Approximate Linear Programming Research Articles

A robust omnichannel pricing and ordering optimization approach with return policies based on data-driven support vector clustering

This work considers an omnichannel retailer selling a product with market demand uncertainties to customers in different regions through an e-store and multiple brick-and-mortar stores. The retailer also manages online product returns by selecting appropriate return policies. A robust omnichannel pricing and ordering optimization model is proposed with two, i.e., a full-refund and a no-refund, return policies. The online demand and the offline demand depend on the prices of the e-store and the brick-and-mortar stores, where the online demand also depends on the refund of the e-store. A linearization technique is adopted to deal with nonlinearity of the model. A data-driven robust optimization approach is used to construct uncertainty sets based on available historical data using support vector clustering to handle demand uncertainties. Furthermore, the proposed model is transformed into an approximate mixed integer linear programming model which can be solved by using commercial software. An electronics retailer in China is used as a case study to illustrate the effectiveness and practicality of the proposed model and the solution method. A comparison with the box uncertainty set reveals that the data-driven uncertainty set is less conservative and performs better by obtaining higher profit for the retailer. Furthermore, sensitive analysis results indicate that return policies and the return rate of the product affect the optimal pricing and ordering decisions and the total profit.

Profit maximization via capacity control for distribution logistics problems

We consider a distribution logistics scenario where a shipping operator, managing a limited amount of resources, receives a stream of service requests, issued by a set of customers along a booking time-horizon, that are referred to a future operational period. The shipping operator must then decide about accepting or rejecting each incoming request at the time it is issued, accounting for revenues, but also considering resource consumptions. In this context, the decision process is based on dynamically finding the best trade-off between the immediate return of accepting the request and the convenience of preserving capacity to possibly exploit more valuable future requests. We give a dynamic formulation of the problem aimed at maximizing the operator revenues, accounting also for the operational distribution costs. Due to the “curse of dimensionality”, the dynamic program cannot be solved optimally. For this reason, we propose a mixed-integer linear programming approximation, whose exact or approximate solutions provide the relevant information to apply some commonplace revenue management policies in the real-time decision-making. Adopting a capacitated vehicle routing problem as an underlying distribution application, we analyze the computational behavior of the proposed techniques on a set of academic test problems.

Data Collection in Multihop Mobile Sink-Aided Backscatter IoT Networks

This article studies a novel wireless-powered Internet of Things (IoT) network that consists of: 1) a hybrid access point (HAP) that charges devices and also helps facilitate backscattering transmissions; 2) devices that use active radio frequency (RF) and backscattering transmissions; and 3) a mobile data collector. Our aim is to maximize the amount of data received by the HAP and data collector. The main problem is to determine the charging duration of the HAP and link activation schedule of devices. We formulate a novel mixed-integer linear program (MILP) and also propose a heuristic algorithm named reduced-set linear program approximation (RS-LPA). The results show that: 1) throughput increases with the number of backscatter transmission sets; 2) smaller amount of data is uploaded to the mobile collector when sampling cost is low; and 3) the throughput of RS-LPA is on average 10.55% lower than MILP.

Cost Partitioning Heuristics for Stochastic Shortest Path Problems

In classical planning, cost partitioning is a powerful method which allows to combine multiple admissible heuristics while retaining an admissible bound. In this paper, we extend the theory of cost partitioning to probabilistic planning by generalizing from deterministic transition systems to stochastic shortest path problems (SSPs). We show that fundamental results related to cost partitioning still hold in our extended theory. We also investigate how to optimally partition costs for a large class of abstraction heuristics for SSPs. Lastly, we analyze occupation measure heuristics for SSPs as well as the theory of approximate linear programming for reward-oriented Markov decision processes. All of these fit our framework and can be seen as cost-partitioned heuristics.

Technical Note—On the Strength of Relaxations of Weakly Coupled Stochastic Dynamic Programs

Many sequential decision problems have a weakly coupled structure in that a set of linking constraints couples an otherwise independent collection of subproblems. This structure arises in a wide variety of applications, such as network revenue management, online advertising, assortment planning, interactive marketing, optimization of power systems, and multilocation inventory management to name only a few. Such problems can be modeled as dynamic programs but are quite difficult to solve. Two widely studied approximation methods are approximate linear programs, which involve finding a best approximation of total value that is additive across the subsystems, and Lagrangian relaxations, which involve relaxing the linking constraints. It is well known that both of these approaches provide upper bounds to the optimal value, and the approximate linear programming approach is a better bound but also, more difficult to compute. In this paper, we provide a detailed theoretical analysis of these two approximations and show that, under fairly broad conditions, these two approximations lead to upper bounds that are very close and often identical. Our theory suggests that, between these two approximations, Lagrangian relaxations should usually be the preferred choice for researchers studying applications involving weakly coupled dynamic programs.

A Distributionally Robust Scheduling Approach for Uncertain Steelmaking and Continuous Casting Processes

This article presents a new model to handle the cast break problem caused by small daily disruptions in the processing time of the steelmaking and continuous casting (SCC) production process. In this model, the exact distribution of the uncertain parameters is unknown, and support set, mean, and covariance information is used to describe the uncertain processing time. The problem aims to determine the assignments, sequences, and time points of the charges to be processed on corresponding machines. The main goal is to minimize the expected value of the production objective while reducing the number of cast break occurrences. The problem is solved in two steps. First, a subproblem is developed by fixing the sequences and the assignments of the charges. This subproblem is formulated as a distributionally robust chance-constrained (DRCC) model, in which the constraints are established with certain probabilities even when the uncertain processing times are in their worst cases. A dual approximation method is proposed to convert the model into a semidefinite programming problem so that it can be solved by standard solvers. Additionally, a linear programming approximation method is used to accelerate the solving procedure. A Tabu search algorithm incorporated with a speed-up strategy is also designed to determine the assignments and sequences of the charges. Both simulated data generated from different distributions and actual production data are used to test the efficacy of our model. Results of the numerical experiments show that the schedule obtained from the DRCC model is more robust, i.e., it causes fewer cast breaks than the nominal schedule obtained from a deterministic model.

Finite‐horizon approximate linear programs for capacity allocation over a rolling horizon

Approximate linear programs (ALPs) have been used extensively to approximately solve stochastic dynamic programs that suffer from the well‐known curse of dimensionality. Due to canonical results establishing the optimality of stationary value functions and policies for infinite‐horizon dynamic programs, the literature has largely focused on approximation architectures that are stationary over time. In a departure from this literature, we apply a nonstationary approximation architecture to an infinite‐dimensional linear programming formulation of the stochastic dynamic programs. We solve the resulting problems using a finite‐horizon approximation. Such finite‐horizon approximations are common in the theoretical analysis of infinite‐horizon linear programs, but have not been considered in the approximate linear programming literature. We illustrate the approach on a rolling‐horizon capacity allocation problem using an affine approximation architecture. We obtain three main results. First, nonstationary approximations can substantially improve upper bounds on the optimal revenue. Second, the upper bounds from the finite‐horizon approximation monotonically decrease as the horizon length increases, and converge to the upper bound from the infinite‐horizon approximation. Finally, the improvement does not come at the expense of tractability, as the resulting ALPs admit compact representations and can be solved efficiently. The resulting approximations also produce strong heuristic policies and significantly reduce optimality gaps in numerical experiments.

Mathematical programming formulations for the alternating current optimal power flow problem

Power flow refers to the injection of power on the lines of an electrical grid, so that all the injections at the nodes form a consistent flow within the network. Optimality, in this setting, is usually intended as the minimization of the cost of generating power. Current can either be direct or alternating: while the former yields approximate linear programming formulations, the latter yields formulations of a much more interesting sort: namely, nonconvex nonlinear programs in complex numbers. In this technical survey, we derive formulation variants and relaxations of the alternating current optimal power flow problem.

An Exact Sequential Linear Programming Algorithm for the Optimal Power Flow Problem

Despite major advancements in nonlinear programming (NLP) and convex relaxations, most system operators around the world still predominantly use some form of linear programming (LP) approximation of the AC power flow equations. This is largely due to LP technology's superior reliability and computational efficiency, especially in real-time market applications, security-constrained applications, and extensions involving integer variables, in addition to its ability to readily generate locational marginal prices (LMP) for market applications. In the aim of leveraging the advantages of LP while retaining the accuracy of NLP interior-point methods (IPMs), this paper proposes a sequential linear programming (SLP) approach consisting of a sequence of carefully constructed <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">supporting hyperplanes and halfspaces</i> . The algorithm is numerically demonstrated to converge on 138 test cases with up the 3375 buses to feasible high-quality solutions (i) <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">without</i> AC feasibility restoration (i.e., using LP solvers exclusively), (ii) in computation times generally within the same order of magnitude as those from a state-of-the-art NLP solver, and (iii) with robustness against the choice of starting point. In particular, the (relative) optimality gaps and the mean constraint violations are on average around <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$10^{-3}$</tex-math></inline-formula> % and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$10^{-7}$</tex-math></inline-formula> , respectively, under a <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">single</i> parameter setting for all the 138 test cases. To the best of our knowledge, the proposed SLP approach is the first to use LP exclusively to reach <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">feasible</i> and high-quality solutions to the nonconvex AC OPF in a reliable way, which paves the way for system and market operators to keep using their LP solvers but now with the ability to accurately capture transmission losses, price reactive power (Q-LMP), and obtain more accurate LMP.

Managing a Hybrid RDC‐DC Inventory System

In this study, we study a hybrid RDC‐DC serial inventory system where the regional distribution center (RDC) replenishes its stock from an outside supplier (OS), while the distribution center (DC) faces random demand and replenishes its stock from the RDC. Unlike in the traditional serial system, the DC itself can replenish its inventory from outside as well. We firstly derive structural properties for the optimal long‐run average cost and the optimal stationary policy by vanishing discount approach, and then propose two simple and easy‐to‐implement policies. The first policy, which we call the three‐index policy, combines the characteristics of the echelon‐base‐stock policy for the serial system (Clark and Scarf. 1960. Management Sci.6(4): 475–490) and the dual‐index policy for the dual‐sourcing system (Veeraraghavan and Scheller‐Wolf. 2008. Oper. Res.56(4): 850–864). We show that the order‐up‐to level of the DC from the RDC can be computed by a newsboy fractile. A simulation‐based optimization procedure for the policy is provided. We then develop the approximate linear programming ( ALP) policy based on the three‐index policy and the multimodularity of the problem. This policy applies the linear programming approach to approximately solve the value function of the dynamic programming formulation. Numerical results show that both the three‐index policy and the ALP policy are comparable to the optimal policy computed via dynamic programming, and the latter performs slightly better. Moreover, the OS of the DC can draw considerable cost savings under both policies. We also conduct a numerical study with problem parameters calibrated using actual data from a consumer goods company in China to glen insights on the management of the system.

Integrated Multiresource Capacity Planning and Multitype Patient Scheduling

The joint optimization problem of multiresource capacity planning and multitype patient scheduling under uncertain demands and random capacity consumption poses a significant computational challenge. The common practice in solving this problem is to first identify capacity levels and then determine patient scheduling decisions separately, which typically leads to suboptimal decisions that often result in ineffective outcomes of care. In order to overcome these inefficiencies, in this paper, we propose a novel two-stage stochastic optimization model that integrates these two decisions, which can lower costs by exploring the coupling relationship between patient scheduling and capacity configuration. The patient scheduling problem is modeled as a Markov decision process. We first analyze the properties for the multitype patient case under specific assumptions and then establish structural properties of the optimal scheduling policy for the one-type patient case. Based on these findings, we propose optimal solution algorithms to solve the joint optimization problem for this special case. Because it is intractable to solve the original two-stage problem for a general multitype system with large state space, we propose a heuristic policy and a two-stage stochastic mixed-integer programming model solved by the Benders decomposition algorithm, which is further improved by combining an approximate linear program and the look-ahead strategy. To illustrate the efficiency of our approaches and draw managerial insights, we apply our solutions to a data set from the day surgery center of a large public hospital in Shanghai, China. The results show that the joint optimization of capacity planning and patient scheduling could significantly improve the performance. Furthermore, our model can be applied to a rolling-horizon framework to optimize dynamic patient scheduling decisions. Through extensive numerical analyses, we demonstrate that our approaches yield good performances, as measured by the gap against an upper bound, and that these approaches outperform several benchmark policies. Summary of Contribution: First, this paper investigates the joint optimization problem of multiresource capacity planning and multitype patient scheduling under uncertain demands and random capacity consumption, which poses a significant computational challenge. It belongs to the scope of computing and operations research. Second, this paper formulates a mathematical model, establishes optimality properties, proposes solution algorithms, and performs extensive numerical experiments using real-world data. This work includes aspects of dynamic stochastic control, computing algorithms, and experiments. Moreover, this paper is motivated by a practical problem (joint management of capacity planning and patient scheduling in the day surgery center) in our cooperative hospital, which is also key to numerous other applications, for example, the make-to-order manufacturing systems and computing facility systems. By using the optimality properties, solution algorithms, and management insights derived in this paper, the practitioners can be equipped with a decision support tool for efficient and effective operation decisions.

On the Strength of Relaxations of Weakly Coupled Stochastic Dynamic Programs

Many stochastic dynamic programs (DPs) have a weakly coupled structure in that a set of linking constraints in each period couple an otherwise independent collection of subproblems. Two widely studied approximations of such problems are approximate linear programs (ALP), which involve optimizing value function approximations that additively separate across subproblems, and Lagrangian relaxations, which involve relaxing the linking constraints. It is well-known that both of these approximations provide upper bounds on the optimal value function in all states and that the ALP provides a tighter upper bound in the initial state. The purpose of this short paper is to provide theoretical justification for the fact that these upper bounds are often close if not identical. We show: (i) For any weakly coupled DP, the difference between these two upper bounds --- the relaxation gap --- is bounded from above in terms of the integrality gap of the constraint separation problems within the ALP; (ii) If subproblem rewards are uniformly bounded and some broadly applicable conditions on the linking constraints hold, the relaxation gap is bounded from above by a constant that is independent of the number of subproblems; and (iii) When subproblem actions are binary and the linking constraints have a unimodular structure, the relaxation gap equals zero. The conditions for (iii) hold in several widely studied problems: restless bandit problems, online stochastic matching problems, network revenue management problems, and price-directed control of relocating resources. These findings generalize and unify existing results.

Mathematical programming formulations for the alternating current optimal power flow problem

Power flow refers to the injection of power on the lines of an electrical grid, so that all the injections at the nodes form a consistent flow within the network. Optimality, in this setting, is usually intended as the minimization of the cost of generating power. Current can either be direct or alternating: while the former yields approximate linear programming formulations, the latter yields formulations of a much more interesting sort: namely, nonconvex nonlinear programs in complex numbers. In this technical survey, we derive formulation variants and relaxations of the alternating current optimal power flow problem.

Charge-and-Activate Policies for Targets Monitoring in RF-Harvesting Sensor Networks

In this paper, we consider a Hybrid Access Point (HAP) that supplies energy to sensor devices tasked with monitoring one or more mobile targets with a known trajectory. The HAP's goal is to maximize a Quality of Monitoring (QoM) metric that is a ratio of the following quantities: (i) distance between a sensor device and a target, and (ii) duration in which a target is monitored by a sensor device. We formulate a Mixed Integer Linear Program (MILP) and use it to determine the subset of sensor devices to be charged in each time slot, their activation time, and the transmission or charging power used by the HAP. We also propose a Cross-Entropy (CE) approach and a heuristic algorithm called Energy Reallocation Linear Programming Approximation (ERLPA) to select sensor devices for charging in large-scale networks. Our results show that (i) QoM is affected by the energy requirement of sensor devices, energy storage capacity, number of channels available to the HAP, sensor sensing radius and energy conversion efficiency of sensor devices, and (ii) both the CE method and ERLPA are capable of producing schedules that are near optimal.

Optimized Integration of a Set of Small Renewable Sources Into a Bulk Power System

Electric power systems are increasingly relying on small renewable plants that must be connected to a bulk grid. This paper proposes a strategy to solve the transmission line planning problem for such an integration. The model minimizes the costs of investment and annual energy losses, considering as planning alternatives the building of transmission lines, the collector substation allocation and the voltage magnitude levels of the transmission line while ensuring the operational constraints of the network and the voltage magnitude at connection points with the bulk power system (BPS). Linearization and approximation techniques are used to obtain an approximate mixed-integer linear programming (MILP) model from the original mixed-integer nonlinear programming model. A search-space reduction procedure is proposed to obtain the most qualified sites to accommodate collector substations. The combined strategy leads to a reduction in the computational effort, which allows full-scale studies of the optimal integration of a set of small renewable sources. For intermittent renewable sources, the proposed approach is used to obtain the Pareto front considering the total planning cost and the annual distributed energy. The results of two scenarios, using a real-world Brazilian case, certify the benefits of the proposals.

A decision-making analysis in UAV-enabled wireless power transfer for IoT networks

We consider an IoT network with energy-harvesting capabilities. To extend the network lifetime, we propose a novel unmanned aerial vehicle (UAV)- enabled wireless power transfer (WPT) system, where UAVs move among IoT devices and act as data aggregators and wireless power providers. This paper addresses the decision-making problem since the limited buffer and energy resources constrain all nodes. Each IoT node must decide on whether to request a data transmission, to ask for a wireless energy transfer or to abstain and not take any action. When a UAV receives a request from an IoT device, either for data reception or wireless energy transmission, it has to accept or decline. In this paper, we aim to find a proper packet delivery and energy transfer policy according to the system state that maximizes the data transmission efficiency of the system. We first formulate the problem as a Markov Decision Process (MDP) to tackle the successive decision issues, to optimize a utility for each node upon a casual environment. As the MDP formalism achieves its limits when the interactions between different nodes are considered, we formulate the problem as a Graph-based MDP (GMDP). The transition functions and rewards are then decomposed into local functions, and a graph illustrates the dependency’ relations among the nodes. To obtain the optimal policy despite the system’s variations, Mean-Field Approximation (MFA) and Approximate linear-programming (ALP) algorithms were proposed to solve the GMDP problem.

A Linear Programming Approximation of Distributionally Robust Chance-Constrained Dispatch With Wasserstein Distance

This paper proposes a data-driven distributionally robust chance constrained real-time dispatch (DRCC-RTD) considering renewable generation forecasting errors. The proposed DRCC-RTD model minimizes the expected quadratic cost function and guarantees that the two-sided chance constraints are satisfied for any distribution in the ambiguity set. The Wasserstein-distance-based ambiguity set, which is a family of distributions centered at an empirical distribution, is employed to hedge against data perturbations. By applying the reformulation linearization technique (RLT) to relax the quadratic constraints of the worst-case costs and constructing linear reformulations of the DRCCs, the proposed DRCC-RTD model is cast into a deterministic linear programming (LP) problem, which can be solved efficiently by off-the-shelf solvers. Case studies are carried out on a 6-bus system and the IEEE 118-bus system to validate the effectiveness and efficiency of the proposed approach.

Transmission-Line Switching for Load Shed Prevention via an Accelerated Linear Programming Approximation of AC Power Flows

Topology control has garnered recent consideration for its value in leveraging the flexiblity of the power system. However, using even the latest technology, the AC optimal transmission switching (ACOTS) problem remains intractable. As such, most research addressing optimal topology control is based on the DCOPF, which has been shown to be particularly inaccurate for transmission switching decisions in the AC power system. This paper accelerates an existing mixed-integer linear power flow model which has been shown to be very accurate in approximating AC power flows. The aim of the new model is to identify transmission switching candidates to prevent load shed caused by contingency events. The methodology was tested on the IEEE 118-bus test case system and presents promising results. The model proposed herein was found to almost always identify AC optimal switching actions while significantly reducing the necessary solution time in comparison with the base model.

Self-guided Approximate Linear Programs

Approximate linear programs (ALPs) are well-known models based on value function approximations (VFAs) to obtain policies and lower bounds on the optimal policy cost of discounted-cost Markov decision processes (MDPs). Formulating an ALP requires (i) basis functions, the linear combination of which defines the VFA, and (ii) a state-relevance distribution, which determines the relative importance of different states in the ALP objective for the purpose of minimizing VFA error. Both these choices are typically heuristic: basis function selection relies on domain knowledge while the state-relevance distribution is specified using the frequency of states visited by a heuristic policy. We propose a self-guided sequence of ALPs that embeds random basis functions obtained via inexpensive sampling and uses the known VFA from the previous iteration to guide VFA computation in the current iteration. Self-guided ALPs mitigate the need for domain knowledge during basis function selection as well as the impact of the initial choice of the state-relevance distribution, thus significantly reducing the ALP implementation burden. We establish high probability error bounds on the VFAs from this sequence and show that a worst-case measure of policy performance is improved. We find that these favorable implementation and theoretical properties translate to encouraging numerical results on perishable inventory control and options pricing applications, where self-guided ALP policies improve upon policies from problem-specific methods. More broadly, our research takes a meaningful step toward application-agnostic policies and bounds for MDPs.

An approximation algorithm for solving standard quadratic optimization problems

Standard quadratic optimization problems (StQPs) are NP-hard in computational complexity theory when the matrix is indefinite. This paper describes an approximate algorithm of finding inner optimal values of StQPs. The approximate algorithm fuzzifies variable x ∈ Rn with normalized possibility distributions and simplifies the solving of StQPs. The approximation ratio is discussed and determined. Numerical results show that (1) the new algorithm achieves higher accuracy than the semidefinite programming method and linear programming approximation method; (2) the novel algorithm consumes less than one out of fourth computational time that is consumed by linear programming approximation method; (3) the computational time of the new algorithm does not correlate with the matrix densities whereas the computational times of the branch-and-bound and heuristic algorithms do.