Mixed-integer Linear Fractional Programming Research Articles

User Association for Reconfigurable Intelligent Surfaces Aided Cell-Free Networks

Cell-free (CF) network can efficiently suppress inter-cell interference and enable multiple base stations (BSs) to serve users cooperatively without being affected by cell boundaries. However, to further increase the network capacity, a large number of BSs need to be deployed, resulting in high cost and power consumption. To tackle this problem, reconfigurable intelligent surface (RIS) has emerged as a cost-effective technology to enhance energy and spectral efficiency of wireless communications. In this paper, we consider an RIS-aided CF network by exploiting their complementary advantages where each user assisted by a dedicated RIS can associate with multiple BSs. We first characterize each user's average signal-to-interference-plus-noise ratio (SINR) in the RIS-CF network. Then, we formulate an SINR balancing problem to maximize the minimum SINR among all users by designing the BS-user associations. Despite of the non-convexity and combinatorial feature of this problem, a two-step fractional linearization algorithm is developed to solve it optimally. Specifically, we transform a nested mixed-integer linear fractional program (MILFP) into a solvable mixed-integer linear programming (MILP) problem through the generalized fractional programming and reformulation-linearization method. Finally, we provide numerical results to validate the effectiveness of our proposed algorithm compared to other benchmark schemes.

Optimisation frameworks for integrated planning with allocation of transportation resources for industrial gas supply chains

This work addresses the integrated optimisation of production-distribution planning and allocation of transportation resources for industrial gas supply chains. The production-distribution planning decisions include the production plan, purchasing plan for both a liquefied product and raw material from external suppliers, distribution plan by railcars and trucks, and demand allocation. In contrast, the allocating decisions of transportation resources involve the number of trucks and railcars at each plant, depot, and third-party supplier. First, we propose a mixed-integer nonlinear programming (MINLP) model, and then the MINLP model is reformulated as a mixed-integer linear fractional programming (MILFP) model. Furthermore, we present a multi-objective optimisation (MOO) model as an alternative approach. As solution strategies, we adopt Dinkelbachs algorithm and the reformulation-linearisation method for the MILFP model, whereas the ε-constraint method is used for the MOO model. Finally, industry-relevant case studies illustrate the applicability and performance of the proposed models and solution methods.

Mixed-integer linear-fractional programming model and it’s linear analogue for reducing inconsistency of pairwise comparison matrices☆

Pairwise comparison matrices are often used in multicriteria decision making (MCDM). The most critical part of this technique is the inconsistency, which emerges for logical reasons but can cause significant problems during the decision making process. Hence it is necessary to keep inconsistency below an acceptable threshold. In order to support the decision maker (DM) in making a rational decision, we must keep in mind the following: Our suggestion should be as close to the DM’s original result as possible, moreover it should have as low inconsistency as possible. We have studied various linear programming (LP) models that are used for reducing the inconsistency of pairwise comparison matrices (PCM) (Bozóki et al., 2011, 2015). These models, however aim at fulfilling only one of the previously mentioned two goals at a time. Therefore, the optimal solutions given may differ from each in the other respect, which is not taken as an objective but as a constraint in the model. So, they cannot be considered as equally good optimal solutions from a wider perspective. Based on our experiences concerning these models, we have developed a mixed-integer linear-fractional programming (MILFP) model that takes both mentioned goals as objectives by combining them into a linear-fractional objective function. We also provide the linear analogue (LA) of our MILFP model using an appropriate adaptation and combination of the Charnes-Cooper transformation and Glover’s linearization scheme.

Reformulation of bilevel linear fractional/linear programming problem into a mixed integer programming problem via complementarity problem

The bilevel programming problem is a static version of the Stackelberg's leader follower game in which Stackelberg strategy is used by the higher level decision maker called the leader given the rational reaction of the lower decision maker called the follower. The bilevel programming problem (BLPP) is a two-level hierarchical optimisation problem and is non-convex. This paper deals with finding links between the bilevel linear fractional/linear programming problem (BF/LP), the generalised linear fractional complementarity problem (GFCP) and mixed integer linear fractional programming problem (MIFP). The (BF/LP) is reformulated as a (GFCP) which in turn is reformulated as an (MIFP). The method is supported with the help of a numerical example.

An Energy Efficient Uplink Scheduling and Resource Allocation for M2M Communications in SC-FDMA Based LTE-A Networks

In future wireless communication, a large number of devices equipped with several different types of sensors will require access networks with diverse quality-of-service constraints. In cellular network evolution, the long term evolution advanced (LTE-A) networks has standardized Machine-to-Machine (M2M) features. Such M2M technology can provide a promising infrastructure for Internet of things (IoT) sensing applications, which usually require real-time data reporting. However, LTE-A is not designed for directly supporting such low-data-rate devices with optimized energy efficiency since it depends on core technologies of LTE that are originally designed for high-data-rate services. This paper investigate the maximum energy efficient data packets M2M transmission with uplink channels in LTE-A network. We formulate it into a joint problem of Modulation-and-Coding Scheme (MCS) assignment, resource allocation and power control, which can be expressed as a non-deterministic polynomial hard (NP-hard) mixed-integer linear fractional programming problem. Then we propose a global optimization scheme with Charnes-Cooper transformation and Glover linearization. The numerical experiment results show that with limited resource blocks, our algorithm can maintain low data packets dropping ratios while achieving optimal energy efficiency for a large number of M2M nodes, comparing with other typical counterparts.

Analysing Shale Gas Energy Systems using Dynamic Material Flow Analysis

This work develops a novel dynamic material flow analysis (MFA)-based optimisation modelling framework for sustainable design of shale gas energy systems. This dynamic MFA-based framework provides high-fidelity modelling of complex material flow networks with recycling options, and it enables detailed accounting of time-dependent life cycle material flow profiles. Moreover, by incorporating a dimension of resource sustainability, the proposed modelling framework facilitates the sustainable supply chain design and operations with a more comprehensive perspective. The resulting optimisation problem is formulated as a mixed-integer linear fractional program and solved by an efficient parametric algorithm. To illustrate the applicability of the proposed modelling framework and solution algorithm, a case study of Marcellus shale gas supply chain is presented. The optimisation results help to identify clear trade-offs among economic, environmental, and resource performances in the shale gas energy system, and the MFA-based framework offers an effective way to find an optimal solution with well-balanced sustainability perfor

Dynamic Material Flow Analysis-Based Life Cycle Optimization Framework and Application to Sustainable Design of Shale Gas Energy Systems

We propose a novel modeling framework integrating the dynamic material flow analysis (MFA) approach with life cycle optimization (LCO) methodology for sustainable design of energy systems. This dynamic MFA-based LCO framework provides high-fidelity modeling of complex material flow networks with recycling options, and it enables detailed accounting of time-dependent life cycle material flow profiles. The decisions regarding input, output, and stock of materials are seamlessly linked to their environmental impacts for rigorous quantification of environmental consequences. Moreover, by incorporating an additional dimension of resource sustainability, the proposed modeling framework facilitates the sustainable supply chain design and operations with a more comprehensive perspective. The resulting optimization problem is formulated as a mixed-integer linear fractional program and solved by an efficient parametric algorithm. To illustrate the applicability of the proposed modeling framework and solution algori...

Modeling framework and computational algorithm for hedging against uncertainty in sustainable supply chain design using functional-unit-based life cycle optimization

In this work, we address the life cycle economic and environmental optimization of a supply chain network considering both design and operational decisions under uncertainty. A modeling framework is proposed that integrates the functional-unit-based life cycle optimization methodology and the two-stage stochastic programming approach for sustainable supply chain optimization under uncertainty. We develop a stochastic mixed-integer linear fractional programming (SMILFP) model to tackle multiple uncertainties regarding feedstock supply and product demand. To address the computational challenge of solving the resulting large-scale SMILFP problems, an efficient solution algorithm is developed that takes advantage of the efficiency of parametric algorithm and the decomposition-based multi-cut L-shaped method. We present a case study based on a spatially explicit model for the optimal design and operations of a county-level hydrocarbon biofuel supply chain in Illinois to demonstrate the applicability of the proposed modeling framework and the efficiency of the solution algorithm.

Optimization-based framework for resin selection strategies in biopharmaceutical purification process development.

This work addresses rapid resin selection for integrated chromatographic separations when conducted as part of a high‐throughput screening exercise during the early stages of purification process development. An optimization‐based decision support framework is proposed to process the data generated from microscale experiments to identify the best resins to maximize key performance metrics for a biopharmaceutical manufacturing process, such as yield and purity. A multiobjective mixed integer nonlinear programming model is developed and solved using the ε‐constraint method. Dinkelbach's algorithm is used to solve the resulting mixed integer linear fractional programming model. The proposed framework is successfully applied to an industrial case study of a process to purify recombinant Fc Fusion protein from low molecular weight and high molecular weight product related impurities, involving two chromatographic steps with eight and three candidate resins for each step, respectively. The computational results show the advantage of the proposed framework in terms of computational efficiency and flexibility. © 2017 The Authors Biotechnology Progress published by Wiley Periodicals, Inc. on behalf of American Institute of Chemical Engineers Biotechnol. Prog., 33:1116–1126, 2017

A fuzzy-stochastic power system planning model: Reflection of dual objectives and dual uncertainties

In this study, a fuzzy stochastic dynamic fractional programming (FSDFP) method is proposed for supporting sustainable management of electric power system (EPS) under dual uncertainties. As an improvement upon the mixed-integer linear fractional programming, FSDFP can not only tackle multi-objective issues effectively without setting weights, but also can deal with uncertain parameters which have both stochastic and fuzzy characteristics. Thus, the developed method can help provide valuable information for supporting capacity-expansion planning and in-depth policy analysis of EPS management problems. For demonstrating these advantages, FSDFP has been applied to a case study of a typical regional EPS planning, where the decision makers have to deal with conflicts between economic development that maximizes the system profit and environmental protection that minimizes the carbon dioxide emissions. The obtained results can be analyzed to generate several decision alternatives, and can then help decision makers make suitable decisions under different input scenarios. Furthermore, comparisons of the solution from FSDFP method with that from fuzzy stochastic dynamic linear programming, linear fractional programming and dynamic stochastic fractional programming methods are undertaken. The contrastive analysis reveals that FSDFP is a more effective approach that can better characterize the complexities and uncertainties of real EPS management problems.

Optimisation approaches for the synthesis of water treatment plants

Efficient water treatment design has progressively been growing in importance as the usage of water resources increases with population rise and industrial development. Their availability has been reduced with the more evident effects of climate change. Addressing this challenge necessitates more and efficient purification plants which can be realised by optimal design at conceptual stage. In this work, a mixed integer nonlinear programming (MINLP) model for the synthesis and optimisation of water treatment processes is proposed. Due to its numerous non-linearities and consequently, its non-stability, various linearisation, approximation and reformulation techniques have been implemented. Consequently, two improved formulations are derived, i.e. a partially linearised MINLP (plMINLP) and a mixed integer linear fractional programming (MILFP) models. The applicability of the mathematical formulations are investigated in case studies of seawater desalination and surface water treatment for the production of potable water. Finally, the models performance is analysed and compared against each other.

The Effects of Coordination Mechanism on Vendor Buyer System of a Supply Chain

In this study, a supply chain consists of vendor and buyer, where the vendor and buyer are two distinct independent logistic organizations have been investigated. Supply chain management has the difficulties for the disconnected and independent economic individuals. Further, rapid technological changes and high competitiveness make supply chain more complex. Various companies have realized the potential for achieving a competitive advantage through effectively coordinating different logistics members in the supply chain. In this research, it is assumed that vendor and buyer are coordinated by sharing their status. Further, it is assumed the coordination mechanisms that can align their objectives and coordinate their activities by sharing the information between vendor and buyer. This plays an important role in decision making in order to mitigate the current challenges and to increase the system performance and individual profitability of the supply chain. The coordinated mechanism among the members of supply chain has been proposed to achieve the optimal solution. This study formulates mixed integer linear fractional programming models that maximize the ratio of return on investment. Numerical example with the sensitivity of different parameters has been deployed to validate the models. Results show that by coordination, the individual profit over and above joint profit could be increased and consumers purchasing price could be decreased.

Mixed Integer Linear Fractional Programming for Conjunctive Use of Surface Water and Groundwater

AbstractA conjunctive-use model is developed for management of groundwater and surface water resources via mixed integer linear fractional programming (MILFP). The objective of the conjunctive-use model is to maximize the ratio of groundwater usage to surface water usage through a water supply network. A conditional head constraint is imposed to the conjunctive-use model to maintain aquifer sustainability. A transformation approach is introduced to transform the conditional head constraint into a set of mixed integer linear constraints in terms of groundwater head. Groundwater head is further linearized with respect to pumping rates that are decision variables. Eventually, the conjunctive-use model is to solve a successive MILFP problem by updating the response matrix in each iteration. To make an MILFP problem tractable, the study develops a transformation technique along with the Charnes–Cooper transformation approach to transform an MILFP problem into an equivalent problem of mixed integer linear progr...

Conjunctive management of surface and groundwater resources under projected future climate change scenarios

This study introduces a mixed integer linear fractional programming (MILFP) method to optimize conjunctive use of future surface water and groundwater resources under projected climate change scenarios. The conjunctive management model maximizes the ratio of groundwater usage to reservoir water usage. Future inflows to the reservoirs were estimated from the future runoffs projected through hydroclimate modeling considering the Variable Infiltration Capacity model, and 11 sets of downscaled Coupled Model Intercomparison Project phase 5 global climate model projections. Bayesian model averaging was adopted to quantify uncertainty in future runoff projections and reservoir inflow projections due to uncertain future climate projections. Optimized conjunctive management solutions were investigated for a water supply network in northern Louisiana which includes the Sparta aquifer. Runoff projections under climate change scenarios indicate that runoff will likely decrease in winter and increase in other seasons. Results from the developed conjunctive management model with MILFP indicate that the future reservoir water, even at 2.5% low inflow cumulative probability level, could counterbalance groundwater pumping reduction to satisfy demands while improving the Sparta aquifer through conditional groundwater head constraints.

Vendor-buyer coordination and supply chain optimization with deterministic demand function

This paper presents a model that deals with a vendor-buyer multi-product, multi-facility and multi-customer location selection problem, which subsume a set of manufacturer with limited production capacities situated within a geographical area. We assume that the vendor and the buyer are coordinated by mutually sharing information. We formulate Mixed Integer Linear Fractional Programming (MILFP) model that maximize the ratio of return on investment of the distribution network, and a Mixed Integer Program (MIP), used for the comparison. The performance of the model is illustrated by a numerical example. In addition, product distribution and allocation of different customers along with the sensitivity of the key parameters are analyzed. It can be observed that the increment of the opening cost decreases the profit in both MILFP and MIP models. If the opening cost of a location decreases or increases, the demand and the capacity of that location changes accordingly.

Optimal design and operations of supply chain networks for water management in shale gas production: MILFP model and algorithms for the water‐energy nexus

The optimal design and operations of water supply chain networks for shale gas production is addressed. A mixed‐integer linear fractional programming (MILFP) model is developed with the objective to maximize profit per unit freshwater consumption, such that both economic performance and water‐use efficiency are optimized. The model simultaneously accounts for the design and operational decisions for freshwater source selection, multiple transportation modes, and water management options. Water management options include disposal, commercial centralized wastewater treatment, and onsite treatment (filtration, lime softening, thermal distillation). To globally optimize the resulting MILFP problem efficiently, three tailored solution algorithms are presented: a parametric approach, a reformulation‐linearization method, and a novel Branch‐and‐Bound and Charnes–Cooper transformation method. The proposed models and algorithms are illustrated through two case studies based on Marcellus shale play, in which onsite treatment shows its superiority in improving freshwater conservancy, maintaining a stable water flow, and reducing transportation burden. © 2014 American Institute of Chemical Engineers AIChE J, 61: 1184–1208, 2015

Mathematical programming approaches for downstream processing optimisation of biopharmaceuticals

This work addresses the optimal downstream chromatography sequencing and column sizing strategies in the manufacturing processes of monoclonal antibodies (mAbs). A mixed integer linear fractional programming (MILFP) model is developed to achieve continuous bed height values. In addition, to ease the computational expense of the literature MILFP model for discrete bed height values, two efficient hierarchical solution approaches are developed involving the two MILFP models, in which, based on its optimal solution of the newly developed MILFP model, the reduced MILFP model is solved with smaller decision region to find the final solution. A Dinkelbach-based algorithm is used as the solution approach of the MILFP models. Finally, a case study with different upstream processing (USP) and downstream processing (DSP) ratios are investigated, and the results show that all proposed approaches have high computational efficiency to satisfy different needs of the decision makers.

Optimising chromatography strategies of antibody purification processes by mixed integer fractional programming techniques

The strategies employed in chromatography steps play a key role in downstream processes for monoclonal antibody (mAb) manufacture. This work addresses the integrated optimisation of chromatography step sequencing and column sizing in mAb purification processes. Chromatography sequencing decisions include the resin selection at each typical step, while the column sizing decisions include the number of columns, the column diameter and bed height, and number of cycles per batch. A mixed integer nonlinear programming (MINLP) model was developed and then reformulated as a mixed integer linear fractional programming (MILFP) model. A literature approach, the Dinkelbach algorithm, was adopted as the solution method for the MILFP model. Finally, an industrially-relevant case study was investigated for the applicability of the proposed models and approaches.

Robust design and operations of hydrocarbon biofuel supply chain integrating with existing petroleum refineries considering unit cost objective

This paper addresses the optimal design and planning of the advanced hydrocarbon biofuel supply chain with the unit cost objective. Benefited from the drop-in properties of advanced hydrocarbon biofuels, the supply chain takes advantage of the existing petroleum infrastructure, which may lead to significant capital and transportation savings. A mixed-integer linear programming model is proposed to simultaneously consider the supply chain design, integration strategy selection, and production planning. A robust optimization approach which tradeoffs the performance and conservatism is adopted to deal with the demand and supply uncertainty. Moreover, the unit cost objective makes the final products more cost-competitive. The resulting mixed-integer linear fractional programming model is solved by tailored optimization algorithm. County level cases in Illinois are analyzed and compared to show the advantage of the proposed optimization framework. The results show that the preconversion to petroleum-upgrading pathway is more economical when applying the unit cost objective.

Sustainable Design and Operation of Cellulosic Bioelectricity Supply Chain Networks with Life Cycle Economic, Environmental, and Social Optimization

In this work, we propose a novel multiobjective optimization model for the sustainable design and operation of bioelectricity supply chain networks, which simultaneously accounts for the associated economic, environmental, and social impacts. The proposed model covers the cradle-to-gate life cycle of bioelectricity including biomass cultivation and harvesting, feedstock pretreatment, energy conversion, and biopower generation, as well as transportation and storage. We formulate the problem as a multiobjective mixed-integer linear fractional programming (MILFP) problem following the functional-unit-based life cycle optimization approach. The geographical dispersion and seasonality of biomass supply are captured and handled by the spatially explicit and multiperiod features of the model. The multiobjective optimization is accomplished via the ε-constraint method to obtain the approximate Pareto frontiers, which reveal the trade-off between economic performance and concerns about environmental and social impacts. Tailored solution methods are proposed for the effective global optimization of the resulting MILFP problem. An illustrative example and a county-level case study on the potential bioelectricity supply chain in the state of Illinois are provided to demonstrate the application of both the modeling framework and solution methods.