Dept. of Industrial Engineering - Ph.D. / Sc.D.

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https://hdl.handle.net/11693/13886

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Open Access
Markov decision process formulations for management of pumped hydro energy storage systems
(Bilkent University, 2023-06) Toufani, Parinaz
Renewable energy sources have received much attention to mitigate the high dependence on fossil fuels and the resulting environmental impacts. Since the variability and intermittency of such renewable sources lower the reliability and security of energy systems, they should often be accompanied by efficient and flexible storage units. This dissertation focuses on pumped hydro energy storage (PHES) facilities, which are one of the most commonly used large-scale storage technologies. We study the energy generation and storage problem for PHES facilities with two connected reservoirs, where water is pumped from the lower reservoir to the upper reservoir to store energy during low-demand/low-electricity price periods, and released back to the lower reservoir to generate energy during high-demand/high-electricity price periods. The first part of this dissertation investigates the potential benefits of transforming conventional cascading hydropower stations into PHES facilities by replacing turbines with reversible ones. The second part compares the short-term cash flows obtained from different PHES configurations (cascading vs. non-cascading facilities, upstream vs. downstream inflows, and closed-loop facilities). We formulate both problems as Markov decision processes under uncertainty in the streamflow rate and electricity price. We include the streamflow rate and electricity price as exogenous state variables in our formulation. We analytically derive bounds on the profit improvement obtained from PHES transformation in the first part and bounds on the revenue differences obtained from different configurations in the second part. In the last part, we establish several structural properties of the optimal profit function for general two-reservoir PHES systems. We show the optimality of a state-dependent threshold policy for non-cascading PHES facilities when the electricity price is always positive. Leveraging our structural results, we construct a heuristic solution method for more general settings when the electricity price can also be negative. In this dissertation, we also conduct comprehensive numerical experiments with data-calibrated time series models to provide insights into the optimal operation of PHES facilities, considering distinct seasons with different streamflow rates, different negative electricity price occurrence frequencies, and different system parameters.
Open Access
Energy operations management for renewable power producers in electricity markets
(Bilkent University, 2023-05) Karakoyun, Ece Çiğdem
Renewable energy generation has grown dramatically around the world in recent years, and policies targeted at reducing greenhouse gas emissions that cause global warming are expected to ensure a consistent expansion of renewable power generation in the electricity sector. With the increasing contribution of renewable sources to the overall energy supply, renewable power producers participate in electricity markets where they are imposed to make advance commitment decisions for energy delivery and purchase. Making advance commitments, however, is a complex task due to the inherent intermittency of renewable sources, increasingly volatile electricity prices, and penalties incurred for possible energy imbalances in electricity markets. Integrating renewable sources with energy storage units is among the most effective methods to address this challenging task. Motivated by the recent trends of paired renewable energy generators and storage units, we study the energy commitment, generation and storage problem of a wind power producer who owns a battery and participates in a spot market operating with hourly commitments and settlements. In each time period, the producer decides how much energy to commit to selling to or purchasing from the market in the next time period, how much energy to generate in the wind power plant, and how much energy to charge into or discharge from the battery. The existence of the battery not only helps smooth out imbalances caused by the fluctuating wind output but also enables the producer to respond to price changes in the market. We formulate the wind power producer's problem as a Markov decision process by taking into account the uncertainties in wind speed and electricity price. In the first part of this dissertation, we consider two different problem settings: In the first setting, the producer may choose to deviate from her commitments based on the latest available information, using the battery to support such deviations. In the second setting, the producer is required to fulfill her commitments, using the battery as a back-up source. We numerically examine the effects of system components, imbalance pricing parameters, and negative prices on the producer's profits, curtailment decisions, and imbalance tendencies in each problem setting. We provide managerial insights to renewable power producers in their assessment of energy storage adoption decisions and to power system operators in their understanding of the producers' behavior in the market with their storage capabilities. In the second part of this dissertation, we establish several multi-dimensional structural properties of the optimal profit function such as supermodularity and joint concavity. This enables us to prove the optimality of a state-dependent threshold policy for the storage and commitment decisions under the assumptions of a perfectly efficient system and positive electricity prices. Leveraging this policy structure, we construct two heuristic solution methods for solving the more general problem in which the battery and transmission line can be imperfectly efficient and the price can also be negative. Numerical experiments with data-calibrated instances have revealed the high efficiency and scalability of our solution procedure. In the third part of this dissertation, we characterize the optimal policy structure by taking into account the battery and transmission line efficiency losses and showing the joint concavity of the optimal profit function. In the last part of this dissertation, we consider an alternative problem setting that allows for real-time trading without making any advance commitment. We analytically compare the total cash flows of this setting to those of our original problem setting. We conclude with a numerical investigation of the effect of advance commitment decisions on the producer's energy storage and generation decisions.
Open Access
Three essays in the interface of optimization with mechanism design, nonexclusive competition, and prophet inequalities
(Bilkent University, 2022-09) Bayrak, Halil İbrahim
Mechanism Design. We consider the mechanism design problem of a principal allocating a single good to one of several agents without monetary transfers. Each agent desires the good and uses it to create value for the principal. We designate this value as the agent's private type. Even though the principal does not know the agents' types, she can verify them at a cost. The allocation of the good thus depends on the agents' self-declared types and the results of any verification performed, and the principal's payoff matches her allocation value minus the verification costs. It is known that when the agents' types are independent, a favored-agent mechanism maximizes her expected payoff. However, this result relies on the unrealistic assumptions that the agents' types follow known independent probability distributions. We assume that the agents' types are governed by an ambiguous joint probability distribution belonging to a commonly known ambiguity set and that the principal maximizes her worst-case expected payoff. We consider three types of ambiguity sets: (i) support-only ambiguity sets, which contain all distributions supported on a rectangle, (ii) Markov ambiguity sets, characterized through first-order moment bounds, and (iii) Markov with independence ambiguity sets. For each of these ambiguity sets, we show that a favored-agent mechanism, which we characterize implicitly, is optimal and also Pareto-robustly optimal. The optimal choices of the favored agent and the threshold do not depend on the verification costs in all three cases. Nonexclusive Competition. A freelancer with a time constraint faces offers from multiple identical parties. The quality of the service provided by the freelancer can be high or low and is only known by the freelancer. The freelancer's time cost is strictly increasing and convex. We show that a pure-strategy equilibrium exists if and only if the preferences of the high-type freelancer satisfy one of the following two distinct conditions: (i) the high-type freelancer does {not} prefer providing his services for a price equal to the expected quality at the no-trade point; (ii) the high-type freelancer prefers providing his services for a price equal to the expected quality at any feasible trade point. If (i) holds, then in equilibrium, the high-type freelancer does not trade, whereas the low-type may not trade, trade efficiently, or exhaust all of his capacity. Moreover, the buyers make zero profit from each of their traded contracts. If (ii) holds, then both types of the freelancer trade at the capacity in equilibrium. Furthermore, the buyers make zero expected profit with cross-subsidization. In any equilibrium, the aggregate equilibrium trades are unique. Prophet Inequalities. Prophet inequalities bound the expected reward obtained in a class of stopping problems by the optimal reward of the corresponding offline problem. We show how to obtain prophet inequalities for a large class of stopping problems associated with selecting a point in a polyhedron. Our approach utilizes linear programming tools and is based on a reduced form representation of the stopping problem. We illustrate the usefulness of our approach by re-establishing three different prophet inequality results from the literature. (i) For polymatroids with nonnegative coefficients in their unique Minkowski sum of simplices, we prove the 1/2-prophet inequality. (ii) We prove the 1/n-prophet inequality when there are n stages, the stages have dependently distributed rewards, and we are restricted to choosing a strategy from an arbitrary polyhedron. (iii) When the feasible set of strategies can be described via K different constraints, we obtain the 1/(K+1)-prophet inequality.
Open Access
Norm minimization-based convex vector optimization algorithms
(Bilkent University, 2022-08) Umer, Muhammad
This thesis is concerned with convex vector optimization problems (CVOP). We propose an outer approximation algorithm (Algorithm 1) for solving CVOPs. In each iteration, the algorithm solves a norm-minimizing scalarization for a reference point in the objective space. The idea is inspired by some Benson-type algorithms in the literature that are based on Pascoletti-Seraﬁni scalarization. Since this scalarization needs a direction parameter, the eﬃciency of these algorithms depend on the selection of the direction parameter. In contrast, our algorithm is free of direction biasedness since it solves a scalarization that is based on minimizing a norm. However, the structure of such algorithms, including ours, has some built-in limitation which makes it diﬃcult to perform convergence analysis. To overcome this, we modify the algorithm by introducing a suitable compact subset of the upper image. After the modiﬁcation, we have Algorithm 2 in which norm-minimizing scalarizations are solved for points in the compact set. To the best of our knowledge, Algorithm 2 is the ﬁrst algorithm for CVOPs, which is proven to be ﬁnite. Finally, we propose a third algorithm for the purposes of con-vergence analysis (Algorithm 3), where a modiﬁed norm-minimizing scalarization is solved in each iteration. This scalarization includes an additional constraint which ensures that the algorithm deals with only a compact subset of the upper image from the beginning. Besides having the ﬁniteness result, Algorithm 3 is the ﬁrst CVOP algorithm with an estimate of a convergence rate. The experimental results, obtained using some benchmark test problems, show comparable performance of our algorithms with respect to an existing CVOP algorithm based on Pascoletti-Seraﬁni scalarization.
Open Access
Risk-averse multi-armed bandit problem
(Bilkent University, 2021-08) Malekipirbazari, Milad
In classical multi-armed bandit problem, the aim is to ﬁnd a policy maximizing the expected total reward, implicitly assuming that the decision maker is risk-neutral. On the other hand, the decision makers are risk-averse in some real life applications. In this study, we design a new setting for the classical multi-armed bandit problem (MAB) based on the concept of dynamic risk measures, where the aim is to ﬁnd a policy with the best risk adjusted total discounted outcome. We provide theoretical analysis of MAB with respect to this novel setting, and propose two diﬀerent priority-index heuristics giving risk-averse allocation indices with structures similar to Gittins index. The ﬁrst proposed heuristic is based on Lagrangian duality and the indices are expressed as the Lagrangian multiplier corresponding to the activation constraint. In the second part, we present a theoretical analysis based on Whittle’s retirement problem and propose a gener-alized version of restart-in-state formulation of the Gittins index to compute the proposed risk-averse allocation indices. Finally, as a practical application of the proposed methods, we focus on optimal design of clinical trials and we apply our risk-averse MAB approach to perform risk-averse treatment allocation based on a Bayesian Bernoulli model. We evaluate the performance of our approach against other allocation rules, including ﬁxed randomization.
Open Access
Deterministic and stochastic team formation problems
(Bilkent University, 2021-01) Berktaş, Nihal
In various organizations, physical or virtual teams are formed to perform jobs that require diﬀerent skills. The success of a team depends on the technical capabilities of the team members as well as the quality of communication among the team members. We study diﬀerent variants of the team formation problem where the goal is to build the best team with respect to given criteria. First, we study a deterministic team formation problem which aims to construct a capable team that can communicate and collaborate eﬀectively. To measure the quality of communication, we assume the candidates constitute a social network and we deﬁne a cost of communication using the proximity of people in the social network. We minimize the sum of all pairwise communication costs, and we impose an upper bound on the largest communication cost. This problem is formulated as a constrained quadratic set covering problem. Our experiments show that a general-purpose solver is capable of solving small and medium-sized instances to optimality. We propose a branch-and-bound algorithm to solve larger sizes: we reformulate the problem and relax it in such a way that it decomposes into a series of linear set covering problems, and we impose the relaxed constraints through branching. Our computational experiments show that the algorithm is capable of solving large-sized instances, which are intractable for the solver. Second, we consider a two-stage stochastic team formation problem where the objective is to minimize the expected communication cost of the team. We as-sume that for a subset of pairs the communication costs are uncertain but they have a known discrete distribution. The ﬁrst stage is a trial stage where the decision-maker chooses a limited number of pairs from this subset. The actual cost values of the chosen pairs are realized before the second stage. Hence, the uncertainty in this problem is decision-dependent, also called endogenous, be-cause the ﬁrst stage decisions determine for which parameters the uncertainty will resolve. For this problem, we give two formulations, the ﬁrst one contains a set of non-anticipativity constraints similar to the models in the related lit-erature. In the second, we are able to eliminate these constraints by changing the objective function into a quadratic one, which is linearized by a set of extra binary variables. We show that the size of instances we can solve with these for-mulations using a commercial solver is limited. Therefore, we develop a Benders’ decomposition-based branch-and-cut algorithm that exploits decision-dependent nature to partition scenarios and use tight linear relaxations to obtain strong cuts. We show the eﬃciency of the algorithm presenting results of experiments conducted with randomly generated instances. Finally, we study a multi-stage team formation problem where the objective is to minimize the monetary cost including hiring and outsourcing costs. In this problem, stages correspond to projects which are carried out consecutively. Each project consists of several tasks each of which requires a human resource. We assume that due to incomplete information there is uncertainty in people’s performances and consequently the time a person needs to complete a task is random for some person-task pairs. When a person is assigned to a task, we learn how long it takes for this person to ﬁnish the task. Hence, the uncertainty is again decision-dependent. If the duration of a task exceeds the allowable time for a project then the manager must hire an external resource to speed up the process. We present an integer programming formulation to this problem and explain that the size of the formulation strongly depends on the number of random parameters and scenarios. While this deterministic equivalent formulation can be solved with a commercial solver for small-sized instances, it easily becomes intractable when the number of random parameters increases by one. For such cases where exact methods are not promising, we investigate heuristic methods to obtain tight bounds and near-optimal solutions. In the related literature, diﬀerent Lagrangian decomposition methods are developed for such stochastic problems. In this study, we show that the convergence of existing methods is very slow, and we propose an alternative method where a relaxation of the formulation is solved by a decomposition-based branch-and-bound algorithm.
Open Access
Radio communications interdiction problem
(Bilkent University, 2020-01) Tanergüçlü, Türker
Tactical communications have always played a pivotal role in maintaining eﬀective command and control of troops operating in hostile, extremely fragile and dynamic battleﬁeld environments. Radio communications, in particular, have served as the backbone of the tactical communications over the years and have proven to be very useful in meeting the information exchange needs of widely dispersed and highly mobile military units, especially in the rugged area. Considering the complexity of today’s modern warfare, and in particular the emerging threats from the latest electronic warfare technologies, the need for optimally designed radio communications networks is more critical than ever. Optimized communication network planning can minimize network vulnerabilities to modern threats and provide additional assurance of continued availability and reliability of tactical communications. To do so, we present the Radio Communications Interdiction Problem (RCIP) to identify the optimal locations of transmitters on the battleﬁeld that will lead to a robust radio communications network by anticipating the degrading eﬀects of intentional radio jamming attacks used by an adversary during electronic warfare. We formulate RCIP as a binary bilevel (max–min) programming problem, present the equivalent single level formulation, and propose an exact solution method using a decomposition scheme. We enhance the performance of the algorithm by utilizing dominance relations, preprocessing, and initial starting heuristics. To reﬂect a more realistic jamming representation, we introduce the probabilistic version of RCIP (P-RCIP) where a jamming probability is associated at each receiver site as a function of the prevalent jamming to signal ratios leading to an expected coverage of receivers as an objective function. We approximate the nonlinearity in the jamming probability function using a piecewise linear convex function and solve this version by adapting the decomposition algorithm constructed for RCIP. Our extensive computational results on realistic scenarios that reﬂect diﬀerent phases of a military conﬂict show the eﬃcacy of the proposed solution methods. We provide valuable tactical insights by analyzing optimal solutions on these scenarios under varying parameters. Finally, we investigate the incorporation of limited artillery assets into communications planning by formulizing RCIP with Artillery (RCIP-A) as a trilevel optimization problem and propose a nested decomposition method as an exact solution methodology. Additionally, we present computational results and tactical insights obtained from the solution of RCIP-A on predeﬁned scenarios.
Open Access
Essays on bilateral trade with discrete types
(Bilkent University, 2019-10) Mohammadinezhad, Kamyar Kargar
Bilateral trade is probably the most common market interaction problem and can be considered as the simplest form of two sided markets where a seller and a buyer bargain over an indivisible object subject to incomplete information on the reservation values of participants. We treat this problem as a combinatorial optimization problem and re-establish some results of economic theory that are well-known under continuous valuations assumptions for the case of discrete valuations using linear programming techniques. First, we propose mathematical formulation for the problem under dominant strategy incentive compatibility (DIC) and ex-post individual rationality (EIR) properties. Then we derive necessary and sufficient conditions under which ex-post efficiency can be obtained together with DIC and EIR. We also define a new property called Allocation Maximality and prove that the Posted Price mechanism is the only mechanism that satisfies DIC, EIR and allocation maximality. In the final part we consider ambiguity in the problem framework originating from different sets of priors for agents types and derive robust counterparts. Next, we study the bilateral trade problem with an intermediary who wants to maximize her expected gains. Using network programming we transform the initial linear program into one from which the structure of mechanism is transparent. We then relax the risk-neutrality assumption of the intermediary and consider the problem from the perspective of risk-averse intermediary. The effects of risk-averse approach are presented using computational experiments. Finally, we broaden the scope of the problem and discuss the case in which the seller is also a producer at the same time and consider benefit and cost functions for the respective parties. Starting by a non-convex optimization problem, we obtain an equivalent convex optimization problem from which the problem is solved easily. We also reconsider the same problem under dominant strategy incentive compatibility and ex-post individual rationality constraints to preserve the practicality of all obtained solutions.
Open Access
Optimizing airline operations under uncertainty
(Bilkent University, 2019-06) Aydıner, Özge Şafak
Fluctuations in passenger demand, airport congestion, and high fuel costs are the main threats to airlines' profit, thereby need to be carefully addressed in airline scheduling problems. This study takes an advantage of aircraft cruise speed control in several scheduling problems to keep the cost of fuel manageable. We first generate a flight schedule by integrating strategic departure time decisions, tactical eeting and routing decisions and more operational flight timing decisions under stochastic demand and non-cruise times. Our model differs from the existing studies by including aircraft cruise speed decisions to compensate for increase in non-cruise time variations due to the airport congestion. To e ciently solve the problem, we provide a scenario group-wise decomposition algorithm. Then, we consider a new problem which aims to accommodate new flights into an existing flight schedule in a short time. We suggest some operational changes such as controlling the aircraft cruise speed, re-timing flight departures and swapping aircraft to open up time for new flights. However, nonlinear fuel cost function, and binary assignment and swapping decisions significantly increase the computational burden of solving scheduling problems. In this thesis, we propose strong mixed integer conic quadratic formulations. Finally, we extend the problem by including a strategic decision to lease an aircraft for introducing new flights. More importantly, we consider the effects of departure time decisions on the probability distribution of random demand. We propose a bounding method based on scenario group-wise decomposition for stochastic programs with decision dependent probabilities.
Open Access
Network design problems and value of control mechanisms in power systems
(Bilkent University, 2019-05) Peker Sarhan, Meltem
Power systems planning and operations is one of the most challenging problems in energy field due to its complex, large-scale and nonlinear nature. Operating power systems with uncertainties and disturbances such as failure of system components increases complexity and causes difficulties in sustaining a supplydemand balance in power systems without jeopardizing grid reliability. To handle with the uncertainties and operate power systems without endangering grid reliability, utilities and system operators implement various control mechanisms such as energy storage, transmission switching, renewable energy curtailment and demand-side management. In this thesis, we first propose a multi-period mathematical programming model to discuss the effect of transmission switching decisions on power systems expansion planning problems. We then explore the value of control mechanisms for integrating renewable energy sources into power systems. We develop a two-stage stochastic programming model that cooptimizes investment decisions and transmission switching operations. Later, we analyze the effect of demand-side management programs on peak load management. We provide a conceptual framework for quantifying the incentives paid to the consumers to reshape their load profiles while taking hourly electrical power generation costs as reference points. Finally, we study reliability aspect of the power system planning and consider unexpected failures of components. We provide a two-stage stochastic programming model and discuss value of transmission switching on grid reliability.
Open Access
Green location and routing problems with conventional vehicles and drones
(Bilkent University, 2019-05) Dükkancı, Okan
Green Location and Routing Problems extend the network design problems that consider location and routing decisions by explicitly accounting environmental impacts such as CO2 emissions caused by fuel or energy consumption of delivery vehicles. These environmental impacts estimated by fuel or energy consumption models are a ected by several factors including payload and speed of delivery vehicles. We present four new green location and routing problems where we consider these factors while calculating the environmental impacts. We rst introduce the Green Location-Routing Problem, in which vehicle payload and speed decisions are incorporated to a location-routing problem and the fuel consumption of trucks is estimated and minimized. Second, we study the Green Hub Location Problem, where we minimize the fuel consumption by optimizing truck payload and speed decisions on a hub network. Third, we present a freight transportation problem called the Drone Delivery Problem, where the integration of trucks and drones is used to make deliveries. Drone speed is considered as a decision of the problem in order to minimize energy consumption of drones while not exceeding the drone range. Fourth, we study an extension of the Drone Delivery Problem, called the Stochastic Drone Delivery Problem, where uncertainty of wind speed and its impact on the drone speed are considered.
Open Access
Risk-averse multi-stage mixed-integer stochastic programming problems
(Bilkent University, 2019-01) Mahmutoğulları, Ali İrfan
Risk-averse multi-stage mixed-integer stochastic programming problems form a class of extremely challenging problems since the problem size grows exponentially with the number of stages, they are non-convex due to integrality restrictions, and their objective functions are nonlinear in general. In this thesis, we first focus on such problems with an objective of dynamic mean conditional value-at-risk. We propose a scenario tree decomposition approach to obtain lower and upper bounds for their optimal values and then use these bounds in an evaluate-and-cut procedure which serves as an exact solution algorithm for such problems with integer first-stage decisions. Later, we consider a risk-averse day-ahead scheduling of electricity generation or unit commitment problem where the objective is a dynamic coherent risk measure. We consider two different versions of the problem: adaptive and non-adaptive. In the adaptive model, the commitment decisions are updated in each stage, whereas in the non-adaptive model, the commitment decisions are fixed in the first-stage. We provide theoretical and empirical analyses on the benefit of using an adaptive multi-stage stochastic model. Finally, we investigate the trade off between the adaptivity of the model and the computational effort to solve it for risk-averse multi-stage production planning problems with an objective of dynamic coherent risk measure. We also conduct computational experiments in order to verify the theoretical findings and discuss the results of these experiments.
Open Access
Age and lifetime based policies for perishable items
(Bilkent University, 2018-10) Poormoaied, Saeed
Many inventory systems hold items which perish after a specific time. Upon perishing, the inventory level falls down to zero which may incur irreparable costs to the system. Therefore, developing a genius control policy for managing such inventories is a crucial task. Since the lifetime of items are now affecting the inventory level, applying the traditional inventory policies which are based only on the stock level causes some shortcomings. The traditional inventory policies lack the information regarding the lifetime of items. On the other hand, the optimal policy for perishable items is known to be a periodic review policy keeping the complete information regarding the remaining lead times of orders, inventory onhand, and lifetimes of items. Optimal control policy class for continuous review is still an open question. In this regard, we attempt to contribute the remaining lifetime of items into the inventory policy for perishable items with positive lead time and fixed lifetime under a continuous review with a service level constraint. We develop a class of hybrid control policies which utilize the remaining lifetimes of items in addition to stock levels. We study a stochastic single item inventory system where demand follows a Poisson process and unmet demand is lost. The aging process of a new batch starts when it joins the inventories. We provide an exact analytic model by using an embedded Markov chain process to derive the stationary distribution of the effective lifetimes in the presence of both one and more than one outstanding orders assumptions. Operating characteristics of the system are derived using the renewal reward theorem. Additionally, we propose some control policies based on only the remaining lifetime of items. Our results reveal that the hybrid policies consistently outperform the stock level and remaining lifetime-based polices, especially when demand during the lifetime is sufficiently small and unit perishing cost is high. It is observed that the dominance relations among these two policy classes depend on the particular parameter setting. In particular, when the lifetime of items is long enough, the stock level based policy performs very well. Finally, we present our methodology for finding the optimal solution thorough a heuristic algorithm derived by considering the structure of the objective function and service level constraint, and a sensitivity analysis is performed to evaluate the impact of the key input parameters.
Open Access
Concepts and analysis in facility location under uncertainty : applications to 1-median problem
(Bilkent University, 2001-08) Demir, Muhittin Hakan
Open Access
Terrain visibility and guarding problems
(Bilkent University, 2017-10) Eliş, Haluk
Watchtowers are located on terrains to detect fires, military units are deployed to watch the terrain to prevent infiltration, and relay stations are placed such that no dead zone is present on the terrain to maintain uninterrupted communication. In this thesis, any entity that is capable of observing or sensing a piece of land or an object on the land is referred to as a guard. Thus, watchtowers, military units and relay stations are guards and so are sensors, observers (human beings), cameras and the like. Observing, seeing, covering and guarding will mean the same. The viewshed of a given guard on a terrain is defined to be those portions of the terrain visible to the guard and the calculation of the viewshed of the guard is referred to as the viewshed problem. Locating minimum number of guards on a terrain (T) such that every point on the terrain is guarded by at least one of the guards is known as terrain guarding problem (TGP). Terrains are generally represented as regular square grids (RSG) or triangulated irregular networks (TIN). In this thesis, we study the terrain guarding problem and the viewshed problem on both representations. The first problem we deal with is the 1.5 dimensional terrain guarding problem (1.5D TGP). 1.5D terrain is a cross-section of a TIN and is characterized by a piecewise linear curve. The problem has been shown to be NP-Hard. To solve the problem to optimality, a finite dominating set (FDS) of size O(n2) and a witness set of size O(n2) have been presented earlier, where n is the number of vertices on T. An FDS is a finite set of points that contains an optimal solution to an optimization problem possibly with an uncountable feasible set. A witness set is a discretization of the terrain, and thus a finite set, such that guarding of the elements of the witness set implies guarding of T. We show that there exists an FDS, composed of convex points and dip points, with cardinality O(n). We also prove that there exist witness sets of cardinality O(n), which are smaller than O(n2) found earlier. The existence of smaller FDSs and witness sets leads to the reduction of decision variables and constraints respectively in the zero-one integer programming (ZOIP) formulation of the problem. Next, we discuss the viewshed problem and TGP on TINs, also known as 2.5D terrain guarding problem. No FDS has been proposed for this problem yet. To solve the problem to optimality the viewshed problem must also be solved. Hidden surface removal algorithms that claim to solve the viewshed problem do not provide analytical solutions and present some ambiguities regarding implementation. Other studies that make use of the horizon information of the terrain to calculate viewshed do so by projecting the vertices of the horizon onto the supporting plane of the triangle of interest and then by connecting the projections of the vertices to find the visible region on the triangle. We show that this approach is erroneous and present an alternative projection model in 3D space. The invisible region on a given triangle caused by another traingle is shown to be characterized by a system of nonlinear equations, which are linearized to obtain a polyhedral set. Finally, a realistic example of the terrain guarding problem is studied, which involves the surveillance of a rugged geographical terrain approximated by RSG by means of thermal cameras. A number of issues related to the terrain-guarding problem on RSGs are addressed with integer-programming models proposed to solve the problem. Next, a sensitivity analysis is carried out in which two fictitious terrains are created to see the effect of the resolution of a terrain, and of terrain characteristics, on coverage optimization. Also, a new problem, called the blocking path problem, is introduced and solved by an integer-programming formulation based on a network paradigm.
Open Access
Exact solution approaches for non-Hamiltonian vehicle routing problems
(Bilkent University, 2017-07) Özbaygın, Amine Gizem
In this thesis, we study di erent non-Hamiltonian vehicle routing problem variants and concentrate on developing e cient optimization algorithms to solve them. First, we consider the split delivery vehicle routing problem (SDVRP).We provide a vehicle-indexed ow formulation for the problem, and then, a relaxation obtained by aggregating the vehicle-indexed variables over all vehicles. This relaxation may have optimal solutions where several vehicles exchange loads at some customers. We cut-o such solutions either by extending the formulation locally with vehicle-indexed variables or by node splitting. We compare these approaches using instances from the literature and new randomly generated instances. Additionally, we introduce two new extensions of the SDVRP by restricting the number of splits and by relaxing the depot return requirement, and modify our algorithms to handle these extensions. Second, we focus on a problem unifying the notion of coverage and routing. In some real-life applications, it may not be viable to visit every single customer separately due to resource limitations or e ciency concerns. In such cases, utilizing the notion of coverage; i.e., satisfying the demand of multiple customers by visiting a single customer location, may be advantageous. With this motivation, we study the time constrained maximal covering salesman problem (TCMCSP) in which the aim is to nd a tour visiting a subset of customers so that the amount of demand covered within a limited time is maximized. We provide ow and cut formulations and derive valid inequalities. Since the connectivity constraints and the proposed valid inequalities are exponential in the size of the problem, we devise di erent branch-and-cut schemes. Computational experiments performed on a set of problem instances demonstrate the e ectiveness of the proposed valid inequalities in terms of strengthening the linear relaxation bounds as well as speeding up the solution procedure. Moreover, the results indicate the superiority of using a branch-and-cut methodology over a ow-based formulation. Finally, we discuss the relation between the problem parameters and the structure of optimal solutions based on the results of our experiments. Third, we study the vehicle routing problem with roaming delivery locations (VRPRDL) in which a customer order has to be delivered to the trunk of the customer's car during the time that the car is parked at one of the locations in the (known) customer's travel itinerary. We formulate the problem as a set covering problem and develop a branch-and-price algorithm for its solution. The algorithm can also be used for solving a more general variant in which a hybrid delivery strategy is considered that allows a delivery to either a customer's home or to the trunk of the customer's car. We evaluate the e ectiveness of the many algorithmic features incorporated in the algorithm in an extensive computational study and analyze the bene ts of these innovative delivery strategies. The computational results show that employing the hybrid delivery strategy results in average cost savings of nearly 20% for the instances in our test set.Finally, we consider the dynamic version of the VRPRDL in which customer itineraries may change during the execution of the planned delivery schedule, which can become infeasible or suboptimal as a result. We refer to this problem as the dynamic VRPRDL (D-VRPRDL) and propose an iterative solution framework in which the previously planned vehicle routes are re-optimized whenever an itinerary update is revealed. We use the branch-and-price algorithm developed for the static VRPRDL both for solving the planning problem (to obtain an initial delivery schedule) and for solving the re-optimization problems. Since many re-optimization problems may have to be solved during the execution stage, it is critical to produce solutions to these problems quickly. To this end, we devise heuristic procedures through which the columns generated during the previous branch-and-price executions can be utilized when solving a re-optimization problem. In this way, we may be able to save time that would otherwise be spent in generating columns which have already been (partially) generated when solving the previous problems, and nd optimal solutions or at least solutions of good quality reasonably quickly. We perform preliminary computational experiments and report the results.
Open Access
Designing intervention strategy for public-interest goods
(Bilkent University, 2016-09) Demirci, Ece Zeliha
Public-interest goods, which are also referred as goods with positive externalities, create benefits to individual consumers as well as non-paying third parties. Some significant examples include health related products such as vaccines and products with less carbon emissions. When positive externalities exist, the good may be under-produced or under-supplied due to incorrect pricing policies or failing to value external benefits and that is why a need for intervention arises. A central authority such as government or social planner intervenes into the system of these goods so that their adoption levels are increased towards socially desirable levels. The central authority seeks to design and finance an intervention strategy that will impact the decisions of the channel in line with the good of the society, specified as social welfare. A key issue in designing an intervention mechanism is choosing the intervention tools to incorporate. The intervention tools can target the supply or demand of the good. One option for the intervention tool is investment in demand-increasing strategies, which affects the level of stochastic demand in the market. Second option is investment in strategies that will improve supply of the good. Alternatives for this option include registering rebates or subsidies and investment in yield-improving strategies when production process faces imperfect yield. As several real life cases indicate, central authority operates under a limited budget in this environment. Thus, we introduce and analyze social welfare maximization models with the emphasis on optimal budget allocation. We model the lower level problem, which represents the channel as a newsvendor problem. We then utilize bilevel programming for modeling the environment incorporating the role of central authority. After obtaining single level equivalent formulations of the problems, we analyze and solve them as non-linear programs. Our first problem is to analyze an intervention strategy, which uses only subsidy issued per unit order quantity. We explore the subsidy design problem for single retailer and n retailers cases. We show that all of the budget will be used under mild conditions and present structural results. Also, we analyze subsidy design problem for two echelon setting, where the central authority gives subsidy both to retailer and manufacturer. We consider centralized and manufacturer-driven problems and present numerical results. In the remaining part of the thesis, we focus on joint intervention mechanisms in which two intervention tools are applied simultaneously. First, we study a joint mechanism composed of demand-increasing strategy and rebate. We present two models and associated structural properties. First model aims to find optimal budget and allocation of it among intervention tools. We deduce that rebate amount may be independent of investment made in demand-increasing strategies and improvement pattern of demand. Second model decides on the optimal allocation of a given budget between intervention tools. We show that central authority will allocate all budget under mild conditions. Furthermore, we use real-life data and information of California electric vehicle market in order to verify the proposed models and show benefits of taking such an approach. We also explore the application of the joint mechanism under a given budget for exponentially distributed demand family and fully characterize the optimal solution. The analysis of the solution reveals that designing an intervention scheme without considering an explicit budget constraint will result in budget deficit and excess money transfers to the retailer. As the second modeling environment we consider a joint mechanism consisting of demand-increasing strategy and yield-improving strategy in a setting where yield uncertainty exists. We introduce lognormal demand and yield models that take into account the investments made for improving them. We test the suggested model with a case study relying on the available estimates of US influenza market. The results indicate that addressing both demand and yield issues by the proposed mechanism will increase vaccination percentages remarkably.
Open Access
Robust portfolio optimization with risk measures under distributional uncertainty
(Bilkent University, 2016-07) Paç, A. Burak
In this study, we consider the portfolio selection problem with different risk measures and different perspectives regarding distributional uncertainty. First, we consider the problem of optimal portfolio choice using the first and second lower partial moment risk measures, for a market consisting of n risky assets and a riskless asset, with short positions allowed. We derive closed-form robust portfolio rules minimizing the worst case risk measure under uncertainty of the return distribution given the mean/covariance information. A criticism levelled against distributionally robust portfolios is sensitivity to uncertainties or estimation errors in the mean return data, i.e., Mean Return Ambiguity. Modeling ambiguity in mean return via an ellipsoidal set, we derive results for a setting with mean return and distributional uncertainty combined. Using the adjustable robustness paradigm we extend the single period results to multiple periods in discrete time, and derive closed-form dynamic portfolio policies. Next, we consider the problem of optimal portfolio choice minimizing the Conditional Value-at-Risk (CVaR) and Value-at-Risk (VaR) measures under the minimum expected return constraint. We derive the optimal portfolio rules for the ellipsoidal mean return vector and distributional ambiguity setting. In the presence of a riskless asset, the robust CVaR and VaR measures, coupled with a minimum mean return constraint, yield simple, mean-variance efficient optimal portfolio rules. In a market without the riskless asset, we obtain a closed-form portfolio rule that generalizes earlier results, without a minimum mean return restriction. In the final problem, we have a change of perspective regarding uncertainty. Rather than the information on first and second moments, knowledge of a nominal distribution of asset returns is assumed, and the actual distribution is considered to be within a ball around this nominal distribution. The metric choice on the probability space is the Kantorovich distance. We investigate convergence of the risky investment to uniform portfolio when a riskless asset is available. While uniform investment to risky assets becomes optimal, it is shown that as the uncertainty radius increases, the total allocation to risky assets diminishes. Hence, as uncertainty increases, the risk averse investor is driven out of the risky market.
Open Access
Scheduling in flexible robotic manufacturing cells
(Bilkent University, 2006) Gültekin, Hakan
The focus of this thesis is the scheduling problems arising in robotic cells which consist of a number of machines and a material handling robot. The machines used in such systems for metal cutting industries are highly flexible CNC machines. Although flexibility is the key term that affects the performance of these systems, the current literature ignores this. As a consequence, the problems considered in the current literature are either too limiting or the provided solutions are suboptimal for the flexible systems. This thesis analyzes different robotic cell configurations with different sources of flexibility. This study is the first one to consider operation allocation problems and controllable processing times as well as some design problems and bicriteria models in the context of robotic cell scheduling. Also, a new class of robot move cycles is defined, which is overlooked in the existing literature. Optimal solutions are provided for solvable cases, whereas complexity analyses and efficient heuristic algorithms are provided for the remaining problems.
Open Access
Quadratic assignment problem : linearizations and polynomial time solvable cases
(Bilkent University, 2006) Erdoğan, Güneş
The Quadratic Assignment Problem (QAP) is one of the hardest combinatorial optimization problems known. Exact solution attempts proposed for instances of size larger than 15 have been generally unsuccessful even though successful implementations have been reported on some test problems from the QAPLIB up to size 36. In this dissertation, we analyze the binary structure of the QAP and present new IP formulations. We focus on “flow-based” formulations, strengthen the formulations with valid inequalities, and report computational experience with a branch-and-cut algorithm. Next, we present new classes of instances of the QAP that can be completely or partially reduced to the Linear Assignment Problem and give procedures to check whether or not an instance is an element of one of these classes. We also identify classes of instances of the Koopmans-Beckmann form of the QAP that are solvable in polynomial time. Lastly, we present a strong lower bound based on Bender’s decomposition.