Robust Linear Programming Research Articles

RETRACTED: Robust optimization based price-taker retailer bidding strategy under pool market price uncertainty

Abstract In the restructured electricity markets, retailers purchase the required demand of its consumers from different energy resources such as self-generating facilities, bilateral contracts and pool market. In this process, the pool market price uncertainty modeling is important for obtaining the maximum profit. Therefore, in this paper, a robust optimization approach is proposed to obtain the optimal bidding strategy of retailer, which should be submitted to pool market. By the proposed method, a collection of robust mixed-integer linear programming problem (RMILP) is solved to build optimal bidding strategy for retailer. For pool market price uncertainty modeling, upper and lower limits of pool prices are considered instead of the forecasted prices. The range of pool prices are sequentially partitioned into a successive of nested subintervals, which permit formulating a collection of RMILP problems. The results of these problems give sufficient data to obtain optimal bidding strategy for submit to pool market by retailer. A detailed analysis is utilized to delineate the proposed method.

A multi-objective robust optimization model for a facility location-allocation problem in a supply chain under uncertainty

This paper presents a mathematical model for a facility location-allocation problem in order to design an integrated supply chain. We consider a supply chain consisting of multiple suppliers, multiple products, multiple plants, multiple transportation alternatives and multiple customers. The problem is to determine a number and capacity level of plants, allocation of customers demand, and selection and order allocation of suppliers. A scenario approach is used to handle the uncertainty of demand and cost parameters. The formulation is a robust multi-objective mixed-integer linear programming (MOMILP) that considers two conflicting objectives simultaneously. The first objective is to minimize the total costs of a supply chain including raw material costs, transportation costs and establishment costs of plants. The second objective function aims to minimize the total deterioration rate occurred by transportation alternatives. Then, the problem transformed into a linear one. Finally, by applying the LP-metric method, the model is solved as a single objective mixed-integer programming model. An experiment study shows that the proposed procedure can provide a promising result to design an efficient supply chain. DOI: http://dx.doi.org/10.5755/j01.ee.26.3.4287

Evaluating Contractors for Bonding: DEA Decision Making Model for Surety Underwriters

AbstractIn several countries, it is a common practice for construction owners to require contractors to submit several types of surety bonds shifting the risk of contractors default to sureties. The process of evaluating contractors’ applications for bonding is complicated and time-consuming. Surety underwriters are faced with the challenge of considering tens of objective and subjective criteria to make the bonding recommendation. These criteria include a contractor’s financial strength, past performance, and other related aspects. Literature review revealed that there is little guidance to surety underwriters in this regard. As a result, the purpose of this paper is to equip surety underwriters with a decision making model for evaluating contractors’ applications for bonding. The proposed model is based on data envelopment analysis (DEA). The DEA is a robust nonparametric linear programming approach that is widely used for benchmarking, performance measurement, and decision making. The proposed DEA mode...

Optimal bidding strategy of electricity retailers using robust optimisation approach considering time‐of‐use rate demand response programs under market price uncertainties

In restructured electricity markets, the electricity retailers try to obtain the consumers' electricity demand at the minimum cost of different resources such as self-generating facilities, bilateral contracts and pool market purchases. Hence, more attention should be paid to the demand response programs (DRPs) which aim to electricity procurement cost reduction. Owing to the uncertain nature of pool prices and the price fluctuation in the pool markets, the uncertainty modelling is inevitable for retailers. In this study, a robust optimisation approach is proposed for decision making of electricity retailers. Meanwhile, considering the effect of DRP on total procurement cost, an optimal bidding strategy is proposed of electricity retailers with the time-based model of DRP in the electricity market. For this purpose, a collection of robust mixed-integer linear programming (RMILP) problem should be solved in the proposed method. Rather than using the forecasted prices as inputs, the upper and lower limits of pool prices are considered for the uncertainty modelling. The range of pool prices is sequentially partitioned into successive nested subintervals, which permits formulating the RMILP problems. The results of these problems give sufficient data to obtain an optimal bidding strategy for electricity retailers considering DRP. Detailed analysis is performed to delineate the proposed method.

Production Planning with Remanufacturing under Uncertain Demand and Returns

In this study, a production planning problem in which a producer remanufactures returned products into serviceable products besides manufacturing serviceable products is considered. The main challenge in this planning problem is the uncertain demand for the serviceable products and the uncertain returns of the used products. The probability distributions of the uncertain demand and uncertain returns are not known. Only the means of uncertain parameters and maximum deviations from these means are known or can be estimated. The problem is to determine the quantity of serviceable products that are manufactured, the quantity of returned products that are remanufactured, and the quantity of returned products that are disposed over a multi-period planning horizon such that total cost composed of production, inventory and disposal costs is minimized. A new robust linear programming model that yields a feasible production-disposal policy regardless of the realization of demand and returns is proposed and compared with a robust linear programming model existing in the literature. The computational results reveal that the proposed model significantly outperforms the one existing in literature in terms of the actual cost savings

The robust linear programming technique for multi-dimensional analysis of preferences

The linear programming technique for multi–dimensional analysis of preferences (LINMAP) is one of the noted multi–attributes decision making (MADM) techniques and has been implemented in crisp and fuzzy environments. Robust optimisation attempts to obtain a solution which is feasible in all circumstances arising due to the uncertainty of parameters. The purpose of this study is to extend the LINMAP method for addressing robustness in MADM problems. In this methodology, robust optimisation concepts are used to describe robustness in decision information and decision making processes. Each alternative is evaluated based on its weighted distance to a robust positive ideal solution (RPIS). The RPIS and the robust weights of attributes are estimated using a new robust linear programming technique. Finally, Monte Carlo simulation is applied to test the robustness of the solution. A numerical example is provided to illustrate the effectiveness of the methodology.

A fuzzy robust linear programming problem with hybrid variables

This paper proposes a fuzzy robust linear programming (FRLP) problem where the coefficients of objective function and constraints contain both fuzziness and randomness. It is shown that the original FRLP problem is transformed into deterministic linear programming problems by new methods based on the chance–constrained programming. In the presented new methods, we use an extended chance–constrained programming based on possibility and necessity measures for fuzzy stochastic constraints in FRLP problem. Finally, a numerical example is also given to show the applicability of the methods discussed in this paper.

Water Use Disaggregation Based on Classification of Feature Vectors Extracted from Smart Meter Data

This work studies a feasibility of water use disaggregation from commercial smart meters, with volume quantization on the level of 1 litre. New water use characterisation is performed by extraction of predefined features from smart meter records and deter- mination of a region from a given pre-calculated region partition to which the corresponding features belong. Region adherence corresponds to a particular fixture type actuation in the household. Regions partition is computed by employing multicategory robust linear programming technique. Performance of the water use disaggregation model is shown when a separate model is tuned for each household based on sole household data.

On the integration of row and column uncertainty in robust linear programming

The usual formulation of a linear program is max $$c\cdot x{:}Ax \le b, x \ge 0$$c·x:Ax≤b,xź0. The core part of this linear program is the $$A$$A matrix since the columns define the variables and the rows define the constraints. The $$A$$A matrix is constructed by populating columns or populating rows, or some of both, depending on the nature of the data and how it is collected. This paper addresses the construction of the $$A$$A matrix and solution procedures when there are separate data sources for the columns and for the rows and, moreover, the data is uncertain. The $$A$$A matrices which are "realizable" are only those which are considered possible from both sources. These realizable matrices then form an uncertainty set $$U$$U for a robust linear program. We show how to formulate and solve linear programs which provide lower and upper bounds to the robust linear program defined by $$U$$U. We also show how to use ordinary linear programming duality to share and divide the "credit/responsibility" of the optimal value of the robust linear program between the two alternative data sources.

Robust Beamforming by Linear Programming

In this paper, a robust linear programming beamformer (RLPB) is proposed for non-Gaussian signals in the presence of steering vector uncertainties. Unlike most of the existing beamforming techniques based on the minimum variance criterion, the proposed RLPB minimizes the l∞-norm of the output to exploit the non-Gaussianity. We make use of a new definition of the lp-norm (1 ≤ p ≤ ∞) of a complex-valued vector, which is based on the lp-modulus of complex numbers. To achieve robustness against steering vector mismatch, the proposed method constrains the l∞-modulus of the response of any steering vector within a specified uncertainty set to exceed unity. The uncertainty set is modeled as a rhombus, which differs from the spherical or ellipsoidal uncertainty region widely adopted in the literature. The resulting optimization problem is cast as a linear programming and hence can be solved efficiently. The proposed RLPB is computationally simpler than its robust counterparts requiring solution to a second-order cone programming. We also address the issue of appropriately choosing the uncertainty region size. Simulation results demonstrate the superiority of the proposed RLPB over several state-of-the-art robust beamformers and show that its performance can approach the optimal performance bounds.

Robust capacitated facility location model for acquisitions under uncertainty

We study a new robust formulation for strategic location and capacity planning considering potential company acquisitions under uncertainty. Long-term logistics network planning is among the most difficult decisions for supply-chain managers. While costs, demands, etc. may be known or estimated well for the short-term, their future development is uncertain and difficult to predict.A new model formulation for the robust capacitated facility location problem is presented to cope with uncertainty in planning. Minimizing the expectation of the relative regrets across scenarios over multiple periods is the objective. It is achieved by dynamically assigning multi-level production allocations, locations and capacity adjustments for uncertain parameter development over time. Considering acquisitions for profit maximization and its supply-chain impact is new as well as the simultaneous decision of capacity adjustment and facility location over time. The solution of the novel robust formulation provides a single setup where good results can be achieved for any realized scenario. Hence, the solution may not be optimal for one particular scenario but may be good, i.e. the highest expected profit to gain, for any highly probable future realization. We show that robust mixed-integer linear programming model achieves superior results to the deterministic configurations in exhaustive computational tests. This dynamic robust formulation allows the supply-chain to favorably adapt to acquisitions and uncertain developments of revenue, demand and costs and hence reduces the potential negative impacts of uncertainty on supply-chain operations.

An Inexact Dynamic Optimization Model for CO2 Emission Reduction in Subei Region, Northeast China

In this study, an inexact mixed-integer fuzzy robust linear programming model for coupled management of coal and power with consideration of CO2 emissions mitigation system planning (IMIFLP-CCPM) was developed under uncertainty. This model could reach into the closed relationship and interactive characteristics of China’s coal production, electric power generation, and CO2 emissions in coupled coal and power management system and thus explore the applicability of the decarburization facilities and mechanism incorporated in the system through scenario analysis. Based on the integration of interval linear programming, fuzzy robust linear programming, and mixed-integer linear programming, the IMIFLP-CCPM could effectively incorporate and handle uncertainties presented in terms of interval values and fuzzy sets. Also, dynamic analysis of capacity expansion, facility improvement, and inventory planning within a multi-period and multi-option context could be facilitated in this model. The developed IMIFLP-CCPM was applied to a long-term coupled coal and power management with CO2 reduction systems in Subei region, Northeast China. One base scenario and four CO2 reduction scenarios were presented and analyzed to examine the optimal coal-flow allocation patterns and carbon mitigation schemes for the studied system when forced to comply with a given CO2 emission limit. The results indicated that the IMIFLP-CCPM model could provide in-depth analysis of tradeoffs between system costs, energy security, and CO2 emission reduction, thus helping investigate interactive relationships among multiple economical, environmental, and energy structural targets within the study system. Moreover, the attempt of planning coupled coal and power management with CO2 mitigation under uncertainty would provide an effective reference to cope with the dilemma of energy development and CO2 mitigation under the climate change situation in China.

Robust Optimization for Process Synthesis and Design of Multifunctional Energy Systems with Uncertainties

Generally, synthesis and design of an optimal process is a challenging task. The procedure includes specifying and optimizing system configurations in order to achieve a certain aspect, such as maximizing economic performance, minimizing environmental impact, etc. However, uncertainties of the design parameters (e.g., volatility of raw materials and products price, variability of feedstock supply and product demand, etc.) may undermine the effectiveness of such systematic design approaches. In response to this issue, various optimization works have been presented to address such problems. In this paper, a robust mixed integer linear programming (MILP) with input–output model is presented to aid decision-makers in addressing process synthesis problems due to uncertainties that arise from variation in feedstock supply and product demand. This work primarily encompasses multifunctional energy systems that can be described by a system of linear equations, which entails “black box” modeling. The robust model helps to determine the design capacity of each process unit in a flexible network which involves sizing of equipment. This network is assumed to be able to operate in all uncertain scenarios considered, with a minimum cost of the plant. In addition, the intended model also helps to determine the requirement to operate additional equipment in a plant in the presence of uncertainties. This is especially true with designs of plants that are incapable of meeting any increase in demand. These aspects of the work are novel and an important contribution as such analysis is not available in the literature, to date. Three case studies that include a polygeneration plant and palm oil based integrated biorefinery are presented to demonstrate the proposed novel approach in a more descriptive manner.

Robust Linear Programming with Norm Uncertainty

We consider the linear programming problem with uncertainty set described byp,w-norm. We suggest that the robust counterpart of this problem is equivalent to a computationally convex optimization problem. We provide probabilistic guarantees on the feasibility of an optimal robust solution when the uncertain coefficients obey independent and identically distributed normal distributions.

A ROBUST INEXACT TYPE-2 FUZZY SETS LINEAR OPTIMIZATION PROGRAMMING FOR IRRIGATION WATER SYSTEM MANAGEMENT UNDER UNCERTAINTY

This research introduces a new Robust Interval Type-2 Fuzzy Set Linear Programming (R-IT2FSLP) model to manage irrigation water resources. This method improves upon previous interval fuzzy bound models by allowing uncertainties presented as multiple fuzzy boundaries. It creates additional degrees of freedom and enable direct modeling of uncertainties within an optimization framework. It more explicitly reflects the system s uncertainties under the fuzzy membership function. The more of R-IT2FSLP method improves upon previous solution by embedding a Robust Two-Step algorithm (RTSM), which guarantee the feasibility of solutions. Compared with formal optimal methods of water systems, the solution of this optimal methodology is to see how applicable it is to irrigation water systems under uncertainty. In case study of this paper, it is clear that the R-IT2FSLP model can provide improved results to support Decision Makers (DMs). It solves problems of fresh water scarcity in irrigation systems.

Robust system design using dynamic small signal stability and linear programming for high-performance mechatronics

All realistic engineering systems operate in uncertain environments, and in some cases even small levels of noise can severely compromise their performance and stability. In this paper, we propose a robust design approach to improve the control performance of dynamic systems. Our proposed approach first specifies a set of linear constraints describing dynamic performance requirements based on pole placement, and develops a robust linear programming model which is solved to yield a parameter design that achieves the performance requirements under uncertainties. The advantage of our proposed approach is its computational efficiency since it reduces the problem to solving a small sequence of linear programming problems, and the proposed optimisation model can be embedded easily in a local search framework. The effectiveness of our proposed approach is verified with theoretical developments and computational studies on a typical high-performance mechatronic system; the hard disc drive.

Robust Systems Design using Dynamic Small Signal Stability and Linear Programming for High-Performance Mechatronics

All realistic engineering systems operate in uncertain environments, and in some cases even small levels of noise can severely compromise their performance and stability. In this paper, we propose a robust design approach to improve the control performance of dynamic systems. Our proposed approach first specifies a set of linear constraints describing dynamic performance requirements based on pole placement, and develops a robust linear programming model which is solved to yield a parameter design that achieves the performance requirements under uncertainties. The advantage of our proposed approach is its computational efficiency since it reduces the problem to solving a small sequence of linear programming problems, and the proposed optimisation model can be embedded easily in a local search framework. The effectiveness of our proposed approach is verified with theoretical developments and computational studies on a typical high-performance mechatronic system; the hard disc drive.

A robust optimization model for agile and build-to-order supply chain planning under uncertainties

Supply chain planning as one of the most important processes within the supply chain management concept, has a great impact on firms’ success or failure. This paper considers a supply chain planning problem of an agile manufacturing company operating in a build-to-order environment under various kinds of uncertainty. An integrated optimization approach of procurement, production and distribution costs associated with the supply chain members has been taken into account. A robust optimization scenario-based approach is used to absorb the influence of uncertain parameters and variables. The formulation is a robust optimization model with the objective of minimizing the expected total supply chain cost while maintaining customer service level. The developed multi-product, multi-period, multi-echelon robust mixed-integer linear programming model is then solved using the CPLEX optimization studio and guidance related to future areas of research is given.

Robust models for the synthesis of flexible palm oil-based regional bioenergy supply chain

EFB (Empty fruit bunch) is among the palm oil biomass residues that have been identified as the most abundant bioenergy resources in Southeast Asia. EFB has a good potential to meet the targeted share of renewable energy in this region. This work presents robust LP (linear programming) model to allow for the synthesis of EFB allocation networks that exhibit operational flexibility under multiple biomass supply scenarios. Such multiple scenarios may arise due to many uncertain factors, such as anticipated closure or expansion of mills. A MILP (mixed integer linear programming) extension of the model is then developed. Real life-based case studies of an industrial application are presented to illustrate the use of both variants of the model.

Robust combinatorial optimization with variable budgeted uncertainty

We introduce a new model for robust combinatorial optimization where the uncertain parameters belong to the image of multifunctions of the problem variables. In particular, we study the variable budgeted uncertainty, an extension of the budgeted uncertainty introduced by Bertsimas and Sim. Variable budgeted uncertainty can provide the same probabilistic guarantee as the budgeted uncertainty while being less conservative for vectors with few non-zero components. The feasibility set of the resulting optimization problem is in general non-convex so that we propose a mixed-integer programming reformulation for the problem, based on the dualization technique often used in robust linear programming. We show how to extend these results to non-binary variables and to more general multifunctions involving uncertainty set defined by conic constraints that are affine in the problem variables. We present a computational comparison of the budgeted uncertainty and the variable budgeted uncertainty on the robust knapsack problem. The experiments show a reduction of the price of robustness by an average factor of 18 %.