines. The tight linear programming relaxation that is induced by this formulation is then embedded in a globally convergent branch-and-bound algorithm. Furthermore, we design another novel formulation for the job-shop scheduling problem that possesses a tight continuous relaxation, where the non-overlapping job sequencing constraints on machines are modeled via a lifted asymmetric traveling salesman problem (ATSP) construct, and specific sets of valid inequalities and RLT-based enhancements are incorporated to further tighten the resulting mathematical program. The efficacy of our enhanced models is demonstrated by an extensive computational experiment using classical benchmark problems from the literature. Our results reveal that the LP relaxations produced by the lifted ATSP-based models provide very tight lower bounds, and directly yield a 0\% optimality gap for many benchmark problems, thereby substantially dominating other alternative mixed-integer programming models available for this class of problems. Notably, our lifted ATSP-based formulation produced a 0\% optimality gap via the root node LP relaxation for 50\% of the classical problem instances due to Lawrence. We also investigate enhanced model formulations and specialized, efficient solution methodologies for applications arising in four particular industrial and sports scheduling settings. The first of these was posed to us by a major trucking company (Volvo Logistics North America), and concerns an integrated assembly and routing problem, which is a unique study of its kind in the literature. In this context, we examine the general class of logistical systems where it is desirable to appropriately ascertain the joint composition of the sequences of vehicles that are to be physically connected along with determining their delivery routes. Such assembly-routing problems occur in the truck manufacturing industry where different models of vehicles designed for a network of customers need to be composed into compatible groups (assemblies) and subsequently dispatched via appropriately optimized delivery routes that are restricted by the particular sequence in which the trucks are connected. A similar structure is exhibited in the business of shipping goods via boat-towed barges along inland waterways, or via trains through railroad networks. We present a novel modeling framework and column generation-based optimization approach for this challenging class of joint vehicle assembly-routing problems. In addition, we suggest several extensions to accommodate particular industrial restrictions where assembly sequence-dependent delivery routes are necessary, as well as those where limited driver- and equipment-related resources are available. Computational experience is provided using large-scale realistic data to demonstrate the applicability of our suggested methodology in practice. The second application addressed pertains to a production planning problem faced by a major motorcycle manufacturing firm (Harley-Davidson Motor Company). We consider the problem of partitioning and sequencing the production of different manufactured items in mixed-model assembly lines, where each model has various specific options and designated destinations. We propose a mixed-integer programming formulation (MPSP1) for this problem that sequences the manufactured goods within production batches in order to balance the motorcycle model and destination outputs as well as the load demands on material and labor resources. An alternative (relaxed) formulation (MPSP2) is also presented to model a closely related case where all production decisions and outputs are monitored within a common sequence of batches, which permits an enhanced tighter representation via an additional set of hierarchical symmetry-defeating constraints that impart specific identities amongst batches of products under composition. The latter model inspires a third set partitioning-based formulation in concert with an efficient column generation approach that directly achieves the joint partitioning of jobs into batches along with ascertaining the sequence of jobs within each composed batch. Finally, we investigate a subgradient-based optimization strategy that exploits a non-differentiable optimization formulation, which is prompted by the flexibility in the production process as reflected in the model via several soft-constraints, thereby providing a real-time decision-making tool. Computational experience is presented to demonstrate the relative effectiveness of the different proposed formulations and the associated optimization strategies for solving a set of realistic problem instances. The third application pertains to the problem of matching or assigning subassembly parts in assembly lines, where we seek to minimize the total deviation of the resulting final assemblies from a vector of nominal and mean quality characteristic values. We introduce three symmetry-defeating enhancements for an existing assignment-based model, and highlight the critical importance of using particular types of symmetry-defeating hierarchical constraints that preserve the model structure. We also develop an alternative set partitioning-based formulation in concert with a column generation approach that efficiently exploits the structure of the problem. A special complementary column generation feature is proposed, and we provide insights into its vital role for the proposed column generation strategy, as well as highlight its benefits in the broader context of set partitioning-based formulations that are characterized by columns having relatively dense non-zero values. In addition, we develop several heuristic procedures. Computational experience is presented to demonstrate the relative effectiveness of the different adopted strategies for solving a set of realistic problem instances. Finally, we analyze a doubles tennis scheduling problem in the context of a training tournament as prompted by a tennis club in Virginia, and develop two alternative 0-1 mixed-integer programming models, each with three different objective functions that attempt to balance the partnership and the opponentship pairings among the players. Our analysis and computational experience demonstrate the superiority of one of these models over the other, and reflect the importance of model structure in formulating discrete optimization problems. Furthermore, we design effective symmetry-defeating strategies that impose certain decision hierarchies within the models, which serve to significantly enhance algorithmic performance. In particular, our study provides the insight that the special structure of the mathematical program to which specific tailored symmetry-defeating constraints are appended can greatly influence their pruning effect. We also propose a novel nonpreemptive multi-objective programming strategy in concert with decision hierarchies, and highlight its effectiveness and conceptual value in enhancing problem solvability. Finally, four specialized heuristics are devised and are computationally evaluated along with the exact solution schemes using a set of realistic practical test problems. Aside from the development of specialized effective models and algorithms for particular interesting and challenging applications arising in different assembly, routing, and scheduling contexts, this dissertation makes several broader contributions that emerge from the foregoing studies, which are generally applicable to solving formidable combinatorial optimization problems. First, we have shown that it is of utmost importance to enforce symmetry-defeating constraints that preserve the structure of mathematical programs to which they are adjoined, so that their pruning effects are most efficiently coupled with the branch-and-bound strategies that are orchestrated within mathematical programming software packages. In addition, our work provides the insight that the concept of symmetry compatible formulations plays a crucial role in the effectiveness of implementing any particular symmetry-defeating constraints. In essence, if the root node LP solution of the original formulation does not conform relatively well with the proposed symmetry-defeating hierarchical constraints, then a significant branching effort might be required to identify a good solution that is compatible with the pattern induced by the selected symmetry-defeating constraints. Therefore, it is advisable to enforce decision hierarchies that conform as much as possible with the problem structure as well as with the initial LP relaxation. Second, we have introduced an alternative concept for defeating symmetry via augmented objective functions. This concept prompts the incorporation of objective perturbation terms that discriminate amongst subsets of originally undistinguishable solution structures and, in particular, leads to the development of a nonpreemptive multiobjective programming approach based on, and combined with, symmetry-defeating constraints. Interestingly, nonpreemptive multiobjective programming approaches that accommodate symmetry-defeating hierarchical objective terms induce a root node solution that is compatible with the imposed symmetry-defeating constraints, and hence affords an automated alternative to the aforementioned concept of symmetry compatible formulations. Third, we have proposed a new idea of complementary column generation in the context of column generation approaches that generally provide a versatile framework for analyzing industrial-related, integrated problems that involve the joint optimization of multiple operational decisions, such as assembly and routing, or partitioning and scheduling. In such situations, we have reinforced the insight that assignment-related problems that involve collections of objects (production batches, final assemblies, etc.) whose permutation yields equivalent symmetric solutions may be judiciously formulated as set partitioning models. The latter can then be effectively tackled via column generation approaches, thereby implicitly obviating the foregoing combinatorial symmetric reflections through the dynamic generation of attractive patterns or columns. The complementary column generation feature we have proposed and investigated in this dissertation proves to be particularly valuable for such set partitioning formulations that involve columns having relatively dense non-zero values. The incorporation of this feature guarantees that every LP iteration (involving the solution of a restricted master program and its associated subproblem) systematically produces a consistent set of columns that collectively qualify as a feasible solution to the problem under consideration. Upon solving the problem to optimality as a linear program, the resultant formulation encompasses multiple feasible solutions that generally include optimal or near-optimal solutions to the original integer-restricted set partitioning formulation, thereby yielding a useful representation for designing heuristic methods as well as exact branch-and-price algorithms. In addition, using duality theory and considering set partitioning problems where the number of patterns needed to collectively compose a feasible solution is bounded, we have derived a lower bound on the objective value that is updated at every LP phase iteration. By virtue of this sequence of lower bounds and the availability of upper bounds via the restricted master program at every LP phase iteration, the LP relaxation of the set partitioning problem is efficiently solved as using a pre-specified optimality tolerance. This yields enhanced algorithmic performance due to early termination strategies that successfully mitigate the tailing-off effect that is commonly witnessed for simplex-based column generation approaches. / Ph. D.

The purpose of this research is to develop a model and solution methodology for selecting and scheduling timely and cost-effective maintenance, rehabilitation, and reconstruction activities (M & R) for each pavement section in a highway network and allocating the funding levels through a finite multi-period horizon within the constraints imposed by budget availability in each period, frequency availability of activities, and specified minimum pavement quality requirements. M & R is defined as a chronological sequence of reconstruction, rehabilitation, and major/minor maintenance, including a "do nothing" activity. A procedure is developed for selecting an M & R activity for each pavement section in each period of a specified extended planning horizon. Each activity in the sequence consumes a known amount of capital and generates a known amount of effectiveness measured in pavement quality. The effectiveness of an activity is the expected value of the overall gains in pavement quality rating due to the activity performed on a highway network over an analysis period. It is assumed that the unused portion of the budget for one period can be carried over to subsequent periods. Dynamic Programming (DP) and Branch-and-Bound (B-and-B) approaches are combined to produce a hybrid algorithm for solving the problem under consideratioin. The algorithm is essentially a DP approach in the sense that the problem is divided into smaller subproblems corresponding to each single period problem. However, the idea of fathoming partial solutions that could not lead to an optimal solution is incorporated within the algorithm to reduce storage and computational requirements in the DP frame using the B-and-B approach. The imbedded-state approach is used to reduce a multi-dimensional DP to a one-dimensional DP. For bounding at each stage, the problem is relaxed in a Lagrangean fashion so that it separates into longest-path network model subproblems. The values of the Lagrangean multipliers are found by a subgradient optimization method, while the Ford-Bellman network algorithm is employed at each iteration of the subgradient optimization procedure to solve the longest-path network problem as well as to obtain an improved lower and upper bound. If the gap between lower and upper bound is sufficiently small, then we may choose to accept the best known solutions as being sufficiently close to optimal and terminate the algorithm rather than continue to the final stage.

In this study, a Lagrangean heuristic based on Lagrangean relaxation and subgradient optimization is proposed for the two-stage modular capacitated facility location problem. The objective is to minimize the cost of locating and operating plants and warehouses, plus the cost of transporting goods at both echelons to satisfy the demand of customers. The difference of our study from the two-stage capacitated facility location problem is the existence of multiple capacity levels as a candidate for each plant in the problem. Each capacity level has a minimum production capacity which has to be satisfied to open the relevant capacity level. Obviously, a single capacity level can be selected for an opened facility location. In the second echelon, the warehouses are capacitated and have unique fixed and variable costs for opening and operating. Multiple sourcing is allowed in both transportation echelons. Firstly, we develop a mixed integer linear programming model for the two-stage modular capacitated facility location problem. Then we develop a Lagrangean heuristic to solve the problem efficiently. Our Lagrangean heuristic consists of three main components: Lagrangean relaxation, subgradient optimization and a primal heuristic. Lagrangean relaxation is employed for obtaining the lower bound, subgradient optimization is used for updating the Lagrange multipliers at each iteration, and finally a three-stage primal heuristic is created for generating the upper bound solutions. At the first stage of the upper bound heuristic, global feasibility of the plants and warehouses is inspected and a greedy heuristic is executed, if there is a global infeasibility. At the next stage, an allocation heuristic is used to assign customers to warehouses and warehouses to plants sequentially. At the final stage of the upper bound heuristic, local feasibilities of the plants are investigated and infeasible capacity levels are adjusted if necessary. In order to show the efficiency of the developed heuristic, we have tested our heuristic on 280 problem instances generated randomly but systematically. The results of the experiments show that the developed heuristic is efficient and effective in terms of solution quality and computational effort especially for large instances.

The purpose of this research is to develop a model and solution methodology for selecting and scheduling timely and cost-effective maintenance, rehabilitation, and reconstruction activities (M & R) for each pavement section in a highway network and allocating the funding levels through a finite multi-period horizon within the constraints imposed by budget availability in each period, frequency availability of activities, and specified minimum pavement quality requirements. M & R is defined as a chronological sequence of reconstruction, rehabilitation, and major/minor maintenance, including a "do nothing" activity. A procedure is developed for selecting an M & R activity for each pavement section in each period of a specified extended planning horizon. Each activity in the sequence consumes a known amount of capital and generates a known amount of effectiveness measured in pavement quality. The effectiveness of an activity is the expected value of the overall gains in pavement quality rating due to the activity performed on a highway network over an analysis period. It is assumed that the unused portion of the budget for one period can be carried over to subsequent periods. Dynamic Programming (DP) and Branch-and-Bound (B-and-B) approaches are combined to produce a hybrid algorithm for solving the problem under consideratioin. The algorithm is essentially a DP approach in the sense that the problem is divided into smaller subproblems corresponding to each single period problem. However, the idea of fathoming partial solutions that could not lead to an optimal solution is incorporated within the algorithm to reduce storage and computational requirements in the DP frame using the B-and-B approach. The imbedded-state approach is used to reduce a multi-dimensional DP to a one-dimensional DP. For bounding at each stage, the problem is relaxed in a Lagrangean fashion so that it separates into longest-path network model subproblems. The values of the Lagrangean multipliers are found by a subgradient optimization method, while the Ford-Bellman network algorithm is employed at each iteration of the subgradient optimization procedure to solve the longest-path network problem as well as to obtain an improved lower and upper bound. If the gap between lower and upper bound is sufficiently small, then we may choose to accept the best known solutions as being sufficiently close to optimal and terminate the algorithm rather than continue to the final stage.

The air transportation market has been growing steadily for the past three decades since the airline deregulation in 1978. With competition also becoming more intense, airline companies have been trying to enhance their market shares and profit margins by composing favorable flight schedules and by efficiently allocating their resources of aircraft and crews so as to reduce operational costs. In practice, this is achieved based on demand forecasts and resource availabilities through a structured airline scheduling process that is comprised of four decision stages: schedule planning, fleet assignment, aircraft routing, and crew scheduling. The outputs of this process are flight schedules along with associated assignments of aircraft and crews that maximize the total expected profit. Traditionally, airlines deal with these four operational scheduling stages in a sequential manner. However, there exist obvious interdependencies among these stages so that restrictive solutions from preceding stages are likely to limit the scope of decisions for succeeding stages, thus leading to suboptimal results and even infeasibilities. To overcome this drawback, we first study the aircraft routing problem, and develop some novel modeling foundations based on which we construct and analyze an integrated model that incorporates fleet assignment, aircraft routing, and crew pairing within a single framework. Given a set of flights to be covered by a specific fleet type, the aircraft routing problem (ARP) determines a flight sequence for each individual aircraft in this fleet, while incorporating specific considerations of minimum turn-time and maintenance checks, as well as restrictions on the total accumulated flying time, the total number of takeoffs, and the total number of days between two consecutive maintenance operations. This stage is significant to airline companies as it directly assigns routes and maintenance breaks for each aircraft in service. Most approaches for solving this problem adopt set partitioning formulations that include exponentially many variables, thus requiring the design of specialized column generation or branch-and-price algorithms. In this dissertation, however, we present a novel compact polynomially sized representation for the ARP, which is then linearized and lifted using the Reformulation-Linearization Technique (RLT). The resulting formulation remains polynomial in size, and we show that it can be solved very efficiently by commercial software without complicated algorithmic implementations. Our numerical experiments using real data obtained from United Airlines demonstrate significant savings in computational effort; for example, for a daily network involving 344 flights, our approach required only about 10 CPU seconds for deriving an optimal solution. We next extend Model ARP to incorporate its preceding and succeeding decision stages, i.e., fleet assignment and crew pairing, within an integrated framework. We formulate a suitable representation for the integrated fleeting, routing, and crew pairing problem (FRC), which accommodates a set of fleet types in a compact manner similar to that used for constructing the aforementioned aircraft routing model, and we generate eligible crew pairings on-the-fly within a set partitioning framework. Furthermore, to better represent industrial practice, we incorporate itinerary-based passenger demands for different fare-classes. The large size of the resulting model obviates a direct solution using off-the-shelf software; hence, we design a solution approach based on Benders decomposition and column generation using several acceleration techniques along with a branch-and-price heuristic for effectively deriving a solution to this model. In order to demonstrate the efficacy of the proposed model and solution approach and to provide insights for the airline industry, we generated several test instances using historical data obtained from United Airlines. Computational results reveal that the massively-sized integrated model can be effectively solved in reasonable times ranging from several minutes to about ten hours, depending on the size and structure of the instance. Moreover, our benchmark results demonstrate an average of 2.73% improvement in total profit (which translates to about 43 million dollars per year) over a partially integrated approach that combines the fleeting and routing decisions, but solves the crew pairing problem sequentially. This improvement is observed to accrue due to the fact that the fully integrated model effectively explores alternative fleet assignment decisions that better utilize available resources and yield significantly lower crew costs. / Ph. D.