Proposed Mixed-integer Nonlinear Programming Model Research Articles

Sustainable forward-reverse logistics for multi-product delivery and pickup in B2C E-commerce towards the circular economy

The circular economy is an important part of sustainable business around the world. Accordingly, this study focuses on developing a sustainable framework for multi-echelon multiple electronics products' closed-loop supply chain in the e-commerce industry toward the circular economy. The proposed framework captures the complexities associated with forward and reverse logistics. The order is picked up from the supplier and delivered to the customer in the forward flow. Return products are picked up from customer locations and delivered to the inspection centre in the reverse flow. During the inspection process, products are classified into three categories: Reselling, Refurbishing, and Recycling (3R). The products that can be resold, refurbished, and recycled are then delivered to their respective final destinations, i.e., the supplier's warehouse, the refurbishing centre, and the recycling centre, respectively. A mixed-integer nonlinear programming (MINLP) model is developed to reduce total cost and maximize revenue associated with the forward and reverse flow of goods in a closed-loop supply chain while prioritising sustainability. To achieve this, a global solver in the LINGO 19 package software is used to solve and generate the exact solutions of the model. We have conducted 15 computational experiments for small-to-medium-to-large-sized problems to test the model. A sensitivity analysis is also performed to analyze the effect on total expected revenue from sustainable Closed-loop Supply Chain (CLSC) of variations in the major parameters of the model. Policymakers can use the obtained results and perform sensitivity analysis to make effective and efficient strategies favouring consumers and companies that could help enhance the country's economy. • Developed a sustainable multi-echelon forward and reverse logistics network design for an e-commerce platform. • Considered fresh product, returned resellable, refurbishable, and recyclable products for study. • Proposed MINLP model helps to maximize the total revenue with minimizing the transportation cost with carbon emission tax. • Sensitivity analysis is performed to see the effects in outcome with variations in main parameters of the model.

Designing the drone based end-to-end local supply chain distribution network

People in rural areas suffer due to lack of proper health care specifically during the emergency condition. Ontime delivery of emergency lifesaving medicine at the remote area is a one of the major concerns and one of the major reasons for this cause is the negligence and lack of proper transportation and road infrastructure. In this paper we have proposed a mixed integer non-linear programming (MINLP) model that is based on drone delivery for end-to-end local supply chain distribution network to ensure the fastest possible delivery of lifesaving medicine at the remote area. Moreover, we have used the Dijkstra algorithm to find the shortest path to deliver the lifesaving medicine at the patient home location using drone. The objective of this paper is to design an end-to-end local supply chain network to minimize the total delivery lead time of the lifesaving medicines under emergencies conditions. Also, the proposed MINLP model help to select the optimal healthcare facility location to minimize the total delivery lead time.

On the optimal reconfiguration of radial AC distribution networks using an MINLP formulation: A GAMS-based approach

This paper deals with the problem of the optimal reconfiguration of medium voltage distribution networks by proposing a mixed-integer nonlinear programming (MINLP) model. This optimization model has as objective function the minimization of the total power losses in all the branches of the network constrained by active and reactive power balance equations, voltage regulation bounds and device capabilities, among others. The proposed MINLP formulation works with branch-to-node incidence that allows representing the active and reactive power flow in branches as a function of the real and imaginary parts of the voltages and currents. The solution of the MINLP model is reached through the general algebraic modeling system widely know as GAMS package by presenting it in a tutorial form. This software allows implementing in compact form the proposed model and solve it via branch and bound methods. Two test feeders composed by 5 and 14 nodes permits demonstrating the fidelity of the proposed MINLP model regarding power losses minimization when compared with literature reports.

Multi-Energy Flow Calculation Considering the Convexification Network Constraints for the Integrated Energy System

To alleviate environmental pollution and improve the energy efficiency of end-user utilization, the integrated energy systems (IESs) have become an important direction of energy structure adjustment over the world. The widespread application of the coupling units, such as gas-fired generators, gas-fired boilers, and combined heat and power (CHP), increases the connection among electrical, natural gas, and heating systems in IESs. This study proposes a mixed-integer nonlinear programming (MINLP) model combining electrical, natural gas, and heating systems, as well as the coupling components, such as CHP and gas-fired generators. The proposed model is applicable for either the radial multi-energy network or the meshed multi-energy network. Since the proposed MINLP model is difficult to be solved, the second-order cone and linearized techniques are used to transform the non-convex fundamental matrix formulation of multi-energy network equations to a mixed-integer convex multi-energy flow model, which can improve the computational efficiency significantly. Moreover, the potential convergence problem of the original model can also be avoided. A simulation of IEEE 14-node electrical system, 6-node natural gas system, and 23-node heating system are studied to verify the accuracy and computational rapidity of the proposed method.

A Mixed-Integer Nonlinear Programming Model for Optimal Reconfiguration of DC Distribution Feeders

This paper deals with the optimal reconfiguration problem of DC distribution networks by proposing a new mixed-integer nonlinear programming (MINLP) formulation. This MINLP model focuses on minimising the power losses in the distribution lines by reformulating the classical power balance equations through a branch-to-node incidence matrix. The general algebraic modelling system (GAMS) is chosen as a solution tool, showing in tutorial form the implementation of the proposed MINLP model in a 6-nodes test feeder with 10 candidate lines. The validation of the MINLP formulation is performed in two classical 10-nodes DC test feeders. These are typically used for power flow and optimal power flow analyses. Numerical results demonstrate that power losses are reduced by about 16% when the optimal reconfiguration plan is found. The numerical validations are made in the GAMS software licensed by Universidad Tecnológica de Bolívar.

Formula omitted] Multi-contingency transient stability constrained AC optimal power flow with volt/var controllers

In order to guarantee the continuous operation of electric power systems (EPS), a proper planning of these networks in steady and transient state must be performed. This paper presents an N−1 multi-contingency AC optimal power flow (OPF) that embeds, in the same mathematical optimization model, a set of transient stability constraints (TSC) that guarantee rotor angle and angular velocity variation within technical limits, at given N−1 contingencies. Furthermore, the proposed model considers the operation of volt/var controllers, such as shunt elements and OLTC transformers, to further improve the operation of the EPS, under given levels of demand and generation. Taking advantage of the classical first-order transient stability model of synchronous machines and the implicit trapezoidal integration rule, the proposed model can be formulated as a stand-alone mixed-integer nonlinear programming (MINLP) model. Then, through well-established linearization techniques, the initially proposed MINLP model is transformed into a new mixed-integer linear programming (MILP) model, which can be implemented via algebraic programming languages, such as AMPL, and solved using convex optimization solvers, such as CPLEX. Three systems with dissimilar number of synchronous machines and nodes have been used for tests (i.e., the 9-Bus/3-Generator Western System Coordinating Council (WSCC) system, the 39-Bus/10-Generator New England system and the 68-Bus/16-Generator IEEE system). The efficiency of the proposed linearization techniques and the stability requirements of the solutions have been validated using an exact AC power flow and the transient stability analysis program PSAT-Matlab. Results show the ability of the proposed MILP model to provide stable operating points under N−1 contingencies, at minimum production cost.

Time-Dependent Pricing for High-Speed Railway in China Based on Revenue Management

High-speed railway (HSR) is recognized as a green transportation mode with lower energy consumption and less pollution emission than other transportation. At present, China has the largest HSR network globally, but the maximum revenue of railway transportation corporations has not been realized. In order to make HSR achieve a favorable position within the fierce competition in the market, increase corporate revenue, and achieve the sustainable development of HSR and railway corporations, we introduce the concept of revenue management in HSR operations and propose an innovative model to optimize the price and seat allocation for HSR simultaneously. In the study, we formulate the optimization problem as a mixed-integer nonlinear programming (MINLP) model, which appropriately captures passengers’ choice behavior. To reduce the computational complexity, we further transform the proposed MINLP model into an equivalent model. Finally, the effectiveness of both the proposed model and solution algorithm are tested and validated by numerical experiments. The research results show that the model can flexibly adjust the price and seat allocation of the corresponding ticketing period according to the passenger demand, and increase the total expected revenue by 5.92% without increasing the capacity.

A stochastic programming model for the optimal operation of unbalanced three-phase islanded microgrids

This paper presents a stochastic mixed-integer nonlinear programming (MINLP) model for the optimal operation of islanded microgrids in the presence of stochastic demands and renewable resources. In the proposed formulation, the microgrid is modeled as an unbalanced three-phase electrical distribution system comprising distributed generation (DG) units with droop control, battery systems (BSs) and wind turbines (WTs). The stochastic nature of the consumption and the renewable generation is considered through a scenario-based approach, which determines the optimal values of the decision variables that minimize the average operational cost of the microgrid. A set of efficient linearizations are used to transform the proposed MINLP model into an approximated convex model that can be solved via commercial solvers. In order to assess the effectiveness of the obtained solution, Monte Carlo simulations (MCS) are carried out. Results show that the proposed model considers the uncertainty while reducing the average operational costs and load curtailments, when compared with a deterministic model.

Cost optimization of project schedules under constrained resources and alternative production processes by mixed-integer nonlinear programming

PurposeThe purpose of this paper is cost optimization of project schedules under constrained resources and alternative production processes (APPs).Design/methodology/approachThe model contains a cost objective function, generalized precedence relationship constraints, activity duration and start time constraints, lag/lead time constraints, execution mode (EM) constraints, project duration constraints, working time unit assignment constraints and resource constraints. The mixed-integer nonlinear programming (MINLP) superstructure of discrete solutions covers time–cost–resource options related to various EMs for project activities as well as variants for production process implementation.FindingsThe proposed model provides the exact optimal output data for project management, such as network diagrams, Gantt charts, histograms and S-curves. In contrast to classic scheduling approaches, here the optimal project structure is obtained as a model-endogenous decision. The project planner is thus enabled to achieve optimization of the production process simultaneously with resource-constrained scheduling of activities in discrete time units and at a minimum total cost.Practical implicationsA set of application examples are addressed on an actual construction project to display the advantages of proposed model.Originality/valueThe unique value this paper contributes to the body of knowledge reflects through the proposed MINLP model, which is capable of performing the exact cost optimization of production process (where presence and number of activities including their mutual relations are dealt as feasible alternatives, meaning not as fixed parameters) simultaneously with the associated resource-constrained project scheduling, whereby that is achieved within a uniform procedure.

A MINLP model for the optimal waterflooding strategy and operation control of surface waterflooding pipeline network considering reservoir characteristics

Waterflooding is one of the most effective methods for oilfield secondary recovery. For the optimal waterflooding strategy and surface waterflooding pipeline network operation control plan, this paper proposes a mixed integer nonlinear programming (MINLP) model with the total profit of waterflooding development as the objection function, considering the oil reservoir characteristics. Based on the reservoir numerical simulation, the relationships between the water injection and oil production of four reservoir types are acquired. In the proposed model, various technical and operational constraints are considered for the safe and efficient operation of the pipeline network. Due to the nonlinearity of partial equations in the model, a piecewise linear approximation method is adopted in this paper, and the proposed MINLP model is converted into a mixed integer linear programming (MILP) model, which can be solved by the commercial solver. Finally, taking a real-world waterflooding oilfield in Daqing, China as an example, the applicability of the model are testified, and the superiority of considering the reservoir characteristics is detailedly illustrated.

Mixed Integer Nonlinear Programming for Aircraft Conflict Avoidance by Applying Velocity and Altitude Changes

The maneuvers of aircraft conflict avoidance are studied for air traffic management systems. In the scenario, aircraft are considered to operate flight within a shared airspace and cannot violate the minimum safe separation that they must keep in flight in order to avoid various possible conflicts. This paper proposes a formulation of the conflict avoidance problem as a mixed integer nonlinear programming (MINLP) with the maneuver that aircraft are allowed to change velocity and altitude to maintain the separation. Numerical experiments are performed to demonstrate the validity of the proposed MINLP model by comparison with the previous conflict avoidance model with only velocity change, and the illustration indicated a significant improvement in the efficiency of the conflict resolution as well as the magnitude of velocity variation. The optimal solutions of the problem are easily obtained by using global optimization solver in a shorter computational time even for large-scale aircraft.

Optimal Operation of Unbalanced Three-Phase Islanded Droop-Based Microgrids

This paper presents a new mixed-integer nonlinear programming (MINLP) model for the optimal operation of unbalanced three-phase droop-based microgrids. The proposed MINLP model can be seen as an extension of an optimal power flow for microgrids operating in islanded mode, that aims to minimize the total amount of unsupplied demand and the total distributed generator (DG) generation cost. Since the slack bus concept is not longer valid, the proposed model considers the frequency and voltage magnitude reference as variables. In this case, DGs units operate with droop control to balance the system and provide a frequency and voltage magnitude reference. Additionally, a set of efficient linearizations are introduced in order to approximate the original MINLP problem into a mixed-integer linear programming (MILP) model that can be solved using commercial solvers. The proposed model has been tested in a 25-bus unbalanced three-phase microgrid and a large 124-node grid, considering different operational and time-coupling constraints for the DGs and the battery systems (BSs). Load curtailment and different modes of operation for the wind turbines have also been tested. Finally, an error assessment between the original MINLP and the approximated MILP model has been conducted.

Optimal Restoration/Maintenance Switching Sequence of Unbalanced Three-Phase Distribution Systems

This paper presents a mixed-integer non-linear programming (MINLP) model for the optimal restoration/maintenance switching sequence of unbalanced three-phase electrical distribution systems. Once the protection coordination has identified and cleared a faulty zone, the proposed MINLP model determines the status of remotely controlled switches and the dispatchable distributed generation (DG) units, used to de-energized the troubled section of the network and supply as much load as possible. The restoration considers the switching sequence over a discrete horizon, guaranteeing that the operational constraints of the distribution system are not violated in every step of the sequence. Furthermore, a set of linearization strategies are presented to transform the proposed MINLP model into a mixed-integer linear programming (MILP) model. The use of MILP models guarantees convergence to optimality by applying convex optimization techniques. Tests are performed on an unbalanced three-phase radial distribution system consisting of 123 nodes, 12 switches, and three dispatchable DG units. The obtained results show that the proposed optimization model is a holistic procedure that can be used to efficiently manage power restoration or to minimize isolated areas in case of scheduled maintenance in modern electrical distribution systems.

Stencil Printing Process Optimization to Control Solder Paste Volume Transfer Efficiency

This research aims to optimize the stencil printing parameters to control the solder paste volume TE and increase the first-pass-yields of printed circuit boards (PCBs). Stencil printing process (SPP) factors considered in this research are related to solder paste composition, product configuration, printer setup, and stencil design. Support vector regression (SVR) with different kernels and regression tree (RT) are used to estimate the solder paste volume transfer efficiency (TE) and capture the nonlinear relationships between the SPP factors and volume TE. A novel mixed-integer nonlinear programming (MINLP) model is proposed to optimize the printing parameters for different PCB configurations of pads. The $\epsilon$ -constraint approach is used to solve the proposed MINLP model using metaheuristics, such as simulated annealing (SA) and evolutionary strategies (ES). The experimental results indicate relatively high ${R}^{2}$ values of 81% and 72% for average and standard deviation volume TE models that use SVR and RT, respectively. Then, the optimal printing parameters for certain pads’ configuration are retrieved for different volume TE spec limits. The results show that SA and ES can find optimal solutions for the spec limits of 50%–150%, 70%–130%, and 80%–120%.

Simultaneous Utility and Heat Exchanger Area Targeting for Integrated Process Synthesis and Heat Integration

We propose a mixed-integer nonlinear programming (MINLP) model for simultaneous utility and heat exchanger area targeting with variable stream conditions. The model represents the composite-curve-based area targeting method by constructing the hot and cold composite curves mathematically. We introduce a “dynamic” enthalpy grid onto which the stream inlet/outlet temperatures and enthalpies are mapped. By calculating the temperatures at each grid point and the stream heat duties at each interval, the utility consumption and heat exchanger areas are simultaneously optimized using an economic criterion. We discuss preprocessing methods tailored to aid the solution of the proposed MINLP model. The model is applied to two illustrative examples as well as an example where it is integrated with a process synthesis model.

Operational strategy and planning for raw natural gas refining complexes: Process modeling and global optimization

Optimal operational strategy and planning of a raw natural gas refining complex (RNGRC) is very challenging since it involves highly nonlinear processes, complex thermodynamics, blending, and utility systems. In this article, we first propose a superstructure integrating a utility system for the RNGRC, involving multiple gas feedstocks, and different product specifications. Then, we develop a large‐scale nonconvex mixed‐integer nonlinear programming (MINLP) optimization model. The model incorporates rigorous process models for input and output relations based on fundamentals of thermodynamics and unit operations and accurate models for utility systems. To reduce the noncovex items in the proposed MINLP model, equivalent reformulation techniques are introduced. Finally, the reformulated nonconvex MINLP model is solved to global optimality using state of the art deterministic global optimization approaches. The computational results demonstrate that a significant profit increase is achieved using the proposed approach compared to that from the real operation. © 2016 American Institute of Chemical Engineers AIChE J, 63: 652–668, 2017

Energy optimization of water supply system scheduling: Novel MINLP model and efficient global optimization algorithm

This article is concerned with global optimization of water supply system scheduling with pump operations to minimize total energy cost. The scheduling problem is first formulated as a non‐convex mixed‐integer nonlinear programming (MINLP) problem, accounting for flow rates in pipes, operation profiles of pumps, water levels of tanks, and customer demand. Binary variables denote on–off switch operations for pumps and flow directions in pipes, and nonlinear terms originate from characteristic functions for pumps and hydraulic functions for pipes. The proposed MINLP model is verified with EPANET, which is a leading software package for water distribution system modeling. We further develop a novel global optimization algorithm for solving the non‐convex MINLP problem. To demonstrate the applicability of the proposed model and the efficiency of the tailored global optimization algorithm, we present results of two case studies with up to 4 tanks, 5 pumps, 5 check valves, and 21 pipes. © 2016 American Institute of Chemical Engineers AIChE J, 62: 4277–4296, 2016

Quantity discount decisions considering multiple suppliers with capacity and quality restrictions

This article considers a supply chain inventory problem for a particular type of raw material with multiple suppliers, where each supplier offers all-unit quantity discounts as a motivation mechanism to increase the placed order quantities, and hence reduce the average the replenishment cost. In addition, multiple orders to the selected suppliers with different frequencies are allowed during a repeating order cycle. In this research, we propose a mixed integer nonlinear programming (MINLP) model to find the optimal number of orders and corresponding order quantities for the selected suppliers that minimise the total replenishment and inventory cost per time unit under suppliers' capacity and quality constraints. A numerical example is provided to illustrate the proposed MINLP model. We also compare the results of our model with those from other researchers, where at most one order can be allocated to each supplier per repeating order cycle.

Mathematical programming model for heat exchanger design through optimization of partial objectives

Mathematical programming can be used for the optimal design of shell-and-tube heat exchangers (STHEs). This paper proposes a mixed integer non-linear programming (MINLP) model for the design of STHEs, following rigorously the standards of the Tubular Exchanger Manufacturers Association (TEMA). Bell–Delaware Method is used for the shell-side calculations. This approach produces a large and non-convex model that cannot be solved to global optimality with the current state of the art solvers. Notwithstanding, it is proposed to perform a sequential optimization approach of partial objective targets through the division of the problem into sets of related equations that are easier to solve. For each one of these problems a heuristic objective function is selected based on the physical behavior of the problem. The global optimal solution of the original problem cannot be ensured even in the case in which each of the sub-problems is solved to global optimality, but at least a very good solution is always guaranteed. Three cases extracted from the literature were studied. The results showed that in all cases the values obtained using the proposed MINLP model containing multiple objective functions improved the values presented in the literature.

MINLP optimization model for the nonlinear discrete time–cost trade-off problem

This paper presents the mixed-integer nonlinear programming (MINLP) optimization model for the nonlinear discrete time-cost trade-off problem (NDTCTP). The nonlinear total project cost objective function of the proposed MINLP optimization model is subjected to a rigorous system of generalized precedence relationship constraints between project activities, project duration constraints, logical constraints, and a budget constraint. By means of the proposed MINLP optimization model, one can obtain the minimum total project cost, the project schedule with the optimal discrete start times and the optimal discrete durations of the activities, as well as the optimal time-cost curves of the project. The proposed model yields the exact optimum solution of the NDTCTP. Solving the NDTCTP, using the proposed MINLP model, avoids the need for (piece-wise) linear approximation of the nonlinear expressions. The MINLP model handles the discrete variables explicitly and requires no rounding of the continuous solution into an integer solution. The applicability of the proposed optimization model is not limited to weakly NDTCTPs. A numerical example from the literature and an example of the project time-cost trade-off analysis are presented at the end of the paper in order to show the advantages of the proposed model.