This thesis is a survey on the approximability of the asymmetric traveling salesmanproblem with triangle inequality (ATSP).In the ATSP we are given a set of cities and a function that gives the cost of travelingbetween any pair of cities. The cost function must satisfy the triangle inequality, i.e.the cost of traveling from city A to city B cannot be larger than the cost of travelingfrom A to some other city C and then to B. However, we allow the cost function tobe asymmetric, i.e. the cost of traveling from city A to city B may not equal the costof traveling from B to A. The problem is then to find the cheapest tour that visit eachcity exactly once. This problem is NP-hard, and thus we are mainly interested in approximationalgorithms. We study the repeated cycle cover heuristic by Frieze et al. We alsostudy the Held-Karp heuristic, including the recent result by Asadpour et al. that givesa new upper bound on the integrality gap. Finally we present the result ofPapadimitriou and Vempala which shows that it is NP-hard to approximate the ATSP with a ratio better than 117/116.

This dissertation is devoted to the development of new flow-based formulations for the asymmetric traveling salesman problem (ATSP) and to the demonstration of their applicability in effectively solving some scheduling problems. The ATSP is commonly encountered in the areas of manufacturing planning and scheduling, and transportation logistics. The integration of decisions pertaining to production and shipping, in the supply chain context, has given rise to an additional and practical relevance to this problem especially in situations involving sequence-dependent setups and routing of vehicles. Our objective is to develop new ATSP formulations so that algorithms can be built by taking advantage of their relaxations (of integer variables, thereby, resulting in linear programs) to effectively solve large-size problems. In view of our objective, it is essential to have a formulation that is amenable to the development of an effective solution procedure for the underlying problem. One characteristic of a formulation that is helpful in this regard is its tightness. The tightness of a formulation usually refers to the quality of its approximation to the convex hull of integer feasible solutions. Another characteristic is its compactness. The compactness of a formulation is measured by the number of variables and constraints that are used to formulate a given problem. Our formulations for the ATSP and the scheduling problems that we address are both tight and compact. We present a new class of polynomial length formulations for the asymmetric traveling salesman problem (ATSP) by lifting an ordered path-based model using logical restrictions in concert with the Reformulation-Linearization Technique (RLT). We show that a relaxed version of this formulation is equivalent to a flow-based ATSP model, which, in turn, is tighter than the formulation based on the exponential number of Dantzig-Fulkerson-Johnson (DFJ) subtour elimination constraints. The proposed lifting idea is applied to derive a variety of new formulations for the ATSP, and a detailed analysis of these formulations is carried out to show that some of these formulations are the tightest among those presented in the literature. Computational results are presented to exhibit the relative tightness of our formulations and the efficacy of the proposed lifting process.> While the computational results demonstrate the efficacy of employing the proposed theoretical RLT and logical lifting ideas, yet it remains of practical interest to take due advantage of the tightest formulations. The key requirement to accomplish this is the ability to solve the underlying LP relaxations more effectively. One approach, to that end, is to solve these LP relaxations to (near-) optimality by using deflected subgradient methods on Lagrangian dual formulations. We solve the LP relaxation of our tightest formulation, ATSP6, to (near-) optimality by using a deflected subgradient algorithm with average direction strategy (SA_ADS) (see Sherali and Ulular [69]). We also use two nondifferentiable optimization (NDO) methods, namely, the variable target value method (VTVM) presented by Sherali et al. [66] and the trust region target value method (TRTV) presented by Lim and Sherali [46], on the Lagrangian dual formulation of ATSP6. The preliminary results show that the near-optimal values obtained by the VTVM on solving the problem in the canonical format are the closest to the target optimal values. Another approach that we use is to derive a set of strong valid inequalities based on our tighter formulations through a suitable surrogation process for inclusion within the more compact manageable formulations. Our computational results show that, when the dual optimal solution is available, the associated strong valid inequalities generated from our procedure can successfully lift the LP relaxation of a less tight formulation, such as ATSP2R¯, to be as tight as the tightest formulation, such as ATSP6. We extend our new formulations to include precedence constraints in order to enforce a partial order on the sequence of cities to be visited in a tour. The presence of precedence constraints within the ATSP framework is encountered quite often in practice. Examples include: disassembly optimization (see Sarin et al. [62]), and scheduling of wafers/ ICs on automated testing equipments in a semiconductor manufacturing facility (see Chen and Hsia [17]); among others. Our flow-based ATSP formulation can very conveniently capture these precedence constraints. We also present computational results to depict the tightness of our precedence-constrained asymmetric traveling salesman problem (PCATSP) formulations. We, then, apply our formulations to the hot strip rolling scheduling problem, which involves the processing of hot steel slabs, in a pre-specified precedence order, on one or more rollers. The single-roller hot strip rolling scheduling problem can be directly formulated as a PCATSP. We also consider the multiple-roller hot strip rolling scheduling problem. This gives rise to the multiple-asymmetric traveling salesman problem (mATSP). Not many formulations have been presented in the literature for the mATSP, and there are none for the mATSP formulations involving a precedence order among the cities to be visited by the salesmen, which is the case for the multiple-roller hot strip rolling scheduling problem. To begin with, we develop new formulations for the mATSP and show the validity of our formulations, and present computational results to depict their tightness. Then, we extend these mATSP formulations to include a pre-specified, special type of precedence order in which to process the slabs, and designate the resulting formulations as the restricted precedence-constrained multiple-asymmetric traveling salesman problem (rPCmATSP) formulations. We directly formulate the multiple-roller hot strip rolling scheduling problem as a rPCmATSP. Furthermore, we consider the hot strip rolling scheduling problem with slab selection in which not all slabs need to be processed. We model the single-roller hot strip rolling scheduling problem with slab selection as a multiple-asymmetric traveling salesman problem with exactly two traveling salesmen. Similarly, the multiple-roller hot strip rolling scheduling problem with slab selection is modeled as a multiple-asymmetric traveling salesman problem with (m+1) traveling salesmen. A series of computational experiments are conducted to exhibit the effectiveness of our formulations for the solution of hot strip rolling scheduling problems. Furthermore, we develop two mixed-integer programming algorithms to solve our formulations. These are based on Benders΄ decomposition [13] and are designated Benders΄ decomposition and Modified Benders΄ methods. In concert with a special type of precedence order presented in the hot strip rolling scheduling problems, we further introduce an adjustable density ratio of the associated precedence network and we use randomly generated test problems to study the effect of various density ratios in solving these scheduling problems. Our experimentation shows the efficacy of our methods over CPLEX. Finally, we present a compact formulation for the job shop scheduling problem, designated as JSCD (job shop conjunctive-disjunctive) formulation, which is an extension of our ATSP formulations. We use two test problems given in Muth and Thompson [53] to demonstrate the optimal schedule and the lower bound values obtained by solving the LP relaxations of our formulations. However, we observe that the lower bound values obtained by solving the LP relaxations of all variations of our JSCD formulation equal to the maximum total processing time among the jobs in the problem. / Ph. D.

This dissertation focuses on supply chain design and logistics problems with emphasis on biomass logistics and routing problems. In biomass logistics, we have studied problems arising in a switchgrass-based bio-ethanol supply chain encountered in the Southeast, and a corn stover harvest scheduling problem faced in the Midwest Unites States, both pertaining to the production of cellulosic ethanol. The main contributions of our work have been in introducing new problems to the literature that lie at the interface of the lot-sizing and routing problems, and in developing effective exact algorithms for their solution. In the routing area, we have addressed extensions of the well-known traveling salesman and vehicle routing problems. We have proposed new formulations and have developed exact algorithms for the single and multiple asymmetric traveling salesmen problems (ATSP and mATP), the high-multiplicity asymmetric traveling salesman problem (HMATSP) and its extensions, and the fixed-destination multi-depot traveling salesman problem with load balancing (FD-MTSPB). Furthermore, we have introduced a new strategy to reduce routing cost in the classical vehicle routing problem (VRP). / Ph. D.

The rapid technological advancement in the past century resulted in a decreased life cycle of a large number of products and, consequently, increased the rate of technological obsolescence. The disposal of obsolete products has resulted in rapid landfilling and now poses a major environmental threat. The governments in many countries around the world have started imposing regulations to curb uncontrolled product disposal. The consumers, too, are now aware of adverse effects of product disposal on environment and increasingly favor environmentally benign products. In the wake of imminent stringent government regulations and the consumer awareness about ecosystem-friendly products, the manufacturers need to think about the alternatives to product disposal. One way to deal with this problem is to disassemble an obsolete product and utilize some of its components/subassemblies in the manufacturing of new products. This seems to be a promising solution because products now-a-days are made in accordance with the highest quality standards and, although an obsolete product may not be in the required functional state as a whole, it is possible that several of its components or subassemblies are still in near perfect condition. However, product disassembly is a complex task requiring human labor as well as automated processes and, consequently, a huge amount of monetary investment. This research addresses a disassembly optimization problem, which aims at minimizing the costs associated with the disassembly process (namely, the costs of breaking the joints and the sequence dependent set-up cost associated with disassembly operations), while maximizing the benefits resulting from recovery of components/subassemblies from a product. We provide a mathematical abstraction of the disassembly optimization problem in the form of integer-programming models. One of our formulations includes a new way of modeling the subtour elimination constraints (SECs), which are usually encountered in the well-known traveling salesman problems. Based on these SECs, a new valid formulation for asymmetric traveling salesman problem (ATSP) was developed. The ATSP formulation was further extended to obtain a valid formulation for the precedence constrained ATSP. A detailed experimentation was conducted to compare the performance of the proposed formulations with that of other well-known formulations discussed in the literature. Our results indicate that in comparison to other well-known formulations, the proposed formulations are quite promising in terms of the LP relaxation bounds obtained and the number of branch and bound nodes explored to reach an optimal integer solution. These new formulations along with the results of experimentation are presented in Appendix A. To solve the disassembly optimization problem, a three-phase iterative solution procedure was developed that can determine optimal or near optimal disassembly plans for complex assemblies. The first phase helps in obtaining an upper bound on our maximization problem through an application of a Lagrangian relaxation scheme. The second phase helps to further improve this bound through addition of a few valid inequalities in our models. In the third phase, we fix some of our decision variables based on the solutions obtained in the iterations of phases 1 and 2 and then implement a branch and bound scheme to obtain the final solution. We test our procedure on several randomly generated data sets and identify the factors that render a problem to be computationally difficult. Also, we establish the practical usefulness of our approach through case studies on the disassembly of a computer processor and a laser printer. / Master of Science

Recently, there has been a surge in the research and application of "Green energy" in the United States. This has been driven by the following three objectives: (1) to reduce the nation's reliance on foreign oil, (2) to mitigate emission of greenhouse gas, and (3) to create an economic stimulus within the United States. Switchgrass is the biomass of choice for the Southeastern United States. In this dissertation, we address a feedstock logistics problem associated with the delivery of switchgrass for conversion into biofuel. In order to satisfy the continual demand of biomass at a bioenergy plant, production fields within a 48-km radius of its location are assumed to be attracted into production. The bioenergy plant is expected to receive as many as 50-400 loads of biomass per day. As a result, an industrialized transportation system must be introduced as early as possible in order to remove bottlenecks and reduce the total system cost. Additionally, we assume locating multiple bioenergy plants within a given region for the production of biofuel. We develop mixed integer programming formulations for the feedstock logistics problem that we address and for some related problems, and we solve them either through the use of decomposition-based methods or directly through the use of CPLEX 12.1.0. The feedstock logistics problem that we address spans the entire system-from the growing of switchgrass to the transporting of bio-crude oil, a high energy density intermediate product, to a refinery for conversion into a final product. To facilitate understanding, we present the reader with a case study that includes a preliminary cost analysis of a real-life-based instance in order to provide the reader appropriate insights of the logistics system before applying optimization techniques for its solution. First, we consider the benefits of active versus passive ownership of the production fields. This is followed by a discussion on the selection of baler type, and then, a discussion of contracts between various business entities. The advantages of storing biomass at a satellite storage location (SSL) and interactions between the operations performed at the production field with those performed at the storage locations are then established. We also provide a detailed description of the operations performed at a SSL. Three potential equipment options are presented for transporting biomass from the SSLs to a utilization point, defined in this study as a Bio-crude Plant (BcP). The details of the entire logistics chain are presented in order to highlight the need for making decisions in view of the entire chain rather than basing them on its segments. We model the feedstock logistics problem as a combination of a 2-level facility location-allocation problem and a multiple traveling salesmen problem (mATSP). The 2-level facility location-allocation problem pertains to the allocation of production fields to SSLs and SSLs to one of the multiple bioenergy plants. The mATSP arises because of the need for scheduling unloading operations at the SSLs. To this end, we provide a detailed study of 13 formulations of the mATSP and their reformulations as ATSPs. First, we assume that the SSLs are always full, regardless of when they are scheduled to be unloaded. We, then, relax this assumption by providing precedence constraints on the availability of the SSLs. This precedence is defined in two different ways and, is then, effectively modeled utilizing all the formulations for the mATSP and ATSP. Given the location of a BcP for the conversion of biomass to bio-crude oil, we develop a feedstock logistics system that relies on the use of SSLs for temporary storage and loading of round bales. Three equipment systems are considered for handling biomass at the SSLs, and they are either placed permanently or are mobile, and thereby, travel from one SSL to another. We use a mathematical programming-based approach to determine SSLs and equipment routes in order to minimize the total cost incurred. The mathematical program is applied to a real-life production region in South-central Virginia (Gretna, VA), and it clearly reveals the benefits of using SSLs as a part of the logistics system. Finally, we provide a sensitivity analysis on the input parameters that we used. This analysis highlights the key cost factors in the model, and it emphasizes areas where biggest gains can be achieved for further cost reduction. For a more general scenario, where multiple BcPs have to be located, we use a nested Benders' decomposition-based method. First, we prove the validity of using this method. We, then, employ this method for the solution of a potential real-life instance. Moreover, we successfully solve problems that are more than an order of magnitude larger than those solved directly by CPLEX 12.1.0. Finally, we develop a Benders' decomposition-based method for the solution of a problem that gives rise to a binary sub-problem. The difficulty arises because of the sub-problem being an integer program for which the dual solution is not readily available. Our approach consists of first solving the integer sub-problem, and then, generating the convex hull at the optimal integer point. We illustrate this approach for an instance for which such a convex hull is readily available, but otherwise, it is too expensive to generate for the entire problem. This special instance is the solution of the mATSP (using Benders' decomposition) for which each of the sub-problems is an ATSP. The convex hull for the ATSP is given by the Dantzig, Fulkerson, and Johnson constraints. These constraints at a given integer solution point are only polynomial in number. With the inclusion of these constraints, a linear programming solution and its corresponding dual solution can now be obtained at the optimal integer points. We have proven the validity of using this method. However, the success of our algorithm is limited because of a large number of integer problems that must be solved at every iteration. While the algorithm is theoretically promising, the advantages of the decomposition do not seem to outweigh the additional cost resulting from solving a larger number of decomposed problems. / Ph. D.